Group #90308 — subgroups

Subgroups — #90308

Order 448.

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

normal subgroups only

This group has 250 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×#121	C₂	1	1	yes	maximal, characteristic, complemented
2	#6502	C₂	1	2	yes	maximal, complemented
3	#6500	C₂	1	2	yes	maximal, complemented
4	D₄×Dic₇	C₂	1	2	yes	maximal, complemented
5	C₂×(C₂×D₇)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	#6487	C₂	1	1	yes	maximal, characteristic, complemented
7	#6461	C₂	1	2	yes	maximal, complemented
8	#6449	C₂	1	2	yes	maximal, complemented
9	#6448	C₂	1	2	yes	maximal, complemented
10	C₂³×D₇	C₂²	1	1	yes	characteristic
11	C₂×C₇×D₄	C₂²	1	2	yes	—
12	C₂×C₇×D₄	C₂²	1	1	yes	characteristic, complemented
13	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊D₇	C₂²	1	2	yes	—
15	C₂×C₂²⋊D₇	C₂²	1	2	yes	—
16	C₂²×Dic₇	C₂²	1	1	yes	characteristic
17	C₂²×Dic₇	C₂²	1	2	yes	complemented
18	C₂×Dic₁₄	C₂²	1	1	yes	characteristic, complemented
19	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
20	C₇×C₂²⋊C₄	C₂²	1	2	yes	complemented
21	C₂²⋊Dic₇	C₂²	1	2	yes	—
22	C₂²⋊Dic₇	C₂²	1	2	yes	complemented
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	(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
27	C₄⋊Dic₇	C₂²	1	1	yes	characteristic, complemented
28	C₄⋊Dic₇	C₂²	1	2	yes	complemented
29	Dic₇⋊C₄	C₂²	1	2	yes	complemented
30	C₄×Dic₇	C₂²	1	2	yes	complemented
31	#538	—	7	1	no	maximal, Sylow
32	C₂³×C₇	C₂³	1	2	yes	—
33	C₂³×C₇	C₂³	1	1	yes	characteristic
34	C₂²×D₇	—	2	2	no	—
35	C₂²×D₇	—	4	1	no	—
36	C₂²×D₇	—	4	1	no	—
37	C₂²×D₇	C₂³	1	2	yes	—
38	C₇×D₄	—	2	4	no	—
39	C₇×D₄	—	4	1	no	—
40	C₂×C₄×C₇	—	4	1	no	—
41	C₂×C₄×C₇	C₂³	1	2	yes	—
42	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
43	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
44	C₂²⋊D₇	—	4	2	no	—
45	C₂²⋊D₇	—	4	2	no	—
46	C₂×Dic₇	—	4	1	no	—
47	C₂×Dic₇	—	4	2	no	—
48	C₂×Dic₇	C₂³	1	2	yes	—
49	C₂×Dic₇	C₂³	1	2	yes	—
50	C₂×Dic₇	C₂³	1	2	yes	—
51	Dic₁₄	—	4	1	no	—
52	#126	—	7	2	no	—
53	#124	—	7	2	no	—
54	#123	—	7	2	no	—
55	C₂²⋊Q₈	—	7	1	no	—
56	#121	—	7	2	no	—
57	#121	D₇	1	1	yes	characteristic, complemented
58	#120	—	7	2	no	—
59	C₄×D₄	—	7	2	no	—
60	C₂×C₂²⋊C₄	—	7	1	no	—
61	C₂²×C₇	—	2	1	no	—
62	C₂²×C₇	—	2	1	no	—
63	C₂²×C₇	—	2	2	no	—
64	C₂²×C₇	—	2	2	no	—
65	C₂²×C₇	—	2	2	no	—
66	C₂²×C₇	#41	1	2	yes	—
67	C₂²×C₇	C₂⁴	1	1	yes	characteristic
68	C₂×D₇	—	2	2	no	—
69	C₂×D₇	—	2	2	no	—
70	C₂×D₇	—	2	2	no	—
71	C₂×D₇	—	4	4	no	—
72	C₄×C₇	—	2	2	no	—
73	C₄×C₇	—	2	1	no	—
74	C₄×C₇	—	2	1	no	—
75	Dic₇	—	2	2	no	—
76	Dic₇	—	2	2	no	—
77	Dic₇	—	2	2	no	—
78	C₂⁴	—	7	1	no	—
79	C₂×Q₈	—	7	1	no	—
80	C₂×D₄	—	7	2	no	—
81	C₂×D₄	—	7	2	no	—
82	C₂×D₄	C₂×D₇	1	2	yes	—
83	C₂×D₄	C₂×D₇	1	1	yes	characteristic, complemented
84	C₂²×C₄	—	7	1	no	—
85	C₂²×C₄	—	7	2	no	—
86	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
87	C₄⋊C₄	—	7	2	no	—
88	C₄⋊C₄	—	7	2	no	—
89	C₄⋊C₄	—	7	1	no	—
90	C₄⋊C₄	C₂×D₇	1	1	yes	characteristic, complemented
91	C₂²⋊C₄	—	7	2	no	—
92	C₂²⋊C₄	—	7	2	no	—
93	C₂²⋊C₄	—	7	2	no	—
94	C₂²⋊C₄	—	7	2	no	—
95	C₂²⋊C₄	—	7	2	no	—
96	C₂²⋊C₄	—	7	2	no	—
97	C₂²⋊C₄	C₂×D₇	1	2	yes	complemented
98	C₄²	—	7	2	no	—
99	C₂×C₇	—	2	2	no	—
100	C₂×C₇	—	4	2	no	—
101	C₂×C₇	C₂×#41	1	1	yes	characteristic
102	C₂×C₇	H(5,𝔽₂)	1	1	yes	characteristic
103	C₂×C₇	H(5,𝔽₂)	1	1	yes	characteristic
104	D₇	—	4	2	no	—
105	C₂³	—	7	2	no	—
106	C₂³	—	14	2	no	—
107	C₂³	—	28	1	no	—
108	C₂³	—	28	1	no	—
109	C₂³	C₂²×D₇	1	2	yes	—
110	C₂³	C₂²×D₇	1	1	yes	characteristic
111	Q₈	—	28	1	no	—
112	D₄	—	2	4	no	—
113	D₄	—	4	1	no	—
114	D₄	—	28	2	no	—
115	D₄	—	28	2	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	7	2	no	—
118	C₂×C₄	—	7	2	no	—
119	C₂×C₄	—	7	2	no	—
120	C₂×C₄	—	28	1	no	—
121	C₂×C₄	—	28	2	no	—
122	C₂×C₄	C₂²×D₇	1	2	yes	—
123	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
124	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
125	C₇	#538	1	1	yes	characteristic, complemented, Sylow
126	C₂²	—	2	1	no	—
127	C₂²	—	2	1	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	2	2	no	—
130	C₂²	—	2	2	no	—
131	C₂²	—	14	2	no	—
132	C₂²	—	14	2	no	—
133	C₂²	—	14	2	no	—
134	C₂²	—	28	4	no	—
135	C₂²	#1159	1	2	yes	—
136	C₂²	C₂³×D₇	1	1	yes	characteristic
137	C₄	—	2	2	no	—
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	14	2	no	—
141	C₄	—	14	2	no	—
142	C₄	—	14	2	no	—
143	C₂	—	2	2	no	—
144	C₂	—	4	2	no	—
145	C₂	—	28	2	no	—
146	C₂	C₂×#1159	1	1	yes	characteristic
147	C₂	#6548	1	1	yes	characteristic
148	C₂	#6553	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#121

C₂

yes

maximal, characteristic, complemented

#6502

C₂

yes

maximal, complemented

#6500

C₂

yes

maximal, complemented

D₄×Dic₇

C₂

yes

maximal, complemented

C₂×(C₂×D₇)⋊C₄

C₂

yes

maximal, characteristic, complemented

#6487

C₂

yes

maximal, characteristic, complemented

#6461

C₂

yes

maximal, complemented

#6449

C₂

yes

maximal, complemented

#6448

C₂

yes

maximal, complemented

C₂³×D₇

C₂²

yes

characteristic

C₂×C₇×D₄

C₂²

yes

—

C₂×C₇×D₄

C₂²

yes

characteristic, complemented

C₂²×C₄×C₇

C₂²

yes

characteristic

C₂×C₂²⋊D₇

C₂²

yes

—

C₂×C₂²⋊D₇

C₂²

yes

—

C₂²×Dic₇

C₂²

yes

characteristic

C₂²×Dic₇

C₂²

yes

complemented

C₂×Dic₁₄

C₂²

yes

characteristic, complemented

C₇×C₄⋊C₄

C₂²

yes

characteristic, complemented

C₇×C₂²⋊C₄

C₂²

yes

complemented

C₂²⋊Dic₇

C₂²

yes

—

C₂²⋊Dic₇

C₂²

yes

complemented

(C₂×D₇)⋊C₄

C₂²

yes

—

(C₂×D₇)⋊C₄

C₂²

yes

—

(C₂×D₇)⋊C₄

C₂²

yes

—

(C₂×D₇)⋊C₄

C₂²

yes

—

C₄⋊Dic₇

C₂²

yes

characteristic, complemented

C₄⋊Dic₇

C₂²

yes

complemented

Dic₇⋊C₄

C₂²

yes

complemented

C₄×Dic₇

C₂²

yes

complemented

#538

—

maximal, Sylow

C₂³×C₇

C₂³

yes

—

C₂³×C₇

C₂³

yes

characteristic

C₂²×D₇

—

C₂²×D₇

—

C₂²×D₇

—

C₂²×D₇

C₂³

yes

—

C₇×D₄

—

C₇×D₄

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂³

yes

—

C₂×C₄×C₇

C₂³

yes

characteristic

C₂×C₄×C₇

C₂³

yes

characteristic

C₂²⋊D₇

—

C₂²⋊D₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

C₂³

yes

—

C₂×Dic₇

C₂³

yes

—

C₂×Dic₇

C₂³

yes

—

Dic₁₄

—

#126

—

#124

—

#123

—

C₂²⋊Q₈

—

#121

—

#121

D₇

yes

characteristic, complemented

#120

—

C₄×D₄

—

C₂×C₂²⋊C₄

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

#41

yes

—

C₂²×C₇

C₂⁴

yes

characteristic

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

Dic₇

—

Dic₇

—

Dic₇

—

C₂⁴

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₇

yes

—

C₂×D₄

C₂×D₇

yes

characteristic, complemented

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₇

yes

characteristic, complemented

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₇

yes

complemented

C₄²

—

C₂×C₇

—

100

C₂×C₇

—

101

C₂×C₇

C₂×#41

yes

characteristic

102

C₂×C₇

H(5,𝔽₂)

yes

characteristic

103

C₂×C₇

H(5,𝔽₂)

yes

characteristic

104

D₇

—

105

C₂³

—

106

C₂³

—

107

C₂³

—

108

C₂³

—

109

C₂³

C₂²×D₇

yes

—

110

C₂³

C₂²×D₇

yes

characteristic

111

Q₈

—

112

D₄

—

113

D₄

—

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

C₂²×D₇

yes

—

123

C₂×C₄

C₂²×D₇

yes

characteristic

124

C₂×C₄

C₂²×D₇

yes

characteristic

125

C₇

#538

yes

characteristic, complemented, Sylow

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

#1159

yes

—

136

C₂²

C₂³×D₇

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₂

—

144

C₂

—

145

C₂

—

146

C₂

C₂×#1159

yes

characteristic

147

C₂

#6548

yes

characteristic

148

C₂

#6553

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	#538
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. #6500	C₂	semidirect
4. D₄×Dic₇	C₂	semidirect
5. C₂×(C₂×D₇)⋊C₄	C₂	semidirect
6. #6487	C₂	semidirect
7. #6461	C₂	semidirect
8. #6449	C₂	semidirect
9. #6448	C₂	semidirect
12. C₂×C₇×D₄	C₂²	semidirect
17. C₂²×Dic₇	C₂²	semidirect
18. C₂×Dic₁₄	C₂²	semidirect
19. C₇×C₄⋊C₄	C₂²	semidirect
20. C₇×C₂²⋊C₄	C₂²	semidirect
22. C₂²⋊Dic₇	C₂²	semidirect
27. C₄⋊Dic₇	C₂²	semidirect
28. C₄⋊Dic₇	C₂²	semidirect
29. Dic₇⋊C₄	C₂²	semidirect
30. C₄×Dic₇	C₂²	semidirect
57. #121	D₇	semidirect
83. C₂×D₄	C₂×D₇	semidirect
90. C₄⋊C₄	C₂×D₇	semidirect
97. C₂²⋊C₄	C₂×D₇	semidirect
125. C₇	#538	semidirect

H ≅ G/N

type

1. C₇×#121

C₂

semidirect

2. #6502

C₂

semidirect

3. #6500

C₂

semidirect

4. D₄×Dic₇

C₂

semidirect

5. C₂×(C₂×D₇)⋊C₄

C₂

semidirect

6. #6487

C₂

semidirect

7. #6461

C₂

semidirect

8. #6449

C₂

semidirect

9. #6448

C₂

semidirect

12. C₂×C₇×D₄

C₂²

semidirect

17. C₂²×Dic₇

C₂²

semidirect

18. C₂×Dic₁₄

C₂²

semidirect

19. C₇×C₄⋊C₄

C₂²

semidirect

20. C₇×C₂²⋊C₄

C₂²

semidirect

22. C₂²⋊Dic₇

C₂²

semidirect

27. C₄⋊Dic₇

C₂²

semidirect

28. C₄⋊Dic₇

C₂²

semidirect

29. Dic₇⋊C₄

C₂²

semidirect

30. C₄×Dic₇

C₂²

semidirect

57. #121

D₇

semidirect

83. C₂×D₄

C₂×D₇

semidirect

90. C₄⋊C₄

C₂×D₇

semidirect

97. C₂²⋊C₄

C₂×D₇

semidirect

125. C₇

#538

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
#6500	2
D₄×Dic₇	2
C₂×(C₂×D₇)⋊C₄	2
#6487	2
#6461	2
#6449	2
#6448	2
#538	7

subgroup

index

C₇×#121

#6502

#6500

D₄×Dic₇

C₂×(C₂×D₇)⋊C₄

Subgroups — #90308

Conjugacy classes of subgroups

Complements

Maximal subgroups