Group #27185949 — subgroups

Subgroups — #27185949

Order 12928.

202 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 54 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	#20266043	C₂	1	2	yes	maximal, complemented
2	#20265078	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11717581	—	2	2	no	—
4	C₂²×D₄₀₄	C₄	1	1	yes	characteristic
5	C₂×#13108051	—	2	2	no	—
6	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₄×C₁₀₁	C₄	1	1	yes	characteristic
8	C₂×#11717581	—	2	4	no	—
9	C₂×#11717581	—	4	2	no	—
10	C₂×#11717581	—	4	2	no	—
11	C₂³×D₁₀₁	—	2	1	no	—
12	C₂×D₄₀₄	—	2	1	no	—
13	C₂×D₄₀₄	—	4	1	no	—
14	C₂×D₄₀₄	—	4	1	no	—
15	C₂×D₄₀₄	C₈	1	2	yes	complemented
16	C₂²×Dic₁₀₁	D₄	1	2	yes	—
17	#13108051	—	4	2	no	—
18	#13108051	—	4	2	no	—
19	C₄×Dic₁₀₁	—	4	1	no	—
20	C₂²×C₄×C₁₀₁	—	2	1	no	—
21	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
22	C₄⋊C₄×C₁₀₁	—	4	1	no	—
23	C₂³×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
24	C₂²×D₁₀₁	—	2	2	no	—
25	C₂²×D₁₀₁	—	4	1	no	—
26	C₂²×D₁₀₁	—	4	1	no	—
27	C₂²×D₁₀₁	—	4	2	no	—
28	C₂²×D₁₀₁	—	4	2	no	—
29	C₂×C₄×C₁₀₁	—	2	1	no	—
30	C₂×C₄×C₁₀₁	—	2	1	no	—
31	C₂×C₄×C₁₀₁	—	4	1	no	—
32	C₂×C₄×C₁₀₁	—	4	2	no	—
33	C₂×C₄×C₁₀₁	C₂×C₈	1	1	yes	characteristic
34	C₂×C₄×C₁₀₁	M₁₆	1	1	yes	characteristic
35	C₂×Dic₁₀₁	—	2	2	no	—
36	C₂×Dic₁₀₁	—	2	2	no	—
37	C₂×Dic₁₀₁	D₈	1	2	yes	—
38	C₂×Dic₁₀₁	SD₁₆	1	2	yes	—
39	D₄₀₄	—	4	2	no	—
40	D₄₀₄	—	4	2	no	—
41	#11717581	—	4	4	no	—
42	C₂²×C₁₀₁	C₂²⋊C₈	1	1	yes	characteristic
43	C₂²×C₁₀₁	#99	1	1	yes	characteristic
44	C₂²×C₁₀₁	#100	1	1	yes	characteristic
45	C₂²×C₁₀₁	#102	1	2	yes	—
46	C₂²×C₁₀₁	#104	1	2	yes	—
47	C₂×D₁₀₁	—	4	2	no	—
48	C₂×D₁₀₁	—	4	4	no	—
49	C₂×D₁₀₁	—	8	2	no	—
50	C₂×D₁₀₁	—	8	1	no	—
51	C₂×D₁₀₁	—	8	1	no	—
52	C₄×C₁₀₁	—	2	1	no	—
53	C₄×C₁₀₁	—	2	1	no	—
54	C₄×C₁₀₁	—	8	1	no	—
55	Dic₁₀₁	—	2	2	no	—
56	Dic₁₀₁	—	2	2	no	—
57	C₂×C₁₀₁	#323	1	1	yes	characteristic
58	C₂×C₁₀₁	D₄⋊C₈	1	2	yes	—
59	C₂×C₁₀₁	#327	1	2	yes	—
60	C₂×C₁₀₁	#329	1	1	yes	characteristic
61	C₂×C₁₀₁	#331	1	1	yes	characteristic
62	D₁₀₁	—	8	2	no	—
63	#1270	—	101	1	no	maximal, Sylow
64	C₁₀₁	#1270	1	1	yes	characteristic, complemented, Sylow
65	#390	—	101	1	no	—
66	#336	—	101	2	no	—
67	C₂²×D₄	—	101	1	no	—
68	C₂²×C₈	—	202	2	no	—
69	C₂×C₄⋊C₄	#87938	1	1	yes	characteristic
70	C₂×C₂²⋊C₄	—	202	2	no	—
71	C₂×C₄²	—	101	1	no	—
72	C₂⁴	—	202	1	no	—
73	C₂×D₄	—	101	2	no	—
74	C₂×D₄	—	404	1	no	—
75	C₂×D₄	—	404	1	no	—
76	C₂×D₄	—	202	1	no	—
77	C₂²×C₄	—	2	1	no	—
78	C₂²×C₄	—	101	2	no	—
79	C₂²×C₄	C₂×#87938	1	1	yes	characteristic
80	C₂×C₈	—	404	2	no	—
81	C₂×C₈	—	404	2	no	—
82	C₂×C₈	—	202	4	no	—
83	C₄⋊C₄	—	4	1	no	—
84	C₂²⋊C₄	—	404	2	no	—
85	C₂²⋊C₄	—	404	2	no	—
86	C₄²	—	404	1	no	—
87	C₂³	—	404	1	no	—
88	C₂³	—	404	1	no	—
89	C₂³	—	404	2	no	—
90	C₂³	—	404	2	no	—
91	C₂³	—	202	2	no	—
92	C₂³	C₂²⋊#87938	1	1	yes	characteristic
93	D₄	—	404	2	no	—
94	D₄	—	404	2	no	—
95	C₂×C₄	—	2	1	no	—
96	C₂×C₄	—	2	1	no	—
97	C₂×C₄	—	4	1	no	—
98	C₂×C₄	—	4	2	no	—
99	C₂×C₄	—	101	2	no	—
100	C₂×C₄	—	101	2	no	—
101	C₂×C₄	—	202	2	no	—
102	C₂×C₄	—	202	2	no	—
103	C₂×C₄	D₁₀₁⋊C₈	1	1	yes	characteristic
104	C₂×C₄	#13108084	1	1	yes	characteristic
105	C₈	—	404	4	no	—
106	C₂²	—	808	2	no	—
107	C₂²	—	808	1	no	—
108	C₂²	—	808	1	no	—
109	C₂²	—	404	2	no	—
110	C₂²	—	404	4	no	—
111	C₂²	#15692312	1	1	yes	characteristic
112	C₂²	#15692314	1	1	yes	characteristic
113	C₂²	#15692315	1	1	yes	characteristic
114	C₂²	#15692317	1	2	yes	—
115	C₂²	#15692320	1	2	yes	—
116	C₄	—	2	1	no	—
117	C₄	—	2	1	no	—
118	C₄	—	8	1	no	—
119	C₄	—	202	2	no	—
120	C₄	—	202	2	no	—
121	C₂	—	808	2	no	—
122	C₂	#20266021	1	1	yes	characteristic
123	C₂	#20266025	1	2	yes	—
124	C₂	#20266029	1	2	yes	—
125	C₂	#20266033	1	1	yes	characteristic
126	C₂	#20266035	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20266043

C₂

yes

maximal, complemented

#20265078

C₂

yes

maximal, characteristic

C₂²×#11717581

—

C₂²×D₄₀₄

C₄

yes

characteristic

C₂×#13108051

—

C₂×C₄×Dic₁₀₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₀₁

C₄

yes

—

—

—

C₂³×D₁₀₁

—

C₂×D₄₀₄

—

C₂×D₄₀₄

—

C₂×D₄₀₄

—

C₂×D₄₀₄

C₈

yes

complemented

C₂²×Dic₁₀₁

D₄

yes

—

#13108051

—

#13108051

—

C₄×Dic₁₀₁

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

characteristic

C₄⋊C₄×C₁₀₁

—

C₂³×C₁₀₁

C₂²⋊C₄

yes

characteristic

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₁

M₁₆

yes

characteristic

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

D₈

yes

—

C₂×Dic₁₀₁

SD₁₆

yes

—

D₄₀₄

—

D₄₀₄

—

#11717581

—

C₂²×C₁₀₁

C₂²⋊C₈

yes

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

yes

—

C₂²×C₁₀₁

#104

yes

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

—

C₂×C₁₀₁

#323

yes

characteristic

C₂×C₁₀₁

D₄⋊C₈

yes

—

C₂×C₁₀₁

#327

yes

—

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

D₁₀₁

—

#1270

—

101

maximal, Sylow

C₁₀₁

#1270

yes

characteristic, complemented, Sylow

#390

—

101

—

#336

—

101

—

C₂²×D₄

—

101

—

C₂²×C₈

—

202

—

C₂×C₄⋊C₄

#87938

yes

characteristic

C₂×C₂²⋊C₄

—

202

—

C₂×C₄²

—

101

—

C₂⁴

—

202

—

C₂×D₄

—

101

—

C₂×D₄

—

404

—

C₂×D₄

—

404

—

C₂×D₄

—

202

—

C₂²×C₄

—

C₂²×C₄

—

101

—

C₂²×C₄

C₂×#87938

yes

characteristic

C₂×C₈

—

404

—

C₂×C₈

—

404

—

C₂×C₈

—

202

—

C₄⋊C₄

—

C₂²⋊C₄

—

404

—

C₂²⋊C₄

—

404

—

C₄²

—

404

—

C₂³

—

404

—

C₂³

—

404

—

C₂³

—

404

—

C₂³

—

404

—

C₂³

—

202

—

C₂³

C₂²⋊#87938

yes

characteristic

D₄

—

404

—

D₄

—

404

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

101

—

100

C₂×C₄

—

101

—

101

C₂×C₄

—

202

—

102

C₂×C₄

—

202

—

103

C₂×C₄

D₁₀₁⋊C₈

yes

characteristic

104

C₂×C₄

#13108084

yes

characteristic

105

C₈

—

404

—

106

C₂²

—

808

—

107

C₂²

—

808

—

108

C₂²

—

808

—

109

C₂²

—

404

—

110

C₂²

—

404

—

111

C₂²

#15692312

yes

characteristic

112

C₂²

#15692314

yes

characteristic

113

C₂²

#15692315

yes

characteristic

114

C₂²

#15692317

yes

—

115

C₂²

#15692320

yes

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

202

—

120

C₄

—

202

—

121

C₂

—

808

—

122

C₂

#20266021

yes

characteristic

123

C₂

#20266025

yes

—

124

C₂

#20266029

yes

—

125

C₂

#20266033

yes

characteristic

126

C₂

#20266035

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²⋊#87938
commutator	C₂×C₄×C₁₀₁	C₂×C₈
Frattini	C₂²×C₄	C₂×#87938
Fitting	C₂×C₄⋊C₄×C₁₀₁	C₄
socle	C₂³×C₁₀₁	C₂²⋊C₄
prime radical	C₂²×D₄₀₄	C₄
cosocle	C₂×C₄×Dic₁₀₁	C₂²
hypercenter	C₂×C₄⋊C₄	#87938
hypocenter	C₁₀₁	#1270
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂²⋊#87938

commutator

C₂×C₄×C₁₀₁

C₂×C₈

Frattini

C₂²×C₄

C₂×#87938

Fitting

C₂×C₄⋊C₄×C₁₀₁

C₄

socle

C₂³×C₁₀₁

C₂²⋊C₄

prime radical

C₂²×D₄₀₄

C₄

cosocle

C₂×C₄×Dic₁₀₁

C₂²

C₂×C₄⋊C₄

C₁₀₁

G ▷ C₂×C₄×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
1. #20266043	C₂	semidirect
15. C₂×D₄₀₄	C₈	semidirect
64. C₁₀₁	#1270	semidirect

H ≅ G/N

type

1. #20266043

C₂

semidirect

15. C₂×D₄₀₄

C₈

semidirect

64. C₁₀₁

#1270

semidirect

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

subgroup	index
#20266043	2
#20265078	2
#1270	101

subgroup

index

#20266043

#20265078

#1270

101

Subgroups — #27185949

Conjugacy classes of subgroups

Complements

Maximal subgroups