Group #11729871 — subgroups

Subgroups — #11729871

Order 896.

266 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 67 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×#575	C₂	1	1	yes	maximal, characteristic
2	#90009	C₂	1	2	yes	maximal, complemented
3	C₂×#6411	C₂	1	2	yes	maximal, complemented
4	#89890	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇⋊#338	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₇×#41	C₂²	1	2	yes	—
7	C₇×#137	—	2	2	no	—
8	C₇×AGL(1,ℤ/8)	—	2	2	no	—
9	C₇×#135	—	2	2	no	—
10	C₂×C₇×M₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₇	C₂²	1	1	yes	characteristic
12	C₂×#1136	C₂²	1	2	yes	—
13	C₂×#1136	C₂²	1	1	yes	characteristic
14	#6411	—	2	2	no	—
15	#6411	—	2	2	no	—
16	#6410	—	2	2	no	—
17	#6407	—	2	2	no	—
18	Dic₇⋊C₈	—	2	2	no	—
19	#2050	—	7	1	no	maximal, Sylow
20	C₇×#41	—	2	2	no	—
21	C₇×#41	—	2	2	no	—
22	C₇×#41	—	2	2	no	—
23	C₇×#41	—	2	2	no	—
24	C₂×C₇×Q₈	D₄	1	2	yes	—
25	C₂×C₇×D₄	—	2	2	no	—
26	C₂×C₇×D₄	D₄	1	2	yes	—
27	C₂²×C₄×C₇	—	2	2	no	—
28	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
29	C₂²×Dic₇	—	2	1	no	—
30	C₇×Q₁₆	—	2	2	no	—
31	C₇×SD₁₆	—	2	4	no	—
32	C₇×D₈	—	2	2	no	—
33	C₇×M₁₆	—	2	2	no	—
34	C₂×C₇×C₈	D₄	1	2	yes	—
35	C₄×Dic₇	—	2	1	no	—
36	C₄×Dic₇	D₄	1	1	yes	characteristic
37	C₄×Dic₇	D₄	1	1	yes	characteristic
38	#1136	—	2	2	no	—
39	#1136	—	2	2	no	—
40	#1136	—	2	2	no	—
41	C₂×C₇⋊C₈	—	2	1	no	—
42	C₂×C₇⋊C₈	—	2	2	no	—
43	#575	D₇	1	1	yes	characteristic
44	#423	—	7	1	no	—
45	C₂×#104	—	7	2	no	—
46	#413	—	7	2	no	—
47	#338	—	7	1	no	—
48	C₂³×C₇	—	2	2	no	—
49	C₂³×C₇	#41	1	1	yes	characteristic
50	C₇×Q₈	—	2	2	no	—
51	C₇×Q₈	—	2	2	no	—
52	C₇×D₄	—	2	2	no	—
53	C₇×D₄	—	2	2	no	—
54	C₇×D₄	—	4	2	no	—
55	C₇×D₄	—	4	2	no	—
56	C₂×C₄×C₇	—	2	2	no	—
57	C₂×C₄×C₇	—	4	2	no	—
58	C₂×C₄×C₇	—	4	2	no	—
59	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₇	#41	1	2	yes	—
63	C₂×Dic₇	—	2	1	no	—
64	C₂×Dic₇	—	2	1	no	—
65	C₂×Dic₇	—	4	2	no	—
66	C₇×C₈	—	2	2	no	—
67	C₇⋊C₈	—	4	1	no	—
68	C₇⋊C₈	—	4	2	no	—
69	C₂×#41	C₂×D₇	1	2	yes	—
70	#137	—	2	2	no	—
71	AGL(1,ℤ/8)	—	2	2	no	—
72	#135	—	2	2	no	—
73	C₂×M₁₆	—	7	2	no	—
74	C₂×M₁₆	—	7	1	no	—
75	C₂×M₁₆	C₂×D₇	1	1	yes	characteristic
76	C₂×C₄²	—	7	1	no	—
77	C₄⋊C₈	—	14	2	no	—
78	#104	—	14	2	no	—
79	#104	—	14	2	no	—
80	#101	—	14	2	no	—
81	#100	—	14	2	no	—
82	C₂²×C₇	—	4	1	no	—
83	C₂²×C₇	—	4	2	no	—
84	C₂²×C₇	—	8	2	no	—
85	C₂²×C₇	#121	1	1	yes	characteristic
86	C₂²×C₇	#123	1	2	yes	—
87	C₄×C₇	—	4	2	no	—
88	C₄×C₇	#120	1	1	yes	characteristic
89	C₄×C₇	#121	1	2	yes	—
90	C₄×C₇	#124	1	1	yes	characteristic
91	Dic₇	—	2	2	no	—
92	Dic₇	—	4	1	no	—
93	#41	—	2	2	no	—
94	#41	—	2	2	no	—
95	#41	—	2	2	no	—
96	#41	—	2	2	no	—
97	C₂×Q₈	C₂²⋊D₇	1	2	yes	—
98	C₂×D₄	—	2	2	no	—
99	C₂×D₄	C₂²⋊D₇	1	2	yes	—
100	C₂²×C₄	—	2	2	no	—
101	C₂²×C₄	—	14	1	no	—
102	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
103	Q₁₆	—	2	2	no	—
104	SD₁₆	—	2	4	no	—
105	D₈	—	2	2	no	—
106	M₁₆	—	2	2	no	—
107	M₁₆	—	14	2	no	—
108	M₁₆	—	14	2	no	—
109	M₁₆	—	14	2	no	—
110	C₂×C₈	—	14	2	no	—
111	C₂×C₈	—	14	1	no	—
112	C₂×C₈	C₂²⋊D₇	1	2	yes	—
113	C₄²	—	7	1	no	—
114	C₄²	—	7	1	no	—
115	C₄²	—	14	1	no	—
116	C₂×C₇	—	2	1	no	—
117	C₂×C₇	—	2	2	no	—
118	C₂×C₇	—	8	2	no	—
119	C₂×C₇	#394	1	1	yes	characteristic
120	C₂³	—	2	2	no	—
121	C₂³	#1159	1	1	yes	characteristic
122	Q₈	—	2	2	no	—
123	Q₈	—	2	2	no	—
124	D₄	—	2	2	no	—
125	D₄	—	2	2	no	—
126	D₄	—	4	2	no	—
127	D₄	—	4	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	4	2	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	14	1	no	—
132	C₂×C₄	—	14	1	no	—
133	C₂×C₄	—	28	2	no	—
134	C₂×C₄	D₄×D₇	1	1	yes	characteristic
135	C₂×C₄	#1159	1	2	yes	—
136	C₂×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
137	C₂×C₄	C₂×C₂²⋊D₇	1	2	yes	—
138	C₈	—	2	2	no	—
139	C₈	—	28	2	no	—
140	C₈	—	28	1	no	—
141	C₇	#2050	1	1	yes	characteristic, complemented, Sylow
142	C₂²	—	4	1	no	—
143	C₂²	—	4	2	no	—
144	C₂²	—	8	2	no	—
145	C₂²	#6498	1	2	yes	—
146	C₂²	#6501	1	1	yes	characteristic
147	C₄	—	4	2	no	—
148	C₄	—	14	2	no	—
149	C₄	—	28	1	no	—
150	C₄	#6499	1	1	yes	characteristic
151	C₄	#6502	1	2	yes	—
152	C₄	#6516	1	1	yes	characteristic
153	C₂	—	2	1	no	—
154	C₂	—	2	2	no	—
155	C₂	—	8	2	no	—
156	C₂	#89994	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#575

C₂

yes

maximal, characteristic

#90009

C₂

yes

maximal, complemented

C₂×#6411

C₂

yes

maximal, complemented

#89890

C₂

yes

maximal, characteristic, complemented

C₇⋊#338

C₂

yes

maximal, characteristic, complemented

C₂×C₇×#41

C₂²

yes

—

C₇×#137

—

C₇×AGL(1,ℤ/8)

—

C₇×#135

—

C₂×C₇×M₁₆

C₂²

yes

characteristic

C₂×C₄×Dic₇

C₂²

yes

characteristic

C₂×#1136

C₂²

yes

—

C₂×#1136

C₂²

yes

—

—

—

—

Dic₇⋊C₈

—

#2050

—

maximal, Sylow

C₇×#41

—

C₇×#41

—

C₇×#41

—

C₇×#41

—

C₂×C₇×Q₈

D₄

yes

—

C₂×C₇×D₄

—

C₂×C₇×D₄

D₄

yes

—

C₂²×C₄×C₇

—

C₂²×C₄×C₇

C₂³

yes

characteristic

C₂²×Dic₇

—

C₇×Q₁₆

—

C₇×SD₁₆

—

C₇×D₈

—

C₇×M₁₆

—

C₂×C₇×C₈

D₄

yes

—

C₄×Dic₇

—

C₄×Dic₇

D₄

yes

characteristic

C₄×Dic₇

D₄

yes

—

—

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

#575

D₇

yes

—

—

—

—

C₂³×C₇

—

C₂³×C₇

#41

yes

characteristic

C₇×Q₈

—

C₇×Q₈

—

C₇×D₄

—

C₇×D₄

—

C₇×D₄

—

C₇×D₄

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

C₂×D₄

yes

—

C₂×C₄×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

#41

yes

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₇×C₈

—

C₇⋊C₈

—

C₇⋊C₈

—

C₂×#41

C₂×D₇

yes

—

#137

—

AGL(1,ℤ/8)

—

#135

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₇

yes

characteristic

C₂×C₄²

—

C₄⋊C₈

—

—

—

—

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

#121

yes

characteristic

C₂²×C₇

#123

yes

—

C₄×C₇

—

C₄×C₇

#120

yes

characteristic

C₄×C₇

#121

yes

—

C₄×C₇

#124

yes

characteristic

Dic₇

—

Dic₇

—

—

—

—

—

C₂×Q₈

C₂²⋊D₇

yes

—

C₂×D₄

—

C₂×D₄

C₂²⋊D₇

yes

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

C₂²×D₇

yes

characteristic

103

Q₁₆

—

104

SD₁₆

—

105

D₈

—

106

M₁₆

—

107

M₁₆

—

108

M₁₆

—

109

M₁₆

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

C₂²⋊D₇

yes

—

113

C₄²

—

114

C₄²

—

115

C₄²

—

116

C₂×C₇

—

117

C₂×C₇

—

118

C₂×C₇

—

119

C₂×C₇

#394

yes

characteristic

120

C₂³

—

121

C₂³

#1159

yes

characteristic

122

Q₈

—

123

Q₈

—

124

D₄

—

125

D₄

—

126

D₄

—

127

D₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

D₄×D₇

yes

characteristic

135

C₂×C₄

#1159

yes

—

136

C₂×C₄

C₂×C₂²⋊D₇

yes

characteristic

137

C₂×C₄

C₂×C₂²⋊D₇

yes

—

138

C₈

—

139

C₈

—

140

C₈

—

141

C₇

#2050

yes

characteristic, complemented, Sylow

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

#6498

yes

—

146

C₂²

#6501

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

#6499

yes

characteristic

151

C₄

#6502

yes

—

152

C₄

#6516

yes

characteristic

153

C₂

—

154

C₂

—

155

C₂

—

156

C₂

#89994

yes

characteristic

	subgroup	G/H
center	C₄	#6499
commutator, cosocle	C₂²×C₄×C₇	C₂³
Frattini	C₂²×C₄	C₂²×D₇
Fitting, prime radical	C₇×#575	C₂
socle	C₂×C₇	#394
hypercenter	#575	D₇
hypocenter	C₇	#2050
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #575

subgroup

G/H

center

C₄

#6499

commutator, cosocle

C₂²×C₄×C₇

C₂³

Frattini

C₂²×C₄

C₂²×D₇

Fitting, prime radical

C₂

C₂×C₇

D₇

C₇

G ▷ C₂²×C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #575

N	H ≅ G/N	type
2. #90009	C₂	semidirect
3. C₂×#6411	C₂	semidirect
4. #89890	C₂	semidirect
5. C₇⋊#338	C₂	semidirect
141. C₇	#2050	semidirect

H ≅ G/N

type

2. #90009

C₂

semidirect

3. C₂×#6411

C₂

semidirect

4. #89890

C₂

semidirect

5. C₇⋊#338

C₂

semidirect

141. C₇

#2050

semidirect

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

subgroup	index
C₇×#575	2
#90009	2
C₂×#6411	2
#89890	2
C₇⋊#338	2
#2050	7

subgroup

index

Subgroups — #11729871

Conjugacy classes of subgroups

Complements

Maximal subgroups