Subgroups — #27186140

Order 12928.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20266100	C₂	1	2	yes	maximal, complemented
2	#20265553	C₂	1	1	yes	maximal, characteristic
3	#15692329	—	2	2	no	—
4	C₂×#13108059	—	2	2	no	—
5	#15692252	C₄	1	1	yes	characteristic
6	C₄×#11717579	C₂²	1	1	yes	characteristic
7	#127×C₁₀₁	C₄	1	1	yes	characteristic
8	#13108077	—	4	2	no	—
9	C₂×D₄₀₄	—	2	1	no	—
10	C₂×D₄₀₄	—	4	1	no	—
11	#13108059	—	4	2	no	—
12	#13108059	—	4	2	no	—
13	C₂×#11717579	D₄	1	2	yes	—
14	C₂×D₄×C₁₀₁	—	2	1	no	—
15	C₂×D₄×C₁₀₁	—	4	1	no	—
16	C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
17	C₂³×C₁₀₁	—	4	1	no	—
18	C₂²×D₁₀₁	—	4	1	no	—
19	D₄×C₁₀₁	—	4	2	no	—
20	D₄×C₁₀₁	—	8	2	no	—
21	C₂×C₄×C₁₀₁	—	2	1	no	—
22	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
23	D₄₀₄	—	4	2	no	—
24	D₄₀₄	—	8	2	no	—
25	#11717579	D₈	1	2	yes	—
26	#11717579	SD₁₆	1	2	yes	—
27	C₂²×C₁₀₁	—	8	2	no	—
28	C₂²×C₁₀₁	—	8	1	no	—
29	C₂²×C₁₀₁	#100	1	1	yes	characteristic
30	C₂×D₁₀₁	—	8	2	no	—
31	C₂×D₁₀₁	—	8	1	no	—
32	C₄×C₁₀₁	—	2	2	no	—
33	C₄×C₁₀₁	#102	1	2	yes	—
34	C₂×C₁₀₁	—	2	1	no	—
35	C₂×C₁₀₁	—	16	1	no	—
36	C₂×C₁₀₁	#331	1	1	yes	characteristic
37	D₁₀₁	—	16	1	no	—
38	#1357	—	101	1	no	maximal, Sylow
39	C₁₀₁	#1357	1	1	yes	characteristic, complemented, Sylow
40	#493	—	101	1	no	—
41	#364	—	101	2	no	—
42	C₂×D₈	—	202	2	no	—
43	#127	—	101	1	no	—
44	#127	#87938	1	1	yes	characteristic
45	M₃₂	—	202	2	no	—
46	C₄×C₈	—	101	1	no	—
47	C₂×D₄	—	2	1	no	—
48	C₂×D₄	—	4	1	no	—
49	C₂×D₄	—	404	1	no	—
50	C₂×D₄	—	202	1	no	—
51	D₈	—	404	2	no	—
52	D₈	—	404	2	no	—
53	C₂×C₈	—	101	2	no	—
54	C₄²	C₂×#87938	1	1	yes	characteristic
55	C₁₆	—	404	2	no	—
56	C₂³	—	4	1	no	—
57	C₂³	—	404	1	no	—
58	D₄	—	4	2	no	—
59	D₄	—	8	2	no	—
60	D₄	—	808	2	no	—
61	D₄	—	404	2	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	C₂²⋊#87938	1	1	yes	characteristic
64	C₈	—	101	2	no	—
65	C₈	—	101	2	no	—
66	C₂²	—	8	2	no	—
67	C₂²	—	8	1	no	—
68	C₂²	—	808	2	no	—
69	C₂²	—	808	1	no	—
70	C₂²	#15692315	1	1	yes	characteristic
71	C₄	—	2	2	no	—
72	C₄	#15692317	1	2	yes	—
73	C₂	—	2	1	no	—
74	C₂	—	16	1	no	—
75	C₂	—	1616	1	no	—
76	C₂	#20266035	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20266035
commutator	C₄²×C₁₀₁	C₂×C₄
Frattini	C₄²	C₂×#87938
Fitting	#127×C₁₀₁	C₄
socle	C₂×C₁₀₁	#331
prime radical	#20265553	C₂
cosocle	C₄×#11717579	C₂²
hypercenter	#127	#87938
hypocenter	C₁₀₁	#1357
LCS	G ▷ C₄²×C₁₀₁ ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄² ◁ #127

Complements

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

N	H ≅ G/N	type
1. #20266100	C₂	semidirect
39. C₁₀₁	#1357	semidirect

Maximal subgroups

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

subgroup	index
#20266100	2
#20265553	2
#1357	101