Group #11740672 — subgroups

Subgroups — #11740672

Order 896.

516 conjugacy classes of subgroups; only the 237 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 237 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×Q₈×Dic₇	C₂	1	2	yes	maximal
3	#90002	C₂	1	2	yes	maximal
4	#89753	C₂	1	8	yes	maximal
5	#89708	C₂	1	1	yes	maximal, characteristic
6	#89704	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₇×Q₈	C₂²	1	2	yes	—
8	C₇×C₄⋊Q₈	C₄	1	8	yes	complemented
9	C₂×C₇×C₄⋊C₄	C₂²	1	4	yes	—
10	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
11	Q₈×Dic₇	—	2	8	no	—
12	C₂×C₄⋊Dic₇	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₇	C₂²	1	4	yes	—
14	C₂×C₄×Dic₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×Dic₇	C₂²	1	4	yes	—
16	C₂×C₄×Dic₇	C₂²	1	1	yes	characteristic
17	C₇⋊#95	C₂²	1	4	yes	—
18	C₇⋊#95	C₂²	1	4	yes	—
19	C₇⋊#95	C₂²	1	8	yes	—
20	#2327	—	7	1	no	maximal, Sylow
21	C₂×C₇×Q₈	—	2	4	no	—
22	C₂×C₇×Q₈	—	2	4	no	—
23	C₂×C₇×Q₈	C₂×C₄	1	8	yes	—
24	C₂²×C₄×C₇	C₂³	1	2	yes	—
25	C₂²×C₄×C₇	C₂³	1	4	yes	—
26	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
27	C₂²×Dic₇	C₂³	1	2	yes	—
28	C₂²×Dic₇	C₂³	1	4	yes	—
29	C₂²×Dic₇	C₂³	1	2	yes	—
30	C₇×C₄⋊C₄	C₂×C₄	1	16	yes	—
31	C₄²×C₇	C₂×C₄	1	4	yes	—
32	C₄⋊Dic₇	—	2	4	no	—
33	C₄⋊Dic₇	—	2	8	no	—
34	C₄×Dic₇	—	2	2	no	—
35	C₄×Dic₇	—	2	8	no	—
36	C₄×Dic₇	D₄	1	4	yes	—
37	C₂×C₄⋊Q₈	D₇	1	1	yes	characteristic
38	C₂×C₄×Q₈	—	7	2	no	—
39	#391	—	7	2	no	—
40	#387	—	7	8	no	—
41	#382	—	7	1	no	—
42	#376	—	7	1	no	—
43	C₂³×C₇	C₂⁴	1	1	yes	characteristic
44	C₇×Q₈	—	2	16	no	—
45	C₂×C₄×C₇	—	2	4	no	—
46	C₂×C₄×C₇	—	2	4	no	—
47	C₂×C₄×C₇	—	2	4	no	—
48	C₂×C₄×C₇	C₂²×C₄	1	4	yes	—
49	C₂×C₄×C₇	C₂²×C₄	1	8	yes	—
50	C₂×C₄×C₇	C₂²×C₄	1	2	yes	—
51	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₇	#41	1	2	yes	—
53	C₂×Dic₇	—	2	4	no	—
54	C₂×Dic₇	—	2	4	no	—
55	C₂×Dic₇	—	2	2	no	—
56	C₂×Dic₇	—	2	8	no	—
57	C₂×Dic₇	—	2	4	no	—
58	C₂×Dic₇	C₂×D₄	1	4	yes	—
59	C₂²×Q₈	C₂×D₇	1	2	yes	—
60	C₄⋊Q₈	Dic₇	1	8	yes	complemented
61	C₄×Q₈	—	14	8	no	—
62	C₂×C₄⋊C₄	—	7	2	no	—
63	C₂×C₄⋊C₄	—	7	4	no	—
64	C₂×C₄⋊C₄	C₂×D₇	1	4	yes	—
65	C₂×C₄²	—	7	1	no	—
66	C₂×C₄²	—	7	4	no	—
67	C₂×C₄²	—	7	1	no	—
68	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
69	#95	—	7	4	no	—
70	#95	—	7	4	no	—
71	#95	—	7	8	no	—
72	C₂²×C₇	C₂³×C₄	1	1	yes	characteristic
73	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
74	C₂²×C₇	C₂×#41	1	1	yes	characteristic
75	C₂²×C₇	2₋¹⁺⁴	1	4	yes	—
76	C₄×C₇	—	2	8	no	—
77	C₄×C₇	—	2	4	no	—
78	C₄×C₇	C₄×D₄	1	4	yes	—
79	Dic₇	—	2	4	no	—
80	Dic₇	—	4	4	no	—
81	Dic₇	—	4	2	no	—
82	C₂×Q₈	—	2	4	no	—
83	C₂×Q₈	—	2	4	no	—
84	C₂×Q₈	C₂×Dic₇	1	8	yes	—
85	C₂²×C₄	—	7	2	no	—
86	C₂²×C₄	—	7	2	no	—
87	C₂²×C₄	—	7	4	no	—
88	C₂²×C₄	C₂²×D₇	1	2	yes	—
89	C₂²×C₄	C₂²×D₇	1	4	yes	—
90	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
91	C₄⋊C₄	—	14	8	no	—
92	C₄⋊C₄	—	14	4	no	—
93	C₄⋊C₄	C₂×Dic₇	1	16	yes	—
94	C₄²	—	7	4	no	—
95	C₄²	—	14	8	no	—
96	C₄²	—	14	2	no	—
97	C₄²	C₂×Dic₇	1	4	yes	—
98	C₂×C₇	C₂×C₄×D₄	1	1	yes	characteristic
99	C₂×C₇	#519	1	2	yes	—
100	C₂×C₇	#536	1	2	yes	—
101	C₂×C₇	#541	1	2	yes	—
102	C₂³	C₂³×D₇	1	1	yes	characteristic
103	Q₈	—	2	16	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	7	4	no	—
108	C₂×C₄	—	14	4	no	—
109	C₂×C₄	—	14	4	no	—
110	C₂×C₄	—	14	2	no	—
111	C₂×C₄	—	14	8	no	—
112	C₂×C₄	—	14	4	no	—
113	C₂×C₄	D₄×D₇	1	2	yes	—
114	C₂×C₄	#1159	1	2	yes	—
115	C₂×C₄	C₂²×Dic₇	1	4	yes	—
116	C₂×C₄	C₂²×Dic₇	1	8	yes	—
117	C₂×C₄	C₂²×Dic₇	1	2	yes	—
118	C₇	#2327	1	1	yes	characteristic, complemented, Sylow
119	C₂²	C₂×D₄×D₇	1	1	yes	characteristic
120	C₂²	C₂×#1159	1	1	yes	characteristic
121	C₂²	#6551	1	4	yes	—
122	C₂²	C₂³×Dic₇	1	1	yes	characteristic
123	C₄	—	2	8	no	—
124	C₄	—	2	4	no	—
125	C₄	—	14	4	no	—
126	C₄	—	28	4	no	—
127	C₄	—	28	2	no	—
128	C₄	D₄×Dic₇	1	4	yes	—
129	C₂	#90415	1	2	yes	—
130	C₂	#90422	1	2	yes	—
131	C₂	C₂×D₄×Dic₇	1	1	yes	characteristic
132	C₂	#90503	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₇×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂×Q₈×Dic₇

C₂

yes

maximal

#90002

C₂

yes

maximal

#89753

C₂

yes

maximal

#89708

C₂

yes

maximal, characteristic

#89704

C₂

yes

maximal, characteristic

C₂²×C₇×Q₈

C₂²

yes

—

C₇×C₄⋊Q₈

C₄

yes

complemented

C₂×C₇×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₇

C₂²

yes

characteristic

Q₈×Dic₇

—

C₂×C₄⋊Dic₇

C₂²

yes

—

C₂×C₄⋊Dic₇

C₂²

yes

—

C₂×C₄×Dic₇

C₂²

yes

characteristic

C₂×C₄×Dic₇

C₂²

yes

—

C₂×C₄×Dic₇

C₂²

yes

characteristic

C₇⋊#95

C₂²

yes

—

C₇⋊#95

C₂²

yes

—

C₇⋊#95

C₂²

yes

—

#2327

—

maximal, Sylow

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂×C₇×Q₈

C₂×C₄

yes

—

C₂²×C₄×C₇

C₂³

yes

—

C₂²×C₄×C₇

C₂³

yes

—

C₂²×C₄×C₇

C₂³

yes

characteristic

C₂²×Dic₇

C₂³

yes

—

C₂²×Dic₇

C₂³

yes

—

C₂²×Dic₇

C₂³

yes

—

C₇×C₄⋊C₄

C₂×C₄

yes

—

C₄²×C₇

C₂×C₄

yes

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄×Dic₇

—

C₄×Dic₇

—

C₄×Dic₇

D₄

yes

—

C₂×C₄⋊Q₈

D₇

yes

characteristic

C₂×C₄×Q₈

—

—

—

—

—

C₂³×C₇

C₂⁴

yes

characteristic

C₇×Q₈

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂²×C₄

yes

—

C₂×C₄×C₇

C₂²×C₄

yes

—

C₂×C₄×C₇

C₂²×C₄

yes

—

C₂×C₄×C₇

C₂×D₄

yes

—

C₂×C₄×C₇

#41

yes

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

C₂×D₄

yes

—

C₂²×Q₈

C₂×D₇

yes

—

C₄⋊Q₈

Dic₇

yes

complemented

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₇

yes

—

—

—

C₂²×C₇

C₂³×C₄

yes

characteristic

C₂²×C₇

C₂²×D₄

yes

characteristic

C₂²×C₇

C₂×#41

yes

characteristic

C₂²×C₇

2₋¹⁺⁴

yes

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

C₄×D₄

yes

—

Dic₇

—

Dic₇

—

Dic₇

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₇

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₇

yes

—

C₂²×C₄

C₂²×D₇

yes

—

C₂²×C₄

C₂²×D₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×Dic₇

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×Dic₇

yes

—

C₂×C₇

C₂×C₄×D₄

yes

characteristic

C₂×C₇

#519

yes

—

100

C₂×C₇

#536

yes

—

101

C₂×C₇

#541

yes

—

102

C₂³

C₂³×D₇

yes

characteristic

103

Q₈

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

D₄×D₇

yes

—

114

C₂×C₄

#1159

yes

—

115

C₂×C₄

C₂²×Dic₇

yes

—

116

C₂×C₄

C₂²×Dic₇

yes

—

117

C₂×C₄

C₂²×Dic₇

yes

—

118

C₇

#2327

yes

characteristic, complemented, Sylow

119

C₂²

C₂×D₄×D₇

yes

characteristic

120

C₂²

C₂×#1159

yes

characteristic

121

C₂²

#6551

yes

—

122

C₂²

C₂³×Dic₇

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

D₄×Dic₇

yes

—

129

C₂

#90415

yes

—

130

C₂

#90422

yes

—

131

C₂

C₂×D₄×Dic₇

yes

characteristic

132

C₂

#90503

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇
commutator	C₂²×C₇	C₂³×C₄
Fitting	C₂×C₇×C₄⋊Q₈	C₂
socle, prime radical, cosocle	C₂³×C₇	C₂⁴
hypercenter	C₂×C₄⋊Q₈	D₇
hypocenter	C₇	#2327
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₇

commutator

C₂²×C₇

C₂³×C₄

Fitting

C₂×C₇×C₄⋊Q₈

C₂

socle, prime radical, cosocle

C₂³×C₇

C₂⁴

hypercenter

C₂×C₄⋊Q₈

D₇

C₇

G ▷ C₂²×C₇ ▷ C₇

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
8. C₇×C₄⋊Q₈	C₄	semidirect
60. C₄⋊Q₈	Dic₇	semidirect
118. C₇	#2327	semidirect

H ≅ G/N

type

8. C₇×C₄⋊Q₈

C₄

semidirect

60. C₄⋊Q₈

Dic₇

semidirect

118. C₇

#2327

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×Q₈×Dic₇	2
#90002	2
#89753	2
#89708	2
#89704	2
#2327	7

subgroup

index

C₂×C₇×C₄⋊Q₈

C₂×Q₈×Dic₇

Subgroups — #11740672

Conjugacy classes of subgroups

Complements

Maximal subgroups