Subgroups — #27173406

Order 12928.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20265581	C₂	1	2	yes	maximal
2	#20265003	C₂	1	1	yes	maximal, characteristic
3	#20265002	C₂	1	2	yes	maximal
4	C₁₆×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic
5	#366×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	#15692254	—	2	2	no	—
7	C₂×#13108049	C₂²	1	1	yes	characteristic
8	#15692189	C₂²	1	2	yes	—
9	#15692188	—	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×Dic₂₀₂	—	4	2	no	—
14	C₂×#11717579	D₄	1	1	yes	characteristic
15	C₄⋊Dic₁₀₁	—	2	2	no	—
16	#13108053	—	4	2	no	—
17	C₄×Dic₁₀₁	D₄	1	1	yes	characteristic
18	#13108049	Q₈	1	2	yes	complemented
19	C₂×C₈×C₁₀₁	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₁₀₁	—	2	2	no	—
21	C₁₆×C₁₀₁	Q₈	1	2	yes	complemented
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	1	yes	characteristic
26	C₂×Dic₁₀₁	D₈	1	1	yes	characteristic
27	Dic₂₀₂	—	8	4	no	—
28	C₈×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
29	C₈×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
30	#11717579	Q₁₆	1	2	yes	—
31	C₂²×C₁₀₁	C₂×D₈	1	1	yes	characteristic
32	C₄×C₁₀₁	—	8	2	no	—
33	C₄×C₁₀₁	C₄⋊Q₈	1	1	yes	characteristic
34	C₄×C₁₀₁	C₂×Q₁₆	1	1	yes	characteristic
35	Dic₁₀₁	—	8	2	no	—
36	Dic₁₀₁	D₁₆	1	2	yes	—
37	Dic₁₀₁	Q₃₂	1	2	yes	—
38	C₂×C₁₀₁	#500	1	1	yes	characteristic
39	C₂×C₁₀₁	C₂×D₁₆	1	1	yes	characteristic
40	C₂×C₁₀₁	C₂×Q₃₂	1	1	yes	characteristic
41	#2252	—	101	1	no	maximal, Sylow
42	C₁₀₁	#2252	1	1	yes	characteristic, complemented, Sylow
43	#500	—	101	2	no	—
44	#366	—	101	1	no	—
45	#366	—	101	2	no	—
46	#366	D₁₀₁	1	1	yes	characteristic
47	C₄×C₁₆	—	101	1	no	—
48	C₄⋊Q₈	—	202	2	no	—
49	C₂×C₁₆	—	101	1	no	—
50	C₂×C₁₆	C₂×D₁₀₁	1	1	yes	characteristic
51	#107	—	101	2	no	—
52	#107	—	202	2	no	—
53	#107	C₂×D₁₀₁	1	2	yes	—
54	C₄×C₈	—	101	1	no	—
55	C₂×Q₈	—	404	2	no	—
56	C₂×C₈	—	101	1	no	—
57	C₂×C₈	C₂²×D₁₀₁	1	1	yes	characteristic
58	C₄⋊C₄	—	2	2	no	—
59	C₄⋊C₄	—	404	2	no	—
60	C₄⋊C₄	—	202	2	no	—
61	C₄²	—	101	1	no	—
62	C₁₆	—	101	2	no	—
63	C₁₆	Dic₂₀₂	1	2	yes	complemented
64	Q₈	—	808	4	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	—	101	1	no	—
67	C₂×C₄	—	101	1	no	—
68	C₂×C₄	—	404	2	no	—
69	C₂×C₄	D₄×D₁₀₁	1	1	yes	characteristic
70	C₈	—	101	2	no	—
71	C₈	Q₈×D₁₀₁	1	1	yes	characteristic
72	C₈	C₂×Dic₂₀₂	1	1	yes	characteristic
73	C₂²	D₈×D₁₀₁	1	1	yes	characteristic
74	C₄	—	8	2	no	—
75	C₄	—	808	2	no	—
76	C₄	—	101	2	no	—
77	C₄	—	101	2	no	—
78	C₄	#15692254	1	1	yes	characteristic
79	C₄	Q₁₆×D₁₀₁	1	1	yes	characteristic
80	C₂	#20265581	1	1	yes	characteristic
81	C₂	D₁₆×D₁₀₁	1	1	yes	characteristic
82	C₂	Q₃₂×D₁₀₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×D₁₀₁
commutator, cosocle	C₂×C₈×C₁₀₁	C₂³
Frattini	C₂×C₈	C₂²×D₁₀₁
Fitting	#366×C₁₀₁	C₂
socle, prime radical	C₂²×C₁₀₁	C₂×D₈
hypercenter	#366	D₁₀₁
hypocenter	C₁₀₁	#2252
LCS	G ▷ C₂×C₈×C₁₀₁ ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

Complements

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

N	H ≅ G/N	type
18. #13108049	Q₈	semidirect
21. C₁₆×C₁₀₁	Q₈	semidirect
42. C₁₀₁	#2252	semidirect
63. C₁₆	Dic₂₀₂	semidirect

Maximal subgroups

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

subgroup	index
#20265581	2
#20265003	2
#20265002	2
C₁₆×Dic₁₀₁	2
#366×C₁₀₁	2
#2252	101