Subgroups — #27186099

Order 12928.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#13108077	C₂	1	1	yes	maximal, characteristic
2	C₄×#13108077	C₂	1	1	yes	maximal, characteristic
3	#20265301	C₂	1	1	yes	maximal, characteristic
4	C₂×#13108077	—	2	1	no	—
5	C₂×#13108077	—	2	1	no	—
6	C₄×Dic₂₀₂	C₄	1	1	yes	characteristic
7	#15692184	—	2	1	no	—
8	C₄×#11717579	C₂²	1	1	yes	characteristic
9	C₈×Dic₁₀₁	—	2	1	no	—
10	C₄⋊C₈×C₁₀₁	C₄	1	1	yes	characteristic
11	#13108077	—	2	2	no	—
12	#13108077	—	4	1	no	—
13	C₂×Dic₂₀₂	C₈	1	1	yes	characteristic
14	C₂×#11717579	D₄	1	1	yes	characteristic
15	C₂×#11717579	D₄	1	1	yes	characteristic
16	C₄⋊Dic₁₀₁	C₈	1	1	yes	characteristic
17	#13108053	—	2	1	no	—
18	C₄×Dic₁₀₁	—	2	1	no	—
19	C₂×C₈×C₁₀₁	—	2	1	no	—
20	C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×C₄×C₁₀₁	C₂×C₈	1	1	yes	characteristic
23	C₂×C₄×C₁₀₁	M₁₆	1	1	yes	characteristic
24	C₂×Dic₁₀₁	—	2	1	no	—
25	C₂×Dic₁₀₁	—	2	1	no	—
26	Dic₂₀₂	—	2	1	no	—
27	Dic₂₀₂	C₁₆	1	2	yes	complemented
28	C₈×C₁₀₁	—	4	1	no	—
29	#11717579	—	2	1	no	—
30	#11717579	SD₁₆	1	1	yes	characteristic
31	#11717579	Q₁₆	1	1	yes	characteristic
32	C₂²×C₁₀₁	C₂²⋊C₈	1	1	yes	characteristic
33	C₄×C₁₀₁	—	2	1	no	—
34	C₄×C₁₀₁	#103	1	1	yes	characteristic
35	C₄×C₁₀₁	#104	1	1	yes	characteristic
36	C₄×C₁₀₁	C₂×C₁₆	1	1	yes	characteristic
37	C₄×C₁₀₁	M₃₂	1	1	yes	characteristic
38	Dic₁₀₁	—	2	2	no	—
39	Dic₁₀₁	—	4	1	no	—
40	C₂×C₁₀₁	Q₈⋊C₈	1	1	yes	characteristic
41	C₂×C₁₀₁	C₂²⋊C₁₆	1	1	yes	characteristic
42	C₂×C₁₀₁	#350	1	1	yes	characteristic
43	Q₈⋊C₁₆	—	101	1	no	maximal, Sylow
44	C₁₀₁	Q₈⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
45	C₈×Q₈	—	101	1	no	—
46	C₄⋊C₁₆	—	101	1	no	—
47	C₄×C₁₆	—	101	1	no	—
48	C₄×Q₈	—	101	1	no	—
49	C₂×C₁₆	—	202	1	no	—
50	C₂×C₁₆	—	202	1	no	—
51	C₄⋊C₈	—	202	1	no	—
52	C₄⋊C₈	#87938	1	1	yes	characteristic
53	C₄×C₈	—	101	1	no	—
54	C₄×C₈	—	202	1	no	—
55	C₂×Q₈	—	101	1	no	—
56	C₂×C₈	—	2	1	no	—
57	C₂×C₈	—	101	1	no	—
58	C₂×C₈	—	101	1	no	—
59	C₄⋊C₄	—	101	1	no	—
60	C₄⋊C₄	—	202	1	no	—
61	C₄²	—	202	1	no	—
62	C₄²	C₂×#87938	1	1	yes	characteristic
63	C₁₆	—	404	1	no	—
64	C₁₆	—	202	2	no	—
65	Q₈	—	101	2	no	—
66	Q₈	—	202	1	no	—
67	C₂×C₄	—	202	1	no	—
68	C₂×C₄	—	202	1	no	—
69	C₂×C₄	C₂²⋊#87938	1	1	yes	characteristic
70	C₂×C₄	D₁₀₁⋊C₈	1	1	yes	characteristic
71	C₂×C₄	#13108084	1	1	yes	characteristic
72	C₈	—	4	1	no	—
73	C₈	—	101	1	no	—
74	C₈	—	101	1	no	—
75	C₈	—	202	1	no	—
76	C₂²	#15692312	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	—	404	1	no	—
79	C₄	—	202	2	no	—
80	C₄	#15692318	1	1	yes	characteristic
81	C₄	#15692319	1	1	yes	characteristic
82	C₄	D₁₀₁⋊C₁₆	1	1	yes	characteristic
83	C₄	#15692330	1	1	yes	characteristic
84	C₂	#20266027	1	1	yes	characteristic
85	C₂	#20266072	1	1	yes	characteristic
86	C₂	#20266076	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊#87938
commutator	C₄×C₁₀₁	C₂×C₁₆
Frattini	C₄²	C₂×#87938
Fitting	C₄⋊C₈×C₁₀₁	C₄
socle, prime radical	C₂²×C₁₀₁	C₂²⋊C₈
cosocle	C₄×#11717579	C₂²
hypercenter	C₄⋊C₈	#87938
hypocenter	C₁₀₁	Q₈⋊C₁₆
LCS	G ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
27. Dic₂₀₂	C₁₆	semidirect
44. C₁₀₁	Q₈⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#13108077	2
C₄×#13108077	2
#20265301	2
Q₈⋊C₁₆	101