Subgroups — #10611690

Order 640.

348 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#65912	C₂	1	1	yes	maximal, characteristic
2	C₅×#446	C₂	1	1	yes	maximal, characteristic, complemented
3	#65313	C₂	1	1	yes	maximal, characteristic, complemented
4	#65128	C₂	1	1	yes	maximal, characteristic
5	#65128	C₂	1	1	yes	maximal, characteristic
6	#65126	C₂	1	1	yes	maximal, characteristic
7	#65120	C₂	1	1	yes	maximal, characteristic, complemented
8	#65117	C₂	1	1	yes	maximal, characteristic, complemented
9	#65117	C₂	1	1	yes	maximal, characteristic, complemented
10	#64997	C₂	1	1	yes	maximal, characteristic
11	D₅×#97	C₂	1	1	yes	maximal, characteristic
12	#64990	C₂	1	1	yes	maximal, characteristic, complemented
13	#64990	C₂	1	1	yes	maximal, characteristic, complemented
14	#64978	C₂	1	1	yes	maximal, characteristic
15	#64968	C₂	1	1	yes	maximal, characteristic, complemented
16	C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
17	#4092	C₄	1	1	yes	characteristic
18	#4090	C₄	1	1	yes	characteristic
19	C₂×#676	C₂²	1	1	yes	characteristic
20	C₂×#676	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₅	C₂²	1	1	yes	characteristic
22	C₂×C₈×D₅	C₂²	1	1	yes	characteristic
23	#4043	C₄	1	1	yes	characteristic
24	#4042	C₄	1	1	yes	characteristic
25	#4042	C₄	1	1	yes	characteristic
26	#4035	C₄	1	1	yes	characteristic
27	#4035	C₄	1	1	yes	characteristic
28	#4034	C₄	1	1	yes	characteristic
29	#4018	C₂²	1	1	yes	characteristic
30	#4018	C₂²	1	1	yes	characteristic
31	C₄²×D₅	C₂²	1	1	yes	characteristic
32	C₄×Dic₁₀	C₂²	1	1	yes	characteristic
33	C₄×Dic₁₀	C₂²	1	1	yes	characteristic
34	C₄×Dic₁₀	C₂²	1	1	yes	characteristic
35	C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
36	C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
37	C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
38	C₅×#97	C₂²	1	1	yes	characteristic
39	C₅×#97	C₂²	1	1	yes	characteristic
40	C₄×C₅×C₈	C₂²	1	1	yes	characteristic
41	#3947	C₂²	1	1	yes	characteristic
42	#3947	C₂²	1	1	yes	characteristic
43	#3947	C₂²	1	1	yes	characteristic
44	#3947	C₂²	1	1	yes	characteristic
45	#3946	C₂²	1	1	yes	characteristic
46	#3946	C₂²	1	1	yes	characteristic
47	#3946	C₂²	1	1	yes	characteristic
48	#3946	C₂²	1	1	yes	characteristic
49	#3946	C₂²	1	1	yes	characteristic
50	#3946	C₂²	1	1	yes	characteristic
51	C₈×Dic₅	C₂²	1	1	yes	characteristic
52	C₈×Dic₅	C₂²	1	1	yes	characteristic
53	C₄⋊#168	C₂²	1	1	yes	characteristic
54	C₄⋊#168	C₂²	1	1	yes	characteristic
55	C₄⋊#168	C₂²	1	1	yes	characteristic
56	#3935	C₂²	1	1	yes	characteristic
57	#3935	C₂²	1	1	yes	characteristic
58	C₄×#168	C₂²	1	1	yes	characteristic
59	#2961	—	5	1	no	maximal, Sylow
60	C₂×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
64	C₂×Dic₁₀	C₂×C₄	1	1	yes	characteristic
65	C₂×Dic₁₀	C₂×C₄	1	1	yes	characteristic
66	C₂×Dic₁₀	C₂×C₄	1	1	yes	characteristic
67	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
68	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
69	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
70	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
71	C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
72	C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
73	C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
74	C₄²×C₅	C₂³	1	1	yes	characteristic
75	C₄²×C₅	C₂³	1	1	yes	characteristic
76	C₄²×C₅	C₂³	1	1	yes	characteristic
77	#685	C₂×C₄	1	1	yes	characteristic
78	#685	C₂×C₄	1	1	yes	characteristic
79	#685	C₂×C₄	1	1	yes	characteristic
80	#685	C₂×C₄	1	1	yes	characteristic
81	C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
82	C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
83	C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
84	#683	C₂×C₄	1	1	yes	characteristic
85	#683	C₂×C₄	1	1	yes	characteristic
86	#683	C₂×C₄	1	1	yes	characteristic
87	#683	C₂×C₄	1	1	yes	characteristic
88	#683	C₂×C₄	1	1	yes	characteristic
89	#683	C₂×C₄	1	1	yes	characteristic
90	#683	C₂×C₄	1	1	yes	characteristic
91	#683	C₂×C₄	1	1	yes	characteristic
92	#683	C₂×C₄	1	1	yes	characteristic
93	#683	C₂×C₄	1	1	yes	characteristic
94	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
95	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
96	C₄×Dic₅	C₂³	1	1	yes	characteristic
97	C₄×Dic₅	C₂³	1	1	yes	characteristic
98	C₄×Dic₅	C₂³	1	1	yes	characteristic
99	C₂×#168	C₂³	1	1	yes	characteristic
100	C₂×#168	C₂³	1	1	yes	characteristic
101	C₂×#168	C₂³	1	1	yes	characteristic
102	C₂×#168	C₂³	1	1	yes	characteristic
103	#676	—	2	2	no	—
104	#676	—	2	2	no	—
105	C₈×D₅	—	2	2	no	—
106	C₈×D₅	Q₈	1	4	yes	complemented
107	#533	—	5	1	no	—
108	#446	—	5	1	no	—
109	#446	—	5	1	no	—
110	#446	—	5	1	no	—
111	#446	—	5	1	no	—
112	#446	—	5	1	no	—
113	#446	—	5	1	no	—
114	#446	—	5	1	no	—
115	#446	D₅	1	1	yes	characteristic, complemented
116	#424	—	5	1	no	—
117	#424	—	5	1	no	—
118	#423	—	5	1	no	—
119	#405	—	5	1	no	—
120	#405	—	5	1	no	—
121	C₂×#97	—	5	1	no	—
122	C₂²×D₅	C₂²×C₄	1	1	yes	characteristic
123	C₅×Q₈	—	4	1	no	—
124	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
126	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
131	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
132	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
133	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
134	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
135	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
136	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
137	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
138	C₄×D₅	—	4	1	no	—
139	C₄×D₅	—	4	1	no	—
140	C₄×D₅	C₂×Q₈	1	2	yes	—
141	C₄×D₅	#41	1	2	yes	—
142	Dic₁₀	—	4	1	no	—
143	Dic₁₀	—	4	1	no	—
144	Dic₁₀	—	4	1	no	—
145	C₅×C₈	—	2	1	no	—
146	C₅×C₈	—	2	1	no	—
147	C₅×C₈	—	2	1	no	—
148	C₅×C₈	C₂×Q₈	1	2	yes	—
149	#168	—	2	1	no	—
150	#168	—	2	1	no	—
151	#168	—	2	1	no	—
152	#168	C₂×Q₈	1	2	yes	—
153	C₂×M₁₆	—	5	1	no	—
154	C₂×M₁₆	—	5	1	no	—
155	C₂²×C₈	—	5	1	no	—
156	C₂²×C₈	—	5	1	no	—
157	C₄⋊Q₈	—	5	1	no	—
158	C₄⋊Q₈	—	5	1	no	—
159	#125	—	5	1	no	—
160	#125	—	5	1	no	—
161	C₂²⋊Q₈	—	5	1	no	—
162	C₂²⋊Q₈	—	5	1	no	—
163	C₂²⋊Q₈	—	5	1	no	—
164	C₂²⋊Q₈	—	5	1	no	—
165	C₄×Q₈	—	5	1	no	—
166	C₄×Q₈	—	5	1	no	—
167	C₄×Q₈	—	5	1	no	—
168	C₄×Q₈	C₂×D₅	1	1	yes	characteristic
169	#117	—	5	1	no	—
170	#117	—	5	1	no	—
171	C₂×C₄²	—	5	1	no	—
172	C₄⋊C₈	—	5	1	no	—
173	C₄⋊C₈	—	5	1	no	—
174	C₄⋊C₈	—	5	1	no	—
175	C₄⋊C₈	—	5	1	no	—
176	C₄⋊C₈	—	5	1	no	—
177	C₄⋊C₈	—	5	1	no	—
178	C₄⋊C₈	—	5	1	no	—
179	C₄⋊C₈	—	5	1	no	—
180	C₄⋊C₈	—	5	1	no	—
181	C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
182	C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
183	C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
184	#97	—	5	1	no	—
185	#97	—	5	1	no	—
186	#97	—	5	1	no	—
187	#97	—	5	1	no	—
188	#97	—	5	1	no	—
189	#97	—	5	1	no	—
190	#97	C₂×D₅	1	1	yes	characteristic
191	#97	C₂×D₅	1	1	yes	characteristic
192	C₄×C₈	—	5	1	no	—
193	C₄×C₈	—	5	1	no	—
194	C₄×C₈	—	5	1	no	—
195	C₄×C₈	C₂×D₅	1	1	yes	characteristic
196	C₂²×C₅	C₂³×C₄	1	1	yes	characteristic
197	C₂×D₅	—	2	1	no	—
198	C₂×D₅	—	2	1	no	—
199	C₂×D₅	C₄×Q₈	1	2	yes	—
200	C₄×C₅	—	2	1	no	—
201	C₄×C₅	—	2	1	no	—
202	C₄×C₅	—	2	1	no	—
203	C₄×C₅	—	2	1	no	—
204	C₄×C₅	—	2	1	no	—
205	C₄×C₅	—	2	1	no	—
206	C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
207	C₄×C₅	C₂×#41	1	1	yes	characteristic
208	Dic₅	—	2	1	no	—
209	Dic₅	—	2	1	no	—
210	Dic₅	—	2	1	no	—
211	Dic₅	—	2	1	no	—
212	Dic₅	—	2	1	no	—
213	Dic₅	—	2	1	no	—
214	Dic₅	C₄×Q₈	1	2	yes	—
215	C₂×Q₈	—	5	1	no	—
216	C₂×Q₈	—	5	1	no	—
217	C₂×Q₈	—	5	1	no	—
218	C₂×Q₈	C₄×D₅	1	1	yes	characteristic
219	C₂²×C₄	—	5	1	no	—
220	C₂²×C₄	—	5	1	no	—
221	C₂²×C₄	—	5	1	no	—
222	M₁₆	—	10	2	no	—
223	M₁₆	—	10	2	no	—
224	C₂×C₈	—	5	1	no	—
225	C₂×C₈	—	5	4	no	—
226	C₂×C₈	—	5	1	no	—
227	C₂×C₈	—	5	1	no	—
228	C₂×C₈	—	5	1	no	—
229	C₂×C₈	—	10	2	no	—
230	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
231	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
232	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
233	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
234	C₄⋊C₄	—	5	1	no	—
235	C₄⋊C₄	—	5	1	no	—
236	C₄⋊C₄	—	5	1	no	—
237	C₄⋊C₄	—	5	1	no	—
238	C₄⋊C₄	—	5	1	no	—
239	C₄⋊C₄	—	5	1	no	—
240	C₄⋊C₄	—	5	1	no	—
241	C₄⋊C₄	—	5	1	no	—
242	C₄⋊C₄	—	5	1	no	—
243	C₄⋊C₄	—	5	1	no	—
244	C₄⋊C₄	—	5	1	no	—
245	C₄⋊C₄	—	5	1	no	—
246	C₄⋊C₄	—	5	1	no	—
247	C₄⋊C₄	C₄×D₅	1	1	yes	characteristic
248	C₄⋊C₄	C₄×D₅	1	1	yes	characteristic
249	C₄⋊C₄	C₄×D₅	1	1	yes	characteristic
250	C₂²⋊C₄	—	5	1	no	—
251	C₂²⋊C₄	—	5	1	no	—
252	C₂²⋊C₄	—	5	1	no	—
253	C₂²⋊C₄	—	5	1	no	—
254	C₄²	—	5	1	no	—
255	C₄²	—	5	1	no	—
256	C₄²	—	5	1	no	—
257	C₄²	—	5	1	no	—
258	C₄²	—	5	1	no	—
259	C₄²	C₂²×D₅	1	1	yes	characteristic
260	C₄²	C₂²×D₅	1	1	yes	characteristic
261	C₄²	C₂²×D₅	1	1	yes	characteristic
262	C₂×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
263	C₂×C₅	#568	1	1	yes	characteristic
264	C₂×C₅	#568	1	1	yes	characteristic
265	D₅	—	2	2	no	—
266	C₂³	—	5	1	no	—
267	Q₈	—	4	1	no	—
268	Q₈	—	20	1	no	—
269	Q₈	—	20	1	no	—
270	Q₈	—	20	1	no	—
271	C₂×C₄	—	5	1	no	—
272	C₂×C₄	—	5	2	no	—
273	C₂×C₄	—	5	2	no	—
274	C₂×C₄	—	5	1	no	—
275	C₂×C₄	—	5	1	no	—
276	C₂×C₄	—	5	1	no	—
277	C₂×C₄	—	5	1	no	—
278	C₂×C₄	—	5	1	no	—
279	C₂×C₄	—	5	1	no	—
280	C₂×C₄	—	20	1	no	—
281	C₂×C₄	—	20	1	no	—
282	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
283	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
284	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
285	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
286	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
287	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
288	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
289	C₈	—	2	1	no	—
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	—	5	2	no	—
293	C₈	—	10	1	no	—
294	C₈	—	10	1	no	—
295	C₈	—	10	1	no	—
296	C₈	Q₈×D₅	1	2	yes	—
297	C₅	#2961	1	1	yes	characteristic, complemented, Sylow
298	C₂²	—	5	2	no	—
299	C₂²	—	10	1	no	—
300	C₂²	—	10	1	no	—
301	C₂²	C₂²×C₄×D₅	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	5	2	no	—
309	C₄	—	10	1	no	—
310	C₄	—	10	1	no	—
311	C₄	—	10	1	no	—
312	C₄	—	10	1	no	—
313	C₄	—	10	1	no	—
314	C₄	—	10	1	no	—
315	C₄	C₂×Q₈×D₅	1	1	yes	characteristic
316	C₄	D₅×#41	1	1	yes	characteristic
317	C₂	—	10	2	no	—
318	C₂	C₄×Q₈×D₅	1	1	yes	characteristic
319	C₂	#66079	1	1	yes	characteristic
320	C₂	#66084	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅
commutator, socle	C₂²×C₅	C₂³×C₄
Fitting	C₅×#446	C₂
prime radical	C₂²×D₅	C₂²×C₄
cosocle	C₂×C₄×C₅	C₂⁴
hypercenter	#446	D₅
hypocenter	C₅	#2961
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. C₅×#446	C₂	semidirect
3. #65313	C₂	semidirect
7. #65120	C₂	semidirect
8. #65117	C₂	semidirect
9. #65117	C₂	semidirect
12. #64990	C₂	semidirect
13. #64990	C₂	semidirect
15. #64968	C₂	semidirect
106. C₈×D₅	Q₈	semidirect
115. #446	D₅	semidirect
297. C₅	#2961	semidirect

Maximal subgroups

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

subgroup	index
#65912	2
C₅×#446	2
#65313	2
#65128	2
#65126	2
#65120	2
#65117	2
#64997	2
D₅×#97	2
#64990	2
#64978	2
#64968	2
#2961	5