Group #11740721 — subgroups

Subgroups — #11740721

Order 896.

484 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 221 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×#532	C₂	1	1	yes	maximal, characteristic
2	C₂×#6511	C₂	1	2	yes	maximal, complemented
3	#90006	C₂	1	2	yes	maximal, complemented
4	#89783	C₂	1	8	yes	maximal, complemented
5	#89696	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#6377	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₇×#41	C₂²	1	2	yes	—
8	C₇×C₄⋊Q₈	C₄	1	1	yes	characteristic
9	C₇×#127	C₄	1	1	yes	characteristic
10	C₇×#124	C₄	1	2	yes	—
11	C₇×#121	C₄	1	4	yes	—
12	C₄×C₇×D₄	C₂²	1	4	yes	—
13	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
14	#6511	—	2	8	no	—
15	C₂×#1136	C₂²	1	2	yes	—
16	C₂×#1136	C₂²	1	4	yes	—
17	C₂²×C₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₂²×C₇⋊C₈	C₂²	1	4	yes	—
19	C₂²×C₇⋊C₈	C₂²	1	1	yes	characteristic
20	C₂²⋊(C₇⋊C₈)	C₂²	1	8	yes	—
21	C₄⋊(C₇⋊C₈)	C₂²	1	4	yes	—
22	#6377	C₂²	1	4	yes	—
23	#2932	—	7	1	no	maximal, Sylow
24	C₇×#41	—	2	4	no	—
25	C₇×#41	—	2	4	no	—
26	C₂×C₇×Q₈	C₂×C₄	1	2	yes	—
27	C₂×C₇×D₄	C₂×C₄	1	4	yes	—
28	C₂×C₇×D₄	C₂×C₄	1	2	yes	—
29	C₂×C₇×D₄	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₇	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₇	C₂³	1	4	yes	—
32	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
33	C₇×C₄⋊C₄	C₂×C₄	1	4	yes	—
34	C₇×C₂²⋊C₄	C₂×C₄	1	8	yes	—
35	C₄²×C₇	C₂×C₄	1	1	yes	characteristic
36	C₄²×C₇	C₂×C₄	1	1	yes	characteristic
37	C₄²×C₇	C₂³	1	2	yes	—
38	#1136	—	2	4	no	—
39	#1136	—	2	8	no	—
40	C₂×C₇⋊C₈	—	2	2	no	—
41	C₂×C₇⋊C₈	—	2	8	no	—
42	C₂×C₇⋊C₈	D₄	1	4	yes	complemented
43	C₂×C₇⋊C₈	C₂³	1	2	yes	—
44	C₂×C₇⋊C₈	C₂³	1	4	yes	—
45	C₂×C₇⋊C₈	C₂³	1	2	yes	—
46	C₂×#131	—	7	2	no	—
47	#532	D₇	1	1	yes	characteristic
48	#435	—	7	8	no	—
49	#423	—	7	1	no	—
50	#408	—	7	2	no	—
51	C₂×#97	—	7	1	no	—
52	C₂³×C₇	C₂²×C₄	1	4	yes	—
53	C₂³×C₇	C₂²×C₄	1	1	yes	characteristic
54	C₇×Q₈	—	2	4	no	—
55	C₇×D₄	—	2	8	no	—
56	C₇×D₄	—	2	4	no	—
57	C₇×D₄	—	4	4	no	—
58	C₂×C₄×C₇	—	2	4	no	—
59	C₂×C₄×C₇	—	2	4	no	—
60	C₂×C₄×C₇	—	4	2	no	—
61	C₂×C₄×C₇	C₂²×C₄	1	2	yes	—
62	C₂×C₄×C₇	C₂²×C₄	1	2	yes	—
63	C₂×C₄×C₇	C₂²×C₄	1	4	yes	—
64	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
66	C₂×C₄×C₇	#41	1	2	yes	—
67	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
68	C₇⋊C₈	—	2	4	no	—
69	C₇⋊C₈	—	2	2	no	—
70	C₇⋊C₈	C₂×D₄	1	4	yes	—
71	C₂×#41	C₂×D₇	1	2	yes	—
72	#131	—	14	8	no	—
73	C₂×M₁₆	—	7	2	no	—
74	C₂×M₁₆	—	7	4	no	—
75	C₂²×C₈	—	7	1	no	—
76	C₂²×C₈	—	7	4	no	—
77	C₂²×C₈	—	7	1	no	—
78	C₄⋊Q₈	Dic₇	1	1	yes	characteristic
79	#127	Dic₇	1	1	yes	characteristic
80	#124	Dic₇	1	2	yes	—
81	#121	Dic₇	1	4	yes	—
82	C₄×D₄	C₂×D₇	1	4	yes	—
83	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
84	C₄⋊C₈	—	7	4	no	—
85	C₂²⋊C₈	—	7	8	no	—
86	#97	—	7	4	no	—
87	C₂²×C₇	—	2	2	no	—
88	C₂²×C₇	—	2	4	no	—
89	C₂²×C₇	—	2	8	no	—
90	C₂²×C₇	C₄×D₄	1	2	yes	—
91	C₂²×C₇	C₂³×C₄	1	1	yes	characteristic
92	C₄×C₇	—	2	4	no	—
93	C₄×C₇	—	2	2	no	—
94	C₄×C₇	—	2	2	no	—
95	C₄×C₇	C₄×D₄	1	2	yes	—
96	C₄×C₇	C₂²×D₄	1	1	yes	characteristic
97	C₄×C₇	C₂×#41	1	1	yes	characteristic
98	#41	—	2	4	no	—
99	#41	—	2	4	no	—
100	C₂×Q₈	C₂×Dic₇	1	2	yes	—
101	C₂×D₄	C₂×Dic₇	1	4	yes	—
102	C₂×D₄	C₂×Dic₇	1	2	yes	—
103	C₂×D₄	C₂×Dic₇	1	4	yes	—
104	C₂²×C₄	C₂×Dic₇	1	2	yes	—
105	C₂²×C₄	C₂²×D₇	1	4	yes	—
106	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
107	M₁₆	—	14	8	no	—
108	M₁₆	—	14	4	no	—
109	C₂×C₈	—	7	4	no	—
110	C₂×C₈	—	7	2	no	—
111	C₂×C₈	—	7	2	no	—
112	C₂×C₈	—	7	4	no	—
113	C₂×C₈	—	14	8	no	—
114	C₂×C₈	—	14	2	no	—
115	C₄⋊C₄	C₂×Dic₇	1	4	yes	—
116	C₂²⋊C₄	C₂×Dic₇	1	8	yes	—
117	C₄²	C₂×Dic₇	1	1	yes	characteristic
118	C₄²	C₂×Dic₇	1	1	yes	characteristic
119	C₄²	C₂²×D₇	1	2	yes	—
120	C₂×C₇	—	2	2	no	—
121	C₂×C₇	—	4	4	no	—
122	C₂×C₇	C₂×C₄×D₄	1	1	yes	characteristic
123	C₂×C₇	#568	1	2	yes	—
124	C₂³	C₂²×Dic₇	1	4	yes	—
125	C₂³	C₂²×Dic₇	1	1	yes	characteristic
126	Q₈	—	2	4	no	—
127	D₄	—	2	8	no	—
128	D₄	—	2	4	no	—
129	D₄	—	4	4	no	—
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	4	2	no	—
133	C₂×C₄	D₄×D₇	1	2	yes	—
134	C₂×C₄	#1159	1	2	yes	—
135	C₂×C₄	C₂²×Dic₇	1	2	yes	—
136	C₂×C₄	C₂²×Dic₇	1	2	yes	—
137	C₂×C₄	C₂²×Dic₇	1	4	yes	—
138	C₂×C₄	C₂²×Dic₇	1	1	yes	characteristic
139	C₂×C₄	C₂³×D₇	1	1	yes	characteristic
140	C₈	—	7	4	no	—
141	C₈	—	14	4	no	—
142	C₈	—	14	2	no	—
143	C₇	#2932	1	1	yes	characteristic, complemented, Sylow
144	C₂²	—	2	2	no	—
145	C₂²	—	2	4	no	—
146	C₂²	—	2	8	no	—
147	C₂²	D₄×Dic₇	1	2	yes	—
148	C₂²	C₂³×Dic₇	1	1	yes	characteristic
149	C₄	—	2	4	no	—
150	C₄	—	2	2	no	—
151	C₄	—	2	2	no	—
152	C₄	D₄×Dic₇	1	2	yes	—
153	C₄	C₂×D₄×D₇	1	1	yes	characteristic
154	C₄	C₂×#1159	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	—	4	4	no	—
157	C₂	C₂×D₄×Dic₇	1	1	yes	characteristic
158	C₂	#90510	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#532

C₂

yes

maximal, characteristic

C₂×#6511

C₂

yes

maximal, complemented

#90006

C₂

yes

maximal, complemented

#89783

C₂

yes

maximal, complemented

#89696

C₂

yes

maximal, characteristic, complemented

C₂×#6377

C₂

yes

maximal, characteristic, complemented

C₂×C₇×#41

C₂²

yes

—

C₇×C₄⋊Q₈

C₄

yes

characteristic

C₇×#127

C₄

yes

characteristic

C₇×#124

C₄

yes

—

C₇×#121

C₄

yes

—

C₄×C₇×D₄

C₂²

yes

—

C₂×C₄²×C₇

C₂²

yes

characteristic

#6511

—

C₂×#1136

C₂²

yes

—

C₂×#1136

C₂²

yes

—

C₂²×C₇⋊C₈

C₂²

yes

characteristic

C₂²×C₇⋊C₈

C₂²

yes

—

C₂²×C₇⋊C₈

C₂²

yes

characteristic

C₂²⋊(C₇⋊C₈)

C₂²

yes

—

C₄⋊(C₇⋊C₈)

C₂²

yes

—

#6377

C₂²

yes

—

#2932

—

maximal, Sylow

C₇×#41

—

C₇×#41

—

C₂×C₇×Q₈

C₂×C₄

yes

—

C₂×C₇×D₄

C₂×C₄

yes

—

C₂×C₇×D₄

C₂×C₄

yes

—

C₂×C₇×D₄

C₂×C₄

yes

—

C₂²×C₄×C₇

C₂×C₄

yes

—

C₂²×C₄×C₇

C₂³

yes

—

C₂²×C₄×C₇

C₂³

yes

characteristic

C₇×C₄⋊C₄

C₂×C₄

yes

—

C₇×C₂²⋊C₄

C₂×C₄

yes

—

C₄²×C₇

C₂×C₄

yes

characteristic

C₄²×C₇

C₂×C₄

yes

characteristic

C₄²×C₇

C₂³

yes

—

#1136

—

#1136

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

D₄

yes

complemented

C₂×C₇⋊C₈

C₂³

yes

—

C₂×C₇⋊C₈

C₂³

yes

—

C₂×C₇⋊C₈

C₂³

yes

—

C₂×#131

—

#532

D₇

yes

—

—

—

C₂×#97

—

C₂³×C₇

C₂²×C₄

yes

—

C₂³×C₇

C₂²×C₄

yes

characteristic

C₇×Q₈

—

C₇×D₄

—

C₇×D₄

—

C₇×D₄

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂²×C₄

yes

—

C₂×C₄×C₇

C₂²×C₄

yes

—

C₂×C₄×C₇

C₂²×C₄

yes

—

C₂×C₄×C₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₇

C₂×D₄

yes

—

C₂×C₄×C₇

#41

yes

—

C₂×C₄×C₇

C₂⁴

yes

characteristic

C₇⋊C₈

—

C₇⋊C₈

—

C₇⋊C₈

C₂×D₄

yes

—

C₂×#41

C₂×D₇

yes

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

Dic₇

yes

characteristic

#127

Dic₇

yes

characteristic

#124

Dic₇

yes

—

#121

Dic₇

yes

—

C₄×D₄

C₂×D₇

yes

—

C₂×C₄²

C₂×D₇

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

C₄×D₄

yes

—

C₂²×C₇

C₂³×C₄

yes

characteristic

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

C₄×D₄

yes

—

C₄×C₇

C₂²×D₄

yes

characteristic

C₄×C₇

C₂×#41

yes

characteristic

#41

—

#41

—

100

C₂×Q₈

C₂×Dic₇

yes

—

101

C₂×D₄

C₂×Dic₇

yes

—

102

C₂×D₄

C₂×Dic₇

yes

—

103

C₂×D₄

C₂×Dic₇

yes

—

104

C₂²×C₄

C₂×Dic₇

yes

—

105

C₂²×C₄

C₂²×D₇

yes

—

106

C₂²×C₄

C₂²×D₇

yes

characteristic

107

M₁₆

—

108

M₁₆

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₄⋊C₄

C₂×Dic₇

yes

—

116

C₂²⋊C₄

C₂×Dic₇

yes

—

117

C₄²

C₂×Dic₇

yes

characteristic

118

C₄²

C₂×Dic₇

yes

characteristic

119

C₄²

C₂²×D₇

yes

—

120

C₂×C₇

—

121

C₂×C₇

—

122

C₂×C₇

C₂×C₄×D₄

yes

characteristic

123

C₂×C₇

#568

yes

—

124

C₂³

C₂²×Dic₇

yes

—

125

C₂³

C₂²×Dic₇

yes

characteristic

126

Q₈

—

127

D₄

—

128

D₄

—

129

D₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

D₄×D₇

yes

—

134

C₂×C₄

#1159

yes

—

135

C₂×C₄

C₂²×Dic₇

yes

—

136

C₂×C₄

C₂²×Dic₇

yes

—

137

C₂×C₄

C₂²×Dic₇

yes

—

138

C₂×C₄

C₂²×Dic₇

yes

characteristic

139

C₂×C₄

C₂³×D₇

yes

characteristic

140

C₈

—

141

C₈

—

142

C₈

—

143

C₇

#2932

yes

characteristic, complemented, Sylow

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

D₄×Dic₇

yes

—

148

C₂²

C₂³×Dic₇

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

D₄×Dic₇

yes

—

153

C₄

C₂×D₄×D₇

yes

characteristic

154

C₄

C₂×#1159

yes

characteristic

155

C₂

—

156

C₂

—

157

C₂

C₂×D₄×Dic₇

yes

characteristic

158

C₂

#90510

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₇
commutator, socle	C₂²×C₇	C₂³×C₄
Fitting, prime radical	C₇×#532	C₂
cosocle	C₂×C₄×C₇	C₂⁴
hypercenter	#532	D₇
hypocenter	C₇	#2932
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ #532

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₇

commutator, socle

C₂²×C₇

C₂³×C₄

Fitting, prime radical

C₇×#532

C₂

cosocle

C₂×C₄×C₇

C₂⁴

D₇

C₇

G ▷ C₂²×C₇ ▷ C₇

C₁ ◁ C₂×C₄ ◁ #532

N	H ≅ G/N	type
2. C₂×#6511	C₂	semidirect
3. #90006	C₂	semidirect
4. #89783	C₂	semidirect
5. #89696	C₂	semidirect
6. C₂×#6377	C₂	semidirect
42. C₂×C₇⋊C₈	D₄	semidirect
143. C₇	#2932	semidirect

H ≅ G/N

type

2. C₂×#6511

C₂

semidirect

3. #90006

C₂

semidirect

4. #89783

C₂

semidirect

5. #89696

C₂

semidirect

6. C₂×#6377

C₂

semidirect

42. C₂×C₇⋊C₈

D₄

semidirect

143. C₇

#2932

semidirect

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

subgroup	index
C₇×#532	2
C₂×#6511	2
#90006	2
#89783	2
#89696	2
C₂×#6377	2
#2932	7

subgroup

index

Subgroups — #11740721

Conjugacy classes of subgroups

Complements

Maximal subgroups