Group #13244243 — subgroups

Subgroups — #13244243

Order 1792.

330 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 139 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×#1452	C₂	1	1	yes	maximal, characteristic
2	#11727503	C₂	1	1	yes	maximal, characteristic
3	#11727480	C₂	1	1	yes	maximal, characteristic
4	#11725546	C₂	1	2	yes	maximal
5	#11725543	C₂	1	2	yes	maximal
6	C₇×#431	C₂²	1	2	yes	—
7	C₂×C₄×C₇×C₈	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×#6377	C₂²	1	1	yes	characteristic
13	C₇×#322	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	1	yes	characteristic
17	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
19	C₂×C₄×Dic₇	C₂×C₄	1	2	yes	—
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	2	yes	—
24	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
25	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
26	C₇⋊#95	—	2	2	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	#6377	—	2	2	no	—
32	C₄×C₇⋊C₈	C₂×C₄	1	4	yes	—
33	#1726	—	7	1	no	—
34	#1452	—	7	1	no	—
35	#1452	D₇	1	1	yes	characteristic
36	#1275	—	7	2	no	—
37	#1274	—	7	2	no	—
38	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₇	#41	1	2	yes	—
40	C₂²×Dic₇	—	2	2	no	—
41	C₂×C₇×C₈	—	2	2	no	—
42	C₂×C₇×C₈	—	2	2	no	—
43	C₂×C₇×C₈	—	2	4	no	—
44	C₂×C₇×C₈	C₄²	1	2	yes	—
45	C₂×C₇×C₈	C₄²	1	2	yes	—
46	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
47	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
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	M₁₆	1	4	yes	—
55	#431	—	7	2	no	—
56	#431	C₂×D₇	1	2	yes	—
57	C₂×#97	—	7	2	no	—
58	C₂×#97	—	7	1	no	—
59	C₂×C₄×C₈	—	7	2	no	—
60	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
61	#376	—	7	1	no	—
62	#322	—	7	4	no	—
63	#322	Dic₇	1	4	yes	complemented
64	C₂³×C₇	#117	1	1	yes	characteristic
65	C₂×C₄×C₇	—	4	2	no	—
66	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
67	C₂×C₄×C₇	#117	1	2	yes	—
68	C₂×C₄×C₇	#117	1	1	yes	characteristic
69	C₂×C₄×C₇	#117	1	2	yes	—
70	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
71	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
72	C₂×Dic₇	—	2	4	no	—
73	C₂×Dic₇	—	4	4	no	—
74	C₇×C₈	—	2	2	no	—
75	C₇×C₈	—	2	2	no	—
76	C₇×C₈	—	4	4	no	—
77	C₇⋊C₈	—	2	4	no	—
78	C₇⋊C₈	—	4	4	no	—
79	C₂²×C₈	—	7	2	no	—
80	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
81	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
82	C₂×C₄²	—	7	2	no	—
83	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
84	C₄⋊C₈	—	2	2	no	—
85	C₄⋊C₈	—	14	2	no	—
86	C₂²⋊C₈	—	2	2	no	—
87	C₂²⋊C₈	—	14	2	no	—
88	#97	—	14	4	no	—
89	#97	—	14	2	no	—
90	C₄×C₈	—	7	4	no	—
91	C₄×C₈	—	14	4	no	—
92	C₄×C₈	C₄×D₇	1	2	yes	—
93	C₄×C₈	C₂×Dic₇	1	4	yes	—
94	C₄×C₈	C₂×Dic₇	1	2	yes	—
95	#95	—	14	2	no	—
96	C₂²×C₇	—	2	2	no	—
97	C₂²×C₇	#376	1	1	yes	characteristic
98	C₂²×C₇	#432	1	2	yes	—
99	C₄×C₇	—	2	2	no	—
100	C₄×C₇	—	2	2	no	—
101	C₄×C₇	C₄×M₁₆	1	1	yes	characteristic
102	C₄×C₇	C₄×M₁₆	1	1	yes	characteristic
103	C₄×C₇	#432	1	2	yes	—
104	Dic₇	—	4	4	no	—
105	C₂²×C₄	—	14	2	no	—
106	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
107	C₂²×C₄	#1157	1	2	yes	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	2	4	no	—
111	C₂×C₈	—	7	4	no	—
112	C₂×C₈	—	14	2	no	—
113	C₂×C₈	—	14	2	no	—
114	C₂×C₈	—	14	4	no	—
115	C₂×C₈	C₄×Dic₇	1	2	yes	—
116	C₂×C₈	C₄×Dic₇	1	2	yes	—
117	C₄²	—	7	4	no	—
118	C₄²	—	14	2	no	—
119	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
120	C₄²	#1157	1	2	yes	—
121	C₄²	C₂²×Dic₇	1	1	yes	characteristic
122	C₂×C₇	—	2	2	no	—
123	C₂×C₇	#1745	1	1	yes	characteristic
124	C₂×C₇	#1747	1	2	yes	—
125	C₂³	#6435	1	1	yes	characteristic
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	—	14	4	no	—
128	C₂×C₄	—	28	4	no	—
129	C₂×C₄	#6435	1	2	yes	—
130	C₂×C₄	#6435	1	1	yes	characteristic
131	C₂×C₄	D₇×M₁₆	1	2	yes	—
132	C₂×C₄	D₇×M₁₆	1	2	yes	—
133	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
134	C₂×C₄	#6489	1	2	yes	—
135	C₈	—	2	2	no	—
136	C₈	—	2	2	no	—
137	C₈	—	4	4	no	—
138	C₈	—	14	4	no	—
139	C₈	—	28	4	no	—
140	C₇	#8139	1	1	yes	characteristic, complemented, Sylow
141	C₂²	—	2	2	no	—
142	C₂²	#89477	1	2	yes	—
143	C₂²	#89704	1	1	yes	characteristic
144	C₄	—	2	2	no	—
145	C₄	—	2	2	no	—
146	C₄	—	28	4	no	—
147	C₄	#89477	1	2	yes	—
148	C₄	M₁₆×Dic₇	1	1	yes	characteristic
149	C₄	M₁₆×Dic₇	1	1	yes	characteristic
150	C₂	—	2	2	no	—
151	C₂	#11727559	1	1	yes	characteristic
152	C₂	#11727649	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#1452

C₂

yes

maximal, characteristic

#11727503

C₂

yes

maximal, characteristic

#11727480

C₂

yes

maximal, characteristic

#11725546

C₂

yes

maximal

#11725543

C₂

yes

maximal

C₇×#431

C₂²

yes

—

C₂×C₄×C₇×C₈

C₂²

yes

characteristic

C₂×#6389

C₄

yes

—

C₂×C₈×Dic₇

C₄

yes

—

#89704

C₂²

yes

characteristic

#89697

C₂²

yes

—

C₂×#6377

C₂²

yes

characteristic

C₇×#322

C₄

yes

complemented

#89249

C₄

yes

complemented

#8139

—

maximal, Sylow

C₂²×C₇×C₈

C₂×C₄

yes

characteristic

C₂²×C₇×C₈

C₂×C₄

yes

characteristic

C₂×C₄²×C₇

C₂³

yes

characteristic

C₂×C₄×Dic₇

C₂×C₄

yes

—

C₂²×C₇⋊C₈

C₂×C₄

yes

—

C₇×C₄⋊C₈

—

C₇×C₂²⋊C₈

—

C₄×C₇×C₈

C₂×C₄

yes

—

C₄×C₇×C₈

C₂×C₄

yes

—

C₄×C₇×C₈

C₂×C₄

yes

—

C₇⋊#95

—

C₂²⋊(C₇⋊C₈)

—

#6389

—

C₈×Dic₇

—

C₄⋊(C₇⋊C₈)

—

#6377

—

C₄×C₇⋊C₈

C₂×C₄

yes

—

#1726

—

#1452

—

#1452

D₇

yes

characteristic

#1275

—

#1274

—

C₂²×C₄×C₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇

#41

yes

—

C₂²×Dic₇

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

C₄²

yes

—

C₂×C₇×C₈

C₄²

yes

—

C₄²×C₇

C₂²×C₄

yes

characteristic

C₄²×C₇

C₂²×C₄

yes

characteristic

C₄²×C₇

#41

yes

—

C₄×Dic₇

—

C₄×Dic₇

M₁₆

yes

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

M₁₆

yes

—

#431

—

#431

C₂×D₇

yes

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₇

yes

—

—

Dic₇

yes

complemented

C₂³×C₇

#117

yes

characteristic

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

—

C₇×C₈

—

C₇×C₈

—

C₇×C₈

—

C₇⋊C₈

—

C₇⋊C₈

—

C₂²×C₈

—

C₂²×C₈

C₄×D₇

yes

characteristic

C₂²×C₈

C₄×D₇

yes

characteristic

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×D₇

yes

—

C₄×C₈

C₂×Dic₇

yes

—

C₄×C₈

C₂×Dic₇

yes

—

#95

—

C₂²×C₇

—

C₂²×C₇

#376

yes

characteristic

C₂²×C₇

#432

yes

—

C₄×C₇

—

100

C₄×C₇

—

101

C₄×C₇

C₄×M₁₆

yes

characteristic

102

C₄×C₇

C₄×M₁₆

yes

characteristic

103

C₄×C₇

#432

yes

—

104

Dic₇

—

105

C₂²×C₄

—

106

C₂²×C₄

C₂×C₄×D₇

yes

characteristic

107

C₂²×C₄

#1157

yes

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

C₄×Dic₇

yes

—

116

C₂×C₈

C₄×Dic₇

yes

—

117

C₄²

—

118

C₄²

—

119

C₄²

C₂×C₄×D₇

yes

characteristic

120

C₄²

#1157

yes

—

121

C₄²

C₂²×Dic₇

yes

characteristic

122

C₂×C₇

—

123

C₂×C₇

#1745

yes

characteristic

124

C₂×C₇

#1747

yes

—

125

C₂³

#6435

yes

characteristic

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

#6435

yes

—

130

C₂×C₄

#6435

yes

characteristic

131

C₂×C₄

D₇×M₁₆

yes

—

132

C₂×C₄

D₇×M₁₆

yes

—

133

C₂×C₄

C₂×C₄×Dic₇

yes

characteristic

134

C₂×C₄

#6489

yes

—

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

—

139

C₈

—

140

C₇

#8139

yes

characteristic, complemented, Sylow

141

C₂²

—

142

C₂²

#89477

yes

—

143

C₂²

#89704

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

#89477

yes

—

148

C₄

M₁₆×Dic₇

yes

characteristic

149

C₄

M₁₆×Dic₇

yes

characteristic

150

C₂

—

151

C₂

#11727559

yes

characteristic

152

C₂

#11727649

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₇
commutator	C₂×C₄×C₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#1452	C₂
socle	C₂²×C₇	#376
prime radical	C₂³×C₇	#117
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	#1452	D₇
hypocenter	C₇	#8139
LCS	G ▷ C₂×C₄×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ #1452

subgroup

G/H

center

C₂×C₄

C₂×C₄×Dic₇

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 ◁ #1452

N	H ≅ G/N	type
13. C₇×#322	C₄	semidirect
14. #89249	C₄	semidirect
63. #322	Dic₇	semidirect
140. C₇	#8139	semidirect

H ≅ G/N

type

13. C₇×#322

C₄

semidirect

14. #89249

C₄

semidirect

63. #322

Dic₇

semidirect

140. C₇

#8139

semidirect

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

subgroup	index
C₇×#1452	2
#11727503	2
#11727480	2
#11725546	2
#11725543	2
#8139	7

subgroup

index

Subgroups — #13244243

Conjugacy classes of subgroups

Complements

Maximal subgroups