Group #27180234 — subgroups

Subgroups — #27180234

Order 12928.

330 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 205 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20265308	C₂	1	4	yes	maximal, complemented
2	Q₈×#11717579	C₂	1	4	yes	maximal, complemented
3	#20265238	C₂	1	4	yes	maximal
4	#20265230	C₂	1	2	yes	maximal
5	#533×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	C₄⋊#11717579	C₂²	1	2	yes	—
7	C₄⋊#11717579	C₂²	1	2	yes	—
8	C₄⋊#11717579	C₂²	1	8	yes	—
9	C₂²⋊#11717579	C₂²	1	4	yes	—
10	#15692166	C₂²	1	4	yes	—
11	C₄×#11717579	C₂²	1	4	yes	—
12	C₄×#11717579	C₂²	1	4	yes	—
13	C₄⋊Q₈×C₁₀₁	C₄	1	2	yes	—
14	#125×C₁₀₁	C₄	1	2	yes	—
15	C₂²⋊Q₈×C₁₀₁	C₄	1	4	yes	—
16	C₄×Q₈×C₁₀₁	C₂²	1	4	yes	—
17	#117×C₁₀₁	C₂²	1	2	yes	—
18	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
19	C₂×#11717579	C₂³	1	4	yes	—
20	C₂×#11717579	C₂³	1	4	yes	—
21	C₂×Q₈×C₁₀₁	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₁₀₁	C₂×C₄	1	4	yes	—
25	C₄⋊C₄×C₁₀₁	C₂×C₄	1	8	yes	—
26	C₄⋊C₄×C₁₀₁	C₂×C₄	1	4	yes	—
27	C₂²⋊C₄×C₁₀₁	C₂×C₄	1	4	yes	—
28	C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₁₀₁	C₂³	1	2	yes	—
31	C₄²×C₁₀₁	C₂³	1	4	yes	—
32	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
33	Q₈×C₁₀₁	—	2	8	no	—
34	C₂×C₄×C₁₀₁	—	2	4	no	—
35	C₂×C₄×C₁₀₁	—	2	2	no	—
36	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
38	C₂×C₄×C₁₀₁	C₂²×C₄	1	8	yes	—
39	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
40	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
41	#11717579	—	2	4	no	—
42	#11717579	—	2	4	no	—
43	C₂²×C₁₀₁	—	2	1	no	—
44	C₂²×C₁₀₁	—	2	2	no	—
45	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
46	C₄×C₁₀₁	—	2	1	no	—
47	C₄×C₁₀₁	—	2	2	no	—
48	C₄×C₁₀₁	—	2	8	no	—
49	C₄×C₁₀₁	#131	1	4	yes	—
50	C₄×C₁₀₁	2₋¹⁺⁴	1	2	yes	—
51	C₂×C₁₀₁	—	4	1	no	—
52	C₂×C₁₀₁	#519	1	1	yes	characteristic
53	C₂×C₁₀₁	C₂×#131	1	2	yes	—
54	#2986	—	101	1	no	maximal, Sylow
55	C₁₀₁	#2986	1	1	yes	characteristic, complemented, Sylow
56	#533	D₁₀₁	1	1	yes	characteristic
57	#446	—	101	4	no	—
58	C₈×Q₈	—	101	4	no	—
59	#433	—	101	4	no	—
60	#431	—	101	2	no	—
61	C₄⋊Q₈	Dic₁₀₁	1	2	yes	—
62	#125	Dic₁₀₁	1	2	yes	—
63	C₂²⋊Q₈	Dic₁₀₁	1	4	yes	—
64	C₄×Q₈	C₂×D₁₀₁	1	4	yes	—
65	#117	C₂×D₁₀₁	1	2	yes	—
66	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
67	C₄⋊C₈	—	101	2	no	—
68	C₄⋊C₈	—	101	2	no	—
69	C₄⋊C₈	—	101	8	no	—
70	C₂²⋊C₈	—	101	4	no	—
71	#97	—	101	4	no	—
72	C₄×C₈	—	101	4	no	—
73	C₄×C₈	—	101	4	no	—
74	C₂×Q₈	C₂×Dic₁₀₁	1	4	yes	—
75	C₂²×C₄	C₂×Dic₁₀₁	1	2	yes	—
76	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
77	C₂×C₈	—	101	4	no	—
78	C₂×C₈	—	101	4	no	—
79	C₄⋊C₄	C₂×Dic₁₀₁	1	4	yes	—
80	C₄⋊C₄	C₂×Dic₁₀₁	1	8	yes	—
81	C₄⋊C₄	C₂×Dic₁₀₁	1	4	yes	—
82	C₂²⋊C₄	C₂×Dic₁₀₁	1	4	yes	—
83	C₄²	C₂×Dic₁₀₁	1	1	yes	characteristic
84	C₄²	C₂×Dic₁₀₁	1	1	yes	characteristic
85	C₄²	C₂²×D₁₀₁	1	2	yes	—
86	C₄²	C₂²×D₁₀₁	1	4	yes	—
87	C₂³	C₂²×Dic₁₀₁	1	1	yes	characteristic
88	Q₈	—	2	8	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	2	4	no	—
91	C₂×C₄	C₂²×Dic₁₀₁	1	1	yes	characteristic
92	C₂×C₄	C₂²×Dic₁₀₁	1	2	yes	—
93	C₂×C₄	C₂²×Dic₁₀₁	1	8	yes	—
94	C₂×C₄	C₂²×Dic₁₀₁	1	2	yes	—
95	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
96	C₈	—	202	4	no	—
97	C₈	—	202	4	no	—
98	C₂²	—	2	1	no	—
99	C₂²	—	2	2	no	—
100	C₂²	C₂³×Dic₁₀₁	1	1	yes	characteristic
101	C₄	—	2	1	no	—
102	C₄	—	2	2	no	—
103	C₄	—	2	8	no	—
104	C₄	#15692262	1	4	yes	—
105	C₄	#15692303	1	2	yes	—
106	C₂	—	4	1	no	—
107	C₂	#20265655	1	1	yes	characteristic
108	C₂	C₂×#15692262	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20265308

C₂

yes

maximal, complemented

Q₈×#11717579

C₂

yes

maximal, complemented

#20265238

C₂

yes

maximal

#20265230

C₂

yes

maximal

#533×C₁₀₁

C₂

yes

maximal, characteristic

C₄⋊#11717579

C₂²

yes

—

C₄⋊#11717579

C₂²

yes

—

C₄⋊#11717579

C₂²

yes

—

C₂²⋊#11717579

C₂²

yes

—

#15692166

C₂²

yes

—

C₄×#11717579

C₂²

yes

—

C₄×#11717579

C₂²

yes

—

C₄⋊Q₈×C₁₀₁

C₄

yes

—

#125×C₁₀₁

C₄

yes

—

C₂²⋊Q₈×C₁₀₁

C₄

yes

—

C₄×Q₈×C₁₀₁

C₂²

yes

—

#117×C₁₀₁

C₂²

yes

—

C₂×C₄²×C₁₀₁

C₂²

yes

characteristic

C₂×#11717579

C₂³

yes

—

C₂×#11717579

C₂³

yes

—

C₂×Q₈×C₁₀₁

C₂×C₄

yes

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₁

C₂×C₄

yes

—

C₄⋊C₄×C₁₀₁

C₂×C₄

yes

—

C₄⋊C₄×C₁₀₁

C₂×C₄

yes

—

C₂²⋊C₄×C₁₀₁

C₂×C₄

yes

—

C₄²×C₁₀₁

C₂×C₄

yes

characteristic

C₄²×C₁₀₁

C₂×C₄

yes

characteristic

C₄²×C₁₀₁

C₂³

yes

—

C₄²×C₁₀₁

C₂³

yes

—

C₂³×C₁₀₁

C₂²×C₄

yes

characteristic

Q₈×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂²×C₄

yes

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

—

C₂×C₄×C₁₀₁

C₂⁴

yes

characteristic

#11717579

—

#11717579

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂³×C₄

yes

characteristic

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

#131

yes

—

C₄×C₁₀₁

2₋¹⁺⁴

yes

—

C₂×C₁₀₁

—

C₂×C₁₀₁

#519

yes

characteristic

C₂×C₁₀₁

C₂×#131

yes

—

#2986

—

101

maximal, Sylow

C₁₀₁

#2986

yes

characteristic, complemented, Sylow

#533

D₁₀₁

yes

characteristic

#446

—

101

—

C₈×Q₈

—

101

—

#433

—

101

—

#431

—

101

—

C₄⋊Q₈

Dic₁₀₁

yes

—

#125

Dic₁₀₁

yes

—

C₂²⋊Q₈

Dic₁₀₁

yes

—

C₄×Q₈

C₂×D₁₀₁

yes

—

#117

C₂×D₁₀₁

yes

—

C₂×C₄²

C₂×D₁₀₁

yes

characteristic

C₄⋊C₈

—

101

—

C₄⋊C₈

—

101

—

C₄⋊C₈

—

101

—

C₂²⋊C₈

—

101

—

#97

—

101

—

C₄×C₈

—

101

—

C₄×C₈

—

101

—

C₂×Q₈

C₂×Dic₁₀₁

yes

—

C₂²×C₄

C₂×Dic₁₀₁

yes

—

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

C₂×C₈

—

101

—

C₂×C₈

—

101

—

C₄⋊C₄

C₂×Dic₁₀₁

yes

—

C₄⋊C₄

C₂×Dic₁₀₁

yes

—

C₄⋊C₄

C₂×Dic₁₀₁

yes

—

C₂²⋊C₄

C₂×Dic₁₀₁

yes

—

C₄²

C₂×Dic₁₀₁

yes

characteristic

C₄²

C₂×Dic₁₀₁

yes

characteristic

C₄²

C₂²×D₁₀₁

yes

—

C₄²

C₂²×D₁₀₁

yes

—

C₂³

C₂²×Dic₁₀₁

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×Dic₁₀₁

yes

characteristic

C₂×C₄

C₂²×Dic₁₀₁

yes

—

C₂×C₄

C₂²×Dic₁₀₁

yes

—

C₂×C₄

C₂²×Dic₁₀₁

yes

—

C₂×C₄

C₂³×D₁₀₁

yes

characteristic

C₈

—

202

—

C₈

—

202

—

C₂²

—

C₂²

—

100

C₂²

C₂³×Dic₁₀₁

yes

characteristic

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

#15692262

yes

—

105

C₄

#15692303

yes

—

106

C₂

—

107

C₂

#20265655

yes

characteristic

108

C₂

C₂×#15692262

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₀₁
commutator, socle	C₂²×C₁₀₁	C₂³×C₄
Fitting	#533×C₁₀₁	C₂
prime radical	C₂³×C₁₀₁	C₂²×C₄
cosocle	C₂×C₄×C₁₀₁	C₂⁴
hypercenter	#533	D₁₀₁
hypocenter	C₁₀₁	#2986
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₄ ◁ #533

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₀₁

commutator, socle

C₂²×C₁₀₁

C₂³×C₄

Fitting

#533×C₁₀₁

C₂

prime radical

C₂³×C₁₀₁

C₂²×C₄

cosocle

C₂×C₄×C₁₀₁

C₂⁴

D₁₀₁

C₁₀₁

G ▷ C₂²×C₁₀₁ ▷ C₁₀₁

C₁ ◁ C₂×C₄ ◁ #533

N	H ≅ G/N	type
1. #20265308	C₂	semidirect
2. Q₈×#11717579	C₂	semidirect
55. C₁₀₁	#2986	semidirect

H ≅ G/N

type

1. #20265308

C₂

semidirect

2. Q₈×#11717579

C₂

semidirect

55. C₁₀₁

#2986

semidirect

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

subgroup	index
#20265308	2
Q₈×#11717579	2
#20265238	2
#20265230	2
#533×C₁₀₁	2
#2986	101

subgroup

index

#533×C₁₀₁

101

Subgroups — #27180234

Conjugacy classes of subgroups

Complements

Maximal subgroups