Group #11740697 — subgroups

Subgroups — #11740697

Order 896.

504 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 197 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×#6432	C₂	1	1	yes	maximal, characteristic
3	#90002	C₂	1	4	yes	maximal
4	#89749	C₂	1	4	yes	maximal
5	#89746	C₂	1	4	yes	maximal
6	C₄²×Dic₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₇×Q₈	C₂²	1	2	yes	—
8	C₂²×Dic₁₄	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	2	yes	—
14	C₂×C₄⋊Dic₇	C₂²	1	2	yes	—
15	C₂×C₄×Dic₇	C₂²	1	2	yes	—
16	C₂×C₄×Dic₇	C₂²	1	2	yes	—
17	C₂×C₄×Dic₇	C₂²	1	2	yes	—
18	#6432	—	2	4	no	—
19	C₇⋊#95	C₂²	1	16	yes	—
20	#2862	—	7	1	no	maximal, Sylow
21	C₂×C₇×Q₈	—	2	4	no	—
22	C₂×C₇×Q₈	—	2	8	no	—
23	C₂²×C₄×C₇	C₂³	1	4	yes	—
24	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
27	C₂²×Dic₇	C₂³	1	4	yes	—
28	C₂²×Dic₇	C₂³	1	4	yes	—
29	C₂×Dic₁₄	—	2	4	no	—
30	C₂×Dic₁₄	—	2	8	no	—
31	C₇×C₄⋊C₄	—	2	4	no	—
32	C₇×C₄⋊C₄	—	2	4	no	—
33	C₄²×C₇	D₄	1	4	yes	—
34	C₄⋊Dic₇	—	2	4	no	—
35	C₄⋊Dic₇	—	2	4	no	—
36	C₄×Dic₇	—	2	4	no	—
37	C₄×Dic₇	—	2	4	no	—
38	C₄×Dic₇	Q₈	1	8	yes	complemented
39	C₂×C₄⋊Q₈	—	7	1	no	—
40	C₂×C₄⋊Q₈	D₇	1	1	yes	characteristic
41	#391	—	7	4	no	—
42	#391	—	7	4	no	—
43	#382	—	7	4	no	—
44	C₄³	—	7	1	no	—
45	C₂³×C₇	C₂⁴	1	1	yes	characteristic
46	C₇×Q₈	—	4	8	no	—
47	C₂×C₄×C₇	—	2	4	no	—
48	C₂×C₄×C₇	—	2	8	no	—
49	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₇	C₂×Q₈	1	4	yes	—
53	C₂×C₄×C₇	#41	1	4	yes	—
54	C₂×C₄×C₇	#41	1	4	yes	—
55	C₂×Dic₇	—	2	4	no	—
56	C₂×Dic₇	—	2	4	no	—
57	C₂×Dic₇	—	2	4	no	—
58	C₂×Dic₇	—	2	8	no	—
59	C₂×Dic₇	C₂×Q₈	1	8	yes	—
60	Dic₁₄	—	4	8	no	—
61	C₂²×Q₈	—	7	2	no	—
62	C₂²×Q₈	C₂×D₇	1	2	yes	—
63	C₄⋊Q₈	—	2	4	no	—
64	C₄⋊Q₈	—	14	4	no	—
65	C₂×C₄⋊C₄	—	7	2	no	—
66	C₂×C₄⋊C₄	—	7	2	no	—
67	C₂×C₄⋊C₄	C₂×D₇	1	2	yes	—
68	C₂×C₄⋊C₄	C₂×D₇	1	2	yes	—
69	C₂×C₄²	—	7	2	no	—
70	C₂×C₄²	—	7	2	no	—
71	C₂×C₄²	—	7	2	no	—
72	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
73	#95	—	7	16	no	—
74	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
75	C₂²×C₇	C₂²×Q₈	1	2	yes	—
76	C₂²×C₇	C₂×#41	1	2	yes	—
77	C₂²×C₇	C₂×#41	1	2	yes	—
78	C₄×C₇	—	4	4	no	—
79	C₄×C₇	#124	1	4	yes	—
80	C₄×C₇	#124	1	4	yes	—
81	C₄×C₇	C₄⋊Q₈	1	4	yes	—
82	Dic₇	—	2	8	no	—
83	Dic₇	—	4	4	no	—
84	C₂×Q₈	—	2	8	no	—
85	C₂×Q₈	—	2	4	no	—
86	C₂×Q₈	—	14	8	no	—
87	C₂×Q₈	—	14	4	no	—
88	C₂²×C₄	—	7	4	no	—
89	C₂²×C₄	—	7	4	no	—
90	C₂²×C₄	C₂²×D₇	1	4	yes	—
91	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
92	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
93	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
94	C₄⋊C₄	—	2	4	no	—
95	C₄⋊C₄	—	2	4	no	—
96	C₄⋊C₄	—	14	4	no	—
97	C₄⋊C₄	—	14	4	no	—
98	C₄²	—	7	8	no	—
99	C₄²	—	14	4	no	—
100	C₄²	—	14	4	no	—
101	C₄²	C₂²⋊D₇	1	4	yes	—
102	C₂×C₇	C₂×#124	1	1	yes	characteristic
103	C₂×C₇	C₂×#124	1	1	yes	characteristic
104	C₂×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
105	C₂×C₇	#533	1	4	yes	—
106	C₂³	C₂³×D₇	1	1	yes	characteristic
107	Q₈	—	4	8	no	—
108	Q₈	—	28	8	no	—
109	C₂×C₄	—	2	8	no	—
110	C₂×C₄	—	2	4	no	—
111	C₂×C₄	—	7	8	no	—
112	C₂×C₄	—	14	4	no	—
113	C₂×C₄	—	14	4	no	—
114	C₂×C₄	—	14	8	no	—
115	C₂×C₄	—	14	4	no	—
116	C₂×C₄	#1159	1	4	yes	—
117	C₂×C₄	Q₈×D₇	1	4	yes	—
118	C₂×C₄	#1161	1	4	yes	—
119	C₂×C₄	C₂×C₂²⋊D₇	1	2	yes	—
120	C₂×C₄	C₂×C₂²⋊D₇	1	2	yes	—
121	C₂×C₄	C₂×C₂²⋊D₇	1	2	yes	—
122	C₇	#2862	1	1	yes	characteristic, complemented, Sylow
123	C₂²	C₂×#1159	1	2	yes	—
124	C₂²	C₂×Q₈×D₇	1	2	yes	—
125	C₂²	C₂×#1161	1	2	yes	—
126	C₂²	C₂²×C₂²⋊D₇	1	1	yes	characteristic
127	C₄	—	4	4	no	—
128	C₄	—	14	8	no	—
129	C₄	—	28	4	no	—
130	C₄	#6499	1	4	yes	—
131	C₄	#6507	1	4	yes	—
132	C₄	#6510	1	4	yes	—
133	C₂	#90419	1	4	yes	—
134	C₂	C₂×#6499	1	1	yes	characteristic
135	C₂	C₂×#6507	1	1	yes	characteristic
136	C₂	C₂×#6510	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₇×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂×#6432

C₂

yes

maximal, characteristic

#90002

C₂

yes

maximal

#89749

C₂

yes

maximal

#89746

C₂

yes

maximal

C₄²×Dic₇

C₂

yes

maximal, characteristic

C₂²×C₇×Q₈

C₂²

yes

—

C₂²×Dic₁₄

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

—

C₂×C₄⋊Dic₇

C₂²

yes

—

C₂×C₄×Dic₇

C₂²

yes

—

C₂×C₄×Dic₇

C₂²

yes

—

C₂×C₄×Dic₇

C₂²

yes

—

#6432

—

C₇⋊#95

C₂²

yes

—

#2862

—

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

—

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₄²×C₇

D₄

yes

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄×Dic₇

—

C₄×Dic₇

—

C₄×Dic₇

Q₈

yes

complemented

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

D₇

yes

—

—

—

C₄³

—

C₂³×C₇

C₂⁴

yes

characteristic

C₇×Q₈

—

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×C₄×C₇

#41

yes

—

C₂×C₄×C₇

#41

yes

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

C₂×Q₈

yes

—

Dic₁₄

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₇

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇

yes

—

C₂×C₄⋊C₄

C₂×D₇

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₇

yes

characteristic

#95

—

C₂²×C₇

C₂²×D₄

yes

characteristic

C₂²×C₇

C₂²×Q₈

yes

—

C₂²×C₇

C₂×#41

yes

—

C₂²×C₇

C₂×#41

yes

—

C₄×C₇

—

C₄×C₇

#124

yes

—

C₄×C₇

#124

yes

—

C₄×C₇

C₄⋊Q₈

yes

—

Dic₇

—

Dic₇

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₇

yes

—

C₂²×C₄

C₂²×D₇

yes

characteristic

C₂²×C₄

C₂²×D₇

yes

characteristic

C₂²×C₄

C₂²×D₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

100

C₄²

—

101

C₄²

C₂²⋊D₇

yes

—

102

C₂×C₇

C₂×#124

yes

characteristic

103

C₂×C₇

C₂×#124

yes

characteristic

104

C₂×C₇

C₂×C₄⋊Q₈

yes

characteristic

105

C₂×C₇

#533

yes

—

106

C₂³

C₂³×D₇

yes

characteristic

107

Q₈

—

108

Q₈

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

#1159

yes

—

117

C₂×C₄

Q₈×D₇

yes

—

118

C₂×C₄

#1161

yes

—

119

C₂×C₄

C₂×C₂²⋊D₇

yes

—

120

C₂×C₄

C₂×C₂²⋊D₇

yes

—

121

C₂×C₄

C₂×C₂²⋊D₇

yes

—

122

C₇

#2862

yes

characteristic, complemented, Sylow

123

C₂²

C₂×#1159

yes

—

124

C₂²

C₂×Q₈×D₇

yes

—

125

C₂²

C₂×#1161

yes

—

126

C₂²

C₂²×C₂²⋊D₇

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

#6499

yes

—

131

C₄

#6507

yes

—

132

C₄

#6510

yes

—

133

C₂

#90419

yes

—

134

C₂

C₂×#6499

yes

characteristic

135

C₂

C₂×#6507

yes

characteristic

136

C₂

C₂×#6510

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	#2862
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
38. C₄×Dic₇	Q₈	semidirect
122. C₇	#2862	semidirect

H ≅ G/N

type

38. C₄×Dic₇

Q₈

semidirect

122. C₇

#2862

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×#6432	2
#90002	2
#89749	2
#89746	2
C₄²×Dic₇	2
#2862	7

subgroup

index

C₂×C₇×C₄⋊Q₈

C₄²×Dic₇

Subgroups — #11740697

Conjugacy classes of subgroups

Complements

Maximal subgroups