Group #90459 — subgroups

Subgroups — #90459

Order 448.

262 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 99 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	D₇×#41	C₂	1	2	yes	maximal, complemented
2	C₇×#135	C₂	1	1	yes	maximal, characteristic, complemented
3	#6513	C₂	1	2	yes	maximal, complemented
4	#6481	C₂	1	2	yes	maximal, complemented
5	#6478	C₂	1	2	yes	maximal, complemented
6	#6477	C₂	1	2	yes	maximal, complemented
7	#6474	C₂	1	2	yes	maximal, complemented
8	#6467	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#1131	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#41	C₂²	1	2	yes	complemented
11	#1161	C₂²	1	2	yes	—
12	Q₈×D₇	C₂²	1	2	yes	complemented
13	#1159	—	2	2	no	—
14	#1159	C₂²	1	2	yes	—
15	D₄×D₇	—	2	2	no	—
16	D₄×D₇	C₂²	1	2	yes	complemented
17	#1157	—	2	2	no	—
18	#1157	C₂²	1	2	yes	complemented
19	C₂×C₄×D₇	—	2	2	no	—
20	C₂×C₄×D₇	C₂²	1	1	yes	characteristic
21	C₇×Q₁₆	C₂²	1	1	yes	characteristic, complemented
22	C₇×SD₁₆	C₂²	1	2	yes	complemented
23	C₇×D₈	C₂²	1	1	yes	characteristic, complemented
24	C₂×C₇×C₈	C₂²	1	1	yes	characteristic, complemented
25	#1144	C₂²	1	2	yes	complemented
26	#1143	C₂²	1	2	yes	complemented
27	#1142	C₂²	1	2	yes	complemented
28	#1141	C₂²	1	2	yes	complemented
29	C₂×C₇⋊C₈	C₂²	1	1	yes	characteristic, complemented
30	Dic₂₈	C₂²	1	1	yes	characteristic, complemented
31	D₅₆	C₂²	1	1	yes	characteristic, complemented
32	#1132	C₂²	1	2	yes	complemented
33	#1131	C₂²	1	2	yes	complemented
34	#1131	C₂²	1	2	yes	complemented
35	#575	—	7	1	no	maximal, Sylow
36	C₂²×D₇	—	2	2	no	—
37	C₂²×D₇	D₄	1	1	yes	characteristic
38	C₇×Q₈	C₂³	1	2	yes	—
39	C₇×D₄	—	2	2	no	—
40	C₇×D₄	C₂³	1	2	yes	—
41	C₂×C₄×C₇	—	2	2	no	—
42	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
43	C₂²⋊D₇	—	2	2	no	—
44	C₂²⋊D₇	—	2	2	no	—
45	C₂²⋊D₇	—	2	2	no	—
46	C₂×Dic₇	—	2	2	no	—
47	C₂×Dic₇	D₄	1	1	yes	characteristic
48	D₂₈	—	2	2	no	—
49	D₂₈	C₂³	1	2	yes	—
50	C₄×D₇	—	2	2	no	—
51	C₄×D₇	—	2	2	no	—
52	C₄×D₇	—	2	2	no	—
53	C₄×D₇	D₄	1	2	yes	—
54	C₄×D₇	C₂³	1	2	yes	—
55	Dic₁₄	—	2	2	no	—
56	Dic₁₄	C₂³	1	2	yes	—
57	C₇×C₈	C₂³	1	1	yes	characteristic
58	C₇×C₈	C₂³	1	1	yes	characteristic
59	C₇⋊C₈	C₂³	1	2	yes	—
60	C₂×#41	—	7	2	no	—
61	#137	—	7	2	no	—
62	#137	—	7	2	no	—
63	AGL(1,ℤ/8)	—	7	2	no	—
64	AGL(1,ℤ/8)	—	7	2	no	—
65	#135	—	7	2	no	—
66	#135	—	7	1	no	—
67	#135	D₇	1	1	yes	characteristic, complemented
68	C₂×M₁₆	—	7	1	no	—
69	C₂²×C₇	—	2	2	no	—
70	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
71	C₂×D₇	—	2	2	no	—
72	C₂×D₇	—	4	1	no	—
73	C₂×D₇	—	4	4	no	—
74	C₂×D₇	C₂×D₄	1	2	yes	—
75	C₄×C₇	—	2	2	no	—
76	C₄×C₇	C₂×D₄	1	1	yes	characteristic
77	C₄×C₇	C₂⁴	1	1	yes	characteristic
78	Dic₇	—	2	2	no	—
79	Dic₇	C₂×D₄	1	2	yes	—
80	#41	—	7	2	no	—
81	#41	—	7	2	no	—
82	#41	—	7	2	no	—
83	#41	—	14	2	no	—
84	#41	—	14	2	no	—
85	#41	C₂×D₇	1	2	yes	complemented
86	C₂×Q₈	—	7	2	no	—
87	C₂×D₄	—	7	2	no	—
88	C₂×D₄	—	14	2	no	—
89	C₂²×C₄	—	7	1	no	—
90	C₂²×C₄	—	14	2	no	—
91	Q₁₆	—	7	2	no	—
92	Q₁₆	—	7	1	no	—
93	Q₁₆	C₂×D₇	1	1	yes	characteristic, complemented
94	SD₁₆	—	7	2	no	—
95	SD₁₆	—	7	2	no	—
96	SD₁₆	—	7	2	no	—
97	SD₁₆	C₂×D₇	1	2	yes	complemented
98	D₈	—	7	2	no	—
99	D₈	—	7	1	no	—
100	D₈	C₂×D₇	1	1	yes	characteristic, complemented
101	M₁₆	—	7	2	no	—
102	M₁₆	—	7	2	no	—
103	C₂×C₈	—	7	1	no	—
104	C₂×C₈	C₂×D₇	1	1	yes	characteristic, complemented
105	C₂×C₇	—	2	1	no	—
106	C₂×C₇	—	4	2	no	—
107	C₂×C₇	C₂²×D₄	1	1	yes	characteristic
108	D₇	—	2	2	no	—
109	D₇	—	4	2	no	—
110	C₂³	—	7	1	no	—
111	C₂³	—	14	2	no	—
112	Q₈	—	7	2	no	—
113	Q₈	—	14	2	no	—
114	Q₈	C₂²×D₇	1	2	yes	—
115	D₄	—	2	2	no	—
116	D₄	—	7	2	no	—
117	D₄	—	14	2	no	—
118	D₄	—	14	2	no	—
119	D₄	—	14	2	no	—
120	D₄	—	14	2	no	—
121	D₄	C₂²×D₇	1	2	yes	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	7	1	no	—
124	C₂×C₄	—	7	2	no	—
125	C₂×C₄	—	7	2	no	—
126	C₂×C₄	—	14	2	no	—
127	C₂×C₄	—	14	2	no	—
128	C₂×C₄	—	14	2	no	—
129	C₂×C₄	—	14	2	no	—
130	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
131	C₈	—	7	2	no	—
132	C₈	C₂²×D₇	1	1	yes	characteristic
133	C₈	C₂²×D₇	1	1	yes	characteristic
134	C₇	#575	1	1	yes	characteristic, complemented, Sylow
135	C₂²	—	2	2	no	—
136	C₂²	—	7	2	no	—
137	C₂²	—	14	2	no	—
138	C₂²	—	28	1	no	—
139	C₂²	—	28	4	no	—
140	C₂²	D₄×D₇	1	1	yes	characteristic
141	C₄	—	2	2	no	—
142	C₄	—	7	2	no	—
143	C₄	—	14	2	no	—
144	C₄	D₄×D₇	1	1	yes	characteristic
145	C₄	C₂³×D₇	1	1	yes	characteristic
146	C₂	—	2	1	no	—
147	C₂	—	4	2	no	—
148	C₂	—	14	2	no	—
149	C₂	—	28	2	no	—
150	C₂	C₂×D₄×D₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₇×#41

C₂

yes

maximal, complemented

C₇×#135

C₂

yes

maximal, characteristic, complemented

#6513

C₂

yes

maximal, complemented

#6481

C₂

yes

maximal, complemented

#6478

C₂

yes

maximal, complemented

#6477

C₂

yes

maximal, complemented

#6474

C₂

yes

maximal, complemented

#6467

C₂

yes

maximal, characteristic, complemented

C₂×#1131

C₂

yes

maximal, characteristic, complemented

C₇×#41

C₂²

yes

complemented

#1161

C₂²

yes

—

Q₈×D₇

C₂²

yes

complemented

#1159

—

#1159

C₂²

yes

—

D₄×D₇

—

D₄×D₇

C₂²

yes

complemented

#1157

—

#1157

C₂²

yes

complemented

C₂×C₄×D₇

—

C₂×C₄×D₇

C₂²

yes

characteristic

C₇×Q₁₆

C₂²

yes

characteristic, complemented

C₇×SD₁₆

C₂²

yes

complemented

C₇×D₈

C₂²

yes

characteristic, complemented

C₂×C₇×C₈

C₂²

yes

characteristic, complemented

#1144

C₂²

yes

complemented

#1143

C₂²

yes

complemented

#1142

C₂²

yes

complemented

#1141

C₂²

yes

complemented

C₂×C₇⋊C₈

C₂²

yes

characteristic, complemented

Dic₂₈

C₂²

yes

characteristic, complemented

D₅₆

C₂²

yes

characteristic, complemented

#1132

C₂²

yes

complemented

#1131

C₂²

yes

complemented

#1131

C₂²

yes

complemented

#575

—

maximal, Sylow

C₂²×D₇

—

C₂²×D₇

D₄

yes

characteristic

C₇×Q₈

C₂³

yes

—

C₇×D₄

—

C₇×D₄

C₂³

yes

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂³

yes

characteristic

C₂²⋊D₇

—

C₂²⋊D₇

—

C₂²⋊D₇

—

C₂×Dic₇

—

C₂×Dic₇

D₄

yes

characteristic

D₂₈

—

D₂₈

C₂³

yes

—

C₄×D₇

—

C₄×D₇

—

C₄×D₇

—

C₄×D₇

D₄

yes

—

C₄×D₇

C₂³

yes

—

Dic₁₄

—

Dic₁₄

C₂³

yes

—

C₇×C₈

C₂³

yes

characteristic

C₇×C₈

C₂³

yes

characteristic

C₇⋊C₈

C₂³

yes

—

C₂×#41

—

—

—

—

—

—

—

D₇

yes

characteristic, complemented

C₂×M₁₆

—

C₂²×C₇

—

C₂²×C₇

C₂×D₄

yes

characteristic

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₂×D₇

C₂×D₄

yes

—

C₄×C₇

—

C₄×C₇

C₂×D₄

yes

characteristic

C₄×C₇

C₂⁴

yes

characteristic

Dic₇

—

Dic₇

C₂×D₄

yes

—

—

—

—

—

—

C₂×D₇

yes

complemented

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

Q₁₆

—

Q₁₆

—

Q₁₆

C₂×D₇

yes

characteristic, complemented

SD₁₆

—

SD₁₆

—

SD₁₆

—

SD₁₆

C₂×D₇

yes

complemented

D₈

—

D₈

—

100

D₈

C₂×D₇

yes

characteristic, complemented

101

M₁₆

—

102

M₁₆

—

103

C₂×C₈

—

104

C₂×C₈

C₂×D₇

yes

characteristic, complemented

105

C₂×C₇

—

106

C₂×C₇

—

107

C₂×C₇

C₂²×D₄

yes

characteristic

108

D₇

—

109

D₇

—

110

C₂³

—

111

C₂³

—

112

Q₈

—

113

Q₈

—

114

Q₈

C₂²×D₇

yes

—

115

D₄

—

116

D₄

—

117

D₄

—

118

D₄

—

119

D₄

—

120

D₄

—

121

D₄

C₂²×D₇

yes

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

C₂²×D₇

yes

characteristic

131

C₈

—

132

C₈

C₂²×D₇

yes

characteristic

133

C₈

C₂²×D₇

yes

characteristic

134

C₇

#575

yes

characteristic, complemented, Sylow

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

D₄×D₇

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

D₄×D₇

yes

characteristic

145

C₄

C₂³×D₇

yes

characteristic

146

C₂

—

147

C₂

—

148

C₂

—

149

C₂

—

150

C₂

C₂×D₄×D₇

yes

characteristic

	subgroup	G/H
center	C₄	D₄×D₇
commutator, cosocle	C₄×C₇	C₂⁴
Frattini	C₄	C₂³×D₇
Fitting	C₇×#135	C₂
socle	C₂×C₇	C₂²×D₄
hypercenter	#135	D₇
hypocenter	C₇	#575
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #135

subgroup

G/H

center

C₄

D₄×D₇

commutator, cosocle

C₄×C₇

C₂⁴

Frattini

C₄

C₂³×D₇

Fitting

C₇×#135

C₂

socle

C₂×C₇

C₂²×D₄

D₇

C₇

G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ #135

N	H ≅ G/N	type
1. D₇×#41	C₂	semidirect
2. C₇×#135	C₂	semidirect
3. #6513	C₂	semidirect
4. #6481	C₂	semidirect
5. #6478	C₂	semidirect
6. #6477	C₂	semidirect
7. #6474	C₂	semidirect
8. #6467	C₂	semidirect
9. C₂×#1131	C₂	semidirect
10. C₇×#41	C₂²	semidirect
12. Q₈×D₇	C₂²	semidirect
16. D₄×D₇	C₂²	semidirect
18. #1157	C₂²	semidirect
21. C₇×Q₁₆	C₂²	semidirect
22. C₇×SD₁₆	C₂²	semidirect
23. C₇×D₈	C₂²	semidirect
24. C₂×C₇×C₈	C₂²	semidirect
25. #1144	C₂²	semidirect
26. #1143	C₂²	semidirect
27. #1142	C₂²	semidirect
28. #1141	C₂²	semidirect
29. C₂×C₇⋊C₈	C₂²	semidirect
30. Dic₂₈	C₂²	semidirect
31. D₅₆	C₂²	semidirect
32. #1132	C₂²	semidirect
33. #1131	C₂²	semidirect
34. #1131	C₂²	semidirect
67. #135	D₇	semidirect
85. #41	C₂×D₇	semidirect
93. Q₁₆	C₂×D₇	semidirect
97. SD₁₆	C₂×D₇	semidirect
100. D₈	C₂×D₇	semidirect
104. C₂×C₈	C₂×D₇	semidirect
134. C₇	#575	semidirect

H ≅ G/N

type

1. D₇×#41

C₂

semidirect

2. C₇×#135

C₂

semidirect

3. #6513

C₂

semidirect

4. #6481

C₂

semidirect

5. #6478

C₂

semidirect

6. #6477

C₂

semidirect

7. #6474

C₂

semidirect

8. #6467

C₂

semidirect

9. C₂×#1131

C₂

semidirect

10. C₇×#41

C₂²

semidirect

12. Q₈×D₇

C₂²

semidirect

16. D₄×D₇

C₂²

semidirect

18. #1157

C₂²

semidirect

21. C₇×Q₁₆

C₂²

semidirect

22. C₇×SD₁₆

C₂²

semidirect

23. C₇×D₈

C₂²

semidirect

24. C₂×C₇×C₈

C₂²

semidirect

25. #1144

C₂²

semidirect

26. #1143

C₂²

semidirect

27. #1142

C₂²

semidirect

28. #1141

C₂²

semidirect

29. C₂×C₇⋊C₈

C₂²

semidirect

30. Dic₂₈

C₂²

semidirect

31. D₅₆

C₂²

semidirect

32. #1132

C₂²

semidirect

33. #1131

C₂²

semidirect

34. #1131

C₂²

semidirect

67. #135

D₇

semidirect

85. #41

C₂×D₇

semidirect

93. Q₁₆

C₂×D₇

semidirect

97. SD₁₆

C₂×D₇

semidirect

100. D₈

C₂×D₇

semidirect

104. C₂×C₈

C₂×D₇

semidirect

134. C₇

#575

semidirect

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

subgroup	index
D₇×#41	2
C₇×#135	2
#6513	2
#6481	2
#6478	2
#6477	2
#6474	2
#6467	2
C₂×#1131	2
#575	7

subgroup

index

D₇×#41

Subgroups — #90459

Conjugacy classes of subgroups

Complements

Maximal subgroups