Subgroups — #27169112

Order 12928.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×Dic₂₀₂	C₂	1	1	yes	maximal, characteristic
2	C₄⋊#13108049	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₆×C₁₀₁	C₂	1	1	yes	maximal, characteristic
4	C₄×Dic₂₀₂	C₄	1	1	yes	characteristic
5	C₂×#13108049	—	2	1	no	—
6	C₄⋊#11717579	C₄	1	1	yes	characteristic
7	#15692184	—	2	1	no	—
8	C₈×Dic₁₀₁	—	2	1	no	—
9	C₂×C₁₆×C₁₀₁	—	2	1	no	—
10	C₄×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
11	C₂×Dic₂₀₂	C₈	1	1	yes	characteristic
12	C₂×#11717579	—	2	1	no	—
13	C₄⋊Dic₁₀₁	C₈	1	1	yes	characteristic
14	#13108053	—	2	1	no	—
15	C₄×Dic₁₀₁	—	2	1	no	—
16	#13108049	—	4	1	no	—
17	C₂×C₈×C₁₀₁	D₄	1	1	yes	characteristic
18	C₂×C₈×C₁₀₁	D₄	1	1	yes	characteristic
19	C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
20	C₁₆×C₁₀₁	—	2	2	no	—
21	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×C₄×C₁₀₁	C₂×C₈	1	1	yes	characteristic
23	C₂×C₄×C₁₀₁	M₁₆	1	1	yes	characteristic
24	C₂×Dic₁₀₁	—	2	1	no	—
25	C₂×Dic₁₀₁	—	2	1	no	—
26	Dic₂₀₂	—	2	1	no	—
27	Dic₂₀₂	C₁₆	1	2	yes	complemented
28	C₈×C₁₀₁	—	2	1	no	—
29	C₈×C₁₀₁	SD₁₆	1	1	yes	characteristic
30	C₈×C₁₀₁	Q₁₆	1	1	yes	characteristic
31	#11717579	—	4	1	no	—
32	C₂²×C₁₀₁	C₂²⋊C₈	1	1	yes	characteristic
33	C₄×C₁₀₁	—	2	1	no	—
34	C₄×C₁₀₁	#103	1	1	yes	characteristic
35	C₄×C₁₀₁	#104	1	1	yes	characteristic
36	C₄×C₁₀₁	C₂×C₁₆	1	1	yes	characteristic
37	C₄×C₁₀₁	M₃₂	1	1	yes	characteristic
38	Dic₁₀₁	—	2	2	no	—
39	Dic₁₀₁	—	4	1	no	—
40	C₂×C₁₀₁	Q₈⋊C₈	1	1	yes	characteristic
41	C₂×C₁₀₁	C₂²⋊C₁₆	1	1	yes	characteristic
42	C₂×C₁₀₁	#350	1	1	yes	characteristic
43	Q₈⋊C₁₆	—	101	1	no	maximal, Sylow
44	C₁₀₁	Q₈⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
45	C₈×Q₈	—	101	1	no	—
46	C₄⋊C₁₆	—	101	1	no	—
47	C₄×C₁₆	D₁₀₁	1	1	yes	characteristic
48	C₄×Q₈	—	101	1	no	—
49	C₂×C₁₆	—	2	1	no	—
50	C₂×C₁₆	—	202	1	no	—
51	C₄⋊C₈	—	101	1	no	—
52	C₄⋊C₈	—	202	1	no	—
53	C₄×C₈	—	202	1	no	—
54	C₄×C₈	C₂×D₁₀₁	1	1	yes	characteristic
55	C₂×Q₈	—	101	1	no	—
56	C₂×C₈	—	202	1	no	—
57	C₂×C₈	D₄₀₄	1	1	yes	characteristic
58	C₂×C₈	C₂²⋊D₁₀₁	1	1	yes	characteristic
59	C₄⋊C₄	—	101	1	no	—
60	C₄⋊C₄	—	202	1	no	—
61	C₄²	—	202	1	no	—
62	C₄²	C₄×D₁₀₁	1	1	yes	characteristic
63	C₁₆	—	2	2	no	—
64	C₁₆	—	404	1	no	—
65	Q₈	—	101	2	no	—
66	Q₈	—	202	1	no	—
67	C₂×C₄	—	202	1	no	—
68	C₂×C₄	—	202	1	no	—
69	C₂×C₄	#13108051	1	1	yes	characteristic
70	C₂×C₄	C₈×D₁₀₁	1	1	yes	characteristic
71	C₂×C₄	#13108057	1	1	yes	characteristic
72	C₈	—	2	1	no	—
73	C₈	—	404	1	no	—
74	C₈	#13108063	1	1	yes	characteristic
75	C₈	Dic₄₀₄	1	1	yes	characteristic
76	C₂²	#15692167	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	—	404	1	no	—
79	C₄	—	202	2	no	—
80	C₄	#15692180	1	1	yes	characteristic
81	C₄	#15692183	1	1	yes	characteristic
82	C₄	C₁₆×D₁₀₁	1	1	yes	characteristic
83	C₄	#15692194	1	1	yes	characteristic
84	C₂	#20264909	1	1	yes	characteristic
85	C₂	#20264956	1	1	yes	characteristic
86	C₂	#20264960	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄₀₄
commutator	C₄×C₁₀₁	C₂×C₁₆
Frattini	C₄×C₈	C₂×D₁₀₁
Fitting	C₄×C₁₆×C₁₀₁	C₂
socle, prime radical	C₂²×C₁₀₁	C₂²⋊C₈
cosocle	C₄×C₈×C₁₀₁	C₂²
hypercenter	C₄×C₁₆	D₁₀₁
hypocenter	C₁₀₁	Q₈⋊C₁₆
LCS	G ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
27. Dic₂₀₂	C₁₆	semidirect
44. C₁₀₁	Q₈⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₈×Dic₂₀₂	2
C₄⋊#13108049	2
C₄×C₁₆×C₁₀₁	2
Q₈⋊C₁₆	101