Group #13069764 — subgroups

Subgroups — #13069764

Order 1408.

258 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 87 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×#125	C₂	1	1	yes	maximal, characteristic
2	C₂×#66874	C₂	1	2	yes	maximal
3	#10622542	C₂	1	1	yes	maximal, characteristic
4	#10622167	C₂	1	2	yes	maximal
5	#10622162	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₂	C₂²	1	1	yes	characteristic
7	C₁₁×#125	—	2	4	no	—
8	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
9	C₂×C₁₁×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂×C₄²×C₁₁	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₁₁	C₂²	1	1	yes	characteristic
12	C₂²×C₁₁⋊C₈	C₂²	1	2	yes	—
13	C₁₁⋊#95	—	2	2	no	—
14	#66874	—	2	4	no	—
15	C₂²×C₄×C₁₁	—	2	2	no	—
16	C₂²×C₄×C₁₁	D₄	1	2	yes	—
17	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
18	C₂²×Dic₁₁	—	2	1	no	—
19	C₂²×Dic₁₁	—	2	1	no	—
20	C₂×Dic₂₂	—	2	2	no	—
21	C₂×Dic₂₂	—	2	2	no	—
22	C₂×Dic₂₂	—	2	1	no	—
23	C₂×Dic₂₂	Q₈	1	2	yes	—
24	C₁₁×C₄⋊C₄	—	2	8	no	—
25	C₁₁×C₄⋊C₄	—	2	2	no	—
26	C₁₁×C₄⋊C₄	D₄	1	4	yes	—
27	C₄²×C₁₁	—	2	2	no	—
28	C₄⋊Dic₁₁	—	2	2	no	—
29	C₂×C₁₁⋊C₈	—	2	2	no	—
30	C₂×C₁₁⋊C₈	—	2	2	no	—
31	C₂×C₁₁⋊C₈	—	2	2	no	—
32	#2026	—	11	1	no	maximal, Sylow
33	C₂³×C₁₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₁	—	2	2	no	—
35	C₂×C₄×C₁₁	—	2	4	no	—
36	C₂×C₄×C₁₁	—	2	2	no	—
37	C₂×C₄×C₁₁	—	2	2	no	—
38	C₂×C₄×C₁₁	—	4	2	no	—
39	C₂×C₄×C₁₁	—	4	2	no	—
40	C₂×C₄×C₁₁	C₂×D₄	1	2	yes	—
41	C₂×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₁₁	#41	1	2	yes	—
43	C₂×C₄×C₁₁	#41	1	1	yes	characteristic
44	C₂×Dic₁₁	—	2	2	no	—
45	C₂×Dic₁₁	—	4	1	no	—
46	C₂×Dic₁₁	—	4	1	no	—
47	C₂×Dic₁₁	—	4	1	no	—
48	C₂×Dic₁₁	—	4	2	no	—
49	Dic₂₂	—	2	4	no	—
50	Dic₂₂	—	4	2	no	—
51	C₁₁⋊C₈	—	4	2	no	—
52	C₂×#125	D₁₁	1	1	yes	characteristic
53	C₂×#103	—	11	2	no	—
54	#391	—	11	1	no	—
55	#340	—	11	2	no	—
56	#336	—	11	1	no	—
57	C₂²×C₁₁	#120	1	1	yes	characteristic
58	C₂²×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
59	C₂²×C₁₁	#124	1	1	yes	characteristic
60	C₂²×C₁₁	#135	1	2	yes	—
61	C₂²×C₁₁	#137	1	2	yes	—
62	C₄×C₁₁	—	4	2	no	—
63	C₄×C₁₁	—	4	4	no	—
64	C₄×C₁₁	C₂²⋊Q₈	1	2	yes	—
65	C₄×C₁₁	#124	1	2	yes	—
66	Dic₁₁	—	4	2	no	—
67	Dic₁₁	—	8	1	no	—
68	C₂²×Q₈	—	11	1	no	—
69	C₂²×C₈	—	11	2	no	—
70	#125	—	2	4	no	—
71	C₂×C₄⋊C₄	—	2	2	no	—
72	C₂×C₄⋊C₄	—	11	1	no	—
73	C₂×C₄⋊C₄	C₂×D₁₁	1	2	yes	—
74	C₂×C₄²	C₂×D₁₁	1	1	yes	characteristic
75	#103	—	22	4	no	—
76	#95	—	22	2	no	—
77	C₂×C₁₁	#393	1	1	yes	characteristic
78	C₂×C₁₁	#452	1	2	yes	—
79	C₂×C₁₁	#479	1	2	yes	—
80	C₂×C₁₁	#488	1	1	yes	characteristic
81	C₂×C₁₁	#491	1	1	yes	characteristic
82	C₂×Q₈	—	11	2	no	—
83	C₂×Q₈	—	22	1	no	—
84	C₂×Q₈	—	22	2	no	—
85	C₂×Q₈	—	22	2	no	—
86	C₂²×C₄	—	2	2	no	—
87	C₂²×C₄	—	22	1	no	—
88	C₂²×C₄	—	22	1	no	—
89	C₂²×C₄	C₂²⋊D₁₁	1	2	yes	—
90	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
91	C₂×C₈	—	22	2	no	—
92	C₂×C₈	—	22	2	no	—
93	C₂×C₈	—	22	2	no	—
94	C₄⋊C₄	—	2	8	no	—
95	C₄⋊C₄	—	2	2	no	—
96	C₄⋊C₄	—	22	2	no	—
97	C₄⋊C₄	C₂²⋊D₁₁	1	4	yes	—
98	C₄²	—	2	2	no	—
99	C₁₁	#2026	1	1	yes	characteristic, complemented, Sylow
100	C₂³	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
101	Q₈	—	22	4	no	—
102	Q₈	—	44	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	22	2	no	—
110	C₂×C₄	—	44	1	no	—
111	C₂×C₄	—	44	1	no	—
112	C₂×C₄	—	44	1	no	—
113	C₂×C₄	—	44	2	no	—
114	C₂×C₄	Q₈×D₁₁	1	1	yes	characteristic
115	C₂×C₄	#4343	1	2	yes	—
116	C₂×C₄	#4343	1	1	yes	characteristic
117	C₂×C₄	C₂×C₂²⋊D₁₁	1	2	yes	—
118	C₈	—	44	2	no	—
119	C₂²	#66999	1	1	yes	characteristic
120	C₂²	#67000	1	1	yes	characteristic
121	C₂²	#67003	1	2	yes	—
122	C₂²	#67004	1	2	yes	—
123	C₂²	#67006	1	1	yes	characteristic
124	C₄	—	4	2	no	—
125	C₄	—	4	4	no	—
126	C₄	—	44	2	no	—
127	C₄	—	88	1	no	—
128	C₄	#66999	1	2	yes	—
129	C₄	#67000	1	2	yes	—
130	C₂	#10622676	1	2	yes	—
131	C₂	#10622681	1	1	yes	characteristic
132	C₂	#10622682	1	1	yes	characteristic
133	C₂	#10622844	1	1	yes	characteristic
134	C₂	#10622852	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₁×#125

C₂

yes

maximal, characteristic

C₂×#66874

C₂

yes

maximal

#10622542

C₂

yes

maximal, characteristic

#10622167

C₂

yes

maximal

#10622162

C₂

yes

maximal, characteristic

C₂²×Dic₂₂

C₂²

yes

characteristic

C₁₁×#125

—

C₂×C₁₁×C₄⋊C₄

—

C₂×C₁₁×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₁₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₁

C₂²

yes

characteristic

C₂²×C₁₁⋊C₈

C₂²

yes

—

C₁₁⋊#95

—

#66874

—

C₂²×C₄×C₁₁

—

C₂²×C₄×C₁₁

D₄

yes

—

C₂²×C₄×C₁₁

C₂³

yes

characteristic

C₂²×Dic₁₁

—

C₂²×Dic₁₁

—

C₂×Dic₂₂

—

C₂×Dic₂₂

—

C₂×Dic₂₂

—

C₂×Dic₂₂

Q₈

yes

—

C₁₁×C₄⋊C₄

—

C₁₁×C₄⋊C₄

—

C₁₁×C₄⋊C₄

D₄

yes

—

C₄²×C₁₁

—

C₄⋊Dic₁₁

—

C₂×C₁₁⋊C₈

—

C₂×C₁₁⋊C₈

—

C₂×C₁₁⋊C₈

—

#2026

—

maximal, Sylow

C₂³×C₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

C₂×D₄

yes

—

C₂×C₄×C₁₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₁

#41

yes

—

C₂×C₄×C₁₁

#41

yes

characteristic

C₂×Dic₁₁

—

C₂×Dic₁₁

—

C₂×Dic₁₁

—

C₂×Dic₁₁

—

C₂×Dic₁₁

—

Dic₂₂

—

Dic₂₂

—

C₁₁⋊C₈

—

C₂×#125

D₁₁

yes

—

—

—

—

C₂²×C₁₁

#120

yes

characteristic

C₂²×C₁₁

C₂²⋊Q₈

yes

C₂²×C₁₁

yes

C₂²×C₁₁

yes

—

C₂²×C₁₁

#137

yes

—

C₄×C₁₁

—

C₄×C₁₁

—

C₄×C₁₁

C₂²⋊Q₈

yes

—

C₄×C₁₁

#124

yes

—

Dic₁₁

—

Dic₁₁

—

C₂²×Q₈

—

C₂²×C₈

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₁

yes

—

C₂×C₄²

C₂×D₁₁

yes

characteristic

#103

—

#95

—

C₂×C₁₁

#393

yes

characteristic

C₂×C₁₁

#452

yes

—

C₂×C₁₁

#479

yes

—

C₂×C₁₁

yes

C₂×C₁₁

yes

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₁

yes

—

C₂²×C₄

C₂²×D₁₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²⋊D₁₁

yes

—

C₄²

—

C₁₁

#2026

yes

characteristic, complemented, Sylow

100

C₂³

C₂×C₂²⋊D₁₁

yes

characteristic

101

Q₈

—

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

Q₈×D₁₁

yes

characteristic

115

C₂×C₄

#4343

yes

—

116

C₂×C₄

#4343

yes

characteristic

117

C₂×C₄

C₂×C₂²⋊D₁₁

yes

—

118

C₈

—

119

C₂²

#66999

yes

characteristic

120

C₂²

#67000

yes

characteristic

121

C₂²

#67003

yes

—

122

C₂²

#67004

yes

—

123

C₂²

#67006

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

#66999

yes

—

129

C₄

#67000

yes

—

130

C₂

#10622676

yes

—

131

C₂

#10622681

yes

characteristic

132

C₂

#10622682

yes

characteristic

133

C₂

#10622844

yes

characteristic

134

C₂

#10622852

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₁
commutator, cosocle	C₂²×C₄×C₁₁	C₂³
Frattini	C₂²×C₄	C₂²×D₁₁
Fitting	C₂×C₁₁×#125	C₂
socle, prime radical	C₂³×C₁₁	C₂×D₄
hypercenter	C₂×#125	D₁₁
hypocenter	C₁₁	#2026
LCS	G ▷ C₂²×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₁

commutator, cosocle

C₂²×C₄×C₁₁

C₂³

Frattini

C₂²×C₄

C₂²×D₁₁

Fitting

C₂×C₁₁×#125

C₂

socle, prime radical

C₂³×C₁₁

C₂×D₄

D₁₁

C₁₁

G ▷ C₂²×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
99. C₁₁	#2026	semidirect

H ≅ G/N

type

99. C₁₁

#2026

semidirect

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

subgroup	index
C₂×C₁₁×#125	2
C₂×#66874	2
#10622542	2
#10622167	2
#10622162	2
#2026	11

subgroup

index

C₂×C₁₁×#125

Subgroups — #13069764

Conjugacy classes of subgroups

Complements

Maximal subgroups