Group #13258607 — subgroups

Subgroups — #13258607

Order 1792.

294 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 85 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×#1838	C₂	1	1	yes	maximal, characteristic
2	#11728260	C₂	1	2	yes	maximal
3	#11727576	C₂	1	1	yes	maximal, characteristic
4	#11725519	C₂	1	2	yes	maximal
5	#11725467	C₂	1	1	yes	maximal, characteristic
6	C₂×#6432	C₂²	1	1	yes	characteristic
7	C₇×#423	C₂²	1	2	yes	—
8	C₂×C₇×#97	C₂²	1	1	yes	characteristic
9	C₂×#6390	—	2	2	no	—
10	#89708	C₂²	1	1	yes	characteristic
11	#89697	C₂²	1	2	yes	—
12	C₇×#341	—	2	2	no	—
13	#89277	—	2	4	no	—
14	#10635	—	7	1	no	maximal, Sylow
15	C₂²×Dic₁₄	—	2	1	no	—
16	C₂×C₇×M₁₆	—	2	2	no	—
17	C₂²×C₇×C₈	D₄	1	2	yes	—
18	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
19	C₂×C₄⋊Dic₇	—	2	2	no	—
20	C₂×C₄⋊Dic₇	—	2	1	no	—
21	#6432	—	2	2	no	—
22	#6432	—	2	1	no	—
23	#6432	D₄	1	2	yes	—
24	C₇×C₄⋊C₈	—	2	2	no	—
25	C₇×C₄⋊C₈	—	2	2	no	—
26	C₇×#97	—	2	2	no	—
27	C₇⋊#95	—	2	2	no	—
28	C₂²⋊(C₇⋊C₈)	—	2	2	no	—
29	#6390	—	4	4	no	—
30	C₄⋊(C₇⋊C₈)	—	2	2	no	—
31	C₄×C₇⋊C₈	—	2	2	no	—
32	#1960	—	7	1	no	—
33	#1838	D₇	1	1	yes	characteristic
34	#1546	—	7	2	no	—
35	#1286	—	7	2	no	—
36	#1281	—	7	1	no	—
37	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₇	#41	1	2	yes	—
39	C₂²×Dic₇	—	4	1	no	—
40	C₂²×Dic₇	—	4	1	no	—
41	C₂×Dic₁₄	—	2	2	no	—
42	C₂×Dic₁₄	—	4	2	no	—
43	C₂×Dic₁₄	—	4	2	no	—
44	C₂×Dic₁₄	—	4	1	no	—
45	C₇×M₁₆	—	4	2	no	—
46	C₇×M₁₆	—	4	2	no	—
47	C₂×C₇×C₈	—	2	2	no	—
48	C₂×C₇×C₈	—	2	2	no	—
49	C₂×C₇×C₈	—	2	2	no	—
50	C₂×C₇×C₈	—	4	2	no	—
51	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
52	C₄²×C₇	#41	1	2	yes	—
53	C₄²×C₇	#41	1	1	yes	characteristic
54	C₄⋊Dic₇	—	2	4	no	—
55	C₄⋊Dic₇	—	4	2	no	—
56	C₄⋊Dic₇	—	4	2	no	—
57	C₂×C₇⋊C₈	—	4	2	no	—
58	C₂×C₄⋊Q₈	—	7	1	no	—
59	#431	—	7	2	no	—
60	#423	C₂×D₇	1	2	yes	—
61	C₂×#103	—	14	2	no	—
62	C₂×#97	C₂×D₇	1	1	yes	characteristic
63	#382	—	7	1	no	—
64	#341	—	2	2	no	—
65	#332	—	14	4	no	—
66	C₂³×C₇	#124	1	1	yes	characteristic
67	C₂×C₄×C₇	—	4	2	no	—
68	C₂×C₄×C₇	#121	1	1	yes	characteristic
69	C₂×C₄×C₇	#123	1	2	yes	—
70	C₂×C₄×C₇	#123	1	1	yes	characteristic
71	C₂×C₄×C₇	#126	1	2	yes	—
72	C₂×C₄×C₇	#135	1	2	yes	—
73	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
74	C₂×Dic₇	—	4	2	no	—
75	C₂×Dic₇	—	8	1	no	—
76	C₂×Dic₇	—	8	1	no	—
77	C₂×Dic₇	—	8	2	no	—
78	C₂×Dic₇	—	8	1	no	—
79	Dic₁₄	—	4	4	no	—
80	Dic₁₄	—	8	2	no	—
81	C₇×C₈	—	4	2	no	—
82	C₇×C₈	—	8	2	no	—
83	C₇⋊C₈	—	8	2	no	—
84	C₂²×Q₈	—	14	1	no	—
85	C₂×M₁₆	—	2	2	no	—
86	C₂²×C₈	C₂²⋊D₇	1	2	yes	—
87	C₄⋊Q₈	—	7	2	no	—
88	C₄⋊Q₈	—	14	2	no	—
89	C₄⋊Q₈	—	14	1	no	—
90	C₂×C₄⋊C₄	—	14	2	no	—
91	C₂×C₄⋊C₄	—	14	1	no	—
92	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
93	C₄⋊C₈	—	2	2	no	—
94	C₄⋊C₈	—	2	2	no	—
95	C₄⋊C₈	—	14	2	no	—
96	#103	—	28	4	no	—
97	C₂²⋊C₈	—	14	2	no	—
98	#97	—	2	2	no	—
99	C₄×C₈	—	14	2	no	—
100	#95	—	14	2	no	—
101	C₂²×C₇	—	2	2	no	—
102	C₂²×C₇	#397	1	1	yes	characteristic
103	C₂²×C₇	#488	1	1	yes	characteristic
104	C₂²×C₇	#490	1	1	yes	characteristic
105	C₄×C₇	—	2	2	no	—
106	C₄×C₇	—	2	2	no	—
107	C₄×C₇	#463	1	2	yes	—
108	C₄×C₇	#482	1	2	yes	—
109	Dic₇	—	8	2	no	—
110	Dic₇	—	16	1	no	—
111	C₂×Q₈	—	14	2	no	—
112	C₂×Q₈	—	28	1	no	—
113	C₂×Q₈	—	28	2	no	—
114	C₂×Q₈	—	28	2	no	—
115	C₂²×C₄	—	28	1	no	—
116	C₂²×C₄	—	28	1	no	—
117	C₂²×C₄	#1161	1	2	yes	—
118	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
119	M₁₆	—	4	2	no	—
120	M₁₆	—	4	2	no	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	4	2	no	—
125	C₂×C₈	—	28	2	no	—
126	C₄⋊C₄	—	14	4	no	—
127	C₄⋊C₄	—	28	2	no	—
128	C₄⋊C₄	—	28	2	no	—
129	C₄²	#1157	1	2	yes	—
130	C₄²	D₄×D₇	1	1	yes	characteristic
131	C₄²	#1159	1	1	yes	characteristic
132	C₂×C₇	—	2	1	no	—
133	C₂×C₇	—	2	1	no	—
134	C₂×C₇	#2073	1	1	yes	characteristic
135	C₂×C₇	#2082	1	2	yes	—
136	C₂³	#6510	1	1	yes	characteristic
137	Q₈	—	28	4	no	—
138	Q₈	—	56	2	no	—
139	C₂×C₄	—	4	2	no	—
140	C₂×C₄	—	28	2	no	—
141	C₂×C₄	—	56	1	no	—
142	C₂×C₄	—	56	1	no	—
143	C₂×C₄	—	56	2	no	—
144	C₂×C₄	—	56	1	no	—
145	C₂×C₄	#6457	1	2	yes	—
146	C₂×C₄	#6461	1	2	yes	—
147	C₂×C₄	#6492	1	1	yes	characteristic
148	C₂×C₄	#6501	1	1	yes	characteristic
149	C₂×C₄	#6512	1	2	yes	—
150	C₂×C₄	#6513	1	2	yes	—
151	C₈	—	4	2	no	—
152	C₈	—	8	2	no	—
153	C₈	—	56	2	no	—
154	C₇	#10635	1	1	yes	characteristic, complemented, Sylow
155	C₂²	—	2	2	no	—
156	C₂²	#89765	1	1	yes	characteristic
157	C₂²	#89839	1	1	yes	characteristic
158	C₂²	#89840	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	2	2	no	—
161	C₄	—	56	2	no	—
162	C₄	—	112	1	no	—
163	C₄	#89775	1	2	yes	—
164	C₄	#89829	1	2	yes	—
165	C₂	—	2	1	no	—
166	C₂	—	2	1	no	—
167	C₂	#11727311	1	1	yes	characteristic
168	C₂	#11728487	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#1838

C₂

yes

maximal, characteristic

#11728260

C₂

yes

maximal

#11727576

C₂

yes

maximal, characteristic

#11725519

C₂

yes

maximal

#11725467

C₂

yes

maximal, characteristic

C₂×#6432

C₂²

yes

characteristic

C₇×#423

C₂²

yes

—

C₂×C₇×#97

C₂²

yes

characteristic

C₂×#6390

—

#89708

C₂²

yes

characteristic

#89697

C₂²

yes

—

—

—

—

C₂²×Dic₁₄

—

C₂×C₇×M₁₆

—

C₂²×C₇×C₈

D₄

yes

—

C₂×C₄²×C₇

C₂³

yes

characteristic

C₂×C₄⋊Dic₇

—

C₂×C₄⋊Dic₇

—

#6432

—

#6432

—

#6432

D₄

yes

—

C₇×C₄⋊C₈

—

C₇×C₄⋊C₈

—

C₇×#97

—

C₇⋊#95

—

C₂²⋊(C₇⋊C₈)

—

#6390

—

C₄⋊(C₇⋊C₈)

—

C₄×C₇⋊C₈

—

#1960

—

#1838

D₇

yes

—

—

—

C₂²×C₄×C₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₇

#41

yes

—

C₂²×Dic₇

—

C₂²×Dic₇

—

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₇×M₁₆

—

C₇×M₁₆

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₄²×C₇

C₂×D₄

yes

characteristic

C₄²×C₇

#41

yes

—

C₄²×C₇

#41

yes

characteristic

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₂×C₇⋊C₈

—

C₂×C₄⋊Q₈

—

#431

—

#423

C₂×D₇

yes

—

C₂×#103

—

C₂×#97

C₂×D₇

yes

—

—

—

C₂³×C₇

#124

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

#121

yes

characteristic

C₂×C₄×C₇

#123

yes

—

C₂×C₄×C₇

#123

yes

characteristic

C₂×C₄×C₇

#126

yes

—

C₂×C₄×C₇

#135

yes

—

C₂×C₄×C₇

AGL(1,ℤ/8)

yes

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

Dic₁₄

—

Dic₁₄

—

C₇×C₈

—

C₇×C₈

—

C₇⋊C₈

—

C₂²×Q₈

—

C₂×M₁₆

—

C₂²×C₈

C₂²⋊D₇

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

100

#95

—

101

C₂²×C₇

—

102

C₂²×C₇

#397

yes

characteristic

103

C₂²×C₇

#488

yes

characteristic

104

C₂²×C₇

#490

yes

characteristic

105

C₄×C₇

—

106

C₄×C₇

—

107

C₄×C₇

#463

yes

—

108

C₄×C₇

#482

yes

—

109

Dic₇

—

110

Dic₇

—

111

C₂×Q₈

—

112

C₂×Q₈

—

113

C₂×Q₈

—

114

C₂×Q₈

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

#1161

yes

—

118

C₂²×C₄

C₂×C₂²⋊D₇

yes

characteristic

119

M₁₆

—

120

M₁₆

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄²

#1157

yes

—

130

C₄²

D₄×D₇

yes

characteristic

131

C₄²

#1159

yes

characteristic

132

C₂×C₇

—

133

C₂×C₇

—

134

C₂×C₇

#2073

yes

characteristic

135

C₂×C₇

#2082

yes

—

136

C₂³

#6510

yes

characteristic

137

Q₈

—

138

Q₈

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

#6457

yes

—

146

C₂×C₄

#6461

yes

—

147

C₂×C₄

#6492

yes

characteristic

148

C₂×C₄

#6501

yes

characteristic

149

C₂×C₄

#6512

yes

—

150

C₂×C₄

#6513

yes

—

151

C₈

—

152

C₈

—

153

C₈

—

154

C₇

#10635

yes

characteristic, complemented, Sylow

155

C₂²

—

156

C₂²

#89765

yes

characteristic

157

C₂²

#89839

yes

characteristic

158

C₂²

#89840

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

112

—

163

C₄

#89775

yes

—

164

C₄

#89829

yes

—

165

C₂

—

166

C₂

—

167

C₂

#11727311

yes

characteristic

168

C₂

#11728487

yes

—

	subgroup	G/H
center	C₂²	#89765
commutator, cosocle	C₂×C₄²×C₇	C₂³
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#1838	C₂
socle	C₂²×C₇	#397
prime radical	C₂³×C₇	#124
hypercenter	#1838	D₇
hypocenter	C₇	#10635
LCS	G ▷ C₂×C₄²×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1838

subgroup

G/H

center

C₂²

#89765

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 ◁ C₂² ◁ C₂×C₄² ◁ #1838

N	H ≅ G/N	type
154. C₇	#10635	semidirect

H ≅ G/N

type

154. C₇

#10635

semidirect

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

subgroup	index
C₇×#1838	2
#11728260	2
#11727576	2
#11725519	2
#11725467	2
#10635	7

subgroup

index

Subgroups — #13258607

Conjugacy classes of subgroups

Complements

Maximal subgroups