Group #13196927 — subgroups

Subgroups — C₇×#19711 #13196927

Order 1792.

636 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 636 conjugacy classes of subgroups — too many to draw. Only its 230 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×#3231	C₂	1	4	yes	maximal
2	C₇×#3170	C₂	1	2	yes	maximal
3	C₇×#1723	C₂	1	4	yes	maximal, complemented
4	C₇×#1718	C₂	1	4	yes	maximal, complemented
5	C₇×#1452	C₂	1	1	yes	maximal, characteristic
6	C₇×#533	C₂²	1	4	yes	—
7	C₇×#500	C₂²	1	2	yes	—
8	C₇×#500	C₂²	1	4	yes	—
9	C₇×#498	C₂²	1	4	yes	—
10	C₇×#494	C₂²	1	2	yes	—
11	C₇×#487	C₂²	1	4	yes	—
12	C₇×#487	C₂²	1	4	yes	—
13	C₇×#487	C₂²	1	4	yes	—
14	C₇×#485	—	2	8	no	—
15	C₇×#479	—	2	8	no	—
16	C₇×#467	—	2	4	no	—
17	C₄×C₇×Q₁₆	—	2	4	no	—
18	C₇×#431	C₂²	1	2	yes	—
19	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
20	C₇×#322	C₂²	1	4	yes	—
21	#19711	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂×C₇×Q₁₆	—	2	4	no	—
23	C₂²×C₇×C₈	D₄	1	2	yes	—
24	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
25	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
26	C₇×#125	—	2	4	no	—
27	C₇×C₂²⋊Q₈	—	2	4	no	—
28	C₇×C₂²⋊Q₈	—	2	4	no	—
29	C₄×C₇×Q₈	—	2	4	no	—
30	C₄×C₇×Q₈	—	2	4	no	—
31	C₇×#117	—	2	4	no	—
32	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
33	C₇×#107	—	2	4	no	—
34	C₇×#107	—	2	8	no	—
35	C₇×#106	—	2	8	no	—
36	C₇×C₄⋊C₈	D₄	1	4	yes	—
37	C₇×#103	—	2	8	no	—
38	C₇×#103	—	2	8	no	—
39	C₇×#103	—	2	8	no	—
40	C₇×C₂²⋊C₈	D₄	1	4	yes	—
41	C₄×C₇×C₈	D₄	1	2	yes	—
42	C₄×C₇×C₈	C₂³	1	2	yes	—
43	C₄×C₇×C₈	C₂³	1	4	yes	—
44	#3231	C₂×C₇	1	4	yes	—
45	#3170	C₂×C₇	1	2	yes	—
46	#1723	C₂×C₇	1	4	yes	complemented
47	#1718	C₂×C₇	1	4	yes	complemented
48	#1452	C₂×C₇	1	1	yes	characteristic
49	C₂×C₇×Q₈	—	2	4	no	—
50	C₂×C₇×Q₈	—	2	4	no	—
51	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
53	C₇×Q₁₆	—	4	8	no	—
54	C₂×C₇×C₈	—	4	2	no	—
55	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
56	C₂×C₇×C₈	C₂×D₄	1	8	yes	—
57	C₇×C₄⋊C₄	—	2	4	no	—
58	C₇×C₄⋊C₄	—	2	4	no	—
59	C₇×C₄⋊C₄	—	2	4	no	—
60	C₇×C₄⋊C₄	—	2	4	no	—
61	C₇×C₄⋊C₄	—	4	4	no	—
62	C₇×C₄⋊C₄	—	4	4	no	—
63	C₇×C₂²⋊C₄	—	4	4	no	—
64	C₄²×C₇	—	4	4	no	—
65	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₇	C₂×D₄	1	2	yes	—
67	C₄²×C₇	C₂⁴	1	1	yes	characteristic
68	#533	C₂²×C₇	1	4	yes	—
69	#500	C₂²×C₇	1	2	yes	—
70	#500	C₂²×C₇	1	4	yes	—
71	#498	C₂²×C₇	1	4	yes	—
72	#494	C₂²×C₇	1	2	yes	—
73	#487	C₂²×C₇	1	4	yes	—
74	#487	C₂²×C₇	1	4	yes	—
75	#487	C₂²×C₇	1	4	yes	—
76	#485	—	2	8	no	—
77	#479	—	2	8	no	—
78	#467	—	2	4	no	—
79	C₄×Q₁₆	—	2	4	no	—
80	#431	C₂²×C₇	1	2	yes	—
81	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
82	#322	C₂²×C₇	1	4	yes	—
83	C₂³×C₇	#127	1	1	yes	characteristic
84	C₇×Q₈	—	4	8	no	—
85	C₇×Q₈	—	4	8	no	—
86	C₂×C₄×C₇	—	2	2	no	—
87	C₂×C₄×C₇	—	2	4	no	—
88	C₂×C₄×C₇	—	4	4	no	—
89	C₂×C₄×C₇	—	4	4	no	—
90	C₂×C₄×C₇	#127	1	2	yes	—
91	C₂×C₄×C₇	#127	1	1	yes	characteristic
92	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
93	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
94	C₇×C₈	—	2	4	no	—
95	C₇×C₈	—	2	8	no	—
96	C₂×Q₁₆	—	2	4	no	—
97	C₂²×C₈	C₇×D₄	1	2	yes	—
98	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
99	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
100	#125	—	2	4	no	—
101	C₂²⋊Q₈	—	2	4	no	—
102	C₂²⋊Q₈	—	2	4	no	—
103	C₄×Q₈	—	2	4	no	—
104	C₄×Q₈	—	2	4	no	—
105	#117	—	2	4	no	—
106	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
107	#107	—	2	4	no	—
108	#107	—	2	8	no	—
109	#106	—	2	8	no	—
110	C₄⋊C₈	C₇×D₄	1	4	yes	—
111	#103	—	2	8	no	—
112	#103	—	2	8	no	—
113	#103	—	2	8	no	—
114	C₂²⋊C₈	C₇×D₄	1	4	yes	—
115	C₄×C₈	C₇×D₄	1	2	yes	—
116	C₄×C₈	C₂³×C₇	1	2	yes	—
117	C₄×C₈	C₂³×C₇	1	4	yes	—
118	C₂²×C₇	—	2	1	no	—
119	C₂²×C₇	—	2	2	no	—
120	C₂²×C₇	C₂×#127	1	1	yes	characteristic
121	C₄×C₇	—	2	1	no	—
122	C₄×C₇	—	2	2	no	—
123	C₄×C₇	—	8	4	no	—
124	C₄×C₇	—	8	4	no	—
125	C₄×C₇	#575	1	2	yes	—
126	C₄×C₇	#578	1	4	yes	—
127	C₂×Q₈	—	2	4	no	—
128	C₂×Q₈	—	2	4	no	—
129	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
130	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
131	Q₁₆	—	4	8	no	—
132	C₂×C₈	—	4	2	no	—
133	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
134	C₂×C₈	C₂×C₇×D₄	1	8	yes	—
135	C₄⋊C₄	—	2	4	no	—
136	C₄⋊C₄	—	2	4	no	—
137	C₄⋊C₄	—	2	4	no	—
138	C₄⋊C₄	—	2	4	no	—
139	C₄⋊C₄	—	4	4	no	—
140	C₄⋊C₄	—	4	4	no	—
141	C₂²⋊C₄	—	4	4	no	—
142	C₄²	—	4	4	no	—
143	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
144	C₄²	C₂×C₇×D₄	1	2	yes	—
145	C₄²	C₂⁴×C₇	1	1	yes	characteristic
146	C₂×C₇	—	4	1	no	—
147	C₂×C₇	#3150	1	1	yes	characteristic
148	C₂×C₇	#3156	1	2	yes	—
149	C₂³	C₇×#127	1	1	yes	characteristic
150	Q₈	—	4	8	no	—
151	Q₈	—	4	8	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	4	4	no	—
155	C₂×C₄	—	4	4	no	—
156	C₂×C₄	C₇×#127	1	2	yes	—
157	C₂×C₄	C₇×#127	1	1	yes	characteristic
158	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
159	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
160	C₈	—	2	4	no	—
161	C₈	—	2	8	no	—
162	C₇	#19711	1	1	yes	characteristic, direct product factor, Sylow
163	C₂²	—	2	2	no	—
164	C₂²	—	2	1	no	—
165	C₂²	C₂×C₇×#127	1	1	yes	characteristic
166	C₄	—	2	2	no	—
167	C₄	—	2	1	no	—
168	C₄	—	8	4	no	—
169	C₄	—	8	4	no	—
170	C₄	C₇×#575	1	2	yes	—
171	C₄	C₇×#578	1	4	yes	—
172	C₂	—	4	1	no	—
173	C₂	C₇×#3150	1	1	yes	characteristic
174	C₂	C₇×#3156	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#3231

C₂

yes

maximal

C₇×#3170

C₂

yes

maximal

C₇×#1723

C₂

yes

maximal, complemented

C₇×#1718

C₂

yes

maximal, complemented

C₇×#1452

C₂

yes

maximal, characteristic

C₇×#533

C₂²

yes

—

C₇×#500

C₂²

yes

—

C₇×#500

C₂²

yes

—

C₇×#498

C₂²

yes

—

C₇×#494

C₂²

yes

—

C₇×#487

C₂²

yes

—

C₇×#487

C₂²

yes

—

C₇×#487

C₂²

yes

—

C₇×#485

—

C₇×#479

—

C₇×#467

—

C₄×C₇×Q₁₆

—

C₇×#431

C₂²

yes

—

C₂×C₄×C₇×C₈

C₂²

yes

characteristic

C₇×#322

C₂²

yes

—

#19711

C₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇×Q₁₆

—

C₂²×C₇×C₈

D₄

yes

—

C₇×C₄⋊Q₈

C₂³

yes

—

C₇×C₄⋊Q₈

C₂³

yes

—

C₇×#125

—

C₇×C₂²⋊Q₈

—

C₇×C₂²⋊Q₈

—

C₄×C₇×Q₈

—

C₄×C₇×Q₈

—

C₇×#117

—

C₂×C₄²×C₇

C₂³

yes

—

—

—

C₇×C₄⋊C₈

D₄

yes

—

C₇×#103

—

C₇×#103

—

C₇×#103

—

C₇×C₂²⋊C₈

D₄

yes

—

C₄×C₇×C₈

D₄

yes

—

C₄×C₇×C₈

C₂³

yes

—

C₄×C₇×C₈

C₂³

yes

—

#3231

C₂×C₇

yes

—

#3170

C₂×C₇

yes

—

#1723

C₂×C₇

yes

complemented

#1718

C₂×C₇

yes

complemented

#1452

C₂×C₇

yes

characteristic

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂²×C₄×C₇

C₂×D₄

yes

—

C₂²×C₄×C₇

C₂×D₄

yes

characteristic

C₇×Q₁₆

—

C₂×C₇×C₈

—

C₂×C₇×C₈

C₂×D₄

yes

—

C₂×C₇×C₈

C₂×D₄

yes

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₂²⋊C₄

—

C₄²×C₇

—

C₄²×C₇

C₂×D₄

yes

characteristic

C₄²×C₇

C₂×D₄

yes

—

C₄²×C₇

C₂⁴

yes

characteristic

#533

C₂²×C₇

yes

—

#500

C₂²×C₇

yes

—

#500

C₂²×C₇

yes

—

#498

C₂²×C₇

yes

—

#494

C₂²×C₇

yes

—

#487

C₂²×C₇

yes

—

#487

C₂²×C₇

yes

—

#487

C₂²×C₇

yes

—

#485

—

#479

—

#467

—

C₄×Q₁₆

—

#431

C₂²×C₇

yes

—

C₂×C₄×C₈

C₂²×C₇

yes

characteristic

#322

C₂²×C₇

yes

—

C₂³×C₇

#127

yes

characteristic

C₇×Q₈

—

C₇×Q₈

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

#127

yes

—

C₂×C₄×C₇

#127

yes

characteristic

C₂×C₄×C₇

C₂²×D₄

yes

characteristic

C₂×C₄×C₇

C₂²×D₄

yes

—

C₇×C₈

—

C₇×C₈

—

C₂×Q₁₆

—

C₂²×C₈

C₇×D₄

yes

—

C₄⋊Q₈

C₂³×C₇

yes

—

C₄⋊Q₈

C₂³×C₇

yes

—

100

#125

—

101

C₂²⋊Q₈

—

102

C₂²⋊Q₈

—

103

C₄×Q₈

—

104

C₄×Q₈

—

105

#117

—

106

C₂×C₄²

C₂³×C₇

yes

characteristic

107

#107

—

108

#107

—

109

#106

—

110

C₄⋊C₈

C₇×D₄

yes

—

111

#103

—

112

#103

—

113

#103

—

114

C₂²⋊C₈

C₇×D₄

yes

—

115

C₄×C₈

C₇×D₄

yes

—

116

C₄×C₈

C₂³×C₇

yes

—

117

C₄×C₈

C₂³×C₇

yes

—

118

C₂²×C₇

—

119

C₂²×C₇

—

120

C₂²×C₇

C₂×#127

yes

characteristic

121

C₄×C₇

—

122

C₄×C₇

—

123

C₄×C₇

—

124

C₄×C₇

—

125

C₄×C₇

#575

yes

—

126

C₄×C₇

#578

yes

—

127

C₂×Q₈

—

128

C₂×Q₈

—

129

C₂²×C₄

C₂×C₇×D₄

yes

—

130

C₂²×C₄

C₂×C₇×D₄

yes

characteristic

131

Q₁₆

—

132

C₂×C₈

—

133

C₂×C₈

C₂×C₇×D₄

yes

—

134

C₂×C₈

C₂×C₇×D₄

yes

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₂²⋊C₄

—

142

C₄²

—

143

C₄²

C₂×C₇×D₄

yes

characteristic

144

C₄²

C₂×C₇×D₄

yes

—

145

C₄²

C₂⁴×C₇

yes

characteristic

146

C₂×C₇

—

147

C₂×C₇

#3150

yes

characteristic

148

C₂×C₇

#3156

yes

—

149

C₂³

C₇×#127

yes

characteristic

150

Q₈

—

151

Q₈

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

C₇×#127

yes

—

157

C₂×C₄

C₇×#127

yes

characteristic

158

C₂×C₄

C₂²×C₇×D₄

yes

characteristic

159

C₂×C₄

C₂²×C₇×D₄

yes

—

160

C₈

—

161

C₈

—

162

C₇

#19711

yes

characteristic, direct product factor, Sylow

163

C₂²

—

164

C₂²

—

165

C₂²

C₂×C₇×#127

yes

characteristic

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

—

170

C₄

C₇×#575

yes

—

171

C₄

C₇×#578

yes

—

172

C₂

—

173

C₂

C₇×#3150

yes

characteristic

174

C₂

C₇×#3156

yes

—

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₂³×C₇	#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₇

C₂×#127

commutator, Frattini, cosocle

C₄²

C₂⁴×C₇

C₂³×C₇

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

N	H ≅ G/N	type
3. C₇×#1723	C₂	semidirect
4. C₇×#1718	C₂	semidirect
21. #19711	C₇	direct (N × H)
46. #1723	C₂×C₇	semidirect
47. #1718	C₂×C₇	semidirect
162. C₇	#19711	direct (N × H)

H ≅ G/N

type

3. C₇×#1723

C₂

semidirect

4. C₇×#1718

C₂

semidirect

21. #19711

C₇

direct (N × H)

46. #1723

C₂×C₇

semidirect

47. #1718

C₂×C₇

semidirect

162. C₇

#19711

direct (N × H)

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

subgroup	index
C₇×#3231	2
C₇×#3170	2
C₇×#1723	2
C₇×#1718	2
C₇×#1452	2
#19711	7

subgroup

index

Subgroups — C₇×#19711 #13196927

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C7×#19711 #13196927

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19711 #13196927