Group #13201021 — subgroups

Subgroups — C₇×#23805 #13201021

Order 1792.

974 conjugacy classes of subgroups; only the 286 normal subgroups are shown.

normal subgroups only

This group has 974 conjugacy classes of subgroups — too many to draw. Only its 286 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	C₂×C₇×#441	C₂	1	4	yes	maximal
2	C₄×C₇×C₄⋊Q₈	C₂	1	2	yes	maximal
3	C₇×#2096	C₂	1	4	yes	maximal
4	C₇×#2016	C₂	1	4	yes	maximal
5	C₇×#1745	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇×Q₁₆	C₂²	1	4	yes	—
7	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×C₄×C₇×Q₈	C₂²	1	4	yes	—
9	C₇×#441	—	2	16	no	—
10	C₂×C₇×#106	C₂²	1	4	yes	—
11	C₂×C₇×#103	C₂²	1	8	yes	—
12	C₂×C₇×#97	C₂²	1	2	yes	—
13	C₇×#391	C₂²	1	4	yes	—
14	C₇×#389	—	2	4	no	—
15	C₇×#384	C₂²	1	2	yes	—
16	C₄×C₇×C₄⋊C₄	—	2	4	no	—
17	C₄³×C₇	C₂²	1	1	yes	characteristic
18	C₇×#336	C₂²	1	4	yes	—
19	#23805	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂²×C₇×Q₈	C₂³	1	4	yes	—
21	C₂×C₇×Q₁₆	—	2	16	no	—
22	C₂×C₇×Q₁₆	—	2	8	no	—
23	C₂²×C₇×C₈	C₂³	1	4	yes	—
24	C₇×C₄⋊Q₈	—	2	8	no	—
25	C₄×C₇×Q₈	—	2	16	no	—
26	C₂×C₇×C₄⋊C₄	—	2	4	no	—
27	C₂×C₇×C₄⋊C₄	—	2	4	no	—
28	C₂×C₇×C₄⋊C₄	—	2	4	no	—
29	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
30	C₂×C₄²×C₇	—	2	4	no	—
31	C₂×C₄²×C₇	D₄	1	4	yes	—
32	C₂×C₄²×C₇	C₂³	1	2	yes	—
33	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
34	C₇×#106	—	2	8	no	—
35	C₇×#103	—	2	16	no	—
36	C₇×#97	D₄	1	8	yes	—
37	C₇×#95	—	2	8	no	—
38	C₂×#441	C₂×C₇	1	4	yes	—
39	C₄×C₄⋊Q₈	C₂×C₇	1	2	yes	—
40	#2096	C₂×C₇	1	4	yes	—
41	#2016	C₂×C₇	1	4	yes	—
42	#1745	C₂×C₇	1	1	yes	characteristic
43	C₂×C₇×Q₈	—	2	4	no	—
44	C₂×C₇×Q₈	—	2	8	no	—
45	C₂×C₇×Q₈	—	2	8	no	—
46	C₂×C₇×Q₈	#41	1	8	yes	—
47	C₂²×C₄×C₇	—	2	4	no	—
48	C₂²×C₄×C₇	—	2	4	no	—
49	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
51	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
52	C₇×Q₁₆	—	2	32	no	—
53	C₂×C₇×C₈	—	2	8	no	—
54	C₂×C₇×C₈	C₂×D₄	1	8	yes	—
55	C₇×C₄⋊C₄	—	2	16	no	—
56	C₇×C₄⋊C₄	—	2	8	no	—
57	C₇×C₄⋊C₄	—	4	8	no	—
58	C₇×C₄⋊C₄	—	4	8	no	—
59	C₄²×C₇	—	2	8	no	—
60	C₄²×C₇	—	2	4	no	—
61	C₄²×C₇	—	4	8	no	—
62	C₄²×C₇	C₂×D₄	1	4	yes	—
63	C₂²×Q₁₆	C₂²×C₇	1	4	yes	—
64	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
65	C₂×C₄×Q₈	C₂²×C₇	1	4	yes	—
66	#441	—	2	16	no	—
67	C₂×#106	C₂²×C₇	1	4	yes	—
68	C₂×#103	C₂²×C₇	1	8	yes	—
69	C₂×#97	C₂²×C₇	1	2	yes	—
70	#391	C₂²×C₇	1	4	yes	—
71	#389	—	2	4	no	—
72	#384	C₂²×C₇	1	2	yes	—
73	C₄×C₄⋊C₄	—	2	4	no	—
74	C₄³	C₂²×C₇	1	1	yes	characteristic
75	#336	C₂²×C₇	1	4	yes	—
76	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
77	C₇×Q₈	—	2	16	no	—
78	C₇×Q₈	—	4	8	no	—
79	C₂×C₄×C₇	—	2	4	no	—
80	C₂×C₄×C₇	—	2	4	no	—
81	C₂×C₄×C₇	—	2	4	no	—
82	C₂×C₄×C₇	—	2	8	no	—
83	C₂×C₄×C₇	—	4	4	no	—
84	C₂×C₄×C₇	—	4	4	no	—
85	C₂×C₄×C₇	—	4	4	no	—
86	C₂×C₄×C₇	—	4	8	no	—
87	C₂×C₄×C₇	#127	1	4	yes	—
88	C₂×C₄×C₇	#137	1	8	yes	—
89	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
90	C₂×C₄×C₇	C₂×#41	1	4	yes	—
91	C₇×C₈	—	2	8	no	—
92	C₂²×Q₈	C₂³×C₇	1	4	yes	—
93	C₂×Q₁₆	—	2	16	no	—
94	C₂×Q₁₆	—	2	8	no	—
95	C₂²×C₈	C₂³×C₇	1	4	yes	—
96	C₄⋊Q₈	—	2	8	no	—
97	C₄×Q₈	—	2	16	no	—
98	C₂×C₄⋊C₄	—	2	4	no	—
99	C₂×C₄⋊C₄	—	2	4	no	—
100	C₂×C₄⋊C₄	—	2	4	no	—
101	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
102	C₂×C₄²	—	2	4	no	—
103	C₂×C₄²	C₇×D₄	1	4	yes	—
104	C₂×C₄²	C₂³×C₇	1	2	yes	—
105	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
106	#106	—	2	8	no	—
107	#103	—	2	16	no	—
108	#97	C₇×D₄	1	8	yes	—
109	#95	—	2	8	no	—
110	C₂²×C₇	#525	1	2	yes	—
111	C₂²×C₇	C₂×#127	1	1	yes	characteristic
112	C₂²×C₇	C₂×#137	1	4	yes	—
113	C₄×C₇	—	2	8	no	—
114	C₄×C₇	—	2	4	no	—
115	C₄×C₇	—	4	8	no	—
116	C₄×C₇	—	8	4	no	—
117	C₄×C₇	#525	1	4	yes	—
118	C₂×Q₈	—	2	4	no	—
119	C₂×Q₈	—	2	8	no	—
120	C₂×Q₈	—	2	8	no	—
121	C₂×Q₈	C₇×#41	1	8	yes	—
122	C₂²×C₄	—	2	4	no	—
123	C₂²×C₄	—	2	4	no	—
124	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
125	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
126	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
127	Q₁₆	—	2	32	no	—
128	C₂×C₈	—	2	8	no	—
129	C₂×C₈	C₂×C₇×D₄	1	8	yes	—
130	C₄⋊C₄	—	2	16	no	—
131	C₄⋊C₄	—	2	8	no	—
132	C₄⋊C₄	—	4	8	no	—
133	C₄⋊C₄	—	4	8	no	—
134	C₄²	—	2	8	no	—
135	C₄²	—	2	4	no	—
136	C₄²	—	4	8	no	—
137	C₄²	C₂×C₇×D₄	1	4	yes	—
138	C₂×C₇	#2397	1	1	yes	characteristic
139	C₂×C₇	#3113	1	4	yes	—
140	C₂×C₇	#3152	1	2	yes	—
141	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
142	Q₈	—	2	16	no	—
143	Q₈	—	4	8	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	—	2	8	no	—
148	C₂×C₄	—	4	4	no	—
149	C₂×C₄	—	4	4	no	—
150	C₂×C₄	—	4	4	no	—
151	C₂×C₄	—	4	8	no	—
152	C₂×C₄	C₇×#127	1	4	yes	—
153	C₂×C₄	C₇×#137	1	8	yes	—
154	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
155	C₂×C₄	C₂×C₇×#41	1	4	yes	—
156	C₈	—	2	8	no	—
157	C₇	#23805	1	1	yes	characteristic, direct product factor, Sylow
158	C₂²	C₇×#525	1	2	yes	—
159	C₂²	C₂×C₇×#127	1	1	yes	characteristic
160	C₂²	C₂×C₇×#137	1	4	yes	—
161	C₄	—	2	8	no	—
162	C₄	—	2	4	no	—
163	C₄	—	4	8	no	—
164	C₄	—	8	4	no	—
165	C₄	C₇×#525	1	4	yes	—
166	C₂	C₇×#2397	1	1	yes	characteristic
167	C₂	C₇×#3113	1	4	yes	—
168	C₂	C₇×#3152	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₇×#441

C₂

yes

maximal

C₄×C₇×C₄⋊Q₈

C₂

yes

maximal

C₇×#2096

C₂

yes

maximal

C₇×#2016

C₂

yes

maximal

C₇×#1745

C₂

yes

maximal, characteristic

C₂²×C₇×Q₁₆

C₂²

yes

—

C₂×C₇×C₄⋊Q₈

C₂²

yes

—

C₂×C₄×C₇×Q₈

C₂²

yes

—

C₇×#441

—

C₂×C₇×#106

C₂²

yes

—

C₂×C₇×#103

C₂²

yes

—

C₂×C₇×#97

C₂²

yes

—

C₇×#391

C₂²

yes

—

C₇×#389

—

C₇×#384

C₂²

yes

—

C₄×C₇×C₄⋊C₄

—

C₄³×C₇

C₂²

yes

characteristic

C₇×#336

C₂²

yes

—

#23805

C₇

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₇×Q₈

C₂³

yes

—

C₂×C₇×Q₁₆

—

C₂×C₇×Q₁₆

—

C₂²×C₇×C₈

C₂³

yes

—

C₇×C₄⋊Q₈

—

C₄×C₇×Q₈

—

C₂×C₇×C₄⋊C₄

—

C₂×C₇×C₄⋊C₄

—

C₂×C₇×C₄⋊C₄

—

C₂×C₇×C₄⋊C₄

C₂³

yes

—

C₂×C₄²×C₇

—

C₂×C₄²×C₇

D₄

yes

—

C₂×C₄²×C₇

C₂³

yes

—

C₂×C₄²×C₇

C₂³

yes

characteristic

C₇×#106

—

C₇×#103

—

C₇×#97

D₄

yes

—

C₇×#95

—

C₂×#441

C₂×C₇

yes

—

C₄×C₄⋊Q₈

C₂×C₇

yes

—

#2096

C₂×C₇

yes

—

#2016

C₂×C₇

yes

—

#1745

C₂×C₇

yes

characteristic

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂×C₇×Q₈

#41

yes

—

C₂²×C₄×C₇

—

C₂²×C₄×C₇

—

C₂²×C₄×C₇

C₂×D₄

yes

—

C₂²×C₄×C₇

C₂×D₄

yes

—

C₂²×C₄×C₇

C₂⁴

yes

characteristic

C₇×Q₁₆

—

C₂×C₇×C₈

—

C₂×C₇×C₈

C₂×D₄

yes

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₄²×C₇

—

C₄²×C₇

—

C₄²×C₇

—

C₄²×C₇

C₂×D₄

yes

—

C₂²×Q₁₆

C₂²×C₇

yes

—

C₂×C₄⋊Q₈

C₂²×C₇

yes

—

C₂×C₄×Q₈

C₂²×C₇

yes

—

#441

—

C₂×#106

C₂²×C₇

yes

—

C₂×#103

C₂²×C₇

yes

—

C₂×#97

C₂²×C₇

yes

—

#391

C₂²×C₇

yes

—

#389

—

#384

C₂²×C₇

yes

—

C₄×C₄⋊C₄

—

C₄³

C₂²×C₇

yes

characteristic

#336

C₂²×C₇

yes

—

C₂³×C₇

C₂²×D₄

yes

characteristic

C₇×Q₈

—

C₇×Q₈

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

#127

yes

—

C₂×C₄×C₇

#137

yes

—

C₂×C₄×C₇

C₂²×D₄

yes

—

C₂×C₄×C₇

C₂×#41

yes

—

C₇×C₈

—

C₂²×Q₈

C₂³×C₇

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

C₂³×C₇

yes

—

C₄⋊Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

100

C₂×C₄⋊C₄

—

101

C₂×C₄⋊C₄

C₂³×C₇

yes

—

102

C₂×C₄²

—

103

C₂×C₄²

C₇×D₄

yes

—

104

C₂×C₄²

C₂³×C₇

yes

—

105

C₂×C₄²

C₂³×C₇

yes

characteristic

106

#106

—

107

#103

—

108

#97

C₇×D₄

yes

—

109

#95

—

110

C₂²×C₇

#525

yes

—

111

C₂²×C₇

C₂×#127

yes

characteristic

112

C₂²×C₇

C₂×#137

yes

—

113

C₄×C₇

—

114

C₄×C₇

—

115

C₄×C₇

—

116

C₄×C₇

—

117

C₄×C₇

#525

yes

—

118

C₂×Q₈

—

119

C₂×Q₈

—

120

C₂×Q₈

—

121

C₂×Q₈

C₇×#41

yes

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

C₂×C₇×D₄

yes

—

125

C₂²×C₄

C₂×C₇×D₄

yes

—

126

C₂²×C₄

C₂⁴×C₇

yes

characteristic

127

Q₁₆

—

128

C₂×C₈

—

129

C₂×C₈

C₂×C₇×D₄

yes

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄²

—

135

C₄²

—

136

C₄²

—

137

C₄²

C₂×C₇×D₄

yes

—

138

C₂×C₇

#2397

yes

characteristic

139

C₂×C₇

#3113

yes

—

140

C₂×C₇

#3152

yes

—

141

C₂³

C₂²×C₇×D₄

yes

characteristic

142

Q₈

—

143

Q₈

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

C₇×#127

yes

—

153

C₂×C₄

C₇×#137

yes

—

154

C₂×C₄

C₂²×C₇×D₄

yes

—

155

C₂×C₄

C₂×C₇×#41

yes

—

156

C₈

—

157

C₇

#23805

yes

characteristic, direct product factor, Sylow

158

C₂²

C₇×#525

yes

—

159

C₂²

C₂×C₇×#127

yes

characteristic

160

C₂²

C₂×C₇×#137

yes

—

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

C₇×#525

yes

—

166

C₂

C₇×#2397

yes

characteristic

167

C₂

C₇×#3113

yes

—

168

C₂

C₇×#3152

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₄³×C₇ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₇

C₂²×D₄

commutator, Frattini, cosocle

C₂²×C₄

C₂⁴×C₇

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₇ ◁ C₄³×C₇ ◁ G

N	H ≅ G/N	type
19. #23805	C₇	direct (N × H)
157. C₇	#23805	direct (N × H)

H ≅ G/N

type

19. #23805

C₇

direct (N × H)

157. C₇

#23805

direct (N × H)

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

subgroup	index
C₂×C₇×#441	2
C₄×C₇×C₄⋊Q₈	2
C₇×#2096	2
C₇×#2016	2
C₇×#1745	2
#23805	7

subgroup

index

C₂×C₇×#441

C₄×C₇×C₄⋊Q₈

Subgroups — C₇×#23805 #13201021

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C7×#23805 #13201021

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#23805 #13201021