Group #11739379 — subgroups

Subgroups — #11739379

Order 896.

368 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 133 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	#90419	C₂	1	2	yes	maximal
2	C₇×#500	C₂	1	1	yes	maximal, characteristic, complemented
3	#89865	C₂	1	2	yes	maximal, complemented
4	#89659	C₂	1	4	yes	maximal
5	#89645	C₂	1	4	yes	maximal, complemented
6	#89471	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₈×D₇	C₂	1	1	yes	maximal, characteristic
8	C₇×C₄⋊Q₈	C₂²	1	2	yes	—
9	#6509	—	2	2	no	—
10	Q₈×Dic₇	—	2	2	no	—
11	C₂×Dic₂₈	C₂²	1	4	yes	—
12	C₂×C₈×D₇	C₂²	1	2	yes	—
13	#6460	C₂²	1	4	yes	—
14	#6455	—	2	2	no	—
15	#6453	—	2	2	no	—
16	#6450	C₂²	1	4	yes	—
17	C₄²×D₇	C₂²	1	1	yes	characteristic
18	#6432	C₂²	1	2	yes	—
19	C₇×#107	C₂²	1	4	yes	—
20	C₄×C₇×C₈	C₂²	1	1	yes	characteristic
21	C₈×Dic₇	C₂²	1	2	yes	—
22	#6384	C₂²	1	8	yes	—
23	C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
24	#3170	—	7	1	no	maximal, Sylow
25	C₂×C₇×Q₈	—	2	2	no	—
26	C₂×C₄×D₇	D₄	1	1	yes	characteristic
27	C₂×C₄×D₇	D₄	1	1	yes	characteristic
28	C₂×C₄×D₇	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₄	C₂³	1	4	yes	—
30	C₂×C₇×C₈	C₂³	1	2	yes	—
31	C₇×C₄⋊C₄	—	2	2	no	—
32	C₇×C₄⋊C₄	C₂³	1	4	yes	—
33	C₄²×C₇	C₂³	1	1	yes	characteristic
34	(C₂×D₇)⋊C₄	—	2	4	no	—
35	C₄⋊Dic₇	—	2	2	no	—
36	C₄⋊Dic₇	—	2	2	no	—
37	C₄⋊Dic₇	—	2	4	no	—
38	Dic₇⋊C₄	—	2	4	no	—
39	C₄×Dic₇	—	2	4	no	—
40	C₄×Dic₇	D₄	1	1	yes	characteristic
41	C₄×Dic₇	D₄	1	1	yes	characteristic
42	C₄×Dic₇	C₂³	1	1	yes	characteristic
43	C₂×C₇⋊C₈	C₂³	1	2	yes	—
44	Dic₂₈	—	2	8	no	—
45	C₈×D₇	—	2	4	no	—
46	#533	—	7	2	no	—
47	#500	D₇	1	1	yes	characteristic, complemented
48	#494	—	7	1	no	—
49	#487	—	7	2	no	—
50	#467	—	7	4	no	—
51	C₄×Q₁₆	—	7	4	no	—
52	C₂×C₄×C₈	—	7	1	no	—
53	C₂²×D₇	C₂×D₄	1	1	yes	characteristic
54	C₇×Q₈	—	4	4	no	—
55	C₂×C₄×C₇	—	2	4	no	—
56	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
59	C₂×Dic₇	—	2	4	no	—
60	C₂×Dic₇	C₂×D₄	1	1	yes	characteristic
61	C₂×Dic₇	C₂×D₄	1	1	yes	characteristic
62	C₂×Dic₇	C₂×D₄	1	1	yes	characteristic
63	C₄×D₇	—	2	2	no	—
64	C₄×D₇	—	2	2	no	—
65	C₄×D₇	Q₁₆	1	4	yes	complemented
66	Dic₁₄	—	2	8	no	—
67	C₇×C₈	#41	1	4	yes	—
68	C₇⋊C₈	—	2	2	no	—
69	C₂×Q₁₆	—	7	4	no	—
70	C₂²×C₈	—	7	2	no	—
71	C₄⋊Q₈	—	7	2	no	—
72	C₄⋊Q₈	C₂×D₇	1	2	yes	—
73	#125	—	14	2	no	—
74	C₂²⋊Q₈	—	7	4	no	—
75	C₂²⋊Q₈	—	14	2	no	—
76	C₄×Q₈	—	7	4	no	—
77	C₄×Q₈	—	14	2	no	—
78	#117	—	14	2	no	—
79	C₂×C₄²	—	7	1	no	—
80	#107	C₂×D₇	1	4	yes	—
81	#103	—	7	8	no	—
82	C₄×C₈	—	7	2	no	—
83	C₄×C₈	—	7	1	no	—
84	C₄×C₈	C₂×D₇	1	1	yes	characteristic
85	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
86	C₂×D₇	—	2	1	no	—
87	C₂×D₇	—	2	1	no	—
88	C₂×D₇	C₂×Q₁₆	1	2	yes	—
89	C₄×C₇	—	4	4	no	—
90	C₄×C₇	C₂×Q₁₆	1	2	yes	—
91	C₄×C₇	#135	1	2	yes	—
92	C₄×C₇	C₂×#41	1	2	yes	—
93	Dic₇	—	2	1	no	—
94	Dic₇	—	2	1	no	—
95	Dic₇	—	4	4	no	—
96	Dic₇	C₂×Q₁₆	1	2	yes	—
97	C₂×Q₈	—	2	2	no	—
98	C₂×Q₈	—	7	4	no	—
99	C₂²×C₄	—	7	1	no	—
100	C₂²×C₄	—	7	1	no	—
101	C₂²×C₄	—	7	1	no	—
102	Q₁₆	—	14	8	no	—
103	C₂×C₈	—	7	2	no	—
104	C₂×C₈	—	14	4	no	—
105	C₂×C₈	C₂²×D₇	1	2	yes	—
106	C₄⋊C₄	—	2	2	no	—
107	C₄⋊C₄	—	14	4	no	—
108	C₄⋊C₄	—	14	4	no	—
109	C₄⋊C₄	—	14	2	no	—
110	C₄⋊C₄	—	14	2	no	—
111	C₄⋊C₄	C₂²×D₇	1	4	yes	—
112	C₂²⋊C₄	—	14	4	no	—
113	C₄²	—	7	1	no	—
114	C₄²	—	7	1	no	—
115	C₄²	—	7	1	no	—
116	C₄²	—	14	4	no	—
117	C₄²	C₂²×D₇	1	1	yes	characteristic
118	C₂×C₇	#532	1	1	yes	characteristic
119	C₂×C₇	C₂²×Q₁₆	1	1	yes	characteristic
120	C₂×C₇	C₂×#135	1	1	yes	characteristic
121	D₇	—	2	2	no	—
122	C₂³	—	7	1	no	—
123	Q₈	—	4	4	no	—
124	Q₈	—	14	8	no	—
125	C₂×C₄	—	2	4	no	—
126	C₂×C₄	—	7	1	no	—
127	C₂×C₄	—	7	4	no	—
128	C₂×C₄	—	7	1	no	—
129	C₂×C₄	—	7	1	no	—
130	C₂×C₄	—	14	2	no	—
131	C₂×C₄	—	14	2	no	—
132	C₂×C₄	—	14	4	no	—
133	C₂×C₄	D₄×D₇	1	1	yes	characteristic
134	C₂×C₄	D₄×D₇	1	1	yes	characteristic
135	C₂×C₄	C₂³×D₇	1	1	yes	characteristic
136	C₈	—	14	2	no	—
137	C₈	#1161	1	4	yes	—
138	C₇	#3170	1	1	yes	characteristic, complemented, Sylow
139	C₂²	—	7	2	no	—
140	C₂²	—	14	1	no	—
141	C₂²	—	14	1	no	—
142	C₂²	C₂×D₄×D₇	1	1	yes	characteristic
143	C₄	—	4	4	no	—
144	C₄	—	7	2	no	—
145	C₄	—	14	1	no	—
146	C₄	—	14	1	no	—
147	C₄	—	28	4	no	—
148	C₄	#6475	1	2	yes	—
149	C₄	D₇×Q₁₆	1	2	yes	—
150	C₄	C₂×#1161	1	2	yes	—
151	C₂	—	14	2	no	—
152	C₂	#90416	1	1	yes	characteristic
153	C₂	C₂×#6475	1	1	yes	characteristic
154	C₂	C₂×D₇×Q₁₆	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#90419

C₂

yes

maximal

C₇×#500

C₂

yes

maximal, characteristic, complemented

#89865

C₂

yes

maximal, complemented

#89659

C₂

yes

maximal

#89645

C₂

yes

maximal, complemented

#89471

C₂

yes

maximal, characteristic, complemented

C₄×C₈×D₇

C₂

yes

maximal, characteristic

C₇×C₄⋊Q₈

C₂²

yes

—

#6509

—

Q₈×Dic₇

—

C₂×Dic₂₈

C₂²

yes

—

C₂×C₈×D₇

C₂²

yes

—

#6460

C₂²

yes

—

#6455

—

#6453

—

#6450

C₂²

yes

—

C₄²×D₇

C₂²

yes

characteristic

#6432

C₂²

yes

—

C₇×#107

C₂²

yes

—

C₄×C₇×C₈

C₂²

yes

characteristic

C₈×Dic₇

C₂²

yes

—

#6384

C₂²

yes

—

C₄×C₇⋊C₈

C₂²

yes

characteristic

#3170

—

maximal, Sylow

C₂×C₇×Q₈

—

C₂×C₄×D₇

D₄

yes

characteristic

C₂×C₄×D₇

D₄

yes

characteristic

C₂×C₄×D₇

C₂³

yes

characteristic

C₂×Dic₁₄

C₂³

yes

—

C₂×C₇×C₈

C₂³

yes

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

C₂³

yes

—

C₄²×C₇

C₂³

yes

characteristic

(C₂×D₇)⋊C₄

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

Dic₇⋊C₄

—

C₄×Dic₇

—

C₄×Dic₇

D₄

yes

characteristic

C₄×Dic₇

D₄

yes

characteristic

C₄×Dic₇

C₂³

yes

characteristic

C₂×C₇⋊C₈

C₂³

yes

—

Dic₂₈

—

C₈×D₇

—

#533

—

#500

D₇

yes

characteristic, complemented

#494

—

#487

—

#467

—

C₄×Q₁₆

—

C₂×C₄×C₈

—

C₂²×D₇

C₂×D₄

yes

characteristic

C₇×Q₈

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

C₂⁴

yes

characteristic

C₂×Dic₇

—

C₂×Dic₇

C₂×D₄

yes

characteristic

C₂×Dic₇

C₂×D₄

yes

characteristic

C₂×Dic₇

C₂×D₄

yes

characteristic

C₄×D₇

—

C₄×D₇

—

C₄×D₇

Q₁₆

yes

complemented

Dic₁₄

—

C₇×C₈

#41

yes

—

C₇⋊C₈

—

C₂×Q₁₆

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₇

yes

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

#107

C₂×D₇

yes

—

#103

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₇

yes

characteristic

C₂²×C₇

C₂²×D₄

yes

characteristic

C₂×D₇

—

C₂×D₇

—

C₂×D₇

C₂×Q₁₆

yes

—

C₄×C₇

—

C₄×C₇

C₂×Q₁₆

yes

—

C₄×C₇

#135

yes

—

C₄×C₇

C₂×#41

yes

—

Dic₇

—

Dic₇

—

Dic₇

—

Dic₇

C₂×Q₁₆

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

Q₁₆

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

C₂²×D₇

yes

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

C₂²×D₇

yes

—

112

C₂²⋊C₄

—

113

C₄²

—

114

C₄²

—

115

C₄²

—

116

C₄²

—

117

C₄²

C₂²×D₇

yes

characteristic

118

C₂×C₇

#532

yes

characteristic

119

C₂×C₇

C₂²×Q₁₆

yes

characteristic

120

C₂×C₇

C₂×#135

yes

characteristic

121

D₇

—

122

C₂³

—

123

Q₈

—

124

Q₈

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

D₄×D₇

yes

characteristic

134

C₂×C₄

D₄×D₇

yes

characteristic

135

C₂×C₄

C₂³×D₇

yes

characteristic

136

C₈

—

137

C₈

#1161

yes

—

138

C₇

#3170

yes

characteristic, complemented, Sylow

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

C₂×D₄×D₇

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

#6475

yes

—

149

C₄

D₇×Q₁₆

yes

—

150

C₄

C₂×#1161

yes

—

151

C₂

—

152

C₂

#90416

yes

characteristic

153

C₂

C₂×#6475

yes

characteristic

154

C₂

C₂×D₇×Q₁₆

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₇
commutator, cosocle	C₂×C₄×C₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₇
Fitting	C₇×#500	C₂
socle	C₂²×C₇	C₂²×D₄
prime radical	C₂²×D₇	C₂×D₄
hypercenter	#500	D₇
hypocenter	C₇	#3170
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

C₂×D₄×D₇

commutator, cosocle

C₂×C₄×C₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₇

Fitting

C₇×#500

C₂

socle

C₂²×C₇

C₂²×D₄

prime radical

C₂²×D₇

C₂×D₄

D₇

C₇

G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
2. C₇×#500	C₂	semidirect
3. #89865	C₂	semidirect
5. #89645	C₂	semidirect
6. #89471	C₂	semidirect
47. #500	D₇	semidirect
65. C₄×D₇	Q₁₆	semidirect
138. C₇	#3170	semidirect

H ≅ G/N

type

2. C₇×#500

C₂

semidirect

3. #89865

C₂

semidirect

5. #89645

C₂

semidirect

6. #89471

C₂

semidirect

47. #500

D₇

semidirect

65. C₄×D₇

Q₁₆

semidirect

138. C₇

#3170

semidirect

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

subgroup	index
#90419	2
C₇×#500	2
#89865	2
#89659	2
#89645	2
#89471	2
C₄×C₈×D₇	2
#3170	7

subgroup

index

C₄×C₈×D₇

Subgroups — #11739379

Conjugacy classes of subgroups

Complements

Maximal subgroups