Group #13244244 — subgroups

Subgroups — #13244244

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	#11727558	C₂	1	1	yes	maximal, characteristic
3	C₇×#1275	C₂	1	1	yes	maximal, characteristic
4	#11725543	C₂	1	2	yes	maximal
5	#11725466	C₂	1	1	yes	maximal, characteristic
6	C₇×#431	C₂²	1	2	yes	—
7	C₂×C₇×#97	C₂²	1	1	yes	characteristic
8	C₂×#6389	C₄	1	2	yes	—
9	C₂×C₈×Dic₇	C₄	1	2	yes	—
10	#89704	C₂²	1	1	yes	characteristic
11	#89697	C₂²	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	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
17	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
18	C₂×C₄×Dic₇	C₂×C₄	1	2	yes	—
19	C₂²×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₇×C₄⋊C₈	—	2	2	no	—
22	C₇×C₂²⋊C₈	—	2	2	no	—
23	C₇×#97	—	2	2	no	—
24	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
25	C₇⋊#95	—	2	1	no	—
26	C₇⋊#95	—	2	1	no	—
27	C₂²⋊(C₇⋊C₈)	—	2	2	no	—
28	#6389	—	2	4	no	—
29	C₈×Dic₇	—	2	4	no	—
30	C₄⋊(C₇⋊C₈)	—	2	2	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	1	no	—
37	#1275	D₇	1	1	yes	characteristic
38	#1274	—	7	2	no	—
39	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₇	#41	1	1	yes	characteristic
41	C₂²×C₄×C₇	#41	1	1	yes	characteristic
42	C₂²×Dic₇	—	2	2	no	—
43	C₂×C₇×C₈	—	2	2	no	—
44	C₂×C₇×C₈	—	2	2	no	—
45	C₂×C₇×C₈	—	2	4	no	—
46	C₂×C₇×C₈	M₁₆	1	4	yes	—
47	C₄²×C₇	C₂²×C₄	1	2	yes	—
48	C₄²×C₇	#41	1	2	yes	—
49	C₄×Dic₇	—	2	2	no	—
50	C₄×Dic₇	M₁₆	1	4	yes	—
51	C₂×C₇⋊C₈	—	2	2	no	—
52	C₂×C₇⋊C₈	—	2	2	no	—
53	C₂×C₇⋊C₈	—	2	4	no	—
54	C₂×C₇⋊C₈	C₄²	1	1	yes	characteristic
55	C₂×C₇⋊C₈	C₄²	1	1	yes	characteristic
56	C₂×C₇⋊C₈	C₄²	1	2	yes	—
57	#431	—	7	2	no	—
58	#431	C₂×D₇	1	2	yes	—
59	C₂×#97	—	7	2	no	—
60	C₂×#97	C₂×D₇	1	1	yes	characteristic
61	C₂×C₄×C₈	—	7	1	no	—
62	C₂×C₄×C₈	—	7	2	no	—
63	#376	—	7	1	no	—
64	#322	—	7	4	no	—
65	#322	—	7	4	no	—
66	C₂³×C₇	#117	1	1	yes	characteristic
67	C₂×C₄×C₇	—	4	1	no	—
68	C₂×C₄×C₇	—	4	1	no	—
69	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
70	C₂×C₄×C₇	#117	1	2	yes	—
71	C₂×C₄×C₇	#117	1	1	yes	characteristic
72	C₂×C₄×C₇	#117	1	2	yes	—
73	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
74	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
75	C₂×Dic₇	—	2	4	no	—
76	C₂×Dic₇	—	4	2	no	—
77	C₂×Dic₇	—	4	2	no	—
78	C₇×C₈	—	2	4	no	—
79	C₇×C₈	—	4	4	no	—
80	C₇⋊C₈	—	2	2	no	—
81	C₇⋊C₈	—	2	1	no	—
82	C₇⋊C₈	—	2	1	no	—
83	C₇⋊C₈	—	4	4	no	—
84	C₂²×C₈	—	7	1	no	—
85	C₂²×C₈	—	7	1	no	—
86	C₂²×C₈	C₄×D₇	1	2	yes	—
87	C₂×C₄²	—	7	2	no	—
88	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
89	C₄⋊C₈	—	2	2	no	—
90	C₄⋊C₈	—	14	2	no	—
91	C₂²⋊C₈	—	2	2	no	—
92	C₂²⋊C₈	—	14	2	no	—
93	#97	—	2	2	no	—
94	#97	—	14	4	no	—
95	C₄×C₈	—	7	2	no	—
96	C₄×C₈	—	7	2	no	—
97	C₄×C₈	—	7	4	no	—
98	C₄×C₈	—	14	4	no	—
99	C₄×C₈	C₄×D₇	1	4	yes	—
100	#95	—	14	1	no	—
101	#95	—	14	1	no	—
102	C₂²×C₇	—	2	1	no	—
103	C₂²×C₇	—	2	1	no	—
104	C₂²×C₇	#376	1	1	yes	characteristic
105	C₂²×C₇	#432	1	2	yes	—
106	C₄×C₇	—	2	2	no	—
107	C₄×C₇	—	2	2	no	—
108	C₄×C₇	C₄×M₁₆	1	1	yes	characteristic
109	C₄×C₇	C₄×M₁₆	1	1	yes	characteristic
110	C₄×C₇	#432	1	2	yes	—
111	Dic₇	—	4	4	no	—
112	C₂²×C₄	—	14	2	no	—
113	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
114	C₂²×C₄	#1159	1	1	yes	characteristic
115	C₂²×C₄	#1161	1	1	yes	characteristic
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	4	no	—
119	C₂×C₈	—	7	2	no	—
120	C₂×C₈	—	7	1	no	—
121	C₂×C₈	—	7	1	no	—
122	C₂×C₈	—	14	2	no	—
123	C₂×C₈	—	14	2	no	—
124	C₂×C₈	—	14	4	no	—
125	C₂×C₈	#1131	1	4	yes	—
126	C₄²	—	7	4	no	—
127	C₄²	—	14	2	no	—
128	C₄²	C₂×C₄×D₇	1	2	yes	—
129	C₄²	#1159	1	2	yes	—
130	C₂×C₇	—	2	2	no	—
131	C₂×C₇	#1745	1	1	yes	characteristic
132	C₂×C₇	#1747	1	1	yes	characteristic
133	C₂×C₇	#1747	1	1	yes	characteristic
134	C₂³	#6455	1	1	yes	characteristic
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	14	4	no	—
138	C₂×C₄	—	28	2	no	—
139	C₂×C₄	—	28	2	no	—
140	C₂×C₄	C₄²×D₇	1	1	yes	characteristic
141	C₂×C₄	#6440	1	2	yes	—
142	C₂×C₄	#6440	1	1	yes	characteristic
143	C₂×C₄	#6455	1	2	yes	—
144	C₂×C₄	C₂×#1131	1	2	yes	—
145	C₂×C₄	D₇×M₁₆	1	2	yes	—
146	C₈	—	2	4	no	—
147	C₈	—	4	4	no	—
148	C₈	—	14	2	no	—
149	C₈	—	14	1	no	—
150	C₈	—	14	1	no	—
151	C₈	—	28	4	no	—
152	C₇	#8139	1	1	yes	characteristic, complemented, Sylow
153	C₂²	—	2	1	no	—
154	C₂²	—	2	1	no	—
155	C₂²	#89420	1	1	yes	characteristic
156	C₂²	#89607	1	2	yes	—
157	C₄	—	2	2	no	—
158	C₄	—	2	2	no	—
159	C₄	—	28	4	no	—
160	C₄	C₄×#1131	1	1	yes	characteristic
161	C₄	#89479	1	1	yes	characteristic
162	C₄	#89491	1	2	yes	—
163	C₂	—	2	2	no	—
164	C₂	#11726351	1	1	yes	characteristic
165	C₂	#11726451	1	1	yes	characteristic
166	C₂	#11726471	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11728496

C₂

yes

maximal

#11727558

C₂

yes

maximal, characteristic

C₇×#1275

C₂

yes

maximal, characteristic

#11725543

C₂

yes

maximal

#11725466

C₂

yes

maximal, characteristic

C₇×#431

C₂²

yes

—

C₂×C₇×#97

C₂²

yes

characteristic

C₂×#6389

C₄

yes

—

C₂×C₈×Dic₇

C₄

yes

—

#89704

C₂²

yes

characteristic

#89697

C₂²

yes

—

C₂×C₄×C₇⋊C₈

C₂²

yes

characteristic

#89250

C₄

yes

complemented

#89249

C₄

yes

complemented

#8139

—

maximal, Sylow

C₂²×C₇×C₈

C₂×C₄

yes

—

C₂×C₄²×C₇

C₂³

yes

characteristic

C₂×C₄×Dic₇

C₂×C₄

yes

—

C₂²×C₇⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₇⋊C₈

C₂×C₄

yes

characteristic

C₇×C₄⋊C₈

—

C₇×C₂²⋊C₈

—

C₇×#97

—

C₄×C₇×C₈

C₂×C₄

yes

—

C₇⋊#95

—

C₇⋊#95

—

C₂²⋊(C₇⋊C₈)

—

#6389

—

C₈×Dic₇

—

C₄⋊(C₇⋊C₈)

—

C₄×C₇⋊C₈

C₂×C₄

yes

—

C₄×C₇⋊C₈

C₂×C₄

yes

—

C₄×C₇⋊C₈

C₂×C₄

yes

—

—

—

—

D₇

yes

characteristic

#1274

—

C₂²×C₄×C₇

C₂²×C₄

yes

C₂²×C₄×C₇

yes

C₂²×C₄×C₇

yes

C₂²×Dic₇

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

M₁₆

yes

—

C₄²×C₇

C₂²×C₄

yes

—

C₄²×C₇

#41

yes

—

C₄×Dic₇

—

C₄×Dic₇

M₁₆

yes

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

C₄²

yes

characteristic

C₂×C₇⋊C₈

C₄²

yes

characteristic

C₂×C₇⋊C₈

C₄²

yes

—

#431

—

#431

C₂×D₇

yes

—

C₂×#97

—

C₂×#97

C₂×D₇

yes

characteristic

C₂×C₄×C₈

—

C₂×C₄×C₈

—

#376

—

#322

—

#322

—

C₂³×C₇

#117

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₇

#117

yes

—

C₂×C₄×C₇

#117

yes

characteristic

C₂×C₄×C₇

#117

yes

—

C₂×C₄×C₇

C₂×M₁₆

yes

—

C₂×C₄×C₇

C₂×M₁₆

yes

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₇×C₈

—

C₇×C₈

—

C₇⋊C₈

—

C₇⋊C₈

—

C₇⋊C₈

—

C₇⋊C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₄×D₇

yes

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄×D₇

yes

—

100

#95

—

101

#95

—

102

C₂²×C₇

—

103

C₂²×C₇

—

104

C₂²×C₇

#376

yes

characteristic

105

C₂²×C₇

#432

yes

—

106

C₄×C₇

—

107

C₄×C₇

—

108

C₄×C₇

C₄×M₁₆

yes

characteristic

109

C₄×C₇

C₄×M₁₆

yes

characteristic

110

C₄×C₇

#432

yes

—

111

Dic₇

—

112

C₂²×C₄

—

113

C₂²×C₄

C₂×C₄×D₇

yes

characteristic

114

C₂²×C₄

#1159

yes

characteristic

115

C₂²×C₄

#1161

yes

characteristic

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

#1131

yes

—

126

C₄²

—

127

C₄²

—

128

C₄²

C₂×C₄×D₇

yes

—

129

C₄²

#1159

yes

—

130

C₂×C₇

—

131

C₂×C₇

#1745

yes

characteristic

132

C₂×C₇

#1747

yes

characteristic

133

C₂×C₇

#1747

yes

characteristic

134

C₂³

#6455

yes

characteristic

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

C₄²×D₇

yes

characteristic

141

C₂×C₄

#6440

yes

—

142

C₂×C₄

#6440

yes

characteristic

143

C₂×C₄

#6455

yes

—

144

C₂×C₄

C₂×#1131

yes

—

145

C₂×C₄

D₇×M₁₆

yes

—

146

C₈

—

147

C₈

—

148

C₈

—

149

C₈

—

150

C₈

—

151

C₈

—

152

C₇

#8139

yes

characteristic, complemented, Sylow

153

C₂²

—

154

C₂²

—

155

C₂²

#89420

yes

characteristic

156

C₂²

#89607

yes

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

C₄×#1131

yes

characteristic

161

C₄

#89479

yes

characteristic

162

C₄

#89491

yes

—

163

C₂

—

164

C₂

#11726351

yes

characteristic

165

C₂

#11726451

yes

characteristic

166

C₂

#11726471

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₄²×D₇
commutator	C₂×C₄×C₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#1275	C₂
socle	C₂²×C₇	#376
prime radical	C₂³×C₇	#117
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	#1275	D₇
hypocenter	C₇	#8139
LCS	G ▷ C₂×C₄×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ #1275

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 ◁ #1275

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

H ≅ G/N

type

13. #89250

C₄

semidirect

14. #89249

C₄

semidirect

152. C₇

#8139

semidirect

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

subgroup	index
#11728496	2
#11727558	2
C₇×#1275	2
#11725543	2
#11725466	2
#8139	7

subgroup

index

Subgroups — #13244244

Conjugacy classes of subgroups

Complements

Maximal subgroups