Group #90307 — subgroups

Subgroups — #90307

Order 448.

236 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 97 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×#121	C₂	1	1	yes	maximal, characteristic, complemented
2	#6502	C₂	1	2	yes	maximal, complemented
3	#6501	C₂	1	1	yes	maximal, characteristic, complemented
4	#6499	C₂	1	2	yes	maximal, complemented
5	D₄×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×C₂²⋊D₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#6489	C₂	1	1	yes	maximal, characteristic, complemented
8	#6455	C₂	1	1	yes	maximal, characteristic, complemented
9	#6451	C₂	1	1	yes	maximal, characteristic, complemented
10	#6448	C₂	1	2	yes	maximal, complemented
11	#6440	C₂	1	2	yes	maximal, complemented
12	C₂×C₇×D₄	C₂²	1	1	yes	characteristic
13	C₂×C₇×D₄	C₂²	1	2	yes	—
14	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊D₇	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊D₇	C₂²	1	2	yes	—
17	C₂²×Dic₇	C₂²	1	2	yes	—
18	C₂×C₄×D₇	C₂²	1	1	yes	characteristic
19	C₂×Dic₁₄	C₂²	1	2	yes	—
20	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₇×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₂²⋊Dic₇	C₂²	1	2	yes	—
23	C₂²⋊Dic₇	C₂²	1	2	yes	—
24	C₂²⋊Dic₇	C₂²	1	2	yes	—
25	C₂²⋊Dic₇	C₂²	1	1	yes	characteristic
26	(C₂×D₇)⋊C₄	C₂²	1	1	yes	characteristic
27	(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
28	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
29	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
30	Dic₇⋊C₄	C₂²	1	1	yes	characteristic
31	Dic₇⋊C₄	C₂²	1	2	yes	—
32	C₄×Dic₇	C₂²	1	2	yes	—
33	C₄×Dic₇	C₂²	1	2	yes	—
34	C₄×Dic₇	C₂²	1	1	yes	characteristic
35	#555	—	7	1	no	maximal, Sylow
36	C₂³×C₇	C₂³	1	1	yes	characteristic
37	C₂³×C₇	C₂³	1	2	yes	—
38	C₂²×D₇	C₂³	1	1	yes	characteristic
39	C₇×D₄	—	2	2	no	—
40	C₇×D₄	—	4	2	no	—
41	C₂×C₄×C₇	—	2	2	no	—
42	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
43	C₂×C₄×C₇	C₂³	1	2	yes	—
44	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
45	C₂²⋊D₇	—	2	2	no	—
46	C₂²⋊D₇	—	4	2	no	—
47	C₂×Dic₇	—	2	4	no	—
48	C₂×Dic₇	C₂³	1	1	yes	characteristic
49	C₂×Dic₇	C₂³	1	2	yes	—
50	C₂×Dic₇	C₂³	1	2	yes	—
51	C₂×Dic₇	C₂³	1	1	yes	characteristic
52	C₂×Dic₇	C₂³	1	1	yes	characteristic
53	C₄×D₇	—	2	2	no	—
54	Dic₁₄	—	4	2	no	—
55	C₄⋊Q₈	—	7	1	no	—
56	#124	—	7	2	no	—
57	#124	—	7	2	no	—
58	#121	—	7	1	no	—
59	#121	—	7	2	no	—
60	#121	D₇	1	1	yes	characteristic, complemented
61	C₄×D₄	—	7	1	no	—
62	C₄×D₄	—	7	1	no	—
63	#117	—	7	1	no	—
64	#117	—	7	1	no	—
65	#117	—	7	2	no	—
66	C₂²×C₇	—	2	2	no	—
67	C₂²×C₇	—	2	1	no	—
68	C₂²×C₇	—	2	1	no	—
69	C₂²×C₇	—	2	2	no	—
70	C₂²×C₇	—	2	2	no	—
71	C₂²×C₇	—	2	1	no	—
72	C₂²×C₇	C₂⁴	1	1	yes	characteristic
73	C₂×D₇	—	2	1	no	—
74	C₂×D₇	—	2	1	no	—
75	C₂×D₇	—	2	1	no	—
76	C₄×C₇	—	2	1	no	—
77	C₄×C₇	—	2	2	no	—
78	C₄×C₇	#41	1	2	yes	—
79	Dic₇	—	2	1	no	—
80	Dic₇	—	2	2	no	—
81	Dic₇	—	2	1	no	—
82	Dic₇	—	2	2	no	—
83	Dic₇	#41	1	2	yes	—
84	C₂×Q₈	—	7	2	no	—
85	C₂×D₄	—	7	1	no	—
86	C₂×D₄	—	7	2	no	—
87	C₂×D₄	C₂×D₇	1	1	yes	characteristic
88	C₂×D₄	C₂×D₇	1	2	yes	—
89	C₂²×C₄	—	7	2	no	—
90	C₂²×C₄	—	7	1	no	—
91	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
92	C₄⋊C₄	—	7	1	no	—
93	C₄⋊C₄	—	7	2	no	—
94	C₄⋊C₄	—	7	1	no	—
95	C₄⋊C₄	—	7	1	no	—
96	C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
97	C₂²⋊C₄	—	7	2	no	—
98	C₂²⋊C₄	—	7	2	no	—
99	C₂²⋊C₄	—	7	1	no	—
100	C₂²⋊C₄	—	7	2	no	—
101	C₂²⋊C₄	—	7	2	no	—
102	C₂²⋊C₄	—	7	1	no	—
103	C₂²⋊C₄	C₂×D₇	1	2	yes	—
104	C₄²	—	7	2	no	—
105	C₄²	—	7	2	no	—
106	C₄²	—	7	1	no	—
107	C₂×C₇	—	4	2	no	—
108	C₂×C₇	—	4	1	no	—
109	C₂×C₇	C₂×#41	1	1	yes	characteristic
110	C₂×C₇	C₂×#41	1	1	yes	characteristic
111	C₂×C₇	H(5,𝔽₂)	1	1	yes	characteristic
112	D₇	—	4	1	no	—
113	C₂³	—	7	1	no	—
114	C₂³	C₂²×D₇	1	1	yes	characteristic
115	C₂³	C₂²×D₇	1	2	yes	—
116	Q₈	—	28	2	no	—
117	D₄	—	2	2	no	—
118	D₄	—	4	2	no	—
119	D₄	—	14	2	no	—
120	D₄	—	28	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	7	1	no	—
123	C₂×C₄	—	7	1	no	—
124	C₂×C₄	—	7	2	no	—
125	C₂×C₄	—	7	1	no	—
126	C₂×C₄	—	7	2	no	—
127	C₂×C₄	—	14	4	no	—
128	C₂×C₄	—	14	2	no	—
129	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
130	C₂×C₄	C₂²×D₇	1	2	yes	—
131	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
132	C₇	#555	1	1	yes	characteristic, complemented, Sylow
133	C₂²	—	2	1	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	2	1	no	—
136	C₂²	—	2	2	no	—
137	C₂²	—	2	2	no	—
138	C₂²	—	2	1	no	—
139	C₂²	—	14	1	no	—
140	C₂²	—	14	1	no	—
141	C₂²	—	14	1	no	—
142	C₂²	C₂³×D₇	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	2	no	—
145	C₄	—	7	2	no	—
146	C₄	—	14	1	no	—
147	C₄	—	14	2	no	—
148	C₄	—	14	1	no	—
149	C₄	—	14	2	no	—
150	C₄	#1159	1	2	yes	—
151	C₂	—	4	1	no	—
152	C₂	—	4	2	no	—
153	C₂	—	28	1	no	—
154	C₂	C₂×#1159	1	1	yes	characteristic
155	C₂	#6548	1	1	yes	characteristic
156	C₂	D₇×#41	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#121

C₂

yes

maximal, characteristic, complemented

#6502

C₂

yes

maximal, complemented

#6501

C₂

yes

maximal, characteristic, complemented

#6499

C₂

yes

maximal, complemented

D₄×Dic₇

C₂

yes

maximal, characteristic, complemented

C₄×C₂²⋊D₇

C₂

yes

maximal, characteristic, complemented

#6489

C₂

yes

maximal, characteristic, complemented

#6455

C₂

yes

maximal, characteristic, complemented

#6451

C₂

yes

maximal, characteristic, complemented

#6448

C₂

yes

maximal, complemented

#6440

C₂

yes

maximal, complemented

C₂×C₇×D₄

C₂²

yes

characteristic

C₂×C₇×D₄

C₂²

yes

—

C₂²×C₄×C₇

C₂²

yes

characteristic

C₂×C₂²⋊D₇

C₂²

yes

characteristic

C₂×C₂²⋊D₇

C₂²

yes

—

C₂²×Dic₇

C₂²

yes

—

C₂×C₄×D₇

C₂²

yes

characteristic

C₂×Dic₁₄

C₂²

yes

—

C₇×C₄⋊C₄

C₂²

yes

characteristic

C₇×C₂²⋊C₄

C₂²

yes

—

C₂²⋊Dic₇

C₂²

yes

—

C₂²⋊Dic₇

C₂²

yes

—

C₂²⋊Dic₇

C₂²

yes

—

C₂²⋊Dic₇

C₂²

yes

characteristic

(C₂×D₇)⋊C₄

C₂²

yes

characteristic

(C₂×D₇)⋊C₄

C₂²

yes

—

C₄⋊Dic₇

C₂²

yes

characteristic

C₄⋊Dic₇

C₂²

yes

characteristic

Dic₇⋊C₄

C₂²

yes

characteristic

Dic₇⋊C₄

C₂²

yes

—

C₄×Dic₇

C₂²

yes

—

C₄×Dic₇

C₂²

yes

—

C₄×Dic₇

C₂²

yes

characteristic

#555

—

maximal, Sylow

C₂³×C₇

C₂³

yes

characteristic

C₂³×C₇

C₂³

yes

—

C₂²×D₇

C₂³

yes

characteristic

C₇×D₄

—

C₇×D₄

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂³

yes

characteristic

C₂×C₄×C₇

C₂³

yes

—

C₂×C₄×C₇

C₂³

yes

characteristic

C₂²⋊D₇

—

C₂²⋊D₇

—

C₂×Dic₇

—

C₂×Dic₇

C₂³

yes

characteristic

C₂×Dic₇

C₂³

yes

—

C₂×Dic₇

C₂³

yes

—

C₂×Dic₇

C₂³

yes

characteristic

C₂×Dic₇

C₂³

yes

characteristic

C₄×D₇

—

Dic₁₄

—

C₄⋊Q₈

—

—

—

—

—

D₇

yes

characteristic, complemented

C₄×D₄

—

C₄×D₄

—

#117

—

#117

—

#117

—

C₂²×C₇

—

C₂²×C₇

—

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

#41

yes

—

Dic₇

—

Dic₇

—

Dic₇

—

Dic₇

—

Dic₇

#41

yes

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₇

yes

characteristic

C₂×D₄

C₂×D₇

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₇

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

100

C₂²⋊C₄

—

101

C₂²⋊C₄

—

102

C₂²⋊C₄

—

103

C₂²⋊C₄

C₂×D₇

yes

—

104

C₄²

—

105

C₄²

—

106

C₄²

—

107

C₂×C₇

—

108

C₂×C₇

—

109

C₂×C₇

C₂×#41

yes

characteristic

110

C₂×C₇

C₂×#41

yes

characteristic

111

C₂×C₇

H(5,𝔽₂)

yes

characteristic

112

D₇

—

113

C₂³

—

114

C₂³

C₂²×D₇

yes

characteristic

115

C₂³

C₂²×D₇

yes

—

116

Q₈

—

117

D₄

—

118

D₄

—

119

D₄

—

120

D₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₂²×D₇

yes

characteristic

130

C₂×C₄

C₂²×D₇

yes

—

131

C₂×C₄

C₂²×D₇

yes

characteristic

132

C₇

#555

yes

characteristic, complemented, Sylow

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

C₂³×D₇

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

#1159

yes

—

151

C₂

—

152

C₂

—

153

C₂

—

154

C₂

C₂×#1159

yes

characteristic

155

C₂

#6548

yes

characteristic

156

C₂

D₇×#41

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇
commutator, socle, cosocle	C₂²×C₇	C₂⁴
Fitting	C₇×#121	C₂
hypercenter	#121	D₇
hypocenter	C₇	#555
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #121

subgroup

G/H

center, Frattini

C₂²

C₂³×D₇

commutator, socle, cosocle

C₂²×C₇

C₂⁴

C₂

D₇

C₇

G ▷ C₂²×C₇ ▷ C₇

C₁ ◁ C₂² ◁ #121

N	H ≅ G/N	type
1. C₇×#121	C₂	semidirect
2. #6502	C₂	semidirect
3. #6501	C₂	semidirect
4. #6499	C₂	semidirect
5. D₄×Dic₇	C₂	semidirect
6. C₄×C₂²⋊D₇	C₂	semidirect
7. #6489	C₂	semidirect
8. #6455	C₂	semidirect
9. #6451	C₂	semidirect
10. #6448	C₂	semidirect
11. #6440	C₂	semidirect
60. #121	D₇	semidirect
132. C₇	#555	semidirect

H ≅ G/N

type

1. C₇×#121

C₂

semidirect

2. #6502

C₂

semidirect

3. #6501

C₂

semidirect

4. #6499

C₂

semidirect

5. D₄×Dic₇

C₂

semidirect

6. C₄×C₂²⋊D₇

C₂

semidirect

7. #6489

C₂

semidirect

8. #6455

C₂

semidirect

9. #6451

C₂

semidirect

10. #6448

C₂

semidirect

11. #6440

C₂

semidirect

60. #121

D₇

semidirect

132. C₇

#555

semidirect

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

subgroup	index
C₇×#121	2
#6502	2
#6501	2
#6499	2
D₄×Dic₇	2
C₄×C₂²⋊D₇	2
#6489	2
#6455	2
#6451	2
#6448	2
#6440	2
#555	7

subgroup

index

D₄×Dic₇

C₄×C₂²⋊D₇

Subgroups — #90307

Conjugacy classes of subgroups

Complements

Maximal subgroups