Subgroups — #27173410

Order 12928.

116 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20265578	C₂	1	2	yes	maximal
2	#20265005	C₂	1	1	yes	maximal, characteristic
3	#20265002	C₂	1	2	yes	maximal
4	C₁₆×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic
5	#367×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	#15692246	—	2	2	no	—
7	C₂×#13108049	C₂²	1	1	yes	characteristic
8	#15692189	C₂²	1	2	yes	—
9	#15692186	—	2	2	no	—
10	C₈×Dic₁₀₁	C₂²	1	1	yes	characteristic
11	C₂×C₁₆×C₁₀₁	C₂²	1	1	yes	characteristic
12	#107×C₁₀₁	C₂²	1	2	yes	—
13	C₂×#11717579	D₄	1	1	yes	characteristic
14	C₄⋊Dic₁₀₁	—	2	2	no	—
15	C₄⋊Dic₁₀₁	—	4	2	no	—
16	#13108053	—	4	2	no	—
17	C₄×Dic₁₀₁	D₄	1	1	yes	characteristic
18	#13108049	Q₈	1	2	yes	—
19	C₂×C₈×C₁₀₁	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₁₀₁	—	2	2	no	—
21	C₁₆×C₁₀₁	Q₈	1	2	yes	—
22	C₂×C₄×C₁₀₁	—	4	2	no	—
23	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
24	C₂×Dic₁₀₁	—	4	2	no	—
25	C₂×Dic₁₀₁	D₈	1	2	yes	—
26	C₈×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
27	C₈×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
28	#11717579	Q₁₆	1	2	yes	—
29	C₂²×C₁₀₁	C₂×D₈	1	1	yes	characteristic
30	C₄×C₁₀₁	—	8	2	no	—
31	C₄×C₁₀₁	C₄⋊Q₈	1	1	yes	characteristic
32	C₄×C₁₀₁	C₂×Q₁₆	1	1	yes	characteristic
33	Dic₁₀₁	—	2	2	no	—
34	Dic₁₀₁	—	8	2	no	—
35	C₂×C₁₀₁	#500	1	1	yes	characteristic
36	C₂×C₁₀₁	#508	1	2	yes	—
37	#2253	—	101	1	no	maximal, Sylow
38	C₁₀₁	#2253	1	1	yes	characteristic, complemented, Sylow
39	#499	—	101	2	no	—
40	#367	—	101	1	no	—
41	#367	D₁₀₁	1	1	yes	characteristic
42	#366	—	101	2	no	—
43	C₄×C₁₆	—	101	1	no	—
44	#125	—	202	2	no	—
45	C₂×C₁₆	—	101	1	no	—
46	C₂×C₁₆	C₂×D₁₀₁	1	1	yes	characteristic
47	#107	—	101	2	no	—
48	#107	C₂×D₁₀₁	1	2	yes	—
49	#106	—	202	2	no	—
50	C₄×C₈	—	101	1	no	—
51	C₂×C₈	—	101	1	no	—
52	C₂×C₈	C₂²×D₁₀₁	1	1	yes	characteristic
53	C₄⋊C₄	—	2	2	no	—
54	C₄⋊C₄	—	404	2	no	—
55	C₄⋊C₄	—	404	2	no	—
56	C₄⋊C₄	—	202	2	no	—
57	C₄²	—	101	1	no	—
58	C₁₆	—	101	2	no	—
59	C₁₆	Dic₂₀₂	1	2	yes	—
60	C₂×C₄	—	4	2	no	—
61	C₂×C₄	—	101	2	no	—
62	C₂×C₄	—	404	2	no	—
63	C₂×C₄	D₄×D₁₀₁	1	1	yes	characteristic
64	C₈	—	101	2	no	—
65	C₈	Q₈×D₁₀₁	1	1	yes	characteristic
66	C₈	C₂×Dic₂₀₂	1	1	yes	characteristic
67	C₂²	D₈×D₁₀₁	1	1	yes	characteristic
68	C₄	—	8	2	no	—
69	C₄	—	808	2	no	—
70	C₄	—	202	2	no	—
71	C₄	#15692254	1	1	yes	characteristic
72	C₄	Q₁₆×D₁₀₁	1	1	yes	characteristic
73	C₂	#20265581	1	1	yes	characteristic
74	C₂	#20265608	1	2	yes	—

	subgroup	G/H
center	C₂²	D₈×D₁₀₁
commutator, cosocle	C₂×C₈×C₁₀₁	C₂³
Frattini	C₂×C₈	C₂²×D₁₀₁
Fitting	#367×C₁₀₁	C₂
socle, prime radical	C₂²×C₁₀₁	C₂×D₈
hypercenter	#367	D₁₀₁
hypocenter	C₁₀₁	#2253
LCS	G ▷ C₂×C₈×C₁₀₁ ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

Complements

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

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

Maximal subgroups

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

subgroup	index
#20265578	2
#20265005	2
#20265002	2
C₁₆×Dic₁₀₁	2
#367×C₁₀₁	2
#2253	101