Subgroups — #13099548

Order 1600.

232 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#87799	C₂	1	1	yes	maximal, characteristic
2	C₅²×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#11716679	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×C₅⋊D₅	C₄	1	1	yes	characteristic
5	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
6	C₂×#6029	C₄	1	1	yes	characteristic
7	#87799	—	2	1	no	—
8	#87799	—	2	1	no	—
9	C₅²×C₁₆	—	2	1	no	—
10	#87756	—	2	1	no	—
11	#64734	—	5	6	no	maximal
12	C₂²×C₅⋊D₅	C₈	1	1	yes	characteristic
13	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
14	C₂×#1039	C₈	1	1	yes	characteristic
15	C₄×C₅⋊D₅	—	2	1	no	—
16	C₄×C₅⋊D₅	—	2	1	no	—
17	C₅²×C₈	D₄	1	1	yes	characteristic
18	C₅²×C₈	D₄	1	1	yes	characteristic
19	#6029	—	2	1	no	—
20	C₂×#676	—	5	6	no	—
21	C₅×M₃₂	D₅	1	6	yes	complemented
22	#3944	—	5	6	no	—
23	C₂²×C₅²	C₂×C₈	1	1	yes	characteristic
24	C₂×C₅⋊D₅	—	2	1	no	—
25	C₂×C₅⋊D₅	—	4	1	no	—
26	C₄×C₅²	C₂²⋊C₄	1	1	yes	characteristic
27	C₄×C₅²	M₁₆	1	1	yes	characteristic
28	#1039	—	2	1	no	—
29	C₂×C₄×D₅	—	5	6	no	—
30	C₂×C₅×C₈	C₂×D₅	1	6	yes	—
31	C₂×#168	—	5	6	no	—
32	#676	—	10	6	no	—
33	#676	—	10	6	no	—
34	C₅×C₁₆	—	2	6	no	—
35	#672	—	10	6	no	—
36	#349	—	25	1	no	Sylow
37	C₂×C₅²	—	2	1	no	—
38	C₂×C₅²	C₂²⋊C₈	1	1	yes	characteristic
39	C₅⋊D₅	—	4	1	no	—
40	C₂²×D₅	—	5	6	no	—
41	C₂×C₄×C₅	C₄×D₅	1	6	yes	—
42	C₂×Dic₅	—	5	6	no	—
43	C₄×D₅	—	10	6	no	—
44	C₄×D₅	—	10	6	no	—
45	C₅×C₈	D₂₀	1	6	yes	—
46	C₅×C₈	C₂²⋊D₅	1	6	yes	—
47	#168	—	10	6	no	—
48	C₂×M₁₆	—	25	1	no	—
49	M₃₂	—	25	1	no	—
50	M₃₂	C₅⋊D₅	1	1	yes	characteristic, complemented
51	C₅²	#349	1	1	yes	characteristic, complemented, Sylow
52	C₂²×C₅	C₈×D₅	1	6	yes	—
53	C₂×D₅	—	10	6	no	—
54	C₂×D₅	—	20	6	no	—
55	C₄×C₅	#676	1	6	yes	—
56	C₄×C₅	#685	1	6	yes	—
57	Dic₅	—	10	6	no	—
58	C₂²×C₄	—	25	1	no	—
59	M₁₆	—	50	1	no	—
60	M₁₆	—	50	1	no	—
61	C₂×C₈	—	25	1	no	—
62	C₂×C₈	C₂×C₅⋊D₅	1	1	yes	characteristic
63	C₁₆	—	2	1	no	—
64	C₁₆	—	50	1	no	—
65	C₂×C₅	—	2	6	no	—
66	C₂×C₅	#3952	1	6	yes	—
67	D₅	—	20	6	no	—
68	C₂³	—	25	1	no	—
69	C₂×C₄	—	25	1	no	—
70	C₂×C₄	—	50	1	no	—
71	C₂×C₄	—	50	1	no	—
72	C₂×C₄	C₄×C₅⋊D₅	1	1	yes	characteristic
73	C₈	—	50	1	no	—
74	C₈	C₄⋊(C₅⋊D₅)	1	1	yes	characteristic
75	C₈	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
76	C₅	#64734	1	6	yes	complemented
77	C₂²	—	50	1	no	—
78	C₂²	—	100	1	no	—
79	C₂²	C₈×C₅⋊D₅	1	1	yes	characteristic
80	C₄	—	50	1	no	—
81	C₄	#87799	1	1	yes	characteristic
82	C₄	#87808	1	1	yes	characteristic
83	C₂	—	2	1	no	—
84	C₂	—	100	1	no	—
85	C₂	#11716687	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#87808
commutator	C₂²×C₅²	C₂×C₈
Frattini	C₂×C₈	C₂×C₅⋊D₅
Fitting	C₅²×M₃₂	C₂
socle	C₂×C₅²	C₂²⋊C₈
prime radical	C₂²×C₅⋊D₅	C₈
cosocle	C₂×C₅²×C₈	C₂²
hypercenter	M₃₂	C₅⋊D₅
hypocenter	C₅²	#349
LCS	G ▷ C₂²×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
2. C₅²×M₃₂	C₂	semidirect
3. #11716679	C₂	semidirect
21. C₅×M₃₂	D₅	semidirect
50. M₃₂	C₅⋊D₅	semidirect
51. C₅²	#349	semidirect
76. C₅	#64734	semidirect

Maximal subgroups

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

subgroup	index
C₂×#87799	2
C₅²×M₃₂	2
#11716679	2
#64734	5