Subgroups — #27173412

Order 12928.

128 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	#20265004	C₂	1	4	yes	maximal
3	C₄×#13108049	C₂	1	1	yes	maximal, characteristic
4	#500×C₁₀₁	C₂	1	1	yes	maximal, characteristic
5	#15692255	—	2	1	no	—
6	C₂×#13108049	C₂²	1	2	yes	—
7	#15692189	—	2	1	no	—
8	#15692189	C₂²	1	2	yes	—
9	C₄⋊Q₈×C₁₀₁	—	2	1	no	—
10	#107×C₁₀₁	—	2	1	no	—
11	#107×C₁₀₁	C₂²	1	2	yes	—
12	C₄×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
13	C₂×Dic₂₀₂	—	4	1	no	—
14	C₄⋊Dic₁₀₁	—	2	2	no	—
15	C₄⋊Dic₁₀₁	—	4	1	no	—
16	#13108049	Q₈	1	4	yes	complemented
17	C₂×Q₈×C₁₀₁	—	4	1	no	—
18	C₂×C₈×C₁₀₁	D₄	1	1	yes	characteristic
19	C₂×C₈×C₁₀₁	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₁₀₁	—	2	2	no	—
21	C₄⋊C₄×C₁₀₁	—	4	1	no	—
22	C₄²×C₁₀₁	D₄	1	1	yes	characteristic
23	Q₈×C₁₀₁	—	8	2	no	—
24	C₂×C₄×C₁₀₁	—	4	2	no	—
25	C₂×C₄×C₁₀₁	D₈	1	1	yes	characteristic
26	C₂×C₄×C₁₀₁	D₈	1	1	yes	characteristic
27	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
28	C₂×Dic₁₀₁	—	4	2	no	—
29	Dic₂₀₂	—	8	2	no	—
30	C₈×C₁₀₁	Q₁₆	1	2	yes	—
31	C₈×C₁₀₁	C₂×Q₈	1	2	yes	—
32	C₂²×C₁₀₁	C₂×D₈	1	1	yes	characteristic
33	C₄×C₁₀₁	—	8	2	no	—
34	C₄×C₁₀₁	SD₃₂	1	2	yes	—
35	C₄×C₁₀₁	SD₃₂	1	2	yes	—
36	C₄×C₁₀₁	C₄⋊Q₈	1	1	yes	characteristic
37	C₄×C₁₀₁	C₂×Q₁₆	1	1	yes	characteristic
38	Dic₁₀₁	—	8	2	no	—
39	C₂×C₁₀₁	#500	1	1	yes	characteristic
40	C₂×C₁₀₁	C₂×SD₃₂	1	1	yes	characteristic
41	C₂×C₁₀₁	C₂×SD₃₂	1	1	yes	characteristic
42	#2254	—	101	1	no	maximal, Sylow
43	C₁₀₁	#2254	1	1	yes	characteristic, complemented, Sylow
44	#500	—	101	1	no	—
45	#500	D₁₀₁	1	1	yes	characteristic
46	#367	—	101	4	no	—
47	C₄×C₁₆	—	101	1	no	—
48	C₄⋊Q₈	—	2	1	no	—
49	C₄⋊Q₈	—	202	1	no	—
50	C₂×C₁₆	—	101	2	no	—
51	#107	—	2	1	no	—
52	#107	—	101	2	no	—
53	#107	—	202	1	no	—
54	#107	C₂×D₁₀₁	1	2	yes	—
55	C₄×C₈	C₂×D₁₀₁	1	1	yes	characteristic
56	C₂×Q₈	—	4	1	no	—
57	C₂×Q₈	—	404	1	no	—
58	C₂×C₈	C₂²⋊D₁₀₁	1	1	yes	characteristic
59	C₂×C₈	C₂²×D₁₀₁	1	1	yes	characteristic
60	C₄⋊C₄	—	2	2	no	—
61	C₄⋊C₄	—	4	1	no	—
62	C₄⋊C₄	—	404	1	no	—
63	C₄⋊C₄	—	202	2	no	—
64	C₄²	C₂²⋊D₁₀₁	1	1	yes	characteristic
65	C₁₆	—	101	4	no	—
66	Q₈	—	8	2	no	—
67	Q₈	—	808	2	no	—
68	C₂×C₄	—	4	2	no	—
69	C₂×C₄	—	404	2	no	—
70	C₂×C₄	#13108059	1	1	yes	characteristic
71	C₂×C₄	#13108059	1	1	yes	characteristic
72	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
73	C₈	#13108064	1	2	yes	—
74	C₈	Q₈×D₁₀₁	1	2	yes	—
75	C₂²	C₂×#13108059	1	1	yes	characteristic
76	C₄	—	8	2	no	—
77	C₄	—	808	2	no	—
78	C₄	#15692198	1	2	yes	—
79	C₄	#15692199	1	2	yes	—
80	C₄	#15692253	1	1	yes	characteristic
81	C₄	C₂×#13108064	1	1	yes	characteristic
82	C₂	#20265580	1	1	yes	characteristic
83	C₂	C₂×#15692198	1	1	yes	characteristic
84	C₂	C₂×#15692199	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13108059
commutator, cosocle	C₂×C₈×C₁₀₁	C₂³
Frattini	C₂×C₈	C₂²×D₁₀₁
Fitting	#500×C₁₀₁	C₂
socle, prime radical	C₂²×C₁₀₁	C₂×D₈
hypercenter	#500	D₁₀₁
hypocenter	C₁₀₁	#2254
LCS	G ▷ C₂×C₈×C₁₀₁ ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

Complements

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

N	H ≅ G/N	type
16. #13108049	Q₈	semidirect
43. C₁₀₁	#2254	semidirect

Maximal subgroups

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

subgroup	index
#20265582	2
#20265004	2
C₄×#13108049	2
#500×C₁₀₁	2
#2254	101