Group #90355 — subgroups

Subgroups — #90355

Order 448.

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

normal subgroups only

This group has 236 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×#123	C₂	1	1	yes	maximal, characteristic, complemented
2	#6503	C₂	1	1	yes	maximal, characteristic
3	#6498	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₂²⋊D₇	C₂	1	2	yes	maximal, complemented
5	#6457	C₂	1	2	yes	maximal, complemented
6	#6450	C₂	1	2	yes	maximal, complemented
7	#6448	C₂	1	2	yes	maximal, complemented
8	#6448	C₂	1	2	yes	maximal, complemented
9	(C₂²⋊D₇)⋊C₄	C₂	1	2	yes	maximal, complemented
10	C₂×C₇×D₄	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₇	C₂²	1	2	yes	—
13	C₂×C₂²⋊D₇	C₂²	1	2	yes	—
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	C₂×Dic₁₄	C₂²	1	2	yes	—
18	C₇×C₄⋊C₄	C₂²	1	2	yes	—
19	C₇×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
21	C₂²⋊Dic₇	C₂²	1	1	yes	characteristic
22	C₂²⋊Dic₇	C₂²	1	2	yes	—
23	(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
24	(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
25	(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
26	Dic₇⋊C₄	C₂²	1	2	yes	—
27	Dic₇⋊C₄	C₂²	1	2	yes	—
28	C₄×Dic₇	C₂²	1	2	yes	—
29	C₄×Dic₇	C₂²	1	2	yes	—
30	C₄×Dic₇	C₂²	1	1	yes	characteristic
31	#559	—	7	1	no	maximal, Sylow
32	C₂³×C₇	C₂³	1	1	yes	characteristic
33	C₂³×C₇	C₂³	1	1	yes	characteristic
34	C₂²×D₇	C₂³	1	2	yes	—
35	C₇×D₄	—	4	1	no	—
36	C₂×C₄×C₇	—	4	1	no	—
37	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₇	C₂³	1	2	yes	—
39	C₂×C₄×C₇	C₂³	1	2	yes	—
40	C₂²⋊D₇	—	2	4	no	—
41	C₂²⋊D₇	—	2	4	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	Dic₁₄	—	2	4	no	—
49	#127	—	7	1	no	—
50	#124	—	7	2	no	—
51	#124	—	7	2	no	—
52	#123	—	7	1	no	—
53	#123	—	7	2	no	—
54	#123	D₇	1	1	yes	characteristic, complemented
55	C₄×Q₈	—	7	2	no	—
56	C₄×D₄	—	7	2	no	—
57	C₄×D₄	—	7	2	no	—
58	C₂²×C₇	—	2	2	no	—
59	C₂²×C₇	—	2	2	no	—
60	C₂²×C₇	—	2	1	no	—
61	C₂²×C₇	—	2	1	no	—
62	C₂²×C₇	C₂⁴	1	1	yes	characteristic
63	C₂×D₇	—	2	2	no	—
64	C₂×D₇	—	2	2	no	—
65	C₂×D₇	—	2	2	no	—
66	C₄×C₇	—	2	2	no	—
67	C₄×C₇	—	2	2	no	—
68	C₄×C₇	—	2	1	no	—
69	Dic₇	—	2	2	no	—
70	Dic₇	—	2	2	no	—
71	Dic₇	#41	1	4	yes	—
72	C₂×Q₈	—	7	2	no	—
73	C₂×D₄	—	7	2	no	—
74	C₂×D₄	—	7	2	no	—
75	C₂×D₄	—	7	1	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	C₂×D₇	1	1	yes	characteristic
80	C₄⋊C₄	—	7	2	no	—
81	C₄⋊C₄	—	7	2	no	—
82	C₄⋊C₄	C₂×D₇	1	2	yes	—
83	C₂²⋊C₄	—	7	2	no	—
84	C₂²⋊C₄	—	7	1	no	—
85	C₂²⋊C₄	—	7	2	no	—
86	C₂²⋊C₄	—	7	2	no	—
87	C₂²⋊C₄	—	7	2	no	—
88	C₂²⋊C₄	C₂×D₇	1	2	yes	—
89	C₂²⋊C₄	C₂×D₇	1	1	yes	characteristic
90	C₄²	—	7	2	no	—
91	C₄²	—	7	2	no	—
92	C₄²	—	7	1	no	—
93	C₂×C₇	—	4	1	no	—
94	C₂×C₇	—	4	1	no	—
95	C₂×C₇	C₂×#41	1	2	yes	—
96	C₂×C₇	H(5,𝔽₂)	1	1	yes	characteristic
97	D₇	—	4	2	no	—
98	C₂³	—	7	2	no	—
99	C₂³	C₂²×D₇	1	1	yes	characteristic
100	C₂³	C₂²×D₇	1	1	yes	characteristic
101	Q₈	—	14	4	no	—
102	D₄	—	4	1	no	—
103	D₄	—	14	4	no	—
104	D₄	—	14	4	no	—
105	D₄	—	28	1	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	7	2	no	—
108	C₂×C₄	—	7	2	no	—
109	C₂×C₄	—	7	2	no	—
110	C₂×C₄	—	28	2	no	—
111	C₂×C₄	—	28	1	no	—
112	C₂×C₄	C₂²×D₇	1	2	yes	—
113	C₂×C₄	C₂²×D₇	1	2	yes	—
114	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
115	C₇	#559	1	1	yes	characteristic, complemented, Sylow
116	C₂²	—	2	1	no	—
117	C₂²	—	2	1	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	2	2	no	—
120	C₂²	—	14	2	no	—
121	C₂²	—	14	2	no	—
122	C₂²	—	14	2	no	—
123	C₂²	C₂³×D₇	1	1	yes	characteristic
124	C₄	—	2	2	no	—
125	C₄	—	2	1	no	—
126	C₄	—	2	2	no	—
127	C₄	—	7	4	no	—
128	C₄	—	14	2	no	—
129	C₄	—	14	2	no	—
130	C₂	—	4	1	no	—
131	C₂	—	4	1	no	—
132	C₂	—	28	2	no	—
133	C₂	#6548	1	1	yes	characteristic
134	C₂	D₇×#41	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#123

C₂

yes

maximal, characteristic, complemented

#6503

C₂

yes

maximal, characteristic

#6498

C₂

yes

maximal, characteristic, complemented

C₄×C₂²⋊D₇

C₂

yes

maximal, complemented

#6457

C₂

yes

maximal, complemented

#6450

C₂

yes

maximal, complemented

#6448

C₂

yes

maximal, complemented

#6448

C₂

yes

maximal, complemented

(C₂²⋊D₇)⋊C₄

C₂

yes

maximal, complemented

C₂×C₇×D₄

C₂²

yes

characteristic

C₂²×C₄×C₇

C₂²

yes

characteristic

C₂×C₂²⋊D₇

C₂²

yes

—

C₂×C₂²⋊D₇

C₂²

yes

—

C₂²×Dic₇

C₂²

yes

characteristic

C₂×D₂₈

C₂²

yes

characteristic

C₂×C₄×D₇

C₂²

yes

—

C₂×Dic₁₄

C₂²

yes

—

C₇×C₄⋊C₄

C₂²

yes

—

C₇×C₂²⋊C₄

C₂²

yes

—

C₇×C₂²⋊C₄

C₂²

yes

characteristic

C₂²⋊Dic₇

C₂²

yes

characteristic

C₂²⋊Dic₇

C₂²

yes

—

(C₂×D₇)⋊C₄

C₂²

yes

—

(C₂×D₇)⋊C₄

C₂²

yes

—

(C₂×D₇)⋊C₄

C₂²

yes

—

Dic₇⋊C₄

C₂²

yes

—

Dic₇⋊C₄

C₂²

yes

—

C₄×Dic₇

C₂²

yes

—

C₄×Dic₇

C₂²

yes

—

C₄×Dic₇

C₂²

yes

characteristic

#559

—

maximal, Sylow

C₂³×C₇

C₂³

yes

characteristic

C₂³×C₇

C₂³

yes

characteristic

C₂²×D₇

C₂³

yes

—

C₇×D₄

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂³

yes

characteristic

C₂×C₄×C₇

C₂³

yes

—

C₂×C₄×C₇

C₂³

yes

—

C₂²⋊D₇

—

C₂²⋊D₇

—

C₂×Dic₇

—

C₂×Dic₇

C₂³

yes

—

C₂×Dic₇

C₂³

yes

—

C₂×Dic₇

C₂³

yes

—

D₂₈

—

C₄×D₇

—

Dic₁₄

—

—

—

—

—

—

D₇

yes

characteristic, complemented

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

C₂⁴

yes

characteristic

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

Dic₇

—

Dic₇

—

Dic₇

#41

yes

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₇

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₇

yes

—

C₂²⋊C₄

C₂×D₇

yes

characteristic

C₄²

—

C₄²

—

C₄²

—

C₂×C₇

—

C₂×C₇

—

C₂×C₇

C₂×#41

yes

—

C₂×C₇

H(5,𝔽₂)

yes

characteristic

D₇

—

C₂³

—

C₂³

C₂²×D₇

yes

characteristic

100

C₂³

C₂²×D₇

yes

characteristic

101

Q₈

—

102

D₄

—

103

D₄

—

104

D₄

—

105

D₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

C₂²×D₇

yes

—

113

C₂×C₄

C₂²×D₇

yes

—

114

C₂×C₄

C₂²×D₇

yes

characteristic

115

C₇

#559

yes

characteristic, complemented, Sylow

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

C₂³×D₇

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₂

—

131

C₂

—

132

C₂

—

133

C₂

#6548

yes

characteristic

134

C₂

D₇×#41

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇
commutator, socle, cosocle	C₂²×C₇	C₂⁴
Fitting	C₇×#123	C₂
prime radical	#6503	C₂
hypercenter	#123	D₇
hypocenter	C₇	#559
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #123

subgroup

G/H

center, Frattini

C₂²

C₂³×D₇

commutator, socle, cosocle

C₂²×C₇

C₂⁴

C₂

C₂

D₇

C₇

G ▷ C₂²×C₇ ▷ C₇

C₁ ◁ C₂² ◁ #123

N	H ≅ G/N	type
1. C₇×#123	C₂	semidirect
3. #6498	C₂	semidirect
4. C₄×C₂²⋊D₇	C₂	semidirect
5. #6457	C₂	semidirect
6. #6450	C₂	semidirect
7. #6448	C₂	semidirect
8. #6448	C₂	semidirect
9. (C₂²⋊D₇)⋊C₄	C₂	semidirect
54. #123	D₇	semidirect
115. C₇	#559	semidirect

H ≅ G/N

type

1. C₇×#123

C₂

semidirect

3. #6498

C₂

semidirect

4. C₄×C₂²⋊D₇

C₂

semidirect

5. #6457

C₂

semidirect

6. #6450

C₂

semidirect

7. #6448

C₂

semidirect

8. #6448

C₂

semidirect

9. (C₂²⋊D₇)⋊C₄

C₂

semidirect

54. #123

D₇

semidirect

115. C₇

#559

semidirect

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

subgroup	index
C₇×#123	2
#6503	2
#6498	2
C₄×C₂²⋊D₇	2
#6457	2
#6450	2
#6448	2
(C₂²⋊D₇)⋊C₄	2
#559	7

subgroup

index

C₄×C₂²⋊D₇

(C₂²⋊D₇)⋊C₄

Subgroups — #90355

Conjugacy classes of subgroups

Complements

Maximal subgroups