Subgroups — #27186136

Order 12928.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20266069	C₂	1	2	yes	maximal, complemented
2	#20265543	C₂	1	1	yes	maximal, characteristic
3	#15692329	—	2	2	no	—
4	#15692252	C₄	1	1	yes	characteristic
5	#15692175	—	2	2	no	—
6	#15692166	C₂²	1	1	yes	characteristic
7	#125×C₁₀₁	C₄	1	1	yes	characteristic
8	#13108077	—	2	4	no	—
9	C₂×D₄₀₄	—	2	1	no	—
10	C₂×D₄₀₄	—	4	1	no	—
11	C₂×#11717579	D₄	1	2	yes	—
12	C₄⋊C₄×C₁₀₁	—	2	1	no	—
13	C₄⋊C₄×C₁₀₁	—	4	1	no	—
14	C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
15	C₂²×D₁₀₁	—	4	1	no	—
16	C₂×C₄×C₁₀₁	—	2	1	no	—
17	C₂×C₄×C₁₀₁	—	4	1	no	—
18	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
19	D₄₀₄	—	4	2	no	—
20	D₄₀₄	—	8	2	no	—
21	#11717579	—	2	2	no	—
22	C₂²×C₁₀₁	#100	1	1	yes	characteristic
23	C₂×D₁₀₁	—	8	2	no	—
24	C₂×D₁₀₁	—	8	1	no	—
25	C₄×C₁₀₁	—	4	1	no	—
26	C₄×C₁₀₁	—	8	1	no	—
27	C₄×C₁₀₁	#104	1	2	yes	—
28	C₂×C₁₀₁	—	2	1	no	—
29	C₂×C₁₀₁	#329	1	1	yes	characteristic
30	D₁₀₁	—	16	1	no	—
31	#1355	—	101	1	no	maximal, Sylow
32	C₁₀₁	#1355	1	1	yes	characteristic, complemented, Sylow
33	#490	—	101	1	no	—
34	#347	—	101	2	no	—
35	#127	—	101	1	no	—
36	#125	#87938	1	1	yes	characteristic
37	M₃₂	—	202	2	no	—
38	#102	—	202	2	no	—
39	#97	—	101	1	no	—
40	C₂×D₄	—	404	1	no	—
41	C₂×D₄	—	202	1	no	—
42	C₂×C₈	—	101	2	no	—
43	C₄⋊C₄	—	2	1	no	—
44	C₄⋊C₄	—	4	1	no	—
45	C₄²	C₂×#87938	1	1	yes	characteristic
46	C₁₆	—	202	4	no	—
47	C₂³	—	404	1	no	—
48	D₄	—	808	2	no	—
49	D₄	—	404	2	no	—
50	C₂×C₄	—	2	1	no	—
51	C₂×C₄	—	4	1	no	—
52	C₂×C₄	C₂²⋊#87938	1	1	yes	characteristic
53	C₈	—	202	2	no	—
54	C₂²	—	808	2	no	—
55	C₂²	—	808	1	no	—
56	C₂²	#15692315	1	1	yes	characteristic
57	C₄	—	4	1	no	—
58	C₄	—	8	1	no	—
59	C₄	#15692320	1	2	yes	—
60	C₂	—	2	1	no	—
61	C₂	—	1616	1	no	—
62	C₂	#20266033	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20266033
commutator	C₄²×C₁₀₁	C₂×C₄
Frattini	C₄²	C₂×#87938
Fitting	#125×C₁₀₁	C₄
socle	C₂×C₁₀₁	#329
prime radical	#15692252	C₄
cosocle	#15692166	C₂²
hypercenter	#125	#87938
hypocenter	C₁₀₁	#1355
LCS	G ▷ C₄²×C₁₀₁ ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
1. #20266069	C₂	semidirect
32. C₁₀₁	#1355	semidirect

Maximal subgroups

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

subgroup	index
#20266069	2
#20265543	2
#1355	101