Group #13246792 — subgroups

Subgroups — #13246792

Order 1792.

320 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 119 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×#2960	C₂	1	1	yes	maximal, characteristic
2	#11729577	C₂	1	2	yes	maximal
3	#11727486	C₂	1	2	yes	maximal
4	#11725522	C₂	1	2	yes	maximal
5	C₇×#533	C₂²	1	1	yes	characteristic
6	C₇×#446	—	2	2	no	—
7	C₇×C₈×Q₈	—	2	2	no	—
8	C₇×#424	C₄	1	2	yes	—
9	C₇×#423	C₂²	1	1	yes	characteristic
10	C₇×#405	C₄	1	2	yes	—
11	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
12	#89698	C₂²	1	4	yes	—
13	#89338	—	2	4	no	—
14	#89253	—	2	2	no	—
15	#89252	—	2	2	no	—
16	#8631	—	7	1	no	maximal, Sylow
17	C₂×C₇×M₁₆	C₂×C₄	1	2	yes	—
18	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
20	C₇×C₄⋊Q₈	Q₈	1	2	yes	—
21	C₇×#125	D₄	1	2	yes	—
22	C₇×C₂²⋊Q₈	—	2	2	no	—
23	C₄×C₇×Q₈	—	2	2	no	—
24	C₇×#117	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
26	C₇×C₄⋊C₈	—	2	4	no	—
27	C₇×C₄⋊C₈	—	2	2	no	—
28	C₇×#97	—	2	2	no	—
29	C₄×C₇×C₈	—	2	2	no	—
30	C₄×C₇×C₈	D₄	1	2	yes	—
31	C₄×C₇×C₈	D₄	1	2	yes	—
32	C₂²⋊(C₇⋊C₈)	—	2	4	no	—
33	C₄⋊(C₇⋊C₈)	—	2	4	no	—
34	#6377	—	2	4	no	—
35	#2960	D₇	1	1	yes	characteristic
36	#1568	—	7	2	no	—
37	#1537	—	7	2	no	—
38	#1307	—	7	2	no	—
39	C₂×C₇×Q₈	—	2	2	no	—
40	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₇	#41	1	2	yes	—
42	C₇×M₁₆	—	4	2	no	—
43	C₂×C₇×C₈	—	2	2	no	—
44	C₂×C₇×C₈	—	2	2	no	—
45	C₂×C₇×C₈	—	4	1	no	—
46	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
48	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
49	C₇×C₄⋊C₄	—	2	4	no	—
50	C₇×C₄⋊C₄	—	2	2	no	—
51	C₇×C₄⋊C₄	C₄⋊C₄	1	4	yes	—
52	C₇×C₂²⋊C₄	—	2	2	no	—
53	C₄²×C₇	—	2	2	no	—
54	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
55	C₄²×C₇	C₂×D₄	1	2	yes	—
56	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₇⋊C₈	—	4	4	no	—
58	#533	C₂×D₇	1	1	yes	characteristic
59	#446	—	2	2	no	—
60	C₈×Q₈	—	2	2	no	—
61	#432	—	7	4	no	—
62	#424	Dic₇	1	2	yes	—
63	#423	C₂×D₇	1	1	yes	characteristic
64	#405	Dic₇	1	2	yes	—
65	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
66	#335	—	14	2	no	—
67	#334	—	14	2	no	—
68	#330	—	14	4	no	—
69	C₂³×C₇	#117	1	1	yes	characteristic
70	C₇×Q₈	—	4	4	no	—
71	C₂×C₄×C₇	—	2	1	no	—
72	C₂×C₄×C₇	—	2	1	no	—
73	C₂×C₄×C₇	—	2	4	no	—
74	C₂×C₄×C₇	—	4	2	no	—
75	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₇	#121	1	2	yes	—
78	C₂×C₄×C₇	C₂²⋊Q₈	1	2	yes	—
79	C₇×C₈	—	2	1	no	—
80	C₇×C₈	—	2	2	no	—
81	C₇×C₈	—	4	2	no	—
82	C₇×C₈	#101	1	2	yes	—
83	C₇⋊C₈	—	8	4	no	—
84	C₂×M₁₆	C₂×Dic₇	1	2	yes	—
85	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
86	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
87	C₄⋊Q₈	Dic₁₄	1	2	yes	—
88	#125	D₂₈	1	2	yes	—
89	C₂²⋊Q₈	—	2	2	no	—
90	C₄×Q₈	—	2	2	no	—
91	#117	C₂×Dic₇	1	2	yes	—
92	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
93	C₄⋊C₈	—	2	4	no	—
94	C₄⋊C₈	—	2	2	no	—
95	C₄⋊C₈	—	14	4	no	—
96	C₂²⋊C₈	—	14	4	no	—
97	#97	—	2	2	no	—
98	#97	—	14	4	no	—
99	C₄×C₈	—	2	2	no	—
100	C₄×C₈	C₂²⋊D₇	1	2	yes	—
101	C₄×C₈	C₂²⋊D₇	1	2	yes	—
102	C₂²×C₇	—	2	1	no	—
103	C₂²×C₇	—	2	2	no	—
104	C₂²×C₇	#380	1	1	yes	characteristic
105	C₄×C₇	—	2	1	no	—
106	C₄×C₇	—	2	2	no	—
107	C₄×C₇	—	2	2	no	—
108	C₄×C₇	—	4	4	no	—
109	C₄×C₇	C₂×#101	1	1	yes	characteristic
110	C₄×C₇	#413	1	1	yes	characteristic
111	C₂×Q₈	—	2	2	no	—
112	C₂²×C₄	#1157	1	2	yes	—
113	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
114	M₁₆	—	4	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	4	1	no	—
118	C₂×C₈	—	28	4	no	—
119	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
120	C₂×C₈	C₂²⋊Dic₇	1	2	yes	—
121	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
122	C₄⋊C₄	—	2	2	no	—
123	C₄⋊C₄	—	2	4	no	—
124	C₄⋊C₄	C₄⋊Dic₇	1	4	yes	—
125	C₂²⋊C₄	—	2	2	no	—
126	C₄²	—	2	2	no	—
127	C₄²	C₂×Dic₁₄	1	1	yes	characteristic
128	C₄²	C₂×D₂₈	1	1	yes	characteristic
129	C₄²	C₂×C₂²⋊D₇	1	2	yes	—
130	C₂×C₇	—	4	1	no	—
131	C₂×C₇	#1801	1	1	yes	characteristic
132	C₂×C₇	#1814	1	2	yes	—
133	C₂³	#6489	1	1	yes	characteristic
134	Q₈	—	4	4	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	4	2	no	—
139	C₂×C₄	#6487	1	2	yes	—
140	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
141	C₂×C₄	#6493	1	2	yes	—
142	C₂×C₄	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
143	C₈	—	2	1	no	—
144	C₈	—	2	2	no	—
145	C₈	—	4	2	no	—
146	C₈	—	56	4	no	—
147	C₈	#6410	1	2	yes	—
148	C₇	#8631	1	1	yes	characteristic, complemented, Sylow
149	C₂²	—	2	2	no	—
150	C₂²	—	2	1	no	—
151	C₂²	#89984	1	1	yes	characteristic
152	C₄	—	2	2	no	—
153	C₄	—	2	2	no	—
154	C₄	—	2	1	no	—
155	C₄	—	4	4	no	—
156	C₄	C₂×#6410	1	1	yes	characteristic
157	C₄	#90009	1	1	yes	characteristic
158	C₂	—	4	1	no	—
159	C₂	#11729391	1	1	yes	characteristic
160	C₂	#11729767	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#2960

C₂

yes

maximal, characteristic

#11729577

C₂

yes

maximal

#11727486

C₂

yes

maximal

#11725522

C₂

yes

maximal

C₇×#533

C₂²

yes

characteristic

C₇×#446

—

C₇×C₈×Q₈

—

C₇×#424

C₄

yes

—

C₇×#423

C₂²

yes

characteristic

C₇×#405

C₄

yes

—

C₂×C₄×C₇×C₈

C₂²

yes

characteristic

#89698

C₂²

yes

—

—

—

—

—

C₂×C₇×M₁₆

C₂×C₄

yes

—

C₂²×C₇×C₈

C₂×C₄

yes

characteristic

C₂²×C₇×C₈

C₂×C₄

yes

characteristic

C₇×C₄⋊Q₈

Q₈

yes

—

C₇×#125

D₄

yes

—

C₇×C₂²⋊Q₈

—

C₄×C₇×Q₈

—

C₇×#117

C₂×C₄

yes

—

C₂×C₄²×C₇

C₂³

yes

characteristic

C₇×C₄⋊C₈

—

C₇×C₄⋊C₈

—

C₇×#97

—

C₄×C₇×C₈

—

C₄×C₇×C₈

D₄

yes

—

C₄×C₇×C₈

D₄

yes

—

C₂²⋊(C₇⋊C₈)

—

C₄⋊(C₇⋊C₈)

—

#6377

—

#2960

D₇

yes

—

—

—

C₂×C₇×Q₈

—

C₂²×C₄×C₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇

#41

yes

—

C₇×M₁₆

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₇×C₈

C₂²⋊C₄

yes

—

C₂×C₇×C₈

C₂²⋊C₄

yes

characteristic

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

C₄⋊C₄

yes

—

C₇×C₂²⋊C₄

—

C₄²×C₇

—

C₄²×C₇

C₂×D₄

yes

characteristic

C₄²×C₇

C₂×D₄

yes

—

C₄²×C₇

C₂×Q₈

yes

characteristic

C₂×C₇⋊C₈

—

#533

C₂×D₇

yes

characteristic

#446

—

C₈×Q₈

—

#432

—

#424

Dic₇

yes

—

#423

C₂×D₇

yes

characteristic

#405

Dic₇

yes

—

C₂×C₄×C₈

C₂×D₇

yes

—

—

—

C₂³×C₇

#117

yes

characteristic

C₇×Q₈

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇

#121

yes

—

C₂×C₄×C₇

C₂²⋊Q₈

yes

—

C₇×C₈

—

C₇×C₈

—

C₇×C₈

—

C₇×C₈

#101

yes

—

C₇⋊C₈

—

C₂×M₁₆

C₂×Dic₇

yes

—

C₂²×C₈

C₂×Dic₇

yes

characteristic

C₂²×C₈

C₂×Dic₇

yes

characteristic

C₄⋊Q₈

Dic₁₄

yes

—

#125

D₂₈

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×Dic₇

yes

—

C₂×C₄²

C₂²×D₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

C₄×C₈

—

100

C₄×C₈

C₂²⋊D₇

yes

—

101

C₄×C₈

C₂²⋊D₇

yes

—

102

C₂²×C₇

—

103

C₂²×C₇

—

104

C₂²×C₇

#380

yes

characteristic

105

C₄×C₇

—

106

C₄×C₇

—

107

C₄×C₇

—

108

C₄×C₇

—

109

C₄×C₇

C₂×#101

yes

characteristic

110

C₄×C₇

#413

yes

characteristic

111

C₂×Q₈

—

112

C₂²×C₄

#1157

yes

—

113

C₂²×C₄

C₂²×Dic₇

yes

characteristic

114

M₁₆

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

C₂²⋊Dic₇

yes

characteristic

120

C₂×C₈

C₂²⋊Dic₇

yes

—

121

C₂×C₈

C₂²⋊Dic₇

yes

characteristic

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

C₄⋊Dic₇

yes

—

125

C₂²⋊C₄

—

126

C₄²

—

127

C₄²

C₂×Dic₁₄

yes

characteristic

128

C₄²

C₂×D₂₈

yes

characteristic

129

C₄²

C₂×C₂²⋊D₇

yes

—

130

C₂×C₇

—

131

C₂×C₇

#1801

yes

characteristic

132

C₂×C₇

#1814

yes

—

133

C₂³

#6489

yes

characteristic

134

Q₈

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

#6487

yes

—

140

C₂×C₄

C₂×C₄⋊Dic₇

yes

characteristic

141

C₂×C₄

#6493

yes

—

142

C₂×C₄

C₂×C₂²⋊Dic₇

yes

characteristic

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

—

147

C₈

#6410

yes

—

148

C₇

#8631

yes

characteristic, complemented, Sylow

149

C₂²

—

150

C₂²

—

151

C₂²

#89984

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

C₂×#6410

yes

characteristic

157

C₄

#90009

yes

characteristic

158

C₂

—

159

C₂

#11729391

yes

characteristic

160

C₂

#11729767

yes

—

	subgroup	G/H
center	C₂²	#89984
commutator	C₂²×C₄×C₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#2960	C₂
socle	C₂²×C₇	#380
prime radical	C₂³×C₇	#117
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	#2960	D₇
hypocenter	C₇	#8631
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960

subgroup

G/H

center

C₂²

#89984

commutator

C₂²×C₄×C₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇

C₂

C₂²×C₇

C₂³×C₇

C₂×C₄²×C₇

C₂³

D₇

C₇

G ▷ C₂²×C₄×C₇ ▷ C₂²×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960

N	H ≅ G/N	type
148. C₇	#8631	semidirect

H ≅ G/N

type

148. C₇

#8631

semidirect

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

subgroup	index
C₇×#2960	2
#11729577	2
#11727486	2
#11725522	2
#8631	7

subgroup

index

Subgroups — #13246792

Conjugacy classes of subgroups

Complements

Maximal subgroups