Subgroups — #27186275

Order 12928.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20266088	C₂	1	2	yes	maximal
2	#20265547	C₂	1	1	yes	maximal, characteristic
3	#15692316	—	2	2	no	—
4	#15692255	C₂²	1	1	yes	characteristic
5	#15692178	—	2	2	no	—
6	#15692166	C₄	1	1	yes	characteristic
7	#125×C₁₀₁	C₄	1	1	yes	characteristic
8	#13108083	—	4	2	no	—
9	C₂×Dic₂₀₂	D₄	1	2	yes	—
10	C₂×#11717579	—	2	1	no	—
11	C₄⋊Dic₁₀₁	—	4	1	no	—
12	C₄⋊C₄×C₁₀₁	—	2	1	no	—
13	C₄⋊C₄×C₁₀₁	—	4	1	no	—
14	C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₁₀₁	—	2	1	no	—
16	C₂×C₄×C₁₀₁	—	4	1	no	—
17	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
18	C₂×Dic₁₀₁	—	2	2	no	—
19	Dic₂₀₂	—	4	2	no	—
20	#11717581	—	8	2	no	—
21	#11717579	—	4	1	no	—
22	C₂²×C₁₀₁	#99	1	1	yes	characteristic
23	C₄×C₁₀₁	—	2	1	no	—
24	C₄×C₁₀₁	—	4	1	no	—
25	C₄×C₁₀₁	—	8	1	no	—
26	Dic₁₀₁	—	4	2	no	—
27	C₂×C₁₀₁	—	2	1	no	—
28	C₂×C₁₀₁	#354	1	1	yes	characteristic
29	#1413	—	101	1	no	maximal, Sylow
30	C₁₀₁	#1413	1	1	yes	characteristic, complemented, Sylow
31	#491	—	101	1	no	—
32	#356	—	101	2	no	—
33	C₄⋊Q₈	—	101	1	no	—
34	#125	#87938	1	1	yes	characteristic
35	#103	—	202	2	no	—
36	#101	—	202	2	no	—
37	#97	—	101	1	no	—
38	C₂×Q₈	—	101	2	no	—
39	M₁₆	—	404	2	no	—
40	C₂×C₈	—	202	1	no	—
41	C₄⋊C₄	—	2	1	no	—
42	C₄⋊C₄	—	4	1	no	—
43	C₄⋊C₄	—	404	1	no	—
44	C₄²	C₂×#87938	1	1	yes	characteristic
45	Q₈	—	404	2	no	—
46	C₂×C₄	—	2	1	no	—
47	C₂×C₄	—	4	1	no	—
48	C₂×C₄	—	202	2	no	—
49	C₂×C₄	C₂²⋊#87938	1	1	yes	characteristic
50	C₈	—	808	2	no	—
51	C₈	—	404	1	no	—
52	C₂²	#15692313	1	1	yes	characteristic
53	C₄	—	2	1	no	—
54	C₄	—	4	1	no	—
55	C₄	—	8	1	no	—
56	C₄	—	404	2	no	—
57	C₂	—	2	1	no	—
58	C₂	#20266083	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20266083
commutator	C₄²×C₁₀₁	C₂×C₄
Frattini	C₄²	C₂×#87938
Fitting	#125×C₁₀₁	C₄
socle	C₂×C₁₀₁	#354
prime radical	C₂²×C₁₀₁	#99
cosocle	#15692255	C₂²
hypercenter	#125	#87938
hypocenter	C₁₀₁	#1413
LCS	G ▷ C₄²×C₁₀₁ ▷ C₂×C₄×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
30. C₁₀₁	#1413	semidirect

Maximal subgroups

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

subgroup	index
#20266088	2
#20265547	2
#1413	101