Group #11728413 — subgroups

Subgroups — #11728413

Order 896.

232 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 117 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	C₇×#427	C₂	1	1	yes	maximal, characteristic
2	C₂×#6393	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×Dic₇	C₂	1	1	yes	maximal, characteristic
4	#89769	C₂	1	2	yes	maximal
5	#89327	C₂	1	2	yes	maximal
6	C₂²×C₇×C₈	C₂²	1	1	yes	characteristic
7	C₇×#117	C₂²	1	2	yes	—
8	C₂×C₄×Dic₇	C₂²	1	1	yes	characteristic
9	C₂×#1136	C₂²	1	2	yes	—
10	C₂²×C₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₇×#107	C₄	1	2	yes	complemented
12	C₇×#106	C₄	1	2	yes	complemented
13	#6393	C₄	1	4	yes	complemented
14	C₈×Dic₇	C₄	1	4	yes	complemented
15	#6377	—	2	2	no	—
16	C₄×C₇⋊C₈	—	2	2	no	—
17	#1772	—	7	1	no	maximal, Sylow
18	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
19	C₂²×Dic₇	—	2	1	no	—
20	C₂×C₇×C₈	C₂×C₄	1	2	yes	—
21	C₂×C₇×C₈	C₂×C₄	1	2	yes	—
22	C₂×C₇×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₇×C₈	C₂×C₄	1	1	yes	characteristic
24	C₇×C₄⋊C₄	C₂×C₄	1	4	yes	—
25	C₇×C₂²⋊C₄	—	2	2	no	—
26	C₄²×C₇	—	2	2	no	—
27	C₄×Dic₇	—	2	1	no	—
28	C₄×Dic₇	C₂×C₄	1	2	yes	—
29	#1136	—	2	2	no	—
30	#1136	C₂×C₄	1	4	yes	—
31	C₂×C₇⋊C₈	—	2	2	no	—
32	C₂×C₇⋊C₈	C₂×C₄	1	2	yes	—
33	C₂×C₇⋊C₈	D₄	1	1	yes	characteristic
34	C₂×C₇⋊C₈	D₄	1	1	yes	characteristic
35	C₂×C₇⋊C₈	Q₈	1	2	yes	—
36	C₂×#108	—	7	1	no	—
37	#427	D₇	1	1	yes	characteristic
38	#405	—	7	2	no	—
39	C₂×C₄×C₈	—	7	1	no	—
40	#339	—	7	2	no	—
41	C₂³×C₇	#41	1	1	yes	characteristic
42	C₂×C₄×C₇	—	2	4	no	—
43	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₇	#41	1	1	yes	characteristic
49	C₂×Dic₇	—	2	4	no	—
50	C₂×Dic₇	—	2	2	no	—
51	C₇×C₈	C₄²	1	2	yes	—
52	C₇×C₈	C₄²	1	2	yes	—
53	C₇⋊C₈	—	2	4	no	—
54	C₇⋊C₈	C₄⋊C₄	1	2	yes	—
55	C₇⋊C₈	C₄⋊C₄	1	2	yes	—
56	C₂×M₁₆	—	7	2	no	—
57	C₂²×C₈	—	7	1	no	—
58	C₂²×C₈	C₂×D₇	1	1	yes	characteristic
59	#117	C₂×D₇	1	2	yes	—
60	C₂×C₄²	—	7	1	no	—
61	#108	—	7	4	no	—
62	#107	Dic₇	1	2	yes	complemented
63	#106	Dic₇	1	2	yes	complemented
64	#97	—	14	2	no	—
65	C₄×C₈	—	7	4	no	—
66	C₄×C₈	—	14	2	no	—
67	C₂²×C₇	—	2	2	no	—
68	C₂²×C₇	#117	1	1	yes	characteristic
69	C₂²×C₇	C₄×D₄	1	1	yes	characteristic
70	C₂²×C₇	C₄×Q₈	1	1	yes	characteristic
71	C₄×C₇	—	4	4	no	—
72	C₄×C₇	C₂×C₄²	1	1	yes	characteristic
73	C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₄×C₇	C₄×D₄	1	1	yes	characteristic
75	C₄×C₇	C₄×Q₈	1	1	yes	characteristic
76	Dic₇	—	2	4	no	—
77	C₂²×C₄	—	14	1	no	—
78	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
79	M₁₆	—	7	4	no	—
80	M₁₆	—	14	2	no	—
81	C₂×C₈	—	7	2	no	—
82	C₂×C₈	—	7	1	no	—
83	C₂×C₈	—	7	1	no	—
84	C₂×C₈	—	7	2	no	—
85	C₂×C₈	—	14	2	no	—
86	C₂×C₈	C₄×D₇	1	2	yes	—
87	C₂×C₈	C₄×D₇	1	2	yes	—
88	C₂×C₈	C₂×Dic₇	1	1	yes	characteristic
89	C₂×C₈	C₂×Dic₇	1	1	yes	characteristic
90	C₄⋊C₄	C₂×Dic₇	1	4	yes	—
91	C₂²⋊C₄	—	2	2	no	—
92	C₄²	—	2	2	no	—
93	C₄²	—	7	2	no	—
94	C₄²	—	14	1	no	—
95	C₂×C₇	—	2	1	no	—
96	C₂×C₇	—	2	1	no	—
97	C₂×C₇	C₄×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₇	#443	1	2	yes	—
99	C₂³	#1159	1	1	yes	characteristic
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	14	4	no	—
102	C₂×C₄	—	14	2	no	—
103	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
104	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
105	C₂×C₄	D₄×D₇	1	1	yes	characteristic
106	C₂×C₄	Q₈×D₇	1	1	yes	characteristic
107	C₂×C₄	#1161	1	1	yes	characteristic
108	C₂×C₄	C₂²×Dic₇	1	1	yes	characteristic
109	C₈	—	7	2	no	—
110	C₈	—	7	2	no	—
111	C₈	—	14	4	no	—
112	C₈	C₄×Dic₇	1	2	yes	—
113	C₈	C₄×Dic₇	1	2	yes	—
114	C₇	#1772	1	1	yes	characteristic, complemented, Sylow
115	C₂²	—	2	2	no	—
116	C₂²	#6450	1	1	yes	characteristic
117	C₂²	#6455	1	1	yes	characteristic
118	C₂²	D₄×Dic₇	1	1	yes	characteristic
119	C₄	—	4	4	no	—
120	C₄	—	14	4	no	—
121	C₄	D₇×C₄⋊C₄	1	1	yes	characteristic
122	C₄	#6456	1	1	yes	characteristic
123	C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
124	C₄	Q₈×Dic₇	1	1	yes	characteristic
125	C₂	—	2	1	no	—
126	C₂	—	2	1	no	—
127	C₂	C₄⋊C₄×Dic₇	1	1	yes	characteristic
128	C₂	#89968	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#427

C₂

yes

maximal, characteristic

C₂×#6393

C₂

yes

maximal, characteristic

C₂×C₈×Dic₇

C₂

yes

maximal, characteristic

#89769

C₂

yes

maximal

#89327

C₂

yes

maximal

C₂²×C₇×C₈

C₂²

yes

characteristic

C₇×#117

C₂²

yes

—

C₂×C₄×Dic₇

C₂²

yes

characteristic

C₂×#1136

C₂²

yes

—

C₂²×C₇⋊C₈

C₂²

yes

characteristic

C₇×#107

C₄

yes

complemented

C₇×#106

C₄

yes

complemented

#6393

C₄

yes

complemented

C₈×Dic₇

C₄

yes

complemented

#6377

—

C₄×C₇⋊C₈

—

#1772

—

maximal, Sylow

C₂²×C₄×C₇

C₂³

yes

characteristic

C₂²×Dic₇

—

C₂×C₇×C₈

C₂×C₄

yes

—

C₂×C₇×C₈

C₂×C₄

yes

—

C₂×C₇×C₈

C₂×C₄

yes

characteristic

C₂×C₇×C₈

C₂×C₄

yes

characteristic

C₇×C₄⋊C₄

C₂×C₄

yes

—

C₇×C₂²⋊C₄

—

C₄²×C₇

—

C₄×Dic₇

—

C₄×Dic₇

C₂×C₄

yes

—

#1136

—

#1136

C₂×C₄

yes

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

C₂×C₄

yes

—

C₂×C₇⋊C₈

D₄

yes

characteristic

C₂×C₇⋊C₈

D₄

yes

characteristic

C₂×C₇⋊C₈

Q₈

yes

—

C₂×#108

—

#427

D₇

yes

characteristic

#405

—

C₂×C₄×C₈

—

#339

—

C₂³×C₇

#41

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₇

#41

yes

characteristic

C₂×Dic₇

—

C₂×Dic₇

—

C₇×C₈

C₄²

yes

—

C₇×C₈

C₄²

yes

—

C₇⋊C₈

—

C₇⋊C₈

C₄⋊C₄

yes

—

C₇⋊C₈

C₄⋊C₄

yes

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₇

yes

characteristic

#117

C₂×D₇

yes

—

C₂×C₄²

—

#108

—

#107

Dic₇

yes

complemented

#106

Dic₇

yes

complemented

#97

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₇

—

C₂²×C₇

#117

yes

characteristic

C₂²×C₇

C₄×D₄

yes

characteristic

C₂²×C₇

C₄×Q₈

yes

characteristic

C₄×C₇

—

C₄×C₇

C₂×C₄²

yes

characteristic

C₄×C₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₇

C₄×D₄

yes

characteristic

C₄×C₇

C₄×Q₈

yes

characteristic

Dic₇

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₇

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₇

yes

—

C₂×C₈

C₄×D₇

yes

—

C₂×C₈

C₂×Dic₇

yes

characteristic

C₂×C₈

C₂×Dic₇

yes

characteristic

C₄⋊C₄

C₂×Dic₇

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₂×C₇

—

C₂×C₇

—

C₂×C₇

C₄×C₄⋊C₄

yes

characteristic

C₂×C₇

#443

yes

—

C₂³

#1159

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

C₂×C₄×D₇

yes

characteristic

104

C₂×C₄

C₂×C₄×D₇

yes

characteristic

105

C₂×C₄

D₄×D₇

yes

characteristic

106

C₂×C₄

Q₈×D₇

yes

characteristic

107

C₂×C₄

#1161

yes

characteristic

108

C₂×C₄

C₂²×Dic₇

yes

characteristic

109

C₈

—

110

C₈

—

111

C₈

—

112

C₈

C₄×Dic₇

yes

—

113

C₈

C₄×Dic₇

yes

—

114

C₇

#1772

yes

characteristic, complemented, Sylow

115

C₂²

—

116

C₂²

#6450

yes

characteristic

117

C₂²

#6455

yes

characteristic

118

C₂²

D₄×Dic₇

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

D₇×C₄⋊C₄

yes

characteristic

122

C₄

#6456

yes

characteristic

123

C₄

C₂×C₄×Dic₇

yes

characteristic

124

C₄

Q₈×Dic₇

yes

characteristic

125

C₂

—

126

C₂

—

127

C₂

C₄⋊C₄×Dic₇

yes

characteristic

128

C₂

#89968

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₇
commutator	C₄×C₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₇
Fitting	C₇×#427	C₂
socle	C₂²×C₇	C₄×D₄
prime radical	C₂³×C₇	#41
cosocle	C₂²×C₄×C₇	C₂³
hypercenter	#427	D₇
hypocenter	C₇	#1772
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₇

commutator

C₄×C₇

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₇

Fitting

C₇×#427

C₂

socle

C₂²×C₇

C₄×D₄

prime radical

C₂³×C₇

#41

cosocle

C₂²×C₄×C₇

C₂³

D₇

C₇

G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
11. C₇×#107	C₄	semidirect
12. C₇×#106	C₄	semidirect
13. #6393	C₄	semidirect
14. C₈×Dic₇	C₄	semidirect
62. #107	Dic₇	semidirect
63. #106	Dic₇	semidirect
114. C₇	#1772	semidirect

H ≅ G/N

type

11. C₇×#107

C₄

semidirect

12. C₇×#106

C₄

semidirect

13. #6393

C₄

semidirect

14. C₈×Dic₇

C₄

semidirect

62. #107

Dic₇

semidirect

63. #106

Dic₇

semidirect

114. C₇

#1772

semidirect

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

subgroup	index
C₇×#427	2
C₂×#6393	2
C₂×C₈×Dic₇	2
#89769	2
#89327	2
#1772	7

subgroup

index

C₂×C₈×Dic₇

Subgroups — #11728413

Conjugacy classes of subgroups

Complements

Maximal subgroups