Group #27186611 — subgroups

Subgroups — #27186611

Order 12928.

248 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 58 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	#20266197	C₂	1	2	yes	maximal, complemented
2	#20266032	C₂	1	4	yes	maximal, complemented
3	C₂×#15692242	C₂	1	1	yes	maximal, characteristic
4	#15692322	—	2	2	no	—
5	C₂²⋊#11717581	—	2	2	no	—
6	#15692310	C₂²	1	4	yes	complemented
7	C₂²×Dic₂₀₂	C₄	1	1	yes	characteristic
8	#15692242	—	2	2	no	—
9	#15692242	C₄	1	2	yes	—
10	#15692242	C₂²	1	2	yes	—
11	C₂×C₂²⋊Dic₁₀₁	—	2	2	no	—
12	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
13	C₂²×D₄×C₁₀₁	C₄	1	1	yes	characteristic
14	C₂×#11717581	—	2	4	no	—
15	C₂×Dic₂₀₂	—	2	1	no	—
16	C₂×Dic₂₀₂	—	4	2	no	—
17	C₂×Dic₂₀₂	C₂×C₄	1	2	yes	—
18	C₂²×Dic₁₀₁	—	2	1	no	—
19	C₂²×Dic₁₀₁	D₄	1	2	yes	—
20	C₂²⋊Dic₁₀₁	—	2	4	no	—
21	C₂²⋊Dic₁₀₁	—	2	4	no	—
22	C₄×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
23	C₄×Dic₁₀₁	D₄	1	2	yes	—
24	C₄×Dic₁₀₁	C₂³	1	1	yes	characteristic
25	C₂⁴×C₁₀₁	—	2	1	no	—
26	C₂×D₄×C₁₀₁	—	2	1	no	—
27	C₂×D₄×C₁₀₁	—	4	2	no	—
28	C₂×D₄×C₁₀₁	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₁₀₁	—	2	2	no	—
31	C₂³×C₁₀₁	—	4	4	no	—
32	C₂³×C₁₀₁	—	4	2	no	—
33	C₂³×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
34	D₄×C₁₀₁	—	4	2	no	—
35	D₄×C₁₀₁	—	4	2	no	—
36	C₂×C₄×C₁₀₁	—	2	2	no	—
37	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
39	C₂×Dic₁₀₁	—	2	2	no	—
40	C₂×Dic₁₀₁	—	4	2	no	—
41	C₂×Dic₁₀₁	—	4	2	no	—
42	C₂×Dic₁₀₁	C₂²⋊C₄	1	2	yes	—
43	C₂×Dic₁₀₁	C₂×D₄	1	2	yes	—
44	Dic₂₀₂	—	4	2	no	—
45	Dic₂₀₂	—	4	2	no	—
46	#11717581	—	4	4	no	—
47	C₂²×C₁₀₁	—	2	2	no	—
48	C₂²×C₁₀₁	—	4	2	no	—
49	C₂²×C₁₀₁	—	4	4	no	—
50	C₂²×C₁₀₁	—	8	4	no	—
51	C₂²×C₁₀₁	#100	1	2	yes	—
52	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₄×C₁₀₁	—	2	1	no	—
54	C₄×C₁₀₁	—	2	1	no	—
55	Dic₁₀₁	—	2	2	no	—
56	Dic₁₀₁	—	2	2	no	—
57	Dic₁₀₁	—	4	2	no	—
58	C₂×C₁₀₁	—	2	2	no	—
59	C₂×C₁₀₁	—	8	2	no	—
60	C₂×C₁₀₁	C₂×#100	1	1	yes	characteristic
61	C₂×C₁₀₁	#421	1	2	yes	—
62	#1533	—	101	1	no	maximal, Sylow
63	C₁₀₁	#1533	1	1	yes	characteristic, complemented, Sylow
64	C₂×#124	—	101	1	no	—
65	#432	—	101	2	no	—
66	#329	—	101	4	no	—
67	C₂²×Q₈	—	101	1	no	—
68	C₂²×D₄	#87938	1	1	yes	characteristic
69	#124	—	101	2	no	—
70	#124	—	101	2	no	—
71	#124	—	202	2	no	—
72	C₂×C₂²⋊C₄	—	202	2	no	—
73	C₂×C₄²	—	101	1	no	—
74	C₄⋊C₈	—	202	2	no	—
75	C₂²⋊C₈	—	202	2	no	—
76	#97	—	101	4	no	—
77	C₂⁴	—	2	1	no	—
78	C₂×Q₈	—	101	2	no	—
79	C₂×Q₈	—	404	2	no	—
80	C₂×Q₈	—	202	1	no	—
81	C₂×D₄	—	2	1	no	—
82	C₂×D₄	—	4	2	no	—
83	C₂×D₄	C₂×#87938	1	2	yes	—
84	C₂²×C₄	—	101	2	no	—
85	C₂²×C₄	—	202	1	no	—
86	C₂²×C₄	C₂×#87938	1	1	yes	characteristic
87	C₂×C₈	—	202	4	no	—
88	C₂²⋊C₄	—	202	4	no	—
89	C₂²⋊C₄	—	202	4	no	—
90	C₄²	—	101	2	no	—
91	C₄²	—	101	1	no	—
92	C₄²	—	101	1	no	—
93	C₂³	—	2	2	no	—
94	C₂³	—	4	4	no	—
95	C₂³	—	4	2	no	—
96	C₂³	C₂²⋊#87938	1	1	yes	characteristic
97	Q₈	—	404	2	no	—
98	Q₈	—	404	2	no	—
99	D₄	—	4	2	no	—
100	D₄	—	4	2	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	101	2	no	—
103	C₂×C₄	—	101	2	no	—
104	C₂×C₄	—	404	2	no	—
105	C₂×C₄	—	404	2	no	—
106	C₂×C₄	—	202	2	no	—
107	C₂×C₄	C₂²⋊#87938	1	1	yes	characteristic
108	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
109	C₈	—	404	4	no	—
110	C₂²	—	2	2	no	—
111	C₂²	—	4	2	no	—
112	C₂²	—	4	4	no	—
113	C₂²	—	8	4	no	—
114	C₂²	#15692315	1	2	yes	—
115	C₂²	C₂×C₂²⋊#87938	1	1	yes	characteristic
116	C₄	—	2	1	no	—
117	C₄	—	2	1	no	—
118	C₄	—	404	2	no	—
119	C₄	—	202	2	no	—
120	C₄	—	202	2	no	—
121	C₂	—	2	2	no	—
122	C₂	—	8	2	no	—
123	C₂	C₂×#15692315	1	1	yes	characteristic
124	C₂	#20266175	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20266197

C₂

yes

maximal, complemented

#20266032

C₂

yes

maximal, complemented

C₂×#15692242

C₂

yes

maximal, characteristic

#15692322

—

C₂²⋊#11717581

—

#15692310

C₂²

yes

complemented

C₂²×Dic₂₀₂

C₄

yes

characteristic

#15692242

—

#15692242

C₄

yes

—

#15692242

C₂²

yes

—

C₂×C₂²⋊Dic₁₀₁

—

C₂×C₄×Dic₁₀₁

C₂²

yes

characteristic

C₂²×D₄×C₁₀₁

C₄

yes

characteristic

C₂×#11717581

—

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

C₂×C₄

yes

—

C₂²×Dic₁₀₁

—

C₂²×Dic₁₀₁

D₄

yes

—

C₂²⋊Dic₁₀₁

—

C₂²⋊Dic₁₀₁

—

C₄×Dic₁₀₁

C₂×C₄

yes

characteristic

C₄×Dic₁₀₁

D₄

yes

—

C₄×Dic₁₀₁

C₂³

yes

characteristic

C₂⁴×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

C₂×C₄

yes

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

characteristic

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

C₂²⋊C₄

yes

characteristic

D₄×C₁₀₁

—

D₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

C₂²⋊C₄

yes

—

C₂×Dic₁₀₁

C₂×D₄

yes

—

Dic₂₀₂

—

Dic₂₀₂

—

#11717581

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

#100

yes

—

C₂²×C₁₀₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₀₁

—

C₄×C₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

C₂×#100

yes

characteristic

C₂×C₁₀₁

#421

yes

—

#1533

—

101

maximal, Sylow

C₁₀₁

#1533

yes

characteristic, complemented, Sylow

C₂×#124

—

101

—

#432

—

101

—

#329

—

101

—

C₂²×Q₈

—

101

—

C₂²×D₄

#87938

yes

characteristic

#124

—

101

—

#124

—

101

—

#124

—

202

—

C₂×C₂²⋊C₄

—

202

—

C₂×C₄²

—

101

—

C₄⋊C₈

—

202

—

C₂²⋊C₈

—

202

—

#97

—

101

—

C₂⁴

—

C₂×Q₈

—

101

—

C₂×Q₈

—

404

—

C₂×Q₈

—

202

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×#87938

yes

—

C₂²×C₄

—

101

—

C₂²×C₄

—

202

—

C₂²×C₄

C₂×#87938

yes

characteristic

C₂×C₈

—

202

—

C₂²⋊C₄

—

202

—

C₂²⋊C₄

—

202

—

C₄²

—

101

—

C₄²

—

101

—

C₄²

—

101

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²⋊#87938

yes

characteristic

Q₈

—

404

—

Q₈

—

404

—

D₄

—

100

D₄

—

101

C₂×C₄

—

102

C₂×C₄

—

101

—

103

C₂×C₄

—

101

—

104

C₂×C₄

—

404

—

105

C₂×C₄

—

404

—

106

C₂×C₄

—

202

—

107

C₂×C₄

C₂²⋊#87938

yes

characteristic

108

C₂×C₄

C₂²×#87938

yes

characteristic

109

C₈

—

404

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

#15692315

yes

—

115

C₂²

C₂×C₂²⋊#87938

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

404

—

119

C₄

—

202

—

120

C₄

—

202

—

121

C₂

—

122

C₂

—

123

C₂

C₂×#15692315

yes

characteristic

124

C₂

#20266175

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#87938
commutator	C₂×C₄×C₁₀₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×#87938
Fitting, prime radical	C₂²×D₄×C₁₀₁	C₄
socle	C₂²×C₁₀₁	C₂×C₂²⋊C₄
cosocle	C₄×Dic₁₀₁	C₂³
hypercenter	C₂²×D₄	#87938
hypocenter	C₁₀₁	#1533
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×D₄

subgroup

G/H

center

C₂²

C₂×C₂²⋊#87938

commutator

C₂×C₄×C₁₀₁

C₂²×C₄

Frattini

C₂×C₄

C₂²×#87938

Fitting, prime radical

C₂²×D₄×C₁₀₁

C₄

socle

C₂²×C₁₀₁

C₂×C₂²⋊C₄

cosocle

C₄×Dic₁₀₁

C₂³

C₂²×D₄

C₁₀₁

G ▷ C₂×C₄×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×D₄

N	H ≅ G/N	type
1. #20266197	C₂	semidirect
2. #20266032	C₂	semidirect
6. #15692310	C₂²	semidirect
63. C₁₀₁	#1533	semidirect

H ≅ G/N

type

1. #20266197

C₂

semidirect

2. #20266032

C₂

semidirect

6. #15692310

C₂²

semidirect

63. C₁₀₁

#1533

semidirect

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

subgroup	index
#20266197	2
#20266032	2
C₂×#15692242	2
#1533	101

subgroup

index

101

Subgroups — #27186611

Conjugacy classes of subgroups

Complements

Maximal subgroups