Subgroups — #27170730

Order 12928.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20265573	C₂	1	2	yes	maximal
2	#20265557	C₂	1	1	yes	maximal, characteristic
3	#20265527	C₂	1	2	yes	maximal
4	C₈×#11717579	C₂	1	1	yes	maximal, characteristic
5	#500×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₀₄	—	2	2	no	—
7	#15692255	C₂²	1	2	yes	—
8	#15692186	—	2	4	no	—
9	#15692179	—	2	4	no	—
10	C₄×#11717579	C₂²	1	1	yes	characteristic
11	C₄×#11717579	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₁₀₁	C₂²	1	2	yes	—
13	#107×C₁₀₁	—	2	2	no	—
14	C₄×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₀₂	—	2	2	no	—
16	Dic₄₀₄	—	4	4	no	—
17	C₂×#11717579	D₄	1	2	yes	—
18	C₂×#11717579	D₄	1	2	yes	—
19	C₄⋊Dic₁₀₁	—	2	2	no	—
20	C₄⋊Dic₁₀₁	—	2	2	no	—
21	C₂×Q₈×C₁₀₁	—	2	2	no	—
22	C₂×C₈×C₁₀₁	D₄	1	2	yes	—
23	C₄⋊C₄×C₁₀₁	—	2	2	no	—
24	C₄⋊C₄×C₁₀₁	—	2	2	no	—
25	C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
26	Q₈×C₁₀₁	—	4	4	no	—
27	C₂×C₄×C₁₀₁	—	4	2	no	—
28	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
31	C₂×Dic₁₀₁	—	4	2	no	—
32	Dic₂₀₂	—	4	4	no	—
33	C₈×C₁₀₁	SD₁₆	1	4	yes	—
34	#11717579	—	2	2	no	—
35	#11717579	Q₁₆	1	4	yes	complemented
36	C₂²×C₁₀₁	#127	1	1	yes	characteristic
37	C₄×C₁₀₁	—	8	2	no	—
38	C₄×C₁₀₁	C₂×SD₁₆	1	2	yes	—
39	C₄×C₁₀₁	C₂×Q₁₆	1	2	yes	—
40	C₄×C₁₀₁	#135	1	2	yes	—
41	Dic₁₀₁	—	8	2	no	—
42	C₂×C₁₀₁	#492	1	1	yes	characteristic
43	C₂×C₁₀₁	#494	1	1	yes	characteristic
44	C₂×C₁₀₁	#495	1	1	yes	characteristic
45	#1707	—	101	1	no	maximal, Sylow
46	C₁₀₁	#1707	1	1	yes	characteristic, complemented, Sylow
47	#500	D₁₀₁	1	1	yes	characteristic
48	#498	—	101	2	no	—
49	#494	—	101	1	no	—
50	#487	—	101	2	no	—
51	C₈²	—	101	1	no	—
52	C₂×Q₁₆	—	202	2	no	—
53	C₄⋊Q₈	—	101	2	no	—
54	C₄⋊Q₈	C₂×D₁₀₁	1	2	yes	—
55	#107	—	2	2	no	—
56	#106	—	202	4	no	—
57	#103	—	202	4	no	—
58	C₄×C₈	—	101	1	no	—
59	C₄×C₈	—	101	1	no	—
60	C₄×C₈	C₂×D₁₀₁	1	1	yes	characteristic
61	C₂×Q₈	—	2	2	no	—
62	C₂×Q₈	—	202	2	no	—
63	Q₁₆	—	404	4	no	—
64	C₂×C₈	—	101	2	no	—
65	C₂×C₈	—	101	2	no	—
66	C₂×C₈	C₂²⋊D₁₀₁	1	2	yes	—
67	C₄⋊C₄	—	2	2	no	—
68	C₄⋊C₄	—	2	2	no	—
69	C₄⋊C₄	—	202	2	no	—
70	C₄⋊C₄	—	202	2	no	—
71	C₄²	C₂²×D₁₀₁	1	1	yes	characteristic
72	Q₈	—	4	4	no	—
73	Q₈	—	404	4	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	404	2	no	—
76	C₂×C₄	D₄×D₁₀₁	1	1	yes	characteristic
77	C₂×C₄	D₄×D₁₀₁	1	1	yes	characteristic
78	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
79	C₈	—	101	4	no	—
80	C₈	—	202	2	no	—
81	C₈	#13108061	1	4	yes	—
82	C₂²	#15692251	1	1	yes	characteristic
83	C₄	—	8	2	no	—
84	C₄	—	808	2	no	—
85	C₄	C₂×#13108061	1	2	yes	—
86	C₄	Q₁₆×D₁₀₁	1	2	yes	—
87	C₄	#15692274	1	2	yes	—
88	C₂	#20265551	1	1	yes	characteristic
89	C₂	#20265555	1	1	yes	characteristic
90	C₂	#20265560	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15692251
commutator, cosocle	C₄²×C₁₀₁	C₂³
Frattini	C₄²	C₂²×D₁₀₁
Fitting	#500×C₁₀₁	C₂
socle, prime radical	C₂²×C₁₀₁	#127
hypercenter	#500	D₁₀₁
hypocenter	C₁₀₁	#1707
LCS	G ▷ C₄²×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

Complements

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

N	H ≅ G/N	type
35. #11717579	Q₁₆	semidirect
46. C₁₀₁	#1707	semidirect

Maximal subgroups

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

subgroup	index
#20265573	2
#20265557	2
#20265527	2
C₈×#11717579	2
#500×C₁₀₁	2
#1707	101