Subgroups — C₅×#23472 #11946192

Order 1280.

1148 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3458	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#421	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#421	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#410	C₂	1	1	yes	maximal, characteristic
5	C₅×#2031	C₂	1	8	yes	maximal, complemented
6	C₂×C₅×#344	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1733	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₅×#41	C₂²	1	1	yes	characteristic
10	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
12	C₅×#544	—	2	2	no	—
13	C₅×#544	—	2	2	no	—
14	C₅×#544	—	2	2	no	—
15	C₅×#543	—	2	2	no	—
16	C₅×#533	—	2	2	no	—
17	C₅×#533	—	2	2	no	—
18	C₅×#533	C₂²	1	4	yes	complemented
19	C₂×C₅×#117	C₂²	1	1	yes	characteristic
20	C₂×C₅×#117	C₂²	1	1	yes	characteristic
21	C₂×C₅×#117	C₂²	1	1	yes	characteristic
22	C₂×C₅×#117	C₂²	1	1	yes	characteristic
23	C₅×#421	—	2	2	no	—
24	C₅×#421	—	2	2	no	—
25	C₅×#421	—	2	2	no	—
26	C₅×#421	C₂²	1	4	yes	—
27	C₂×C₅×#104	—	2	2	no	—
28	C₂×C₅×#104	C₂²	1	4	yes	—
29	C₅×#410	—	2	2	no	—
30	C₅×#410	C₂²	1	4	yes	—
31	C₂×C₅×#99	—	2	2	no	—
32	C₅×#344	C₂²	1	4	yes	complemented
33	C₅×#339	C₂²	1	4	yes	complemented
34	C₅×H(3,ℤ/4)	C₂²	1	4	yes	complemented
35	#23472	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂×C₅×#41	—	2	1	no	—
37	C₂×C₅×#41	—	2	1	no	—
38	C₂×C₅×#41	—	2	1	no	—
39	C₂×C₅×#41	—	2	1	no	—
40	C₂×C₅×#41	—	2	1	no	—
41	C₂×C₅×#41	—	2	1	no	—
42	C₂×C₅×#41	—	2	1	no	—
43	C₂×C₅×#41	—	2	1	no	—
44	C₂×C₅×#41	—	2	1	no	—
45	C₂×C₅×#41	—	2	1	no	—
46	C₂×C₅×#41	Q₈	1	2	yes	complemented
47	C₂×C₅×#41	C₂³	1	2	yes	—
48	C₂²×C₅×Q₈	Q₈	1	1	yes	characteristic, complemented
49	C₂²×C₅×D₄	—	2	1	no	—
50	C₂²×C₅×D₄	Q₈	1	1	yes	characteristic, complemented
51	C₂³×C₄×C₅	—	2	1	no	—
52	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
53	C₂×C₅×M₁₆	—	2	1	no	—
54	C₂×C₅×M₁₆	—	2	1	no	—
55	C₂×C₅×M₁₆	—	2	1	no	—
56	C₂×C₅×M₁₆	—	2	1	no	—
57	C₂×C₅×M₁₆	—	2	1	no	—
58	C₂×C₅×M₁₆	—	2	1	no	—
59	C₂×C₅×M₁₆	—	2	1	no	—
60	C₂×C₅×M₁₆	—	2	1	no	—
61	C₂×C₅×M₁₆	—	2	1	no	—
62	C₂×C₅×M₁₆	—	2	1	no	—
63	C₂×C₅×M₁₆	C₂³	1	2	yes	—
64	C₂×C₅×M₁₆	C₂³	1	2	yes	—
65	C₂²×C₅×C₈	—	2	1	no	—
66	C₂²×C₅×C₈	—	2	1	no	—
67	C₅×C₄⋊Q₈	—	2	4	no	—
68	C₅×C₄⋊Q₈	—	2	4	no	—
69	C₅×C₄⋊Q₈	—	2	4	no	—
70	C₅×#125	—	2	4	no	—
71	C₅×#125	—	2	4	no	—
72	C₅×#125	—	2	4	no	—
73	C₅×C₂²⋊Q₈	—	2	8	no	—
74	C₅×C₂²⋊Q₈	—	2	8	no	—
75	C₅×C₂²⋊Q₈	—	2	8	no	—
76	C₄×C₅×Q₈	—	2	8	no	—
77	C₅×#117	—	2	1	no	—
78	C₅×#117	—	2	1	no	—
79	C₅×#117	—	2	1	no	—
80	C₅×#117	—	2	1	no	—
81	C₅×#117	—	2	1	no	—
82	C₅×#117	—	2	1	no	—
83	C₅×#117	—	2	1	no	—
84	C₅×#117	—	2	1	no	—
85	C₅×#117	—	2	1	no	—
86	C₅×#117	D₄	1	2	yes	—
87	C₅×#117	D₄	1	2	yes	—
88	C₅×#117	D₄	1	2	yes	—
89	C₅×#117	D₄	1	2	yes	—
90	C₅×#117	C₂³	1	2	yes	—
91	C₅×#117	C₂³	1	2	yes	—
92	C₅×#117	C₂³	1	2	yes	—
93	C₂×C₅×C₄⋊C₄	—	2	1	no	—
94	C₂×C₅×C₄⋊C₄	—	2	1	no	—
95	C₂×C₅×C₄⋊C₄	—	2	1	no	—
96	C₂×C₅×C₄⋊C₄	—	2	1	no	—
97	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
98	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
99	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
100	C₂×C₅×C₂²⋊C₄	D₄	1	2	yes	—
101	C₂×C₄²×C₅	—	2	1	no	—
102	C₂×C₄²×C₅	—	2	1	no	—
103	C₂×C₄²×C₅	D₄	1	2	yes	—
104	C₂×C₄²×C₅	C₂³	1	2	yes	—
105	C₅×#104	—	2	8	no	—
106	C₅×#104	—	2	8	no	—
107	C₅×#99	—	2	8	no	—
108	#3458	C₂×C₅	1	1	yes	characteristic, complemented
109	C₂×#421	C₂×C₅	1	1	yes	characteristic
110	C₂×#421	C₂×C₅	1	1	yes	characteristic
111	C₂×#410	C₂×C₅	1	1	yes	characteristic
112	#2031	C₂×C₅	1	8	yes	complemented
113	C₂×#344	C₂×C₅	1	1	yes	characteristic, complemented
114	#1733	C₂×C₅	1	1	yes	characteristic, complemented
115	C₂×H(3,ℤ/4)	C₂×C₅	1	1	yes	characteristic, complemented
116	C₂⁴×C₅	—	2	1	no	—
117	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
118	C₅×#41	—	2	2	no	—
119	C₅×#41	—	2	2	no	—
120	C₅×#41	—	2	2	no	—
121	C₅×#41	—	2	2	no	—
122	C₅×#41	—	2	2	no	—
123	C₅×#41	—	2	2	no	—
124	C₅×#41	—	2	2	no	—
125	C₅×#41	—	2	2	no	—
126	C₅×#41	—	4	1	no	—
127	C₅×#41	—	4	1	no	—
128	C₅×#41	—	4	1	no	—
129	C₅×#41	—	4	1	no	—
130	C₅×#41	—	4	1	no	—
131	C₅×#41	—	4	1	no	—
132	C₅×#41	—	4	1	no	—
133	C₅×#41	—	4	1	no	—
134	C₂×C₅×Q₈	—	2	8	no	—
135	C₂×C₅×Q₈	—	2	1	no	—
136	C₂×C₅×Q₈	—	2	1	no	—
137	C₂×C₅×Q₈	—	2	1	no	—
138	C₂×C₅×Q₈	—	2	1	no	—
139	C₂×C₅×Q₈	—	2	1	no	—
140	C₂×C₅×Q₈	C₂×Q₈	1	2	yes	—
141	C₂×C₅×D₄	—	2	2	no	—
142	C₂×C₅×D₄	—	2	2	no	—
143	C₂×C₅×D₄	—	2	1	no	—
144	C₂×C₅×D₄	—	2	1	no	—
145	C₂×C₅×D₄	—	2	1	no	—
146	C₂×C₅×D₄	—	2	1	no	—
147	C₂×C₅×D₄	—	2	1	no	—
148	C₂×C₅×D₄	—	4	1	no	—
149	C₂×C₅×D₄	—	4	1	no	—
150	C₂×C₅×D₄	—	4	1	no	—
151	C₂×C₅×D₄	—	4	1	no	—
152	C₂×C₅×D₄	C₂×Q₈	1	2	yes	—
153	C₂²×C₄×C₅	—	2	1	no	—
154	C₂²×C₄×C₅	—	2	2	no	—
155	C₂²×C₄×C₅	—	2	2	no	—
156	C₂²×C₄×C₅	—	2	2	no	—
157	C₂²×C₄×C₅	—	2	2	no	—
158	C₂²×C₄×C₅	—	2	2	no	—
159	C₂²×C₄×C₅	—	4	1	no	—
160	C₂²×C₄×C₅	—	4	1	no	—
161	C₂²×C₄×C₅	—	4	1	no	—
162	C₂²×C₄×C₅	—	4	1	no	—
163	C₂²×C₄×C₅	—	4	1	no	—
164	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
167	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
168	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
169	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
170	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
171	C₂²×C₄×C₅	#41	1	1	yes	characteristic
172	C₂²×C₄×C₅	#41	1	1	yes	characteristic
173	C₂²×C₄×C₅	#41	1	1	yes	characteristic
174	C₂²×C₄×C₅	#41	1	1	yes	characteristic
175	C₂²×C₄×C₅	#41	1	1	yes	characteristic
176	C₂²×C₄×C₅	#41	1	1	yes	characteristic
177	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
178	C₅×M₁₆	—	2	2	no	—
179	C₅×M₁₆	—	2	2	no	—
180	C₅×M₁₆	—	2	2	no	—
181	C₅×M₁₆	—	2	2	no	—
182	C₅×M₁₆	—	4	1	no	—
183	C₅×M₁₆	—	4	1	no	—
184	C₅×M₁₆	—	4	1	no	—
185	C₅×M₁₆	—	4	1	no	—
186	C₂×C₅×C₈	—	2	2	no	—
187	C₂×C₅×C₈	—	2	2	no	—
188	C₂×C₅×C₈	—	4	1	no	—
189	C₂×C₅×C₈	—	4	1	no	—
190	C₂×C₅×C₈	—	4	1	no	—
191	C₂×C₅×C₈	—	4	1	no	—
192	C₅×C₄⋊C₄	—	2	8	no	—
193	C₅×C₄⋊C₄	—	2	8	no	—
194	C₅×C₄⋊C₄	—	2	8	no	—
195	C₅×C₄⋊C₄	—	2	2	no	—
196	C₅×C₄⋊C₄	—	2	2	no	—
197	C₅×C₄⋊C₄	—	2	2	no	—
198	C₅×C₄⋊C₄	—	2	2	no	—
199	C₅×C₄⋊C₄	—	2	2	no	—
200	C₅×C₄⋊C₄	—	2	2	no	—
201	C₅×C₄⋊C₄	—	2	2	no	—
202	C₅×C₄⋊C₄	—	4	1	no	—
203	C₅×C₂²⋊C₄	—	2	2	no	—
204	C₅×C₂²⋊C₄	—	2	2	no	—
205	C₅×C₂²⋊C₄	—	2	2	no	—
206	C₅×C₂²⋊C₄	—	2	2	no	—
207	C₅×C₂²⋊C₄	—	2	2	no	—
208	C₅×C₂²⋊C₄	—	2	2	no	—
209	C₅×C₂²⋊C₄	—	4	1	no	—
210	C₅×C₂²⋊C₄	C₂×D₄	1	4	yes	—
211	C₄²×C₅	—	2	2	no	—
212	C₄²×C₅	—	2	2	no	—
213	C₄²×C₅	—	2	2	no	—
214	C₄²×C₅	—	2	2	no	—
215	C₄²×C₅	—	2	2	no	—
216	C₄²×C₅	—	4	1	no	—
217	C₄²×C₅	C₂×D₄	1	4	yes	—
218	C₄²×C₅	C₂×D₄	1	4	yes	—
219	C₂²×#41	C₂²×C₅	1	1	yes	characteristic
220	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
221	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
222	#544	—	2	2	no	—
223	#544	—	2	2	no	—
224	#544	—	2	2	no	—
225	#543	—	2	2	no	—
226	#533	—	2	2	no	—
227	#533	—	2	2	no	—
228	#533	C₂²×C₅	1	4	yes	complemented
229	C₂×#117	C₂²×C₅	1	1	yes	characteristic
230	C₂×#117	C₂²×C₅	1	1	yes	characteristic
231	C₂×#117	C₂²×C₅	1	1	yes	characteristic
232	C₂×#117	C₂²×C₅	1	1	yes	characteristic
233	#421	—	2	2	no	—
234	#421	—	2	2	no	—
235	#421	—	2	2	no	—
236	#421	C₂²×C₅	1	4	yes	—
237	C₂×#104	—	2	2	no	—
238	C₂×#104	C₂²×C₅	1	4	yes	—
239	#410	—	2	2	no	—
240	#410	C₂²×C₅	1	4	yes	—
241	C₂×#99	—	2	2	no	—
242	#344	C₂²×C₅	1	4	yes	complemented
243	#339	C₂²×C₅	1	4	yes	complemented
244	H(3,ℤ/4)	C₂²×C₅	1	4	yes	complemented
245	C₂³×C₅	—	2	2	no	—
246	C₂³×C₅	—	4	1	no	—
247	C₂³×C₅	—	4	1	no	—
248	C₂³×C₅	—	8	1	no	—
249	C₂³×C₅	—	8	2	no	—
250	C₂³×C₅	#120	1	1	yes	characteristic
251	C₂³×C₅	#120	1	1	yes	characteristic
252	C₂³×C₅	C₂²⋊Q₈	1	1	yes	characteristic
253	C₂³×C₅	C₂²⋊Q₈	1	1	yes	characteristic
254	C₂³×C₅	#124	1	1	yes	characteristic
255	C₂³×C₅	#124	1	1	yes	characteristic
256	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
257	C₅×Q₈	—	2	2	no	—
258	C₅×Q₈	—	2	2	no	—
259	C₅×Q₈	—	4	1	no	—
260	C₅×Q₈	—	4	1	no	—
261	C₅×Q₈	—	8	8	no	—
262	C₅×D₄	—	2	2	no	—
263	C₅×D₄	—	2	2	no	—
264	C₅×D₄	—	4	2	no	—
265	C₅×D₄	—	4	2	no	—
266	C₅×D₄	—	4	2	no	—
267	C₅×D₄	—	4	2	no	—
268	C₅×D₄	—	4	1	no	—
269	C₅×D₄	—	4	1	no	—
270	C₂×C₄×C₅	—	2	4	no	—
271	C₂×C₄×C₅	—	2	4	no	—
272	C₂×C₄×C₅	—	2	4	no	—
273	C₂×C₄×C₅	—	2	2	no	—
274	C₂×C₄×C₅	—	4	1	no	—
275	C₂×C₄×C₅	—	4	1	no	—
276	C₂×C₄×C₅	—	4	2	no	—
277	C₂×C₄×C₅	—	4	2	no	—
278	C₂×C₄×C₅	—	4	2	no	—
279	C₂×C₄×C₅	—	4	2	no	—
280	C₂×C₄×C₅	—	4	2	no	—
281	C₂×C₄×C₅	—	8	2	no	—
282	C₂×C₄×C₅	—	8	2	no	—
283	C₂×C₄×C₅	—	8	2	no	—
284	C₂×C₄×C₅	—	8	2	no	—
285	C₂×C₄×C₅	#120	1	1	yes	characteristic
286	C₂×C₄×C₅	#120	1	1	yes	characteristic
287	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
288	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
289	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
290	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
291	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
292	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
293	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
294	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
295	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
296	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
297	C₂×C₄×C₅	#124	1	1	yes	characteristic
298	C₂×C₄×C₅	#124	1	1	yes	characteristic
299	C₂×C₄×C₅	#124	1	1	yes	characteristic
300	C₂×C₄×C₅	#124	1	1	yes	characteristic
301	C₂×C₄×C₅	#124	1	1	yes	characteristic
302	C₂×C₄×C₅	#124	1	1	yes	characteristic
303	C₂×C₄×C₅	#124	1	1	yes	characteristic
304	C₂×C₄×C₅	#124	1	1	yes	characteristic
305	C₂×C₄×C₅	#124	1	1	yes	characteristic
306	C₂×C₄×C₅	#124	1	1	yes	characteristic
307	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
308	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
309	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
310	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
311	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
312	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
313	C₅×C₈	—	4	2	no	—
314	C₅×C₈	—	4	2	no	—
315	C₂×#41	—	2	1	no	—
316	C₂×#41	—	2	1	no	—
317	C₂×#41	—	2	1	no	—
318	C₂×#41	—	2	1	no	—
319	C₂×#41	—	2	1	no	—
320	C₂×#41	—	2	1	no	—
321	C₂×#41	—	2	1	no	—
322	C₂×#41	—	2	1	no	—
323	C₂×#41	—	2	1	no	—
324	C₂×#41	—	2	1	no	—
325	C₂×#41	C₅×Q₈	1	2	yes	complemented
326	C₂×#41	C₂³×C₅	1	2	yes	—
327	C₂²×Q₈	C₅×Q₈	1	1	yes	characteristic, complemented
328	C₂²×D₄	—	2	1	no	—
329	C₂²×D₄	C₅×Q₈	1	1	yes	characteristic, complemented
330	C₂³×C₄	—	2	1	no	—
331	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
332	C₂×M₁₆	—	2	1	no	—
333	C₂×M₁₆	—	2	1	no	—
334	C₂×M₁₆	—	2	1	no	—
335	C₂×M₁₆	—	2	1	no	—
336	C₂×M₁₆	—	2	1	no	—
337	C₂×M₁₆	—	2	1	no	—
338	C₂×M₁₆	—	2	1	no	—
339	C₂×M₁₆	—	2	1	no	—
340	C₂×M₁₆	—	2	1	no	—
341	C₂×M₁₆	—	2	1	no	—
342	C₂×M₁₆	C₂³×C₅	1	2	yes	—
343	C₂×M₁₆	C₂³×C₅	1	2	yes	—
344	C₂²×C₈	—	2	1	no	—
345	C₂²×C₈	—	2	1	no	—
346	C₄⋊Q₈	—	2	4	no	—
347	C₄⋊Q₈	—	2	4	no	—
348	C₄⋊Q₈	—	2	4	no	—
349	#125	—	2	4	no	—
350	#125	—	2	4	no	—
351	#125	—	2	4	no	—
352	C₂²⋊Q₈	—	2	8	no	—
353	C₂²⋊Q₈	—	2	8	no	—
354	C₂²⋊Q₈	—	2	8	no	—
355	C₄×Q₈	—	2	8	no	—
356	#117	—	2	1	no	—
357	#117	—	2	1	no	—
358	#117	—	2	1	no	—
359	#117	—	2	1	no	—
360	#117	—	2	1	no	—
361	#117	—	2	1	no	—
362	#117	—	2	1	no	—
363	#117	—	2	1	no	—
364	#117	—	2	1	no	—
365	#117	C₅×D₄	1	2	yes	—
366	#117	C₅×D₄	1	2	yes	—
367	#117	C₅×D₄	1	2	yes	—
368	#117	C₅×D₄	1	2	yes	—
369	#117	C₂³×C₅	1	2	yes	—
370	#117	C₂³×C₅	1	2	yes	—
371	#117	C₂³×C₅	1	2	yes	—
372	C₂×C₄⋊C₄	—	2	1	no	—
373	C₂×C₄⋊C₄	—	2	1	no	—
374	C₂×C₄⋊C₄	—	2	1	no	—
375	C₂×C₄⋊C₄	—	2	1	no	—
376	C₂×C₂²⋊C₄	—	2	1	no	—
377	C₂×C₂²⋊C₄	—	2	1	no	—
378	C₂×C₂²⋊C₄	—	2	1	no	—
379	C₂×C₂²⋊C₄	C₅×D₄	1	2	yes	—
380	C₂×C₄²	—	2	1	no	—
381	C₂×C₄²	—	2	1	no	—
382	C₂×C₄²	C₅×D₄	1	2	yes	—
383	C₂×C₄²	C₂³×C₅	1	2	yes	—
384	#104	—	2	8	no	—
385	#104	—	2	8	no	—
386	#99	—	2	8	no	—
387	C₂²×C₅	—	4	1	no	—
388	C₂²×C₅	—	4	1	no	—
389	C₂²×C₅	—	4	1	no	—
390	C₂²×C₅	—	4	1	no	—
391	C₂²×C₅	—	4	1	no	—
392	C₂²×C₅	—	4	1	no	—
393	C₂²×C₅	—	4	1	no	—
394	C₂²×C₅	—	4	2	no	—
395	C₂²×C₅	—	8	2	no	—
396	C₂²×C₅	—	8	2	no	—
397	C₂²×C₅	—	8	2	no	—
398	C₂²×C₅	#393	1	1	yes	characteristic
399	C₂²×C₅	#393	1	1	yes	characteristic
400	C₂²×C₅	#393	1	1	yes	characteristic
401	C₂²×C₅	#393	1	1	yes	characteristic
402	C₂²×C₅	C₂×#120	1	1	yes	characteristic
403	C₂²×C₅	C₂×C₂²⋊Q₈	1	1	yes	characteristic
404	C₂²×C₅	C₂×#124	1	1	yes	characteristic
405	C₄×C₅	—	4	4	no	—
406	C₄×C₅	—	4	4	no	—
407	C₄×C₅	—	4	4	no	—
408	C₄×C₅	—	4	2	no	—
409	C₄×C₅	—	8	2	no	—
410	C₄×C₅	#393	1	1	yes	characteristic
411	C₄×C₅	#393	1	1	yes	characteristic
412	C₄×C₅	#393	1	1	yes	characteristic
413	C₄×C₅	#393	1	1	yes	characteristic
414	C₄×C₅	C₂×C₂²⋊Q₈	1	1	yes	characteristic
415	C₄×C₅	C₂×C₂²⋊Q₈	1	1	yes	characteristic
416	C₄×C₅	C₂×#124	1	1	yes	characteristic
417	C₄×C₅	C₂×#124	1	1	yes	characteristic
418	C₂⁴	—	2	1	no	—
419	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
420	#41	—	2	2	no	—
421	#41	—	2	2	no	—
422	#41	—	2	2	no	—
423	#41	—	2	2	no	—
424	#41	—	2	2	no	—
425	#41	—	2	2	no	—
426	#41	—	2	2	no	—
427	#41	—	2	2	no	—
428	#41	—	4	1	no	—
429	#41	—	4	1	no	—
430	#41	—	4	1	no	—
431	#41	—	4	1	no	—
432	#41	—	4	1	no	—
433	#41	—	4	1	no	—
434	#41	—	4	1	no	—
435	#41	—	4	1	no	—
436	C₂×Q₈	—	2	8	no	—
437	C₂×Q₈	—	2	1	no	—
438	C₂×Q₈	—	2	1	no	—
439	C₂×Q₈	—	2	1	no	—
440	C₂×Q₈	—	2	1	no	—
441	C₂×Q₈	—	2	1	no	—
442	C₂×Q₈	C₂×C₅×Q₈	1	2	yes	—
443	C₂×D₄	—	2	2	no	—
444	C₂×D₄	—	2	2	no	—
445	C₂×D₄	—	2	1	no	—
446	C₂×D₄	—	2	1	no	—
447	C₂×D₄	—	2	1	no	—
448	C₂×D₄	—	2	1	no	—
449	C₂×D₄	—	2	1	no	—
450	C₂×D₄	—	4	1	no	—
451	C₂×D₄	—	4	1	no	—
452	C₂×D₄	—	4	1	no	—
453	C₂×D₄	—	4	1	no	—
454	C₂×D₄	C₂×C₅×Q₈	1	2	yes	—
455	C₂²×C₄	—	2	1	no	—
456	C₂²×C₄	—	2	2	no	—
457	C₂²×C₄	—	2	2	no	—
458	C₂²×C₄	—	2	2	no	—
459	C₂²×C₄	—	2	2	no	—
460	C₂²×C₄	—	2	2	no	—
461	C₂²×C₄	—	4	1	no	—
462	C₂²×C₄	—	4	1	no	—
463	C₂²×C₄	—	4	1	no	—
464	C₂²×C₄	—	4	1	no	—
465	C₂²×C₄	—	4	1	no	—
466	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
467	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
468	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
469	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
470	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
471	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
472	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
473	C₂²×C₄	C₅×#41	1	1	yes	characteristic
474	C₂²×C₄	C₅×#41	1	1	yes	characteristic
475	C₂²×C₄	C₅×#41	1	1	yes	characteristic
476	C₂²×C₄	C₅×#41	1	1	yes	characteristic
477	C₂²×C₄	C₅×#41	1	1	yes	characteristic
478	C₂²×C₄	C₅×#41	1	1	yes	characteristic
479	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
480	M₁₆	—	2	2	no	—
481	M₁₆	—	2	2	no	—
482	M₁₆	—	2	2	no	—
483	M₁₆	—	2	2	no	—
484	M₁₆	—	4	1	no	—
485	M₁₆	—	4	1	no	—
486	M₁₆	—	4	1	no	—
487	M₁₆	—	4	1	no	—
488	C₂×C₈	—	2	2	no	—
489	C₂×C₈	—	2	2	no	—
490	C₂×C₈	—	4	1	no	—
491	C₂×C₈	—	4	1	no	—
492	C₂×C₈	—	4	1	no	—
493	C₂×C₈	—	4	1	no	—
494	C₄⋊C₄	—	2	8	no	—
495	C₄⋊C₄	—	2	8	no	—
496	C₄⋊C₄	—	2	8	no	—
497	C₄⋊C₄	—	2	2	no	—
498	C₄⋊C₄	—	2	2	no	—
499	C₄⋊C₄	—	2	2	no	—
500	C₄⋊C₄	—	2	2	no	—
501	C₄⋊C₄	—	2	2	no	—
502	C₄⋊C₄	—	2	2	no	—
503	C₄⋊C₄	—	2	2	no	—
504	C₄⋊C₄	—	4	1	no	—
505	C₂²⋊C₄	—	2	2	no	—
506	C₂²⋊C₄	—	2	2	no	—
507	C₂²⋊C₄	—	2	2	no	—
508	C₂²⋊C₄	—	2	2	no	—
509	C₂²⋊C₄	—	2	2	no	—
510	C₂²⋊C₄	—	2	2	no	—
511	C₂²⋊C₄	—	4	1	no	—
512	C₂²⋊C₄	C₂×C₅×D₄	1	4	yes	—
513	C₄²	—	2	2	no	—
514	C₄²	—	2	2	no	—
515	C₄²	—	2	2	no	—
516	C₄²	—	2	2	no	—
517	C₄²	—	2	2	no	—
518	C₄²	—	4	1	no	—
519	C₄²	C₂×C₅×D₄	1	4	yes	—
520	C₄²	C₂×C₅×D₄	1	4	yes	—
521	C₂×C₅	—	2	1	no	—
522	C₂×C₅	—	2	1	no	—
523	C₂×C₅	—	2	1	no	—
524	C₂×C₅	—	2	1	no	—
525	C₂×C₅	—	2	1	no	—
526	C₂×C₅	—	2	1	no	—
527	C₂×C₅	—	2	1	no	—
528	C₂×C₅	—	8	2	no	—
529	C₂×C₅	C₂×#393	1	1	yes	characteristic
530	C₂³	—	2	2	no	—
531	C₂³	—	4	1	no	—
532	C₂³	—	4	1	no	—
533	C₂³	—	8	1	no	—
534	C₂³	—	8	2	no	—
535	C₂³	C₅×#120	1	1	yes	characteristic
536	C₂³	C₅×#120	1	1	yes	characteristic
537	C₂³	C₅×C₂²⋊Q₈	1	1	yes	characteristic
538	C₂³	C₅×C₂²⋊Q₈	1	1	yes	characteristic
539	C₂³	C₅×#124	1	1	yes	characteristic
540	C₂³	C₅×#124	1	1	yes	characteristic
541	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
542	Q₈	—	2	2	no	—
543	Q₈	—	2	2	no	—
544	Q₈	—	4	1	no	—
545	Q₈	—	4	1	no	—
546	Q₈	—	8	8	no	—
547	D₄	—	2	2	no	—
548	D₄	—	2	2	no	—
549	D₄	—	4	2	no	—
550	D₄	—	4	2	no	—
551	D₄	—	4	2	no	—
552	D₄	—	4	2	no	—
553	D₄	—	4	1	no	—
554	D₄	—	4	1	no	—
555	C₂×C₄	—	2	4	no	—
556	C₂×C₄	—	2	4	no	—
557	C₂×C₄	—	2	4	no	—
558	C₂×C₄	—	2	2	no	—
559	C₂×C₄	—	4	1	no	—
560	C₂×C₄	—	4	1	no	—
561	C₂×C₄	—	4	2	no	—
562	C₂×C₄	—	4	2	no	—
563	C₂×C₄	—	4	2	no	—
564	C₂×C₄	—	4	2	no	—
565	C₂×C₄	—	4	2	no	—
566	C₂×C₄	—	8	2	no	—
567	C₂×C₄	—	8	2	no	—
568	C₂×C₄	—	8	2	no	—
569	C₂×C₄	—	8	2	no	—
570	C₂×C₄	C₅×#120	1	1	yes	characteristic
571	C₂×C₄	C₅×#120	1	1	yes	characteristic
572	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
573	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
574	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
575	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
576	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
577	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
578	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
579	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
580	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
581	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
582	C₂×C₄	C₅×#124	1	1	yes	characteristic
583	C₂×C₄	C₅×#124	1	1	yes	characteristic
584	C₂×C₄	C₅×#124	1	1	yes	characteristic
585	C₂×C₄	C₅×#124	1	1	yes	characteristic
586	C₂×C₄	C₅×#124	1	1	yes	characteristic
587	C₂×C₄	C₅×#124	1	1	yes	characteristic
588	C₂×C₄	C₅×#124	1	1	yes	characteristic
589	C₂×C₄	C₅×#124	1	1	yes	characteristic
590	C₂×C₄	C₅×#124	1	1	yes	characteristic
591	C₂×C₄	C₅×#124	1	1	yes	characteristic
592	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
593	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
594	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
595	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
596	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
597	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
598	C₈	—	4	2	no	—
599	C₈	—	4	2	no	—
600	C₅	#23472	1	1	yes	characteristic, direct product factor, Sylow
601	C₂²	—	4	1	no	—
602	C₂²	—	4	1	no	—
603	C₂²	—	4	1	no	—
604	C₂²	—	4	1	no	—
605	C₂²	—	4	1	no	—
606	C₂²	—	4	1	no	—
607	C₂²	—	4	1	no	—
608	C₂²	—	4	2	no	—
609	C₂²	—	8	2	no	—
610	C₂²	—	8	2	no	—
611	C₂²	—	8	2	no	—
612	C₂²	C₅×#393	1	1	yes	characteristic
613	C₂²	C₅×#393	1	1	yes	characteristic
614	C₂²	C₅×#393	1	1	yes	characteristic
615	C₂²	C₅×#393	1	1	yes	characteristic
616	C₂²	C₂×C₅×#120	1	1	yes	characteristic
617	C₂²	C₂×C₅×C₂²⋊Q₈	1	1	yes	characteristic
618	C₂²	C₂×C₅×#124	1	1	yes	characteristic
619	C₄	—	4	4	no	—
620	C₄	—	4	4	no	—
621	C₄	—	4	4	no	—
622	C₄	—	4	2	no	—
623	C₄	—	8	2	no	—
624	C₄	C₅×#393	1	1	yes	characteristic
625	C₄	C₅×#393	1	1	yes	characteristic
626	C₄	C₅×#393	1	1	yes	characteristic
627	C₄	C₅×#393	1	1	yes	characteristic
628	C₄	C₂×C₅×C₂²⋊Q₈	1	1	yes	characteristic
629	C₄	C₂×C₅×C₂²⋊Q₈	1	1	yes	characteristic
630	C₄	C₂×C₅×#124	1	1	yes	characteristic
631	C₄	C₂×C₅×#124	1	1	yes	characteristic
632	C₂	—	2	1	no	—
633	C₂	—	2	1	no	—
634	C₂	—	2	1	no	—
635	C₂	—	2	1	no	—
636	C₂	—	2	1	no	—
637	C₂	—	2	1	no	—
638	C₂	—	2	1	no	—
639	C₂	—	8	2	no	—
640	C₂	C₂×C₅×#393	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	C₂×#124
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂×C₅	C₂×#393
prime radical	C₂²×C₅×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂³×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅×#3458	C₂	semidirect
5. C₅×#2031	C₂	semidirect
6. C₂×C₅×#344	C₂	semidirect
7. C₅×#1733	C₂	semidirect
8. C₂×C₅×H(3,ℤ/4)	C₂	semidirect
18. C₅×#533	C₂²	semidirect
32. C₅×#344	C₂²	semidirect
33. C₅×#339	C₂²	semidirect
34. C₅×H(3,ℤ/4)	C₂²	semidirect
35. #23472	C₅	direct (N × H)
46. C₂×C₅×#41	Q₈	semidirect
48. C₂²×C₅×Q₈	Q₈	semidirect
50. C₂²×C₅×D₄	Q₈	semidirect
108. #3458	C₂×C₅	semidirect
112. #2031	C₂×C₅	semidirect
113. C₂×#344	C₂×C₅	semidirect
114. #1733	C₂×C₅	semidirect
115. C₂×H(3,ℤ/4)	C₂×C₅	semidirect
228. #533	C₂²×C₅	semidirect
242. #344	C₂²×C₅	semidirect
243. #339	C₂²×C₅	semidirect
244. H(3,ℤ/4)	C₂²×C₅	semidirect
325. C₂×#41	C₅×Q₈	semidirect
327. C₂²×Q₈	C₅×Q₈	semidirect
329. C₂²×D₄	C₅×Q₈	semidirect
600. C₅	#23472	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×#3458	2
C₂×C₅×#421	2
C₂×C₅×#410	2
C₅×#2031	2
C₂×C₅×#344	2
C₅×#1733	2
C₂×C₅×H(3,ℤ/4)	2
#23472	5

Subgroups — C5×#23472 #11946192

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#23472 #11946192