Group #13076993 — subgroups

Subgroups — #13076993

Order 1408.

504 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 197 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	C₂×#66922	C₂	1	1	yes	maximal, characteristic
3	#10622843	C₂	1	4	yes	maximal
4	#10622590	C₂	1	4	yes	maximal
5	#10622587	C₂	1	4	yes	maximal
6	C₄²×Dic₁₁	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₈×C₁₁	C₂²	1	2	yes	—
8	C₂²×Dic₂₂	C₂²	1	2	yes	—
9	C₁₁×C₄⋊Q₈	—	2	4	no	—
10	C₂×C₁₁×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂×C₁₁×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₁	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₁₁	C₂²	1	2	yes	—
14	C₂×C₄⋊Dic₁₁	C₂²	1	2	yes	—
15	C₂×C₄×Dic₁₁	C₂²	1	2	yes	—
16	C₂×C₄×Dic₁₁	C₂²	1	2	yes	—
17	C₂×C₄×Dic₁₁	C₂²	1	2	yes	—
18	#66922	—	2	4	no	—
19	C₁₁⋊#95	C₂²	1	16	yes	—
20	C₂×Q₈×C₁₁	—	2	4	no	—
21	C₂×Q₈×C₁₁	—	2	8	no	—
22	C₂²×C₄×C₁₁	C₂³	1	4	yes	—
23	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
24	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
26	C₂²×Dic₁₁	C₂³	1	4	yes	—
27	C₂²×Dic₁₁	C₂³	1	4	yes	—
28	C₂×Dic₂₂	—	2	4	no	—
29	C₂×Dic₂₂	—	2	8	no	—
30	C₁₁×C₄⋊C₄	—	2	4	no	—
31	C₁₁×C₄⋊C₄	—	2	4	no	—
32	C₄²×C₁₁	D₄	1	4	yes	—
33	C₄⋊Dic₁₁	—	2	4	no	—
34	C₄⋊Dic₁₁	—	2	4	no	—
35	C₄×Dic₁₁	—	2	4	no	—
36	C₄×Dic₁₁	—	2	4	no	—
37	C₄×Dic₁₁	Q₈	1	8	yes	complemented
38	#2862	—	11	1	no	maximal, Sylow
39	C₂³×C₁₁	C₂⁴	1	1	yes	characteristic
40	Q₈×C₁₁	—	4	8	no	—
41	C₂×C₄×C₁₁	—	2	4	no	—
42	C₂×C₄×C₁₁	—	2	8	no	—
43	C₂×C₄×C₁₁	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₁₁	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₁₁	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₁	C₂×Q₈	1	4	yes	—
47	C₂×C₄×C₁₁	#41	1	4	yes	—
48	C₂×C₄×C₁₁	#41	1	4	yes	—
49	C₂×Dic₁₁	—	2	4	no	—
50	C₂×Dic₁₁	—	2	4	no	—
51	C₂×Dic₁₁	—	2	4	no	—
52	C₂×Dic₁₁	—	2	8	no	—
53	C₂×Dic₁₁	C₂×Q₈	1	8	yes	—
54	Dic₂₂	—	4	8	no	—
55	C₂×C₄⋊Q₈	—	11	1	no	—
56	C₂×C₄⋊Q₈	D₁₁	1	1	yes	characteristic
57	#391	—	11	4	no	—
58	#391	—	11	4	no	—
59	#382	—	11	4	no	—
60	C₄³	—	11	1	no	—
61	C₂²×C₁₁	C₂²×D₄	1	1	yes	characteristic
62	C₂²×C₁₁	C₂²×Q₈	1	2	yes	—
63	C₂²×C₁₁	C₂×#41	1	2	yes	—
64	C₂²×C₁₁	C₂×#41	1	2	yes	—
65	C₄×C₁₁	—	4	4	no	—
66	C₄×C₁₁	#124	1	4	yes	—
67	C₄×C₁₁	#124	1	4	yes	—
68	C₄×C₁₁	C₄⋊Q₈	1	4	yes	—
69	Dic₁₁	—	2	8	no	—
70	Dic₁₁	—	4	4	no	—
71	C₂²×Q₈	—	11	2	no	—
72	C₂²×Q₈	C₂×D₁₁	1	2	yes	—
73	C₄⋊Q₈	—	2	4	no	—
74	C₄⋊Q₈	—	22	4	no	—
75	C₂×C₄⋊C₄	—	11	2	no	—
76	C₂×C₄⋊C₄	—	11	2	no	—
77	C₂×C₄⋊C₄	C₂×D₁₁	1	2	yes	—
78	C₂×C₄⋊C₄	C₂×D₁₁	1	2	yes	—
79	C₂×C₄²	—	11	2	no	—
80	C₂×C₄²	—	11	2	no	—
81	C₂×C₄²	—	11	2	no	—
82	C₂×C₄²	C₂×D₁₁	1	1	yes	characteristic
83	#95	—	11	16	no	—
84	C₂×C₁₁	C₂×#124	1	1	yes	characteristic
85	C₂×C₁₁	C₂×#124	1	1	yes	characteristic
86	C₂×C₁₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
87	C₂×C₁₁	#533	1	4	yes	—
88	C₂×Q₈	—	2	8	no	—
89	C₂×Q₈	—	2	4	no	—
90	C₂×Q₈	—	22	8	no	—
91	C₂×Q₈	—	22	4	no	—
92	C₂²×C₄	—	11	4	no	—
93	C₂²×C₄	—	11	4	no	—
94	C₂²×C₄	C₂²×D₁₁	1	4	yes	—
95	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
96	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
97	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
98	C₄⋊C₄	—	2	4	no	—
99	C₄⋊C₄	—	2	4	no	—
100	C₄⋊C₄	—	22	4	no	—
101	C₄⋊C₄	—	22	4	no	—
102	C₄²	—	11	8	no	—
103	C₄²	—	22	4	no	—
104	C₄²	—	22	4	no	—
105	C₄²	C₂²⋊D₁₁	1	4	yes	—
106	C₁₁	#2862	1	1	yes	characteristic, complemented, Sylow
107	C₂³	C₂³×D₁₁	1	1	yes	characteristic
108	Q₈	—	4	8	no	—
109	Q₈	—	44	8	no	—
110	C₂×C₄	—	2	8	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	11	8	no	—
113	C₂×C₄	—	22	4	no	—
114	C₂×C₄	—	22	4	no	—
115	C₂×C₄	—	22	8	no	—
116	C₂×C₄	—	22	4	no	—
117	C₂×C₄	#4341	1	4	yes	—
118	C₂×C₄	Q₈×D₁₁	1	4	yes	—
119	C₂×C₄	#4343	1	4	yes	—
120	C₂×C₄	C₂×C₂²⋊D₁₁	1	2	yes	—
121	C₂×C₄	C₂×C₂²⋊D₁₁	1	2	yes	—
122	C₂×C₄	C₂×C₂²⋊D₁₁	1	2	yes	—
123	C₂²	C₂×#4341	1	2	yes	—
124	C₂²	C₂×Q₈×D₁₁	1	2	yes	—
125	C₂²	C₂×#4343	1	2	yes	—
126	C₂²	C₂²×C₂²⋊D₁₁	1	1	yes	characteristic
127	C₄	—	4	4	no	—
128	C₄	—	22	8	no	—
129	C₄	—	44	4	no	—
130	C₄	#66989	1	4	yes	—
131	C₄	#66997	1	4	yes	—
132	C₄	#67000	1	4	yes	—
133	C₂	#10623259	1	4	yes	—
134	C₂	C₂×#66989	1	1	yes	characteristic
135	C₂	C₂×#66997	1	1	yes	characteristic
136	C₂	C₂×#67000	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₁×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂×#66922

C₂

yes

maximal, characteristic

#10622843

C₂

yes

maximal

#10622590

C₂

yes

maximal

#10622587

C₂

yes

maximal

C₄²×Dic₁₁

C₂

yes

maximal, characteristic

C₂²×Q₈×C₁₁

C₂²

yes

—

C₂²×Dic₂₂

C₂²

yes

—

C₁₁×C₄⋊Q₈

—

C₂×C₁₁×C₄⋊C₄

C₂²

yes

—

C₂×C₁₁×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₁₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₁

C₂²

yes

—

C₂×C₄⋊Dic₁₁

C₂²

yes

—

C₂×C₄×Dic₁₁

C₂²

yes

—

C₂×C₄×Dic₁₁

C₂²

yes

—

C₂×C₄×Dic₁₁

C₂²

yes

—

#66922

—

C₁₁⋊#95

C₂²

yes

—

C₂×Q₈×C₁₁

—

C₂×Q₈×C₁₁

—

C₂²×C₄×C₁₁

C₂³

yes

—

C₂²×C₄×C₁₁

C₂³

yes

characteristic

C₂²×C₄×C₁₁

C₂³

yes

characteristic

C₂²×C₄×C₁₁

C₂³

yes

characteristic

C₂²×Dic₁₁

C₂³

yes

—

C₂²×Dic₁₁

C₂³

yes

—

C₂×Dic₂₂

—

C₂×Dic₂₂

—

C₁₁×C₄⋊C₄

—

C₁₁×C₄⋊C₄

—

C₄²×C₁₁

D₄

yes

—

C₄⋊Dic₁₁

—

C₄⋊Dic₁₁

—

C₄×Dic₁₁

—

C₄×Dic₁₁

—

C₄×Dic₁₁

Q₈

yes

complemented

#2862

—

maximal, Sylow

C₂³×C₁₁

C₂⁴

yes

characteristic

Q₈×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

C₂×D₄

yes

—

C₂×C₄×C₁₁

C₂×D₄

yes

—

C₂×C₄×C₁₁

C₂×D₄

yes

—

C₂×C₄×C₁₁

C₂×Q₈

yes

—

C₂×C₄×C₁₁

#41

yes

—

C₂×C₄×C₁₁

#41

yes

—

C₂×Dic₁₁

—

C₂×Dic₁₁

—

C₂×Dic₁₁

—

C₂×Dic₁₁

—

C₂×Dic₁₁

C₂×Q₈

yes

—

Dic₂₂

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

D₁₁

yes

—

—

—

C₄³

—

C₂²×C₁₁

C₂²×D₄

yes

characteristic

C₂²×C₁₁

C₂²×Q₈

yes

—

C₂²×C₁₁

C₂×#41

yes

—

C₂²×C₁₁

C₂×#41

yes

—

C₄×C₁₁

—

C₄×C₁₁

#124

yes

—

C₄×C₁₁

#124

yes

—

C₄×C₁₁

C₄⋊Q₈

yes

—

Dic₁₁

—

Dic₁₁

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₁₁

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₁

yes

—

C₂×C₄⋊C₄

C₂×D₁₁

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₁

yes

characteristic

#95

—

C₂×C₁₁

yes

C₂×C₁₁

yes

C₂×C₁₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₁

#533

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₁

yes

—

C₂²×C₄

C₂²×D₁₁

yes

characteristic

C₂²×C₄

C₂²×D₁₁

yes

characteristic

C₂²×C₄

C₂²×D₁₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄²

—

103

C₄²

—

104

C₄²

—

105

C₄²

C₂²⋊D₁₁

yes

—

106

C₁₁

#2862

yes

characteristic, complemented, Sylow

107

C₂³

C₂³×D₁₁

yes

characteristic

108

Q₈

—

109

Q₈

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

#4341

yes

—

118

C₂×C₄

Q₈×D₁₁

yes

—

119

C₂×C₄

#4343

yes

—

120

C₂×C₄

C₂×C₂²⋊D₁₁

yes

—

121

C₂×C₄

C₂×C₂²⋊D₁₁

yes

—

122

C₂×C₄

C₂×C₂²⋊D₁₁

yes

—

123

C₂²

C₂×#4341

yes

—

124

C₂²

C₂×Q₈×D₁₁

yes

—

125

C₂²

C₂×#4343

yes

—

126

C₂²

C₂²×C₂²⋊D₁₁

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

#66989

yes

—

131

C₄

#66997

yes

—

132

C₄

#67000

yes

—

133

C₂

#10623259

yes

—

134

C₂

C₂×#66989

yes

characteristic

135

C₂

C₂×#66997

yes

characteristic

136

C₂

C₂×#67000

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₁
commutator, socle, prime radical, cosocle	C₂³×C₁₁	C₂⁴
Fitting	C₂×C₁₁×C₄⋊Q₈	C₂
hypercenter	C₂×C₄⋊Q₈	D₁₁
hypocenter	C₁₁	#2862
LCS	G ▷ C₂³×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₁

commutator, socle, prime radical, cosocle

C₂³×C₁₁

C₂⁴

Fitting

C₂×C₁₁×C₄⋊Q₈

C₂

hypercenter

C₂×C₄⋊Q₈

D₁₁

C₁₁

G ▷ C₂³×C₁₁ ▷ C₁₁

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
37. C₄×Dic₁₁	Q₈	semidirect
106. C₁₁	#2862	semidirect

H ≅ G/N

type

37. C₄×Dic₁₁

Q₈

semidirect

106. C₁₁

#2862

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
C₂×#66922	2
#10622843	2
#10622590	2
#10622587	2
C₄²×Dic₁₁	2
#2862	11

subgroup

index

C₂×C₁₁×C₄⋊Q₈

C₄²×Dic₁₁

Subgroups — #13076993

Conjugacy classes of subgroups

Complements

Maximal subgroups