Group #129159 — subgroups

Subgroups — #129159

Order 512.

229 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 229 conjugacy classes of subgroups — too many to draw. Only its 120 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	#12668	C₂	1	1	yes	maximal, characteristic
2	#12667	C₂	1	2	yes	maximal
3	#12002	C₂	1	1	yes	maximal, characteristic
4	C₈×M₃₂	C₂	1	1	yes	maximal, characteristic
5	#7179	C₂	1	2	yes	maximal
6	#2155	C₄	1	2	yes	—
7	C₂×#364	C₂²	1	2	yes	—
8	#2110	—	2	2	no	—
9	C₄×M₃₂	C₂²	1	1	yes	characteristic
10	C₄×M₃₂	C₂²	1	1	yes	characteristic
11	#1772	C₂²	1	2	yes	—
12	C₂×C₈²	C₂²	1	1	yes	characteristic
13	#1385	C₄	1	1	yes	characteristic
14	#1384	C₄	1	1	yes	characteristic
15	#1378	—	2	2	no	—
16	#1369	C₄	1	4	yes	complemented
17	#1311	C₄	1	2	yes	complemented
18	C₈×C₁₆	C₄	1	2	yes	complemented
19	C₂×M₃₂	—	2	2	no	—
20	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
21	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
22	C₂×M₃₂	D₄	1	1	yes	characteristic
23	C₂×M₃₂	D₄	1	1	yes	characteristic
24	C₂×#108	C₂×C₄	1	2	yes	—
25	#427	C₂×C₄	1	2	yes	—
26	#405	—	2	2	no	—
27	C₂×C₄×C₈	—	2	1	no	—
28	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
29	#365	—	2	4	no	—
30	#364	—	2	2	no	—
31	#364	C₂×C₄	1	4	yes	—
32	#347	—	4	4	no	—
33	#346	C₂×C₄	1	2	yes	—
34	#346	Q₈	1	2	yes	—
35	C₄×C₁₆	C₂×C₄	1	2	yes	—
36	C₄×C₁₆	C₂×C₄	1	2	yes	—
37	#339	—	2	2	no	—
38	C₈²	—	2	1	no	—
39	C₈²	C₂×C₄	1	1	yes	characteristic
40	C₈²	C₂×C₄	1	1	yes	characteristic
41	C₂×M₁₆	—	2	2	no	—
42	C₂²×C₈	—	2	2	no	—
43	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
45	#117	—	2	2	no	—
46	C₂×C₄²	#41	1	1	yes	characteristic
47	M₃₂	—	2	2	no	—
48	M₃₂	—	2	2	no	—
49	M₃₂	—	2	4	no	—
50	M₃₂	—	4	2	no	—
51	M₃₂	M₁₆	1	4	yes	—
52	M₃₂	SD₁₆	1	4	yes	—
53	C₂×C₁₆	—	4	2	no	—
54	C₂×C₁₆	C₄²	1	2	yes	—
55	C₂×C₁₆	C₄²	1	2	yes	—
56	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
57	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
58	#108	—	2	2	no	—
59	#108	—	2	2	no	—
60	#107	—	2	2	no	—
61	#106	—	2	2	no	—
62	#97	—	4	2	no	—
63	C₄×C₈	—	2	2	no	—
64	C₄×C₈	—	2	1	no	—
65	C₄×C₈	—	2	1	no	—
66	C₄×C₈	—	4	2	no	—
67	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
68	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
69	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
70	C₄×C₈	#41	1	1	yes	characteristic
71	C₂²×C₄	—	2	1	no	—
72	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
73	M₁₆	—	4	2	no	—
74	M₁₆	—	4	2	no	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	4	4	no	—
78	C₂×C₈	—	4	2	no	—
79	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
80	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₈	C₄×D₄	1	1	yes	characteristic
82	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
83	C₂×C₈	C₂×M₁₆	1	2	yes	—
84	C₂×C₈	#131	1	2	yes	—
85	C₂×C₈	C₂×SD₁₆	1	2	yes	—
86	C₂×C₈	#135	1	2	yes	—
87	C₄⋊C₄	—	4	2	no	—
88	C₂²⋊C₄	—	4	2	no	—
89	C₄²	—	2	1	no	—
90	C₄²	—	4	2	no	—
91	C₄²	#117	1	1	yes	characteristic
92	C₄²	C₄×Q₈	1	1	yes	characteristic
93	C₁₆	—	2	2	no	—
94	C₁₆	—	2	2	no	—
95	C₁₆	—	4	4	no	—
96	C₁₆	#97	1	4	yes	—
97	C₁₆	#106	1	4	yes	—
98	C₂³	#443	1	1	yes	characteristic
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	—	2	1	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	8	2	no	—
103	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄	#436	1	2	yes	—
105	C₂×C₄	C₄×SD₁₆	1	2	yes	—
106	C₂×C₄	#443	1	1	yes	characteristic
107	C₈	—	2	4	no	—
108	C₈	—	4	2	no	—
109	C₈	—	8	2	no	—
110	C₈	C₂×#97	1	1	yes	characteristic
111	C₈	#405	1	1	yes	characteristic
112	C₈	C₂×#106	1	1	yes	characteristic
113	C₈	#427	1	1	yes	characteristic
114	C₈	C₄×SD₁₆	1	2	yes	—
115	C₈	#446	1	2	yes	—
116	C₂²	—	4	1	no	—
117	C₂²	#1590	1	2	yes	—
118	C₂²	#1772	1	1	yes	characteristic
119	C₄	—	2	2	no	—
120	C₄	—	4	1	no	—
121	C₄	—	16	2	no	—
122	C₄	#1590	1	2	yes	—
123	C₄	#1770	1	1	yes	characteristic
124	C₄	C₄×#106	1	1	yes	characteristic
125	C₂	—	2	1	no	—
126	C₂	—	2	2	no	—
127	C₂	#8283	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12668

C₂

yes

maximal, characteristic

#12667

C₂

yes

maximal

#12002

C₂

yes

maximal, characteristic

C₈×M₃₂

C₂

yes

maximal, characteristic

#7179

C₂

yes

maximal

#2155

C₄

yes

—

C₂×#364

C₂²

yes

—

#2110

—

C₄×M₃₂

C₂²

yes

characteristic

C₄×M₃₂

C₂²

yes

characteristic

#1772

C₂²

yes

—

C₂×C₈²

C₂²

yes

characteristic

#1385

C₄

yes

characteristic

#1384

C₄

yes

characteristic

#1378

—

#1369

C₄

yes

complemented

#1311

C₄

yes

complemented

C₈×C₁₆

C₄

yes

complemented

C₂×M₃₂

—

C₂×M₃₂

C₂×C₄

yes

characteristic

C₂×M₃₂

C₂×C₄

yes

characteristic

C₂×M₃₂

D₄

yes

characteristic

C₂×M₃₂

D₄

yes

characteristic

C₂×#108

C₂×C₄

yes

—

#427

C₂×C₄

yes

—

#405

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂³

yes

—

—

C₂×C₄

yes

—

#347

—

#346

C₂×C₄

yes

—

#346

Q₈

yes

—

C₄×C₁₆

C₂×C₄

yes

—

C₄×C₁₆

C₂×C₄

yes

—

#339

—

C₈²

—

C₈²

C₂×C₄

yes

characteristic

C₈²

C₂×C₄

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

#117

—

C₂×C₄²

#41

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

M₁₆

yes

—

M₃₂

SD₁₆

yes

—

C₂×C₁₆

—

C₂×C₁₆

C₄²

yes

—

C₂×C₁₆

C₄²

yes

—

C₂×C₁₆

C₄⋊C₄

yes

—

C₂×C₁₆

C₄⋊C₄

yes

—

—

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×Q₈

yes

characteristic

C₄×C₈

#41

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₄×D₄

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₄²

yes

characteristic

C₂×C₈

C₂×C₄⋊C₄

yes

characteristic

C₂×C₈

C₄×D₄

yes

characteristic

C₂×C₈

C₄×Q₈

yes

characteristic

C₂×C₈

C₂×M₁₆

yes

—

C₂×C₈

#131

yes

—

C₂×C₈

C₂×SD₁₆

yes

—

C₂×C₈

#135

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

#117

yes

characteristic

C₄²

C₄×Q₈

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

#97

yes

—

C₁₆

#106

yes

—

C₂³

#443

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

C₄×C₄⋊C₄

yes

characteristic

104

C₂×C₄

#436

yes

—

105

C₂×C₄

C₄×SD₁₆

yes

—

106

C₂×C₄

#443

yes

characteristic

107

C₈

—

108

C₈

—

109

C₈

—

110

C₈

C₂×#97

yes

characteristic

111

C₈

#405

yes

characteristic

112

C₈

C₂×#106

yes

characteristic

113

C₈

#427

yes

characteristic

114

C₈

C₄×SD₁₆

yes

—

115

C₈

#446

yes

—

116

C₂²

—

117

C₂²

#1590

yes

—

118

C₂²

#1772

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

#1590

yes

—

123

C₄

#1770

yes

characteristic

124

C₄

C₄×#106

yes

characteristic

125

C₂

—

126

C₂

—

127

C₂

#8283

yes

characteristic

	subgroup	G/H
center	C₈	#427
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#8283
prime radical	C₂³	#443
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×M₃₂ ◁ C₄×M₃₂ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₈

#427

commutator

C₂×C₈

C₂×C₄²

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂

C₂³

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₈ ◁ C₂×M₃₂ ◁ C₄×M₃₂ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
16. #1369	C₄	semidirect
17. #1311	C₄	semidirect
18. C₈×C₁₆	C₄	semidirect

H ≅ G/N

type

16. #1369

C₄

semidirect

17. #1311

C₄

semidirect

18. C₈×C₁₆

C₄

semidirect

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

subgroup	index
#12668	2
#12667	2
#12002	2
C₈×M₃₂	2
#7179	2

subgroup

index

C₈×M₃₂

Subgroups — #129159

Conjugacy classes of subgroups

Complements

Maximal subgroups