Subgroups — C₅×#24324 #11947044

Order 1280.

1008 conjugacy classes of subgroups.

The subgroup lattice has 786 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3504	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#3077	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₅×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic
4	C₅×#2566	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#2061	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#2060	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#2027	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#2027	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#1998	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#1986	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#1883	C₂	1	1	yes	maximal, characteristic
12	C₅×#1883	C₂	1	1	yes	maximal, characteristic
13	C₅×#1860	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅×#1859	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅×#1806	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₅×AGL(1,ℤ/8)	C₂²	1	1	yes	characteristic
17	C₅×#557	—	2	1	no	—
18	C₅×#557	—	2	1	no	—
19	C₅×#554	—	2	1	no	—
20	C₅×#554	—	2	1	no	—
21	C₅×#554	—	2	2	no	—
22	C₅×#554	—	2	2	no	—
23	C₅×#544	—	2	1	no	—
24	C₅×#544	—	2	1	no	—
25	C₅×#544	C₂²	1	1	yes	characteristic
26	C₅×#544	C₂²	1	1	yes	characteristic
27	C₅×#533	—	2	1	no	—
28	C₅×#533	C₂²	1	1	yes	characteristic
29	C₂×C₅×#125	—	2	1	no	—
30	C₂×C₅×#125	C₂²	1	1	yes	characteristic
31	C₅×#520	C₂²	1	1	yes	characteristic
32	C₅×#520	C₂²	1	1	yes	characteristic
33	C₄×C₅×#41	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
35	C₅×#478	—	2	1	no	—
36	C₅×#478	—	2	1	no	—
37	C₅×#476	—	2	1	no	—
38	C₅×#474	—	2	1	no	—
39	C₅×#442	—	2	1	no	—
40	C₅×#440	—	2	1	no	—
41	C₄×C₅×SD₁₆	—	2	1	no	—
42	C₄×C₅×D₈	—	2	1	no	—
43	C₅×#428	C₂²	1	1	yes	characteristic
44	C₅×#428	C₂²	1	1	yes	characteristic
45	C₅×#428	C₂²	1	1	yes	characteristic
46	C₅×#423	C₂²	1	1	yes	characteristic
47	C₂×C₅×#104	C₂²	1	1	yes	characteristic
48	C₂×C₅×#104	C₂²	1	1	yes	characteristic
49	C₅×#418	C₂²	1	1	yes	characteristic
50	C₅×#418	C₂²	1	1	yes	characteristic
51	C₅×#418	C₂²	1	1	yes	characteristic
52	C₅×#417	C₂²	1	1	yes	characteristic
53	C₄×C₅×M₁₆	C₂²	1	1	yes	characteristic
54	C₅×#396	—	2	2	no	—
55	C₅×#390	C₂²	1	1	yes	characteristic
56	C₅×#390	C₂²	1	1	yes	characteristic
57	C₅×#385	—	2	2	no	—
58	C₅×#384	C₂²	1	1	yes	characteristic
59	C₅×#382	C₂²	1	1	yes	characteristic
60	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
61	C₅×#343	C₂²	1	1	yes	characteristic
62	C₅×#343	C₂²	1	1	yes	characteristic
63	C₅×#343	C₂²	1	1	yes	characteristic
64	C₅×#343	C₂²	1	1	yes	characteristic
65	C₅×#339	C₂²	1	1	yes	characteristic
66	C₅×#339	C₂²	1	1	yes	characteristic
67	C₅×#339	C₂²	1	1	yes	characteristic
68	C₅×#339	C₂²	1	1	yes	characteristic
69	C₅×#339	C₂²	1	1	yes	characteristic
70	C₅×#339	C₂²	1	1	yes	characteristic
71	#24324	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂×C₅×#41	C₂³	1	1	yes	characteristic
73	C₂²×C₅×D₄	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₅	—	2	1	no	—
75	C₅×AGL(1,ℤ/8)	—	2	2	no	—
76	C₅×AGL(1,ℤ/8)	Q₈	1	4	yes	complemented
77	C₂×C₅×SD₁₆	—	2	1	no	—
78	C₂×C₅×D₈	—	2	1	no	—
79	C₂×C₅×M₁₆	C₂³	1	1	yes	characteristic
80	C₂×C₅×M₁₆	C₂³	1	1	yes	characteristic
81	C₂×C₅×M₁₆	C₂³	1	1	yes	characteristic
82	C₂×C₅×M₁₆	C₂³	1	1	yes	characteristic
83	C₅×C₄⋊Q₈	—	2	2	no	—
84	C₅×C₄⋊Q₈	—	2	2	no	—
85	C₅×C₄⋊Q₈	—	2	1	no	—
86	C₅×C₄⋊Q₈	—	2	1	no	—
87	C₅×C₄⋊Q₈	—	2	1	no	—
88	C₅×C₄⋊Q₈	—	2	1	no	—
89	C₅×C₄⋊Q₈	D₄	1	1	yes	characteristic
90	C₅×C₄⋊Q₈	D₄	1	1	yes	characteristic
91	C₅×#125	—	2	2	no	—
92	C₅×#125	—	2	2	no	—
93	C₅×#125	—	2	2	no	—
94	C₅×#125	—	2	2	no	—
95	C₅×#125	—	2	1	no	—
96	C₅×#125	—	2	1	no	—
97	C₅×#125	—	2	1	no	—
98	C₅×#125	—	2	1	no	—
99	C₅×#125	—	2	1	no	—
100	C₅×#125	—	2	1	no	—
101	C₅×#125	—	2	1	no	—
102	C₅×#125	—	2	1	no	—
103	C₅×#125	D₄	1	1	yes	characteristic
104	C₅×#125	D₄	1	1	yes	characteristic
105	C₅×C₂²⋊Q₈	—	2	2	no	—
106	C₅×C₂²⋊Q₈	—	2	2	no	—
107	C₅×C₂²⋊Q₈	—	2	2	no	—
108	C₅×C₂²⋊Q₈	—	2	2	no	—
109	C₅×C₂²⋊Q₈	—	2	2	no	—
110	C₅×C₂²⋊Q₈	—	2	2	no	—
111	C₅×C₂²⋊Q₈	—	2	1	no	—
112	C₅×C₂²⋊Q₈	—	2	1	no	—
113	C₅×C₂²⋊Q₈	—	2	2	no	—
114	C₅×C₂²⋊Q₈	—	2	2	no	—
115	C₄×C₅×Q₈	—	2	2	no	—
116	C₄×C₅×Q₈	—	2	2	no	—
117	C₄×C₅×Q₈	—	2	1	no	—
118	C₄×C₅×Q₈	—	2	1	no	—
119	C₄×C₅×Q₈	—	2	1	no	—
120	C₄×C₅×Q₈	—	2	1	no	—
121	C₄×C₅×Q₈	D₄	1	1	yes	characteristic
122	C₄×C₅×Q₈	D₄	1	1	yes	characteristic
123	C₄×C₅×D₄	—	2	1	no	—
124	C₄×C₅×D₄	—	2	1	no	—
125	C₄×C₅×D₄	—	2	1	no	—
126	C₄×C₅×D₄	—	2	1	no	—
127	C₄×C₅×D₄	—	2	2	no	—
128	C₄×C₅×D₄	—	2	2	no	—
129	C₄×C₅×D₄	—	2	1	no	—
130	C₄×C₅×D₄	—	2	1	no	—
131	C₄×C₅×D₄	—	2	1	no	—
132	C₄×C₅×D₄	—	2	1	no	—
133	C₄×C₅×D₄	D₄	1	1	yes	characteristic
134	C₄×C₅×D₄	D₄	1	1	yes	characteristic
135	C₅×#117	—	2	1	no	—
136	C₅×#117	—	2	1	no	—
137	C₅×#117	—	2	1	no	—
138	C₅×#117	C₂³	1	1	yes	characteristic
139	C₅×#117	C₂³	1	1	yes	characteristic
140	C₅×#117	C₂³	1	1	yes	characteristic
141	C₂×C₅×C₄⋊C₄	—	2	2	no	—
142	C₂×C₅×C₄⋊C₄	—	2	2	no	—
143	C₂×C₅×C₄⋊C₄	—	2	1	no	—
144	C₂×C₅×C₄⋊C₄	—	2	1	no	—
145	C₂×C₅×C₄⋊C₄	—	2	1	no	—
146	C₂×C₅×C₄⋊C₄	—	2	2	no	—
147	C₂×C₅×C₄⋊C₄	D₄	1	2	yes	—
148	C₂×C₅×C₄⋊C₄	D₄	1	2	yes	—
149	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
150	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
151	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
152	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
153	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
154	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
155	C₂×C₄²×C₅	—	2	1	no	—
156	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
157	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
158	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
159	C₅×#107	—	2	1	no	—
160	C₅×#107	—	2	1	no	—
161	C₅×#107	—	2	1	no	—
162	C₅×#106	—	2	1	no	—
163	C₅×#106	—	2	1	no	—
164	C₅×#106	—	2	1	no	—
165	C₅×C₄⋊C₈	—	2	1	no	—
166	C₅×C₄⋊C₈	—	2	1	no	—
167	C₅×#104	—	2	2	no	—
168	C₅×#104	—	2	2	no	—
169	C₅×#103	—	2	1	no	—
170	C₅×#102	—	2	1	no	—
171	C₅×#102	—	2	1	no	—
172	C₅×#102	—	2	1	no	—
173	C₅×#102	—	2	1	no	—
174	C₅×#102	—	2	1	no	—
175	C₅×#102	—	2	1	no	—
176	C₅×#102	—	2	1	no	—
177	C₅×#97	—	2	1	no	—
178	C₄×C₅×C₈	—	2	1	no	—
179	C₅×#95	—	2	1	no	—
180	C₅×#95	—	2	2	no	—
181	#3504	C₂×C₅	1	1	yes	characteristic, complemented
182	#3077	C₂×C₅	1	1	yes	characteristic, complemented
183	C₄×AGL(1,ℤ/8)	C₂×C₅	1	1	yes	characteristic
184	#2566	C₂×C₅	1	1	yes	characteristic, complemented
185	#2061	C₂×C₅	1	1	yes	characteristic, complemented
186	#2060	C₂×C₅	1	1	yes	characteristic, complemented
187	#2027	C₂×C₅	1	1	yes	characteristic, complemented
188	#2027	C₂×C₅	1	1	yes	characteristic, complemented
189	#1998	C₂×C₅	1	1	yes	characteristic, complemented
190	#1986	C₂×C₅	1	1	yes	characteristic, complemented
191	#1883	C₂×C₅	1	1	yes	characteristic
192	#1883	C₂×C₅	1	1	yes	characteristic
193	#1860	C₂×C₅	1	1	yes	characteristic, complemented
194	#1859	C₂×C₅	1	1	yes	characteristic, complemented
195	#1806	C₂×C₅	1	1	yes	characteristic, complemented
196	C₂⁴×C₅	—	2	1	no	—
197	C₅×#41	—	2	1	no	—
198	C₅×#41	—	2	1	no	—
199	C₅×#41	C₂×Q₈	1	2	yes	—
200	C₅×#41	#41	1	2	yes	—
201	C₂×C₅×Q₈	—	2	2	no	—
202	C₂×C₅×Q₈	—	2	2	no	—
203	C₂×C₅×Q₈	—	2	1	no	—
204	C₂×C₅×Q₈	—	2	1	no	—
205	C₂×C₅×Q₈	C₂×D₄	1	1	yes	characteristic
206	C₂×C₅×D₄	—	2	1	no	—
207	C₂×C₅×D₄	—	2	1	no	—
208	C₂×C₅×D₄	—	2	1	no	—
209	C₂×C₅×D₄	—	2	1	no	—
210	C₂×C₅×D₄	—	2	1	no	—
211	C₂×C₅×D₄	—	2	1	no	—
212	C₂×C₅×D₄	C₂×D₄	1	1	yes	characteristic
213	C₂×C₅×D₄	C₂×Q₈	1	2	yes	—
214	C₂²×C₄×C₅	—	2	1	no	—
215	C₂²×C₄×C₅	—	2	1	no	—
216	C₂²×C₄×C₅	—	2	2	no	—
217	C₂²×C₄×C₅	—	2	2	no	—
218	C₂²×C₄×C₅	—	2	1	no	—
219	C₂²×C₄×C₅	—	2	1	no	—
220	C₂²×C₄×C₅	—	2	1	no	—
221	C₂²×C₄×C₅	—	2	1	no	—
222	C₂²×C₄×C₅	—	4	1	no	—
223	C₂²×C₄×C₅	—	4	1	no	—
224	C₂²×C₄×C₅	—	8	2	no	—
225	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
226	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
227	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
228	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
229	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
230	C₅×SD₁₆	—	2	4	no	—
231	C₅×D₈	—	2	4	no	—
232	C₅×M₁₆	—	2	1	no	—
233	C₅×M₁₆	—	2	1	no	—
234	C₅×M₁₆	—	2	1	no	—
235	C₅×M₁₆	—	2	1	no	—
236	C₅×M₁₆	—	2	1	no	—
237	C₅×M₁₆	—	2	1	no	—
238	C₅×M₁₆	—	2	1	no	—
239	C₅×M₁₆	C₂×Q₈	1	2	yes	—
240	C₂×C₅×C₈	—	2	1	no	—
241	C₂×C₅×C₈	—	2	1	no	—
242	C₂×C₅×C₈	—	2	1	no	—
243	C₂×C₅×C₈	—	2	1	no	—
244	C₅×C₄⋊C₄	—	2	2	no	—
245	C₅×C₄⋊C₄	—	2	2	no	—
246	C₅×C₄⋊C₄	—	2	2	no	—
247	C₅×C₄⋊C₄	—	2	2	no	—
248	C₅×C₄⋊C₄	—	2	2	no	—
249	C₅×C₄⋊C₄	—	2	2	no	—
250	C₅×C₄⋊C₄	—	2	2	no	—
251	C₅×C₄⋊C₄	—	2	2	no	—
252	C₅×C₄⋊C₄	—	2	1	no	—
253	C₅×C₄⋊C₄	—	2	1	no	—
254	C₅×C₄⋊C₄	—	2	2	no	—
255	C₅×C₄⋊C₄	—	2	2	no	—
256	C₅×C₄⋊C₄	—	2	2	no	—
257	C₅×C₄⋊C₄	—	2	2	no	—
258	C₅×C₄⋊C₄	—	2	1	no	—
259	C₅×C₄⋊C₄	—	2	1	no	—
260	C₅×C₄⋊C₄	—	2	1	no	—
261	C₅×C₄⋊C₄	—	2	1	no	—
262	C₅×C₄⋊C₄	—	2	1	no	—
263	C₅×C₄⋊C₄	—	2	1	no	—
264	C₅×C₄⋊C₄	—	2	1	no	—
265	C₅×C₄⋊C₄	—	2	1	no	—
266	C₅×C₄⋊C₄	—	2	1	no	—
267	C₅×C₄⋊C₄	—	2	1	no	—
268	C₅×C₄⋊C₄	—	2	1	no	—
269	C₅×C₄⋊C₄	—	2	1	no	—
270	C₅×C₄⋊C₄	—	4	2	no	—
271	C₅×C₄⋊C₄	—	8	2	no	—
272	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
273	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
274	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
275	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
276	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
277	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
278	C₅×C₂²⋊C₄	—	2	2	no	—
279	C₅×C₂²⋊C₄	—	2	2	no	—
280	C₅×C₂²⋊C₄	—	2	1	no	—
281	C₅×C₂²⋊C₄	—	2	1	no	—
282	C₅×C₂²⋊C₄	—	2	1	no	—
283	C₅×C₂²⋊C₄	—	2	1	no	—
284	C₅×C₂²⋊C₄	—	2	1	no	—
285	C₅×C₂²⋊C₄	—	4	4	no	—
286	C₅×C₂²⋊C₄	—	4	1	no	—
287	C₅×C₂²⋊C₄	—	4	1	no	—
288	C₅×C₂²⋊C₄	—	8	2	no	—
289	C₄²×C₅	—	2	2	no	—
290	C₄²×C₅	—	2	2	no	—
291	C₄²×C₅	—	2	1	no	—
292	C₄²×C₅	—	2	1	no	—
293	C₄²×C₅	—	2	1	no	—
294	C₄²×C₅	—	2	1	no	—
295	C₄²×C₅	—	2	1	no	—
296	C₄²×C₅	—	2	1	no	—
297	C₄²×C₅	—	2	1	no	—
298	C₄²×C₅	—	2	1	no	—
299	C₄²×C₅	—	2	1	no	—
300	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
301	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
302	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
303	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
304	C₂×AGL(1,ℤ/8)	C₂²×C₅	1	1	yes	characteristic
305	#557	—	2	1	no	—
306	#557	—	2	1	no	—
307	#554	—	2	1	no	—
308	#554	—	2	1	no	—
309	#554	—	2	2	no	—
310	#554	—	2	2	no	—
311	#544	—	2	1	no	—
312	#544	—	2	1	no	—
313	#544	C₂²×C₅	1	1	yes	characteristic
314	#544	C₂²×C₅	1	1	yes	characteristic
315	#533	—	2	1	no	—
316	#533	C₂²×C₅	1	1	yes	characteristic
317	C₂×#125	—	2	1	no	—
318	C₂×#125	C₂²×C₅	1	1	yes	characteristic
319	#520	C₂²×C₅	1	1	yes	characteristic
320	#520	C₂²×C₅	1	1	yes	characteristic
321	C₄×#41	C₂²×C₅	1	1	yes	characteristic
322	C₂×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
323	#478	—	2	1	no	—
324	#478	—	2	1	no	—
325	#476	—	2	1	no	—
326	#474	—	2	1	no	—
327	#442	—	2	1	no	—
328	#440	—	2	1	no	—
329	C₄×SD₁₆	—	2	1	no	—
330	C₄×D₈	—	2	1	no	—
331	#428	C₂²×C₅	1	1	yes	characteristic
332	#428	C₂²×C₅	1	1	yes	characteristic
333	#428	C₂²×C₅	1	1	yes	characteristic
334	#423	C₂²×C₅	1	1	yes	characteristic
335	C₂×#104	C₂²×C₅	1	1	yes	characteristic
336	C₂×#104	C₂²×C₅	1	1	yes	characteristic
337	#418	C₂²×C₅	1	1	yes	characteristic
338	#418	C₂²×C₅	1	1	yes	characteristic
339	#418	C₂²×C₅	1	1	yes	characteristic
340	#417	C₂²×C₅	1	1	yes	characteristic
341	C₄×M₁₆	C₂²×C₅	1	1	yes	characteristic
342	#396	—	2	2	no	—
343	#390	C₂²×C₅	1	1	yes	characteristic
344	#390	C₂²×C₅	1	1	yes	characteristic
345	#385	—	2	2	no	—
346	#384	C₂²×C₅	1	1	yes	characteristic
347	#382	C₂²×C₅	1	1	yes	characteristic
348	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
349	#343	C₂²×C₅	1	1	yes	characteristic
350	#343	C₂²×C₅	1	1	yes	characteristic
351	#343	C₂²×C₅	1	1	yes	characteristic
352	#343	C₂²×C₅	1	1	yes	characteristic
353	#339	C₂²×C₅	1	1	yes	characteristic
354	#339	C₂²×C₅	1	1	yes	characteristic
355	#339	C₂²×C₅	1	1	yes	characteristic
356	#339	C₂²×C₅	1	1	yes	characteristic
357	#339	C₂²×C₅	1	1	yes	characteristic
358	#339	C₂²×C₅	1	1	yes	characteristic
359	C₂³×C₅	—	2	1	no	—
360	C₂³×C₅	—	2	2	no	—
361	C₂³×C₅	—	4	2	no	—
362	C₂³×C₅	—	4	2	no	—
363	C₂³×C₅	—	4	1	no	—
364	C₂³×C₅	—	4	1	no	—
365	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
366	C₅×Q₈	—	2	1	no	—
367	C₅×Q₈	—	8	1	no	—
368	C₅×Q₈	—	8	1	no	—
369	C₅×Q₈	—	8	2	no	—
370	C₅×Q₈	—	8	2	no	—
371	C₅×Q₈	C₂²⋊Q₈	1	2	yes	—
372	C₅×D₄	—	2	2	no	—
373	C₅×D₄	—	2	2	no	—
374	C₅×D₄	—	2	2	no	—
375	C₅×D₄	—	2	1	no	—
376	C₅×D₄	—	2	2	no	—
377	C₅×D₄	—	4	1	no	—
378	C₅×D₄	—	4	1	no	—
379	C₅×D₄	C₂²⋊Q₈	1	2	yes	—
380	C₂×C₄×C₅	—	2	1	no	—
381	C₂×C₄×C₅	—	2	2	no	—
382	C₂×C₄×C₅	—	2	2	no	—
383	C₂×C₄×C₅	—	2	2	no	—
384	C₂×C₄×C₅	—	2	1	no	—
385	C₂×C₄×C₅	—	2	1	no	—
386	C₂×C₄×C₅	—	2	1	no	—
387	C₂×C₄×C₅	—	2	1	no	—
388	C₂×C₄×C₅	—	2	2	no	—
389	C₂×C₄×C₅	—	2	2	no	—
390	C₂×C₄×C₅	—	2	1	no	—
391	C₂×C₄×C₅	—	2	1	no	—
392	C₂×C₄×C₅	—	4	1	no	—
393	C₂×C₄×C₅	—	4	1	no	—
394	C₂×C₄×C₅	—	4	1	no	—
395	C₂×C₄×C₅	—	4	1	no	—
396	C₂×C₄×C₅	—	4	2	no	—
397	C₂×C₄×C₅	—	4	2	no	—
398	C₂×C₄×C₅	—	4	1	no	—
399	C₂×C₄×C₅	—	4	1	no	—
400	C₂×C₄×C₅	—	4	2	no	—
401	C₂×C₄×C₅	—	4	1	no	—
402	C₂×C₄×C₅	—	4	1	no	—
403	C₂×C₄×C₅	—	8	1	no	—
404	C₂×C₄×C₅	—	8	1	no	—
405	C₂×C₄×C₅	—	8	2	no	—
406	C₂×C₄×C₅	—	8	2	no	—
407	C₂×C₄×C₅	—	8	2	no	—
408	C₂×C₄×C₅	—	8	2	no	—
409	C₂×C₄×C₅	#120	1	1	yes	characteristic
410	C₂×C₄×C₅	#120	1	1	yes	characteristic
411	C₂×C₄×C₅	#120	1	1	yes	characteristic
412	C₂×C₄×C₅	#120	1	1	yes	characteristic
413	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
414	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
415	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
416	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
417	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
418	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
419	C₅×C₈	—	2	2	no	—
420	C₅×C₈	—	4	1	no	—
421	C₅×C₈	—	4	1	no	—
422	C₅×C₈	—	4	1	no	—
423	C₂×#41	C₂³×C₅	1	1	yes	characteristic
424	C₂²×D₄	C₂³×C₅	1	1	yes	characteristic
425	C₂³×C₄	—	2	1	no	—
426	AGL(1,ℤ/8)	—	2	2	no	—
427	AGL(1,ℤ/8)	C₅×Q₈	1	4	yes	complemented
428	C₂×SD₁₆	—	2	1	no	—
429	C₂×D₈	—	2	1	no	—
430	C₂×M₁₆	C₂³×C₅	1	1	yes	characteristic
431	C₂×M₁₆	C₂³×C₅	1	1	yes	characteristic
432	C₂×M₁₆	C₂³×C₅	1	1	yes	characteristic
433	C₂×M₁₆	C₂³×C₅	1	1	yes	characteristic
434	C₄⋊Q₈	—	2	2	no	—
435	C₄⋊Q₈	—	2	2	no	—
436	C₄⋊Q₈	—	2	1	no	—
437	C₄⋊Q₈	—	2	1	no	—
438	C₄⋊Q₈	—	2	1	no	—
439	C₄⋊Q₈	—	2	1	no	—
440	C₄⋊Q₈	C₅×D₄	1	1	yes	characteristic
441	C₄⋊Q₈	C₅×D₄	1	1	yes	characteristic
442	#125	—	2	2	no	—
443	#125	—	2	2	no	—
444	#125	—	2	2	no	—
445	#125	—	2	2	no	—
446	#125	—	2	1	no	—
447	#125	—	2	1	no	—
448	#125	—	2	1	no	—
449	#125	—	2	1	no	—
450	#125	—	2	1	no	—
451	#125	—	2	1	no	—
452	#125	—	2	1	no	—
453	#125	—	2	1	no	—
454	#125	C₅×D₄	1	1	yes	characteristic
455	#125	C₅×D₄	1	1	yes	characteristic
456	C₂²⋊Q₈	—	2	2	no	—
457	C₂²⋊Q₈	—	2	2	no	—
458	C₂²⋊Q₈	—	2	2	no	—
459	C₂²⋊Q₈	—	2	2	no	—
460	C₂²⋊Q₈	—	2	2	no	—
461	C₂²⋊Q₈	—	2	2	no	—
462	C₂²⋊Q₈	—	2	1	no	—
463	C₂²⋊Q₈	—	2	1	no	—
464	C₂²⋊Q₈	—	2	2	no	—
465	C₂²⋊Q₈	—	2	2	no	—
466	C₄×Q₈	—	2	2	no	—
467	C₄×Q₈	—	2	2	no	—
468	C₄×Q₈	—	2	1	no	—
469	C₄×Q₈	—	2	1	no	—
470	C₄×Q₈	—	2	1	no	—
471	C₄×Q₈	—	2	1	no	—
472	C₄×Q₈	C₅×D₄	1	1	yes	characteristic
473	C₄×Q₈	C₅×D₄	1	1	yes	characteristic
474	C₄×D₄	—	2	1	no	—
475	C₄×D₄	—	2	1	no	—
476	C₄×D₄	—	2	1	no	—
477	C₄×D₄	—	2	1	no	—
478	C₄×D₄	—	2	2	no	—
479	C₄×D₄	—	2	2	no	—
480	C₄×D₄	—	2	1	no	—
481	C₄×D₄	—	2	1	no	—
482	C₄×D₄	—	2	1	no	—
483	C₄×D₄	—	2	1	no	—
484	C₄×D₄	C₅×D₄	1	1	yes	characteristic
485	C₄×D₄	C₅×D₄	1	1	yes	characteristic
486	#117	—	2	1	no	—
487	#117	—	2	1	no	—
488	#117	—	2	1	no	—
489	#117	C₂³×C₅	1	1	yes	characteristic
490	#117	C₂³×C₅	1	1	yes	characteristic
491	#117	C₂³×C₅	1	1	yes	characteristic
492	C₂×C₄⋊C₄	—	2	2	no	—
493	C₂×C₄⋊C₄	—	2	2	no	—
494	C₂×C₄⋊C₄	—	2	1	no	—
495	C₂×C₄⋊C₄	—	2	1	no	—
496	C₂×C₄⋊C₄	—	2	1	no	—
497	C₂×C₄⋊C₄	—	2	2	no	—
498	C₂×C₄⋊C₄	C₅×D₄	1	2	yes	—
499	C₂×C₄⋊C₄	C₅×D₄	1	2	yes	—
500	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
501	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
502	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
503	C₂×C₂²⋊C₄	—	2	1	no	—
504	C₂×C₂²⋊C₄	—	2	2	no	—
505	C₂×C₂²⋊C₄	—	2	2	no	—
506	C₂×C₄²	—	2	1	no	—
507	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
508	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
509	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
510	#107	—	2	1	no	—
511	#107	—	2	1	no	—
512	#107	—	2	1	no	—
513	#106	—	2	1	no	—
514	#106	—	2	1	no	—
515	#106	—	2	1	no	—
516	C₄⋊C₈	—	2	1	no	—
517	C₄⋊C₈	—	2	1	no	—
518	#104	—	2	2	no	—
519	#104	—	2	2	no	—
520	#103	—	2	1	no	—
521	#102	—	2	1	no	—
522	#102	—	2	1	no	—
523	#102	—	2	1	no	—
524	#102	—	2	1	no	—
525	#102	—	2	1	no	—
526	#102	—	2	1	no	—
527	#102	—	2	1	no	—
528	#97	—	2	1	no	—
529	C₄×C₈	—	2	1	no	—
530	#95	—	2	1	no	—
531	#95	—	2	2	no	—
532	C₂²×C₅	—	2	1	no	—
533	C₂²×C₅	—	2	1	no	—
534	C₂²×C₅	—	2	2	no	—
535	C₂²×C₅	—	4	4	no	—
536	C₂²×C₅	—	4	1	no	—
537	C₂²×C₅	—	4	1	no	—
538	C₂²×C₅	—	4	2	no	—
539	C₂²×C₅	—	8	2	no	—
540	C₂²×C₅	—	8	1	no	—
541	C₂²×C₅	—	8	1	no	—
542	C₂²×C₅	C₂×#120	1	1	yes	characteristic
543	C₂²×C₅	#548	1	1	yes	characteristic
544	C₂²×C₅	D₄×Q₈	1	1	yes	characteristic
545	C₄×C₅	—	2	2	no	—
546	C₄×C₅	—	2	1	no	—
547	C₄×C₅	—	2	1	no	—
548	C₄×C₅	—	2	1	no	—
549	C₄×C₅	—	2	1	no	—
550	C₄×C₅	—	4	2	no	—
551	C₄×C₅	—	4	2	no	—
552	C₄×C₅	—	4	1	no	—
553	C₄×C₅	—	4	1	no	—
554	C₄×C₅	—	4	1	no	—
555	C₄×C₅	—	4	2	no	—
556	C₄×C₅	—	4	1	no	—
557	C₄×C₅	—	4	1	no	—
558	C₄×C₅	—	4	1	no	—
559	C₄×C₅	—	8	1	no	—
560	C₄×C₅	—	8	1	no	—
561	C₄×C₅	C₂×C₂²⋊Q₈	1	1	yes	characteristic
562	C₄×C₅	#536	1	1	yes	characteristic
563	C₄×C₅	#548	1	1	yes	characteristic
564	C₄×C₅	D₄×Q₈	1	1	yes	characteristic
565	C₂⁴	—	2	1	no	—
566	#41	—	2	1	no	—
567	#41	—	2	1	no	—
568	#41	C₂×C₅×Q₈	1	2	yes	—
569	#41	C₅×#41	1	2	yes	—
570	C₂×Q₈	—	2	2	no	—
571	C₂×Q₈	—	2	2	no	—
572	C₂×Q₈	—	2	1	no	—
573	C₂×Q₈	—	2	1	no	—
574	C₂×Q₈	C₂×C₅×D₄	1	1	yes	characteristic
575	C₂×D₄	—	2	1	no	—
576	C₂×D₄	—	2	1	no	—
577	C₂×D₄	—	2	1	no	—
578	C₂×D₄	—	2	1	no	—
579	C₂×D₄	—	2	1	no	—
580	C₂×D₄	—	2	1	no	—
581	C₂×D₄	C₂×C₅×D₄	1	1	yes	characteristic
582	C₂×D₄	C₂×C₅×Q₈	1	2	yes	—
583	C₂²×C₄	—	2	1	no	—
584	C₂²×C₄	—	2	1	no	—
585	C₂²×C₄	—	2	2	no	—
586	C₂²×C₄	—	2	2	no	—
587	C₂²×C₄	—	2	1	no	—
588	C₂²×C₄	—	2	1	no	—
589	C₂²×C₄	—	2	1	no	—
590	C₂²×C₄	—	2	1	no	—
591	C₂²×C₄	—	4	1	no	—
592	C₂²×C₄	—	4	1	no	—
593	C₂²×C₄	—	8	2	no	—
594	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
595	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
596	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
597	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
598	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
599	SD₁₆	—	2	4	no	—
600	D₈	—	2	4	no	—
601	M₁₆	—	2	1	no	—
602	M₁₆	—	2	1	no	—
603	M₁₆	—	2	1	no	—
604	M₁₆	—	2	1	no	—
605	M₁₆	—	2	1	no	—
606	M₁₆	—	2	1	no	—
607	M₁₆	—	2	1	no	—
608	M₁₆	C₂×C₅×Q₈	1	2	yes	—
609	C₂×C₈	—	2	1	no	—
610	C₂×C₈	—	2	1	no	—
611	C₂×C₈	—	2	1	no	—
612	C₂×C₈	—	2	1	no	—
613	C₄⋊C₄	—	2	2	no	—
614	C₄⋊C₄	—	2	2	no	—
615	C₄⋊C₄	—	2	2	no	—
616	C₄⋊C₄	—	2	2	no	—
617	C₄⋊C₄	—	2	2	no	—
618	C₄⋊C₄	—	2	2	no	—
619	C₄⋊C₄	—	2	2	no	—
620	C₄⋊C₄	—	2	2	no	—
621	C₄⋊C₄	—	2	1	no	—
622	C₄⋊C₄	—	2	1	no	—
623	C₄⋊C₄	—	2	2	no	—
624	C₄⋊C₄	—	2	2	no	—
625	C₄⋊C₄	—	2	2	no	—
626	C₄⋊C₄	—	2	2	no	—
627	C₄⋊C₄	—	2	1	no	—
628	C₄⋊C₄	—	2	1	no	—
629	C₄⋊C₄	—	2	1	no	—
630	C₄⋊C₄	—	2	1	no	—
631	C₄⋊C₄	—	2	1	no	—
632	C₄⋊C₄	—	2	1	no	—
633	C₄⋊C₄	—	2	1	no	—
634	C₄⋊C₄	—	2	1	no	—
635	C₄⋊C₄	—	2	1	no	—
636	C₄⋊C₄	—	2	1	no	—
637	C₄⋊C₄	—	2	1	no	—
638	C₄⋊C₄	—	2	1	no	—
639	C₄⋊C₄	—	4	2	no	—
640	C₄⋊C₄	—	8	2	no	—
641	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
642	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
643	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
644	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
645	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
646	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
647	C₂²⋊C₄	—	2	2	no	—
648	C₂²⋊C₄	—	2	2	no	—
649	C₂²⋊C₄	—	2	1	no	—
650	C₂²⋊C₄	—	2	1	no	—
651	C₂²⋊C₄	—	2	1	no	—
652	C₂²⋊C₄	—	2	1	no	—
653	C₂²⋊C₄	—	2	1	no	—
654	C₂²⋊C₄	—	4	4	no	—
655	C₂²⋊C₄	—	4	1	no	—
656	C₂²⋊C₄	—	4	1	no	—
657	C₂²⋊C₄	—	8	2	no	—
658	C₄²	—	2	2	no	—
659	C₄²	—	2	2	no	—
660	C₄²	—	2	1	no	—
661	C₄²	—	2	1	no	—
662	C₄²	—	2	1	no	—
663	C₄²	—	2	1	no	—
664	C₄²	—	2	1	no	—
665	C₄²	—	2	1	no	—
666	C₄²	—	2	1	no	—
667	C₄²	—	2	1	no	—
668	C₄²	—	2	1	no	—
669	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
670	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
671	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
672	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
673	C₂×C₅	—	2	1	no	—
674	C₂×C₅	—	2	1	no	—
675	C₂×C₅	—	4	2	no	—
676	C₂×C₅	—	8	2	no	—
677	C₂×C₅	#2409	1	1	yes	characteristic
678	C₂×C₅	#3018	1	1	yes	characteristic
679	C₂×C₅	#3021	1	1	yes	characteristic
680	C₂³	—	2	1	no	—
681	C₂³	—	2	2	no	—
682	C₂³	—	4	2	no	—
683	C₂³	—	4	2	no	—
684	C₂³	—	4	1	no	—
685	C₂³	—	4	1	no	—
686	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
687	Q₈	—	2	1	no	—
688	Q₈	—	8	1	no	—
689	Q₈	—	8	1	no	—
690	Q₈	—	8	2	no	—
691	Q₈	—	8	2	no	—
692	Q₈	C₅×C₂²⋊Q₈	1	2	yes	—
693	D₄	—	2	2	no	—
694	D₄	—	2	2	no	—
695	D₄	—	2	2	no	—
696	D₄	—	2	1	no	—
697	D₄	—	2	2	no	—
698	D₄	—	4	1	no	—
699	D₄	—	4	1	no	—
700	D₄	C₅×C₂²⋊Q₈	1	2	yes	—
701	C₂×C₄	—	2	1	no	—
702	C₂×C₄	—	2	2	no	—
703	C₂×C₄	—	2	2	no	—
704	C₂×C₄	—	2	2	no	—
705	C₂×C₄	—	2	1	no	—
706	C₂×C₄	—	2	1	no	—
707	C₂×C₄	—	2	1	no	—
708	C₂×C₄	—	2	1	no	—
709	C₂×C₄	—	2	2	no	—
710	C₂×C₄	—	2	2	no	—
711	C₂×C₄	—	2	1	no	—
712	C₂×C₄	—	2	1	no	—
713	C₂×C₄	—	4	1	no	—
714	C₂×C₄	—	4	1	no	—
715	C₂×C₄	—	4	1	no	—
716	C₂×C₄	—	4	1	no	—
717	C₂×C₄	—	4	2	no	—
718	C₂×C₄	—	4	2	no	—
719	C₂×C₄	—	4	1	no	—
720	C₂×C₄	—	4	1	no	—
721	C₂×C₄	—	4	2	no	—
722	C₂×C₄	—	4	1	no	—
723	C₂×C₄	—	4	1	no	—
724	C₂×C₄	—	8	1	no	—
725	C₂×C₄	—	8	1	no	—
726	C₂×C₄	—	8	2	no	—
727	C₂×C₄	—	8	2	no	—
728	C₂×C₄	—	8	2	no	—
729	C₂×C₄	—	8	2	no	—
730	C₂×C₄	C₅×#120	1	1	yes	characteristic
731	C₂×C₄	C₅×#120	1	1	yes	characteristic
732	C₂×C₄	C₅×#120	1	1	yes	characteristic
733	C₂×C₄	C₅×#120	1	1	yes	characteristic
734	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
735	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
736	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
737	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
738	C₂×C₄	C₅×2₋¹⁺⁴	1	1	yes	characteristic
739	C₂×C₄	C₅×2₋¹⁺⁴	1	1	yes	characteristic
740	C₈	—	2	2	no	—
741	C₈	—	4	1	no	—
742	C₈	—	4	1	no	—
743	C₈	—	4	1	no	—
744	C₅	#24324	1	1	yes	characteristic, direct product factor, Sylow
745	C₂²	—	2	1	no	—
746	C₂²	—	2	1	no	—
747	C₂²	—	2	2	no	—
748	C₂²	—	4	4	no	—
749	C₂²	—	4	1	no	—
750	C₂²	—	4	1	no	—
751	C₂²	—	4	2	no	—
752	C₂²	—	8	2	no	—
753	C₂²	—	8	1	no	—
754	C₂²	—	8	1	no	—
755	C₂²	C₂×C₅×#120	1	1	yes	characteristic
756	C₂²	C₅×#548	1	1	yes	characteristic
757	C₂²	C₅×D₄×Q₈	1	1	yes	characteristic
758	C₄	—	2	1	no	—
759	C₄	—	2	1	no	—
760	C₄	—	2	1	no	—
761	C₄	—	2	1	no	—
762	C₄	—	2	2	no	—
763	C₄	—	4	2	no	—
764	C₄	—	4	2	no	—
765	C₄	—	4	1	no	—
766	C₄	—	4	1	no	—
767	C₄	—	4	1	no	—
768	C₄	—	4	2	no	—
769	C₄	—	4	1	no	—
770	C₄	—	4	1	no	—
771	C₄	—	4	1	no	—
772	C₄	—	8	1	no	—
773	C₄	—	8	1	no	—
774	C₄	C₂×C₅×C₂²⋊Q₈	1	1	yes	characteristic
775	C₄	C₅×#536	1	1	yes	characteristic
776	C₄	C₅×#548	1	1	yes	characteristic
777	C₄	C₅×D₄×Q₈	1	1	yes	characteristic
778	C₂	—	2	1	no	—
779	C₂	—	2	1	no	—
780	C₂	—	4	2	no	—
781	C₂	—	8	2	no	—
782	C₂	C₅×#2409	1	1	yes	characteristic
783	C₂	C₅×#3018	1	1	yes	characteristic
784	C₂	C₅×#3021	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
prime radical	C₂×C₅×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₅×#3504	C₂	semidirect
2. C₅×#3077	C₂	semidirect
4. C₅×#2566	C₂	semidirect
5. C₅×#2061	C₂	semidirect
6. C₅×#2060	C₂	semidirect
7. C₅×#2027	C₂	semidirect
8. C₅×#2027	C₂	semidirect
9. C₅×#1998	C₂	semidirect
10. C₅×#1986	C₂	semidirect
13. C₅×#1860	C₂	semidirect
14. C₅×#1859	C₂	semidirect
15. C₅×#1806	C₂	semidirect
71. #24324	C₅	direct (N × H)
76. C₅×AGL(1,ℤ/8)	Q₈	semidirect
181. #3504	C₂×C₅	semidirect
182. #3077	C₂×C₅	semidirect
184. #2566	C₂×C₅	semidirect
185. #2061	C₂×C₅	semidirect
186. #2060	C₂×C₅	semidirect
187. #2027	C₂×C₅	semidirect
188. #2027	C₂×C₅	semidirect
189. #1998	C₂×C₅	semidirect
190. #1986	C₂×C₅	semidirect
193. #1860	C₂×C₅	semidirect
194. #1859	C₂×C₅	semidirect
195. #1806	C₂×C₅	semidirect
427. AGL(1,ℤ/8)	C₅×Q₈	semidirect
744. C₅	#24324	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₅×#3504	2
C₅×#3077	2
C₄×C₅×AGL(1,ℤ/8)	2
C₅×#2566	2
C₅×#2061	2
C₅×#2060	2
C₅×#2027	2
C₅×#1998	2
C₅×#1986	2
C₅×#1883	2
C₅×#1860	2
C₅×#1859	2
C₅×#1806	2
#24324	5

Subgroups — C5×#24324 #11947044

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#24324 #11947044