Subgroups — #13098358

Order 1600.

147 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#11716431	C₂	1	1	yes	maximal, characteristic
3	#11716356	C₂	1	4	yes	maximal
4	C₄×#6014	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₈×C₂₅	—	2	1	no	—
6	C₂×Dic₅₀	—	2	1	no	—
7	C₄⋊C₄×C₂₅	—	2	1	no	—
8	C₄⋊C₄×C₂₅	C₂²	1	2	yes	—
9	C₄²×C₂₅	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₂₅	—	2	1	no	—
11	C₄⋊Dic₂₅	C₂²	1	2	yes	—
12	C₂×#6014	C₂²	1	2	yes	—
13	#65368	—	5	1	no	maximal
14	Q₈×C₂₅	—	4	2	no	—
15	C₂×C₄×C₂₅	—	2	2	no	—
16	C₂×C₄×C₂₅	D₄	1	1	yes	characteristic
17	C₂×C₄×C₂₅	D₄	1	1	yes	characteristic
18	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
19	C₂×Dic₂₅	—	2	2	no	—
20	Dic₅₀	—	4	2	no	—
21	#6014	Q₈	1	4	yes	complemented
22	C₅×C₄⋊Q₈	D₅	1	1	yes	characteristic
23	#4015	—	5	1	no	—
24	#3940	—	5	4	no	—
25	C₄×#168	—	5	1	no	—
26	C₂²×C₂₅	C₂×D₄	1	1	yes	characteristic
27	C₄×C₂₅	—	4	2	no	—
28	C₄×C₂₅	SD₁₆	1	2	yes	—
29	C₄×C₂₅	SD₁₆	1	2	yes	—
30	C₄×C₂₅	C₂×Q₈	1	2	yes	—
31	Dic₂₅	—	4	2	no	—
32	C₂×C₅×Q₈	—	2	1	no	—
33	C₂×Dic₁₀	—	10	1	no	—
34	C₅×C₄⋊C₄	—	2	1	no	—
35	C₅×C₄⋊C₄	C₂×D₅	1	2	yes	—
36	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
37	C₄⋊Dic₅	—	5	2	no	—
38	C₄⋊Dic₅	—	10	1	no	—
39	C₂×#168	—	5	2	no	—
40	#498	—	25	1	no	Sylow
41	C₂×C₂₅	C₄⋊Q₈	1	1	yes	characteristic
42	C₂×C₂₅	C₂×SD₁₆	1	1	yes	characteristic
43	C₂×C₂₅	C₂×SD₁₆	1	1	yes	characteristic
44	C₅×Q₈	—	4	2	no	—
45	C₂×C₄×C₅	—	2	2	no	—
46	C₂×C₄×C₅	C₂²⋊D₅	1	1	yes	characteristic
47	C₂×C₄×C₅	C₂²⋊D₅	1	1	yes	characteristic
48	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
49	C₂×Dic₅	—	10	2	no	—
50	Dic₁₀	—	20	2	no	—
51	#168	—	5	4	no	—
52	C₄⋊Q₈	—	25	1	no	—
53	C₄⋊Q₈	D₂₅	1	1	yes	characteristic
54	#106	—	25	4	no	—
55	C₄×C₈	—	25	1	no	—
56	C₂₅	#498	1	1	yes	characteristic, complemented, Sylow
57	C₂²×C₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
58	C₄×C₅	—	4	2	no	—
59	C₄×C₅	#687	1	2	yes	—
60	C₄×C₅	#688	1	2	yes	—
61	C₄×C₅	Q₈×D₅	1	2	yes	—
62	Dic₅	—	20	2	no	—
63	C₂×Q₈	—	2	1	no	—
64	C₂×Q₈	—	50	1	no	—
65	C₂×C₈	—	25	2	no	—
66	C₄⋊C₄	—	2	1	no	—
67	C₄⋊C₄	—	25	2	no	—
68	C₄⋊C₄	—	50	1	no	—
69	C₄⋊C₄	C₂×D₂₅	1	2	yes	—
70	C₄²	C₂×D₂₅	1	1	yes	characteristic
71	C₂×C₅	C₂×#687	1	1	yes	characteristic
72	C₂×C₅	C₂×#688	1	1	yes	characteristic
73	C₂×C₅	#4090	1	1	yes	characteristic
74	Q₈	—	4	2	no	—
75	Q₈	—	100	2	no	—
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	—	50	2	no	—
78	C₂×C₄	C₂²⋊D₂₅	1	1	yes	characteristic
79	C₂×C₄	C₂²⋊D₂₅	1	1	yes	characteristic
80	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
81	C₈	—	25	4	no	—
82	C₅	#65368	1	1	yes	characteristic
83	C₂²	C₂×C₂²⋊D₂₅	1	1	yes	characteristic
84	C₄	—	4	2	no	—
85	C₄	—	100	2	no	—
86	C₄	#87721	1	2	yes	—
87	C₄	#87724	1	2	yes	—
88	C₄	Q₈×D₂₅	1	2	yes	—
89	C₂	C₂×#87721	1	1	yes	characteristic
90	C₂	C₂×#87724	1	1	yes	characteristic
91	C₂	#11716506	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₅
commutator, cosocle	C₂×C₄×C₂₅	C₂³
Frattini	C₂×C₄×C₅	C₂²×D₅
Fitting	C₂₅×C₄⋊Q₈	C₂
socle, prime radical	C₂²×C₅	C₂×C₂²⋊D₅
hypercenter	C₄⋊Q₈	D₂₅
hypocenter	C₂₅	#498
LCS	G ▷ C₂×C₄×C₂₅ ▷ C₂×C₂₅ ▷ C₂₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
21. #6014	Q₈	semidirect
56. C₂₅	#498	semidirect

Maximal subgroups

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

subgroup	index
C₂₅×C₄⋊Q₈	2
#11716431	2
#11716356	2
C₄×#6014	2
#65368	5