Subgroups — #27170746

Order 12928.

280 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20265554	C₂	1	1	yes	maximal, characteristic, complemented
2	#20265553	C₂	1	4	yes	maximal, complemented
3	C₈×#11717579	C₂	1	1	yes	maximal, characteristic, complemented
4	#493×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₈₀₈	—	2	2	no	—
6	C₂×#13108059	—	2	8	no	—
7	#15692252	C₂²	1	2	yes	—
8	C₄×#11717579	C₂²	1	2	yes	—
9	C₂×D₈×C₁₀₁	—	2	2	no	—
10	#127×C₁₀₁	C₂²	1	2	yes	—
11	C₄×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
12	C₂×D₄₀₄	—	2	4	no	—
13	C₂×D₄₀₄	—	2	2	no	—
14	D₈₀₈	—	4	4	no	—
15	#13108059	—	4	16	no	—
16	C₂×#11717579	D₄	1	4	yes	—
17	C₂×D₄×C₁₀₁	—	2	2	no	—
18	C₂×D₄×C₁₀₁	—	2	4	no	—
19	D₈×C₁₀₁	—	4	4	no	—
20	C₂×C₈×C₁₀₁	D₄	1	2	yes	—
21	C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
22	C₂³×C₁₀₁	—	4	2	no	—
23	C₂²×D₁₀₁	—	4	2	no	—
24	D₄×C₁₀₁	—	4	4	no	—
25	D₄×C₁₀₁	—	4	8	no	—
26	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
27	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
28	D₄₀₄	—	4	8	no	—
29	D₄₀₄	—	4	4	no	—
30	C₈×C₁₀₁	D₈	1	4	yes	complemented
31	#11717579	D₈	1	8	yes	complemented
32	C₂²×C₁₀₁	—	8	4	no	—
33	C₂²×C₁₀₁	—	8	2	no	—
34	C₂²×C₁₀₁	#127	1	1	yes	characteristic
35	C₂×D₁₀₁	—	8	4	no	—
36	C₂×D₁₀₁	—	8	2	no	—
37	C₄×C₁₀₁	C₂×D₈	1	4	yes	—
38	C₄×C₁₀₁	C₂×D₈	1	2	yes	—
39	C₂×C₁₀₁	—	16	2	no	—
40	C₂×C₁₀₁	#493	1	2	yes	—
41	C₂×C₁₀₁	#493	1	1	yes	characteristic
42	D₁₀₁	—	16	2	no	—
43	#1712	—	101	1	no	maximal, Sylow
44	C₁₀₁	#1712	1	1	yes	characteristic, complemented, Sylow
45	#493	—	101	4	no	—
46	#493	—	101	1	no	—
47	#493	D₁₀₁	1	1	yes	characteristic, complemented
48	C₈²	—	101	1	no	—
49	C₂×D₈	—	2	2	no	—
50	C₂×D₈	—	202	8	no	—
51	C₂×D₈	—	202	2	no	—
52	#127	—	101	2	no	—
53	#127	C₂×D₁₀₁	1	2	yes	—
54	C₄×C₈	—	101	2	no	—
55	C₄×C₈	C₂×D₁₀₁	1	1	yes	characteristic
56	C₂×D₄	—	2	4	no	—
57	C₂×D₄	—	2	2	no	—
58	C₂×D₄	—	202	4	no	—
59	C₂×D₄	—	202	2	no	—
60	D₈	—	4	4	no	—
61	D₈	—	404	16	no	—
62	D₈	—	404	4	no	—
63	C₂×C₈	—	101	4	no	—
64	C₂×C₈	C₂²⋊D₁₀₁	1	2	yes	—
65	C₄²	C₂²×D₁₀₁	1	1	yes	characteristic
66	C₂³	—	4	2	no	—
67	C₂³	—	404	2	no	—
68	D₄	—	4	8	no	—
69	D₄	—	4	4	no	—
70	D₄	—	404	8	no	—
71	D₄	—	404	4	no	—
72	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
73	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
74	C₈	—	101	8	no	—
75	C₈	#13108059	1	4	yes	complemented
76	C₂²	—	8	4	no	—
77	C₂²	—	8	2	no	—
78	C₂²	—	808	4	no	—
79	C₂²	—	808	2	no	—
80	C₂²	#15692251	1	1	yes	characteristic
81	C₄	D₈×D₁₀₁	1	4	yes	—
82	C₄	C₂×#13108059	1	2	yes	—
83	C₂	—	16	2	no	—
84	C₂	—	1616	2	no	—
85	C₂	#20265552	1	2	yes	—
86	C₂	#20265553	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15692251
commutator, cosocle	C₄²×C₁₀₁	C₂³
Frattini	C₄²	C₂²×D₁₀₁
Fitting	#493×C₁₀₁	C₂
socle	C₂²×C₁₀₁	#127
hypercenter	#493	D₁₀₁
hypocenter	C₁₀₁	#1712
LCS	G ▷ C₄²×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

Complements

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

N	H ≅ G/N	type
1. #20265554	C₂	semidirect
2. #20265553	C₂	semidirect
3. C₈×#11717579	C₂	semidirect
4. #493×C₁₀₁	C₂	semidirect
30. C₈×C₁₀₁	D₈	semidirect
31. #11717579	D₈	semidirect
44. C₁₀₁	#1712	semidirect
47. #493	D₁₀₁	semidirect
75. C₈	#13108059	semidirect

Maximal subgroups

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

subgroup	index
#20265554	2
#20265553	2
C₈×#11717579	2
#493×C₁₀₁	2
#1712	101