Subgroups — #10615519

Order 640.

492 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#131	C₂	1	1	yes	maximal, characteristic, complemented
2	#66166	C₂	1	1	yes	maximal, characteristic
3	C₂×#4053	C₂	1	1	yes	maximal, characteristic, complemented
4	#65523	C₂	1	1	yes	maximal, characteristic, complemented
5	#65417	C₂	1	2	yes	maximal, complemented
6	#65416	C₂	1	2	yes	maximal, complemented
7	M₁₆×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
8	#65400	C₂	1	2	yes	maximal, complemented
9	#65399	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×C₂²⋊D₅	C₂	1	2	yes	maximal, complemented
11	C₂×#3946	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₅×#41	C₂²	1	1	yes	characteristic
13	C₂×#709	C₂²	1	1	yes	characteristic
14	C₅×#131	—	2	4	no	—
15	C₂×C₅×M₁₆	C₂²	1	2	yes	—
16	C₂×C₅×M₁₆	C₂²	1	1	yes	characteristic
17	C₂²×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₂²×C₅×C₈	C₂²	1	2	yes	—
19	#4093	C₄	1	1	yes	characteristic
20	#4090	C₄	1	1	yes	characteristic
21	#4086	C₄	1	1	yes	characteristic
22	#4085	C₄	1	2	yes	—
23	#4085	C₄	1	2	yes	—
24	#4082	C₄	1	1	yes	characteristic
25	C₄×C₂²⋊D₅	C₂²	1	2	yes	—
26	C₄×C₂²⋊D₅	C₂²	1	2	yes	—
27	C₂×C₄×Dic₅	C₂²	1	1	yes	characteristic
28	C₂×#681	C₂²	1	1	yes	characteristic
29	C₂²×#168	C₂²	1	1	yes	characteristic
30	#4053	—	2	4	no	—
31	C₂×#676	C₂²	1	2	yes	—
32	C₂×C₈×D₅	C₂²	1	2	yes	—
33	C₂²⋊#168	C₂²	1	2	yes	—
34	C₂²⋊#168	C₂²	1	2	yes	—
35	#3952	C₂²	1	2	yes	—
36	#3952	C₂²	1	2	yes	—
37	#3947	C₂²	1	2	yes	—
38	#3946	C₂²	1	1	yes	characteristic
39	#3946	C₂²	1	1	yes	characteristic
40	#3946	C₂²	1	1	yes	characteristic
41	#3946	C₂²	1	1	yes	characteristic
42	C₈×Dic₅	C₂²	1	2	yes	—
43	#2935	—	5	1	no	maximal, Sylow
44	C₅×#41	—	2	2	no	—
45	C₅×#41	—	2	2	no	—
46	C₂×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
47	C₂×C₅×D₄	C₂×C₄	1	1	yes	characteristic
48	C₂×C₅×D₄	C₂×C₄	1	2	yes	—
49	C₂²×C₄×C₅	C₂³	1	2	yes	—
50	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
51	C₂×C₂²⋊D₅	C₂×C₄	1	2	yes	—
52	C₂×C₂²⋊D₅	C₂×C₄	1	2	yes	—
53	C₂×C₂²⋊D₅	C₂×C₄	1	2	yes	—
54	C₂²×Dic₅	C₂×C₄	1	1	yes	characteristic
55	C₂²×Dic₅	C₂×C₄	1	1	yes	characteristic
56	#709	—	2	2	no	—
57	#709	—	2	2	no	—
58	C₂×D₂₀	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄×D₅	C₂³	1	2	yes	—
60	C₂×Dic₁₀	C₂×C₄	1	1	yes	characteristic
61	C₅×M₁₆	—	2	4	no	—
62	C₅×M₁₆	D₄	1	4	yes	complemented
63	C₂×C₅×C₈	—	2	2	no	—
64	C₂×C₅×C₈	—	2	4	no	—
65	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
66	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
67	C₂×C₅×C₈	C₂³	1	2	yes	—
68	C₂²⋊Dic₅	C₂×C₄	1	2	yes	—
69	C₂²⋊Dic₅	C₂×C₄	1	2	yes	—
70	#685	C₂×C₄	1	2	yes	—
71	#685	C₂×C₄	1	2	yes	—
72	#683	C₂×C₄	1	2	yes	—
73	#683	C₂×C₄	1	2	yes	—
74	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
75	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
76	C₄×Dic₅	C₂³	1	1	yes	characteristic
77	C₄×Dic₅	C₂³	1	1	yes	characteristic
78	#681	—	2	2	no	—
79	C₂×#168	—	2	2	no	—
80	C₂×#168	C₂³	1	2	yes	—
81	C₂×#168	C₂³	1	1	yes	characteristic
82	C₂×#168	C₂³	1	1	yes	characteristic
83	#676	—	2	4	no	—
84	C₈×D₅	—	2	4	no	—
85	C₂×#131	—	5	1	no	—
86	C₂×#131	D₅	1	1	yes	characteristic, complemented
87	#532	—	5	1	no	—
88	#436	—	5	2	no	—
89	#435	—	5	2	no	—
90	#435	—	5	2	no	—
91	C₈×D₄	—	5	2	no	—
92	C₂×C₄⋊C₈	—	5	1	no	—
93	#408	—	5	1	no	—
94	#408	—	5	1	no	—
95	C₄×M₁₆	—	5	1	no	—
96	C₂³×C₅	C₂²×C₄	1	2	yes	—
97	C₂³×C₅	C₂²×C₄	1	1	yes	characteristic
98	C₂²×D₅	C₂²×C₄	1	2	yes	—
99	C₅×Q₈	—	2	2	no	—
100	C₅×D₄	—	2	4	no	—
101	C₅×D₄	—	2	2	no	—
102	C₂×C₄×C₅	—	2	2	no	—
103	C₂×C₄×C₅	—	2	2	no	—
104	C₂×C₄×C₅	C₂²×C₄	1	2	yes	—
105	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
107	C₂×C₄×C₅	#41	1	2	yes	—
108	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
109	C₂²⋊D₅	—	2	4	no	—
110	C₂²⋊D₅	—	2	4	no	—
111	C₂²⋊D₅	—	4	2	no	—
112	C₂×Dic₅	—	2	2	no	—
113	C₂×Dic₅	—	4	1	no	—
114	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
115	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
116	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
117	C₂×Dic₅	C₂²×C₄	1	2	yes	—
118	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
119	D₂₀	—	2	2	no	—
120	C₄×D₅	—	2	2	no	—
121	C₄×D₅	—	2	2	no	—
122	Dic₁₀	—	2	2	no	—
123	C₅×C₈	—	2	1	no	—
124	C₅×C₈	—	2	1	no	—
125	C₅×C₈	C₂×D₄	1	4	yes	—
126	#168	—	2	1	no	—
127	#168	—	2	1	no	—
128	#168	—	2	2	no	—
129	C₂×#41	—	5	1	no	—
130	C₂×#41	C₂×D₅	1	1	yes	characteristic
131	#131	—	2	4	no	—
132	#131	—	10	4	no	—
133	C₂×M₁₆	—	5	1	no	—
134	C₂×M₁₆	—	5	2	no	—
135	C₂×M₁₆	C₂×D₅	1	2	yes	—
136	C₂×M₁₆	C₂×D₅	1	1	yes	characteristic
137	C₂²×C₈	—	5	1	no	—
138	C₂²×C₈	—	5	2	no	—
139	C₂²×C₈	C₂×D₅	1	1	yes	characteristic
140	C₂²×C₈	C₂×D₅	1	2	yes	—
141	C₄⋊Q₈	—	5	1	no	—
142	#127	—	5	1	no	—
143	#124	—	5	1	no	—
144	#124	—	5	1	no	—
145	#121	—	5	2	no	—
146	#121	—	5	2	no	—
147	C₄×D₄	—	5	2	no	—
148	C₄×D₄	—	5	2	no	—
149	C₂×C₄²	—	5	1	no	—
150	C₄⋊C₈	—	5	1	no	—
151	C₄⋊C₈	—	5	1	no	—
152	C₄⋊C₈	—	5	1	no	—
153	C₄⋊C₈	—	5	1	no	—
154	C₂²⋊C₈	—	5	2	no	—
155	C₂²⋊C₈	—	5	2	no	—
156	C₂²⋊C₈	—	5	2	no	—
157	C₂²⋊C₈	—	5	2	no	—
158	#97	—	5	2	no	—
159	C₄×C₈	—	5	2	no	—
160	C₂²×C₅	—	2	1	no	—
161	C₂²×C₅	—	2	1	no	—
162	C₂²×C₅	—	2	2	no	—
163	C₂²×C₅	—	2	2	no	—
164	C₂²×C₅	—	2	2	no	—
165	C₂²×C₅	C₄×D₄	1	2	yes	—
166	C₂²×C₅	C₂³×C₄	1	1	yes	characteristic
167	C₂×D₅	—	2	2	no	—
168	C₂×D₅	—	2	2	no	—
169	C₂×D₅	—	2	2	no	—
170	C₄×C₅	—	2	2	no	—
171	C₄×C₅	C₄×D₄	1	2	yes	—
172	C₄×C₅	C₂²×D₄	1	1	yes	characteristic
173	C₄×C₅	C₂×#41	1	1	yes	characteristic
174	Dic₅	—	2	2	no	—
175	Dic₅	—	2	1	no	—
176	Dic₅	—	2	1	no	—
177	Dic₅	—	2	1	no	—
178	Dic₅	#131	1	2	yes	complemented
179	#41	—	2	2	no	—
180	#41	—	2	2	no	—
181	#41	—	10	2	no	—
182	#41	—	10	2	no	—
183	C₂×Q₈	—	5	1	no	—
184	C₂×Q₈	C₄×D₅	1	1	yes	characteristic
185	C₂×D₄	—	5	2	no	—
186	C₂×D₄	—	5	2	no	—
187	C₂×D₄	—	5	2	no	—
188	C₂×D₄	—	5	1	no	—
189	C₂×D₄	C₄×D₅	1	2	yes	—
190	C₂×D₄	C₄×D₅	1	1	yes	characteristic
191	C₂²×C₄	—	5	1	no	—
192	C₂²×C₄	—	5	1	no	—
193	C₂²×C₄	—	5	2	no	—
194	C₂²×C₄	C₂²×D₅	1	2	yes	—
195	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
196	M₁₆	—	2	4	no	—
197	M₁₆	—	10	2	no	—
198	M₁₆	—	10	4	no	—
199	M₁₆	C₂²⋊D₅	1	4	yes	complemented
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	—	5	2	no	—
203	C₂×C₈	—	5	1	no	—
204	C₂×C₈	—	5	1	no	—
205	C₂×C₈	—	10	2	no	—
206	C₂×C₈	—	10	4	no	—
207	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
208	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
209	C₂×C₈	C₂²×D₅	1	2	yes	—
210	C₄⋊C₄	—	5	2	no	—
211	C₄⋊C₄	—	5	2	no	—
212	C₂²⋊C₄	—	5	2	no	—
213	C₂²⋊C₄	—	5	2	no	—
214	C₂²⋊C₄	—	5	2	no	—
215	C₂²⋊C₄	—	5	2	no	—
216	C₄²	—	5	1	no	—
217	C₄²	—	5	1	no	—
218	C₄²	—	5	1	no	—
219	C₄²	—	5	1	no	—
220	C₂×C₅	—	2	2	no	—
221	C₂×C₅	—	4	2	no	—
222	C₂×C₅	C₂×C₄×D₄	1	1	yes	characteristic
223	C₂×C₅	C₂×#131	1	1	yes	characteristic
224	C₂×C₅	#568	1	1	yes	characteristic
225	D₅	—	4	2	no	—
226	C₂³	—	5	2	no	—
227	C₂³	C₂×C₄×D₅	1	2	yes	—
228	C₂³	C₂×C₄×D₅	1	1	yes	characteristic
229	Q₈	—	2	2	no	—
230	Q₈	—	10	2	no	—
231	D₄	—	2	4	no	—
232	D₄	—	2	2	no	—
233	D₄	—	10	4	no	—
234	D₄	—	10	4	no	—
235	D₄	—	10	2	no	—
236	D₄	—	20	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	5	1	no	—
240	C₂×C₄	—	5	1	no	—
241	C₂×C₄	—	5	1	no	—
242	C₂×C₄	—	5	1	no	—
243	C₂×C₄	—	5	2	no	—
244	C₂×C₄	—	10	2	no	—
245	C₂×C₄	—	10	2	no	—
246	C₂×C₄	—	10	2	no	—
247	C₂×C₄	—	20	1	no	—
248	C₂×C₄	C₂×C₄×D₅	1	2	yes	—
249	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
250	C₂×C₄	#709	1	2	yes	—
251	C₂×C₄	C₂×C₂²⋊D₅	1	2	yes	—
252	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
253	C₈	—	2	1	no	—
254	C₈	—	2	1	no	—
255	C₈	—	10	2	no	—
256	C₈	—	10	1	no	—
257	C₈	—	10	1	no	—
258	C₈	C₂×C₂²⋊D₅	1	4	yes	—
259	C₅	#2935	1	1	yes	characteristic, complemented, Sylow
260	C₂²	—	2	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	10	2	no	—
266	C₂²	—	10	2	no	—
267	C₂²	—	10	2	no	—
268	C₂²	C₄×C₂²⋊D₅	1	2	yes	—
269	C₂²	C₂²×C₄×D₅	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	5	2	no	—
272	C₄	—	10	1	no	—
273	C₄	—	10	1	no	—
274	C₄	—	10	1	no	—
275	C₄	—	10	2	no	—
276	C₄	C₄×C₂²⋊D₅	1	2	yes	—
277	C₄	C₂×#709	1	1	yes	characteristic
278	C₄	C₂²×C₂²⋊D₅	1	1	yes	characteristic
279	C₂	—	2	2	no	—
280	C₂	—	4	2	no	—
281	C₂	—	20	2	no	—
282	C₂	D₅×#131	1	1	yes	characteristic
283	C₂	#66084	1	1	yes	characteristic
284	C₂	C₂×C₄×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅
commutator, socle	C₂²×C₅	C₂³×C₄
Fitting	C₂×C₅×#131	C₂
prime radical	#66166	C₂
cosocle	C₂×C₄×C₅	C₂⁴
hypercenter	C₂×#131	D₅
hypocenter	C₅	#2935
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#131	C₂	semidirect
3. C₂×#4053	C₂	semidirect
4. #65523	C₂	semidirect
5. #65417	C₂	semidirect
6. #65416	C₂	semidirect
7. M₁₆×Dic₅	C₂	semidirect
8. #65400	C₂	semidirect
9. #65399	C₂	semidirect
10. C₈×C₂²⋊D₅	C₂	semidirect
11. C₂×#3946	C₂	semidirect
62. C₅×M₁₆	D₄	semidirect
86. C₂×#131	D₅	semidirect
178. Dic₅	#131	semidirect
199. M₁₆	C₂²⋊D₅	semidirect
259. C₅	#2935	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#131	2
#66166	2
C₂×#4053	2
#65523	2
#65417	2
#65416	2
M₁₆×Dic₅	2
#65400	2
#65399	2
C₈×C₂²⋊D₅	2
C₂×#3946	2
#2935	5