Group #513161 — subgroups

Subgroups — #513161

Order 512.

303 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 303 conjugacy classes of subgroups — too many to draw. Only its 110 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	#33741	C₂	1	2	yes	maximal
2	#13708	C₂	1	4	yes	maximal, complemented
3	#13698	C₂	1	2	yes	maximal
4	#13695	C₂	1	2	yes	maximal
5	#13688	C₂	1	4	yes	maximal, complemented
6	#13628	C₂	1	1	yes	maximal, characteristic
7	#3177	—	2	2	no	—
8	C₂×Q₆₄	C₂²	1	2	yes	—
9	C₂×Q₆₄	C₂²	1	2	yes	—
10	C₂×M₆₄	C₂²	1	1	yes	characteristic
11	C₂²×C₃₂	C₂²	1	1	yes	characteristic
12	#2253	C₂²	1	2	yes	—
13	#2247	C₂²	1	2	yes	—
14	#2245	—	2	2	no	—
15	#2219	—	2	2	no	—
16	#2216	C₂²	1	4	yes	—
17	#2176	—	2	2	no	—
18	C₄×Q₃₂	C₂²	1	4	yes	—
19	#2108	C₂²	1	1	yes	characteristic
20	#1424	C₂²	1	2	yes	—
21	#1423	C₂²	1	2	yes	—
22	C₄⋊C₃₂	C₂²	1	4	yes	—
23	#1416	C₂²	1	4	yes	—
24	#1416	C₂²	1	4	yes	—
25	#541	—	4	2	no	—
26	C₂×Q₃₂	—	2	2	no	—
27	C₂×Q₃₂	C₂³	1	4	yes	—
28	C₂×M₃₂	D₄	1	1	yes	characteristic
29	C₂²×C₁₆	C₂³	1	1	yes	characteristic
30	#499	—	2	2	no	—
31	#494	—	2	2	no	—
32	#491	—	4	2	no	—
33	#470	—	4	2	no	—
34	#467	—	2	4	no	—
35	#441	—	4	2	no	—
36	C₄×Q₁₆	—	2	4	no	—
37	#405	D₄	1	1	yes	characteristic
38	Q₆₄	—	4	2	no	—
39	Q₆₄	—	4	2	no	—
40	M₆₄	—	4	1	no	—
41	C₂×C₃₂	—	4	1	no	—
42	C₂×C₃₂	C₂³	1	2	yes	—
43	C₂×C₃₂	C₂³	1	2	yes	—
44	#367	—	2	2	no	—
45	#366	C₂³	1	4	yes	—
46	#358	—	2	4	no	—
47	#358	—	2	4	no	—
48	#346	D₄	1	2	yes	—
49	C₄×C₁₆	C₂³	1	2	yes	—
50	C₂×Q₁₆	—	2	4	no	—
51	C₂×Q₁₆	—	4	2	no	—
52	C₂×M₁₆	D₈	1	1	yes	characteristic
53	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
54	C₄⋊Q₈	—	4	2	no	—
55	#126	—	8	4	no	—
56	#125	—	4	2	no	—
57	#125	—	8	4	no	—
58	C₂²⋊Q₈	—	4	4	no	—
59	C₄×Q₈	—	4	4	no	—
60	#117	D₈	1	1	yes	characteristic
61	Q₃₂	—	2	4	no	—
62	Q₃₂	—	2	4	no	—
63	Q₃₂	—	4	4	no	—
64	M₃₂	—	4	1	no	—
65	C₂×C₁₆	—	2	1	no	—
66	C₂×C₁₆	—	2	1	no	—
67	C₂×C₁₆	C₂×D₄	1	2	yes	—
68	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
69	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
70	#107	—	2	4	no	—
71	#106	—	4	2	no	—
72	#103	—	4	4	no	—
73	#103	—	4	4	no	—
74	#97	D₈	1	2	yes	—
75	C₄×C₈	C₂×D₄	1	2	yes	—
76	C₃₂	—	2	2	no	—
77	C₃₂	—	2	2	no	—
78	C₂×Q₈	—	4	4	no	—
79	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
80	Q₁₆	—	4	4	no	—
81	Q₁₆	—	4	4	no	—
82	Q₁₆	—	8	4	no	—
83	M₁₆	—	4	1	no	—
84	C₂×C₈	—	2	1	no	—
85	C₂×C₈	—	2	1	no	—
86	C₂×C₈	C₂×D₈	1	2	yes	—
87	C₂×C₈	C₂×D₈	1	1	yes	characteristic
88	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
89	C₄⋊C₄	—	4	4	no	—
90	C₄⋊C₄	—	8	4	no	—
91	C₄⋊C₄	—	8	4	no	—
92	C₄⋊C₄	—	8	4	no	—
93	C₄⋊C₄	—	8	4	no	—
94	C₄⋊C₄	D₁₆	1	2	yes	—
95	C₂²⋊C₄	—	8	4	no	—
96	C₂²⋊C₄	D₁₆	1	2	yes	—
97	C₄²	—	8	4	no	—
98	C₄²	C₂×D₈	1	2	yes	—
99	C₁₆	—	2	1	no	—
100	C₁₆	—	2	2	no	—
101	C₁₆	H(5,𝔽₂)	1	1	yes	characteristic
102	C₁₆	2₋¹⁺⁴	1	1	yes	characteristic
103	C₂³	C₂×D₁₆	1	1	yes	characteristic
104	Q₈	—	8	4	no	—
105	Q₈	—	8	4	no	—
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	8	4	no	—
109	C₂×C₄	—	8	4	no	—
110	C₂×C₄	C₂×D₁₆	1	4	yes	—
111	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
112	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	—	2	1	no	—
115	C₈	#537	1	1	yes	characteristic
116	C₈	#575	1	1	yes	characteristic
117	C₂²	—	2	1	no	—
118	C₂²	—	2	2	no	—
119	C₂²	C₂²×D₁₆	1	1	yes	characteristic
120	C₄	—	2	1	no	—
121	C₄	—	2	4	no	—
122	C₄	—	16	4	no	—
123	C₄	—	16	4	no	—
124	C₄	#3245	1	1	yes	characteristic
125	C₄	#3414	1	1	yes	characteristic
126	C₂	—	4	1	no	—
127	C₂	#33811	1	1	yes	characteristic
128	C₂	#33984	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#33741

C₂

yes

maximal

#13708

C₂

yes

maximal, complemented

#13698

C₂

yes

maximal

#13695

C₂

yes

maximal

#13688

C₂

yes

maximal, complemented

#13628

C₂

yes

maximal, characteristic

#3177

—

C₂×Q₆₄

C₂²

yes

—

C₂×Q₆₄

C₂²

yes

—

C₂×M₆₄

C₂²

yes

characteristic

C₂²×C₃₂

C₂²

yes

characteristic

#2253

C₂²

yes

—

#2247

C₂²

yes

—

#2245

—

#2219

—

#2216

C₂²

yes

—

#2176

—

C₄×Q₃₂

C₂²

yes

—

#2108

C₂²

yes

characteristic

#1424

C₂²

yes

—

#1423

C₂²

yes

—

C₄⋊C₃₂

C₂²

yes

—

#1416

C₂²

yes

—

#1416

C₂²

yes

—

#541

—

C₂×Q₃₂

—

C₂×Q₃₂

C₂³

yes

—

C₂×M₃₂

D₄

yes

characteristic

C₂²×C₁₆

C₂³

yes

—

—

—

—

—

—

C₄×Q₁₆

—

#405

D₄

yes

characteristic

Q₆₄

—

Q₆₄

—

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

C₂³

yes

—

C₂×C₃₂

C₂³

yes

—

#367

—

#366

C₂³

yes

—

#358

—

#358

—

#346

D₄

yes

—

C₄×C₁₆

C₂³

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

D₈

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₄⋊Q₈

—

#126

—

#125

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

D₈

yes

characteristic

Q₃₂

—

Q₃₂

—

Q₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₄

yes

—

C₂×C₁₆

C₂×D₄

yes

characteristic

C₂×C₁₆

C₂⁴

yes

—

—

—

—

D₈

yes

—

C₄×C₈

C₂×D₄

yes

—

C₃₂

—

C₃₂

—

C₂×Q₈

—

C₂²×C₄

C₂×D₈

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×D₈

yes

—

C₂×C₈

C₂×D₈

yes

characteristic

C₂×C₈

C₂²×D₄

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₁₆

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

D₁₆

yes

—

C₄²

—

C₄²

C₂×D₈

yes

—

C₁₆

—

100

C₁₆

—

101

C₁₆

H(5,𝔽₂)

yes

characteristic

102

C₁₆

2₋¹⁺⁴

yes

characteristic

103

C₂³

C₂×D₁₆

yes

characteristic

104

Q₈

—

105

Q₈

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

C₂×D₁₆

yes

—

111

C₂×C₄

C₂×D₁₆

yes

characteristic

112

C₂×C₄

C₂²×D₈

yes

characteristic

113

C₈

—

114

C₈

—

115

C₈

#537

yes

characteristic

116

C₈

#575

yes

characteristic

117

C₂²

—

118

C₂²

—

119

C₂²

C₂²×D₁₆

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

#3245

yes

characteristic

125

C₄

#3414

yes

characteristic

126

C₂

—

127

C₂

#33811

yes

characteristic

128

C₂

#33984

yes

—

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	C₂³	C₂×D₁₆
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108 ◁ 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

C₂³

C₂×D₁₆

LCS

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108 ◁ G

LECS

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. #13708	C₂	semidirect
5. #13688	C₂	semidirect

H ≅ G/N

type

2. #13708

C₂

semidirect

5. #13688

C₂

semidirect

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

subgroup	index
#33741	2
#13708	2
#13698	2
#13695	2
#13688	2
#13628	2

subgroup

index

Subgroups — #513161

Conjugacy classes of subgroups

Complements

Maximal subgroups