Group #11740691 — subgroups

Subgroups — #11740691

Order 896.

448 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 165 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×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#6433	C₂	1	1	yes	maximal, characteristic
3	#90001	C₂	1	4	yes	maximal
4	#89752	C₂	1	4	yes	maximal
5	#89749	C₂	1	2	yes	maximal
6	#89748	C₂	1	2	yes	maximal
7	#89704	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₇×Q₈	C₂²	1	2	yes	—
9	C₇×C₄⋊Q₈	—	2	4	no	—
10	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₇⋊C₄	C₂²	1	2	yes	—
16	C₂×Dic₇⋊C₄	C₂²	1	2	yes	—
17	C₂×Dic₇⋊C₄	C₂²	1	8	yes	—
18	C₂×C₄×Dic₇	C₂²	1	2	yes	—
19	#6433	—	2	4	no	—
20	C₇⋊#95	C₂²	1	2	yes	—
21	C₇⋊#95	C₂²	1	2	yes	—
22	C₇⋊#95	C₂²	1	8	yes	—
23	#2660	—	7	1	no	maximal, Sylow
24	C₂×C₇×Q₈	—	2	4	no	—
25	C₂×C₇×Q₈	—	4	4	no	—
26	C₂²×C₄×C₇	C₂³	1	4	yes	—
27	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
30	C₂²×Dic₇	C₂³	1	2	yes	—
31	C₂²×Dic₇	C₂³	1	2	yes	—
32	C₂²×Dic₇	C₂³	1	4	yes	—
33	C₇×C₄⋊C₄	—	2	4	no	—
34	C₇×C₄⋊C₄	—	2	4	no	—
35	C₄²×C₇	D₄	1	4	yes	—
36	C₄⋊Dic₇	—	2	2	no	—
37	C₄⋊Dic₇	—	2	2	no	—
38	Dic₇⋊C₄	—	2	4	no	—
39	Dic₇⋊C₄	—	2	4	no	—
40	Dic₇⋊C₄	—	4	8	no	—
41	C₄×Dic₇	Q₈	1	8	yes	complemented
42	C₂×C₄⋊Q₈	D₇	1	1	yes	characteristic
43	C₂×#125	—	7	1	no	—
44	#398	—	7	4	no	—
45	#395	—	7	4	no	—
46	#384	—	7	2	no	—
47	#382	—	7	2	no	—
48	#376	—	7	1	no	—
49	C₂³×C₇	C₂⁴	1	1	yes	characteristic
50	C₇×Q₈	—	4	8	no	—
51	C₂×C₄×C₇	—	2	2	no	—
52	C₂×C₄×C₇	—	2	2	no	—
53	C₂×C₄×C₇	—	2	4	no	—
54	C₂×C₄×C₇	—	2	8	no	—
55	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₇	C₂×Q₈	1	4	yes	—
59	C₂×Dic₇	—	2	4	no	—
60	C₂×Dic₇	—	2	4	no	—
61	C₂×Dic₇	—	2	2	no	—
62	C₂×Dic₇	—	2	2	no	—
63	C₂×Dic₇	—	2	4	no	—
64	C₂×Dic₇	—	2	4	no	—
65	C₂×Dic₇	C₂×Q₈	1	8	yes	—
66	C₂²×Q₈	C₂×D₇	1	2	yes	—
67	C₄⋊Q₈	—	2	4	no	—
68	#125	—	14	4	no	—
69	C₂×C₄⋊C₄	—	7	2	no	—
70	C₂×C₄⋊C₄	—	7	2	no	—
71	C₂×C₄⋊C₄	—	7	1	no	—
72	C₂×C₄⋊C₄	—	7	1	no	—
73	C₂×C₄⋊C₄	—	7	8	no	—
74	C₂×C₄⋊C₄	C₂×D₇	1	2	yes	—
75	C₂×C₄⋊C₄	C₂×D₇	1	2	yes	—
76	C₂×C₄²	—	7	2	no	—
77	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
78	#95	—	7	2	no	—
79	#95	—	7	2	no	—
80	#95	—	7	8	no	—
81	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
82	C₂²×C₇	C₂²×Q₈	1	2	yes	—
83	C₂²×C₇	H(5,𝔽₂)	1	2	yes	—
84	C₂²×C₇	2₋¹⁺⁴	1	2	yes	—
85	C₄×C₇	—	2	2	no	—
86	C₄×C₇	—	2	2	no	—
87	C₄×C₇	—	4	4	no	—
88	C₄×C₇	C₄⋊Q₈	1	4	yes	—
89	Dic₇	—	2	8	no	—
90	Dic₇	—	4	2	no	—
91	Dic₇	—	4	2	no	—
92	C₂×Q₈	—	2	4	no	—
93	C₂×Q₈	—	4	4	no	—
94	C₂²×C₄	—	7	4	no	—
95	C₂²×C₄	—	7	2	no	—
96	C₂²×C₄	—	7	2	no	—
97	C₂²×C₄	C₂²×D₇	1	4	yes	—
98	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
99	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
100	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
101	C₄⋊C₄	—	2	4	no	—
102	C₄⋊C₄	—	2	4	no	—
103	C₄⋊C₄	—	14	4	no	—
104	C₄⋊C₄	—	14	4	no	—
105	C₄⋊C₄	—	14	2	no	—
106	C₄⋊C₄	—	14	2	no	—
107	C₄⋊C₄	—	28	8	no	—
108	C₄²	—	7	8	no	—
109	C₄²	C₂²⋊D₇	1	4	yes	—
110	C₂×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
111	C₂×C₇	#535	1	1	yes	characteristic
112	C₂×C₇	#536	1	1	yes	characteristic
113	C₂×C₇	#544	1	4	yes	—
114	C₂³	C₂³×D₇	1	1	yes	characteristic
115	Q₈	—	4	8	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	8	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	7	8	no	—
121	C₂×C₄	—	14	4	no	—
122	C₂×C₄	—	14	4	no	—
123	C₂×C₄	—	14	4	no	—
124	C₂×C₄	—	14	4	no	—
125	C₂×C₄	—	14	2	no	—
126	C₂×C₄	—	14	2	no	—
127	C₂×C₄	Q₈×D₇	1	4	yes	—
128	C₂×C₄	C₂×C₂²⋊D₇	1	2	yes	—
129	C₂×C₄	C₂×C₂²⋊D₇	1	2	yes	—
130	C₂×C₄	C₂×C₂²⋊D₇	1	2	yes	—
131	C₇	#2660	1	1	yes	characteristic, complemented, Sylow
132	C₂²	#6548	1	2	yes	—
133	C₂²	C₂×Q₈×D₇	1	2	yes	—
134	C₂²	#6551	1	2	yes	—
135	C₂²	C₂²×C₂²⋊D₇	1	1	yes	characteristic
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	4	4	no	—
139	C₄	—	14	8	no	—
140	C₄	—	28	2	no	—
141	C₄	—	28	2	no	—
142	C₄	#6507	1	4	yes	—
143	C₂	#90420	1	4	yes	—
144	C₂	#90496	1	1	yes	characteristic
145	C₂	C₂×#6507	1	1	yes	characteristic
146	C₂	#90507	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₇×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂×#6433

C₂

yes

maximal, characteristic

#90001

C₂

yes

maximal

#89752

C₂

yes

maximal

#89749

C₂

yes

maximal

#89748

C₂

yes

maximal

#89704

C₂

yes

maximal, characteristic

C₂²×C₇×Q₈

C₂²

yes

—

C₇×C₄⋊Q₈

—

C₂×C₇×C₄⋊C₄

C₂²

yes

—

C₂×C₇×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₇

C₂²

yes

characteristic

C₂×Dic₇⋊C₄

C₂²

yes

—

C₂×Dic₇⋊C₄

C₂²

yes

—

C₂×Dic₇⋊C₄

C₂²

yes

—

C₂×C₄×Dic₇

C₂²

yes

—

#6433

—

C₇⋊#95

C₂²

yes

—

C₇⋊#95

C₂²

yes

—

C₇⋊#95

C₂²

yes

—

#2660

—

maximal, Sylow

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂²×C₄×C₇

C₂³

yes

—

C₂²×C₄×C₇

C₂³

yes

characteristic

C₂²×C₄×C₇

C₂³

yes

characteristic

C₂²×C₄×C₇

C₂³

yes

characteristic

C₂²×Dic₇

C₂³

yes

—

C₂²×Dic₇

C₂³

yes

—

C₂²×Dic₇

C₂³

yes

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₄²×C₇

D₄

yes

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

Dic₇⋊C₄

—

Dic₇⋊C₄

—

Dic₇⋊C₄

—

C₄×Dic₇

Q₈

yes

complemented

C₂×C₄⋊Q₈

D₇

yes

—

—

—

—

—

—

C₂³×C₇

C₂⁴

yes

characteristic

C₇×Q₈

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×D₄

yes

—

C₂×C₄×C₇

C₂×D₄

yes

—

C₂×C₄×C₇

C₂×D₄

yes

—

C₂×C₄×C₇

C₂×Q₈

yes

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

C₂×Q₈

yes

—

C₂²×Q₈

C₂×D₇

yes

—

C₄⋊Q₈

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇

yes

—

C₂×C₄⋊C₄

C₂×D₇

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₇

yes

—

—

—

C₂²×C₇

C₂²×D₄

yes

characteristic

C₂²×C₇

C₂²×Q₈

yes

—

C₂²×C₇

H(5,𝔽₂)

yes

—

C₂²×C₇

2₋¹⁺⁴

yes

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

C₄⋊Q₈

yes

—

Dic₇

—

Dic₇

—

Dic₇

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₇

yes

—

C₂²×C₄

C₂²×D₇

yes

characteristic

C₂²×C₄

C₂²×D₇

yes

characteristic

100

C₂²×C₄

C₂²×D₇

yes

characteristic

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄²

—

109

C₄²

C₂²⋊D₇

yes

—

110

C₂×C₇

C₂×C₄⋊Q₈

yes

characteristic

111

C₂×C₇

#535

yes

characteristic

112

C₂×C₇

#536

yes

characteristic

113

C₂×C₇

#544

yes

—

114

C₂³

C₂³×D₇

yes

characteristic

115

Q₈

—

116

C₂×C₄

—

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

Q₈×D₇

yes

—

128

C₂×C₄

C₂×C₂²⋊D₇

yes

—

129

C₂×C₄

C₂×C₂²⋊D₇

yes

—

130

C₂×C₄

C₂×C₂²⋊D₇

yes

—

131

C₇

#2660

yes

characteristic, complemented, Sylow

132

C₂²

#6548

yes

—

133

C₂²

C₂×Q₈×D₇

yes

—

134

C₂²

#6551

yes

—

135

C₂²

C₂²×C₂²⋊D₇

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

#6507

yes

—

143

C₂

#90420

yes

—

144

C₂

#90496

yes

characteristic

145

C₂

C₂×#6507

yes

characteristic

146

C₂

#90507

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇
commutator, socle, prime radical, cosocle	C₂³×C₇	C₂⁴
Fitting	C₂×C₇×C₄⋊Q₈	C₂
hypercenter	C₂×C₄⋊Q₈	D₇
hypocenter	C₇	#2660
LCS	G ▷ C₂³×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₇

commutator, socle, prime radical, cosocle

C₂³×C₇

C₂⁴

Fitting

C₂×C₇×C₄⋊Q₈

C₂

hypercenter

C₂×C₄⋊Q₈

D₇

C₇

G ▷ C₂³×C₇ ▷ C₇

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
41. C₄×Dic₇	Q₈	semidirect
131. C₇	#2660	semidirect

H ≅ G/N

type

41. C₄×Dic₇

Q₈

semidirect

131. C₇

#2660

semidirect

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

subgroup	index
C₂×C₇×C₄⋊Q₈	2
C₂×#6433	2
#90001	2
#89752	2
#89749	2
#89748	2
#89704	2
#2660	7

subgroup

index

C₂×C₇×C₄⋊Q₈

Subgroups — #11740691

Conjugacy classes of subgroups

Complements

Maximal subgroups