Subgroups — C₅×#23049 #11945769

Order 1280.

866 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#471	C₂	1	1	yes	maximal, characteristic
2	C₅×#2949	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2947	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#2946	C₂	1	1	yes	maximal, characteristic
5	C₅×#2913	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1973	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1952	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1931	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#1930	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#1883	C₂	1	1	yes	maximal, characteristic
11	C₅×#1882	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#1857	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅×#1856	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅×#1773	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅×#1765	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₅×#137	C₂²	1	1	yes	characteristic
17	C₂×C₅×AGL(1,ℤ/8)	C₂²	1	1	yes	characteristic
18	C₂²×C₅×SD₁₆	C₂²	1	1	yes	characteristic
19	C₂×C₅×#131	C₂²	1	1	yes	characteristic
20	C₅×#520	C₂²	1	1	yes	characteristic
21	C₅×#520	C₂²	1	1	yes	characteristic
22	C₅×#519	C₂²	1	1	yes	characteristic
23	C₅×#518	C₂²	1	1	yes	characteristic
24	C₅×#471	C₄	1	8	yes	complemented
25	C₅×#442	—	2	1	no	—
26	C₅×#441	—	2	1	no	—
27	C₅×#440	—	2	1	no	—
28	C₅×#440	—	2	1	no	—
29	C₅×#440	—	2	1	no	—
30	C₅×#440	—	2	1	no	—
31	C₄×C₅×Q₁₆	—	2	1	no	—
32	C₄×C₅×SD₁₆	—	2	1	no	—
33	C₄×C₅×SD₁₆	—	2	1	no	—
34	C₄×C₅×SD₁₆	—	2	1	no	—
35	C₄×C₅×SD₁₆	—	2	1	no	—
36	C₄×C₅×D₈	—	2	1	no	—
37	C₂×C₅×#108	C₂²	1	1	yes	characteristic
38	C₅×#428	—	2	1	no	—
39	C₅×#428	—	2	1	no	—
40	C₅×#428	C₂²	1	1	yes	characteristic
41	C₅×#428	C₂²	1	1	yes	characteristic
42	C₅×#428	C₂²	1	1	yes	characteristic
43	C₅×#428	C₂²	1	1	yes	characteristic
44	C₅×#427	—	2	1	no	—
45	C₅×#427	C₂²	1	1	yes	characteristic
46	C₂×C₅×#107	—	2	1	no	—
47	C₂×C₅×#107	C₂²	1	1	yes	characteristic
48	C₂×C₅×#104	C₂²	1	1	yes	characteristic
49	C₅×#419	C₂²	1	1	yes	characteristic
50	C₅×#418	C₂²	1	1	yes	characteristic
51	C₅×#416	C₂²	1	1	yes	characteristic
52	C₂×C₅×#103	C₂²	1	1	yes	characteristic
53	C₂×C₅×#102	C₂²	1	1	yes	characteristic
54	C₂×C₅×#101	C₂²	1	1	yes	characteristic
55	C₂×C₅×#100	C₂²	1	1	yes	characteristic
56	C₅×#405	C₂²	1	1	yes	characteristic
57	C₅×#405	C₂²	1	1	yes	characteristic
58	C₅×#405	C₂²	1	1	yes	characteristic
59	C₂×C₅×#97	C₂²	1	1	yes	characteristic
60	C₅×#391	C₂²	1	1	yes	characteristic
61	C₅×#390	C₂²	1	1	yes	characteristic
62	C₅×#343	C₂²	1	1	yes	characteristic
63	C₅×#343	C₂²	1	1	yes	characteristic
64	C₅×#341	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	#23049	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂×C₅×#41	C₂³	1	1	yes	characteristic
70	C₂²×C₅×Q₈	C₂³	1	1	yes	characteristic
71	C₂²×C₅×D₄	C₂³	1	1	yes	characteristic
72	C₅×#137	—	2	2	no	—
73	C₅×#137	C₂×C₄	1	4	yes	complemented
74	C₅×AGL(1,ℤ/8)	—	2	2	no	—
75	C₅×AGL(1,ℤ/8)	C₂×C₄	1	4	yes	—
76	C₂×C₅×Q₁₆	—	2	1	no	—
77	C₂×C₅×SD₁₆	—	2	1	no	—
78	C₂×C₅×SD₁₆	—	2	1	no	—
79	C₂×C₅×SD₁₆	—	2	1	no	—
80	C₂×C₅×SD₁₆	—	2	1	no	—
81	C₂×C₅×SD₁₆	—	2	2	no	—
82	C₂×C₅×SD₁₆	C₂×C₄	1	4	yes	complemented
83	C₂×C₅×D₈	—	2	1	no	—
84	C₅×#131	—	2	2	no	—
85	C₅×#131	C₂×C₄	1	4	yes	complemented
86	C₂×C₅×M₁₆	—	2	1	no	—
87	C₂×C₅×M₁₆	C₂³	1	1	yes	characteristic
88	C₂×C₅×M₁₆	C₂³	1	1	yes	characteristic
89	C₂×C₅×M₁₆	C₂³	1	1	yes	characteristic
90	C₂×C₅×M₁₆	C₂³	1	1	yes	characteristic
91	C₂²×C₅×C₈	—	2	1	no	—
92	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
93	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
94	C₄×C₅×Q₈	—	2	1	no	—
95	C₄×C₅×Q₈	—	2	1	no	—
96	C₄×C₅×Q₈	—	2	1	no	—
97	C₄×C₅×Q₈	—	2	1	no	—
98	C₄×C₅×Q₈	—	2	1	no	—
99	C₄×C₅×Q₈	—	2	1	no	—
100	C₄×C₅×D₄	—	2	1	no	—
101	C₄×C₅×D₄	—	2	1	no	—
102	C₄×C₅×D₄	—	2	1	no	—
103	C₄×C₅×D₄	—	2	1	no	—
104	C₄×C₅×D₄	—	2	1	no	—
105	C₄×C₅×D₄	—	2	1	no	—
106	C₄×C₅×D₄	—	4	1	no	—
107	C₄×C₅×D₄	—	4	1	no	—
108	C₅×#117	—	2	1	no	—
109	C₅×#117	—	2	1	no	—
110	C₅×#117	—	4	1	no	—
111	C₅×#117	C₂³	1	1	yes	characteristic
112	C₅×#117	C₂³	1	1	yes	characteristic
113	C₅×#117	C₂³	1	1	yes	characteristic
114	C₂×C₅×C₄⋊C₄	—	2	1	no	—
115	C₂×C₅×C₄⋊C₄	—	2	1	no	—
116	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
117	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
118	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
119	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
120	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
121	C₅×#108	C₂×C₄	1	4	yes	complemented
122	C₅×#107	—	2	1	no	—
123	C₅×#107	—	2	1	no	—
124	C₅×#107	—	2	1	no	—
125	C₅×#107	—	2	1	no	—
126	C₅×#107	—	2	1	no	—
127	C₅×#107	—	2	1	no	—
128	C₅×#107	D₄	1	1	yes	characteristic
129	C₅×#107	D₄	1	1	yes	characteristic
130	C₅×#106	—	2	1	no	—
131	C₅×#106	—	2	1	no	—
132	C₅×#106	—	2	1	no	—
133	C₅×#106	—	2	1	no	—
134	C₅×#106	D₄	1	1	yes	characteristic
135	C₅×#106	D₄	1	1	yes	characteristic
136	C₅×#104	—	2	2	no	—
137	C₅×#103	—	2	1	no	—
138	C₅×#103	—	2	1	no	—
139	C₅×#103	—	2	1	no	—
140	C₅×#103	—	2	1	no	—
141	C₅×#103	D₄	1	1	yes	characteristic
142	C₅×#103	D₄	1	1	yes	characteristic
143	C₅×#102	—	2	1	no	—
144	C₅×#102	—	2	1	no	—
145	C₅×#102	—	2	1	no	—
146	C₅×#102	—	2	1	no	—
147	C₅×#102	D₄	1	1	yes	characteristic
148	C₅×#102	D₄	1	1	yes	characteristic
149	C₅×#101	C₂×C₄	1	4	yes	complemented
150	C₅×#100	C₂×C₄	1	4	yes	complemented
151	C₅×#97	—	2	1	no	—
152	C₅×#97	—	2	1	no	—
153	C₅×#97	—	2	1	no	—
154	C₅×#97	—	2	2	no	—
155	C₄×C₅×C₈	—	2	1	no	—
156	C₄×C₅×C₈	—	2	1	no	—
157	C₄×C₅×C₈	—	2	1	no	—
158	C₅×#95	—	4	1	no	—
159	C₂×#471	C₂×C₅	1	1	yes	characteristic
160	#2949	C₂×C₅	1	1	yes	characteristic, complemented
161	#2947	C₂×C₅	1	1	yes	characteristic, complemented
162	#2946	C₂×C₅	1	1	yes	characteristic
163	#2913	C₂×C₅	1	1	yes	characteristic, complemented
164	#1973	C₂×C₅	1	1	yes	characteristic, complemented
165	#1952	C₂×C₅	1	1	yes	characteristic, complemented
166	#1931	C₂×C₅	1	1	yes	characteristic, complemented
167	#1930	C₂×C₅	1	1	yes	characteristic, complemented
168	#1883	C₂×C₅	1	1	yes	characteristic
169	#1882	C₂×C₅	1	1	yes	characteristic, complemented
170	#1857	C₂×C₅	1	1	yes	characteristic, complemented
171	#1856	C₂×C₅	1	1	yes	characteristic, complemented
172	#1773	C₂×C₅	1	1	yes	characteristic, complemented
173	#1765	C₂×C₅	1	1	yes	characteristic, complemented
174	C₂⁴×C₅	—	2	1	no	—
175	C₅×#41	—	2	1	no	—
176	C₅×#41	—	2	1	no	—
177	C₅×#41	C₂²×C₄	1	2	yes	—
178	C₅×#41	#41	1	2	yes	—
179	C₂×C₅×Q₈	—	2	1	no	—
180	C₂×C₅×Q₈	—	2	1	no	—
181	C₂×C₅×Q₈	—	2	1	no	—
182	C₂×C₅×Q₈	—	2	1	no	—
183	C₂×C₅×Q₈	—	2	1	no	—
184	C₂×C₅×Q₈	C₂²×C₄	1	2	yes	—
185	C₂×C₅×Q₈	C₂×D₄	1	1	yes	characteristic
186	C₂×C₅×D₄	—	2	1	no	—
187	C₂×C₅×D₄	—	2	1	no	—
188	C₂×C₅×D₄	—	2	1	no	—
189	C₂×C₅×D₄	—	2	1	no	—
190	C₂×C₅×D₄	—	2	1	no	—
191	C₂×C₅×D₄	—	2	1	no	—
192	C₂×C₅×D₄	C₂²×C₄	1	2	yes	—
193	C₂×C₅×D₄	C₂×D₄	1	1	yes	characteristic
194	C₂²×C₄×C₅	—	2	1	no	—
195	C₂²×C₄×C₅	—	2	1	no	—
196	C₂²×C₄×C₅	—	2	1	no	—
197	C₂²×C₄×C₅	—	2	1	no	—
198	C₂²×C₄×C₅	—	2	1	no	—
199	C₂²×C₄×C₅	—	4	1	no	—
200	C₂²×C₄×C₅	—	4	1	no	—
201	C₂²×C₄×C₅	—	4	1	no	—
202	C₂²×C₄×C₅	—	4	1	no	—
203	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
204	C₅×Q₁₆	—	2	4	no	—
205	C₅×SD₁₆	—	2	4	no	—
206	C₅×SD₁₆	—	2	4	no	—
207	C₅×SD₁₆	—	2	4	no	—
208	C₅×SD₁₆	—	2	4	no	—
209	C₅×D₈	—	2	4	no	—
210	C₅×M₁₆	—	2	1	no	—
211	C₅×M₁₆	—	2	1	no	—
212	C₅×M₁₆	—	2	1	no	—
213	C₅×M₁₆	—	2	4	no	—
214	C₅×M₁₆	—	2	1	no	—
215	C₅×M₁₆	—	2	1	no	—
216	C₅×M₁₆	C₂²×C₄	1	2	yes	—
217	C₅×M₁₆	C₂²×C₄	1	2	yes	—
218	C₅×M₁₆	C₂²×C₄	1	2	yes	—
219	C₂×C₅×C₈	—	2	1	no	—
220	C₂×C₅×C₈	—	2	1	no	—
221	C₂×C₅×C₈	—	2	1	no	—
222	C₂×C₅×C₈	—	2	1	no	—
223	C₂×C₅×C₈	—	2	1	no	—
224	C₂×C₅×C₈	—	2	4	no	—
225	C₂×C₅×C₈	—	2	1	no	—
226	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
227	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
228	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
229	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
230	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
231	C₂×C₅×C₈	#41	1	2	yes	—
232	C₅×C₄⋊C₄	—	2	1	no	—
233	C₅×C₄⋊C₄	—	2	1	no	—
234	C₅×C₄⋊C₄	—	2	1	no	—
235	C₅×C₄⋊C₄	—	2	1	no	—
236	C₅×C₄⋊C₄	—	4	1	no	—
237	C₅×C₄⋊C₄	—	4	1	no	—
238	C₅×C₄⋊C₄	—	4	1	no	—
239	C₅×C₄⋊C₄	—	4	1	no	—
240	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
241	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
242	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
243	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
244	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
245	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
246	C₅×C₂²⋊C₄	—	2	1	no	—
247	C₅×C₂²⋊C₄	—	2	1	no	—
248	C₅×C₂²⋊C₄	—	2	1	no	—
249	C₅×C₂²⋊C₄	—	4	1	no	—
250	C₅×C₂²⋊C₄	—	4	1	no	—
251	C₅×C₂²⋊C₄	—	4	1	no	—
252	C₅×C₂²⋊C₄	—	4	1	no	—
253	C₅×C₂²⋊C₄	—	8	2	no	—
254	C₄²×C₅	—	2	1	no	—
255	C₄²×C₅	—	2	1	no	—
256	C₄²×C₅	—	2	1	no	—
257	C₄²×C₅	—	2	1	no	—
258	C₄²×C₅	—	2	1	no	—
259	C₄²×C₅	—	4	1	no	—
260	C₄²×C₅	—	4	1	no	—
261	C₄²×C₅	—	4	1	no	—
262	C₄²×C₅	—	4	1	no	—
263	C₂×#137	C₂²×C₅	1	1	yes	characteristic
264	C₂×AGL(1,ℤ/8)	C₂²×C₅	1	1	yes	characteristic
265	C₂²×SD₁₆	C₂²×C₅	1	1	yes	characteristic
266	C₂×#131	C₂²×C₅	1	1	yes	characteristic
267	#520	C₂²×C₅	1	1	yes	characteristic
268	#520	C₂²×C₅	1	1	yes	characteristic
269	#519	C₂²×C₅	1	1	yes	characteristic
270	#518	C₂²×C₅	1	1	yes	characteristic
271	#471	C₄×C₅	1	8	yes	complemented
272	#442	—	2	1	no	—
273	#441	—	2	1	no	—
274	#440	—	2	1	no	—
275	#440	—	2	1	no	—
276	#440	—	2	1	no	—
277	#440	—	2	1	no	—
278	C₄×Q₁₆	—	2	1	no	—
279	C₄×SD₁₆	—	2	1	no	—
280	C₄×SD₁₆	—	2	1	no	—
281	C₄×SD₁₆	—	2	1	no	—
282	C₄×SD₁₆	—	2	1	no	—
283	C₄×D₈	—	2	1	no	—
284	C₂×#108	C₂²×C₅	1	1	yes	characteristic
285	#428	—	2	1	no	—
286	#428	—	2	1	no	—
287	#428	C₂²×C₅	1	1	yes	characteristic
288	#428	C₂²×C₅	1	1	yes	characteristic
289	#428	C₂²×C₅	1	1	yes	characteristic
290	#428	C₂²×C₅	1	1	yes	characteristic
291	#427	—	2	1	no	—
292	#427	C₂²×C₅	1	1	yes	characteristic
293	C₂×#107	—	2	1	no	—
294	C₂×#107	C₂²×C₅	1	1	yes	characteristic
295	C₂×#104	C₂²×C₅	1	1	yes	characteristic
296	#419	C₂²×C₅	1	1	yes	characteristic
297	#418	C₂²×C₅	1	1	yes	characteristic
298	#416	C₂²×C₅	1	1	yes	characteristic
299	C₂×#103	C₂²×C₅	1	1	yes	characteristic
300	C₂×#102	C₂²×C₅	1	1	yes	characteristic
301	C₂×#101	C₂²×C₅	1	1	yes	characteristic
302	C₂×#100	C₂²×C₅	1	1	yes	characteristic
303	#405	C₂²×C₅	1	1	yes	characteristic
304	#405	C₂²×C₅	1	1	yes	characteristic
305	#405	C₂²×C₅	1	1	yes	characteristic
306	C₂×#97	C₂²×C₅	1	1	yes	characteristic
307	#391	C₂²×C₅	1	1	yes	characteristic
308	#390	C₂²×C₅	1	1	yes	characteristic
309	#343	C₂²×C₅	1	1	yes	characteristic
310	#343	C₂²×C₅	1	1	yes	characteristic
311	#341	C₂²×C₅	1	1	yes	characteristic
312	#339	C₂²×C₅	1	1	yes	characteristic
313	#339	C₂²×C₅	1	1	yes	characteristic
314	#339	C₂²×C₅	1	1	yes	characteristic
315	C₂³×C₅	—	2	1	no	—
316	C₂³×C₅	—	2	2	no	—
317	C₂³×C₅	—	4	1	no	—
318	C₂³×C₅	—	4	1	no	—
319	C₂³×C₅	—	8	1	no	—
320	C₂³×C₅	—	8	1	no	—
321	C₂³×C₅	H(5,𝔽₂)	1	1	yes	characteristic
322	C₅×Q₈	—	2	1	no	—
323	C₅×Q₈	—	2	2	no	—
324	C₅×Q₈	—	2	2	no	—
325	C₅×Q₈	—	4	1	no	—
326	C₅×Q₈	—	4	1	no	—
327	C₅×Q₈	C₄×D₄	1	2	yes	—
328	C₅×D₄	—	2	2	no	—
329	C₅×D₄	—	2	2	no	—
330	C₅×D₄	—	2	2	no	—
331	C₅×D₄	—	2	2	no	—
332	C₅×D₄	—	2	1	no	—
333	C₅×D₄	—	4	1	no	—
334	C₅×D₄	—	4	1	no	—
335	C₅×D₄	C₄×D₄	1	2	yes	—
336	C₂×C₄×C₅	—	2	1	no	—
337	C₂×C₄×C₅	—	2	2	no	—
338	C₂×C₄×C₅	—	2	2	no	—
339	C₂×C₄×C₅	—	2	2	no	—
340	C₂×C₄×C₅	—	2	1	no	—
341	C₂×C₄×C₅	—	2	1	no	—
342	C₂×C₄×C₅	—	2	1	no	—
343	C₂×C₄×C₅	—	2	1	no	—
344	C₂×C₄×C₅	—	2	1	no	—
345	C₂×C₄×C₅	—	2	1	no	—
346	C₂×C₄×C₅	—	4	1	no	—
347	C₂×C₄×C₅	—	4	1	no	—
348	C₂×C₄×C₅	—	4	1	no	—
349	C₂×C₄×C₅	—	4	1	no	—
350	C₂×C₄×C₅	—	4	1	no	—
351	C₂×C₄×C₅	—	4	1	no	—
352	C₂×C₄×C₅	—	4	1	no	—
353	C₂×C₄×C₅	—	8	1	no	—
354	C₂×C₄×C₅	—	8	1	no	—
355	C₂×C₄×C₅	—	16	1	no	—
356	C₂×C₄×C₅	—	16	1	no	—
357	C₂×C₄×C₅	—	16	1	no	—
358	C₂×C₄×C₅	—	16	1	no	—
359	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
360	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
361	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
362	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
363	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
364	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
365	C₅×C₈	—	2	2	no	—
366	C₅×C₈	—	2	2	no	—
367	C₅×C₈	—	2	2	no	—
368	C₅×C₈	—	2	1	no	—
369	C₅×C₈	—	2	1	no	—
370	C₅×C₈	—	4	1	no	—
371	C₅×C₈	C₄×D₄	1	2	yes	—
372	C₅×C₈	C₄×D₄	1	2	yes	—
373	C₂×#41	C₂³×C₅	1	1	yes	characteristic
374	C₂²×Q₈	C₂³×C₅	1	1	yes	characteristic
375	C₂²×D₄	C₂³×C₅	1	1	yes	characteristic
376	#137	—	2	2	no	—
377	#137	C₂×C₄×C₅	1	4	yes	complemented
378	AGL(1,ℤ/8)	—	2	2	no	—
379	AGL(1,ℤ/8)	C₂×C₄×C₅	1	4	yes	—
380	C₂×Q₁₆	—	2	1	no	—
381	C₂×SD₁₆	—	2	1	no	—
382	C₂×SD₁₆	—	2	1	no	—
383	C₂×SD₁₆	—	2	1	no	—
384	C₂×SD₁₆	—	2	1	no	—
385	C₂×SD₁₆	—	2	2	no	—
386	C₂×SD₁₆	C₂×C₄×C₅	1	4	yes	complemented
387	C₂×D₈	—	2	1	no	—
388	#131	—	2	2	no	—
389	#131	C₂×C₄×C₅	1	4	yes	complemented
390	C₂×M₁₆	—	2	1	no	—
391	C₂×M₁₆	C₂³×C₅	1	1	yes	characteristic
392	C₂×M₁₆	C₂³×C₅	1	1	yes	characteristic
393	C₂×M₁₆	C₂³×C₅	1	1	yes	characteristic
394	C₂×M₁₆	C₂³×C₅	1	1	yes	characteristic
395	C₂²×C₈	—	2	1	no	—
396	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
397	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
398	C₄×Q₈	—	2	1	no	—
399	C₄×Q₈	—	2	1	no	—
400	C₄×Q₈	—	2	1	no	—
401	C₄×Q₈	—	2	1	no	—
402	C₄×Q₈	—	2	1	no	—
403	C₄×Q₈	—	2	1	no	—
404	C₄×D₄	—	2	1	no	—
405	C₄×D₄	—	2	1	no	—
406	C₄×D₄	—	2	1	no	—
407	C₄×D₄	—	2	1	no	—
408	C₄×D₄	—	2	1	no	—
409	C₄×D₄	—	2	1	no	—
410	C₄×D₄	—	4	1	no	—
411	C₄×D₄	—	4	1	no	—
412	#117	—	2	1	no	—
413	#117	—	2	1	no	—
414	#117	—	4	1	no	—
415	#117	C₂³×C₅	1	1	yes	characteristic
416	#117	C₂³×C₅	1	1	yes	characteristic
417	#117	C₂³×C₅	1	1	yes	characteristic
418	C₂×C₄⋊C₄	—	2	1	no	—
419	C₂×C₄⋊C₄	—	2	1	no	—
420	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
421	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
422	C₂×C₂²⋊C₄	—	4	1	no	—
423	C₂×C₂²⋊C₄	—	4	1	no	—
424	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
425	#108	C₂×C₄×C₅	1	4	yes	complemented
426	#107	—	2	1	no	—
427	#107	—	2	1	no	—
428	#107	—	2	1	no	—
429	#107	—	2	1	no	—
430	#107	—	2	1	no	—
431	#107	—	2	1	no	—
432	#107	C₅×D₄	1	1	yes	characteristic
433	#107	C₅×D₄	1	1	yes	characteristic
434	#106	—	2	1	no	—
435	#106	—	2	1	no	—
436	#106	—	2	1	no	—
437	#106	—	2	1	no	—
438	#106	C₅×D₄	1	1	yes	characteristic
439	#106	C₅×D₄	1	1	yes	characteristic
440	#104	—	2	2	no	—
441	#103	—	2	1	no	—
442	#103	—	2	1	no	—
443	#103	—	2	1	no	—
444	#103	—	2	1	no	—
445	#103	C₅×D₄	1	1	yes	characteristic
446	#103	C₅×D₄	1	1	yes	characteristic
447	#102	—	2	1	no	—
448	#102	—	2	1	no	—
449	#102	—	2	1	no	—
450	#102	—	2	1	no	—
451	#102	C₅×D₄	1	1	yes	characteristic
452	#102	C₅×D₄	1	1	yes	characteristic
453	#101	C₂×C₄×C₅	1	4	yes	complemented
454	#100	C₂×C₄×C₅	1	4	yes	complemented
455	#97	—	2	1	no	—
456	#97	—	2	1	no	—
457	#97	—	2	1	no	—
458	#97	—	2	2	no	—
459	C₄×C₈	—	2	1	no	—
460	C₄×C₈	—	2	1	no	—
461	C₄×C₈	—	2	1	no	—
462	#95	—	4	1	no	—
463	C₂²×C₅	—	2	1	no	—
464	C₂²×C₅	—	2	1	no	—
465	C₂²×C₅	—	2	2	no	—
466	C₂²×C₅	—	4	1	no	—
467	C₂²×C₅	—	4	1	no	—
468	C₂²×C₅	—	4	2	no	—
469	C₂²×C₅	—	8	2	no	—
470	C₂²×C₅	—	8	1	no	—
471	C₂²×C₅	—	8	1	no	—
472	C₂²×C₅	—	16	1	no	—
473	C₂²×C₅	#520	1	1	yes	characteristic
474	C₂²×C₅	D₄²	1	1	yes	characteristic
475	C₂²×C₅	#553	1	1	yes	characteristic
476	C₄×C₅	—	2	2	no	—
477	C₄×C₅	—	4	2	no	—
478	C₄×C₅	—	4	1	no	—
479	C₄×C₅	—	4	1	no	—
480	C₄×C₅	—	4	1	no	—
481	C₄×C₅	—	4	1	no	—
482	C₄×C₅	—	4	1	no	—
483	C₄×C₅	—	4	1	no	—
484	C₄×C₅	—	8	1	no	—
485	C₄×C₅	—	8	1	no	—
486	C₄×C₅	—	8	1	no	—
487	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
488	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
489	C₄×C₅	#547	1	1	yes	characteristic
490	C₄×C₅	#548	1	1	yes	characteristic
491	C₂⁴	—	2	1	no	—
492	#41	—	2	1	no	—
493	#41	—	2	1	no	—
494	#41	C₂²×C₄×C₅	1	2	yes	—
495	#41	C₅×#41	1	2	yes	—
496	C₂×Q₈	—	2	1	no	—
497	C₂×Q₈	—	2	1	no	—
498	C₂×Q₈	—	2	1	no	—
499	C₂×Q₈	—	2	1	no	—
500	C₂×Q₈	—	2	1	no	—
501	C₂×Q₈	C₂²×C₄×C₅	1	2	yes	—
502	C₂×Q₈	C₂×C₅×D₄	1	1	yes	characteristic
503	C₂×D₄	—	2	1	no	—
504	C₂×D₄	—	2	1	no	—
505	C₂×D₄	—	2	1	no	—
506	C₂×D₄	—	2	1	no	—
507	C₂×D₄	—	2	1	no	—
508	C₂×D₄	—	2	1	no	—
509	C₂×D₄	C₂²×C₄×C₅	1	2	yes	—
510	C₂×D₄	C₂×C₅×D₄	1	1	yes	characteristic
511	C₂²×C₄	—	2	1	no	—
512	C₂²×C₄	—	2	1	no	—
513	C₂²×C₄	—	2	1	no	—
514	C₂²×C₄	—	2	1	no	—
515	C₂²×C₄	—	2	1	no	—
516	C₂²×C₄	—	4	1	no	—
517	C₂²×C₄	—	4	1	no	—
518	C₂²×C₄	—	4	1	no	—
519	C₂²×C₄	—	4	1	no	—
520	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
521	Q₁₆	—	2	4	no	—
522	SD₁₆	—	2	4	no	—
523	SD₁₆	—	2	4	no	—
524	SD₁₆	—	2	4	no	—
525	SD₁₆	—	2	4	no	—
526	D₈	—	2	4	no	—
527	M₁₆	—	2	1	no	—
528	M₁₆	—	2	1	no	—
529	M₁₆	—	2	1	no	—
530	M₁₆	—	2	4	no	—
531	M₁₆	—	2	1	no	—
532	M₁₆	—	2	1	no	—
533	M₁₆	C₂²×C₄×C₅	1	2	yes	—
534	M₁₆	C₂²×C₄×C₅	1	2	yes	—
535	M₁₆	C₂²×C₄×C₅	1	2	yes	—
536	C₂×C₈	—	2	1	no	—
537	C₂×C₈	—	2	1	no	—
538	C₂×C₈	—	2	1	no	—
539	C₂×C₈	—	2	1	no	—
540	C₂×C₈	—	2	1	no	—
541	C₂×C₈	—	2	4	no	—
542	C₂×C₈	—	2	1	no	—
543	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
544	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
545	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
546	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
547	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
548	C₂×C₈	C₅×#41	1	2	yes	—
549	C₄⋊C₄	—	2	1	no	—
550	C₄⋊C₄	—	2	1	no	—
551	C₄⋊C₄	—	2	1	no	—
552	C₄⋊C₄	—	2	1	no	—
553	C₄⋊C₄	—	4	1	no	—
554	C₄⋊C₄	—	4	1	no	—
555	C₄⋊C₄	—	4	1	no	—
556	C₄⋊C₄	—	4	1	no	—
557	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
558	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
559	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
560	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
561	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
562	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
563	C₂²⋊C₄	—	2	1	no	—
564	C₂²⋊C₄	—	2	1	no	—
565	C₂²⋊C₄	—	2	1	no	—
566	C₂²⋊C₄	—	4	1	no	—
567	C₂²⋊C₄	—	4	1	no	—
568	C₂²⋊C₄	—	4	1	no	—
569	C₂²⋊C₄	—	4	1	no	—
570	C₂²⋊C₄	—	8	2	no	—
571	C₄²	—	2	1	no	—
572	C₄²	—	2	1	no	—
573	C₄²	—	2	1	no	—
574	C₄²	—	2	1	no	—
575	C₄²	—	2	1	no	—
576	C₄²	—	4	1	no	—
577	C₄²	—	4	1	no	—
578	C₄²	—	4	1	no	—
579	C₄²	—	4	1	no	—
580	C₂×C₅	—	2	1	no	—
581	C₂×C₅	—	2	1	no	—
582	C₂×C₅	—	4	2	no	—
583	C₂×C₅	—	8	2	no	—
584	C₂×C₅	#2361	1	1	yes	characteristic
585	C₂×C₅	#3288	1	1	yes	characteristic
586	C₂×C₅	#3289	1	1	yes	characteristic
587	C₂³	—	2	1	no	—
588	C₂³	—	2	2	no	—
589	C₂³	—	4	1	no	—
590	C₂³	—	4	1	no	—
591	C₂³	—	8	1	no	—
592	C₂³	—	8	1	no	—
593	C₂³	C₅×H(5,𝔽₂)	1	1	yes	characteristic
594	Q₈	—	2	1	no	—
595	Q₈	—	2	2	no	—
596	Q₈	—	2	2	no	—
597	Q₈	—	4	1	no	—
598	Q₈	—	4	1	no	—
599	Q₈	C₄×C₅×D₄	1	2	yes	—
600	D₄	—	2	2	no	—
601	D₄	—	2	2	no	—
602	D₄	—	2	2	no	—
603	D₄	—	2	2	no	—
604	D₄	—	2	1	no	—
605	D₄	—	4	1	no	—
606	D₄	—	4	1	no	—
607	D₄	C₄×C₅×D₄	1	2	yes	—
608	C₂×C₄	—	2	1	no	—
609	C₂×C₄	—	2	2	no	—
610	C₂×C₄	—	2	2	no	—
611	C₂×C₄	—	2	2	no	—
612	C₂×C₄	—	2	1	no	—
613	C₂×C₄	—	2	1	no	—
614	C₂×C₄	—	2	1	no	—
615	C₂×C₄	—	2	1	no	—
616	C₂×C₄	—	2	1	no	—
617	C₂×C₄	—	2	1	no	—
618	C₂×C₄	—	4	1	no	—
619	C₂×C₄	—	4	1	no	—
620	C₂×C₄	—	4	1	no	—
621	C₂×C₄	—	4	1	no	—
622	C₂×C₄	—	4	1	no	—
623	C₂×C₄	—	4	1	no	—
624	C₂×C₄	—	4	1	no	—
625	C₂×C₄	—	8	1	no	—
626	C₂×C₄	—	8	1	no	—
627	C₂×C₄	—	16	1	no	—
628	C₂×C₄	—	16	1	no	—
629	C₂×C₄	—	16	1	no	—
630	C₂×C₄	—	16	1	no	—
631	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
632	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
633	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
634	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
635	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
636	C₂×C₄	C₅×2₋¹⁺⁴	1	1	yes	characteristic
637	C₈	—	2	2	no	—
638	C₈	—	2	2	no	—
639	C₈	—	2	2	no	—
640	C₈	—	2	1	no	—
641	C₈	—	2	1	no	—
642	C₈	—	4	1	no	—
643	C₈	C₄×C₅×D₄	1	2	yes	—
644	C₈	C₄×C₅×D₄	1	2	yes	—
645	C₅	#23049	1	1	yes	characteristic, direct product factor, Sylow
646	C₂²	—	2	1	no	—
647	C₂²	—	2	1	no	—
648	C₂²	—	2	2	no	—
649	C₂²	—	4	1	no	—
650	C₂²	—	4	1	no	—
651	C₂²	—	4	2	no	—
652	C₂²	—	8	2	no	—
653	C₂²	—	8	1	no	—
654	C₂²	—	8	1	no	—
655	C₂²	—	16	1	no	—
656	C₂²	C₅×#520	1	1	yes	characteristic
657	C₂²	C₅×D₄²	1	1	yes	characteristic
658	C₂²	C₅×#553	1	1	yes	characteristic
659	C₄	—	2	2	no	—
660	C₄	—	4	1	no	—
661	C₄	—	4	1	no	—
662	C₄	—	4	1	no	—
663	C₄	—	4	1	no	—
664	C₄	—	4	1	no	—
665	C₄	—	4	1	no	—
666	C₄	—	4	2	no	—
667	C₄	—	8	1	no	—
668	C₄	—	8	1	no	—
669	C₄	—	8	1	no	—
670	C₄	C₂×C₄×C₅×D₄	1	1	yes	characteristic
671	C₄	C₂×C₄×C₅×D₄	1	1	yes	characteristic
672	C₄	C₅×#547	1	1	yes	characteristic
673	C₄	C₅×#548	1	1	yes	characteristic
674	C₂	—	2	1	no	—
675	C₂	—	2	1	no	—
676	C₂	—	4	2	no	—
677	C₂	—	8	2	no	—
678	C₂	C₅×#2361	1	1	yes	characteristic
679	C₂	C₅×#3288	1	1	yes	characteristic
680	C₂	C₅×#3289	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	D₄²
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
prime radical	C₂×C₅×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#2949	C₂	semidirect
3. C₅×#2947	C₂	semidirect
5. C₅×#2913	C₂	semidirect
6. C₅×#1973	C₂	semidirect
7. C₅×#1952	C₂	semidirect
8. C₅×#1931	C₂	semidirect
9. C₅×#1930	C₂	semidirect
11. C₅×#1882	C₂	semidirect
12. C₅×#1857	C₂	semidirect
13. C₅×#1856	C₂	semidirect
14. C₅×#1773	C₂	semidirect
15. C₅×#1765	C₂	semidirect
24. C₅×#471	C₄	semidirect
68. #23049	C₅	direct (N × H)
73. C₅×#137	C₂×C₄	semidirect
82. C₂×C₅×SD₁₆	C₂×C₄	semidirect
85. C₅×#131	C₂×C₄	semidirect
121. C₅×#108	C₂×C₄	semidirect
149. C₅×#101	C₂×C₄	semidirect
150. C₅×#100	C₂×C₄	semidirect
160. #2949	C₂×C₅	semidirect
161. #2947	C₂×C₅	semidirect
163. #2913	C₂×C₅	semidirect
164. #1973	C₂×C₅	semidirect
165. #1952	C₂×C₅	semidirect
166. #1931	C₂×C₅	semidirect
167. #1930	C₂×C₅	semidirect
169. #1882	C₂×C₅	semidirect
170. #1857	C₂×C₅	semidirect
171. #1856	C₂×C₅	semidirect
172. #1773	C₂×C₅	semidirect
173. #1765	C₂×C₅	semidirect
271. #471	C₄×C₅	semidirect
377. #137	C₂×C₄×C₅	semidirect
386. C₂×SD₁₆	C₂×C₄×C₅	semidirect
389. #131	C₂×C₄×C₅	semidirect
425. #108	C₂×C₄×C₅	semidirect
453. #101	C₂×C₄×C₅	semidirect
454. #100	C₂×C₄×C₅	semidirect
645. C₅	#23049	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×#471	2
C₅×#2949	2
C₅×#2947	2
C₅×#2946	2
C₅×#2913	2
C₅×#1973	2
C₅×#1952	2
C₅×#1931	2
C₅×#1930	2
C₅×#1883	2
C₅×#1882	2
C₅×#1857	2
C₅×#1856	2
C₅×#1773	2
C₅×#1765	2
#23049	5

Subgroups — C5×#23049 #11945769

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#23049 #11945769