Subgroups — C₅×#3377 #10616972

Order 640.

332 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#557	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#556	C₂	1	1	yes	maximal, characteristic
3	C₅×#491	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#489	C₂	1	1	yes	maximal, characteristic
5	C₅×#488	C₂	1	1	yes	maximal, characteristic
6	C₅×#479	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#478	C₂	1	1	yes	maximal, characteristic
8	C₅×#477	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#464	C₂	1	1	yes	maximal, characteristic
10	C₅×#464	C₂	1	1	yes	maximal, characteristic
11	C₅×#462	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#446	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅×#441	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅×#440	C₂	1	1	yes	maximal, characteristic
15	C₄×C₅×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₅×Q₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₅×Q₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₅×SD₁₆	C₂²	1	1	yes	characteristic
19	C₅×C₄⋊Q₈	—	2	1	no	—
20	C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
21	C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
22	C₅×#126	—	2	1	no	—
23	C₅×#126	—	2	1	no	—
24	C₅×#125	—	2	1	no	—
25	C₅×#125	—	2	1	no	—
26	C₅×#125	C₂²	1	1	yes	characteristic
27	C₅×#125	C₂²	1	1	yes	characteristic
28	C₅×#124	C₂²	1	1	yes	characteristic
29	C₅×#124	C₂²	1	1	yes	characteristic
30	C₅×C₂²⋊Q₈	—	2	1	no	—
31	C₄×C₅×Q₈	—	2	1	no	—
32	C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
33	C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
34	C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
35	C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
36	C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
37	C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
38	C₄×C₅×D₄	C₂²	1	1	yes	characteristic
39	C₅×#117	—	2	1	no	—
40	C₅×#107	C₂²	1	1	yes	characteristic
41	C₅×#107	C₂²	1	1	yes	characteristic
42	C₅×#106	C₂²	1	1	yes	characteristic
43	C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
44	C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
45	C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
46	C₅×#103	C₂²	1	1	yes	characteristic
47	C₅×#103	C₂²	1	1	yes	characteristic
48	C₅×#103	C₂²	1	1	yes	characteristic
49	C₅×#103	C₂²	1	1	yes	characteristic
50	C₅×#103	C₂²	1	1	yes	characteristic
51	C₅×#103	C₂²	1	1	yes	characteristic
52	C₅×#103	C₂²	1	1	yes	characteristic
53	C₅×#102	C₂²	1	1	yes	characteristic
54	C₅×#102	C₂²	1	1	yes	characteristic
55	C₅×#102	C₂²	1	1	yes	characteristic
56	C₅×#97	C₂²	1	1	yes	characteristic
57	C₅×#97	C₂²	1	1	yes	characteristic
58	C₄×C₅×C₈	C₂²	1	1	yes	characteristic
59	#3377	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂×C₅×Q₈	—	2	1	no	—
61	C₂×C₅×Q₈	D₄	1	1	yes	characteristic
62	C₂×C₅×Q₈	C₂³	1	1	yes	characteristic
63	C₂×C₅×Q₈	C₂³	1	1	yes	characteristic
64	C₂×C₅×Q₈	C₂³	1	1	yes	characteristic
65	C₂×C₅×D₄	C₂³	1	1	yes	characteristic
66	C₂²×C₄×C₅	—	2	1	no	—
67	C₅×Q₁₆	—	2	2	no	—
68	C₅×Q₁₆	—	2	2	no	—
69	C₅×SD₁₆	—	2	2	no	—
70	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
71	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
72	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
73	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
74	C₅×C₄⋊C₄	—	2	1	no	—
75	C₅×C₄⋊C₄	—	2	1	no	—
76	C₅×C₄⋊C₄	—	2	1	no	—
77	C₅×C₄⋊C₄	—	2	1	no	—
78	C₅×C₄⋊C₄	—	2	1	no	—
79	C₅×C₄⋊C₄	—	2	1	no	—
80	C₅×C₄⋊C₄	—	2	1	no	—
81	C₅×C₄⋊C₄	—	2	1	no	—
82	C₅×C₄⋊C₄	—	2	1	no	—
83	C₅×C₄⋊C₄	—	2	1	no	—
84	C₅×C₄⋊C₄	—	2	1	no	—
85	C₅×C₄⋊C₄	—	2	1	no	—
86	C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
87	C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
88	C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
89	C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
90	C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
91	C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
92	C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
93	C₅×C₂²⋊C₄	—	2	1	no	—
94	C₅×C₂²⋊C₄	—	2	1	no	—
95	C₅×C₂²⋊C₄	—	2	1	no	—
96	C₅×C₂²⋊C₄	—	2	1	no	—
97	C₅×C₂²⋊C₄	—	2	1	no	—
98	C₄²×C₅	—	2	1	no	—
99	C₄²×C₅	—	2	1	no	—
100	C₄²×C₅	—	2	1	no	—
101	C₄²×C₅	—	2	1	no	—
102	C₄²×C₅	—	2	1	no	—
103	C₄²×C₅	C₂³	1	1	yes	characteristic
104	C₄²×C₅	C₂³	1	1	yes	characteristic
105	C₄²×C₅	C₂³	1	1	yes	characteristic
106	#557	C₂×C₅	1	1	yes	characteristic, complemented
107	#556	C₂×C₅	1	1	yes	characteristic
108	#491	C₂×C₅	1	1	yes	characteristic, complemented
109	#489	C₂×C₅	1	1	yes	characteristic
110	#488	C₂×C₅	1	1	yes	characteristic
111	#479	C₂×C₅	1	1	yes	characteristic, complemented
112	#478	C₂×C₅	1	1	yes	characteristic
113	#477	C₂×C₅	1	1	yes	characteristic, complemented
114	#464	C₂×C₅	1	1	yes	characteristic
115	#464	C₂×C₅	1	1	yes	characteristic
116	#462	C₂×C₅	1	1	yes	characteristic, complemented
117	#446	C₂×C₅	1	1	yes	characteristic, complemented
118	#441	C₂×C₅	1	1	yes	characteristic, complemented
119	#440	C₂×C₅	1	1	yes	characteristic
120	C₄×Q₁₆	C₂×C₅	1	1	yes	characteristic, complemented
121	C₂³×C₅	—	2	1	no	—
122	C₅×Q₈	—	2	1	no	—
123	C₅×Q₈	—	2	1	no	—
124	C₅×Q₈	—	2	1	no	—
125	C₅×Q₈	—	2	1	no	—
126	C₅×Q₈	—	2	1	no	—
127	C₅×Q₈	—	4	1	no	—
128	C₅×Q₈	—	4	2	no	—
129	C₅×Q₈	#41	1	2	yes	—
130	C₅×D₄	—	2	1	no	—
131	C₅×D₄	—	2	1	no	—
132	C₂×C₄×C₅	—	2	1	no	—
133	C₂×C₄×C₅	—	2	1	no	—
134	C₂×C₄×C₅	—	2	1	no	—
135	C₂×C₄×C₅	—	2	1	no	—
136	C₂×C₄×C₅	—	2	1	no	—
137	C₂×C₄×C₅	—	2	1	no	—
138	C₂×C₄×C₅	—	2	1	no	—
139	C₂×C₄×C₅	—	4	2	no	—
140	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
145	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
146	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
147	C₅×C₈	—	2	1	no	—
148	C₅×C₈	—	2	1	no	—
149	C₅×C₈	—	2	1	no	—
150	C₅×C₈	—	2	1	no	—
151	C₂×Q₁₆	C₂²×C₅	1	1	yes	characteristic
152	C₂×Q₁₆	C₂²×C₅	1	1	yes	characteristic
153	C₂×SD₁₆	C₂²×C₅	1	1	yes	characteristic
154	C₄⋊Q₈	—	2	1	no	—
155	C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
156	C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
157	#126	—	2	1	no	—
158	#126	—	2	1	no	—
159	#125	—	2	1	no	—
160	#125	—	2	1	no	—
161	#125	C₂²×C₅	1	1	yes	characteristic
162	#125	C₂²×C₅	1	1	yes	characteristic
163	#124	C₂²×C₅	1	1	yes	characteristic
164	#124	C₂²×C₅	1	1	yes	characteristic
165	C₂²⋊Q₈	—	2	1	no	—
166	C₄×Q₈	—	2	1	no	—
167	C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
168	C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
169	C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
170	C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
171	C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
172	C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
173	C₄×D₄	C₂²×C₅	1	1	yes	characteristic
174	#117	—	2	1	no	—
175	#107	C₂²×C₅	1	1	yes	characteristic
176	#107	C₂²×C₅	1	1	yes	characteristic
177	#106	C₂²×C₅	1	1	yes	characteristic
178	C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
179	C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
180	C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
181	#103	C₂²×C₅	1	1	yes	characteristic
182	#103	C₂²×C₅	1	1	yes	characteristic
183	#103	C₂²×C₅	1	1	yes	characteristic
184	#103	C₂²×C₅	1	1	yes	characteristic
185	#103	C₂²×C₅	1	1	yes	characteristic
186	#103	C₂²×C₅	1	1	yes	characteristic
187	#103	C₂²×C₅	1	1	yes	characteristic
188	#102	C₂²×C₅	1	1	yes	characteristic
189	#102	C₂²×C₅	1	1	yes	characteristic
190	#102	C₂²×C₅	1	1	yes	characteristic
191	#97	C₂²×C₅	1	1	yes	characteristic
192	#97	C₂²×C₅	1	1	yes	characteristic
193	C₄×C₈	C₂²×C₅	1	1	yes	characteristic
194	C₂²×C₅	—	4	1	no	—
195	C₂²×C₅	—	4	1	no	—
196	C₂²×C₅	—	4	1	no	—
197	C₂²×C₅	C₂²×D₄	1	1	yes	characteristic
198	C₄×C₅	—	2	2	no	—
199	C₄×C₅	—	2	1	no	—
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	—	4	1	no	—
206	C₄×C₅	—	4	1	no	—
207	C₄×C₅	—	4	1	no	—
208	C₄×C₅	—	4	1	no	—
209	C₄×C₅	—	4	1	no	—
210	C₄×C₅	—	4	1	no	—
211	C₄×C₅	C₂×#41	1	1	yes	characteristic
212	C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
213	C₂×Q₈	—	2	1	no	—
214	C₂×Q₈	C₅×D₄	1	1	yes	characteristic
215	C₂×Q₈	C₂³×C₅	1	1	yes	characteristic
216	C₂×Q₈	C₂³×C₅	1	1	yes	characteristic
217	C₂×Q₈	C₂³×C₅	1	1	yes	characteristic
218	C₂×D₄	C₂³×C₅	1	1	yes	characteristic
219	C₂²×C₄	—	2	1	no	—
220	Q₁₆	—	2	2	no	—
221	Q₁₆	—	2	2	no	—
222	SD₁₆	—	2	2	no	—
223	C₂×C₈	C₂³×C₅	1	1	yes	characteristic
224	C₂×C₈	C₂³×C₅	1	1	yes	characteristic
225	C₂×C₈	C₂³×C₅	1	1	yes	characteristic
226	C₂×C₈	C₂³×C₅	1	1	yes	characteristic
227	C₄⋊C₄	—	2	1	no	—
228	C₄⋊C₄	—	2	1	no	—
229	C₄⋊C₄	—	2	1	no	—
230	C₄⋊C₄	—	2	1	no	—
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	2	1	no	—
238	C₄⋊C₄	—	2	1	no	—
239	C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
240	C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
241	C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
242	C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
243	C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
244	C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
245	C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
246	C₂²⋊C₄	—	2	1	no	—
247	C₂²⋊C₄	—	2	1	no	—
248	C₂²⋊C₄	—	2	1	no	—
249	C₂²⋊C₄	—	2	1	no	—
250	C₂²⋊C₄	—	2	1	no	—
251	C₄²	—	2	1	no	—
252	C₄²	—	2	1	no	—
253	C₄²	—	2	1	no	—
254	C₄²	—	2	1	no	—
255	C₄²	—	2	1	no	—
256	C₄²	C₂³×C₅	1	1	yes	characteristic
257	C₄²	C₂³×C₅	1	1	yes	characteristic
258	C₄²	C₂³×C₅	1	1	yes	characteristic
259	C₂×C₅	—	8	1	no	—
260	C₂×C₅	#548	1	1	yes	characteristic
261	C₂×C₅	#575	1	1	yes	characteristic
262	C₂×C₅	#578	1	1	yes	characteristic
263	C₂³	—	2	1	no	—
264	Q₈	—	2	1	no	—
265	Q₈	—	2	1	no	—
266	Q₈	—	2	1	no	—
267	Q₈	—	2	1	no	—
268	Q₈	—	2	1	no	—
269	Q₈	—	4	1	no	—
270	Q₈	—	4	2	no	—
271	Q₈	C₅×#41	1	2	yes	—
272	D₄	—	2	1	no	—
273	D₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	C₂×C₅×D₄	1	1	yes	characteristic
283	C₂×C₄	C₂×C₅×D₄	1	1	yes	characteristic
284	C₂×C₄	C₂×C₅×D₄	1	1	yes	characteristic
285	C₂×C₄	C₂×C₅×D₄	1	1	yes	characteristic
286	C₂×C₄	C₂×C₅×D₄	1	1	yes	characteristic
287	C₂×C₄	C₂×C₅×D₄	1	1	yes	characteristic
288	C₂×C₄	C₂⁴×C₅	1	1	yes	characteristic
289	C₈	—	2	1	no	—
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	—	2	1	no	—
293	C₅	#3377	1	1	yes	characteristic, direct product factor, Sylow
294	C₂²	—	4	1	no	—
295	C₂²	—	4	1	no	—
296	C₂²	—	4	1	no	—
297	C₂²	C₂²×C₅×D₄	1	1	yes	characteristic
298	C₄	—	2	2	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	4	1	no	—
306	C₄	—	4	1	no	—
307	C₄	—	4	1	no	—
308	C₄	—	4	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	4	1	no	—
311	C₄	C₂×C₅×#41	1	1	yes	characteristic
312	C₄	C₅×2₋¹⁺⁴	1	1	yes	characteristic
313	C₂	—	8	1	no	—
314	C₂	C₅×#548	1	1	yes	characteristic
315	C₂	C₅×#575	1	1	yes	characteristic
316	C₂	C₅×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₅
prime radical	C₂×C₅×D₄	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄×C₅×Q₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅×#557	C₂	semidirect
3. C₅×#491	C₂	semidirect
6. C₅×#479	C₂	semidirect
8. C₅×#477	C₂	semidirect
11. C₅×#462	C₂	semidirect
12. C₅×#446	C₂	semidirect
13. C₅×#441	C₂	semidirect
15. C₄×C₅×Q₁₆	C₂	semidirect
59. #3377	C₅	direct (N × H)
106. #557	C₂×C₅	semidirect
108. #491	C₂×C₅	semidirect
111. #479	C₂×C₅	semidirect
113. #477	C₂×C₅	semidirect
116. #462	C₂×C₅	semidirect
117. #446	C₂×C₅	semidirect
118. #441	C₂×C₅	semidirect
120. C₄×Q₁₆	C₂×C₅	semidirect
293. C₅	#3377	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×#557	2
C₅×#556	2
C₅×#491	2
C₅×#489	2
C₅×#488	2
C₅×#479	2
C₅×#478	2
C₅×#477	2
C₅×#464	2
C₅×#462	2
C₅×#446	2
C₅×#441	2
C₅×#440	2
C₄×C₅×Q₁₆	2
#3377	5

Subgroups — C5×#3377 #10616972

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#3377 #10616972