Group #11736617 — subgroups

Subgroups — #11736617

Order 896.

340 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 149 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	#90210	C₂	1	2	yes	maximal
2	C₇×#427	C₂	1	1	yes	maximal, characteristic
3	C₂×#6391	C₂	1	1	yes	maximal, characteristic
4	#89873	C₂	1	1	yes	maximal, characteristic
5	#89767	C₂	1	2	yes	maximal
6	#89648	C₂	1	4	yes	maximal, complemented
7	#89630	C₂	1	2	yes	maximal, complemented
8	#89627	C₂	1	2	yes	maximal, complemented
9	C₂²×C₇×C₈	C₂²	1	1	yes	characteristic
10	C₇×#117	C₂²	1	2	yes	—
11	#6489	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₇	C₂²	1	2	yes	—
13	C₂×#1136	C₂²	1	1	yes	characteristic
14	#6453	C₂²	1	4	yes	—
15	#6451	C₂²	1	4	yes	—
16	#6441	—	2	4	no	—
17	#6433	—	2	2	no	—
18	#6432	—	2	2	no	—
19	C₇×#107	C₂²	1	2	yes	—
20	C₇×#106	C₂²	1	1	yes	characteristic
21	C₇×#106	C₂²	1	1	yes	characteristic
22	#6391	C₂²	1	2	yes	—
23	#6391	C₂²	1	2	yes	—
24	#6389	C₂²	1	2	yes	—
25	C₈×Dic₇	C₂²	1	2	yes	—
26	#6382	C₂²	1	4	yes	—
27	#6381	C₂²	1	4	yes	—
28	#3163	—	7	1	no	maximal, Sylow
29	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
30	C₂²×Dic₇	—	2	2	no	—
31	C₂×Dic₁₄	—	2	4	no	—
32	C₂×C₇×C₈	Q₈	1	4	yes	complemented
33	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
34	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
35	C₇×C₄⋊C₄	C₂³	1	2	yes	—
36	C₇×C₄⋊C₄	C₂³	1	2	yes	—
37	C₇×C₂²⋊C₄	—	2	2	no	—
38	C₄²×C₇	—	2	2	no	—
39	C₂²⋊Dic₇	D₄	1	2	yes	—
40	C₄⋊Dic₇	—	2	2	no	—
41	C₄⋊Dic₇	—	2	4	no	—
42	C₄⋊Dic₇	D₄	1	2	yes	—
43	C₄⋊Dic₇	C₂³	1	4	yes	—
44	Dic₇⋊C₄	—	2	4	no	—
45	Dic₇⋊C₄	—	2	4	no	—
46	C₄×Dic₇	C₂³	1	2	yes	—
47	#1136	Q₈	1	4	yes	complemented
48	C₂×C₇⋊C₈	C₂³	1	2	yes	—
49	#544	—	7	2	no	—
50	#501	—	7	4	no	—
51	#499	—	7	2	no	—
52	#498	—	7	2	no	—
53	#428	—	7	2	no	—
54	#427	D₇	1	1	yes	characteristic
55	C₂×#106	—	7	1	no	—
56	#405	—	7	1	no	—
57	C₂³×C₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₇	—	2	2	no	—
59	C₂×C₄×C₇	—	2	2	no	—
60	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
62	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
63	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
64	C₂×Dic₇	—	2	4	no	—
65	C₂×Dic₇	—	8	2	no	—
66	C₂×Dic₇	C₂×D₄	1	4	yes	—
67	Dic₁₄	—	8	4	no	—
68	C₇×C₈	C₂×Q₈	1	2	yes	—
69	C₇×C₈	C₂×Q₈	1	2	yes	—
70	C₇⋊C₈	C₂×Q₈	1	4	yes	—
71	C₂×M₁₆	—	7	1	no	—
72	C₂²×C₈	C₂×D₇	1	1	yes	characteristic
73	C₄⋊Q₈	—	7	4	no	—
74	C₄⋊Q₈	—	14	2	no	—
75	#125	—	7	4	no	—
76	#125	—	14	2	no	—
77	C₂²⋊Q₈	—	14	4	no	—
78	#117	—	7	1	no	—
79	#117	C₂×D₇	1	2	yes	—
80	C₂×C₄⋊C₄	—	7	2	no	—
81	#107	—	7	4	no	—
82	#107	C₂×D₇	1	2	yes	—
83	#106	—	7	2	no	—
84	#106	—	7	2	no	—
85	#106	—	7	4	no	—
86	#106	C₂×D₇	1	1	yes	characteristic
87	#106	C₂×D₇	1	1	yes	characteristic
88	#97	—	7	2	no	—
89	C₄×C₈	—	7	2	no	—
90	C₂²×C₇	—	2	2	no	—
91	C₂²×C₇	C₄⋊Q₈	1	2	yes	—
92	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
93	C₄×C₇	—	4	2	no	—
94	C₄×C₇	—	4	2	no	—
95	C₄×C₇	C₄⋊Q₈	1	2	yes	—
96	C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
97	C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
98	Dic₇	—	2	4	no	—
99	Dic₇	—	4	4	no	—
100	C₂×Q₈	—	14	4	no	—
101	C₂²×C₄	—	14	2	no	—
102	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
103	M₁₆	—	7	4	no	—
104	C₂×C₈	—	7	2	no	—
105	C₂×C₈	Dic₁₄	1	4	yes	complemented
106	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
107	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
108	C₄⋊C₄	—	7	2	no	—
109	C₄⋊C₄	—	7	4	no	—
110	C₄⋊C₄	—	14	4	no	—
111	C₄⋊C₄	—	14	4	no	—
112	C₄⋊C₄	—	14	4	no	—
113	C₄⋊C₄	—	14	2	no	—
114	C₄⋊C₄	C₂²×D₇	1	2	yes	—
115	C₄⋊C₄	C₂²×D₇	1	2	yes	—
116	C₂²⋊C₄	—	2	2	no	—
117	C₂²⋊C₄	—	7	2	no	—
118	C₄²	—	2	2	no	—
119	C₄²	—	7	2	no	—
120	C₂×C₇	—	2	2	no	—
121	C₂×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
122	C₂×C₇	#577	1	2	yes	—
123	C₂³	D₄×D₇	1	1	yes	characteristic
124	Q₈	—	56	4	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	7	4	no	—
128	C₂×C₄	—	14	4	no	—
129	C₂×C₄	—	56	2	no	—
130	C₂×C₄	C₂×Dic₁₄	1	2	yes	—
131	C₂×C₄	D₄×D₇	1	1	yes	characteristic
132	C₂×C₄	Q₈×D₇	1	2	yes	—
133	C₂×C₄	C₂³×D₇	1	1	yes	characteristic
134	C₈	—	7	4	no	—
135	C₈	C₂×Dic₁₄	1	2	yes	—
136	C₈	C₂×Dic₁₄	1	2	yes	—
137	C₇	#3163	1	1	yes	characteristic, complemented, Sylow
138	C₂²	—	2	2	no	—
139	C₂²	#6451	1	2	yes	—
140	C₂²	C₂×D₄×D₇	1	1	yes	characteristic
141	C₄	—	4	2	no	—
142	C₄	—	4	2	no	—
143	C₄	—	14	4	no	—
144	C₄	—	28	4	no	—
145	C₄	#6451	1	2	yes	—
146	C₄	C₂²×Dic₁₄	1	1	yes	characteristic
147	C₄	C₂×Q₈×D₇	1	1	yes	characteristic
148	C₂	—	2	2	no	—
149	C₂	C₂×#6451	1	1	yes	characteristic
150	C₂	#90461	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#90210

C₂

yes

maximal

C₇×#427

C₂

yes

maximal, characteristic

C₂×#6391

C₂

yes

maximal, characteristic

#89873

C₂

yes

maximal, characteristic

#89767

C₂

yes

maximal

#89648

C₂

yes

maximal, complemented

#89630

C₂

yes

maximal, complemented

#89627

C₂

yes

maximal, complemented

C₂²×C₇×C₈

C₂²

yes

characteristic

C₇×#117

C₂²

yes

—

#6489

C₂²

yes

characteristic

C₂×C₄⋊Dic₇

C₂²

yes

—

C₂×#1136

C₂²

yes

characteristic

#6453

C₂²

yes

—

#6451

C₂²

yes

—

—

—

—

C₂²

yes

—

C₇×#106

C₂²

yes

characteristic

C₇×#106

C₂²

yes

characteristic

#6391

C₂²

yes

—

#6391

C₂²

yes

—

#6389

C₂²

yes

—

C₈×Dic₇

C₂²

yes

—

#6382

C₂²

yes

—

#6381

C₂²

yes

—

#3163

—

maximal, Sylow

C₂²×C₄×C₇

C₂³

yes

characteristic

C₂²×Dic₇

—

C₂×Dic₁₄

—

C₂×C₇×C₈

Q₈

yes

complemented

C₂×C₇×C₈

C₂³

yes

characteristic

C₂×C₇×C₈

C₂³

yes

characteristic

C₇×C₄⋊C₄

C₂³

yes

—

C₇×C₄⋊C₄

C₂³

yes

—

C₇×C₂²⋊C₄

—

C₄²×C₇

—

C₂²⋊Dic₇

D₄

yes

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

D₄

yes

—

C₄⋊Dic₇

C₂³

yes

—

Dic₇⋊C₄

—

Dic₇⋊C₄

—

C₄×Dic₇

C₂³

yes

—

#1136

Q₈

yes

complemented

C₂×C₇⋊C₈

C₂³

yes

—

—

—

—

—

—

D₇

yes

characteristic

C₂×#106

—

#405

—

C₂³×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

C₂×Q₈

yes

—

C₂×C₄×C₇

C₂×Q₈

yes

—

C₂×C₄×C₇

C₂⁴

yes

characteristic

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

C₂×D₄

yes

—

Dic₁₄

—

C₇×C₈

C₂×Q₈

yes

—

C₇×C₈

C₂×Q₈

yes

—

C₇⋊C₈

C₂×Q₈

yes

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₇

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

#125

—

C₂²⋊Q₈

—

#117

—

#117

C₂×D₇

yes

—

C₂×C₄⋊C₄

—

#107

—

#107

C₂×D₇

yes

—

—

—

—

C₂×D₇

yes

characteristic

#106

C₂×D₇

yes

characteristic

#97

—

C₄×C₈

—

C₂²×C₇

—

C₂²×C₇

C₄⋊Q₈

yes

—

C₂²×C₇

C₂²×D₄

yes

characteristic

C₄×C₇

—

C₄×C₇

—

C₄×C₇

C₄⋊Q₈

yes

—

C₄×C₇

C₂²×Q₈

yes

characteristic

C₄×C₇

C₂²×Q₈

yes

characteristic

Dic₇

—

Dic₇

—

100

C₂×Q₈

—

101

C₂²×C₄

—

102

C₂²×C₄

C₂²×D₇

yes

characteristic

103

M₁₆

—

104

C₂×C₈

—

105

C₂×C₈

Dic₁₄

yes

complemented

106

C₂×C₈

C₂²×D₇

yes

characteristic

107

C₂×C₈

C₂²×D₇

yes

characteristic

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

C₂²×D₇

yes

—

115

C₄⋊C₄

C₂²×D₇

yes

—

116

C₂²⋊C₄

—

117

C₂²⋊C₄

—

118

C₄²

—

119

C₄²

—

120

C₂×C₇

—

121

C₂×C₇

C₂×C₄⋊Q₈

yes

characteristic

122

C₂×C₇

#577

yes

—

123

C₂³

D₄×D₇

yes

characteristic

124

Q₈

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

C₂×Dic₁₄

yes

—

131

C₂×C₄

D₄×D₇

yes

characteristic

132

C₂×C₄

Q₈×D₇

yes

—

133

C₂×C₄

C₂³×D₇

yes

characteristic

134

C₈

—

135

C₈

C₂×Dic₁₄

yes

—

136

C₈

C₂×Dic₁₄

yes

—

137

C₇

#3163

yes

characteristic, complemented, Sylow

138

C₂²

—

139

C₂²

#6451

yes

—

140

C₂²

C₂×D₄×D₇

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

#6451

yes

—

146

C₄

C₂²×Dic₁₄

yes

characteristic

147

C₄

C₂×Q₈×D₇

yes

characteristic

148

C₂

—

149

C₂

C₂×#6451

yes

characteristic

150

C₂

#90461

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₇
commutator, cosocle	C₂×C₄×C₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₇
Fitting	C₇×#427	C₂
socle	C₂²×C₇	C₂²×D₄
prime radical	C₂³×C₇	C₂×D₄
hypercenter	#427	D₇
hypocenter	C₇	#3163
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂²

C₂×D₄×D₇

commutator, cosocle

C₂×C₄×C₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₇

Fitting

C₇×#427

C₂

socle

C₂²×C₇

C₂²×D₄

prime radical

C₂³×C₇

C₂×D₄

D₇

C₇

G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
6. #89648	C₂	semidirect
7. #89630	C₂	semidirect
8. #89627	C₂	semidirect
32. C₂×C₇×C₈	Q₈	semidirect
47. #1136	Q₈	semidirect
105. C₂×C₈	Dic₁₄	semidirect
137. C₇	#3163	semidirect

H ≅ G/N

type

6. #89648

C₂

semidirect

7. #89630

C₂

semidirect

8. #89627

C₂

semidirect

32. C₂×C₇×C₈

Q₈

semidirect

47. #1136

Q₈

semidirect

105. C₂×C₈

Dic₁₄

semidirect

137. C₇

#3163

semidirect

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

subgroup	index
#90210	2
C₇×#427	2
C₂×#6391	2
#89873	2
#89767	2
#89648	2
#89630	2
#89627	2
#3163	7

subgroup

index

Subgroups — #11736617

Conjugacy classes of subgroups

Complements

Maximal subgroups