Group #13078996 — subgroups

Subgroups — #13078996

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	C₂×C₁₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#10623001	C₂	1	1	yes	maximal, characteristic
3	#10622838	C₂	1	4	yes	maximal
4	#10622695	C₂	1	4	yes	maximal, complemented
5	#10622694	C₂	1	4	yes	maximal, complemented
6	C₄×#4318	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₈×C₁₁	C₂²	1	2	yes	—
8	C₁₁×C₄⋊Q₈	—	2	2	no	—
9	C₁₁×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₁₁×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
12	C₂×C₄²×C₁₁	C₂²	1	1	yes	characteristic
13	#66979	C₂²	1	2	yes	—
14	#66977	—	2	2	no	—
15	C₂×#4318	C₂²	1	2	yes	—
16	#66923	C₂²	1	2	yes	—
17	#66922	C₂²	1	2	yes	—
18	C₄×Dic₂₂	—	2	2	no	—
19	#66899	C₂²	1	16	yes	—
20	#66867	C₂²	1	2	yes	—
21	C₄×C₁₁⋊C₈	C₂²	1	2	yes	—
22	C₂×Q₈×C₁₁	—	2	2	no	—
23	C₂×Q₈×C₁₁	—	2	8	no	—
24	C₂×Q₈×C₁₁	C₂³	1	4	yes	—
25	C₂²×C₄×C₁₁	—	2	2	no	—
26	C₂²×C₄×C₁₁	D₄	1	2	yes	—
27	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
28	C₂×Dic₂₂	—	2	2	no	—
29	C₁₁×C₄⋊C₄	—	2	4	no	—
30	C₁₁×C₄⋊C₄	—	2	4	no	—
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	C₂³	1	4	yes	—
37	Dic₁₁⋊C₄	—	2	4	no	—
38	C₄×Dic₁₁	—	2	2	no	—
39	#4318	—	2	4	no	—
40	C₂×C₁₁⋊C₈	C₂³	1	4	yes	—
41	#3139	—	11	1	no	maximal, Sylow
42	C₂³×C₁₁	C₂×D₄	1	1	yes	characteristic
43	Q₈×C₁₁	—	2	8	no	—
44	Q₈×C₁₁	—	4	4	no	—
45	C₂×C₄×C₁₁	—	2	4	no	—
46	C₂×C₄×C₁₁	—	2	4	no	—
47	C₂×C₄×C₁₁	—	4	4	no	—
48	C₂×C₄×C₁₁	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₁	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₁₁	#41	1	4	yes	—
52	C₂×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
53	C₂×Dic₁₁	—	2	4	no	—
54	Dic₂₂	—	4	4	no	—
55	C₁₁⋊C₈	—	2	4	no	—
56	#533	—	11	1	no	—
57	C₂×C₄⋊Q₈	D₁₁	1	1	yes	characteristic
58	#491	—	11	4	no	—
59	#487	—	11	4	no	—
60	#419	—	11	4	no	—
61	C₄×M₁₆	—	11	1	no	—
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	—	4	4	no	—
67	C₄×C₁₁	#124	1	2	yes	—
68	C₄×C₁₁	#137	1	4	yes	—
69	C₄×C₁₁	C₂×#41	1	2	yes	—
70	Dic₁₁	—	4	4	no	—
71	C₂²×Q₈	C₂×D₁₁	1	2	yes	—
72	C₂×M₁₆	—	11	2	no	—
73	C₄⋊Q₈	—	2	2	no	—
74	C₄⋊Q₈	—	11	2	no	—
75	C₄⋊Q₈	C₂×D₁₁	1	2	yes	—
76	C₄⋊Q₈	C₂×D₁₁	1	2	yes	—
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	—	2	2	no	—
82	C₂×C₄²	C₂×D₁₁	1	1	yes	characteristic
83	#103	—	11	16	no	—
84	#97	—	11	2	no	—
85	C₄×C₈	—	11	2	no	—
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	—	2	2	no	—
90	C₂×Q₈	—	2	8	no	—
91	C₂×Q₈	—	22	2	no	—
92	C₂×Q₈	C₂²×D₁₁	1	4	yes	—
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	C₂²⋊D₁₁	1	2	yes	—
95	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
96	M₁₆	—	22	4	no	—
97	C₂×C₈	—	11	4	no	—
98	C₄⋊C₄	—	2	4	no	—
99	C₄⋊C₄	—	2	4	no	—
100	C₄⋊C₄	—	11	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	C₂²⋊D₁₁	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×C₂²⋊D₁₁	1	1	yes	characteristic
110	Q₈	—	2	8	no	—
111	Q₈	—	4	4	no	—
112	Q₈	—	44	4	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	4	4	no	—
116	C₂×C₄	—	22	4	no	—
117	C₂×C₄	#4341	1	4	yes	—
118	C₂×C₄	C₂×C₂²⋊D₁₁	1	2	yes	—
119	C₂×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
120	C₂×C₄	C₂×C₂²⋊D₁₁	1	2	yes	—
121	C₂×C₄	C₂³×D₁₁	1	1	yes	characteristic
122	C₈	—	22	4	no	—
123	C₂²	—	2	2	no	—
124	C₂²	#66989	1	2	yes	—
125	C₂²	C₂²×C₂²⋊D₁₁	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	4	4	no	—
128	C₄	—	44	4	no	—
129	C₄	#66989	1	2	yes	—
130	C₄	#66995	1	4	yes	—
131	C₄	C₂×#4341	1	2	yes	—
132	C₂	—	2	2	no	—
133	C₂	C₂×#66989	1	1	yes	characteristic
134	C₂	C₂×#66995	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₁×C₄⋊Q₈

C₂

yes

maximal, characteristic

#10623001

C₂

yes

maximal, characteristic

#10622838

C₂

yes

maximal

#10622695

C₂

yes

maximal, complemented

#10622694

C₂

yes

maximal, complemented

C₄×#4318

C₂

yes

maximal, characteristic

C₂²×Q₈×C₁₁

C₂²

yes

—

C₁₁×C₄⋊Q₈

—

C₁₁×C₄⋊Q₈

C₂²

yes

—

C₁₁×C₄⋊Q₈

C₂²

yes

—

C₂×C₁₁×C₄⋊C₄

—

C₂×C₄²×C₁₁

C₂²

yes

characteristic

#66979

C₂²

yes

—

#66977

—

C₂×#4318

C₂²

yes

—

#66923

C₂²

yes

—

#66922

C₂²

yes

—

C₄×Dic₂₂

—

#66899

C₂²

yes

—

#66867

C₂²

yes

—

C₄×C₁₁⋊C₈

C₂²

yes

—

C₂×Q₈×C₁₁

—

C₂×Q₈×C₁₁

—

C₂×Q₈×C₁₁

C₂³

yes

—

C₂²×C₄×C₁₁

—

C₂²×C₄×C₁₁

D₄

yes

—

C₂²×C₄×C₁₁

C₂³

yes

characteristic

C₂×Dic₂₂

—

C₁₁×C₄⋊C₄

—

C₁₁×C₄⋊C₄

—

C₄²×C₁₁

D₄

yes

—

C₄²×C₁₁

C₂³

yes

characteristic

C₄²×C₁₁

C₂³

yes

characteristic

C₂²⋊Dic₁₁

—

C₄⋊Dic₁₁

—

C₄⋊Dic₁₁

C₂³

yes

—

Dic₁₁⋊C₄

—

C₄×Dic₁₁

—

#4318

—

C₂×C₁₁⋊C₈

C₂³

yes

—

#3139

—

maximal, Sylow

C₂³×C₁₁

C₂×D₄

yes

characteristic

Q₈×C₁₁

—

Q₈×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

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

—

Dic₂₂

—

C₁₁⋊C₈

—

#533

—

C₂×C₄⋊Q₈

D₁₁

yes

—

—

—

C₄×M₁₆

—

C₂²×C₁₁

—

C₂²×C₁₁

#124

yes

—

C₂²×C₁₁

C₂²×D₄

yes

characteristic

C₄×C₁₁

—

C₄×C₁₁

—

C₄×C₁₁

#124

yes

—

C₄×C₁₁

#137

yes

—

C₄×C₁₁

C₂×#41

yes

—

Dic₁₁

—

C₂²×Q₈

C₂×D₁₁

yes

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₁₁

yes

—

C₄⋊Q₈

C₂×D₁₁

yes

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₁₁

yes

characteristic

#103

—

#97

—

C₄×C₈

—

C₂×C₁₁

—

C₂×C₁₁

C₂×#124

yes

characteristic

C₂×C₁₁

C₂×#137

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²×D₁₁

yes

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₁

yes

—

C₂²×C₄

C₂²×D₁₁

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₂²⋊C₄

—

104

C₄²

—

105

C₄²

C₂²⋊D₁₁

yes

—

106

C₄²

C₂²×D₁₁

yes

characteristic

107

C₄²

C₂²×D₁₁

yes

characteristic

108

C₁₁

#3139

yes

characteristic, complemented, Sylow

109

C₂³

C₂×C₂²⋊D₁₁

yes

characteristic

110

Q₈

—

111

Q₈

—

112

Q₈

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

#4341

yes

—

118

C₂×C₄

C₂×C₂²⋊D₁₁

yes

—

119

C₂×C₄

C₂×C₂²⋊D₁₁

yes

characteristic

120

C₂×C₄

C₂×C₂²⋊D₁₁

yes

—

121

C₂×C₄

C₂³×D₁₁

yes

characteristic

122

C₈

—

123

C₂²

—

124

C₂²

#66989

yes

—

125

C₂²

C₂²×C₂²⋊D₁₁

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

#66989

yes

—

130

C₄

#66995

yes

—

131

C₄

C₂×#4341

yes

—

132

C₂

—

133

C₂

C₂×#66989

yes

characteristic

134

C₂

C₂×#66995

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₁
commutator, cosocle	C₂×C₄×C₁₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₁
Fitting	C₂×C₁₁×C₄⋊Q₈	C₂
socle	C₂²×C₁₁	C₂²×D₄
prime radical	C₂³×C₁₁	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₁₁
hypocenter	C₁₁	#3139
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₁

commutator, cosocle

C₂×C₄×C₁₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₁

Fitting

C₂×C₁₁×C₄⋊Q₈

C₂

socle

C₂²×C₁₁

C₂²×D₄

prime radical

C₂³×C₁₁

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₁₁

hypocenter

C₁₁

#3139

LCS

G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
4. #10622695	C₂	semidirect
5. #10622694	C₂	semidirect
108. C₁₁	#3139	semidirect

H ≅ G/N

type

4. #10622695

C₂

semidirect

5. #10622694

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
C₂×C₁₁×C₄⋊Q₈	2
#10623001	2
#10622838	2
#10622695	2
#10622694	2
C₄×#4318	2
#3139	11

subgroup

index

C₂×C₁₁×C₄⋊Q₈

Subgroups — #13078996

Conjugacy classes of subgroups

Complements

Maximal subgroups