Group #152859 — subgroups

Subgroups — #152859

Order 512.

170 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 84 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×#1425	C₂	1	1	yes	maximal, characteristic
2	C₂×#1425	C₂	1	1	yes	maximal, characteristic
3	#13549	C₂	1	1	yes	maximal, characteristic
4	#13173	C₂	1	2	yes	maximal
5	#7479	C₂	1	2	yes	maximal
6	C₂×M₆₄	C₂²	1	1	yes	characteristic
7	C₂²×C₃₂	C₂²	1	1	yes	characteristic
8	C₂×#368	C₂²	1	2	yes	—
9	C₂×#368	C₂²	1	2	yes	—
10	#2108	C₂²	1	1	yes	characteristic
11	#1953	—	2	2	no	—
12	#1425	—	2	1	no	—
13	#1425	—	2	1	no	—
14	#1425	C₄	1	4	yes	complemented
15	#1397	—	2	1	no	—
16	C₄×C₃₂	—	2	1	no	—
17	#1388	—	2	2	no	—
18	C₂×M₃₂	D₄	1	1	yes	characteristic
19	C₂²×C₁₆	C₂³	1	1	yes	characteristic
20	#430	—	4	2	no	—
21	C₂×#108	—	2	2	no	—
22	C₂×#108	—	2	2	no	—
23	#405	D₄	1	1	yes	characteristic
24	M₆₄	—	2	1	no	—
25	M₆₄	Q₈	1	2	yes	—
26	C₂×C₃₂	—	2	1	no	—
27	C₂×C₃₂	C₂×C₄	1	2	yes	—
28	C₂×C₃₂	D₄	1	1	yes	characteristic
29	C₂×C₃₂	D₄	1	1	yes	characteristic
30	C₂×C₃₂	Q₈	1	2	yes	—
31	#368	—	2	2	no	—
32	#368	—	2	2	no	—
33	#368	—	2	2	no	—
34	#368	C₂×C₄	1	4	yes	—
35	#346	—	2	1	no	—
36	C₄×C₁₆	—	2	1	no	—
37	#344	—	4	2	no	—
38	C₂×M₁₆	—	4	2	no	—
39	C₂×M₁₆	—	4	2	no	—
40	C₂×M₁₆	D₈	1	1	yes	characteristic
41	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
42	#117	D₈	1	1	yes	characteristic
43	M₃₂	—	2	1	no	—
44	M₃₂	Q₁₆	1	2	yes	—
45	C₂×C₁₆	—	2	1	no	—
46	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
48	C₂×C₁₆	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₁₆	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₁₆	#41	1	1	yes	characteristic
51	C₂×C₁₆	#41	1	1	yes	characteristic
52	#108	—	2	4	no	—
53	#108	—	4	2	no	—
54	#108	—	4	2	no	—
55	#108	—	4	2	no	—
56	#108	—	8	4	no	—
57	#97	—	2	1	no	—
58	C₄×C₈	—	2	1	no	—
59	C₃₂	—	2	2	no	—
60	C₃₂	C₄⋊C₄	1	2	yes	—
61	C₃₂	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
63	M₁₆	—	2	1	no	—
64	M₁₆	—	4	4	no	—
65	M₁₆	—	8	2	no	—
66	M₁₆	—	8	2	no	—
67	M₁₆	—	8	2	no	—
68	M₁₆	D₁₆	1	2	yes	—
69	C₂×C₈	—	2	1	no	—
70	C₂×C₈	—	8	2	no	—
71	C₂×C₈	—	8	2	no	—
72	C₂×C₈	C₄×D₄	1	1	yes	characteristic
73	C₂×C₈	#121	1	1	yes	characteristic
74	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
75	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
76	C₂×C₈	#135	1	1	yes	characteristic
77	C₂×C₈	#135	1	1	yes	characteristic
78	C₄⋊C₄	—	2	1	no	—
79	C₂²⋊C₄	Q₃₂	1	2	yes	—
80	C₄²	—	2	1	no	—
81	C₁₆	—	2	2	no	—
82	C₁₆	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₁₆	C₄×Q₈	1	1	yes	characteristic
84	C₁₆	C₂²⋊Q₈	1	1	yes	characteristic
85	C₁₆	#125	1	1	yes	characteristic
86	C₂³	C₂×Q₃₂	1	1	yes	characteristic
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	C₄×D₈	1	1	yes	characteristic
89	C₂×C₄	#466	1	1	yes	characteristic
90	C₂×C₄	#500	1	1	yes	characteristic
91	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
92	C₂×C₄	#508	1	1	yes	characteristic
93	C₂×C₄	#508	1	1	yes	characteristic
94	C₈	—	2	2	no	—
95	C₈	—	8	4	no	—
96	C₈	—	16	2	no	—
97	C₈	#389	1	1	yes	characteristic
98	C₈	C₄×Q₁₆	1	1	yes	characteristic
99	C₈	#467	1	1	yes	characteristic
100	C₈	#499	1	1	yes	characteristic
101	C₂²	—	4	1	no	—
102	C₂²	C₄×Q₃₂	1	1	yes	characteristic
103	C₂²	#2216	1	1	yes	characteristic
104	C₂²	#2252	1	1	yes	characteristic
105	C₄	—	4	2	no	—
106	C₄	#1942	1	1	yes	characteristic
107	C₄	C₄×D₁₆	1	1	yes	characteristic
108	C₄	#2215	1	1	yes	characteristic
109	C₄	#2253	1	1	yes	characteristic
110	C₂	—	2	1	no	—
111	C₂	—	2	1	no	—
112	C₂	—	2	1	no	—
113	C₂	#13165	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#1425

C₂

yes

maximal, characteristic

C₂×#1425

C₂

yes

maximal, characteristic

#13549

C₂

yes

maximal, characteristic

#13173

C₂

yes

maximal

#7479

C₂

yes

maximal

C₂×M₆₄

C₂²

yes

characteristic

C₂²×C₃₂

C₂²

yes

characteristic

C₂×#368

C₂²

yes

—

C₂×#368

C₂²

yes

—

#2108

C₂²

yes

—

—

—

C₄

yes

complemented

#1397

—

C₄×C₃₂

—

#1388

—

C₂×M₃₂

D₄

yes

characteristic

C₂²×C₁₆

C₂³

yes

—

—

—

D₄

yes

characteristic

M₆₄

—

M₆₄

Q₈

yes

—

C₂×C₃₂

—

C₂×C₃₂

C₂×C₄

yes

—

C₂×C₃₂

D₄

yes

characteristic

C₂×C₃₂

D₄

yes

characteristic

C₂×C₃₂

Q₈

yes

—

—

—

—

C₂×C₄

yes

—

#346

—

C₄×C₁₆

—

#344

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

D₈

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

#117

D₈

yes

characteristic

M₃₂

—

M₃₂

Q₁₆

yes

—

C₂×C₁₆

—

C₂×C₁₆

C₂²×C₄

yes

characteristic

C₂×C₁₆

C₂×D₄

yes

characteristic

C₂×C₁₆

C₂×Q₈

yes

characteristic

C₂×C₁₆

C₂×Q₈

yes

C₂×C₁₆

yes

C₂×C₁₆

yes

—

—

—

—

—

—

C₄×C₈

—

C₃₂

—

C₃₂

C₄⋊C₄

yes

—

C₃₂

C₄⋊C₄

yes

—

C₂²×C₄

C₂×D₈

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

D₁₆

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₄

yes

characteristic

C₂×C₈

#121

yes

characteristic

C₂×C₈

C₄⋊Q₈

yes

characteristic

C₂×C₈

C₂×Q₁₆

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₂²⋊C₄

Q₃₂

yes

—

C₄²

—

C₁₆

—

C₁₆

C₂×C₄⋊C₄

yes

characteristic

C₁₆

C₄×Q₈

yes

characteristic

C₁₆

C₂²⋊Q₈

yes

characteristic

C₁₆

#125

yes

characteristic

C₂³

C₂×Q₃₂

yes

characteristic

C₂×C₄

—

C₂×C₄

C₄×D₈

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂×D₁₆

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

C₈

—

C₈

#389

yes

characteristic

C₈

C₄×Q₁₆

yes

characteristic

C₈

#467

yes

characteristic

100

C₈

#499

yes

characteristic

101

C₂²

—

102

C₂²

C₄×Q₃₂

yes

characteristic

103

C₂²

#2216

yes

characteristic

104

C₂²

#2252

yes

characteristic

105

C₄

—

106

C₄

#1942

yes

characteristic

107

C₄

C₄×D₁₆

yes

characteristic

108

C₄

#2215

yes

characteristic

109

C₄

#2253

yes

characteristic

110

C₂

—

111

C₂

—

112

C₂

—

113

C₂

#13165

yes

characteristic

	subgroup	G/H
center	C₄	#2215
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂	#13165
prime radical	C₂³	C₂×Q₃₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₄

#2215

commutator

C₂×C₁₆

C₂²×C₄

Frattini, cosocle

C₂²×C₁₆

C₂³

socle

C₂

#13165

prime radical

C₂³

C₂×Q₃₂

LCS

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ G

LECS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
14. #1425	C₄	semidirect

H ≅ G/N

type

14. #1425

C₄

semidirect

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

subgroup	index
C₂×#1425	2
#13549	2
#13173	2
#7479	2

subgroup

index

Subgroups — #152859

Conjugacy classes of subgroups

Complements

Maximal subgroups