Subgroups — #10617301

Order 640.

364 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#65983	C₂	1	1	yes	maximal, characteristic
2	#65773	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×#170	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#4004	C₂	1	1	yes	maximal, characteristic, complemented
5	#65748	C₂	1	1	yes	maximal, characteristic, complemented
6	#65708	C₂	1	1	yes	maximal, characteristic, complemented
7	#65707	C₂	1	1	yes	maximal, characteristic, complemented
8	#65706	C₂	1	1	yes	maximal, characteristic, complemented
9	#65703	C₂	1	1	yes	maximal, characteristic, complemented
10	#65697	C₂	1	1	yes	maximal, characteristic, complemented
11	#65696	C₂	1	1	yes	maximal, characteristic, complemented
12	#65695	C₂	1	1	yes	maximal, characteristic, complemented
13	#65693	C₂	1	1	yes	maximal, characteristic, complemented
14	#65692	C₂	1	1	yes	maximal, characteristic, complemented
15	#65692	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#704	C₂²	1	1	yes	characteristic
17	C₂²×#170	C₂²	1	1	yes	characteristic
18	C₂²×#170	C₂²	1	1	yes	characteristic
19	C₂²×#170	C₂²	1	1	yes	characteristic
20	C₂×#700	C₂²	1	1	yes	characteristic
21	C₂×D₅⋊C₈	C₂²	1	1	yes	characteristic
22	C₅×#123	C₄	1	1	yes	characteristic
23	#4085	C₄	1	1	yes	characteristic
24	D₄×Dic₅	C₂²	1	1	yes	characteristic
25	#4075	C₄	1	1	yes	characteristic
26	C₂×C₄×Dic₅	C₂²	1	1	yes	characteristic
27	#4044	C₄	1	1	yes	characteristic
28	#4038	C₂²	1	1	yes	characteristic
29	#4035	C₄	1	1	yes	characteristic
30	#4033	C₂²	1	1	yes	characteristic
31	#4031	C₄	1	1	yes	characteristic
32	#4027	C₂²	1	1	yes	characteristic
33	#4027	C₂²	1	1	yes	characteristic
34	#4025	C₄	1	1	yes	characteristic
35	#4024	C₄	1	1	yes	characteristic
36	#4023	C₂²	1	1	yes	characteristic
37	C₂²⋊#170	C₂²	1	1	yes	characteristic
38	C₂²⋊#170	C₂²	1	1	yes	characteristic
39	C₂²⋊#170	C₂²	1	1	yes	characteristic
40	C₂²⋊#170	C₂²	1	1	yes	characteristic
41	#4004	C₂²	1	1	yes	characteristic
42	#4004	C₂²	1	1	yes	characteristic
43	#4004	C₂²	1	1	yes	characteristic
44	#4004	C₂²	1	1	yes	characteristic
45	#4004	C₂²	1	1	yes	characteristic
46	#4004	C₂²	1	1	yes	characteristic
47	#4003	C₂²	1	1	yes	characteristic
48	#4003	C₂²	1	1	yes	characteristic
49	#4002	C₂²	1	1	yes	characteristic
50	#4002	C₂²	1	1	yes	characteristic
51	C₄⋊#170	C₂²	1	1	yes	characteristic
52	C₄⋊#170	C₂²	1	1	yes	characteristic
53	C₄⋊#170	C₂²	1	1	yes	characteristic
54	C₄⋊#170	C₂²	1	1	yes	characteristic
55	C₄×#170	C₂²	1	1	yes	characteristic
56	C₄×#170	C₂²	1	1	yes	characteristic
57	C₄×#170	C₂²	1	1	yes	characteristic
58	C₄×#170	C₂²	1	1	yes	characteristic
59	#2994	—	5	1	no	maximal, Sylow
60	C₂×C₅×D₄	C₂×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
62	C₂×C₂²⋊D₅	C₂×C₄	1	1	yes	characteristic
63	C₂×C₂²⋊D₅	C₂×C₄	1	1	yes	characteristic
64	C₂²×Dic₅	C₂×C₄	1	1	yes	characteristic
65	C₂²×Dic₅	C₂³	1	1	yes	characteristic
66	C₂²×Dic₅	C₂³	1	1	yes	characteristic
67	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
68	C₂×Dic₁₀	C₂×C₄	1	1	yes	characteristic
69	#704	—	4	1	no	—
70	C₂×#170	—	4	1	no	—
71	C₂×#170	—	4	1	no	—
72	C₂×#170	—	4	1	no	—
73	C₂×#170	C₂³	1	1	yes	characteristic
74	C₂×#170	C₂³	1	1	yes	characteristic
75	C₂×#170	C₂³	1	1	yes	characteristic
76	C₂×#170	C₂³	1	1	yes	characteristic
77	C₂×#170	C₂³	1	1	yes	characteristic
78	C₂×#170	C₂³	1	1	yes	characteristic
79	C₂×#170	C₂³	1	1	yes	characteristic
80	C₂×#170	C₂³	1	1	yes	characteristic
81	#700	—	4	1	no	—
82	D₅⋊C₈	—	4	1	no	—
83	C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
84	C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
85	C₅×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
86	C₅×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
87	C₅×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
88	C₂²⋊Dic₅	C₂×C₄	1	1	yes	characteristic
89	C₂²⋊Dic₅	C₂×C₄	1	1	yes	characteristic
90	C₂²⋊Dic₅	C₂×C₄	1	1	yes	characteristic
91	#685	C₂×C₄	1	1	yes	characteristic
92	#685	C₂×C₄	1	1	yes	characteristic
93	#685	C₂×C₄	1	1	yes	characteristic
94	#685	C₂×C₄	1	1	yes	characteristic
95	C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
96	C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
97	#683	C₂×C₄	1	1	yes	characteristic
98	#683	C₂×C₄	1	1	yes	characteristic
99	#683	C₂×C₄	1	1	yes	characteristic
100	#683	C₂×C₄	1	1	yes	characteristic
101	#683	C₂×C₄	1	1	yes	characteristic
102	#683	C₂×C₄	1	1	yes	characteristic
103	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
104	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
105	C₄×Dic₅	C₂³	1	1	yes	characteristic
106	C₄×Dic₅	C₂³	1	1	yes	characteristic
107	C₄×Dic₅	C₂³	1	1	yes	characteristic
108	C₄×Dic₅	C₂³	1	1	yes	characteristic
109	#529	—	5	1	no	—
110	#446	—	5	1	no	—
111	C₈×Q₈	—	5	1	no	—
112	#436	—	5	1	no	—
113	#435	—	5	1	no	—
114	#435	—	5	1	no	—
115	C₈×D₄	—	5	1	no	—
116	C₈×D₄	—	5	1	no	—
117	C₈×D₄	—	5	1	no	—
118	#433	—	5	1	no	—
119	#433	—	5	1	no	—
120	#431	—	5	1	no	—
121	#424	—	5	1	no	—
122	#424	—	5	1	no	—
123	C₂×C₄⋊C₈	—	5	1	no	—
124	C₂³×C₅	C₂²×C₄	1	1	yes	characteristic
125	C₂³×C₅	C₂²×C₄	1	1	yes	characteristic
126	C₂²×D₅	C₂²×C₄	1	1	yes	characteristic
127	C₅×D₄	—	2	2	no	—
128	C₂×C₄×C₅	—	2	2	no	—
129	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
132	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
133	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
134	C₂²⋊D₅	—	2	2	no	—
135	C₂²⋊D₅	—	2	2	no	—
136	C₂×Dic₅	—	2	2	no	—
137	C₂×Dic₅	—	2	1	no	—
138	C₂×Dic₅	—	2	1	no	—
139	C₂×Dic₅	—	2	1	no	—
140	C₂×Dic₅	—	2	1	no	—
141	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
142	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
143	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
144	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
145	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
146	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
147	C₂×Dic₅	C₂⁴	1	1	yes	characteristic
148	C₄×D₅	—	2	1	no	—
149	C₄×D₅	—	2	1	no	—
150	Dic₁₀	—	2	2	no	—
151	#170	—	2	1	no	—
152	#170	—	2	1	no	—
153	#170	—	2	1	no	—
154	#170	—	2	1	no	—
155	#170	—	2	1	no	—
156	#170	—	2	1	no	—
157	#170	—	2	1	no	—
158	#170	—	2	1	no	—
159	C₂×M₁₆	—	5	1	no	—
160	C₂×M₁₆	—	5	1	no	—
161	C₂²×C₈	—	5	1	no	—
162	C₂²×C₈	—	5	1	no	—
163	C₂²×C₈	—	5	1	no	—
164	C₂²×C₈	—	5	1	no	—
165	#126	—	5	1	no	—
166	#126	—	5	1	no	—
167	#125	—	5	1	no	—
168	#124	—	5	1	no	—
169	#123	—	5	1	no	—
170	#123	AGL(1,𝔽₅)	1	1	yes	characteristic
171	C₂²⋊Q₈	—	5	1	no	—
172	#121	—	5	1	no	—
173	C₄×Q₈	—	5	1	no	—
174	C₄×D₄	—	5	1	no	—
175	C₄×D₄	—	5	1	no	—
176	C₄×D₄	—	5	1	no	—
177	#117	—	5	1	no	—
178	#117	—	5	1	no	—
179	C₂×C₄²	—	5	1	no	—
180	C₄⋊C₈	—	5	1	no	—
181	C₄⋊C₈	—	5	1	no	—
182	C₄⋊C₈	—	5	1	no	—
183	C₄⋊C₈	—	5	1	no	—
184	C₄⋊C₈	—	5	1	no	—
185	C₄⋊C₈	—	5	1	no	—
186	C₄⋊C₈	—	5	1	no	—
187	C₄⋊C₈	—	5	1	no	—
188	C₄⋊C₈	—	5	1	no	—
189	C₄⋊C₈	—	5	1	no	—
190	C₂²⋊C₈	—	5	1	no	—
191	C₂²⋊C₈	—	5	1	no	—
192	C₂²⋊C₈	—	5	1	no	—
193	C₂²⋊C₈	—	5	1	no	—
194	C₂²⋊C₈	—	5	1	no	—
195	C₂²⋊C₈	—	5	1	no	—
196	#97	—	5	1	no	—
197	#97	—	5	1	no	—
198	C₄×C₈	—	5	1	no	—
199	C₄×C₈	—	5	1	no	—
200	C₄×C₈	—	5	1	no	—
201	C₄×C₈	—	5	1	no	—
202	C₂²×C₅	—	2	1	no	—
203	C₂²×C₅	—	2	1	no	—
204	C₂²×C₅	—	2	1	no	—
205	C₂²×C₅	—	2	1	no	—
206	C₂²×C₅	—	2	1	no	—
207	C₂²×C₅	#131	1	2	yes	—
208	C₂²×C₅	C₂³×C₄	1	1	yes	characteristic
209	C₂×D₅	—	2	1	no	—
210	C₂×D₅	—	2	1	no	—
211	C₂×D₅	—	2	1	no	—
212	C₄×C₅	—	2	1	no	—
213	C₄×C₅	—	2	1	no	—
214	C₄×C₅	—	2	1	no	—
215	C₄×C₅	—	2	1	no	—
216	C₄×C₅	—	2	1	no	—
217	Dic₅	—	2	1	no	—
218	Dic₅	—	2	1	no	—
219	Dic₅	—	2	1	no	—
220	Dic₅	—	2	1	no	—
221	Dic₅	—	2	1	no	—
222	Dic₅	#131	1	2	yes	—
223	Dic₅	H(5,𝔽₂)	1	1	yes	characteristic
224	Dic₅	2₋¹⁺⁴	1	1	yes	characteristic
225	C₂×Q₈	—	5	1	no	—
226	C₂×D₄	—	5	1	no	—
227	C₂×D₄	—	5	1	no	—
228	C₂×D₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
229	C₂²×C₄	—	5	1	no	—
230	C₂²×C₄	—	5	1	no	—
231	C₂²×C₄	—	5	1	no	—
232	C₂²×C₄	—	5	1	no	—
233	C₂²×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
234	M₁₆	—	20	1	no	—
235	M₁₆	—	20	1	no	—
236	C₂×C₈	—	5	1	no	—
237	C₂×C₈	—	5	1	no	—
238	C₂×C₈	—	5	1	no	—
239	C₂×C₈	—	5	1	no	—
240	C₂×C₈	—	5	1	no	—
241	C₂×C₈	—	5	1	no	—
242	C₂×C₈	—	5	1	no	—
243	C₂×C₈	—	5	1	no	—
244	C₂×C₈	—	20	1	no	—
245	C₂×C₈	—	20	1	no	—
246	C₂×C₈	—	20	1	no	—
247	C₂×C₈	—	20	1	no	—
248	C₄⋊C₄	—	5	1	no	—
249	C₄⋊C₄	—	5	1	no	—
250	C₄⋊C₄	—	5	1	no	—
251	C₄⋊C₄	—	5	1	no	—
252	C₄⋊C₄	—	5	1	no	—
253	C₄⋊C₄	—	5	1	no	—
254	C₄⋊C₄	—	5	1	no	—
255	C₄⋊C₄	—	5	1	no	—
256	C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
257	C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
258	C₂²⋊C₄	—	5	1	no	—
259	C₂²⋊C₄	—	5	1	no	—
260	C₂²⋊C₄	—	5	1	no	—
261	C₂²⋊C₄	—	5	1	no	—
262	C₂²⋊C₄	—	5	1	no	—
263	C₂²⋊C₄	—	5	1	no	—
264	C₂²⋊C₄	—	5	1	no	—
265	C₂²⋊C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
266	C₂²⋊C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
267	C₂²⋊C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
268	C₄²	—	5	1	no	—
269	C₄²	—	5	1	no	—
270	C₄²	—	5	1	no	—
271	C₄²	—	5	1	no	—
272	C₄²	—	5	1	no	—
273	C₄²	—	5	1	no	—
274	C₂×C₅	—	2	2	no	—
275	C₂×C₅	—	4	1	no	—
276	C₂×C₅	#520	1	1	yes	characteristic
277	C₂×C₅	C₂×#131	1	1	yes	characteristic
278	C₂×C₅	C₂×#131	1	1	yes	characteristic
279	D₅	—	4	1	no	—
280	C₂³	—	5	1	no	—
281	C₂³	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
282	C₂³	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
283	Q₈	—	10	2	no	—
284	D₄	—	2	2	no	—
285	D₄	—	10	2	no	—
286	D₄	—	10	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	5	1	no	—
289	C₂×C₄	—	5	1	no	—
290	C₂×C₄	—	5	1	no	—
291	C₂×C₄	—	5	1	no	—
292	C₂×C₄	—	5	1	no	—
293	C₂×C₄	—	5	1	no	—
294	C₂×C₄	—	5	1	no	—
295	C₂×C₄	—	10	1	no	—
296	C₂×C₄	—	10	1	no	—
297	C₂×C₄	—	10	1	no	—
298	C₂×C₄	—	10	1	no	—
299	C₂×C₄	—	10	2	no	—
300	C₂×C₄	—	10	1	no	—
301	C₂×C₄	—	10	1	no	—
302	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
303	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
304	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
305	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
306	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
307	C₈	—	10	1	no	—
308	C₈	—	10	1	no	—
309	C₈	—	10	1	no	—
310	C₈	—	10	1	no	—
311	C₈	—	10	1	no	—
312	C₈	—	10	1	no	—
313	C₈	—	10	1	no	—
314	C₈	—	10	1	no	—
315	C₅	#2994	1	1	yes	characteristic, complemented, Sylow
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	10	1	no	—
322	C₂²	—	10	1	no	—
323	C₂²	—	10	1	no	—
324	C₂²	#4131	1	2	yes	—
325	C₂²	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	5	2	no	—
332	C₄	—	5	1	no	—
333	C₄	—	5	1	no	—
334	C₄	—	10	1	no	—
335	C₄	—	10	1	no	—
336	C₄	—	10	1	no	—
337	C₄	—	10	1	no	—
338	C₄	—	10	1	no	—
339	C₂	—	2	2	no	—
340	C₂	—	4	1	no	—
341	C₂	—	20	1	no	—
342	C₂	C₂×#4131	1	1	yes	characteristic
343	C₂	D₅⋊#131	1	1	yes	characteristic
344	C₂	#66266	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×AGL(1,𝔽₅)
commutator, socle	C₂²×C₅	C₂³×C₄
Fitting	C₅×#123	C₄
prime radical	#4085	C₄
cosocle	C₂×Dic₅	C₂⁴
hypercenter	#123	AGL(1,𝔽₅)
hypocenter	C₅	#2994
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ #123

Complements

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

N	H ≅ G/N	type
2. #65773	C₂	semidirect
3. D₄×#170	C₂	semidirect
4. C₂×#4004	C₂	semidirect
5. #65748	C₂	semidirect
6. #65708	C₂	semidirect
7. #65707	C₂	semidirect
8. #65706	C₂	semidirect
9. #65703	C₂	semidirect
10. #65697	C₂	semidirect
11. #65696	C₂	semidirect
12. #65695	C₂	semidirect
13. #65693	C₂	semidirect
14. #65692	C₂	semidirect
15. #65692	C₂	semidirect
315. C₅	#2994	semidirect

Maximal subgroups

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

subgroup	index
#65983	2
#65773	2
D₄×#170	2
C₂×#4004	2
#65748	2
#65708	2
#65707	2
#65706	2
#65703	2
#65697	2
#65696	2
#65695	2
#65693	2
#65692	2
#2994	5