Group #13078997 — subgroups

Subgroups — #13078997

Order 1408.

380 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 141 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	#10623001	C₂	1	1	yes	maximal, characteristic
2	C₂×#66922	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₁×M₁₆	C₂	1	1	yes	maximal, characteristic
4	#10622733	C₂	1	4	yes	maximal
5	#10622316	C₂	1	4	yes	maximal, complemented
6	#10622294	C₂	1	4	yes	maximal, complemented
7	C₂²×Dic₂₂	C₂²	1	2	yes	—
8	C₂×C₁₁×M₁₆	C₂²	1	2	yes	—
9	C₂×C₄²×C₁₁	C₂²	1	1	yes	characteristic
10	#66979	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₁₁	—	2	2	no	—
12	#66977	—	2	2	no	—
13	#66923	C₂²	1	2	yes	—
14	#66922	—	2	2	no	—
15	#66922	C₂²	1	2	yes	—
16	#66922	C₂²	1	2	yes	—
17	#66922	C₂²	1	2	yes	—
18	C₄×Dic₂₂	—	2	2	no	—
19	C₁₁×#97	C₂²	1	2	yes	—
20	C₄×C₈×C₁₁	C₂²	1	2	yes	—
21	#66880	C₂²	1	16	yes	—
22	C₂²×C₄×C₁₁	D₄	1	2	yes	—
23	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
24	C₂²×Dic₁₁	—	2	2	no	—
25	C₂×Dic₂₂	—	2	2	no	—
26	C₂×Dic₂₂	—	2	2	no	—
27	C₂×Dic₂₂	—	2	8	no	—
28	C₂×Dic₂₂	C₂³	1	4	yes	—
29	C₁₁×M₁₆	—	2	4	no	—
30	C₂×C₈×C₁₁	C₂³	1	4	yes	—
31	C₄²×C₁₁	D₄	1	2	yes	—
32	C₄²×C₁₁	C₂³	1	1	yes	characteristic
33	C₄²×C₁₁	C₂³	1	1	yes	characteristic
34	C₂²⋊Dic₁₁	—	2	2	no	—
35	C₄⋊Dic₁₁	—	2	2	no	—
36	C₄⋊Dic₁₁	—	2	4	no	—
37	C₄⋊Dic₁₁	—	2	4	no	—
38	C₄⋊Dic₁₁	C₂³	1	4	yes	—
39	Dic₁₁⋊C₄	—	2	4	no	—
40	C₄×Dic₁₁	—	2	2	no	—
41	#3139	—	11	1	no	maximal, Sylow
42	C₂³×C₁₁	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₁₁	—	2	4	no	—
44	C₂×C₄×C₁₁	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₁	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₁₁	#41	1	4	yes	—
48	C₂×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
49	C₂×Dic₁₁	—	2	4	no	—
50	C₂×Dic₁₁	—	2	4	no	—
51	C₂×Dic₁₁	—	4	4	no	—
52	Dic₂₂	—	2	8	no	—
53	Dic₂₂	—	4	4	no	—
54	Dic₂₂	—	4	4	no	—
55	C₈×C₁₁	—	2	4	no	—
56	#533	—	11	1	no	—
57	C₂×C₄⋊Q₈	—	11	1	no	—
58	#491	—	11	4	no	—
59	#487	—	11	4	no	—
60	#419	—	11	4	no	—
61	C₄×M₁₆	D₁₁	1	1	yes	characteristic
62	C₂²×C₁₁	—	2	2	no	—
63	C₂²×C₁₁	#124	1	2	yes	—
64	C₂²×C₁₁	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₁₁	—	2	2	no	—
66	C₄×C₁₁	#124	1	2	yes	—
67	C₄×C₁₁	#137	1	4	yes	—
68	C₄×C₁₁	C₂×#41	1	2	yes	—
69	Dic₁₁	—	4	4	no	—
70	Dic₁₁	—	4	4	no	—
71	C₂²×Q₈	—	11	2	no	—
72	C₂×M₁₆	C₂×D₁₁	1	2	yes	—
73	C₄⋊Q₈	—	11	2	no	—
74	C₄⋊Q₈	—	11	2	no	—
75	C₄⋊Q₈	—	11	2	no	—
76	C₄⋊Q₈	—	22	2	no	—
77	#125	—	11	2	no	—
78	C₂²⋊Q₈	—	22	2	no	—
79	C₄×Q₈	—	22	2	no	—
80	#117	—	11	2	no	—
81	C₂×C₄⋊C₄	—	22	2	no	—
82	C₂×C₄²	C₂×D₁₁	1	1	yes	characteristic
83	#103	—	11	16	no	—
84	#97	C₂×D₁₁	1	2	yes	—
85	C₄×C₈	C₂×D₁₁	1	2	yes	—
86	C₂×C₁₁	—	2	2	no	—
87	C₂×C₁₁	C₂×#124	1	1	yes	characteristic
88	C₂×C₁₁	C₂×#137	1	2	yes	—
89	C₂×Q₈	—	11	4	no	—
90	C₂×Q₈	—	22	2	no	—
91	C₂×Q₈	—	22	2	no	—
92	C₂×Q₈	—	22	8	no	—
93	C₂²×C₄	—	22	2	no	—
94	C₂²×C₄	D₄₄	1	2	yes	—
95	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
96	M₁₆	—	2	4	no	—
97	C₂×C₈	C₂²×D₁₁	1	4	yes	—
98	C₄⋊C₄	—	11	4	no	—
99	C₄⋊C₄	—	22	4	no	—
100	C₄⋊C₄	—	22	4	no	—
101	C₄⋊C₄	—	22	4	no	—
102	C₄⋊C₄	—	22	2	no	—
103	C₂²⋊C₄	—	22	2	no	—
104	C₄²	—	22	2	no	—
105	C₄²	D₄₄	1	2	yes	—
106	C₄²	C₂²×D₁₁	1	1	yes	characteristic
107	C₄²	C₂²×D₁₁	1	1	yes	characteristic
108	C₁₁	#3139	1	1	yes	characteristic, complemented, Sylow
109	C₂³	C₂×D₄₄	1	1	yes	characteristic
110	Q₈	—	22	8	no	—
111	Q₈	—	44	4	no	—
112	Q₈	—	44	4	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	22	4	no	—
115	C₂×C₄	—	22	4	no	—
116	C₂×C₄	—	44	4	no	—
117	C₂×C₄	C₂×D₄₄	1	2	yes	—
118	C₂×C₄	C₂×D₄₄	1	1	yes	characteristic
119	C₂×C₄	C₂×D₄₄	1	2	yes	—
120	C₂×C₄	#4339	1	4	yes	—
121	C₂×C₄	C₂³×D₁₁	1	1	yes	characteristic
122	C₈	—	2	4	no	—
123	C₂²	—	2	2	no	—
124	C₂²	#66928	1	2	yes	—
125	C₂²	C₂²×D₄₄	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	44	4	no	—
128	C₄	—	44	4	no	—
129	C₄	#66928	1	2	yes	—
130	C₄	#66962	1	4	yes	—
131	C₄	C₂×#4339	1	2	yes	—
132	C₂	—	2	2	no	—
133	C₂	C₂×#66928	1	1	yes	characteristic
134	C₂	C₂×#66962	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10623001

C₂

yes

maximal, characteristic

C₂×#66922

C₂

yes

maximal, characteristic

C₄×C₁₁×M₁₆

C₂

yes

maximal, characteristic

#10622733

C₂

yes

maximal

#10622316

C₂

yes

maximal, complemented

#10622294

C₂

yes

maximal, complemented

C₂²×Dic₂₂

C₂²

yes

—

C₂×C₁₁×M₁₆

C₂²

yes

—

C₂×C₄²×C₁₁

C₂²

yes

characteristic

#66979

C₂²

yes

—

C₂×C₄⋊Dic₁₁

—

#66977

—

#66923

C₂²

yes

—

#66922

—

#66922

C₂²

yes

—

#66922

C₂²

yes

—

#66922

C₂²

yes

—

C₄×Dic₂₂

—

C₁₁×#97

C₂²

yes

—

C₄×C₈×C₁₁

C₂²

yes

—

#66880

C₂²

yes

—

C₂²×C₄×C₁₁

D₄

yes

—

C₂²×C₄×C₁₁

C₂³

yes

characteristic

C₂²×Dic₁₁

—

C₂×Dic₂₂

—

C₂×Dic₂₂

—

C₂×Dic₂₂

—

C₂×Dic₂₂

C₂³

yes

—

C₁₁×M₁₆

—

C₂×C₈×C₁₁

C₂³

yes

—

C₄²×C₁₁

D₄

yes

—

C₄²×C₁₁

C₂³

yes

characteristic

C₄²×C₁₁

C₂³

yes

characteristic

C₂²⋊Dic₁₁

—

C₄⋊Dic₁₁

—

C₄⋊Dic₁₁

—

C₄⋊Dic₁₁

—

C₄⋊Dic₁₁

C₂³

yes

—

Dic₁₁⋊C₄

—

C₄×Dic₁₁

—

#3139

—

maximal, Sylow

C₂³×C₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

C₂×D₄

yes

—

C₂×C₄×C₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₁

C₂×D₄

yes

—

C₂×C₄×C₁₁

#41

yes

—

C₂×C₄×C₁₁

C₂⁴

yes

characteristic

C₂×Dic₁₁

—

C₂×Dic₁₁

—

C₂×Dic₁₁

—

Dic₂₂

—

Dic₂₂

—

Dic₂₂

—

C₈×C₁₁

—

#533

—

C₂×C₄⋊Q₈

—

#491

—

#487

—

#419

—

C₄×M₁₆

D₁₁

yes

characteristic

C₂²×C₁₁

—

C₂²×C₁₁

#124

yes

—

C₂²×C₁₁

C₂²×D₄

yes

characteristic

C₄×C₁₁

—

C₄×C₁₁

#124

yes

—

C₄×C₁₁

#137

yes

—

C₄×C₁₁

C₂×#41

yes

—

Dic₁₁

—

Dic₁₁

—

C₂²×Q₈

—

C₂×M₁₆

C₂×D₁₁

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₁₁

yes

characteristic

#103

—

#97

C₂×D₁₁

yes

—

C₄×C₈

C₂×D₁₁

yes

—

C₂×C₁₁

—

C₂×C₁₁

C₂×#124

yes

characteristic

C₂×C₁₁

C₂×#137

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

D₄₄

yes

—

C₂²×C₄

C₂²×D₁₁

yes

characteristic

M₁₆

—

C₂×C₈

C₂²×D₁₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₂²⋊C₄

—

104

C₄²

—

105

C₄²

D₄₄

yes

—

106

C₄²

C₂²×D₁₁

yes

characteristic

107

C₄²

C₂²×D₁₁

yes

characteristic

108

C₁₁

#3139

yes

characteristic, complemented, Sylow

109

C₂³

C₂×D₄₄

yes

characteristic

110

Q₈

—

111

Q₈

—

112

Q₈

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₂×D₄₄

yes

—

118

C₂×C₄

C₂×D₄₄

yes

characteristic

119

C₂×C₄

C₂×D₄₄

yes

—

120

C₂×C₄

#4339

yes

—

121

C₂×C₄

C₂³×D₁₁

yes

characteristic

122

C₈

—

123

C₂²

—

124

C₂²

#66928

yes

—

125

C₂²

C₂²×D₄₄

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

#66928

yes

—

130

C₄

#66962

yes

—

131

C₄

C₂×#4339

yes

—

132

C₂

—

133

C₂

C₂×#66928

yes

characteristic

134

C₂

C₂×#66962

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₄₄
commutator, cosocle	C₂×C₄×C₁₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₁
Fitting	C₄×C₁₁×M₁₆	C₂
socle	C₂²×C₁₁	C₂²×D₄
prime radical	C₂³×C₁₁	C₂×D₄
hypercenter	C₄×M₁₆	D₁₁
hypocenter	C₁₁	#3139
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂²

C₂²×D₄₄

commutator, cosocle

C₂×C₄×C₁₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₁

Fitting

C₄×C₁₁×M₁₆

C₂

socle

C₂²×C₁₁

C₂²×D₄

prime radical

C₂³×C₁₁

C₂×D₄

hypercenter

C₄×M₁₆

D₁₁

hypocenter

C₁₁

#3139

LCS

G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
5. #10622316	C₂	semidirect
6. #10622294	C₂	semidirect
108. C₁₁	#3139	semidirect

H ≅ G/N

type

5. #10622316

C₂

semidirect

6. #10622294

C₂

semidirect

108. C₁₁

#3139

semidirect

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

subgroup	index
#10623001	2
C₂×#66922	2
C₄×C₁₁×M₁₆	2
#10622733	2
#10622316	2
#10622294	2
#3139	11

subgroup

index

C₄×C₁₁×M₁₆

Subgroups — #13078997

Conjugacy classes of subgroups

Complements

Maximal subgroups