Group #90336 — subgroups

Subgroups — #90336

Order 448.

218 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 95 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×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#6509	C₂	1	2	yes	maximal, complemented
3	#6493	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₇⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	#6461	C₂	1	2	yes	maximal, complemented
6	#6457	C₂	1	2	yes	maximal, complemented
7	#6453	C₂	1	2	yes	maximal, complemented
8	#6449	C₂	1	2	yes	maximal, complemented
9	(C₂²⋊D₇)⋊C₄	C₂	1	2	yes	maximal, complemented
10	C₂×C₇×Q₈	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₇	C₂²	1	2	yes	—
13	C₂²×Dic₇	C₂²	1	1	yes	characteristic
14	C₂²×Dic₇	C₂²	1	1	yes	characteristic
15	C₂×D₂₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₇	C₂²	1	2	yes	—
17	C₇×C₄⋊C₄	C₂²	1	2	yes	—
18	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₇×C₂²⋊C₄	C₂²	1	2	yes	—
20	(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
21	(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
22	(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
23	(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
24	C₄⋊Dic₇	C₂²	1	2	yes	—
25	C₄⋊Dic₇	C₂²	1	2	yes	—
26	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
27	Dic₇⋊C₄	C₂²	1	2	yes	—
28	Dic₇⋊C₄	C₂²	1	2	yes	—
29	Dic₇⋊C₄	C₂²	1	2	yes	—
30	C₄×Dic₇	C₂²	1	2	yes	—
31	#539	—	7	1	no	maximal, Sylow
32	C₂³×C₇	C₂³	1	1	yes	characteristic
33	C₂²×D₇	C₂³	1	2	yes	—
34	C₇×Q₈	—	4	1	no	—
35	C₂×C₄×C₇	—	4	1	no	—
36	C₂×C₄×C₇	C₂³	1	2	yes	—
37	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₇	C₂³	1	2	yes	—
40	C₂²⋊D₇	—	2	4	no	—
41	C₂×Dic₇	—	2	2	no	—
42	C₂×Dic₇	—	4	1	no	—
43	C₂×Dic₇	C₂³	1	2	yes	—
44	C₂×Dic₇	C₂³	1	2	yes	—
45	C₂×Dic₇	C₂³	1	2	yes	—
46	D₂₈	—	4	1	no	—
47	C₄×D₇	—	4	2	no	—
48	#126	—	7	2	no	—
49	#125	—	7	2	no	—
50	#123	—	7	2	no	—
51	#123	—	7	2	no	—
52	C₂²⋊Q₈	—	7	2	no	—
53	C₂²⋊Q₈	D₇	1	1	yes	characteristic, complemented
54	#121	—	7	1	no	—
55	C₄×D₄	—	7	2	no	—
56	C₂×C₄⋊C₄	—	7	1	no	—
57	C₂²×C₇	—	2	1	no	—
58	C₂²×C₇	—	2	1	no	—
59	C₂²×C₇	#41	1	2	yes	—
60	C₂²×C₇	C₂⁴	1	1	yes	characteristic
61	C₂×D₇	—	2	2	no	—
62	C₂×D₇	—	2	2	no	—
63	C₂×D₇	—	2	2	no	—
64	C₄×C₇	—	2	2	no	—
65	C₄×C₇	—	2	1	no	—
66	C₄×C₇	—	2	1	no	—
67	C₄×C₇	—	2	2	no	—
68	Dic₇	—	2	2	no	—
69	Dic₇	—	2	2	no	—
70	Dic₇	—	2	2	no	—
71	C₂×Q₈	C₂×D₇	1	1	yes	characteristic
72	C₂×D₄	—	7	2	no	—
73	C₂×D₄	—	7	1	no	—
74	C₂²×C₄	—	7	1	no	—
75	C₂²×C₄	—	7	1	no	—
76	C₂²×C₄	—	7	2	no	—
77	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
78	C₄⋊C₄	—	7	2	no	—
79	C₄⋊C₄	—	7	2	no	—
80	C₄⋊C₄	—	7	2	no	—
81	C₄⋊C₄	—	7	1	no	—
82	C₄⋊C₄	—	7	2	no	—
83	C₄⋊C₄	—	7	2	no	—
84	C₄⋊C₄	C₂×D₇	1	2	yes	—
85	C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
86	C₂²⋊C₄	—	7	2	no	—
87	C₂²⋊C₄	—	7	2	no	—
88	C₂²⋊C₄	—	7	2	no	—
89	C₂²⋊C₄	—	7	2	no	—
90	C₂²⋊C₄	C₂×D₇	1	2	yes	—
91	C₄²	—	7	2	no	—
92	C₂×C₇	—	2	2	no	—
93	C₂×C₇	C₂×#41	1	1	yes	characteristic
94	C₂×C₇	H(5,𝔽₂)	1	1	yes	characteristic
95	C₂×C₇	2₋¹⁺⁴	1	1	yes	characteristic
96	D₇	—	4	2	no	—
97	C₂³	—	7	2	no	—
98	C₂³	C₂²×D₇	1	1	yes	characteristic
99	Q₈	—	4	1	no	—
100	D₄	—	14	4	no	—
101	D₄	—	28	1	no	—
102	C₂×C₄	—	4	1	no	—
103	C₂×C₄	—	7	2	no	—
104	C₂×C₄	—	7	2	no	—
105	C₂×C₄	—	7	2	no	—
106	C₂×C₄	—	14	2	no	—
107	C₂×C₄	—	28	2	no	—
108	C₂×C₄	—	28	1	no	—
109	C₂×C₄	C₂²×D₇	1	2	yes	—
110	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
111	C₂×C₄	C₂²×D₇	1	2	yes	—
112	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
113	C₇	#539	1	1	yes	characteristic, complemented, Sylow
114	C₂²	—	2	1	no	—
115	C₂²	—	2	1	no	—
116	C₂²	—	14	2	no	—
117	C₂²	—	14	2	no	—
118	C₂²	—	14	2	no	—
119	C₂²	#1161	1	2	yes	—
120	C₂²	C₂³×D₇	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	2	1	no	—
123	C₄	—	2	2	no	—
124	C₄	—	2	2	no	—
125	C₄	—	14	2	no	—
126	C₄	—	14	2	no	—
127	C₄	—	14	2	no	—
128	C₂	—	2	2	no	—
129	C₂	—	28	2	no	—
130	C₂	#6548	1	1	yes	characteristic
131	C₂	C₂×#1161	1	1	yes	characteristic
132	C₂	#6554	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×C₂²⋊Q₈

C₂

yes

maximal, characteristic, complemented

#6509

C₂

yes

maximal, complemented

#6493

C₂

yes

maximal, characteristic

C₂×Dic₇⋊C₄

C₂

yes

maximal, characteristic, complemented

#6461

C₂

yes

maximal, complemented

#6457

C₂

yes

maximal, complemented

#6453

C₂

yes

maximal, complemented

#6449

C₂

yes

maximal, complemented

(C₂²⋊D₇)⋊C₄

C₂

yes

maximal, complemented

C₂×C₇×Q₈

C₂²

yes

characteristic

C₂²×C₄×C₇

C₂²

yes

characteristic

C₂×C₂²⋊D₇

C₂²

yes

—

C₂²×Dic₇

C₂²

yes

characteristic

C₂²×Dic₇

C₂²

yes

characteristic

C₂×D₂₈

C₂²

yes

characteristic

C₂×C₄×D₇

C₂²

yes

—

C₇×C₄⋊C₄

C₂²

yes

—

C₇×C₄⋊C₄

C₂²

yes

characteristic

C₇×C₂²⋊C₄

C₂²

yes

—

(C₂×D₇)⋊C₄

C₂²

yes

—

(C₂×D₇)⋊C₄

C₂²

yes

—

(C₂×D₇)⋊C₄

C₂²

yes

—

(C₂×D₇)⋊C₄

C₂²

yes

—

C₄⋊Dic₇

C₂²

yes

—

C₄⋊Dic₇

C₂²

yes

—

C₄⋊Dic₇

C₂²

yes

characteristic

Dic₇⋊C₄

C₂²

yes

—

Dic₇⋊C₄

C₂²

yes

—

Dic₇⋊C₄

C₂²

yes

—

C₄×Dic₇

C₂²

yes

—

#539

—

maximal, Sylow

C₂³×C₇

C₂³

yes

characteristic

C₂²×D₇

C₂³

yes

—

C₇×Q₈

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂³

yes

—

C₂×C₄×C₇

C₂³

yes

characteristic

C₂×C₄×C₇

C₂³

yes

characteristic

C₂×C₄×C₇

C₂³

yes

—

C₂²⋊D₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

C₂³

yes

—

C₂×Dic₇

C₂³

yes

—

C₂×Dic₇

C₂³

yes

—

D₂₈

—

C₄×D₇

—

—

—

—

—

C₂²⋊Q₈

—

C₂²⋊Q₈

D₇

yes

characteristic, complemented

#121

—

C₄×D₄

—

C₂×C₄⋊C₄

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

#41

yes

—

C₂²×C₇

C₂⁴

yes

characteristic

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

Dic₇

—

Dic₇

—

Dic₇

—

C₂×Q₈

C₂×D₇

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₇

yes

—

C₄⋊C₄

C₂×D₇

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₇

yes

—

C₄²

—

C₂×C₇

—

C₂×C₇

C₂×#41

yes

characteristic

C₂×C₇

H(5,𝔽₂)

yes

characteristic

C₂×C₇

2₋¹⁺⁴

yes

characteristic

D₇

—

C₂³

—

C₂³

C₂²×D₇

yes

characteristic

Q₈

—

100

D₄

—

101

D₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₂²×D₇

yes

—

110

C₂×C₄

C₂²×D₇

yes

characteristic

111

C₂×C₄

C₂²×D₇

yes

—

112

C₂×C₄

C₂²×D₇

yes

characteristic

113

C₇

#539

yes

characteristic, complemented, Sylow

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

#1161

yes

—

120

C₂²

C₂³×D₇

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₂

—

129

C₂

—

130

C₂

#6548

yes

characteristic

131

C₂

C₂×#1161

yes

characteristic

132

C₂

#6554

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇
commutator, socle, cosocle	C₂²×C₇	C₂⁴
Fitting	C₇×C₂²⋊Q₈	C₂
prime radical	#6493	C₂
hypercenter	C₂²⋊Q₈	D₇
hypocenter	C₇	#539
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₇

commutator, socle, cosocle

C₂²×C₇

C₂⁴

Fitting

C₇×C₂²⋊Q₈

C₂

prime radical

#6493

C₂

hypercenter

C₂²⋊Q₈

D₇

C₇

G ▷ C₂²×C₇ ▷ C₇

C₁ ◁ C₂² ◁ C₂²⋊Q₈

N	H ≅ G/N	type
1. C₇×C₂²⋊Q₈	C₂	semidirect
2. #6509	C₂	semidirect
4. C₂×Dic₇⋊C₄	C₂	semidirect
5. #6461	C₂	semidirect
6. #6457	C₂	semidirect
7. #6453	C₂	semidirect
8. #6449	C₂	semidirect
9. (C₂²⋊D₇)⋊C₄	C₂	semidirect
53. C₂²⋊Q₈	D₇	semidirect
113. C₇	#539	semidirect

H ≅ G/N

type

1. C₇×C₂²⋊Q₈

C₂

semidirect

2. #6509

C₂

semidirect

4. C₂×Dic₇⋊C₄

C₂

semidirect

5. #6461

C₂

semidirect

6. #6457

C₂

semidirect

7. #6453

C₂

semidirect

8. #6449

C₂

semidirect

9. (C₂²⋊D₇)⋊C₄

C₂

semidirect

53. C₂²⋊Q₈

D₇

semidirect

113. C₇

#539

semidirect

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

subgroup	index
C₇×C₂²⋊Q₈	2
#6509	2
#6493	2
C₂×Dic₇⋊C₄	2
#6461	2
#6457	2
#6453	2
#6449	2
(C₂²⋊D₇)⋊C₄	2
#539	7

subgroup

index

C₇×C₂²⋊Q₈

C₂×Dic₇⋊C₄

(C₂²⋊D₇)⋊C₄

Subgroups — #90336

Conjugacy classes of subgroups

Complements

Maximal subgroups