Group #513235 — subgroups

Subgroups — #513235

Order 512.

338 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 122 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	#33901	C₂	1	2	yes	maximal, complemented
2	#13731	C₂	1	1	yes	maximal, characteristic, complemented
3	#13711	C₂	1	4	yes	maximal, complemented
4	#13703	C₂	1	2	yes	maximal, complemented
5	#13694	C₂	1	2	yes	maximal, complemented
6	C₄×D₃₂	C₂	1	1	yes	maximal, characteristic, complemented
7	D₄×C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#1423	C₂	1	2	yes	maximal, complemented
9	#3334	—	2	2	no	—
10	C₂×D₃₂	C₂²	1	1	yes	characteristic
11	C₂²×C₃₂	C₂²	1	2	yes	—
12	#2252	C₂²	1	2	yes	—
13	#2232	—	2	4	no	—
14	#2224	—	2	2	no	—
15	#2215	C₂²	1	4	yes	—
16	C₄×D₁₆	C₂²	1	2	yes	—
17	D₄×C₁₆	C₂²	1	1	yes	characteristic
18	C₂×#366	C₂²	1	4	yes	—
19	#1423	C₂²	1	1	yes	characteristic
20	#1423	C₂²	1	2	yes	—
21	#1423	C₂²	1	4	yes	—
22	#1423	C₂²	1	2	yes	—
23	C₄⋊C₃₂	C₂²	1	1	yes	characteristic
24	#1415	C₂²	1	2	yes	—
25	#1415	C₂²	1	4	yes	—
26	C₂²⋊C₃₂	C₂²	1	2	yes	—
27	C₄×C₃₂	C₂²	1	1	yes	characteristic
28	#547	—	4	2	no	—
29	C₂×D₁₆	C₂³	1	2	yes	—
30	C₂²×C₁₆	C₂³	1	2	yes	—
31	#500	—	2	2	no	—
32	#480	—	4	4	no	—
33	#474	—	4	2	no	—
34	#466	—	2	4	no	—
35	C₄×D₈	—	2	2	no	—
36	C₈×D₄	D₄	1	1	yes	characteristic
37	C₂×#107	—	2	4	no	—
38	D₃₂	—	2	2	no	—
39	C₂×C₃₂	—	2	4	no	—
40	C₂×C₃₂	C₂³	1	1	yes	characteristic
41	C₂×C₃₂	C₂³	1	2	yes	—
42	C₂×C₃₂	C₂³	1	1	yes	characteristic
43	#366	—	2	4	no	—
44	#366	—	2	2	no	—
45	#366	C₂³	1	2	yes	—
46	#366	C₂³	1	4	yes	—
47	C₄⋊C₁₆	D₄	1	1	yes	characteristic
48	#357	—	2	2	no	—
49	#357	—	2	4	no	—
50	C₂²⋊C₁₆	D₄	1	2	yes	—
51	C₄×C₁₆	C₂³	1	1	yes	characteristic
52	C₂×#41	—	8	2	no	—
53	C₂×D₈	—	2	2	no	—
54	C₂²×C₈	C₂×D₄	1	2	yes	—
55	C₄⋊Q₈	—	4	2	no	—
56	#123	—	8	4	no	—
57	C₂²⋊Q₈	—	8	2	no	—
58	#121	—	4	4	no	—
59	C₄×D₄	—	4	2	no	—
60	C₄×D₄	D₈	1	1	yes	characteristic
61	C₂×C₄⋊C₄	—	4	4	no	—
62	D₁₆	—	2	2	no	—
63	D₁₆	—	2	2	no	—
64	C₂×C₁₆	—	2	2	no	—
65	C₂×C₁₆	—	2	2	no	—
66	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
67	C₂×C₁₆	C₂×D₄	1	2	yes	—
68	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
69	#107	—	2	2	no	—
70	#107	—	2	4	no	—
71	#107	—	4	4	no	—
72	#107	—	4	2	no	—
73	C₄⋊C₈	D₈	1	1	yes	characteristic
74	#102	—	4	2	no	—
75	#102	—	4	4	no	—
76	C₂²⋊C₈	D₈	1	2	yes	—
77	C₄×C₈	C₂×D₄	1	1	yes	characteristic
78	C₃₂	—	2	2	no	—
79	C₃₂	—	2	1	no	—
80	C₃₂	#41	1	2	yes	complemented
81	#41	—	16	8	no	—
82	C₂×Q₈	—	8	2	no	—
83	C₂×D₄	—	4	2	no	—
84	C₂×D₄	—	8	4	no	—
85	C₂×D₄	D₁₆	1	1	yes	characteristic
86	C₂²×C₄	—	8	4	no	—
87	C₂²×C₄	—	8	2	no	—
88	C₂²×C₄	C₂×D₈	1	2	yes	—
89	D₈	—	4	2	no	—
90	D₈	—	4	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	C₂×D₈	1	1	yes	characteristic
94	C₂×C₈	C₂×D₈	1	2	yes	—
95	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
96	C₄⋊C₄	—	4	2	no	—
97	C₄⋊C₄	—	4	4	no	—
98	C₄⋊C₄	—	8	4	no	—
99	C₄⋊C₄	—	8	2	no	—
100	C₄⋊C₄	D₁₆	1	1	yes	characteristic
101	C₂²⋊C₄	—	8	2	no	—
102	C₂²⋊C₄	—	8	4	no	—
103	C₂²⋊C₄	D₁₆	1	2	yes	—
104	C₄²	C₂×D₈	1	1	yes	characteristic
105	C₁₆	—	2	2	no	—
106	C₁₆	—	2	1	no	—
107	C₁₆	C₂×#41	1	1	yes	characteristic
108	C₁₆	2₋¹⁺⁴	1	1	yes	characteristic
109	C₂³	—	8	2	no	—
110	C₂³	C₂×D₁₆	1	2	yes	—
111	Q₈	—	16	4	no	—
112	D₄	—	8	2	no	—
113	D₄	—	8	2	no	—
114	D₄	—	16	8	no	—
115	D₄	D₃₂	1	4	yes	complemented
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	8	2	no	—
119	C₂×C₄	—	8	4	no	—
120	C₂×C₄	—	16	8	no	—
121	C₂×C₄	—	16	4	no	—
122	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
123	C₂×C₄	C₂×D₁₆	1	2	yes	—
124	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
125	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
126	C₈	—	2	2	no	—
127	C₈	—	2	1	no	—
128	C₈	#547	1	1	yes	characteristic
129	C₈	#578	1	1	yes	characteristic
130	C₂²	—	2	2	no	—
131	C₂²	—	2	2	no	—
132	C₂²	—	16	2	no	—
133	C₂²	—	16	2	no	—
134	C₂²	—	16	2	no	—
135	C₂²	C₂×D₃₂	1	4	yes	—
136	C₂²	C₂²×D₁₆	1	1	yes	characteristic
137	C₄	—	2	2	no	—
138	C₄	—	2	1	no	—
139	C₄	—	16	2	no	—
140	C₄	—	16	4	no	—
141	C₄	C₂×D₃₂	1	2	yes	—
142	C₄	#3334	1	1	yes	characteristic
143	C₄	#3417	1	1	yes	characteristic
144	C₂	—	2	4	no	—
145	C₂	—	32	2	no	—
146	C₂	#33901	1	1	yes	characteristic
147	C₂	C₂²×D₃₂	1	1	yes	characteristic
148	C₂	#33984	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#33901

C₂

yes

maximal, complemented

#13731

C₂

yes

maximal, characteristic, complemented

#13711

C₂

yes

maximal, complemented

#13703

C₂

yes

maximal, complemented

#13694

C₂

yes

maximal, complemented

C₄×D₃₂

C₂

yes

maximal, characteristic, complemented

D₄×C₃₂

C₂

yes

maximal, characteristic, complemented

C₂×#1423

C₂

yes

maximal, complemented

#3334

—

C₂×D₃₂

C₂²

yes

characteristic

C₂²×C₃₂

C₂²

yes

—

#2252

C₂²

yes

—

#2232

—

#2224

—

#2215

C₂²

yes

—

C₄×D₁₆

C₂²

yes

—

D₄×C₁₆

C₂²

yes

characteristic

C₂×#366

C₂²

yes

—

#1423

C₂²

yes

characteristic

#1423

C₂²

yes

—

#1423

C₂²

yes

—

#1423

C₂²

yes

—

C₄⋊C₃₂

C₂²

yes

characteristic

#1415

C₂²

yes

—

#1415

C₂²

yes

—

C₂²⋊C₃₂

C₂²

yes

—

C₄×C₃₂

C₂²

yes

characteristic

#547

—

C₂×D₁₆

C₂³

yes

—

C₂²×C₁₆

C₂³

yes

—

—

—

—

—

C₄×D₈

—

C₈×D₄

D₄

yes

characteristic

C₂×#107

—

D₃₂

—

C₂×C₃₂

—

C₂×C₃₂

C₂³

yes

characteristic

C₂×C₃₂

C₂³

yes

—

C₂×C₃₂

C₂³

yes

—

—

C₂³

yes

—

#366

C₂³

yes

—

C₄⋊C₁₆

D₄

yes

characteristic

#357

—

#357

—

C₂²⋊C₁₆

D₄

yes

—

C₄×C₁₆

C₂³

yes

characteristic

C₂×#41

—

C₂×D₈

—

C₂²×C₈

C₂×D₄

yes

—

C₄⋊Q₈

—

#123

—

C₂²⋊Q₈

—

#121

—

C₄×D₄

—

C₄×D₄

D₈

yes

characteristic

C₂×C₄⋊C₄

—

D₁₆

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₄

yes

characteristic

C₂×C₁₆

C₂×D₄

yes

—

C₂×C₁₆

C₂⁴

yes

—

—

—

—

C₄⋊C₈

D₈

yes

characteristic

#102

—

#102

—

C₂²⋊C₈

D₈

yes

—

C₄×C₈

C₂×D₄

yes

characteristic

C₃₂

—

C₃₂

—

C₃₂

#41

yes

complemented

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

D₁₆

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₈

yes

—

D₈

—

D₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×D₈

yes

characteristic

C₂×C₈

C₂×D₈

yes

—

C₂×C₈

C₂²×D₄

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

D₁₆

yes

characteristic

101

C₂²⋊C₄

—

102

C₂²⋊C₄

—

103

C₂²⋊C₄

D₁₆

yes

—

104

C₄²

C₂×D₈

yes

characteristic

105

C₁₆

—

106

C₁₆

—

107

C₁₆

C₂×#41

yes

characteristic

108

C₁₆

2₋¹⁺⁴

yes

characteristic

109

C₂³

—

110

C₂³

C₂×D₁₆

yes

—

111

Q₈

—

112

D₄

—

113

D₄

—

114

D₄

—

115

D₄

D₃₂

yes

complemented

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

C₂×D₁₆

yes

characteristic

123

C₂×C₄

C₂×D₁₆

yes

—

124

C₂×C₄

C₂×D₁₆

yes

characteristic

125

C₂×C₄

C₂²×D₈

yes

characteristic

126

C₈

—

127

C₈

—

128

C₈

#547

yes

characteristic

129

C₈

#578

yes

characteristic

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

C₂×D₃₂

yes

—

136

C₂²

C₂²×D₁₆

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

C₂×D₃₂

yes

—

142

C₄

#3334

yes

characteristic

143

C₄

#3417

yes

characteristic

144

C₂

—

145

C₂

—

146

C₂

#33901

yes

characteristic

147

C₂

C₂²×D₃₂

yes

characteristic

148

C₂

#33984

yes

characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂²×D₁₆

commutator, Frattini, cosocle

C₂×C₁₆

C₂⁴

LCS

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×C₁₆ ◁ G

LECS

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
1. #33901	C₂	semidirect
2. #13731	C₂	semidirect
3. #13711	C₂	semidirect
4. #13703	C₂	semidirect
5. #13694	C₂	semidirect
6. C₄×D₃₂	C₂	semidirect
7. D₄×C₃₂	C₂	semidirect
8. C₂×#1423	C₂	semidirect
80. C₃₂	#41	semidirect
115. D₄	D₃₂	semidirect

H ≅ G/N

type

1. #33901

C₂

semidirect

2. #13731

C₂

semidirect

3. #13711

C₂

semidirect

4. #13703

C₂

semidirect

5. #13694

C₂

semidirect

6. C₄×D₃₂

C₂

semidirect

7. D₄×C₃₂

C₂

semidirect

8. C₂×#1423

C₂

semidirect

80. C₃₂

#41

semidirect

115. D₄

D₃₂

semidirect

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

subgroup	index
#33901	2
#13731	2
#13711	2
#13703	2
#13694	2
C₄×D₃₂	2
D₄×C₃₂	2
C₂×#1423	2

subgroup

index

C₄×D₃₂

D₄×C₃₂

C₂×#1423

Subgroups — #513235

Conjugacy classes of subgroups

Complements

Maximal subgroups