Subgroups — #13099046

Order 1600.

318 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	#11716506	C₂	1	4	yes	maximal
3	#11716450	C₂	1	4	yes	maximal
4	#11716449	C₂	1	4	yes	maximal
5	C₄×Dic₅₀	C₂	1	2	yes	maximal
6	C₂×Q₈×C₂₅	C₂²	1	2	yes	—
7	C₂×Dic₅₀	C₂²	1	4	yes	—
8	C₂×Dic₅₀	C₂²	1	2	yes	—
9	C₄⋊C₄×C₂₅	C₂²	1	4	yes	—
10	C₄²×C₂₅	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₂₅	C₂²	1	2	yes	—
12	#87718	C₂²	1	4	yes	—
13	#87718	C₂²	1	8	yes	—
14	C₄×Dic₂₅	C₂²	1	4	yes	—
15	C₄×Dic₂₅	C₂²	1	4	yes	—
16	#66056	—	5	1	no	maximal
17	Q₈×C₂₅	—	4	2	no	—
18	C₂×C₄×C₂₅	C₂³	1	4	yes	—
19	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
20	C₂×C₄×C₂₅	C₂³	1	2	yes	—
21	C₂×Dic₂₅	C₂³	1	4	yes	—
22	C₂×Dic₂₅	C₂³	1	4	yes	—
23	Dic₅₀	—	4	4	no	—
24	Dic₅₀	Q₈	1	8	yes	complemented
25	C₅×C₄⋊Q₈	D₅	1	1	yes	characteristic
26	#4090	—	5	4	no	—
27	#4034	—	5	4	no	—
28	#4033	—	5	4	no	—
29	C₄×Dic₁₀	—	5	2	no	—
30	C₂²×C₂₅	C₂⁴	1	1	yes	characteristic
31	C₄×C₂₅	—	2	4	no	—
32	C₄×C₂₅	—	2	1	no	—
33	C₄×C₂₅	C₂×Q₈	1	4	yes	—
34	Dic₂₅	—	2	4	no	—
35	Dic₂₅	C₂×Q₈	1	8	yes	—
36	C₂×C₅×Q₈	C₂×D₅	1	2	yes	—
37	C₂×Dic₁₀	—	5	2	no	—
38	C₂×Dic₁₀	—	5	4	no	—
39	C₅×C₄⋊C₄	C₂×D₅	1	4	yes	—
40	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
41	C₄⋊Dic₅	—	5	2	no	—
42	#683	—	5	4	no	—
43	#683	—	5	8	no	—
44	C₄×Dic₅	—	5	4	no	—
45	C₄×Dic₅	—	5	4	no	—
46	Q₈²	—	25	1	no	Sylow
47	C₂×C₂₅	C₂²×Q₈	1	2	yes	—
48	C₂×C₂₅	H(5,𝔽₂)	1	1	yes	characteristic
49	C₅×Q₈	—	4	2	no	—
50	C₂×C₄×C₅	C₂²×D₅	1	4	yes	—
51	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
52	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
53	C₂×Dic₅	—	5	4	no	—
54	C₂×Dic₅	—	5	4	no	—
55	Dic₁₀	—	5	8	no	—
56	Dic₁₀	—	20	4	no	—
57	C₄⋊Q₈	—	25	4	no	—
58	C₄⋊Q₈	—	25	4	no	—
59	C₄⋊Q₈	D₂₅	1	1	yes	characteristic
60	C₄×Q₈	—	25	4	no	—
61	C₄×Q₈	—	25	2	no	—
62	C₂₅	Q₈²	1	1	yes	characteristic, complemented, Sylow
63	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
64	C₄×C₅	—	2	4	no	—
65	C₄×C₅	—	2	1	no	—
66	C₄×C₅	Q₈×D₅	1	4	yes	—
67	Dic₅	—	5	8	no	—
68	Dic₅	—	10	4	no	—
69	C₂×Q₈	—	25	2	no	—
70	C₂×Q₈	—	25	4	no	—
71	C₂×Q₈	C₂×D₂₅	1	2	yes	—
72	C₄⋊C₄	—	25	8	no	—
73	C₄⋊C₄	—	25	4	no	—
74	C₄⋊C₄	—	25	2	no	—
75	C₄⋊C₄	C₂×D₂₅	1	4	yes	—
76	C₄²	—	25	4	no	—
77	C₄²	—	25	4	no	—
78	C₄²	C₂×D₂₅	1	1	yes	characteristic
79	C₂×C₅	#4144	1	1	yes	characteristic
80	C₂×C₅	C₂×Q₈×D₅	1	2	yes	—
81	Q₈	—	4	2	no	—
82	Q₈	—	25	8	no	—
83	Q₈	—	100	4	no	—
84	C₂×C₄	—	25	4	no	—
85	C₂×C₄	—	25	4	no	—
86	C₂×C₄	C₂²×D₂₅	1	4	yes	—
87	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
88	C₂×C₄	C₂²×D₂₅	1	2	yes	—
89	C₅	#66056	1	1	yes	characteristic
90	C₂²	C₂³×D₂₅	1	1	yes	characteristic
91	C₄	—	2	1	no	—
92	C₄	—	2	4	no	—
93	C₄	—	25	8	no	—
94	C₄	—	50	4	no	—
95	C₄	Q₈×D₂₅	1	4	yes	—
96	C₂	#11716563	1	1	yes	characteristic
97	C₂	C₂×Q₈×D₂₅	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂³×D₂₅
commutator, cosocle	C₂²×C₂₅	C₂⁴
Frattini, socle, prime radical	C₂²×C₅	C₂³×D₅
Fitting	C₂₅×C₄⋊Q₈	C₂
hypercenter	C₄⋊Q₈	D₂₅
hypocenter	C₂₅	Q₈²
LCS	G ▷ C₂²×C₂₅ ▷ C₂₅
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
24. Dic₅₀	Q₈	semidirect
62. C₂₅	Q₈²	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
#11716506	2
#11716450	2
#11716449	2
C₄×Dic₅₀	2
#66056	5