Group #513239 — subgroups

Subgroups — #513239

Order 512.

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

normal subgroups only

This group has 306 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	#33905	C₂	1	2	yes	maximal
2	#13731	C₂	1	1	yes	maximal, characteristic, complemented
3	#13715	C₂	1	4	yes	maximal, complemented
4	#13706	C₂	1	2	yes	maximal, complemented
5	#13695	C₂	1	2	yes	maximal, complemented
6	C₄×Q₆₄	C₂	1	1	yes	maximal, characteristic, complemented
7	D₄×C₃₂	C₂	1	1	yes	maximal, characteristic
8	C₂×#1423	C₂	1	2	yes	maximal, complemented
9	#3338	—	2	2	no	—
10	C₂×Q₆₄	C₂²	1	1	yes	characteristic
11	C₂²×C₃₂	C₂²	1	2	yes	—
12	#2252	C₂²	1	2	yes	—
13	#2236	—	2	4	no	—
14	#2227	—	2	2	no	—
15	#2216	C₂²	1	4	yes	—
16	C₄×Q₃₂	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	#1416	C₂²	1	2	yes	—
25	#1416	C₂²	1	4	yes	—
26	C₂²⋊C₃₂	C₂²	1	2	yes	—
27	C₄×C₃₂	C₂²	1	1	yes	characteristic
28	#554	—	4	2	no	—
29	C₂×Q₃₂	C₂³	1	2	yes	—
30	C₂²×C₁₆	C₂³	1	2	yes	—
31	#500	—	2	2	no	—
32	#484	—	4	4	no	—
33	#477	—	4	2	no	—
34	#467	—	2	4	no	—
35	C₄×Q₁₆	—	2	2	no	—
36	C₈×D₄	D₄	1	1	yes	characteristic
37	C₂×#107	—	2	4	no	—
38	Q₆₄	—	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	#358	—	2	2	no	—
49	#358	—	2	4	no	—
50	C₂²⋊C₁₆	D₄	1	2	yes	—
51	C₄×C₁₆	C₂³	1	1	yes	characteristic
52	C₂×Q₁₆	—	2	2	no	—
53	C₂²×C₈	C₂×D₄	1	2	yes	—
54	C₄⋊Q₈	—	4	2	no	—
55	#125	—	8	2	no	—
56	C₂²⋊Q₈	—	4	4	no	—
57	C₂²⋊Q₈	—	8	4	no	—
58	C₄×Q₈	—	4	2	no	—
59	C₄×D₄	—	8	2	no	—
60	C₄×D₄	D₈	1	1	yes	characteristic
61	C₂×C₄⋊C₄	—	4	4	no	—
62	Q₃₂	—	2	2	no	—
63	Q₃₂	—	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	#103	—	4	2	no	—
75	#103	—	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	—
81	C₂×Q₈	—	4	2	no	—
82	C₂×Q₈	—	8	2	no	—
83	C₂×D₄	D₁₆	1	1	yes	characteristic
84	C₂²×C₄	—	8	4	no	—
85	C₂²×C₄	C₂×D₈	1	2	yes	—
86	Q₁₆	—	4	2	no	—
87	Q₁₆	—	4	2	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	C₂×D₈	1	1	yes	characteristic
91	C₂×C₈	C₂×D₈	1	2	yes	—
92	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
93	C₄⋊C₄	—	4	2	no	—
94	C₄⋊C₄	—	4	4	no	—
95	C₄⋊C₄	—	8	4	no	—
96	C₄⋊C₄	—	8	2	no	—
97	C₄⋊C₄	—	8	2	no	—
98	C₄⋊C₄	—	8	4	no	—
99	C₄⋊C₄	D₁₆	1	1	yes	characteristic
100	C₂²⋊C₄	—	8	4	no	—
101	C₂²⋊C₄	D₁₆	1	2	yes	—
102	C₄²	—	8	2	no	—
103	C₄²	C₂×D₈	1	1	yes	characteristic
104	C₁₆	—	2	2	no	—
105	C₁₆	—	2	1	no	—
106	C₁₆	C₂×#41	1	1	yes	characteristic
107	C₁₆	2₋¹⁺⁴	1	1	yes	characteristic
108	C₂³	C₂×D₁₆	1	2	yes	—
109	Q₈	—	8	2	no	—
110	Q₈	—	8	2	no	—
111	Q₈	—	32	2	no	—
112	D₄	Q₆₄	1	4	yes	complemented
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	8	2	no	—
116	C₂×C₄	—	8	2	no	—
117	C₂×C₄	—	8	4	no	—
118	C₂×C₄	—	32	4	no	—
119	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
120	C₂×C₄	C₂×D₁₆	1	2	yes	—
121	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
122	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
123	C₈	—	2	2	no	—
124	C₈	—	2	1	no	—
125	C₈	#547	1	1	yes	characteristic
126	C₈	#578	1	1	yes	characteristic
127	C₂²	—	2	2	no	—
128	C₂²	—	2	2	no	—
129	C₂²	C₂×Q₆₄	1	4	yes	—
130	C₂²	C₂²×D₁₆	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	2	1	no	—
133	C₄	—	16	2	no	—
134	C₄	—	16	2	no	—
135	C₄	—	16	4	no	—
136	C₄	C₂×Q₆₄	1	2	yes	—
137	C₄	#3334	1	1	yes	characteristic
138	C₄	#3417	1	1	yes	characteristic
139	C₂	—	2	4	no	—
140	C₂	#33901	1	1	yes	characteristic
141	C₂	C₂²×Q₆₄	1	1	yes	characteristic
142	C₂	#33982	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#33905

C₂

yes

maximal

#13731

C₂

yes

maximal, characteristic, complemented

#13715

C₂

yes

maximal, complemented

#13706

C₂

yes

maximal, complemented

#13695

C₂

yes

maximal, complemented

C₄×Q₆₄

C₂

yes

maximal, characteristic, complemented

D₄×C₃₂

C₂

yes

maximal, characteristic

C₂×#1423

C₂

yes

maximal, complemented

#3338

—

C₂×Q₆₄

C₂²

yes

characteristic

C₂²×C₃₂

C₂²

yes

—

#2252

C₂²

yes

—

#2236

—

#2227

—

#2216

C₂²

yes

—

C₄×Q₃₂

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

#1416

C₂²

yes

—

#1416

C₂²

yes

—

C₂²⋊C₃₂

C₂²

yes

—

C₄×C₃₂

C₂²

yes

characteristic

#554

—

C₂×Q₃₂

C₂³

yes

—

C₂²×C₁₆

C₂³

yes

—

—

—

—

—

C₄×Q₁₆

—

C₈×D₄

D₄

yes

characteristic

C₂×#107

—

Q₆₄

—

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

#358

—

#358

—

C₂²⋊C₁₆

D₄

yes

—

C₄×C₁₆

C₂³

yes

characteristic

C₂×Q₁₆

—

C₂²×C₈

C₂×D₄

yes

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

D₈

yes

characteristic

C₂×C₄⋊C₄

—

Q₃₂

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₄

yes

characteristic

C₂×C₁₆

C₂×D₄

yes

—

C₂×C₁₆

C₂⁴

yes

—

—

—

—

C₄⋊C₈

D₈

yes

characteristic

#103

—

#103

—

C₂²⋊C₈

D₈

yes

—

C₄×C₈

C₂×D₄

yes

characteristic

C₃₂

—

C₃₂

—

C₃₂

#41

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

D₁₆

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂×D₈

yes

—

Q₁₆

—

Q₁₆

—

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

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₁₆

yes

characteristic

100

C₂²⋊C₄

—

101

C₂²⋊C₄

D₁₆

yes

—

102

C₄²

—

103

C₄²

C₂×D₈

yes

characteristic

104

C₁₆

—

105

C₁₆

—

106

C₁₆

C₂×#41

yes

characteristic

107

C₁₆

2₋¹⁺⁴

yes

characteristic

108

C₂³

C₂×D₁₆

yes

—

109

Q₈

—

110

Q₈

—

111

Q₈

—

112

D₄

Q₆₄

yes

complemented

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

C₂×D₁₆

yes

characteristic

120

C₂×C₄

C₂×D₁₆

yes

—

121

C₂×C₄

C₂×D₁₆

yes

characteristic

122

C₂×C₄

C₂²×D₈

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

#547

yes

characteristic

126

C₈

#578

yes

characteristic

127

C₂²

—

128

C₂²

—

129

C₂²

C₂×Q₆₄

yes

—

130

C₂²

C₂²×D₁₆

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

C₂×Q₆₄

yes

—

137

C₄

#3334

yes

characteristic

138

C₄

#3417

yes

characteristic

139

C₂

—

140

C₂

#33901

yes

characteristic

141

C₂

C₂²×Q₆₄

yes

characteristic

142

C₂

#33982

yes

characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	C₂×D₄	D₁₆
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

prime radical

C₂×D₄

D₁₆

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
2. #13731	C₂	semidirect
3. #13715	C₂	semidirect
4. #13706	C₂	semidirect
5. #13695	C₂	semidirect
6. C₄×Q₆₄	C₂	semidirect
8. C₂×#1423	C₂	semidirect
112. D₄	Q₆₄	semidirect

H ≅ G/N

type

2. #13731

C₂

semidirect

3. #13715

C₂

semidirect

4. #13706

C₂

semidirect

5. #13695

C₂

semidirect

6. C₄×Q₆₄

C₂

semidirect

8. C₂×#1423

C₂

semidirect

112. D₄

Q₆₄

semidirect

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

subgroup	index
#33905	2
#13731	2
#13715	2
#13706	2
#13695	2
C₄×Q₆₄	2
D₄×C₃₂	2
C₂×#1423	2

subgroup

index

C₄×Q₆₄

D₄×C₃₂

C₂×#1423

Subgroups — #513239

Conjugacy classes of subgroups

Complements

Maximal subgroups