Subgroups — #13107251

Order 1600.

848 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#11717264	C₂	1	4	yes	maximal
3	#11717208	C₂	1	4	yes	maximal
4	#11717207	C₂	1	4	yes	maximal
5	C₄×#6045	C₂	1	2	yes	maximal
6	C₂×C₅²×Q₈	C₂²	1	2	yes	—
7	C₂×#6045	C₂²	1	4	yes	—
8	C₂×#6045	C₂²	1	2	yes	—
9	C₅²×C₄⋊C₄	C₂²	1	4	yes	—
10	C₄²×C₅²	C₂²	1	1	yes	characteristic
11	C₄⋊#1039	C₂²	1	2	yes	—
12	#87806	C₂²	1	4	yes	—
13	#87806	C₂²	1	8	yes	—
14	C₄×#1039	C₂²	1	4	yes	—
15	C₄×#1039	C₂²	1	4	yes	—
16	#66056	—	5	6	no	maximal
17	C₅²×Q₈	—	4	2	no	—
18	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
19	C₂×C₄×C₅²	C₂³	1	4	yes	—
20	C₂×C₄×C₅²	C₂³	1	2	yes	—
21	C₂×#1039	C₂³	1	4	yes	—
22	C₂×#1039	C₂³	1	4	yes	—
23	#6045	—	4	4	no	—
24	#6045	Q₈	1	8	yes	complemented
25	C₅×C₄⋊Q₈	D₅	1	6	yes	—
26	#4090	—	5	24	no	—
27	#4034	—	5	24	no	—
28	#4033	—	5	24	no	—
29	C₄×Dic₁₀	—	5	12	no	—
30	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
31	C₄×C₅²	—	2	4	no	—
32	C₄×C₅²	—	2	1	no	—
33	C₄×C₅²	C₂×Q₈	1	4	yes	—
34	#1039	—	2	4	no	—
35	#1039	C₂×Q₈	1	8	yes	—
36	C₂×C₅×Q₈	C₂×D₅	1	12	yes	—
37	C₂×Dic₁₀	—	5	12	no	—
38	C₂×Dic₁₀	—	5	24	no	—
39	C₅×C₄⋊C₄	C₂×D₅	1	24	yes	—
40	C₄²×C₅	C₂×D₅	1	6	yes	—
41	C₄⋊Dic₅	—	5	12	no	—
42	#683	—	5	24	no	—
43	#683	—	5	48	no	—
44	C₄×Dic₅	—	5	24	no	—
45	C₄×Dic₅	—	5	24	no	—
46	Q₈²	—	25	1	no	Sylow
47	C₂×C₅²	C₂²×Q₈	1	2	yes	—
48	C₂×C₅²	H(5,𝔽₂)	1	1	yes	characteristic
49	C₅×Q₈	—	4	12	no	—
50	C₂×C₄×C₅	C₂²×D₅	1	24	yes	—
51	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
52	C₂×C₄×C₅	C₂²×D₅	1	12	yes	—
53	C₂×Dic₅	—	5	24	no	—
54	C₂×Dic₅	—	5	24	no	—
55	Dic₁₀	—	5	48	no	—
56	Dic₁₀	—	20	24	no	—
57	C₄⋊Q₈	—	25	4	no	—
58	C₄⋊Q₈	—	25	4	no	—
59	C₄⋊Q₈	C₅⋊D₅	1	1	yes	characteristic
60	C₄×Q₈	—	25	4	no	—
61	C₄×Q₈	—	25	2	no	—
62	C₅²	Q₈²	1	1	yes	characteristic, complemented, Sylow
63	C₂²×C₅	C₂³×D₅	1	6	yes	—
64	C₄×C₅	—	2	24	no	—
65	C₄×C₅	—	2	6	no	—
66	C₄×C₅	Q₈×D₅	1	24	yes	—
67	Dic₅	—	5	48	no	—
68	Dic₅	—	10	24	no	—
69	C₂×Q₈	—	25	2	no	—
70	C₂×Q₈	—	25	4	no	—
71	C₂×Q₈	C₂×C₅⋊D₅	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×C₅⋊D₅	1	4	yes	—
76	C₄²	—	25	4	no	—
77	C₄²	—	25	4	no	—
78	C₄²	C₂×C₅⋊D₅	1	1	yes	characteristic
79	C₂×C₅	#4144	1	6	yes	—
80	C₂×C₅	C₂×Q₈×D₅	1	12	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×C₅⋊D₅	1	4	yes	—
87	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
88	C₂×C₄	C₂²×C₅⋊D₅	1	2	yes	—
89	C₅	#66056	1	6	yes	complemented
90	C₂²	C₂³×C₅⋊D₅	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×C₅⋊D₅	1	4	yes	—
96	C₂	#11717517	1	1	yes	characteristic
97	C₂	C₂×Q₈×C₅⋊D₅	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×C₅⋊D₅
commutator, socle, prime radical, cosocle	C₂²×C₅²	C₂⁴
Fitting	C₅²×C₄⋊Q₈	C₂
hypercenter	C₄⋊Q₈	C₅⋊D₅
hypocenter	C₅²	Q₈²
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
24. #6045	Q₈	semidirect
62. C₅²	Q₈²	semidirect
89. C₅	#66056	semidirect

Maximal subgroups

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

subgroup	index
C₅²×C₄⋊Q₈	2
#11717264	2
#11717208	2
#11717207	2
C₄×#6045	2
#66056	5