Group #90293 — subgroups

Subgroups — #90293

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×#121	C₂	1	1	yes	maximal, characteristic
2	#6498	C₂	1	2	yes	maximal, complemented
3	#6498	C₂	1	2	yes	maximal, complemented
4	D₄×Dic₇	C₂	1	2	yes	maximal, complemented
5	#6487	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₇⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
7	#6452	C₂	1	2	yes	maximal, complemented
8	#6442	C₂	1	2	yes	maximal, complemented
9	#6441	C₂	1	2	yes	maximal, complemented
10	C₂×C₇×D₄	C₂²	1	2	yes	—
11	C₂×C₇×D₄	C₂²	1	1	yes	characteristic
12	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
13	C₂²×Dic₇	C₂²	1	1	yes	characteristic
14	C₂²×Dic₇	C₂²	1	2	yes	—
15	C₂²×Dic₇	C₂²	1	1	yes	characteristic
16	C₂×Dic₁₄	C₂²	1	1	yes	characteristic
17	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₇×C₂²⋊C₄	C₂²	1	2	yes	—
19	C₂²⋊Dic₇	C₂²	1	2	yes	—
20	C₂²⋊Dic₇	C₂²	1	2	yes	—
21	C₂²⋊Dic₇	C₂²	1	2	yes	—
22	C₂²⋊Dic₇	C₂²	1	2	yes	—
23	C₄⋊Dic₇	C₂²	1	2	yes	—
24	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
25	Dic₇⋊C₄	C₂²	1	2	yes	—
26	Dic₇⋊C₄	C₂²	1	2	yes	—
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	2	yes	—
33	C₂³×C₇	C₂³	1	1	yes	characteristic
34	C₇×D₄	—	2	4	no	—
35	C₇×D₄	—	4	1	no	—
36	C₂×C₄×C₇	—	4	1	no	—
37	C₂×C₄×C₇	C₂³	1	2	yes	—
38	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
40	C₂×Dic₇	—	2	2	no	—
41	C₂×Dic₇	—	4	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	C₂×Dic₇	C₂³	1	2	yes	—
47	Dic₁₄	—	4	1	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	1	no	—
53	C₂²⋊Q₈	—	7	2	no	—
54	#121	D₇	1	1	yes	characteristic
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	—	2	2	no	—
60	C₂²×C₇	—	2	2	no	—
61	C₂²×C₇	—	2	2	no	—
62	C₂²×C₇	#41	1	2	yes	—
63	C₂²×C₇	C₂⁴	1	1	yes	characteristic
64	C₄×C₇	—	2	2	no	—
65	C₄×C₇	—	2	1	no	—
66	C₄×C₇	—	2	1	no	—
67	Dic₇	—	2	2	no	—
68	Dic₇	—	2	2	no	—
69	Dic₇	—	2	2	no	—
70	Dic₇	—	2	2	no	—
71	C₂×Q₈	—	7	1	no	—
72	C₂×D₄	C₂×D₇	1	2	yes	—
73	C₂×D₄	C₂×D₇	1	1	yes	characteristic
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	2	no	—
82	C₄⋊C₄	—	7	2	no	—
83	C₄⋊C₄	—	7	2	no	—
84	C₄⋊C₄	—	7	1	no	—
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	—	4	2	no	—
94	C₂×C₇	C₂×#41	1	1	yes	characteristic
95	C₂×C₇	H(5,𝔽₂)	1	1	yes	characteristic
96	C₂×C₇	2₋¹⁺⁴	1	1	yes	characteristic
97	C₂³	C₂²×D₇	1	2	yes	—
98	C₂³	C₂²×D₇	1	1	yes	characteristic
99	Q₈	—	28	1	no	—
100	D₄	—	2	4	no	—
101	D₄	—	4	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	—	7	2	no	—
107	C₂×C₄	—	14	2	no	—
108	C₂×C₄	—	28	1	no	—
109	C₂×C₄	—	28	2	no	—
110	C₂×C₄	C₂²×D₇	1	2	yes	—
111	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
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	—	2	2	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	2	no	—
119	C₂²	#1159	1	2	yes	—
120	C₂²	C₂³×D₇	1	1	yes	characteristic
121	C₄	—	2	2	no	—
122	C₄	—	2	1	no	—
123	C₄	—	2	1	no	—
124	C₄	—	14	2	no	—
125	C₄	—	14	2	no	—
126	C₄	—	14	2	no	—
127	C₄	—	14	2	no	—
128	C₂	—	2	2	no	—
129	C₂	—	4	2	no	—
130	C₂	C₂×#1159	1	1	yes	characteristic
131	C₂	#6548	1	1	yes	characteristic
132	C₂	#6554	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#121

C₂

yes

maximal, characteristic

#6498

C₂

yes

maximal, complemented

#6498

C₂

yes

maximal, complemented

D₄×Dic₇

C₂

yes

maximal, complemented

#6487

C₂

yes

maximal, characteristic, complemented

C₂×Dic₇⋊C₄

C₂

yes

maximal, characteristic, complemented

#6452

C₂

yes

maximal, complemented

#6442

C₂

yes

maximal, complemented

#6441

C₂

yes

maximal, complemented

C₂×C₇×D₄

C₂²

yes

—

C₂×C₇×D₄

C₂²

yes

characteristic

C₂²×C₄×C₇

C₂²

yes

characteristic

C₂²×Dic₇

C₂²

yes

characteristic

C₂²×Dic₇

C₂²

yes

—

C₂²×Dic₇

C₂²

yes

characteristic

C₂×Dic₁₄

C₂²

yes

characteristic

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

—

C₄⋊Dic₇

C₂²

yes

—

C₄⋊Dic₇

C₂²

yes

characteristic

Dic₇⋊C₄

C₂²

yes

—

Dic₇⋊C₄

C₂²

yes

—

Dic₇⋊C₄

C₂²

yes

—

Dic₇⋊C₄

C₂²

yes

—

Dic₇⋊C₄

C₂²

yes

—

C₄×Dic₇

C₂²

yes

—

#539

—

maximal, Sylow

C₂³×C₇

C₂³

yes

—

C₂³×C₇

C₂³

yes

characteristic

C₇×D₄

—

C₇×D₄

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂³

yes

—

C₂×C₄×C₇

C₂³

yes

characteristic

C₂×C₄×C₇

C₂³

yes

characteristic

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

C₂³

yes

—

C₂×Dic₇

C₂³

yes

—

C₂×Dic₇

C₂³

yes

—

C₂×Dic₇

C₂³

yes

—

Dic₁₄

—

—

—

—

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

#121

D₇

yes

characteristic

C₄×D₄

—

C₂×C₄⋊C₄

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

#41

yes

—

C₂²×C₇

C₂⁴

yes

characteristic

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

Dic₇

—

Dic₇

—

Dic₇

—

Dic₇

—

C₂×Q₈

—

C₂×D₄

C₂×D₇

yes

—

C₂×D₄

C₂×D₇

yes

characteristic

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

—

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×C₇

C₂×#41

yes

characteristic

C₂×C₇

H(5,𝔽₂)

yes

characteristic

C₂×C₇

2₋¹⁺⁴

yes

characteristic

C₂³

C₂²×D₇

yes

—

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

—

110

C₂×C₄

C₂²×D₇

yes

—

111

C₂×C₄

C₂²×D₇

yes

characteristic

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

#1159

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

C₂×#1159

yes

characteristic

131

C₂

#6548

yes

characteristic

132

C₂

#6554

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇
commutator, socle, cosocle	C₂²×C₇	C₂⁴
Fitting, prime radical	C₇×#121	C₂
hypercenter	#121	D₇
hypocenter	C₇	#539
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #121

subgroup

G/H

center, Frattini

C₂²

C₂³×D₇

commutator, socle, cosocle

C₂²×C₇

C₂⁴

Fitting, prime radical

C₂

D₇

C₇

G ▷ C₂²×C₇ ▷ C₇

C₁ ◁ C₂² ◁ #121

N	H ≅ G/N	type
2. #6498	C₂	semidirect
3. #6498	C₂	semidirect
4. D₄×Dic₇	C₂	semidirect
5. #6487	C₂	semidirect
6. C₂×Dic₇⋊C₄	C₂	semidirect
7. #6452	C₂	semidirect
8. #6442	C₂	semidirect
9. #6441	C₂	semidirect
113. C₇	#539	semidirect

H ≅ G/N

type

2. #6498

C₂

semidirect

3. #6498

C₂

semidirect

4. D₄×Dic₇

C₂

semidirect

5. #6487

C₂

semidirect

6. C₂×Dic₇⋊C₄

C₂

semidirect

7. #6452

C₂

semidirect

8. #6442

C₂

semidirect

9. #6441

C₂

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×#121	2
#6498	2
D₄×Dic₇	2
#6487	2
C₂×Dic₇⋊C₄	2
#6452	2
#6442	2
#6441	2
#539	7

subgroup

index

D₄×Dic₇

C₂×Dic₇⋊C₄

Subgroups — #90293

Conjugacy classes of subgroups

Complements

Maximal subgroups