Subgroups — #27169188

Order 12928.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20265582	C₂	1	1	yes	maximal, characteristic
2	C₄⋊#13108049	C₂	1	1	yes	maximal, characteristic
3	#335×C₁₀₁	C₂	1	1	yes	maximal, characteristic
4	#15692255	—	2	1	no	—
5	C₂×#13108049	—	2	1	no	—
6	#15692189	—	2	1	no	—
7	#15692189	C₄	1	2	yes	—
8	C₄⋊C₈×C₁₀₁	—	2	1	no	—
9	C₄×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
10	C₂×Dic₂₀₂	—	4	1	no	—
11	C₄⋊Dic₁₀₁	—	2	2	no	—
12	C₄⋊Dic₁₀₁	—	4	1	no	—
13	#13108049	—	4	1	no	—
14	C₂×C₈×C₁₀₁	—	4	1	no	—
15	C₂×C₈×C₁₀₁	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₁₀₁	D₄	1	1	yes	characteristic
17	C₄²×C₁₀₁	D₄	1	1	yes	characteristic
18	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
19	C₂×C₄×C₁₀₁	D₈	1	1	yes	characteristic
20	C₂×C₄×C₁₀₁	SD₁₆	1	1	yes	characteristic
21	C₂×Dic₁₀₁	—	4	2	no	—
22	Dic₂₀₂	—	8	1	no	—
23	Dic₂₀₂	—	8	1	no	—
24	C₈×C₁₀₁	—	2	1	no	—
25	C₈×C₁₀₁	—	8	1	no	—
26	C₈×C₁₀₁	SD₁₆	1	1	yes	characteristic
27	C₈×C₁₀₁	Q₁₆	1	1	yes	characteristic
28	C₂²×C₁₀₁	#102	1	1	yes	characteristic
29	C₄×C₁₀₁	—	2	1	no	—
30	C₄×C₁₀₁	#101	1	1	yes	characteristic
31	C₄×C₁₀₁	#103	1	1	yes	characteristic
32	C₄×C₁₀₁	D₁₆	1	1	yes	characteristic
33	C₄×C₁₀₁	SD₃₂	1	1	yes	characteristic
34	Dic₁₀₁	—	8	2	no	—
35	C₂×C₁₀₁	#332	1	1	yes	characteristic
36	C₂×C₁₀₁	#357	1	1	yes	characteristic
37	C₂×C₁₀₁	#362	1	1	yes	characteristic
38	#1364	—	101	1	no	maximal, Sylow
39	C₁₀₁	#1364	1	1	yes	characteristic, complemented, Sylow
40	#500	—	101	1	no	—
41	C₄⋊C₁₆	—	101	1	no	—
42	#335	D₁₀₁	1	1	yes	characteristic
43	C₄⋊Q₈	—	202	1	no	—
44	C₂×C₁₆	—	202	1	no	—
45	#107	—	101	2	no	—
46	#107	—	202	1	no	—
47	C₄⋊C₈	—	2	1	no	—
48	C₄×C₈	C₂×D₁₀₁	1	1	yes	characteristic
49	C₂×Q₈	—	404	1	no	—
50	C₂×C₈	—	4	1	no	—
51	C₂×C₈	C₄×D₁₀₁	1	1	yes	characteristic
52	C₂×C₈	D₄₀₄	1	1	yes	characteristic
53	C₄⋊C₄	—	404	1	no	—
54	C₄⋊C₄	—	202	2	no	—
55	C₄²	C₂²⋊D₁₀₁	1	1	yes	characteristic
56	C₁₆	—	404	1	no	—
57	Q₈	—	808	1	no	—
58	Q₈	—	808	1	no	—
59	C₂×C₄	—	404	2	no	—
60	C₂×C₄	#13108051	1	1	yes	characteristic
61	C₂×C₄	#13108059	1	1	yes	characteristic
62	C₂×C₄	#13108061	1	1	yes	characteristic
63	C₈	—	2	1	no	—
64	C₈	—	8	1	no	—
65	C₈	#13108063	1	1	yes	characteristic
66	C₈	Dic₄₀₄	1	1	yes	characteristic
67	C₂²	#15692175	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	808	2	no	—
70	C₄	#15692173	1	1	yes	characteristic
71	C₄	#15692180	1	1	yes	characteristic
72	C₄	#15692196	1	1	yes	characteristic
73	C₄	#15692198	1	1	yes	characteristic
74	C₂	#20264924	1	1	yes	characteristic
75	C₂	#20264977	1	1	yes	characteristic
76	C₂	#20264992	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15692175
commutator	C₂×C₈×C₁₀₁	C₂×C₄
Frattini	C₄×C₈	C₂×D₁₀₁
Fitting	#335×C₁₀₁	C₂
socle, prime radical	C₂²×C₁₀₁	#102
cosocle	C₄×C₈×C₁₀₁	C₂²
hypercenter	#335	D₁₀₁
hypocenter	C₁₀₁	#1364
LCS	G ▷ C₂×C₈×C₁₀₁ ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #335

Complements

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

N	H ≅ G/N	type
39. C₁₀₁	#1364	semidirect

Maximal subgroups

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

subgroup	index
#20265582	2
C₄⋊#13108049	2
#335×C₁₀₁	2
#1364	101