Group #13258829 — subgroups

Subgroups — #13258829

Order 1792.

316 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 101 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	#11728522	C₂	1	2	yes	maximal
2	#11728503	C₂	1	2	yes	maximal
3	#11727195	C₂	1	1	yes	maximal, characteristic
4	#11727193	C₂	1	1	yes	maximal, characteristic
5	C₇×#1274	C₂	1	1	yes	maximal, characteristic
6	#90161	C₂²	1	1	yes	characteristic
7	C₇×#431	C₂²	1	2	yes	—
8	C₇×#376	C₂²	1	1	yes	characteristic
9	#89696	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊(C₇⋊C₈)	—	2	2	no	—
11	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
12	#89318	—	2	2	no	—
13	#89252	—	2	4	no	—
14	#89251	—	2	4	no	—
15	#10670	—	7	1	no	maximal, Sylow
16	C₂×C₄²×C₇	D₄	1	2	yes	—
17	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
18	#6489	—	2	1	no	—
19	#6487	—	2	2	no	—
20	C₂×#1136	—	2	1	no	—
21	C₂²×C₇⋊C₈	—	2	2	no	—
22	#6433	—	2	1	no	—
23	#6432	Q₈	1	2	yes	—
24	C₄×Dic₁₄	—	2	2	no	—
25	C₇×C₄⋊C₈	—	2	2	no	—
26	C₇×C₂²⋊C₈	—	2	2	no	—
27	C₄×C₇×C₈	D₄	1	4	yes	—
28	C₇×#95	—	2	2	no	—
29	C₄⋊(C₇⋊C₈)	—	2	2	no	—
30	C₄⋊(C₇⋊C₈)	—	2	2	no	—
31	C₄⋊(C₇⋊C₈)	—	2	2	no	—
32	C₄⋊(C₇⋊C₈)	—	4	4	no	—
33	#1995	—	7	1	no	—
34	#1843	—	7	1	no	—
35	#1570	—	7	2	no	—
36	#1481	—	7	2	no	—
37	#1274	D₇	1	1	yes	characteristic
38	C₂²×C₄×C₇	—	2	2	no	—
39	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₇	#41	1	2	yes	—
41	C₂×Dic₁₄	—	2	2	no	—
42	C₂×C₇×C₈	—	2	4	no	—
43	C₄²×C₇	—	2	2	no	—
44	C₄²×C₇	SD₁₆	1	4	yes	—
45	C₄²×C₇	C₂×D₄	1	2	yes	—
46	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
47	C₄²×C₇	#41	1	1	yes	characteristic
48	C₂²⋊Dic₇	—	4	1	no	—
49	C₄⋊Dic₇	—	2	2	no	—
50	C₄⋊Dic₇	—	2	2	no	—
51	Dic₇⋊C₄	—	4	2	no	—
52	C₄×Dic₇	—	4	1	no	—
53	#1136	—	4	2	no	—
54	C₂×C₇⋊C₈	—	4	2	no	—
55	C₂×C₇⋊C₈	—	4	1	no	—
56	C₂×C₇⋊C₈	—	4	4	no	—
57	#533	—	7	1	no	—
58	#431	C₂×D₇	1	2	yes	—
59	#423	—	7	1	no	—
60	C₂×C₄⋊C₈	—	7	2	no	—
61	C₂×C₄⋊C₈	—	14	2	no	—
62	#376	C₂×D₇	1	1	yes	characteristic
63	#339	—	14	2	no	—
64	#334	—	14	4	no	—
65	Q₈⋊C₈	—	14	4	no	—
66	C₂³×C₇	#123	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	C₂²⋊Q₈	1	2	yes	—
71	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₇	#123	1	2	yes	—
73	C₂×C₄×C₇	#127	1	1	yes	characteristic
74	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
75	C₂×C₄×C₇	#137	1	2	yes	—
76	C₂×Dic₇	—	4	2	no	—
77	Dic₁₄	—	4	4	no	—
78	C₇×C₈	—	4	4	no	—
79	C₇⋊C₈	—	8	2	no	—
80	C₇⋊C₈	—	8	1	no	—
81	C₂×M₁₆	—	14	1	no	—
82	C₂²×C₈	—	14	2	no	—
83	C₄⋊Q₈	—	7	2	no	—
84	#125	—	14	1	no	—
85	C₂²⋊Q₈	—	14	2	no	—
86	C₄×Q₈	—	14	2	no	—
87	#117	—	14	1	no	—
88	C₂×C₄²	D₂₈	1	2	yes	—
89	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
90	C₄⋊C₈	—	2	2	no	—
91	C₄⋊C₈	—	14	2	no	—
92	C₄⋊C₈	—	14	2	no	—
93	C₄⋊C₈	—	14	2	no	—
94	C₄⋊C₈	—	28	4	no	—
95	C₂²⋊C₈	—	2	2	no	—
96	C₄×C₈	D₂₈	1	4	yes	—
97	#95	—	2	2	no	—
98	C₂²×C₇	—	2	2	no	—
99	C₂²×C₇	#399	1	1	yes	characteristic
100	C₂²×C₇	#483	1	2	yes	—
101	C₄×C₇	—	2	2	no	—
102	C₄×C₇	—	2	2	no	—
103	C₄×C₇	—	4	4	no	—
104	C₄×C₇	#475	1	2	yes	—
105	C₄×C₇	#492	1	1	yes	characteristic
106	C₄×C₇	#497	1	1	yes	characteristic
107	Dic₇	—	8	2	no	—
108	C₂×Q₈	—	14	2	no	—
109	C₂²×C₄	—	2	2	no	—
110	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
111	C₂²×C₄	#1159	1	2	yes	—
112	M₁₆	—	28	2	no	—
113	C₂×C₈	—	2	4	no	—
114	C₂×C₈	—	28	2	no	—
115	C₂×C₈	—	28	1	no	—
116	C₂×C₈	—	28	4	no	—
117	C₄⋊C₄	—	14	2	no	—
118	C₄⋊C₄	—	14	2	no	—
119	C₄⋊C₄	—	28	2	no	—
120	C₂²⋊C₄	—	28	1	no	—
121	C₄²	—	2	2	no	—
122	C₄²	—	28	1	no	—
123	C₄²	#1132	1	4	yes	—
124	C₄²	C₂×D₂₈	1	2	yes	—
125	C₄²	#1159	1	1	yes	characteristic
126	C₄²	Q₈×D₇	1	1	yes	characteristic
127	C₂×C₇	—	2	2	no	—
128	C₂×C₇	#2096	1	1	yes	characteristic
129	C₂×C₇	#2098	1	2	yes	—
130	C₂³	#6449	1	1	yes	characteristic
131	Q₈	—	28	4	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	4	4	no	—
135	C₂×C₄	—	28	2	no	—
136	C₂×C₄	#6437	1	1	yes	characteristic
137	C₂×C₄	#6449	1	2	yes	—
138	C₂×C₄	#6460	1	2	yes	—
139	C₂×C₄	#6460	1	1	yes	characteristic
140	C₂×C₄	C₂×#1132	1	2	yes	—
141	C₂×C₄	#6472	1	2	yes	—
142	C₈	—	4	4	no	—
143	C₈	—	56	2	no	—
144	C₈	—	56	1	no	—
145	C₇	#10670	1	1	yes	characteristic, complemented, Sylow
146	C₂²	—	2	2	no	—
147	C₂²	#89449	1	1	yes	characteristic
148	C₂²	#89493	1	2	yes	—
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	4	4	no	—
152	C₄	—	56	2	no	—
153	C₄	#89465	1	1	yes	characteristic
154	C₄	#89487	1	1	yes	characteristic
155	C₄	#89623	1	2	yes	—
156	C₂	—	2	2	no	—
157	C₂	#11726446	1	1	yes	characteristic
158	C₂	#11726469	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11728522

C₂

yes

maximal

#11728503

C₂

yes

maximal

#11727195

C₂

yes

maximal, characteristic

#11727193

C₂

yes

maximal, characteristic

C₇×#1274

C₂

yes

maximal, characteristic

#90161

C₂²

yes

characteristic

C₇×#431

C₂²

yes

—

C₇×#376

C₂²

yes

characteristic

#89696

C₂²

yes

characteristic

C₂×C₄⋊(C₇⋊C₈)

—

C₂×C₄⋊(C₇⋊C₈)

C₂²

yes

—

—

—

—

—

C₂×C₄²×C₇

D₄

yes

—

C₂×C₄²×C₇

C₂³

yes

—

—

—

C₂²×C₇⋊C₈

—

#6433

—

#6432

Q₈

yes

—

C₄×Dic₁₄

—

C₇×C₄⋊C₈

—

C₇×C₂²⋊C₈

—

C₄×C₇×C₈

D₄

yes

—

C₇×#95

—

C₄⋊(C₇⋊C₈)

—

C₄⋊(C₇⋊C₈)

—

C₄⋊(C₇⋊C₈)

—

C₄⋊(C₇⋊C₈)

—

—

—

—

—

D₇

yes

characteristic

C₂²×C₄×C₇

—

C₂²×C₄×C₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₇

#41

yes

—

C₂×Dic₁₄

—

C₂×C₇×C₈

—

C₄²×C₇

—

C₄²×C₇

SD₁₆

yes

—

C₄²×C₇

C₂×D₄

yes

—

C₄²×C₇

C₂×Q₈

yes

characteristic

C₄²×C₇

#41

yes

characteristic

C₂²⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

Dic₇⋊C₄

—

C₄×Dic₇

—

#1136

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

#533

—

#431

C₂×D₇

yes

—

#423

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

#376

C₂×D₇

yes

characteristic

#339

—

#334

—

Q₈⋊C₈

—

C₂³×C₇

#123

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₇

#123

yes

—

C₂×C₄×C₇

#127

yes

characteristic

C₂×C₄×C₇

C₂×SD₁₆

yes

—

C₂×C₄×C₇

#137

yes

—

C₂×Dic₇

—

Dic₁₄

—

C₇×C₈

—

C₇⋊C₈

—

C₇⋊C₈

—

C₂×M₁₆

—

C₂²×C₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

D₂₈

yes

—

C₂×C₄²

C₂²×D₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

D₂₈

yes

—

#95

—

C₂²×C₇

—

C₂²×C₇

#399

yes

characteristic

100

C₂²×C₇

#483

yes

—

101

C₄×C₇

—

102

C₄×C₇

—

103

C₄×C₇

—

104

C₄×C₇

#475

yes

—

105

C₄×C₇

#492

yes

characteristic

106

C₄×C₇

#497

yes

characteristic

107

Dic₇

—

108

C₂×Q₈

—

109

C₂²×C₄

—

110

C₂²×C₄

C₂×D₂₈

yes

characteristic

111

C₂²×C₄

#1159

yes

—

112

M₁₆

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₂²⋊C₄

—

121

C₄²

—

122

C₄²

—

123

C₄²

#1132

yes

—

124

C₄²

C₂×D₂₈

yes

—

125

C₄²

#1159

yes

characteristic

126

C₄²

Q₈×D₇

yes

characteristic

127

C₂×C₇

—

128

C₂×C₇

#2096

yes

characteristic

129

C₂×C₇

#2098

yes

—

130

C₂³

#6449

yes

characteristic

131

Q₈

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

#6437

yes

characteristic

137

C₂×C₄

#6449

yes

—

138

C₂×C₄

#6460

yes

—

139

C₂×C₄

#6460

yes

characteristic

140

C₂×C₄

C₂×#1132

yes

—

141

C₂×C₄

#6472

yes

—

142

C₈

—

143

C₈

—

144

C₈

—

145

C₇

#10670

yes

characteristic, complemented, Sylow

146

C₂²

—

147

C₂²

#89449

yes

characteristic

148

C₂²

#89493

yes

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

#89465

yes

characteristic

154

C₄

#89487

yes

characteristic

155

C₄

#89623

yes

—

156

C₂

—

157

C₂

#11726446

yes

characteristic

158

C₂

#11726469

yes

—

	subgroup	G/H
center	C₂×C₄	#6437
commutator, cosocle	C₂×C₄²×C₇	C₂³
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#1274	C₂
socle	C₂²×C₇	#399
prime radical	C₂³×C₇	#123
hypercenter	#1274	D₇
hypocenter	C₇	#10670
LCS	G ▷ C₂×C₄²×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1274

subgroup

G/H

center

C₂×C₄

#6437

commutator, cosocle

C₂×C₄²×C₇

C₂³

Frattini

C₂×C₄²

C₂²×D₇

C₂

C₂²×C₇

C₂³×C₇

D₇

C₇

G ▷ C₂×C₄²×C₇ ▷ C₂²×C₇ ▷ C₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1274

N	H ≅ G/N	type
145. C₇	#10670	semidirect

H ≅ G/N

type

145. C₇

#10670

semidirect

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

subgroup	index
#11728522	2
#11728503	2
#11727195	2
#11727193	2
C₇×#1274	2
#10670	7

subgroup

index

Subgroups — #13258829

Conjugacy classes of subgroups

Complements

Maximal subgroups