Group #90365 — subgroups

Subgroups — #90365

Order 448.

334 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 103 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	C₂×#1161	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₄×D₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#124	C₂	1	1	yes	maximal, characteristic, complemented
4	#6510	C₂	1	1	yes	maximal, characteristic, complemented
5	#6503	C₂	1	1	yes	maximal, characteristic, complemented
6	#6447	C₂	1	4	yes	maximal, complemented
7	#6445	C₂	1	4	yes	maximal, complemented
8	#6437	C₂	1	1	yes	maximal, characteristic, complemented
9	#6435	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×D₇	C₂²	1	2	yes	—
11	C₂×C₇×Q₈	C₂²	1	1	yes	characteristic, complemented
12	C₂×C₇×D₄	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊D₇	C₂²	1	2	yes	—
14	C₂×C₂²⋊D₇	C₂²	1	4	yes	complemented
15	#1161	—	2	4	no	—
16	D₄×D₇	—	2	4	no	—
17	C₂×D₂₈	C₂²	1	1	yes	characteristic
18	C₂×D₂₈	C₂²	1	2	yes	—
19	C₂×D₂₈	C₂²	1	1	yes	characteristic, complemented
20	C₂×D₂₈	C₂²	1	4	yes	complemented
21	C₂×C₄×D₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₇	C₂²	1	2	yes	complemented
23	C₇×C₂²⋊C₄	C₂²	1	4	yes	complemented
24	C₄²×C₇	C₂²	1	1	yes	characteristic, complemented
25	(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
26	(C₂×D₇)⋊C₄	C₂²	1	4	yes	complemented
27	Dic₇⋊C₄	C₂²	1	2	yes	complemented
28	C₄×Dic₇	C₂²	1	1	yes	characteristic, complemented
29	#535	—	7	1	no	maximal, Sylow
30	C₂³×C₇	C₂³	1	2	yes	—
31	C₂²×D₇	—	2	4	no	—
32	C₂²×D₇	—	4	4	no	—
33	C₂²×D₇	C₂³	1	2	yes	—
34	C₂²×D₇	C₂³	1	2	yes	—
35	C₂²×D₇	C₂³	1	1	yes	characteristic
36	C₇×Q₈	—	2	2	no	—
37	C₇×D₄	—	2	2	no	—
38	C₂×C₄×C₇	C₂³	1	2	yes	—
39	C₂×C₄×C₇	C₂³	1	2	yes	—
40	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
41	C₂²⋊D₇	—	2	4	no	—
42	C₂²⋊D₇	—	4	4	no	—
43	C₂×Dic₇	C₂³	1	2	yes	—
44	C₂×Dic₇	C₂³	1	1	yes	characteristic
45	D₂₈	—	2	4	no	—
46	D₂₈	—	2	2	no	—
47	D₂₈	—	2	2	no	—
48	D₂₈	—	4	4	no	—
49	C₄×D₇	—	2	4	no	—
50	C₄×D₇	D₄	1	4	yes	complemented
51	C₂×#41	—	7	1	no	—
52	C₂²×D₄	—	7	1	no	—
53	#127	—	7	1	no	—
54	#127	—	7	1	no	—
55	#124	—	7	1	no	—
56	#124	D₇	1	1	yes	characteristic, complemented
57	#121	—	7	4	no	—
58	#120	—	7	4	no	—
59	#117	—	7	1	no	—
60	C₂²×C₇	—	2	2	no	—
61	C₂²×C₇	—	2	4	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	C₂×D₇	—	2	4	no	—
67	C₂×D₇	—	2	4	no	—
68	C₂×D₇	—	4	8	no	—
69	C₂×D₇	C₂×D₄	1	2	yes	—
70	C₄×C₇	—	2	2	no	—
71	C₄×C₇	—	2	2	no	—
72	C₄×C₇	C₂×D₄	1	2	yes	—
73	Dic₇	—	2	2	no	—
74	Dic₇	C₂×D₄	1	2	yes	—
75	C₂⁴	—	7	2	no	—
76	#41	—	14	4	no	—
77	C₂×Q₈	C₂×D₇	1	1	yes	characteristic, complemented
78	C₂×D₄	—	7	2	no	—
79	C₂×D₄	—	7	4	no	—
80	C₂×D₄	—	7	2	no	—
81	C₂×D₄	—	7	1	no	—
82	C₂×D₄	—	7	1	no	—
83	C₂×D₄	—	7	4	no	—
84	C₂×D₄	—	14	4	no	—
85	C₂×D₄	C₂×D₇	1	1	yes	characteristic
86	C₂²×C₄	—	7	1	no	—
87	C₂²×C₄	—	7	2	no	—
88	C₄⋊C₄	—	7	2	no	—
89	C₂²⋊C₄	—	7	2	no	—
90	C₂²⋊C₄	—	7	4	no	—
91	C₂²⋊C₄	C₂×D₇	1	4	yes	complemented
92	C₄²	—	7	1	no	—
93	C₄²	C₂×D₇	1	1	yes	characteristic, complemented
94	C₂×C₇	—	4	2	no	—
95	C₂×C₇	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₇	H(5,𝔽₂)	1	2	yes	—
97	D₇	—	2	2	no	—
98	D₇	—	4	2	no	—
99	D₇	—	4	2	no	—
100	C₂³	—	7	1	no	—
101	C₂³	—	7	2	no	—
102	C₂³	—	7	2	no	—
103	C₂³	—	14	4	no	—
104	C₂³	—	28	4	no	—
105	C₂³	C₂²×D₇	1	2	yes	—
106	Q₈	—	2	2	no	—
107	D₄	—	2	2	no	—
108	D₄	—	14	4	no	—
109	D₄	—	14	4	no	—
110	D₄	—	14	2	no	—
111	D₄	—	14	2	no	—
112	D₄	—	28	4	no	—
113	D₄	—	28	4	no	—
114	C₂×C₄	—	7	1	no	—
115	C₂×C₄	—	7	4	no	—
116	C₂×C₄	—	7	2	no	—
117	C₂×C₄	—	14	4	no	—
118	C₂×C₄	C₂²×D₇	1	2	yes	—
119	C₂×C₄	C₂²×D₇	1	2	yes	—
120	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
121	C₇	#535	1	1	yes	characteristic, complemented, Sylow
122	C₂²	—	2	2	no	—
123	C₂²	—	2	4	no	—
124	C₂²	—	7	2	no	—
125	C₂²	—	14	2	no	—
126	C₂²	—	14	2	no	—
127	C₂²	—	14	2	no	—
128	C₂²	—	14	4	no	—
129	C₂²	—	14	4	no	—
130	C₂²	—	28	8	no	—
131	C₂²	C₂³×D₇	1	1	yes	characteristic
132	C₄	—	2	2	no	—
133	C₄	—	2	2	no	—
134	C₄	—	7	2	no	—
135	C₄	—	14	2	no	—
136	C₄	D₄×D₇	1	2	yes	—
137	C₂	—	4	2	no	—
138	C₂	—	14	2	no	—
139	C₂	—	28	2	no	—
140	C₂	—	28	2	no	—
141	C₂	C₂×D₄×D₇	1	1	yes	characteristic
142	C₂	#6553	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#1161

C₂

yes

maximal, characteristic, complemented

C₂×D₄×D₇

C₂

yes

maximal, characteristic, complemented

C₇×#124

C₂

yes

maximal, characteristic, complemented

#6510

C₂

yes

maximal, characteristic, complemented

#6503

C₂

yes

maximal, characteristic, complemented

#6447

C₂

yes

maximal, complemented

#6445

C₂

yes

maximal, complemented

#6437

C₂

yes

maximal, characteristic, complemented

#6435

C₂

yes

maximal, characteristic, complemented

C₂³×D₇

C₂²

yes

—

C₂×C₇×Q₈

C₂²

yes

characteristic, complemented

C₂×C₇×D₄

C₂²

yes

characteristic

C₂×C₂²⋊D₇

C₂²

yes

—

C₂×C₂²⋊D₇

C₂²

yes

complemented

#1161

—

D₄×D₇

—

C₂×D₂₈

C₂²

yes

characteristic

C₂×D₂₈

C₂²

yes

—

C₂×D₂₈

C₂²

yes

characteristic, complemented

C₂×D₂₈

C₂²

yes

complemented

C₂×C₄×D₇

C₂²

yes

characteristic

C₂×C₄×D₇

C₂²

yes

complemented

C₇×C₂²⋊C₄

C₂²

yes

complemented

C₄²×C₇

C₂²

yes

characteristic, complemented

(C₂×D₇)⋊C₄

C₂²

yes

—

(C₂×D₇)⋊C₄

C₂²

yes

complemented

Dic₇⋊C₄

C₂²

yes

complemented

C₄×Dic₇

C₂²

yes

characteristic, complemented

#535

—

maximal, Sylow

C₂³×C₇

C₂³

yes

—

C₂²×D₇

—

C₂²×D₇

—

C₂²×D₇

C₂³

yes

—

C₂²×D₇

C₂³

yes

—

C₂²×D₇

C₂³

yes

characteristic

C₇×Q₈

—

C₇×D₄

—

C₂×C₄×C₇

C₂³

yes

—

C₂×C₄×C₇

C₂³

yes

—

C₂×C₄×C₇

C₂³

yes

characteristic

C₂²⋊D₇

—

C₂²⋊D₇

—

C₂×Dic₇

C₂³

yes

—

C₂×Dic₇

C₂³

yes

characteristic

D₂₈

—

D₂₈

—

D₂₈

—

D₂₈

—

C₄×D₇

—

C₄×D₇

D₄

yes

complemented

C₂×#41

—

C₂²×D₄

—

—

—

—

D₇

yes

characteristic, complemented

#121

—

#120

—

#117

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

C₂⁴

yes

characteristic

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₂×D₇

C₂×D₄

yes

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

C₂×D₄

yes

—

Dic₇

—

Dic₇

C₂×D₄

yes

—

C₂⁴

—

#41

—

C₂×Q₈

C₂×D₇

yes

characteristic, complemented

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₇

yes

complemented

C₄²

—

C₄²

C₂×D₇

yes

characteristic, complemented

C₂×C₇

—

C₂×C₇

C₂²×D₄

yes

characteristic

C₂×C₇

H(5,𝔽₂)

yes

—

D₇

—

D₇

—

D₇

—

100

C₂³

—

101

C₂³

—

102

C₂³

—

103

C₂³

—

104

C₂³

—

105

C₂³

C₂²×D₇

yes

—

106

Q₈

—

107

D₄

—

108

D₄

—

109

D₄

—

110

D₄

—

111

D₄

—

112

D₄

—

113

D₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

C₂²×D₇

yes

—

119

C₂×C₄

C₂²×D₇

yes

—

120

C₂×C₄

C₂²×D₇

yes

characteristic

121

C₇

#535

yes

characteristic, complemented, Sylow

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

C₂³×D₇

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

D₄×D₇

yes

—

137

C₂

—

138

C₂

—

139

C₂

—

140

C₂

—

141

C₂

C₂×D₄×D₇

yes

characteristic

142

C₂

#6553

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇
commutator, socle, cosocle	C₂²×C₇	C₂⁴
Fitting	C₇×#124	C₂
hypercenter	#124	D₇
hypocenter	C₇	#535
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×D₇

commutator, socle, cosocle

C₂²×C₇

C₂⁴

C₂

D₇

C₇

G ▷ C₂²×C₇ ▷ C₇

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
1. C₂×#1161	C₂	semidirect
2. C₂×D₄×D₇	C₂	semidirect
3. C₇×#124	C₂	semidirect
4. #6510	C₂	semidirect
5. #6503	C₂	semidirect
6. #6447	C₂	semidirect
7. #6445	C₂	semidirect
8. #6437	C₂	semidirect
9. #6435	C₂	semidirect
11. C₂×C₇×Q₈	C₂²	semidirect
14. C₂×C₂²⋊D₇	C₂²	semidirect
19. C₂×D₂₈	C₂²	semidirect
20. C₂×D₂₈	C₂²	semidirect
22. C₂×C₄×D₇	C₂²	semidirect
23. C₇×C₂²⋊C₄	C₂²	semidirect
24. C₄²×C₇	C₂²	semidirect
26. (C₂×D₇)⋊C₄	C₂²	semidirect
27. Dic₇⋊C₄	C₂²	semidirect
28. C₄×Dic₇	C₂²	semidirect
50. C₄×D₇	D₄	semidirect
56. #124	D₇	semidirect
77. C₂×Q₈	C₂×D₇	semidirect
91. C₂²⋊C₄	C₂×D₇	semidirect
93. C₄²	C₂×D₇	semidirect
121. C₇	#535	semidirect

H ≅ G/N

type

1. C₂×#1161

C₂

semidirect

2. C₂×D₄×D₇

C₂

semidirect

3. C₇×#124

C₂

semidirect

4. #6510

C₂

semidirect

5. #6503

C₂

semidirect

6. #6447

C₂

semidirect

7. #6445

C₂

semidirect

8. #6437

C₂

semidirect

9. #6435

C₂

semidirect

11. C₂×C₇×Q₈

C₂²

semidirect

14. C₂×C₂²⋊D₇

C₂²

semidirect

19. C₂×D₂₈

C₂²

semidirect

20. C₂×D₂₈

C₂²

semidirect

22. C₂×C₄×D₇

C₂²

semidirect

23. C₇×C₂²⋊C₄

C₂²

semidirect

24. C₄²×C₇

C₂²

semidirect

26. (C₂×D₇)⋊C₄

C₂²

semidirect

27. Dic₇⋊C₄

C₂²

semidirect

28. C₄×Dic₇

C₂²

semidirect

50. C₄×D₇

D₄

semidirect

56. #124

D₇

semidirect

77. C₂×Q₈

C₂×D₇

semidirect

91. C₂²⋊C₄

C₂×D₇

semidirect

93. C₄²

C₂×D₇

semidirect

121. C₇

#535

semidirect

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

subgroup	index
C₂×#1161	2
C₂×D₄×D₇	2
C₇×#124	2
#6510	2
#6503	2
#6447	2
#6445	2
#6437	2
#6435	2
#535	7

subgroup

index

C₂×D₄×D₇

Subgroups — #90365

Conjugacy classes of subgroups

Complements

Maximal subgroups