Subgroups — #13102956

Order 1600.

392 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	#11717189	C₂	1	1	yes	maximal, characteristic
3	#11716677	C₂	1	4	yes	maximal
4	C₄×#6029	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅²×Q₈	—	2	1	no	—
6	C₂×#6045	C₂²	1	2	yes	—
7	C₅²×C₄⋊C₄	—	2	1	no	—
8	C₅²×C₄⋊C₄	C₂²	1	2	yes	—
9	C₄²×C₅²	C₂²	1	1	yes	characteristic
10	C₄⋊#1039	—	2	1	no	—
11	C₄⋊#1039	—	2	1	no	—
12	C₂×#6029	C₂²	1	2	yes	—
13	#65382	—	5	6	no	maximal
14	C₅²×Q₈	—	4	2	no	—
15	C₂×C₄×C₅²	—	2	2	no	—
16	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
17	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
18	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
19	C₂×#1039	—	2	2	no	—
20	#6045	—	2	4	no	—
21	#6029	—	2	2	no	—
22	C₅×C₄⋊Q₈	D₅	1	6	yes	—
23	#4015	—	5	6	no	—
24	#3942	—	5	24	no	—
25	C₄×#168	—	5	6	no	—
26	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
27	C₄×C₅²	—	4	2	no	—
28	C₄×C₅²	SD₁₆	1	2	yes	—
29	C₄×C₅²	Q₁₆	1	2	yes	—
30	C₄×C₅²	#41	1	2	yes	—
31	#1039	—	4	2	no	—
32	C₂×C₅×Q₈	—	2	6	no	—
33	C₂×Dic₁₀	—	5	12	no	—
34	C₅×C₄⋊C₄	—	2	6	no	—
35	C₅×C₄⋊C₄	C₂×D₅	1	12	yes	—
36	C₄²×C₅	C₂×D₅	1	6	yes	—
37	C₄⋊Dic₅	—	10	6	no	—
38	C₄⋊Dic₅	—	10	6	no	—
39	C₂×#168	—	5	12	no	—
40	#487	—	25	1	no	Sylow
41	C₂×C₅²	#124	1	1	yes	characteristic
42	C₂×C₅²	C₂×SD₁₆	1	1	yes	characteristic
43	C₂×C₅²	C₂×Q₁₆	1	1	yes	characteristic
44	C₅×Q₈	—	4	12	no	—
45	C₂×C₄×C₅	—	2	12	no	—
46	C₂×C₄×C₅	C₂²⋊D₅	1	6	yes	—
47	C₂×C₄×C₅	C₂²⋊D₅	1	6	yes	—
48	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
49	C₂×Dic₅	—	10	12	no	—
50	Dic₁₀	—	10	24	no	—
51	#168	—	10	12	no	—
52	C₄⋊Q₈	—	25	1	no	—
53	C₄⋊Q₈	C₅⋊D₅	1	1	yes	characteristic
54	#103	—	25	4	no	—
55	C₄×C₈	—	25	1	no	—
56	C₅²	#487	1	1	yes	characteristic, complemented, Sylow
57	C₂²×C₅	C₂×C₂²⋊D₅	1	6	yes	—
58	C₄×C₅	—	4	12	no	—
59	C₄×C₅	#687	1	12	yes	—
60	C₄×C₅	#689	1	12	yes	—
61	C₄×C₅	#713	1	12	yes	—
62	Dic₅	—	20	12	no	—
63	C₂×Q₈	—	2	1	no	—
64	C₂×Q₈	—	25	2	no	—
65	C₂×C₈	—	25	2	no	—
66	C₄⋊C₄	—	2	1	no	—
67	C₄⋊C₄	—	50	1	no	—
68	C₄⋊C₄	—	50	1	no	—
69	C₄⋊C₄	C₂×C₅⋊D₅	1	2	yes	—
70	C₄²	C₂×C₅⋊D₅	1	1	yes	characteristic
71	C₂×C₅	C₂×#687	1	6	yes	—
72	C₂×C₅	C₂×#689	1	6	yes	—
73	C₂×C₅	#4093	1	6	yes	—
74	Q₈	—	4	2	no	—
75	Q₈	—	50	4	no	—
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	—	50	2	no	—
78	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
79	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
80	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
81	C₈	—	50	2	no	—
82	C₅	#65382	1	6	yes	complemented
83	C₂²	C₂×C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
84	C₄	—	4	2	no	—
85	C₄	—	100	2	no	—
86	C₄	#87810	1	2	yes	—
87	C₄	#87812	1	2	yes	—
88	C₄	#87904	1	2	yes	—
89	C₂	C₂×#87810	1	1	yes	characteristic
90	C₂	C₂×#87812	1	1	yes	characteristic
91	C₂	#11717267	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊(C₅⋊D₅)
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂²×C₅⋊D₅
Fitting	C₅²×C₄⋊Q₈	C₂
socle, prime radical	C₂²×C₅²	C₂×D₄
hypercenter	C₄⋊Q₈	C₅⋊D₅
hypocenter	C₅²	#487
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
56. C₅²	#487	semidirect
82. C₅	#65382	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
#11717189	2
#11716677	2
C₄×#6029	2
#65382	5