Group #152303 — subgroups

Subgroups — #152303

Order 512.

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

normal subgroups only

This group has 308 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	C₂×#2108	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊C₃₂	C₂	1	2	yes	maximal
3	#7465	C₂	1	4	yes	maximal, complemented
4	C₂²×M₃₂	C₄	1	1	yes	characteristic
5	C₂³×C₁₆	C₂²	1	1	yes	characteristic
6	C₂×#405	C₄	1	1	yes	characteristic
7	C₂²×C₃₂	C₂²	1	4	yes	—
8	#2108	—	2	4	no	—
9	C₂×#346	C₄	1	2	yes	—
10	C₂×C₄×C₁₆	C₂²	1	2	yes	—
11	C₂²⋊C₃₂	—	2	4	no	—
12	C₂²×M₁₆	C₈	1	1	yes	characteristic
13	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
14	C₂×#117	C₈	1	1	yes	characteristic
15	C₂×M₃₂	—	2	2	no	—
16	C₂×M₃₂	—	2	4	no	—
17	C₂²×C₁₆	—	2	4	no	—
18	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₁₆	C₂×C₄	1	2	yes	—
20	C₂²×C₁₆	D₄	1	4	yes	—
21	C₂²×C₁₆	C₂³	1	1	yes	characteristic
22	#405	—	2	2	no	—
23	#405	—	2	2	no	—
24	C₂×#97	C₈	1	2	yes	—
25	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
26	C₂×C₃₂	—	2	4	no	—
27	C₂×C₃₂	—	2	8	no	—
28	#346	—	2	4	no	—
29	C₄×C₁₆	—	2	4	no	—
30	C₂³×C₄	C₂×C₈	1	1	yes	characteristic
31	C₂×M₁₆	—	2	2	no	—
32	C₂×M₁₆	—	2	4	no	—
33	C₂²×C₈	—	2	2	no	—
34	C₂²×C₈	—	2	2	no	—
35	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
36	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₈	C₂×C₈	1	2	yes	—
38	C₂²×C₈	M₁₆	1	2	yes	—
39	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
40	#117	—	2	2	no	—
41	#117	—	2	2	no	—
42	C₂×C₄⋊C₄	C₁₆	1	2	yes	—
43	C₂×C₂²⋊C₄	C₁₆	1	2	yes	—
44	C₂×C₄²	C₂×C₈	1	2	yes	—
45	M₃₂	—	2	8	no	—
46	C₂×C₁₆	—	2	2	no	—
47	C₂×C₁₆	—	2	2	no	—
48	C₂×C₁₆	—	2	2	no	—
49	C₂×C₁₆	—	2	8	no	—
50	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
51	C₂×C₁₆	M₁₆	1	4	yes	—
52	C₂×C₁₆	C₂×D₄	1	2	yes	—
53	C₂×C₁₆	#41	1	4	yes	—
54	#97	—	2	4	no	—
55	C₄×C₈	—	2	2	no	—
56	C₄×C₈	—	2	2	no	—
57	C₃₂	—	4	4	no	—
58	C₂⁴	C₂×C₁₆	1	1	yes	characteristic
59	C₂²×C₄	—	2	2	no	—
60	C₂²×C₄	—	2	2	no	—
61	C₂²×C₄	C₂²⋊C₈	1	2	yes	—
62	C₂²×C₄	C₂×C₁₆	1	4	yes	—
63	C₂²×C₄	C₂×C₁₆	1	1	yes	characteristic
64	C₂²×C₄	M₃₂	1	2	yes	—
65	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
66	M₁₆	—	2	8	no	—
67	C₂×C₈	—	2	2	no	—
68	C₂×C₈	—	2	2	no	—
69	C₂×C₈	—	2	2	no	—
70	C₂×C₈	—	2	2	no	—
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	—	2	4	no	—
73	C₂×C₈	C₂²⋊C₈	1	2	yes	—
74	C₂×C₈	M₃₂	1	4	yes	—
75	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈	#117	1	2	yes	—
77	C₂×C₈	C₂×M₁₆	1	2	yes	—
78	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
79	C₄⋊C₄	—	2	4	no	—
80	C₂²⋊C₄	—	2	4	no	—
81	C₄²	—	2	2	no	—
82	C₄²	—	2	2	no	—
83	C₁₆	—	2	2	no	—
84	C₁₆	—	2	4	no	—
85	C₁₆	#123	1	4	yes	—
86	C₂³	—	2	2	no	—
87	C₂³	—	2	4	no	—
88	C₂³	C₂²⋊C₁₆	1	2	yes	—
89	C₂³	C₂²×C₁₆	1	1	yes	characteristic
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	—	2	8	no	—
96	C₂×C₄	C₂²⋊C₁₆	1	2	yes	—
97	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
98	C₂×C₄	#431	1	2	yes	—
99	C₂×C₄	C₂×M₃₂	1	2	yes	—
100	C₂×C₄	C₂×M₃₂	1	1	yes	characteristic
101	C₈	—	2	2	no	—
102	C₈	—	2	4	no	—
103	C₈	#381	1	1	yes	characteristic
104	C₈	#407	1	1	yes	characteristic
105	C₈	#432	1	2	yes	—
106	C₂²	—	2	2	no	—
107	C₂²	—	2	4	no	—
108	C₂²	—	4	4	no	—
109	C₂²	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
110	C₂²	#2162	1	2	yes	—
111	C₂²	#2258	1	4	yes	—
112	C₄	—	2	2	no	—
113	C₄	—	4	4	no	—
114	C₄	#1817	1	1	yes	characteristic
115	C₄	#2112	1	1	yes	characteristic
116	C₄	#2163	1	2	yes	—
117	C₂	—	4	2	no	—
118	C₂	#12844	1	1	yes	characteristic
119	C₂	#13554	1	2	yes	—
120	C₂	#13641	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#2108

C₂

yes

maximal, characteristic

C₂×C₂²⋊C₃₂

C₂

yes

maximal

#7465

C₂

yes

maximal, complemented

C₂²×M₃₂

C₄

yes

characteristic

C₂³×C₁₆

C₂²

yes

characteristic

C₂×#405

C₄

yes

characteristic

C₂²×C₃₂

C₂²

yes

—

#2108

—

C₂×#346

C₄

yes

—

C₂×C₄×C₁₆

C₂²

yes

—

C₂²⋊C₃₂

—

C₂²×M₁₆

C₈

yes

characteristic

C₂³×C₈

C₂×C₄

yes

characteristic

C₂×#117

C₈

yes

characteristic

C₂×M₃₂

—

C₂×M₃₂

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

—

C₂²×C₁₆

D₄

yes

—

C₂²×C₁₆

C₂³

yes

characteristic

#405

—

#405

—

C₂×#97

C₈

yes

—

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₃₂

—

C₂×C₃₂

—

#346

—

C₄×C₁₆

—

C₂³×C₄

C₂×C₈

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊C₄

yes

—

C₂²×C₈

C₂×C₈

yes

characteristic

C₂²×C₈

C₂×C₈

yes

—

C₂²×C₈

M₁₆

yes

—

C₂²×C₈

C₂²×C₄

yes

characteristic

#117

—

#117

—

C₂×C₄⋊C₄

C₁₆

yes

—

C₂×C₂²⋊C₄

C₁₆

yes

—

C₂×C₄²

C₂×C₈

yes

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊C₄

yes

—

C₂×C₁₆

M₁₆

yes

—

C₂×C₁₆

C₂×D₄

yes

—

C₂×C₁₆

#41

yes

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₃₂

—

C₂⁴

C₂×C₁₆

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊C₈

yes

—

C₂²×C₄

C₂×C₁₆

yes

—

C₂²×C₄

C₂×C₁₆

yes

characteristic

C₂²×C₄

M₃₂

yes

—

C₂²×C₄

C₂²×C₈

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₈

yes

—

C₂×C₈

M₃₂

yes

—

C₂×C₈

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₈

#117

yes

—

C₂×C₈

C₂×M₁₆

yes

—

C₂×C₈

C₂×M₁₆

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₁₆

—

C₁₆

—

C₁₆

#123

yes

—

C₂³

—

C₂³

—

C₂³

C₂²⋊C₁₆

yes

—

C₂³

C₂²×C₁₆

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊C₁₆

yes

—

C₂×C₄

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₄

#431

yes

—

C₂×C₄

C₂×M₃₂

yes

—

100

C₂×C₄

C₂×M₃₂

yes

characteristic

101

C₈

—

102

C₈

—

103

C₈

#381

yes

characteristic

104

C₈

#407

yes

characteristic

105

C₈

#432

yes

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

C₂×C₂²⋊C₁₆

yes

characteristic

110

C₂²

#2162

yes

—

111

C₂²

#2258

yes

—

112

C₄

—

113

C₄

—

114

C₄

#1817

yes

characteristic

115

C₄

#2112

yes

characteristic

116

C₄

#2163

yes

—

117

C₂

—

118

C₂

#12844

yes

characteristic

119

C₂

#13554

yes

—

120

C₂

#13641

yes

—

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

subgroup

G/H

center, Frattini, cosocle

C₂²×C₁₆

C₂³

commutator, socle

C₂³

C₂²×C₁₆

prime radical

C₂⁴

C₂×C₁₆

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₆ ◁ G

LECS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
3. #7465	C₂	semidirect

H ≅ G/N

type

3. #7465

C₂

semidirect

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

subgroup	index
C₂×#2108	2
C₂×C₂²⋊C₃₂	2
#7465	2

subgroup

index

C₂×#2108

C₂×C₂²⋊C₃₂

#7465

Subgroups — #152303

Conjugacy classes of subgroups

Complements

Maximal subgroups