Group #13244242 — subgroups

Subgroups — #13244242

Order 1792.

330 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 131 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	#11728496	C₂	1	2	yes	maximal
2	#11727188	C₂	1	1	yes	maximal, characteristic
3	C₇×#1274	C₂	1	1	yes	maximal, characteristic
4	#11725546	C₂	1	2	yes	maximal
5	#11725420	C₂	1	1	yes	maximal, characteristic
6	C₇×#431	C₂²	1	2	yes	—
7	C₇×#376	C₂²	1	1	yes	characteristic
8	#89697	C₂²	1	2	yes	—
9	C₂×#6377	C₄	1	2	yes	—
10	C₂×#6377	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₇⋊C₈	C₄	1	2	yes	—
12	C₂×C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
13	#89250	C₄	1	4	yes	complemented
14	#89249	C₄	1	4	yes	complemented
15	#8139	—	7	1	no	maximal, Sylow
16	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
17	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
18	C₂²×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₇⋊C₈	C₂×C₄	1	2	yes	—
21	C₇×C₄⋊C₈	—	2	2	no	—
22	C₇×C₂²⋊C₈	—	2	2	no	—
23	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
24	C₇×#95	—	2	2	no	—
25	C₂²⋊(C₇⋊C₈)	—	2	2	no	—
26	C₄⋊(C₇⋊C₈)	—	2	2	no	—
27	#6377	—	2	1	no	—
28	#6377	—	2	1	no	—
29	#6377	—	2	4	no	—
30	C₄×C₇⋊C₈	—	2	4	no	—
31	C₄×C₇⋊C₈	C₂×C₄	1	4	yes	—
32	C₄×C₇⋊C₈	C₂×C₄	1	2	yes	—
33	C₄×C₇⋊C₈	C₂×C₄	1	2	yes	—
34	#1726	—	7	1	no	—
35	#1452	—	7	2	no	—
36	#1275	—	7	2	no	—
37	#1274	—	7	1	no	—
38	#1274	D₇	1	1	yes	characteristic
39	C₂²×C₄×C₇	—	2	2	no	—
40	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₇	#41	1	2	yes	—
42	C₂×C₇×C₈	—	2	4	no	—
43	C₄²×C₇	—	2	2	no	—
44	C₄²×C₇	M₁₆	1	4	yes	—
45	C₄²×C₇	C₂²×C₄	1	2	yes	—
46	C₄²×C₇	#41	1	1	yes	characteristic
47	C₄²×C₇	#41	1	1	yes	characteristic
48	C₂×C₇⋊C₈	—	2	2	no	—
49	C₂×C₇⋊C₈	—	2	2	no	—
50	C₂×C₇⋊C₈	—	2	2	no	—
51	C₂×C₇⋊C₈	—	2	2	no	—
52	C₂×C₇⋊C₈	—	2	4	no	—
53	C₂×C₇⋊C₈	C₄²	1	2	yes	—
54	C₂×C₇⋊C₈	C₄²	1	1	yes	characteristic
55	C₂×C₇⋊C₈	C₄²	1	1	yes	characteristic
56	C₂×C₇⋊C₈	M₁₆	1	4	yes	—
57	#431	—	7	2	no	—
58	#431	C₂×D₇	1	2	yes	—
59	C₂×#97	—	7	2	no	—
60	C₂×#97	—	7	1	no	—
61	C₂×C₄×C₈	—	7	1	no	—
62	C₂×C₄×C₈	—	7	2	no	—
63	#376	C₂×D₇	1	1	yes	characteristic
64	#322	—	7	4	no	—
65	#322	—	7	4	no	—
66	C₂³×C₇	#117	1	1	yes	characteristic
67	C₂×C₄×C₇	—	2	4	no	—
68	C₂×C₄×C₇	—	4	2	no	—
69	C₂×C₄×C₇	—	4	4	no	—
70	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
71	C₂×C₄×C₇	#117	1	2	yes	—
72	C₂×C₄×C₇	#117	1	2	yes	—
73	C₂×C₄×C₇	#117	1	1	yes	characteristic
74	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
75	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
76	C₇×C₈	—	4	4	no	—
77	C₇⋊C₈	—	2	2	no	—
78	C₇⋊C₈	—	2	1	no	—
79	C₇⋊C₈	—	2	1	no	—
80	C₇⋊C₈	—	2	4	no	—
81	C₇⋊C₈	—	4	4	no	—
82	C₂²×C₈	—	7	1	no	—
83	C₂²×C₈	—	7	1	no	—
84	C₂²×C₈	—	7	2	no	—
85	C₂×C₄²	C₄×D₇	1	2	yes	—
86	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
87	C₄⋊C₈	—	2	2	no	—
88	C₄⋊C₈	—	14	2	no	—
89	C₂²⋊C₈	—	2	2	no	—
90	C₂²⋊C₈	—	14	2	no	—
91	#97	—	14	4	no	—
92	#97	—	14	1	no	—
93	#97	—	14	1	no	—
94	C₄×C₈	—	7	2	no	—
95	C₄×C₈	—	7	2	no	—
96	C₄×C₈	—	7	4	no	—
97	C₄×C₈	—	14	4	no	—
98	C₄×C₈	C₄×D₇	1	4	yes	—
99	#95	—	2	2	no	—
100	C₂²×C₇	—	2	2	no	—
101	C₂²×C₇	#376	1	1	yes	characteristic
102	C₂²×C₇	#432	1	2	yes	—
103	C₄×C₇	—	2	2	no	—
104	C₄×C₇	—	2	2	no	—
105	C₄×C₇	—	4	4	no	—
106	C₄×C₇	C₄×M₁₆	1	1	yes	characteristic
107	C₄×C₇	C₄×M₁₆	1	1	yes	characteristic
108	C₄×C₇	#432	1	2	yes	—
109	C₂²×C₄	—	2	2	no	—
110	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
111	C₂²×C₄	#1159	1	2	yes	—
112	C₂×C₈	—	2	4	no	—
113	C₂×C₈	—	7	2	no	—
114	C₂×C₈	—	7	1	no	—
115	C₂×C₈	—	7	1	no	—
116	C₂×C₈	—	7	4	no	—
117	C₂×C₈	—	14	2	no	—
118	C₂×C₈	—	14	2	no	—
119	C₂×C₈	—	14	2	no	—
120	C₂×C₈	—	14	2	no	—
121	C₂×C₈	—	14	4	no	—
122	C₄²	—	2	2	no	—
123	C₄²	#1131	1	4	yes	—
124	C₄²	C₂×C₄×D₇	1	2	yes	—
125	C₄²	#1159	1	1	yes	characteristic
126	C₄²	#1161	1	1	yes	characteristic
127	C₂×C₇	—	2	2	no	—
128	C₂×C₇	#1745	1	1	yes	characteristic
129	C₂×C₇	#1747	1	2	yes	—
130	C₂³	#6440	1	1	yes	characteristic
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	4	2	no	—
133	C₂×C₄	—	4	4	no	—
134	C₂×C₄	C₄²×D₇	1	1	yes	characteristic
135	C₂×C₄	#6440	1	2	yes	—
136	C₂×C₄	#6455	1	2	yes	—
137	C₂×C₄	#6455	1	1	yes	characteristic
138	C₂×C₄	C₂×#1131	1	2	yes	—
139	C₂×C₄	D₇×M₁₆	1	2	yes	—
140	C₈	—	4	4	no	—
141	C₈	—	14	4	no	—
142	C₈	—	14	2	no	—
143	C₈	—	14	1	no	—
144	C₈	—	14	1	no	—
145	C₈	—	28	4	no	—
146	C₇	#8139	1	1	yes	characteristic, complemented, Sylow
147	C₂²	—	2	2	no	—
148	C₂²	#89420	1	1	yes	characteristic
149	C₂²	#89491	1	2	yes	—
150	C₄	—	2	2	no	—
151	C₄	—	2	2	no	—
152	C₄	—	4	4	no	—
153	C₄	C₄×#1131	1	1	yes	characteristic
154	C₄	#89479	1	1	yes	characteristic
155	C₄	#89607	1	2	yes	—
156	C₂	—	2	2	no	—
157	C₂	#11726351	1	1	yes	characteristic
158	C₂	#11726451	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11728496

C₂

yes

maximal

#11727188

C₂

yes

maximal, characteristic

C₇×#1274

C₂

yes

maximal, characteristic

#11725546

C₂

yes

maximal

#11725420

C₂

yes

maximal, characteristic

C₇×#431

C₂²

yes

—

C₇×#376

C₂²

yes

characteristic

#89697

C₂²

yes

—

C₂×#6377

C₄

yes

—

C₂×#6377

C₂²

yes

characteristic

C₂×C₄×C₇⋊C₈

C₄

yes

—

C₂×C₄×C₇⋊C₈

C₂²

yes

characteristic

#89250

C₄

yes

complemented

#89249

C₄

yes

complemented

#8139

—

maximal, Sylow

C₂×C₄²×C₇

C₂×C₄

yes

—

C₂×C₄²×C₇

C₂³

yes

characteristic

C₂²×C₇⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₇⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₇⋊C₈

C₂×C₄

yes

—

C₇×C₄⋊C₈

—

C₇×C₂²⋊C₈

—

C₄×C₇×C₈

C₂×C₄

yes

—

C₇×#95

—

C₂²⋊(C₇⋊C₈)

—

C₄⋊(C₇⋊C₈)

—

#6377

—

#6377

—

#6377

—

C₄×C₇⋊C₈

—

C₄×C₇⋊C₈

C₂×C₄

yes

—

C₄×C₇⋊C₈

C₂×C₄

yes

—

C₄×C₇⋊C₈

C₂×C₄

yes

—

—

—

—

—

D₇

yes

characteristic

C₂²×C₄×C₇

—

C₂²×C₄×C₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇

#41

yes

—

C₂×C₇×C₈

—

C₄²×C₇

—

C₄²×C₇

M₁₆

yes

—

C₄²×C₇

C₂²×C₄

yes

—

C₄²×C₇

yes

C₄²×C₇

yes

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

C₄²

yes

—

C₂×C₇⋊C₈

C₄²

yes

characteristic

C₂×C₇⋊C₈

C₄²

yes

characteristic

C₂×C₇⋊C₈

M₁₆

yes

—

#431

—

#431

C₂×D₇

yes

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

#376

C₂×D₇

yes

characteristic

#322

—

#322

—

C₂³×C₇

#117

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₇

#117

yes

—

C₂×C₄×C₇

#117

yes

—

C₂×C₄×C₇

#117

yes

characteristic

C₂×C₄×C₇

C₂×M₁₆

yes

—

C₂×C₄×C₇

C₂×M₁₆

yes

—

C₇×C₈

—

C₇⋊C₈

—

C₇⋊C₈

—

C₇⋊C₈

—

C₇⋊C₈

—

C₇⋊C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

C₄×D₇

yes

—

C₂×C₄²

C₂²×D₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄×D₇

yes

—

#95

—

100

C₂²×C₇

—

101

C₂²×C₇

#376

yes

characteristic

102

C₂²×C₇

#432

yes

—

103

C₄×C₇

—

104

C₄×C₇

—

105

C₄×C₇

—

106

C₄×C₇

C₄×M₁₆

yes

characteristic

107

C₄×C₇

C₄×M₁₆

yes

characteristic

108

C₄×C₇

#432

yes

—

109

C₂²×C₄

—

110

C₂²×C₄

C₂×C₄×D₇

yes

characteristic

111

C₂²×C₄

#1159

yes

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₄²

—

123

C₄²

#1131

yes

—

124

C₄²

C₂×C₄×D₇

yes

—

125

C₄²

#1159

yes

characteristic

126

C₄²

#1161

yes

characteristic

127

C₂×C₇

—

128

C₂×C₇

#1745

yes

characteristic

129

C₂×C₇

#1747

yes

—

130

C₂³

#6440

yes

characteristic

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

C₄²×D₇

yes

characteristic

135

C₂×C₄

#6440

yes

—

136

C₂×C₄

#6455

yes

—

137

C₂×C₄

#6455

yes

characteristic

138

C₂×C₄

C₂×#1131

yes

—

139

C₂×C₄

D₇×M₁₆

yes

—

140

C₈

—

141

C₈

—

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₇

#8139

yes

characteristic, complemented, Sylow

147

C₂²

—

148

C₂²

#89420

yes

characteristic

149

C₂²

#89491

yes

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

C₄×#1131

yes

characteristic

154

C₄

#89479

yes

characteristic

155

C₄

#89607

yes

—

156

C₂

—

157

C₂

#11726351

yes

characteristic

158

C₂

#11726451

yes

—

	subgroup	G/H
center	C₂×C₄	C₄²×D₇
commutator	C₂×C₄×C₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#1274	C₂
socle	C₂²×C₇	#376
prime radical	C₂³×C₇	#117
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	#1274	D₇
hypocenter	C₇	#8139
LCS	G ▷ C₂×C₄×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ #1274

subgroup

G/H

center

C₂×C₄

C₄²×D₇

commutator

C₂×C₄×C₇

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₇

C₂

C₂²×C₇

C₂³×C₇

C₂×C₄²×C₇

C₂³

D₇

C₇

G ▷ C₂×C₄×C₇ ▷ C₇

C₁ ◁ C₂×C₄ ◁ #1274

N	H ≅ G/N	type
13. #89250	C₄	semidirect
14. #89249	C₄	semidirect
146. C₇	#8139	semidirect

H ≅ G/N

type

13. #89250

C₄

semidirect

14. #89249

C₄

semidirect

146. C₇

#8139

semidirect

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

subgroup	index
#11728496	2
#11727188	2
C₇×#1274	2
#11725546	2
#11725420	2
#8139	7

subgroup

index

Subgroups — #13244242

Conjugacy classes of subgroups

Complements

Maximal subgroups