Subgroups — #27186080

Order 12928.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20266073	C₂	1	2	yes	maximal
2	#20265204	C₂	1	1	yes	maximal, characteristic
3	#15692329	—	2	2	no	—
4	C₂×#13108058	C₂²	1	1	yes	characteristic
5	C₂²×#11717579	C₄	1	1	yes	characteristic
6	C₄⋊#11717579	—	2	2	no	—
7	#117×C₁₀₁	C₄	1	1	yes	characteristic
8	#13108077	—	4	2	no	—
9	#13108058	—	4	1	no	—
10	C₂×#11717579	—	2	1	no	—
11	C₂×#11717579	—	4	1	no	—
12	C₂×#11717579	D₄	1	2	yes	—
13	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
14	C₄⋊C₄×C₁₀₁	—	2	1	no	—
15	C₂²⋊C₄×C₁₀₁	C₈	1	2	yes	—
16	C₄²×C₁₀₁	—	2	1	no	—
17	C₂³×C₁₀₁	C₂×C₈	1	1	yes	characteristic
18	C₂×C₄×C₁₀₁	—	2	1	no	—
19	C₂×C₄×C₁₀₁	—	2	1	no	—
20	C₂×C₄×C₁₀₁	—	2	2	no	—
21	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×C₄×C₁₀₁	M₁₆	1	1	yes	characteristic
23	#11717579	—	2	2	no	—
24	#11717579	—	4	1	no	—
25	C₂²×C₁₀₁	—	2	1	no	—
26	C₂²×C₁₀₁	—	4	1	no	—
27	C₂²×C₁₀₁	C₂²⋊C₈	1	1	yes	characteristic
28	C₄×C₁₀₁	—	2	1	no	—
29	C₄×C₁₀₁	—	4	2	no	—
30	C₄×C₁₀₁	#99	1	1	yes	characteristic
31	C₄×C₁₀₁	#101	1	1	yes	characteristic
32	C₂×C₁₀₁	—	2	1	no	—
33	C₂×C₁₀₁	—	4	1	no	—
34	C₂×C₁₀₁	#324	1	1	yes	characteristic
35	#1328	—	101	1	no	maximal, Sylow
36	C₁₀₁	#1328	1	1	yes	characteristic, complemented, Sylow
37	#424	—	101	1	no	—
38	#349	—	101	2	no	—
39	C₂×M₁₆	—	101	1	no	—
40	C₂²×C₈	—	101	1	no	—
41	#117	#87938	1	1	yes	characteristic
42	M₃₂	—	202	2	no	—
43	C₄⋊C₈	—	202	2	no	—
44	C₂²×C₄	C₂×#87938	1	1	yes	characteristic
45	M₁₆	—	404	1	no	—
46	C₂×C₈	—	101	2	no	—
47	C₂×C₈	—	404	1	no	—
48	C₂×C₈	—	202	1	no	—
49	C₄⋊C₄	—	2	1	no	—
50	C₂²⋊C₄	#11717581	1	2	yes	—
51	C₄²	—	2	1	no	—
52	C₁₆	—	404	2	no	—
53	C₂³	C₂×#11717581	1	1	yes	characteristic
54	C₂×C₄	—	2	1	no	—
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	C₂²⋊#87938	1	1	yes	characteristic
58	C₂×C₄	#13108083	1	1	yes	characteristic
59	C₈	—	404	1	no	—
60	C₈	—	202	2	no	—
61	C₂²	—	2	1	no	—
62	C₂²	—	4	1	no	—
63	C₂²	C₂²⋊#11717581	1	1	yes	characteristic
64	C₄	—	2	1	no	—
65	C₄	—	4	2	no	—
66	C₄	#15692314	1	1	yes	characteristic
67	C₄	#15692316	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	4	1	no	—
70	C₂	#20266022	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#15692314
commutator, prime radical	C₂³×C₁₀₁	C₂×C₈
Frattini	C₂²×C₄	C₂×#87938
Fitting	#117×C₁₀₁	C₄
socle	C₂×C₁₀₁	#324
cosocle	C₂×#13108058	C₂²
hypercenter	#117	#87938
hypocenter	C₁₀₁	#1328
LCS	G ▷ C₂³×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
36. C₁₀₁	#1328	semidirect

Maximal subgroups

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

subgroup	index
#20266073	2
#20265204	2
#1328	101