Group #90366 — subgroups

Subgroups — #90366

Order 448.

270 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 103 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×Q₈×D₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#1159	C₂	1	1	yes	maximal, characteristic
3	C₇×#124	C₂	1	1	yes	maximal, characteristic, complemented
4	#6507	C₂	1	1	yes	maximal, characteristic, complemented
5	#6499	C₂	1	1	yes	maximal, characteristic, complemented
6	#6446	C₂	1	4	yes	maximal, complemented
7	#6441	C₂	1	4	yes	maximal, complemented
8	#6435	C₂	1	1	yes	maximal, characteristic, complemented
9	#6432	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₇×Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₇×D₄	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₇	C₂²	1	2	yes	—
13	C₂²×Dic₇	C₂²	1	2	yes	—
14	Q₈×D₇	—	2	4	no	—
15	#1159	—	2	4	no	—
16	C₂×C₄×D₇	C₂²	1	2	yes	—
17	C₂×C₄×D₇	C₂²	1	1	yes	characteristic
18	C₂×Dic₁₄	C₂²	1	1	yes	characteristic
19	C₂×Dic₁₄	C₂²	1	1	yes	characteristic
20	C₂×Dic₁₄	C₂²	1	2	yes	—
21	C₇×C₂²⋊C₄	C₂²	1	4	yes	—
22	C₄²×C₇	C₂²	1	1	yes	characteristic
23	C₂²⋊Dic₇	C₂²	1	4	yes	—
24	(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
25	C₄⋊Dic₇	C₂²	1	4	yes	—
26	Dic₇⋊C₄	C₂²	1	2	yes	—
27	Dic₇⋊C₄	C₂²	1	4	yes	—
28	C₄×Dic₇	C₂²	1	1	yes	characteristic
29	#536	—	7	1	no	maximal, Sylow
30	C₂³×C₇	C₂³	1	2	yes	—
31	C₂²×D₇	C₂³	1	1	yes	characteristic
32	C₇×Q₈	—	2	2	no	—
33	C₇×D₄	—	2	2	no	—
34	C₂×C₄×C₇	C₂³	1	2	yes	—
35	C₂×C₄×C₇	C₂³	1	2	yes	—
36	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
37	C₂²⋊D₇	—	2	4	no	—
38	C₂×Dic₇	—	2	4	no	—
39	C₂×Dic₇	C₂³	1	2	yes	—
40	C₂×Dic₇	C₂³	1	2	yes	—
41	C₂×Dic₇	C₂³	1	2	yes	—
42	C₂×Dic₇	C₂³	1	1	yes	characteristic
43	C₄×D₇	—	2	4	no	—
44	C₄×D₇	D₄	1	4	yes	—
45	Dic₁₄	—	2	2	no	—
46	Dic₁₄	—	2	2	no	—
47	Dic₁₄	—	2	4	no	—
48	C₂×#41	—	7	1	no	—
49	C₂²×Q₈	—	7	1	no	—
50	C₄⋊Q₈	—	7	1	no	—
51	C₄⋊Q₈	—	7	1	no	—
52	#124	—	7	1	no	—
53	#124	D₇	1	1	yes	characteristic, complemented
54	#123	—	7	4	no	—
55	C₂²⋊Q₈	—	7	4	no	—
56	#117	—	7	1	no	—
57	C₂²×C₇	—	2	2	no	—
58	C₂²×C₇	—	2	4	no	—
59	C₂²×C₇	C₂⁴	1	1	yes	characteristic
60	C₂×D₇	—	2	2	no	—
61	C₂×D₇	C₂×D₄	1	2	yes	—
62	C₄×C₇	—	2	2	no	—
63	C₄×C₇	—	2	2	no	—
64	C₄×C₇	C₂×D₄	1	2	yes	—
65	Dic₇	—	2	2	no	—
66	Dic₇	—	2	2	no	—
67	Dic₇	—	2	2	no	—
68	Dic₇	C₂×D₄	1	2	yes	—
69	#41	—	14	4	no	—
70	C₂×Q₈	—	7	2	no	—
71	C₂×Q₈	—	7	1	no	—
72	C₂×Q₈	—	7	1	no	—
73	C₂×Q₈	—	14	4	no	—
74	C₂×Q₈	C₂×D₇	1	1	yes	characteristic
75	C₂×D₄	—	7	2	no	—
76	C₂×D₄	C₂×D₇	1	1	yes	characteristic
77	C₂²×C₄	—	7	1	no	—
78	C₂²×C₄	—	7	2	no	—
79	C₂²×C₄	—	7	2	no	—
80	C₄⋊C₄	—	7	2	no	—
81	C₄⋊C₄	—	7	4	no	—
82	C₄⋊C₄	—	7	4	no	—
83	C₂²⋊C₄	—	7	4	no	—
84	C₂²⋊C₄	—	7	2	no	—
85	C₂²⋊C₄	C₂×D₇	1	4	yes	—
86	C₄²	—	7	1	no	—
87	C₄²	C₂×D₇	1	1	yes	characteristic
88	C₂×C₇	—	4	2	no	—
89	C₂×C₇	C₂²×D₄	1	1	yes	characteristic
90	C₂×C₇	2₋¹⁺⁴	1	2	yes	—
91	D₇	—	2	2	no	—
92	C₂³	—	7	1	no	—
93	C₂³	C₂²×D₇	1	2	yes	—
94	Q₈	—	2	2	no	—
95	Q₈	—	14	4	no	—
96	Q₈	—	14	2	no	—
97	Q₈	—	14	2	no	—
98	D₄	—	2	2	no	—
99	D₄	—	14	4	no	—
100	C₂×C₄	—	7	1	no	—
101	C₂×C₄	—	7	4	no	—
102	C₂×C₄	—	7	2	no	—
103	C₂×C₄	—	7	2	no	—
104	C₂×C₄	—	7	2	no	—
105	C₂×C₄	—	14	4	no	—
106	C₂×C₄	—	14	4	no	—
107	C₂×C₄	C₂²×D₇	1	2	yes	—
108	C₂×C₄	C₂²×D₇	1	2	yes	—
109	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
110	C₇	#536	1	1	yes	characteristic, complemented, Sylow
111	C₂²	—	2	2	no	—
112	C₂²	—	2	4	no	—
113	C₂²	—	7	2	no	—
114	C₂²	—	14	2	no	—
115	C₂²	C₂³×D₇	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	—	2	2	no	—
118	C₄	—	7	2	no	—
119	C₄	—	14	2	no	—
120	C₄	—	14	2	no	—
121	C₄	—	14	2	no	—
122	C₄	D₄×D₇	1	2	yes	—
123	C₂	—	4	2	no	—
124	C₂	—	14	2	no	—
125	C₂	C₂×D₄×D₇	1	1	yes	characteristic
126	C₂	#6554	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×D₇

C₂

yes

maximal, characteristic, complemented

C₂×#1159

C₂

yes

maximal, characteristic

C₇×#124

C₂

yes

maximal, characteristic, complemented

#6507

C₂

yes

maximal, characteristic, complemented

#6499

C₂

yes

maximal, characteristic, complemented

#6446

C₂

yes

maximal, complemented

#6441

C₂

yes

maximal, complemented

#6435

C₂

yes

maximal, characteristic, complemented

#6432

C₂

yes

maximal, characteristic, complemented

C₂×C₇×Q₈

C₂²

yes

characteristic

C₂×C₇×D₄

C₂²

yes

characteristic

C₂×C₂²⋊D₇

C₂²

yes

—

C₂²×Dic₇

C₂²

yes

—

Q₈×D₇

—

#1159

—

C₂×C₄×D₇

C₂²

yes

—

C₂×C₄×D₇

C₂²

yes

characteristic

C₂×Dic₁₄

C₂²

yes

characteristic

C₂×Dic₁₄

C₂²

yes

characteristic

C₂×Dic₁₄

C₂²

yes

—

C₇×C₂²⋊C₄

C₂²

yes

—

C₄²×C₇

C₂²

yes

characteristic

C₂²⋊Dic₇

C₂²

yes

—

(C₂×D₇)⋊C₄

C₂²

yes

—

C₄⋊Dic₇

C₂²

yes

—

Dic₇⋊C₄

C₂²

yes

—

Dic₇⋊C₄

C₂²

yes

—

C₄×Dic₇

C₂²

yes

characteristic

#536

—

maximal, Sylow

C₂³×C₇

C₂³

yes

—

C₂²×D₇

C₂³

yes

characteristic

C₇×Q₈

—

C₇×D₄

—

C₂×C₄×C₇

C₂³

yes

—

C₂×C₄×C₇

C₂³

yes

—

C₂×C₄×C₇

C₂³

yes

characteristic

C₂²⋊D₇

—

C₂×Dic₇

—

C₂×Dic₇

C₂³

yes

—

C₂×Dic₇

C₂³

yes

—

C₂×Dic₇

C₂³

yes

—

C₂×Dic₇

C₂³

yes

characteristic

C₄×D₇

—

C₄×D₇

D₄

yes

—

Dic₁₄

—

Dic₁₄

—

Dic₁₄

—

C₂×#41

—

C₂²×Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#124

—

#124

D₇

yes

characteristic, complemented

#123

—

C₂²⋊Q₈

—

#117

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

C₂⁴

yes

characteristic

C₂×D₇

—

C₂×D₇

C₂×D₄

yes

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

C₂×D₄

yes

—

Dic₇

—

Dic₇

—

Dic₇

—

Dic₇

C₂×D₄

yes

—

#41

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×D₇

yes

characteristic

C₂×D₄

—

C₂×D₄

C₂×D₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₇

yes

—

C₄²

—

C₄²

C₂×D₇

yes

characteristic

C₂×C₇

—

C₂×C₇

C₂²×D₄

yes

characteristic

C₂×C₇

2₋¹⁺⁴

yes

—

D₇

—

C₂³

—

C₂³

C₂²×D₇

yes

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

D₄

—

D₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₂²×D₇

yes

—

108

C₂×C₄

C₂²×D₇

yes

—

109

C₂×C₄

C₂²×D₇

yes

characteristic

110

C₇

#536

yes

characteristic, complemented, Sylow

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

C₂³×D₇

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

D₄×D₇

yes

—

123

C₂

—

124

C₂

—

125

C₂

C₂×D₄×D₇

yes

characteristic

126

C₂

#6554

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇
commutator, socle, cosocle	C₂²×C₇	C₂⁴
Fitting	C₇×#124	C₂
prime radical	C₂×#1159	C₂
hypercenter	#124	D₇
hypocenter	C₇	#536
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×D₇

commutator, socle, cosocle

C₂²×C₇

C₂⁴

C₂

C₂

D₇

C₇

G ▷ C₂²×C₇ ▷ C₇

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
1. C₂×Q₈×D₇	C₂	semidirect
3. C₇×#124	C₂	semidirect
4. #6507	C₂	semidirect
5. #6499	C₂	semidirect
6. #6446	C₂	semidirect
7. #6441	C₂	semidirect
8. #6435	C₂	semidirect
9. #6432	C₂	semidirect
53. #124	D₇	semidirect
110. C₇	#536	semidirect

H ≅ G/N

type

1. C₂×Q₈×D₇

C₂

semidirect

3. C₇×#124

C₂

semidirect

4. #6507

C₂

semidirect

5. #6499

C₂

semidirect

6. #6446

C₂

semidirect

7. #6441

C₂

semidirect

8. #6435

C₂

semidirect

9. #6432

C₂

semidirect

53. #124

D₇

semidirect

110. C₇

#536

semidirect

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

subgroup	index
C₂×Q₈×D₇	2
C₂×#1159	2
C₇×#124	2
#6507	2
#6499	2
#6446	2
#6441	2
#6435	2
#6432	2
#536	7

subgroup

index

C₂×Q₈×D₇

Subgroups — #90366

Conjugacy classes of subgroups

Complements

Maximal subgroups