Group #13069557 — subgroups

Subgroups — #13069557

Order 1408.

280 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 121 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×C₄×Q₈×C₁₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₁₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×#66867	C₂	1	1	yes	maximal, characteristic
4	#10622162	C₂	1	4	yes	maximal
5	C₂²×Q₈×C₁₁	C₄	1	1	yes	characteristic
6	C₄×Q₈×C₁₁	—	2	2	no	—
7	C₄×Q₈×C₁₁	—	2	2	no	—
8	C₂×C₁₁×C₄⋊C₄	C₄	1	1	yes	characteristic
9	C₂×C₁₁×C₄⋊C₄	C₄	1	2	yes	—
10	C₂×C₄²×C₁₁	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₁	C₂²	1	1	yes	characteristic
12	C₂²×C₁₁⋊C₈	C₂²	1	2	yes	—
13	C₂²×C₁₁⋊C₈	C₂²	1	2	yes	—
14	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
15	#66867	—	2	2	no	—
16	C₂×Q₈×C₁₁	—	2	2	no	—
17	C₂×Q₈×C₁₁	—	2	4	no	—
18	C₂²×C₄×C₁₁	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₁	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
23	C₁₁×C₄⋊C₄	—	2	4	no	—
24	C₁₁×C₄⋊C₄	—	2	2	no	—
25	C₄²×C₁₁	—	2	2	no	—
26	C₄²×C₁₁	—	2	2	no	—
27	C₄²×C₁₁	D₄	1	2	yes	—
28	C₄²×C₁₁	D₄	1	2	yes	—
29	C₂×C₁₁⋊C₈	—	2	2	no	—
30	C₂×C₁₁⋊C₈	—	2	2	no	—
31	C₂×C₁₁⋊C₈	—	2	2	no	—
32	C₂×C₁₁⋊C₈	—	2	4	no	—
33	C₂×C₁₁⋊C₈	Q₈	1	4	yes	complemented
34	#1985	—	11	1	no	maximal, Sylow
35	C₂³×C₁₁	C₂²×C₄	1	1	yes	characteristic
36	Q₈×C₁₁	—	2	8	no	—
37	C₂×C₄×C₁₁	—	2	2	no	—
38	C₂×C₄×C₁₁	—	2	2	no	—
39	C₂×C₄×C₁₁	—	2	2	no	—
40	C₂×C₄×C₁₁	—	2	2	no	—
41	C₂×C₄×C₁₁	—	2	2	no	—
42	C₂×C₄×C₁₁	—	2	4	no	—
43	C₂×C₄×C₁₁	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₁₁	C₂²⋊C₄	1	2	yes	—
45	C₂×C₄×C₁₁	M₁₆	1	4	yes	—
46	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₁	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₁₁	#41	1	2	yes	—
50	C₁₁⋊C₈	—	2	4	no	—
51	C₁₁⋊C₈	—	4	2	no	—
52	C₂×C₄×Q₈	D₁₁	1	1	yes	characteristic
53	C₂×C₄⋊C₈	—	11	1	no	—
54	C₂×#97	—	11	1	no	—
55	#336	—	11	4	no	—
56	C₂²×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₁₁	C₄×Q₈	1	2	yes	—
58	C₂²×C₁₁	C₂×M₁₆	1	2	yes	—
59	C₂²×C₁₁	#131	1	2	yes	—
60	C₄×C₁₁	—	2	2	no	—
61	C₄×C₁₁	—	2	4	no	—
62	C₄×C₁₁	—	2	2	no	—
63	C₄×C₁₁	—	4	2	no	—
64	C₄×C₁₁	C₂²⋊Q₈	1	2	yes	—
65	C₄×C₁₁	#124	1	1	yes	characteristic
66	C₄×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
67	C₂²×Q₈	Dic₁₁	1	1	yes	characteristic
68	C₂²×C₈	—	11	2	no	—
69	C₂²×C₈	—	11	2	no	—
70	C₄×Q₈	—	2	2	no	—
71	C₄×Q₈	—	2	2	no	—
72	C₂×C₄⋊C₄	Dic₁₁	1	1	yes	characteristic
73	C₂×C₄⋊C₄	Dic₁₁	1	2	yes	—
74	C₂×C₄²	C₂×D₁₁	1	2	yes	—
75	C₂×C₄²	C₂×D₁₁	1	1	yes	characteristic
76	C₄⋊C₈	—	22	2	no	—
77	#97	—	22	2	no	—
78	C₂×C₁₁	#391	1	1	yes	characteristic
79	C₂×C₁₁	#407	1	1	yes	characteristic
80	C₂×C₁₁	#408	1	1	yes	characteristic
81	C₂×C₁₁	#446	1	4	yes	—
82	C₂×Q₈	—	2	4	no	—
83	C₂×Q₈	—	2	2	no	—
84	C₂²×C₄	C₂×Dic₁₁	1	2	yes	—
85	C₂²×C₄	C₂×Dic₁₁	1	2	yes	—
86	C₂²×C₄	C₂×Dic₁₁	1	1	yes	characteristic
87	C₂²×C₄	C₂×Dic₁₁	1	1	yes	characteristic
88	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
89	C₂×C₈	—	11	4	no	—
90	C₂×C₈	—	22	2	no	—
91	C₂×C₈	—	22	2	no	—
92	C₂×C₈	—	22	4	no	—
93	C₂×C₈	—	22	2	no	—
94	C₄⋊C₄	—	2	4	no	—
95	C₄⋊C₄	—	2	2	no	—
96	C₄²	—	2	2	no	—
97	C₄²	—	2	2	no	—
98	C₄²	C₂²⋊D₁₁	1	2	yes	—
99	C₄²	C₂²⋊D₁₁	1	2	yes	—
100	C₁₁	#1985	1	1	yes	characteristic, complemented, Sylow
101	C₂³	C₂²×Dic₁₁	1	1	yes	characteristic
102	Q₈	—	2	8	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	#4318	1	4	yes	—
110	C₂×C₄	C₂²⋊Dic₁₁	1	2	yes	—
111	C₂×C₄	C₂²⋊Dic₁₁	1	2	yes	—
112	C₂×C₄	Q₈×D₁₁	1	2	yes	—
113	C₂×C₄	#4343	1	2	yes	—
114	C₂×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
115	C₂×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
116	C₈	—	22	4	no	—
117	C₈	—	44	2	no	—
118	C₂²	C₂×#4318	1	2	yes	—
119	C₂²	Q₈×Dic₁₁	1	2	yes	—
120	C₂²	#67001	1	2	yes	—
121	C₂²	C₂×C₂²⋊Dic₁₁	1	1	yes	characteristic
122	C₄	—	2	4	no	—
123	C₄	—	2	2	no	—
124	C₄	—	2	2	no	—
125	C₄	—	4	2	no	—
126	C₄	#66997	1	1	yes	characteristic
127	C₄	#66999	1	2	yes	—
128	C₄	#67000	1	1	yes	characteristic
129	C₂	#10622637	1	4	yes	—
130	C₂	#10622820	1	1	yes	characteristic
131	C₂	#10622843	1	1	yes	characteristic
132	C₂	#10622847	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Q₈×C₁₁

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₁₁⋊C₈)

C₂

yes

maximal, characteristic

C₂×#66867

C₂

yes

maximal, characteristic

#10622162

C₂

yes

maximal

C₂²×Q₈×C₁₁

C₄

yes

characteristic

C₄×Q₈×C₁₁

—

C₄×Q₈×C₁₁

—

C₂×C₁₁×C₄⋊C₄

C₄

yes

characteristic

C₂×C₁₁×C₄⋊C₄

C₄

yes

—

C₂×C₄²×C₁₁

C₂²

yes

—

C₂×C₄²×C₁₁

C₂²

yes

characteristic

C₂²×C₁₁⋊C₈

C₂²

yes

—

C₂²×C₁₁⋊C₈

C₂²

yes

—

C₄⋊(C₁₁⋊C₈)

—

#66867

—

C₂×Q₈×C₁₁

—

C₂×Q₈×C₁₁

—

C₂²×C₄×C₁₁

C₂×C₄

yes

—

C₂²×C₄×C₁₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₁

C₂×C₄

yes

—

C₂²×C₄×C₁₁

C₂³

yes

characteristic

C₁₁×C₄⋊C₄

—

C₁₁×C₄⋊C₄

—

C₄²×C₁₁

—

C₄²×C₁₁

—

C₄²×C₁₁

D₄

yes

—

C₄²×C₁₁

D₄

yes

—

C₂×C₁₁⋊C₈

—

C₂×C₁₁⋊C₈

—

C₂×C₁₁⋊C₈

—

C₂×C₁₁⋊C₈

—

C₂×C₁₁⋊C₈

Q₈

yes

complemented

#1985

—

maximal, Sylow

C₂³×C₁₁

C₂²×C₄

yes

characteristic

Q₈×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₁

M₁₆

yes

—

C₂×C₄×C₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₁

C₂×Q₈

yes

—

C₂×C₄×C₁₁

#41

yes

—

C₁₁⋊C₈

—

C₁₁⋊C₈

—

C₂×C₄×Q₈

D₁₁

yes

characteristic

C₂×C₄⋊C₈

—

C₂×#97

—

#336

—

C₂²×C₁₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₁

C₄×Q₈

yes

—

C₂²×C₁₁

C₂×M₁₆

yes

—

C₂²×C₁₁

#131

yes

—

C₄×C₁₁

—

C₄×C₁₁

—

C₄×C₁₁

—

C₄×C₁₁

—

C₄×C₁₁

C₂²⋊Q₈

yes

—

C₄×C₁₁

#124

yes

characteristic

C₄×C₁₁

C₄⋊Q₈

yes

characteristic

C₂²×Q₈

Dic₁₁

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

Dic₁₁

yes

characteristic

C₂×C₄⋊C₄

Dic₁₁

yes

—

C₂×C₄²

C₂×D₁₁

yes

—

C₂×C₄²

C₂×D₁₁

yes

characteristic

C₄⋊C₈

—

#97

—

C₂×C₁₁

yes

C₂×C₁₁

yes

C₂×C₁₁

yes

C₂×C₁₁

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₂×Dic₁₁

yes

—

C₂²×C₄

C₂×Dic₁₁

yes

—

C₂²×C₄

C₂×Dic₁₁

yes

characteristic

C₂²×C₄

C₂×Dic₁₁

yes

characteristic

C₂²×C₄

C₂²×D₁₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²⋊D₁₁

yes

—

C₄²

C₂²⋊D₁₁

yes

—

100

C₁₁

#1985

yes

characteristic, complemented, Sylow

101

C₂³

C₂²×Dic₁₁

yes

characteristic

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

#4318

yes

—

110

C₂×C₄

C₂²⋊Dic₁₁

yes

—

111

C₂×C₄

C₂²⋊Dic₁₁

yes

—

112

C₂×C₄

Q₈×D₁₁

yes

—

113

C₂×C₄

#4343

yes

—

114

C₂×C₄

C₂×C₂²⋊D₁₁

yes

characteristic

115

C₂×C₄

C₂×C₂²⋊D₁₁

yes

characteristic

116

C₈

—

117

C₈

—

118

C₂²

C₂×#4318

yes

—

119

C₂²

Q₈×Dic₁₁

yes

—

120

C₂²

#67001

yes

—

121

C₂²

C₂×C₂²⋊Dic₁₁

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

#66997

yes

characteristic

127

C₄

#66999

yes

—

128

C₄

#67000

yes

characteristic

129

C₂

#10622637

yes

—

130

C₂

#10622820

yes

characteristic

131

C₂

#10622843

yes

characteristic

132

C₂

#10622847

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₁
commutator, socle, prime radical	C₂³×C₁₁	C₂²×C₄
Fitting	C₂×C₄×Q₈×C₁₁	C₂
cosocle	C₂²×C₄×C₁₁	C₂³
hypercenter	C₂×C₄×Q₈	D₁₁
hypocenter	C₁₁	#1985
LCS	G ▷ C₂³×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₁

commutator, socle, prime radical

C₂³×C₁₁

C₂²×C₄

Fitting

C₂×C₄×Q₈×C₁₁

C₂

cosocle

C₂²×C₄×C₁₁

C₂³

hypercenter

C₂×C₄×Q₈

D₁₁

C₁₁

G ▷ C₂³×C₁₁ ▷ C₁₁

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
33. C₂×C₁₁⋊C₈	Q₈	semidirect
100. C₁₁	#1985	semidirect

H ≅ G/N

type

33. C₂×C₁₁⋊C₈

Q₈

semidirect

100. C₁₁

#1985

semidirect

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

subgroup	index
C₂×C₄×Q₈×C₁₁	2
C₂×C₄⋊(C₁₁⋊C₈)	2
C₂×#66867	2
#10622162	2
#1985	11

subgroup

index

C₂×C₄×Q₈×C₁₁

C₂×C₄⋊(C₁₁⋊C₈)

C₂×#66867

#10622162

#1985

Subgroups — #13069557

Conjugacy classes of subgroups

Complements

Maximal subgroups