Subgroups — #27186267

Order 12928.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20266088	C₂	1	2	yes	maximal, complemented
2	#20265549	C₂	1	1	yes	maximal, characteristic
3	#15692316	—	2	2	no	—
4	C₂×#13108062	—	2	2	no	—
5	#15692255	C₂²	1	1	yes	characteristic
6	C₄×#11717579	C₄	1	1	yes	characteristic
7	#127×C₁₀₁	C₄	1	1	yes	characteristic
8	#13108083	—	4	2	no	—
9	C₂×Dic₂₀₂	D₄	1	2	yes	—
10	#13108062	—	4	4	no	—
11	C₂×#11717579	—	2	1	no	—
12	C₄⋊Dic₁₀₁	—	4	1	no	—
13	C₂×D₄×C₁₀₁	—	2	1	no	—
14	C₂×D₄×C₁₀₁	—	4	1	no	—
15	C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
16	C₂³×C₁₀₁	—	4	1	no	—
17	D₄×C₁₀₁	—	4	2	no	—
18	D₄×C₁₀₁	—	8	2	no	—
19	C₂×C₄×C₁₀₁	—	2	1	no	—
20	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
21	C₂×Dic₁₀₁	—	2	2	no	—
22	Dic₂₀₂	—	4	2	no	—
23	#11717581	—	8	2	no	—
24	#11717579	—	2	2	no	—
25	C₂²×C₁₀₁	—	8	2	no	—
26	C₂²×C₁₀₁	—	8	1	no	—
27	C₂²×C₁₀₁	#99	1	1	yes	characteristic
28	C₄×C₁₀₁	—	2	2	no	—
29	C₄×C₁₀₁	—	2	1	no	—
30	Dic₁₀₁	—	4	2	no	—
31	C₂×C₁₀₁	—	2	1	no	—
32	C₂×C₁₀₁	—	16	1	no	—
33	C₂×C₁₀₁	#353	1	1	yes	characteristic
34	#1410	—	101	1	no	maximal, Sylow
35	C₁₀₁	#1410	1	1	yes	characteristic, complemented, Sylow
36	#492	—	101	1	no	—
37	#356	—	101	2	no	—
38	C₂×SD₁₆	—	202	2	no	—
39	C₄⋊Q₈	—	101	1	no	—
40	#127	#87938	1	1	yes	characteristic
41	#101	—	202	2	no	—
42	C₄×C₈	—	101	1	no	—
43	C₂×Q₈	—	101	2	no	—
44	C₂×D₄	—	2	1	no	—
45	C₂×D₄	—	4	1	no	—
46	SD₁₆	—	404	4	no	—
47	M₁₆	—	404	2	no	—
48	C₂×C₈	—	202	1	no	—
49	C₄⋊C₄	—	404	1	no	—
50	C₄²	C₂×#87938	1	1	yes	characteristic
51	C₂³	—	4	1	no	—
52	Q₈	—	404	2	no	—
53	D₄	—	4	2	no	—
54	D₄	—	8	2	no	—
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	202	2	no	—
57	C₂×C₄	C₂²⋊#87938	1	1	yes	characteristic
58	C₈	—	808	2	no	—
59	C₈	—	202	2	no	—
60	C₂²	—	8	2	no	—
61	C₂²	—	8	1	no	—
62	C₂²	#15692313	1	1	yes	characteristic
63	C₄	—	2	2	no	—
64	C₄	—	2	1	no	—
65	C₄	—	404	2	no	—
66	C₂	—	2	1	no	—
67	C₂	—	16	1	no	—
68	C₂	#20266081	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20266081
commutator	C₄²×C₁₀₁	C₂×C₄
Frattini	C₄²	C₂×#87938
Fitting, prime radical	#127×C₁₀₁	C₄
socle	C₂×C₁₀₁	#353
cosocle	#15692255	C₂²
hypercenter	#127	#87938
hypocenter	C₁₀₁	#1410
LCS	G ▷ C₄²×C₁₀₁ ▷ C₂×C₄×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ #127

Complements

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

N	H ≅ G/N	type
1. #20266088	C₂	semidirect
35. C₁₀₁	#1410	semidirect

Maximal subgroups

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

subgroup	index
#20266088	2
#20265549	2
#1410	101