Group #13247499 — subgroups

Subgroups — #13247499

Order 1792.

270 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 115 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	#11727545	C₂	1	1	yes	maximal, characteristic
3	#11727520	C₂	1	1	yes	maximal, characteristic
4	#11725544	C₂	1	2	yes	maximal
5	#11725544	C₂	1	2	yes	maximal
6	C₇×#431	C₂²	1	2	yes	—
7	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₂×#6392	—	2	1	no	—
9	C₂×#6391	—	2	1	no	—
10	#89708	C₂²	1	2	yes	—
11	#89696	C₂²	1	1	yes	characteristic
12	#89696	C₂²	1	1	yes	characteristic
13	C₇×#322	C₄	1	4	yes	complemented
14	#89347	—	2	1	no	—
15	#89347	—	2	1	no	—
16	#8795	—	7	1	no	maximal, Sylow
17	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
18	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
19	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
20	C₂×C₄⋊Dic₇	—	2	2	no	—
21	C₂×#1136	—	2	1	no	—
22	C₂×#1136	—	2	1	no	—
23	C₇×C₄⋊C₈	—	2	2	no	—
24	C₇×C₂²⋊C₈	—	2	2	no	—
25	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
26	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
27	C₄×C₇×C₈	Q₈	1	2	yes	—
28	C₇⋊#95	—	2	4	no	—
29	#6392	—	4	2	no	—
30	#6391	—	4	2	no	—
31	C₄⋊(C₇⋊C₈)	—	2	2	no	—
32	C₄⋊(C₇⋊C₈)	—	2	2	no	—
33	#1846	—	7	1	no	—
34	#1837	—	7	1	no	—
35	#1452	D₇	1	1	yes	characteristic
36	#1286	—	7	2	no	—
37	#1286	—	7	2	no	—
38	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₇	#41	1	2	yes	—
40	C₂²×Dic₇	—	4	2	no	—
41	C₂×C₇×C₈	—	2	1	no	—
42	C₂×C₇×C₈	—	2	1	no	—
43	C₂×C₇×C₈	—	2	1	no	—
44	C₂×C₇×C₈	—	2	1	no	—
45	C₂×C₇×C₈	—	2	4	no	—
46	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
47	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
48	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
49	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
50	C₄²×C₇	#41	1	2	yes	—
51	C₄⋊Dic₇	—	4	2	no	—
52	C₄⋊Dic₇	—	4	2	no	—
53	#1136	—	4	1	no	—
54	#1136	—	4	1	no	—
55	#1136	—	4	1	no	—
56	#1136	—	4	1	no	—
57	C₂×C₇⋊C₈	—	4	1	no	—
58	C₂×C₇⋊C₈	—	4	1	no	—
59	#431	C₂×D₇	1	2	yes	—
60	C₂×#107	—	14	1	no	—
61	C₂×#106	—	14	1	no	—
62	#423	—	7	1	no	—
63	#423	—	7	1	no	—
64	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
65	#382	—	7	2	no	—
66	#341	—	14	1	no	—
67	#341	—	14	1	no	—
68	#322	Dic₇	1	4	yes	complemented
69	C₂³×C₇	#124	1	1	yes	characteristic
70	C₂×C₄×C₇	—	4	2	no	—
71	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₇	#117	1	2	yes	—
73	C₂×C₄×C₇	#124	1	1	yes	characteristic
74	C₂×C₄×C₇	#125	1	2	yes	—
75	C₂×C₄×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
76	C₂×C₄×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₂×C₄×C₇	#137	1	1	yes	characteristic
78	C₂×C₄×C₇	#137	1	1	yes	characteristic
79	C₂×Dic₇	—	8	4	no	—
80	C₂×Dic₇	—	8	2	no	—
81	C₇×C₈	—	2	2	no	—
82	C₇×C₈	—	2	2	no	—
83	C₇×C₈	—	4	4	no	—
84	C₇⋊C₈	—	8	1	no	—
85	C₇⋊C₈	—	8	1	no	—
86	C₂×M₁₆	—	14	1	no	—
87	C₂×M₁₆	—	14	1	no	—
88	C₂²×C₈	D₂₈	1	1	yes	characteristic
89	C₂²×C₈	D₂₈	1	1	yes	characteristic
90	C₂×C₄⋊C₄	—	14	2	no	—
91	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
92	#107	—	28	2	no	—
93	#106	—	28	2	no	—
94	C₄⋊C₈	—	2	2	no	—
95	C₄⋊C₈	—	14	2	no	—
96	C₄⋊C₈	—	14	2	no	—
97	C₂²⋊C₈	—	2	2	no	—
98	C₄×C₈	Dic₁₄	1	2	yes	—
99	C₄×C₈	C₂×Dic₇	1	4	yes	—
100	C₄×C₈	C₂×Dic₇	1	2	yes	—
101	#95	—	14	4	no	—
102	C₂²×C₇	—	2	2	no	—
103	C₂²×C₇	#382	1	1	yes	characteristic
104	C₂²×C₇	#490	1	1	yes	characteristic
105	C₂²×C₇	#491	1	1	yes	characteristic
106	C₄×C₇	—	2	2	no	—
107	C₄×C₇	—	2	2	no	—
108	C₄×C₇	#428	1	1	yes	characteristic
109	C₄×C₇	#428	1	1	yes	characteristic
110	C₄×C₇	#489	1	1	yes	characteristic
111	C₄×C₇	#489	1	1	yes	characteristic
112	Dic₇	—	16	2	no	—
113	C₂²×C₄	—	28	2	no	—
114	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
115	C₂²×C₄	#1157	1	2	yes	—
116	M₁₆	—	28	1	no	—
117	M₁₆	—	28	1	no	—
118	M₁₆	—	28	1	no	—
119	M₁₆	—	28	1	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	2	4	no	—
125	C₂×C₈	—	28	1	no	—
126	C₂×C₈	—	28	1	no	—
127	C₂×C₈	C₄⋊Dic₇	1	2	yes	—
128	C₂×C₈	C₄⋊Dic₇	1	2	yes	—
129	C₄⋊C₄	—	28	2	no	—
130	C₄⋊C₄	—	28	2	no	—
131	C₄²	C₂×Dic₁₄	1	1	yes	characteristic
132	C₄²	#1157	1	2	yes	—
133	C₄²	C₂²×Dic₇	1	1	yes	characteristic
134	C₂×C₇	—	2	1	no	—
135	C₂×C₇	—	2	1	no	—
136	C₂×C₇	#1833	1	2	yes	—
137	C₂×C₇	#1836	1	1	yes	characteristic
138	C₂³	#6438	1	1	yes	characteristic
139	C₂×C₄	—	4	2	no	—
140	C₂×C₄	—	56	4	no	—
141	C₂×C₄	—	56	2	no	—
142	C₂×C₄	#6433	1	2	yes	—
143	C₂×C₄	#6438	1	1	yes	characteristic
144	C₂×C₄	#6471	1	1	yes	characteristic
145	C₂×C₄	#6471	1	1	yes	characteristic
146	C₂×C₄	#6472	1	1	yes	characteristic
147	C₂×C₄	#6472	1	1	yes	characteristic
148	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
149	C₂×C₄	#6489	1	2	yes	—
150	C₈	—	2	2	no	—
151	C₈	—	2	2	no	—
152	C₈	—	4	4	no	—
153	C₈	—	56	1	no	—
154	C₈	—	56	1	no	—
155	C₇	#8795	1	1	yes	characteristic, complemented, Sylow
156	C₂²	—	2	2	no	—
157	C₂²	#89475	1	1	yes	characteristic
158	C₂²	#89485	1	1	yes	characteristic
159	C₂²	#89708	1	1	yes	characteristic
160	C₄	—	2	2	no	—
161	C₄	—	2	2	no	—
162	C₄	—	112	2	no	—
163	C₄	#89486	1	1	yes	characteristic
164	C₄	#89486	1	1	yes	characteristic
165	C₄	#89893	1	1	yes	characteristic
166	C₄	#89893	1	1	yes	characteristic
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	#11727569	1	1	yes	characteristic
170	C₂	#11727650	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#1452

C₂

yes

maximal, characteristic

#11727545

C₂

yes

maximal, characteristic

#11727520

C₂

yes

maximal, characteristic

#11725544

C₂

yes

maximal

#11725544

C₂

yes

maximal

C₇×#431

C₂²

yes

—

C₂×C₄×C₇×C₈

C₂²

yes

—

—

C₂²

yes

—

#89696

C₂²

yes

characteristic

#89696

C₂²

yes

characteristic

C₇×#322

C₄

yes

—

—

—

C₂²×C₇×C₈

D₄

yes

characteristic

C₂²×C₇×C₈

D₄

yes

characteristic

C₂×C₄²×C₇

C₂³

yes

characteristic

C₂×C₄⋊Dic₇

—

C₂×#1136

—

C₂×#1136

—

C₇×C₄⋊C₈

—

C₇×C₂²⋊C₈

—

C₄×C₇×C₈

C₂×C₄

yes

—

C₄×C₇×C₈

C₂×C₄

yes

—

C₄×C₇×C₈

Q₈

yes

—

C₇⋊#95

—

#6392

—

#6391

—

C₄⋊(C₇⋊C₈)

—

C₄⋊(C₇⋊C₈)

—

#1846

—

#1837

—

#1452

D₇

yes

characteristic

#1286

—

#1286

—

C₂²×C₄×C₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₇

#41

yes

—

C₂²×Dic₇

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

C₄⋊C₄

yes

—

C₂×C₇×C₈

C₄⋊C₄

yes

—

C₄²×C₇

C₂²×C₄

yes

characteristic

C₄²×C₇

C₂×Q₈

yes

characteristic

C₄²×C₇

#41

yes

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

—

—

—

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

#431

C₂×D₇

yes

—

—

—

—

—

C₂×C₄×C₈

C₂×D₇

yes

—

—

—

Dic₇

yes

complemented

C₂³×C₇

#124

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇

#117

yes

—

C₂×C₄×C₇

#124

yes

characteristic

C₂×C₄×C₇

#125

yes

—

C₂×C₄×C₇

yes

C₂×C₄×C₇

yes

C₂×C₄×C₇

yes

C₂×C₄×C₇

yes

C₂×Dic₇

—

C₂×Dic₇

—

C₇×C₈

—

C₇×C₈

—

C₇×C₈

—

C₇⋊C₈

—

C₇⋊C₈

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

D₂₈

yes

characteristic

C₂²×C₈

D₂₈

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₇

yes

characteristic

#107

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

Dic₁₄

yes

—

C₄×C₈

C₂×Dic₇

yes

—

100

C₄×C₈

C₂×Dic₇

yes

—

101

#95

—

102

C₂²×C₇

—

103

C₂²×C₇

#382

yes

characteristic

104

C₂²×C₇

#490

yes

characteristic

105

C₂²×C₇

#491

yes

characteristic

106

C₄×C₇

—

107

C₄×C₇

—

108

C₄×C₇

#428

yes

characteristic

109

C₄×C₇

#428

yes

characteristic

110

C₄×C₇

#489

yes

characteristic

111

C₄×C₇

#489

yes

characteristic

112

Dic₇

—

113

C₂²×C₄

—

114

C₂²×C₄

C₂×D₂₈

yes

characteristic

115

C₂²×C₄

#1157

yes

—

116

M₁₆

—

117

M₁₆

—

118

M₁₆

—

119

M₁₆

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

C₄⋊Dic₇

yes

—

128

C₂×C₈

C₄⋊Dic₇

yes

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄²

C₂×Dic₁₄

yes

characteristic

132

C₄²

#1157

yes

—

133

C₄²

C₂²×Dic₇

yes

characteristic

134

C₂×C₇

—

135

C₂×C₇

—

136

C₂×C₇

#1833

yes

—

137

C₂×C₇

#1836

yes

characteristic

138

C₂³

#6438

yes

characteristic

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

#6433

yes

—

143

C₂×C₄

#6438

yes

characteristic

144

C₂×C₄

#6471

yes

characteristic

145

C₂×C₄

#6471

yes

characteristic

146

C₂×C₄

#6472

yes

characteristic

147

C₂×C₄

#6472

yes

characteristic

148

C₂×C₄

C₂×C₄⋊Dic₇

yes

characteristic

149

C₂×C₄

#6489

yes

—

150

C₈

—

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

—

155

C₇

#8795

yes

characteristic, complemented, Sylow

156

C₂²

—

157

C₂²

#89475

yes

characteristic

158

C₂²

#89485

yes

characteristic

159

C₂²

#89708

yes

characteristic

160

C₄

—

161

C₄

—

162

C₄

—

112

—

163

C₄

#89486

yes

characteristic

164

C₄

#89486

yes

characteristic

165

C₄

#89893

yes

characteristic

166

C₄

#89893

yes

characteristic

167

C₂

—

168

C₂

—

169

C₂

#11727569

yes

characteristic

170

C₂

#11727650

yes

—

	subgroup	G/H
center	C₂²	#89708
commutator	C₄²×C₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#1452	C₂
socle	C₂²×C₇	#382
prime radical	C₂³×C₇	#124
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	#1452	D₇
hypocenter	C₇	#8795
LCS	G ▷ C₄²×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

subgroup

G/H

center

C₂²

#89708

commutator

C₄²×C₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇

C₂

C₂²×C₇

C₂³×C₇

C₂×C₄²×C₇

C₂³

D₇

C₇

G ▷ C₄²×C₇ ▷ C₂²×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

N	H ≅ G/N	type
13. C₇×#322	C₄	semidirect
68. #322	Dic₇	semidirect
155. C₇	#8795	semidirect

H ≅ G/N

type

13. C₇×#322

C₄

semidirect

68. #322

Dic₇

semidirect

155. C₇

#8795

semidirect

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

subgroup	index
C₇×#1452	2
#11727545	2
#11727520	2
#11725544	2
#8795	7

subgroup

index

Subgroups — #13247499

Conjugacy classes of subgroups

Complements

Maximal subgroups