Subgroups — #13098355

Order 1600.

213 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, complemented
2	C₂×#87724	C₂	1	4	yes	maximal, complemented
3	#11716436	C₂	1	1	yes	maximal, characteristic
4	C₄×#6014	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×Q₈×C₂₅	C₂²	1	2	yes	—
6	C₂×D₁₀₀	—	2	2	no	—
7	C₂×D₁₀₀	C₂²	1	2	yes	—
8	C₄⋊C₄×C₂₅	—	2	2	no	—
9	C₄²×C₂₅	C₂²	1	1	yes	characteristic
10	#87724	—	2	8	no	—
11	C₂×#6014	C₂²	1	2	yes	—
12	#65374	—	5	1	no	maximal
13	C₂²×D₂₅	—	2	2	no	—
14	Q₈×C₂₅	—	2	4	no	—
15	C₂×C₄×C₂₅	—	2	2	no	—
16	C₂×C₄×C₂₅	D₄	1	2	yes	—
17	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
18	D₁₀₀	—	2	4	no	—
19	D₁₀₀	—	4	4	no	—
20	#6014	D₄	1	4	yes	complemented
21	C₅×C₄⋊Q₈	D₅	1	1	yes	characteristic
22	C₂×#688	—	5	4	no	—
23	#4020	—	5	1	no	—
24	C₄×#168	—	5	1	no	—
25	C₂²×C₂₅	C₂×D₄	1	1	yes	characteristic
26	C₂×D₂₅	—	4	2	no	—
27	C₂×D₂₅	—	4	4	no	—
28	C₄×C₂₅	—	4	2	no	—
29	C₄×C₂₅	SD₁₆	1	4	yes	complemented
30	C₄×C₂₅	C₂×D₄	1	2	yes	—
31	C₂×C₅×Q₈	C₂×D₅	1	2	yes	—
32	C₂×D₂₀	—	5	2	no	—
33	C₂×D₂₀	—	10	2	no	—
34	C₅×C₄⋊C₄	—	2	2	no	—
35	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
36	#688	—	10	8	no	—
37	C₂×#168	—	5	2	no	—
38	#492	—	25	1	no	Sylow
39	C₂×C₂₅	#127	1	1	yes	characteristic
40	C₂×C₂₅	C₂×SD₁₆	1	2	yes	—
41	D₂₅	—	8	2	no	—
42	C₂²×D₅	—	10	2	no	—
43	C₅×Q₈	—	2	4	no	—
44	C₂×C₄×C₅	—	2	2	no	—
45	C₂×C₄×C₅	C₂²⋊D₅	1	2	yes	—
46	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
47	D₂₀	—	10	4	no	—
48	D₂₀	—	20	4	no	—
49	#168	—	5	4	no	—
50	C₂×SD₁₆	—	25	4	no	—
51	C₄⋊Q₈	D₂₅	1	1	yes	characteristic, complemented
52	#127	—	25	1	no	—
53	C₄×C₈	—	25	1	no	—
54	C₂₅	#492	1	1	yes	characteristic, complemented, Sylow
55	C₂²×C₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
56	C₂×D₅	—	20	2	no	—
57	C₂×D₅	—	20	4	no	—
58	C₄×C₅	—	4	2	no	—
59	C₄×C₅	#688	1	4	yes	—
60	C₄×C₅	D₄×D₅	1	2	yes	—
61	C₂×Q₈	C₂×D₂₅	1	2	yes	—
62	C₂×D₄	—	25	2	no	—
63	C₂×D₄	—	50	2	no	—
64	SD₁₆	—	50	8	no	—
65	C₂×C₈	—	25	2	no	—
66	C₄⋊C₄	—	2	2	no	—
67	C₄²	C₂×D₂₅	1	1	yes	characteristic
68	C₂×C₅	#4086	1	1	yes	characteristic
69	C₂×C₅	C₂×#688	1	2	yes	—
70	D₅	—	40	2	no	—
71	C₂³	—	50	2	no	—
72	Q₈	—	2	4	no	—
73	D₄	—	50	4	no	—
74	D₄	—	100	4	no	—
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	C₂²⋊D₂₅	1	2	yes	—
77	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
78	C₈	—	25	4	no	—
79	C₅	#65374	1	1	yes	characteristic
80	C₂²	—	100	2	no	—
81	C₂²	—	100	4	no	—
82	C₂²	C₂×C₂²⋊D₂₅	1	1	yes	characteristic
83	C₄	—	4	2	no	—
84	C₄	#87724	1	4	yes	complemented
85	C₄	D₄×D₂₅	1	2	yes	—
86	C₂	—	200	2	no	—
87	C₂	#11716502	1	1	yes	characteristic
88	C₂	C₂×#87724	1	2	yes	—

	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	C₂²×C₅	C₂×C₂²⋊D₅
prime radical	#11716436	C₂
hypercenter	C₄⋊Q₈	D₂₅
hypocenter	C₂₅	#492
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
1. C₂₅×C₄⋊Q₈	C₂	semidirect
2. C₂×#87724	C₂	semidirect
4. C₄×#6014	C₂	semidirect
20. #6014	D₄	semidirect
29. C₄×C₂₅	SD₁₆	semidirect
51. C₄⋊Q₈	D₂₅	semidirect
54. C₂₅	#492	semidirect
84. C₄	#87724	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
C₂×#87724	2
#11716436	2
C₄×#6014	2
#65374	5