Group #13258579 — subgroups

Subgroups — #13258579

Order 1792.

272 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 87 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×#1995	C₂	1	1	yes	maximal, characteristic
2	#11729582	C₂	1	2	yes	maximal
3	#11727191	C₂	1	1	yes	maximal, characteristic
4	#11725520	C₂	1	2	yes	maximal
5	#11725420	C₂	1	1	yes	maximal, characteristic
6	C₇×#533	C₂²	1	1	yes	characteristic
7	C₇×#423	C₂²	1	1	yes	characteristic
8	C₇×#376	C₂²	1	1	yes	characteristic
9	#89697	C₂²	1	2	yes	—
10	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
11	C₇×#339	—	2	2	no	—
12	#89342	—	2	2	no	—
13	#89341	—	2	2	no	—
14	#89319	—	2	2	no	—
15	#10630	—	7	1	no	maximal, Sylow
16	C₂×C₇×M₁₆	—	2	1	no	—
17	C₇×C₄⋊Q₈	D₄	1	2	yes	—
18	C₇×#125	—	2	1	no	—
19	C₇×C₂²⋊Q₈	—	2	2	no	—
20	C₄×C₇×Q₈	—	2	2	no	—
21	C₇×#117	—	2	1	no	—
22	C₂×C₄²×C₇	D₄	1	2	yes	—
23	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
24	C₂²×C₇⋊C₈	—	2	2	no	—
25	C₇×C₄⋊C₈	—	2	2	no	—
26	C₇×#95	—	2	2	no	—
27	C₂²⋊(C₇⋊C₈)	—	2	2	no	—
28	C₄⋊(C₇⋊C₈)	—	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	#1995	D₇	1	1	yes	characteristic
33	#1832	—	7	1	no	—
34	#1542	—	7	2	no	—
35	#1301	—	7	2	no	—
36	#1274	—	7	1	no	—
37	C₂×C₇×Q₈	—	2	2	no	—
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	C₇×M₁₆	—	4	2	no	—
42	C₂×C₇×C₈	—	4	1	no	—
43	C₇×C₄⋊C₄	—	2	2	no	—
44	C₇×C₄⋊C₄	—	2	2	no	—
45	C₇×C₄⋊C₄	—	4	2	no	—
46	C₇×C₂²⋊C₄	—	4	1	no	—
47	C₄²×C₇	—	2	2	no	—
48	C₄²×C₇	—	2	2	no	—
49	C₄²×C₇	—	4	1	no	—
50	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
51	C₄²×C₇	#41	1	1	yes	characteristic
52	C₄²×C₇	#41	1	2	yes	—
53	C₂×C₇⋊C₈	—	4	2	no	—
54	C₂×C₇⋊C₈	—	4	2	no	—
55	C₂×C₇⋊C₈	—	4	2	no	—
56	C₂×C₇⋊C₈	—	4	2	no	—
57	#533	C₂×D₇	1	1	yes	characteristic
58	#431	—	7	2	no	—
59	#423	C₂×D₇	1	1	yes	characteristic
60	C₂×C₄⋊C₈	—	7	2	no	—
61	#376	C₂×D₇	1	1	yes	characteristic
62	#339	—	2	2	no	—
63	#336	—	14	2	no	—
64	#333	—	14	2	no	—
65	#332	—	14	2	no	—
66	C₂³×C₇	#123	1	1	yes	characteristic
67	C₇×Q₈	—	8	2	no	—
68	C₂×C₄×C₇	—	4	2	no	—
69	C₂×C₄×C₇	—	4	4	no	—
70	C₂×C₄×C₇	—	4	2	no	—
71	C₂×C₄×C₇	—	4	2	no	—
72	C₂×C₄×C₇	#121	1	1	yes	characteristic
73	C₂×C₄×C₇	#123	1	2	yes	—
74	C₂×C₄×C₇	#124	1	1	yes	characteristic
75	C₂×C₄×C₇	#126	1	2	yes	—
76	C₂×C₄×C₇	#135	1	2	yes	—
77	C₂×C₄×C₇	#137	1	2	yes	—
78	C₇×C₈	—	8	1	no	—
79	C₇⋊C₈	—	8	2	no	—
80	C₇⋊C₈	—	8	2	no	—
81	C₂×M₁₆	—	2	1	no	—
82	C₂²×C₈	—	14	2	no	—
83	C₄⋊Q₈	D₂₈	1	2	yes	—
84	#125	—	2	1	no	—
85	C₂²⋊Q₈	—	2	2	no	—
86	C₄×Q₈	—	2	2	no	—
87	#117	—	2	1	no	—
88	C₂×C₄²	C₂²⋊D₇	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	C₂²⋊C₈	—	14	2	no	—
95	C₄×C₈	—	14	2	no	—
96	#95	—	2	2	no	—
97	C₂²×C₇	—	2	2	no	—
98	C₂²×C₇	#397	1	1	yes	characteristic
99	C₂²×C₇	#485	1	2	yes	—
100	C₄×C₇	—	2	2	no	—
101	C₄×C₇	—	2	2	no	—
102	C₄×C₇	—	8	2	no	—
103	C₄×C₇	—	8	2	no	—
104	C₄×C₇	#464	1	2	yes	—
105	C₄×C₇	#488	1	1	yes	characteristic
106	C₄×C₇	#491	1	1	yes	characteristic
107	C₂×Q₈	—	2	2	no	—
108	C₂²×C₄	—	2	2	no	—
109	C₂²×C₄	#1159	1	2	yes	—
110	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
111	M₁₆	—	4	2	no	—
112	C₂×C₈	—	4	1	no	—
113	C₂×C₈	—	28	2	no	—
114	C₂×C₈	—	28	2	no	—
115	C₂×C₈	—	28	2	no	—
116	C₂×C₈	—	28	2	no	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	—	2	2	no	—
119	C₄⋊C₄	—	4	2	no	—
120	C₂²⋊C₄	—	4	1	no	—
121	C₄²	—	2	2	no	—
122	C₄²	—	2	2	no	—
123	C₄²	—	4	1	no	—
124	C₄²	C₂×D₂₈	1	1	yes	characteristic
125	C₄²	#1157	1	1	yes	characteristic
126	C₄²	#1159	1	2	yes	—
127	C₂×C₇	—	2	2	no	—
128	C₂×C₇	#2074	1	1	yes	characteristic
129	C₂×C₇	#2080	1	2	yes	—
130	C₂³	#6498	1	1	yes	characteristic
131	Q₈	—	8	2	no	—
132	C₂×C₄	—	4	2	no	—
133	C₂×C₄	—	4	4	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	4	2	no	—
136	C₂×C₄	#6442	1	2	yes	—
137	C₂×C₄	#6449	1	2	yes	—
138	C₂×C₄	#6493	1	1	yes	characteristic
139	C₂×C₄	#6499	1	1	yes	characteristic
140	C₂×C₄	#6505	1	2	yes	—
141	C₂×C₄	#6513	1	2	yes	—
142	C₈	—	8	1	no	—
143	C₈	—	56	2	no	—
144	C₈	—	56	2	no	—
145	C₇	#10630	1	1	yes	characteristic, complemented, Sylow
146	C₂²	—	2	2	no	—
147	C₂²	#89725	1	1	yes	characteristic
148	C₂²	#89817	1	2	yes	—
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	8	2	no	—
152	C₄	—	8	2	no	—
153	C₄	#89800	1	2	yes	—
154	C₄	#89828	1	1	yes	characteristic
155	C₄	#89854	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	#11727425	1	1	yes	characteristic
158	C₂	#11728158	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#1995

C₂

yes

maximal, characteristic

#11729582

C₂

yes

maximal

#11727191

C₂

yes

maximal, characteristic

#11725520

C₂

yes

maximal

#11725420

C₂

yes

maximal, characteristic

C₇×#533

C₂²

yes

characteristic

C₇×#423

C₂²

yes

characteristic

C₇×#376

C₂²

yes

characteristic

#89697

C₂²

yes

—

C₂×C₄⋊(C₇⋊C₈)

C₂²

yes

—

—

—

—

—

—

C₂×C₇×M₁₆

—

C₇×C₄⋊Q₈

D₄

yes

—

C₇×#125

—

C₇×C₂²⋊Q₈

—

C₄×C₇×Q₈

—

C₇×#117

—

C₂×C₄²×C₇

D₄

yes

—

C₂×C₄²×C₇

C₂³

yes

characteristic

C₂²×C₇⋊C₈

—

C₇×C₄⋊C₈

—

C₇×#95

—

C₂²⋊(C₇⋊C₈)

—

C₄⋊(C₇⋊C₈)

—

C₄⋊(C₇⋊C₈)

—

C₄⋊(C₇⋊C₈)

—

C₄×C₇⋊C₈

—

#1995

D₇

yes

—

—

—

—

C₂×C₇×Q₈

—

C₂²×C₄×C₇

—

C₂²×C₄×C₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₇

#41

yes

—

C₇×M₁₆

—

C₂×C₇×C₈

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₂²⋊C₄

—

C₄²×C₇

—

C₄²×C₇

—

C₄²×C₇

—

C₄²×C₇

C₂×D₄

yes

C₄²×C₇

yes

C₄²×C₇

yes

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

#533

C₂×D₇

yes

characteristic

#431

—

#423

C₂×D₇

yes

characteristic

C₂×C₄⋊C₈

—

#376

C₂×D₇

yes

—

—

—

—

C₂³×C₇

#123

yes

characteristic

C₇×Q₈

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

#121

yes

characteristic

C₂×C₄×C₇

#123

yes

—

C₂×C₄×C₇

#124

yes

characteristic

C₂×C₄×C₇

#126

yes

—

C₂×C₄×C₇

#135

yes

—

C₂×C₄×C₇

#137

yes

—

C₇×C₈

—

C₇⋊C₈

—

C₇⋊C₈

—

C₂×M₁₆

—

C₂²×C₈

—

C₄⋊Q₈

D₂₈

yes

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂²⋊D₇

yes

—

C₂×C₄²

C₂²×D₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

#95

—

C₂²×C₇

—

C₂²×C₇

#397

yes

characteristic

C₂²×C₇

#485

yes

—

100

C₄×C₇

—

101

C₄×C₇

—

102

C₄×C₇

—

103

C₄×C₇

—

104

C₄×C₇

#464

yes

—

105

C₄×C₇

#488

yes

characteristic

106

C₄×C₇

#491

yes

characteristic

107

C₂×Q₈

—

108

C₂²×C₄

—

109

C₂²×C₄

#1159

yes

—

110

C₂²×C₄

C₂×C₂²⋊D₇

yes

characteristic

111

M₁₆

—

112

C₂×C₈

—

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

—

124

C₄²

C₂×D₂₈

yes

characteristic

125

C₄²

#1157

yes

characteristic

126

C₄²

#1159

yes

—

127

C₂×C₇

—

128

C₂×C₇

#2074

yes

characteristic

129

C₂×C₇

#2080

yes

—

130

C₂³

#6498

yes

characteristic

131

Q₈

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

#6442

yes

—

137

C₂×C₄

#6449

yes

—

138

C₂×C₄

#6493

yes

characteristic

139

C₂×C₄

#6499

yes

characteristic

140

C₂×C₄

#6505

yes

—

141

C₂×C₄

#6513

yes

—

142

C₈

—

143

C₈

—

144

C₈

—

145

C₇

#10630

yes

characteristic, complemented, Sylow

146

C₂²

—

147

C₂²

#89725

yes

characteristic

148

C₂²

#89817

yes

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

#89800

yes

—

154

C₄

#89828

yes

characteristic

155

C₄

#89854

yes

characteristic

156

C₂

—

157

C₂

#11727425

yes

characteristic

158

C₂

#11728158

yes

—

	subgroup	G/H
center	C₂×C₄	#6499
commutator, cosocle	C₂×C₄²×C₇	C₂³
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#1995	C₂
socle	C₂²×C₇	#397
prime radical	C₂³×C₇	#123
hypercenter	#1995	D₇
hypocenter	C₇	#10630
LCS	G ▷ C₂×C₄²×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1995

subgroup

G/H

center

C₂×C₄

#6499

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 ◁ #1995

N	H ≅ G/N	type
145. C₇	#10630	semidirect

H ≅ G/N

type

145. C₇

#10630

semidirect

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

subgroup	index
C₇×#1995	2
#11729582	2
#11727191	2
#11725520	2
#11725420	2
#10630	7

subgroup

index

Subgroups — #13258579

Conjugacy classes of subgroups

Complements

Maximal subgroups