Subgroups — #13098893

Order 1600.

300 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Q₈×C₂₅	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₂₅	C₂	1	2	yes	maximal
3	#11716452	C₂	1	6	yes	maximal
4	#11716431	C₂	1	3	yes	maximal
5	C₄×Dic₅₀	C₂	1	3	yes	maximal
6	C₂×Q₈×C₂₅	C₂²	1	1	yes	characteristic
7	C₂×Dic₅₀	C₂²	1	3	yes	—
8	C₄⋊C₄×C₂₅	C₂²	1	3	yes	—
9	C₄²×C₂₅	C₂²	1	3	yes	—
10	C₄⋊Dic₂₅	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₂₅	C₂²	1	6	yes	—
12	C₄⋊Dic₂₅	C₂²	1	6	yes	—
13	#87718	C₂²	1	6	yes	—
14	C₄×Dic₂₅	C₂²	1	6	yes	—
15	#65904	—	5	1	no	maximal
16	Q₈×C₂₅	Q₈	1	4	yes	complemented
17	C₂×C₄×C₂₅	C₂³	1	3	yes	—
18	C₂×C₄×C₂₅	C₂³	1	3	yes	—
19	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
20	C₂×Dic₂₅	C₂³	1	2	yes	—
21	C₂×Dic₂₅	C₂³	1	6	yes	—
22	Dic₅₀	—	2	6	no	—
23	C₄×C₅×Q₈	D₅	1	1	yes	characteristic
24	Q₈×Dic₅	—	5	2	no	—
25	#4036	—	5	6	no	—
26	#4015	—	5	3	no	—
27	C₄×Dic₁₀	—	5	3	no	—
28	C₂²×C₂₅	C₂⁴	1	1	yes	characteristic
29	C₄×C₂₅	—	2	3	no	—
30	C₄×C₂₅	C₂×Q₈	1	6	yes	—
31	C₄×C₂₅	#41	1	2	yes	—
32	Dic₂₅	—	2	6	no	—
33	Dic₂₅	—	2	2	no	—
34	C₂×C₅×Q₈	C₂×D₅	1	1	yes	characteristic
35	C₂×Dic₁₀	—	5	3	no	—
36	C₅×C₄⋊C₄	C₂×D₅	1	3	yes	—
37	C₄²×C₅	C₂×D₅	1	3	yes	—
38	C₄⋊Dic₅	—	5	1	no	—
39	C₄⋊Dic₅	—	5	6	no	—
40	C₄⋊Dic₅	—	5	6	no	—
41	#683	—	5	6	no	—
42	C₄×Dic₅	—	5	6	no	—
43	#557	—	25	1	no	Sylow
44	C₂×C₂₅	C₂²×Q₈	1	1	yes	characteristic
45	C₂×C₂₅	C₂×#41	1	1	yes	characteristic
46	C₂×C₂₅	2₋¹⁺⁴	1	1	yes	characteristic
47	C₅×Q₈	Dic₁₀	1	4	yes	—
48	C₂×C₄×C₅	C₂²×D₅	1	3	yes	—
49	C₂×C₄×C₅	C₂²×D₅	1	3	yes	—
50	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
51	C₂×Dic₅	—	5	2	no	—
52	C₂×Dic₅	—	5	6	no	—
53	Dic₁₀	—	10	6	no	—
54	C₄⋊Q₈	—	25	3	no	—
55	#125	—	25	6	no	—
56	C₄×Q₈	—	25	2	no	—
57	C₄×Q₈	—	25	3	no	—
58	C₄×Q₈	D₂₅	1	1	yes	characteristic
59	C₂₅	#557	1	1	yes	characteristic, complemented, Sylow
60	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
61	C₄×C₅	—	2	3	no	—
62	C₄×C₅	C₂×Dic₁₀	1	6	yes	—
63	C₄×C₅	#713	1	2	yes	—
64	Dic₅	—	10	6	no	—
65	Dic₅	—	10	2	no	—
66	C₂×Q₈	—	25	3	no	—
67	C₂×Q₈	C₂×D₂₅	1	1	yes	characteristic
68	C₄⋊C₄	—	25	1	no	—
69	C₄⋊C₄	—	25	6	no	—
70	C₄⋊C₄	—	25	6	no	—
71	C₄⋊C₄	—	25	6	no	—
72	C₄⋊C₄	C₂×D₂₅	1	3	yes	—
73	C₄²	—	25	6	no	—
74	C₄²	C₂×D₂₅	1	3	yes	—
75	C₂×C₅	C₂²×Dic₁₀	1	1	yes	characteristic
76	C₂×C₅	C₂×#713	1	1	yes	characteristic
77	C₂×C₅	#4150	1	1	yes	characteristic
78	Q₈	—	50	6	no	—
79	Q₈	Dic₅₀	1	4	yes	complemented
80	C₂×C₄	—	25	2	no	—
81	C₂×C₄	—	25	6	no	—
82	C₂×C₄	C₂²×D₂₅	1	3	yes	—
83	C₂×C₄	C₂²×D₂₅	1	3	yes	—
84	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
85	C₅	#65904	1	1	yes	characteristic
86	C₂²	C₂³×D₂₅	1	1	yes	characteristic
87	C₄	—	2	3	no	—
88	C₄	—	50	6	no	—
89	C₄	—	50	2	no	—
90	C₄	C₂×Dic₅₀	1	6	yes	—
91	C₄	#87748	1	2	yes	—
92	C₂	C₂²×Dic₅₀	1	1	yes	characteristic
93	C₂	C₂×#87748	1	1	yes	characteristic
94	C₂	#11716569	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂³×D₂₅
commutator, cosocle	C₂²×C₂₅	C₂⁴
Frattini, socle, prime radical	C₂²×C₅	C₂³×D₅
Fitting	C₄×Q₈×C₂₅	C₂
hypercenter	C₄×Q₈	D₂₅
hypocenter	C₂₅	#557
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
16. Q₈×C₂₅	Q₈	semidirect
59. C₂₅	#557	semidirect
79. Q₈	Dic₅₀	semidirect

Maximal subgroups

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

subgroup	index
C₄×Q₈×C₂₅	2
Q₈×Dic₂₅	2
#11716452	2
#11716431	2
C₄×Dic₅₀	2
#65904	5