Group #11728906 — subgroups

Subgroups — #11728906

Order 896.

146 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 53 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	C₇×#477	C₂	1	1	yes	maximal, characteristic
2	#89851	C₂	1	1	yes	maximal, characteristic
3	#89803	C₂	1	1	yes	maximal, characteristic
4	#89796	C₂	1	1	yes	maximal, characteristic
5	#89341	C₂	1	1	yes	maximal, characteristic
6	#89277	C₂	1	1	yes	maximal, characteristic
7	#89274	C₂	1	1	yes	maximal, characteristic
8	C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
10	C₂×#1144	—	2	1	no	—
11	#6432	C₂²	1	1	yes	characteristic
12	C₇×#107	—	2	1	no	—
13	C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₇×#103	—	2	1	no	—
15	#6384	—	2	1	no	—
16	#6382	—	2	1	no	—
17	#6382	—	2	1	no	—
18	C₄⋊(C₇⋊C₈)	—	2	1	no	—
19	C₄⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
20	C₄⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
21	#6377	—	2	1	no	—
22	C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
23	#1692	—	7	1	no	maximal, Sylow
24	C₂×C₇×Q₈	—	2	1	no	—
25	C₂×C₇×Q₈	D₄	1	1	yes	characteristic
26	C₂×Dic₁₄	—	2	1	no	—
27	C₂×C₇×C₈	—	2	1	no	—
28	C₇×C₄⋊C₄	—	2	1	no	—
29	C₇×C₄⋊C₄	—	2	1	no	—
30	C₇×C₄⋊C₄	—	2	1	no	—
31	C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
32	C₄²×C₇	—	2	1	no	—
33	C₄²×C₇	C₂³	1	1	yes	characteristic
34	#1144	—	4	2	no	—
35	C₄⋊Dic₇	—	2	1	no	—
36	C₄⋊Dic₇	—	2	1	no	—
37	C₂×C₇⋊C₈	—	2	1	no	—
38	C₂×C₇⋊C₈	—	2	1	no	—
39	C₂×C₇⋊C₈	D₄	1	1	yes	characteristic
40	C₂×C₇⋊C₈	D₄	1	1	yes	characteristic
41	#498	—	7	1	no	—
42	#477	D₇	1	1	yes	characteristic
43	#462	—	7	1	no	—
44	#446	—	7	1	no	—
45	#335	—	7	1	no	—
46	#333	—	7	1	no	—
47	#332	—	7	1	no	—
48	C₇×Q₈	—	2	1	no	—
49	C₇×Q₈	—	2	1	no	—
50	C₇×Q₈	—	8	1	no	—
51	C₂×C₄×C₇	—	2	1	no	—
52	C₂×C₄×C₇	—	2	1	no	—
53	C₂×C₄×C₇	—	4	1	no	—
54	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₇	#41	1	1	yes	characteristic
57	C₂×Dic₇	—	4	1	no	—
58	Dic₁₄	—	4	1	no	—
59	Dic₁₄	—	4	1	no	—
60	C₇×C₈	—	4	1	no	—
61	C₇⋊C₈	—	2	1	no	—
62	C₇⋊C₈	—	4	1	no	—
63	C₇⋊C₈	—	4	1	no	—
64	C₇⋊C₈	Q₁₆	1	2	yes	complemented
65	C₂×Q₁₆	—	14	1	no	—
66	C₄⋊Q₈	—	7	1	no	—
67	C₄⋊Q₈	C₂×D₇	1	1	yes	characteristic
68	C₄×Q₈	C₂×D₇	1	1	yes	characteristic
69	#107	—	2	1	no	—
70	#106	—	14	1	no	—
71	#106	—	14	1	no	—
72	C₄⋊C₈	—	7	1	no	—
73	C₄⋊C₈	—	7	1	no	—
74	C₄⋊C₈	—	14	1	no	—
75	C₄⋊C₈	C₂×D₇	1	1	yes	characteristic
76	#103	—	2	1	no	—
77	#103	—	14	1	no	—
78	#97	—	14	1	no	—
79	C₄×C₈	—	7	1	no	—
80	C₂²×C₇	#121	1	1	yes	characteristic
81	C₄×C₇	—	2	1	no	—
82	C₄×C₇	—	4	1	no	—
83	C₄×C₇	—	4	1	no	—
84	C₄×C₇	—	8	1	no	—
85	C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
86	C₄×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
87	C₄×C₇	#137	1	1	yes	characteristic
88	C₄×C₇	#137	1	1	yes	characteristic
89	Dic₇	—	8	1	no	—
90	C₂×Q₈	—	2	1	no	—
91	C₂×Q₈	—	14	1	no	—
92	C₂×Q₈	C₂²⋊D₇	1	1	yes	characteristic
93	Q₁₆	—	28	2	no	—
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	—	7	1	no	—
96	C₂×C₈	—	7	1	no	—
97	C₂×C₈	—	14	1	no	—
98	C₂×C₈	—	14	1	no	—
99	C₄⋊C₄	—	2	1	no	—
100	C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	14	1	no	—
103	C₄⋊C₄	—	14	1	no	—
104	C₄⋊C₄	C₂²⋊D₇	1	1	yes	characteristic
105	C₄²	—	2	1	no	—
106	C₄²	C₂²×D₇	1	1	yes	characteristic
107	C₂×C₇	#462	1	1	yes	characteristic
108	C₂×C₇	#468	1	1	yes	characteristic
109	C₂×C₇	#471	1	1	yes	characteristic
110	Q₈	—	2	1	no	—
111	Q₈	—	2	1	no	—
112	Q₈	—	8	1	no	—
113	Q₈	—	28	1	no	—
114	Q₈	—	28	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	28	1	no	—
119	C₂×C₄	D₄×D₇	1	1	yes	characteristic
120	C₂×C₄	#1159	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
122	C₈	—	4	1	no	—
123	C₈	—	7	2	no	—
124	C₈	—	14	1	no	—
125	C₈	—	28	1	no	—
126	C₈	—	28	1	no	—
127	C₇	#1692	1	1	yes	characteristic, complemented, Sylow
128	C₂²	#6502	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	4	1	no	—
131	C₄	—	4	1	no	—
132	C₄	—	8	1	no	—
133	C₄	—	56	1	no	—
134	C₄	D₇×Q₁₆	1	1	yes	characteristic
135	C₄	#6481	1	1	yes	characteristic
136	C₄	#6505	1	1	yes	characteristic
137	C₄	#6512	1	1	yes	characteristic
138	C₂	#89821	1	1	yes	characteristic
139	C₂	#89954	1	1	yes	characteristic
140	C₂	#89972	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#477

C₂

yes

maximal, characteristic

#89851

C₂

yes

maximal, characteristic

#89803

C₂

yes

maximal, characteristic

#89796

C₂

yes

maximal, characteristic

#89341

C₂

yes

maximal, characteristic

#89277

C₂

yes

maximal, characteristic

#89274

C₂

yes

maximal, characteristic

C₇×C₄⋊Q₈

C₂²

yes

characteristic

C₄×C₇×Q₈

C₂²

yes

characteristic

C₂×#1144

—

#6432

C₂²

yes

characteristic

C₇×#107

—

C₇×C₄⋊C₈

C₂²

yes

—

—

—

—

C₄⋊(C₇⋊C₈)

—

C₄⋊(C₇⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₇⋊C₈)

C₂²

yes

characteristic

#6377

—

C₄×C₇⋊C₈

C₂²

yes

characteristic

#1692

—

maximal, Sylow

C₂×C₇×Q₈

—

C₂×C₇×Q₈

D₄

yes

characteristic

C₂×Dic₁₄

—

C₂×C₇×C₈

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

D₄

yes

characteristic

C₄²×C₇

—

C₄²×C₇

C₂³

yes

characteristic

#1144

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

D₄

yes

characteristic

C₂×C₇⋊C₈

D₄

yes

characteristic

#498

—

#477

D₇

yes

—

—

—

—

—

C₇×Q₈

—

C₇×Q₈

—

C₇×Q₈

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

#41

yes

characteristic

C₂×Dic₇

—

Dic₁₄

—

Dic₁₄

—

C₇×C₈

—

C₇⋊C₈

—

C₇⋊C₈

—

C₇⋊C₈

—

C₇⋊C₈

Q₁₆

yes

complemented

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₇

yes

characteristic

C₄×Q₈

C₂×D₇

yes

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₇

yes

—

—

—

C₄×C₈

—

C₂²×C₇

#121

yes

characteristic

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

C₂×Q₁₆

yes

C₄×C₇

yes

C₄×C₇

yes

C₄×C₇

yes

Dic₇

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₇

yes

characteristic

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

C₂²⋊D₇

yes

characteristic

105

C₄²

—

106

C₄²

C₂²×D₇

yes

characteristic

107

C₂×C₇

#462

yes

characteristic

108

C₂×C₇

#468

yes

characteristic

109

C₂×C₇

#471

yes

characteristic

110

Q₈

—

111

Q₈

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

D₄×D₇

yes

characteristic

120

C₂×C₄

#1159

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₇

yes

characteristic

122

C₈

—

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

127

C₇

#1692

yes

characteristic, complemented, Sylow

128

C₂²

#6502

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

D₇×Q₁₆

yes

characteristic

135

C₄

#6481

yes

characteristic

136

C₄

#6505

yes

characteristic

137

C₄

#6512

yes

characteristic

138

C₂

#89821

yes

characteristic

139

C₂

#89954

yes

characteristic

140

C₂

#89972

yes

characteristic

	subgroup	G/H
center	C₂²	#6502
commutator, cosocle	C₄²×C₇	C₂³
Frattini	C₄²	C₂²×D₇
Fitting	C₇×#477	C₂
socle, prime radical	C₂²×C₇	#121
hypercenter	#477	D₇
hypocenter	C₇	#1692
LCS	G ▷ C₄²×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #477

subgroup

G/H

center

C₂²

#6502

commutator, cosocle

C₄²×C₇

C₂³

Frattini

C₄²

C₂²×D₇

C₂

C₂²×C₇

D₇

C₇

G ▷ C₄²×C₇ ▷ C₂²×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #477

N	H ≅ G/N	type
64. C₇⋊C₈	Q₁₆	semidirect
127. C₇	#1692	semidirect

H ≅ G/N

type

64. C₇⋊C₈

Q₁₆

semidirect

127. C₇

#1692

semidirect

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

subgroup	index
C₇×#477	2
#89851	2
#89803	2
#89796	2
#89341	2
#89277	2
#89274	2
#1692	7

subgroup

index

Subgroups — #11728906

Conjugacy classes of subgroups

Complements

Maximal subgroups