Group #11728391 — subgroups

Subgroups — #11728391

Order 896.

256 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 81 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×#107	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×(C₂×D₇)⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#89759	C₂	1	2	yes	maximal
4	#89346	C₂	1	1	yes	maximal, characteristic, complemented
5	#89324	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×D₇	C₂²	1	1	yes	characteristic
7	C₂²×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂×(C₂×D₇)⋊C₄	—	2	2	no	—
10	C₂×C₄⋊Dic₇	C₂²	1	2	yes	—
11	C₂²×C₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₇×#107	—	2	2	no	—
13	C₇⋊#95	—	2	2	no	—
14	(C₂×D₇)⋊C₈	—	2	2	no	—
15	#2076	—	7	1	no	maximal, Sylow
16	C₂³×D₇	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₇	—	2	2	no	—
18	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
19	C₂²×Dic₇	—	2	2	no	—
20	C₂²×Dic₇	D₄	1	1	yes	characteristic
21	C₂×C₄×D₇	—	2	1	no	—
22	C₂×C₄×D₇	—	2	2	no	—
23	C₂×C₄×D₇	—	2	2	no	—
24	C₂×C₄×D₇	—	2	1	no	—
25	C₂×C₇×C₈	—	2	2	no	—
26	C₂×C₇×C₈	D₄	1	2	yes	—
27	C₇×C₄⋊C₄	—	2	2	no	—
28	C₇×C₄⋊C₄	—	2	2	no	—
29	(C₂×D₇)⋊C₄	—	4	4	no	—
30	C₄⋊Dic₇	—	2	2	no	—
31	C₄⋊Dic₇	—	2	2	no	—
32	C₂×C₇⋊C₈	—	2	1	no	—
33	C₂×C₇⋊C₈	—	2	2	no	—
34	C₂×#107	D₇	1	1	yes	characteristic, complemented
35	C₂×C₂²⋊C₈	—	7	1	no	—
36	#380	—	7	2	no	—
37	#340	—	7	1	no	—
38	#340	—	7	2	no	—
39	C₂³×C₇	C₂×D₄	1	1	yes	characteristic
40	C₂²×D₇	—	2	1	no	—
41	C₂²×D₇	—	2	2	no	—
42	C₂²×D₇	—	2	1	no	—
43	C₂²×D₇	—	2	2	no	—
44	C₂²×D₇	Q₁₆	1	2	yes	—
45	C₂×C₄×C₇	—	4	2	no	—
46	C₂×C₄×C₇	—	4	2	no	—
47	C₂×C₄×C₇	—	4	2	no	—
48	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₇	#41	1	2	yes	—
50	C₂×C₄×C₇	#41	1	1	yes	characteristic
51	C₂×C₄×C₇	#41	1	2	yes	—
52	C₂×Dic₇	—	2	1	no	—
53	C₂×Dic₇	—	2	2	no	—
54	C₂×Dic₇	—	4	2	no	—
55	C₂×Dic₇	—	4	2	no	—
56	C₂×Dic₇	—	4	2	no	—
57	C₄×D₇	—	4	1	no	—
58	C₄×D₇	—	4	1	no	—
59	C₄×D₇	—	4	2	no	—
60	C₇×C₈	—	2	2	no	—
61	C₇⋊C₈	—	4	1	no	—
62	C₂³×C₄	—	7	1	no	—
63	C₂²×C₈	—	7	1	no	—
64	C₂²×C₈	C₂×D₇	1	1	yes	characteristic
65	C₂×C₄⋊C₄	—	7	2	no	—
66	C₂×C₄⋊C₄	C₂×D₇	1	2	yes	—
67	C₂×C₂²⋊C₄	—	14	2	no	—
68	#107	—	2	2	no	—
69	C₂²⋊C₈	—	14	2	no	—
70	#95	—	14	2	no	—
71	C₂²×C₇	#121	1	1	yes	characteristic
72	C₂²×C₇	#123	1	2	yes	—
73	C₂²×C₇	C₂×Q₁₆	1	1	yes	characteristic
74	C₂²×C₇	#135	1	1	yes	characteristic
75	C₂²×C₇	AGL(1,ℤ/8)	1	2	yes	—
76	C₂×D₇	—	2	2	no	—
77	C₂×D₇	—	2	4	no	—
78	C₂×D₇	—	4	1	no	—
79	C₂×D₇	—	4	1	no	—
80	C₂×D₇	—	4	2	no	—
81	C₄×C₇	—	8	2	no	—
82	C₄×C₇	#123	1	1	yes	characteristic
83	C₄×C₇	#124	1	1	yes	characteristic
84	C₄×C₇	#126	1	2	yes	—
85	Dic₇	—	4	1	no	—
86	Dic₇	—	8	2	no	—
87	C₂⁴	—	7	1	no	—
88	C₂²×C₄	—	2	2	no	—
89	C₂²×C₄	—	7	1	no	—
90	C₂²×C₄	—	14	2	no	—
91	C₂²×C₄	—	14	1	no	—
92	C₂²×C₄	—	14	2	no	—
93	C₂²×C₄	—	14	1	no	—
94	C₂²×C₄	—	14	2	no	—
95	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	14	2	no	—
98	C₂×C₈	—	14	1	no	—
99	C₂×C₈	C₂²⋊D₇	1	2	yes	—
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	14	2	no	—
103	C₄⋊C₄	—	14	2	no	—
104	C₂²⋊C₄	—	28	4	no	—
105	C₂×C₇	#397	1	1	yes	characteristic
106	C₂×C₇	#467	1	1	yes	characteristic
107	C₂×C₇	#469	1	1	yes	characteristic
108	C₂×C₇	#482	1	2	yes	—
109	C₂×C₇	#484	1	2	yes	—
110	D₇	—	4	2	no	—
111	C₂³	—	7	2	no	—
112	C₂³	—	14	2	no	—
113	C₂³	—	14	2	no	—
114	C₂³	—	14	1	no	—
115	C₂³	—	14	1	no	—
116	C₂³	D₄×D₇	1	1	yes	characteristic
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	14	2	no	—
121	C₂×C₄	—	14	1	no	—
122	C₂×C₄	—	28	2	no	—
123	C₂×C₄	—	28	1	no	—
124	C₂×C₄	—	28	1	no	—
125	C₂×C₄	—	28	2	no	—
126	C₂×C₄	—	28	2	no	—
127	C₂×C₄	—	28	2	no	—
128	C₂×C₄	#1157	1	2	yes	—
129	C₂×C₄	#1159	1	1	yes	characteristic
130	C₂×C₄	#1161	1	2	yes	—
131	C₂×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
132	C₈	—	2	2	no	—
133	C₈	—	28	1	no	—
134	C₇	#2076	1	1	yes	characteristic, complemented, Sylow
135	C₂²	—	14	4	no	—
136	C₂²	—	14	2	no	—
137	C₂²	—	28	2	no	—
138	C₂²	—	28	1	no	—
139	C₂²	—	28	1	no	—
140	C₂²	#6457	1	2	yes	—
141	C₂²	#6474	1	2	yes	—
142	C₂²	D₇×Q₁₆	1	1	yes	characteristic
143	C₂²	#6482	1	1	yes	characteristic
144	C₂²	#6501	1	1	yes	characteristic
145	C₄	—	8	2	no	—
146	C₄	—	28	1	no	—
147	C₄	—	56	2	no	—
148	C₄	#6461	1	2	yes	—
149	C₄	#6492	1	1	yes	characteristic
150	C₄	#6510	1	1	yes	characteristic
151	C₂	—	28	2	no	—
152	C₂	#89656	1	2	yes	—
153	C₂	#89657	1	2	yes	—
154	C₂	#89765	1	1	yes	characteristic
155	C₂	#89931	1	1	yes	characteristic
156	C₂	#89960	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₇×#107

C₂

yes

maximal, characteristic, complemented

C₂×(C₂×D₇)⋊C₈

C₂

yes

maximal, characteristic

#89759

C₂

yes

maximal

#89346

C₂

yes

maximal, characteristic, complemented

#89324

C₂

yes

maximal, complemented

C₂²×C₄×D₇

C₂²

yes

characteristic

C₂²×C₇×C₈

C₂²

yes

characteristic

C₂×C₇×C₄⋊C₄

C₂²

yes

—

C₂×(C₂×D₇)⋊C₄

—

C₂×C₄⋊Dic₇

C₂²

yes

—

C₂²×C₇⋊C₈

C₂²

yes

characteristic

C₇×#107

—

C₇⋊#95

—

(C₂×D₇)⋊C₈

—

#2076

—

maximal, Sylow

C₂³×D₇

D₄

yes

characteristic

C₂²×C₄×C₇

—

C₂²×C₄×C₇

C₂³

yes

characteristic

C₂²×Dic₇

—

C₂²×Dic₇

D₄

yes

characteristic

C₂×C₄×D₇

—

C₂×C₄×D₇

—

C₂×C₄×D₇

—

C₂×C₄×D₇

—

C₂×C₇×C₈

—

C₂×C₇×C₈

D₄

yes

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

(C₂×D₇)⋊C₄

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×#107

D₇

yes

characteristic, complemented

C₂×C₂²⋊C₈

—

#380

—

#340

—

#340

—

C₂³×C₇

C₂×D₄

yes

characteristic

C₂²×D₇

—

C₂²×D₇

—

C₂²×D₇

—

C₂²×D₇

—

C₂²×D₇

Q₁₆

yes

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

#41

yes

—

C₂×C₄×C₇

#41

yes

characteristic

C₂×C₄×C₇

#41

yes

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₄×D₇

—

C₄×D₇

—

C₄×D₇

—

C₇×C₈

—

C₇⋊C₈

—

C₂³×C₄

—

C₂²×C₈

—

C₂²×C₈

C₂×D₇

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇

yes

—

C₂×C₂²⋊C₄

—

#107

—

C₂²⋊C₈

—

#95

—

C₂²×C₇

#121

yes

characteristic

C₂²×C₇

#123

yes

—

C₂²×C₇

C₂×Q₁₆

yes

C₂²×C₇

yes

C₂²×C₇

yes

—

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₄×C₇

—

C₄×C₇

yes

C₄×C₇

yes

C₄×C₇

yes

—

Dic₇

—

Dic₇

—

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₇

yes

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₂²⋊C₄

—

105

C₂×C₇

#397

yes

characteristic

106

C₂×C₇

#467

yes

characteristic

107

C₂×C₇

#469

yes

characteristic

108

C₂×C₇

#482

yes

—

109

C₂×C₇

#484

yes

—

110

D₇

—

111

C₂³

—

112

C₂³

—

113

C₂³

—

114

C₂³

—

115

C₂³

—

116

C₂³

D₄×D₇

yes

characteristic

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

#1157

yes

—

129

C₂×C₄

#1159

yes

characteristic

130

C₂×C₄

#1161

yes

—

131

C₂×C₄

C₂×C₂²⋊D₇

yes

characteristic

132

C₈

—

133

C₈

—

134

C₇

#2076

yes

characteristic, complemented, Sylow

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

#6457

yes

—

141

C₂²

#6474

yes

—

142

C₂²

D₇×Q₁₆

yes

characteristic

143

C₂²

#6482

yes

characteristic

144

C₂²

#6501

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

#6461

yes

—

149

C₄

#6492

yes

characteristic

150

C₄

#6510

yes

characteristic

151

C₂

—

152

C₂

#89656

yes

—

153

C₂

#89657

yes

—

154

C₂

#89765

yes

characteristic

155

C₂

#89931

yes

characteristic

156

C₂

#89960

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₇
commutator, cosocle	C₂²×C₄×C₇	C₂³
Frattini	C₂²×C₄	C₂²×D₇
Fitting	C₂×C₇×#107	C₂
socle	C₂³×C₇	C₂×D₄
prime radical	C₂³×D₇	D₄
hypercenter	C₂×#107	D₇
hypocenter	C₇	#2076
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₇

commutator, cosocle

C₂²×C₄×C₇

C₂³

Frattini

C₂²×C₄

C₂²×D₇

Fitting

C₂×C₇×#107

C₂

socle

C₂³×C₇

C₂×D₄

prime radical

C₂³×D₇

D₄

D₇

C₇

G ▷ C₂²×C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
1. C₂×C₇×#107	C₂	semidirect
4. #89346	C₂	semidirect
5. #89324	C₂	semidirect
34. C₂×#107	D₇	semidirect
134. C₇	#2076	semidirect

H ≅ G/N

type

1. C₂×C₇×#107

C₂

semidirect

4. #89346

C₂

semidirect

5. #89324

C₂

semidirect

34. C₂×#107

D₇

semidirect

134. C₇

#2076

semidirect

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

subgroup	index
C₂×C₇×#107	2
C₂×(C₂×D₇)⋊C₈	2
#89759	2
#89346	2
#89324	2
#2076	7

subgroup

index

C₂×C₇×#107

C₂×(C₂×D₇)⋊C₈

Subgroups — #11728391

Conjugacy classes of subgroups

Complements

Maximal subgroups