Subgroups — #27173408

Order 12928.

116 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20265579	C₂	1	1	yes	maximal, characteristic
2	#20265004	C₂	1	2	yes	maximal
3	#20265002	C₂	1	2	yes	maximal
4	C₄×#13108049	C₂	1	1	yes	maximal, characteristic
5	#499×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	#15692247	—	2	1	no	—
7	C₂×#13108049	C₂²	1	1	yes	characteristic
8	C₂×#13108049	C₂²	1	1	yes	characteristic
9	#15692189	C₂²	1	2	yes	—
10	#15692187	—	2	1	no	—
11	#125×C₁₀₁	—	2	1	no	—
12	#107×C₁₀₁	C₂²	1	2	yes	—
13	#106×C₁₀₁	—	2	1	no	—
14	C₄×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₁₀₁	—	2	2	no	—
16	#13108053	—	4	2	no	—
17	#13108049	Q₈	1	2	yes	—
18	#13108049	Q₈	1	2	yes	—
19	C₂×C₈×C₁₀₁	D₄	1	1	yes	characteristic
20	C₂×C₈×C₁₀₁	C₂³	1	1	yes	characteristic
21	C₄⋊C₄×C₁₀₁	—	2	2	no	—
22	C₄⋊C₄×C₁₀₁	—	4	2	no	—
23	C₄²×C₁₀₁	D₄	1	1	yes	characteristic
24	C₂×C₄×C₁₀₁	—	4	2	no	—
25	C₂×C₄×C₁₀₁	D₈	1	2	yes	—
26	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
27	C₂×Dic₁₀₁	—	4	2	no	—
28	C₈×C₁₀₁	Q₁₆	1	2	yes	—
29	C₈×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
30	C₈×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
31	C₂²×C₁₀₁	C₂×D₈	1	1	yes	characteristic
32	C₄×C₁₀₁	—	2	2	no	—
33	C₄×C₁₀₁	—	8	2	no	—
34	C₄×C₁₀₁	C₄⋊Q₈	1	1	yes	characteristic
35	C₄×C₁₀₁	C₂×Q₁₆	1	1	yes	characteristic
36	Dic₁₀₁	—	8	2	no	—
37	C₂×C₁₀₁	#500	1	1	yes	characteristic
38	C₂×C₁₀₁	#508	1	2	yes	—
39	#2253	—	101	1	no	maximal, Sylow
40	C₁₀₁	#2253	1	1	yes	characteristic, complemented, Sylow
41	#499	—	101	1	no	—
42	#499	D₁₀₁	1	1	yes	characteristic
43	#367	—	101	2	no	—
44	#366	—	101	2	no	—
45	C₄×C₁₆	—	101	1	no	—
46	#125	—	2	1	no	—
47	#125	—	202	1	no	—
48	C₂×C₁₆	—	101	1	no	—
49	C₂×C₁₆	—	101	1	no	—
50	#107	—	101	2	no	—
51	#107	C₂×D₁₀₁	1	2	yes	—
52	#106	—	2	1	no	—
53	#106	—	202	1	no	—
54	C₄×C₈	C₂×D₁₀₁	1	1	yes	characteristic
55	C₂×C₈	C₂²⋊D₁₀₁	1	1	yes	characteristic
56	C₂×C₈	C₂²×D₁₀₁	1	1	yes	characteristic
57	C₄⋊C₄	—	2	2	no	—
58	C₄⋊C₄	—	4	2	no	—
59	C₄⋊C₄	—	404	2	no	—
60	C₄⋊C₄	—	202	2	no	—
61	C₄²	C₂²⋊D₁₀₁	1	1	yes	characteristic
62	C₁₆	—	101	2	no	—
63	C₁₆	—	101	2	no	—
64	C₂×C₄	—	4	2	no	—
65	C₂×C₄	—	404	2	no	—
66	C₂×C₄	#13108059	1	2	yes	—
67	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
68	C₈	#13108064	1	2	yes	—
69	C₈	Q₈×D₁₀₁	1	1	yes	characteristic
70	C₈	Q₈×D₁₀₁	1	1	yes	characteristic
71	C₂²	C₂×#13108059	1	1	yes	characteristic
72	C₄	—	2	2	no	—
73	C₄	—	8	2	no	—
74	C₄	—	808	2	no	—
75	C₄	#15692253	1	1	yes	characteristic
76	C₄	C₂×#13108064	1	1	yes	characteristic
77	C₂	#20265580	1	1	yes	characteristic
78	C₂	#20265607	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#13108059
commutator, cosocle	C₂×C₈×C₁₀₁	C₂³
Frattini	C₂×C₈	C₂²×D₁₀₁
Fitting	#499×C₁₀₁	C₂
socle, prime radical	C₂²×C₁₀₁	C₂×D₈
hypercenter	#499	D₁₀₁
hypocenter	C₁₀₁	#2253
LCS	G ▷ C₂×C₈×C₁₀₁ ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #499

Complements

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

N	H ≅ G/N	type
40. C₁₀₁	#2253	semidirect

Maximal subgroups

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

subgroup	index
#20265579	2
#20265004	2
#20265002	2
C₄×#13108049	2
#499×C₁₀₁	2
#2253	101