Group #128209 — subgroups

Subgroups — #128209

Order 512.

342 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 202 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	#12590	C₂	1	1	yes	maximal, characteristic
2	C₄²×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₈×#97	C₂	1	1	yes	maximal, characteristic
4	#7136	C₂	1	4	yes	maximal
5	C₂×C₄×C₁₆	C₄	1	4	yes	—
6	C₂×C₄×C₁₆	C₂²	1	2	yes	—
7	C₂×C₄×C₁₆	C₂²	1	2	yes	—
8	#1744	C₄	1	1	yes	characteristic
9	C₄×#97	C₄	1	1	yes	characteristic
10	C₄²×C₈	C₂²	1	1	yes	characteristic
11	C₂×#322	C₄	1	2	yes	—
12	C₂×C₈²	C₂²	1	2	yes	—
13	#1374	C₄	1	4	yes	—
14	C₂²×C₁₆	C₂×C₄	1	4	yes	—
15	C₂²×C₁₆	C₂×C₄	1	4	yes	—
16	C₂×#97	C₈	1	2	yes	—
17	C₂×#97	C₈	1	2	yes	—
18	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
19	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
20	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
21	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
22	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
24	C₄³	C₂×C₄	1	1	yes	characteristic
25	C₄×C₁₆	—	2	8	no	—
26	C₄×C₁₆	—	2	4	no	—
27	C₄×C₁₆	—	2	4	no	—
28	#336	C₈	1	4	yes	—
29	#322	—	2	4	no	—
30	C₈²	—	2	4	no	—
31	C₂²×C₈	C₄²	1	2	yes	—
32	C₂²×C₈	C₂×C₈	1	4	yes	—
33	C₂²×C₈	C₂×C₈	1	4	yes	—
34	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
35	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄²	C₄²	1	2	yes	—
37	C₂×C₄²	C₂×C₈	1	2	yes	—
38	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
39	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
40	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₁₆	—	2	4	no	—
42	C₂×C₁₆	—	2	4	no	—
43	C₂×C₁₆	—	2	4	no	—
44	C₂×C₁₆	—	2	8	no	—
45	C₂×C₁₆	M₁₆	1	8	yes	—
46	#97	—	2	4	no	—
47	#97	C₁₆	1	8	yes	complemented
48	C₄×C₈	—	2	4	no	—
49	C₄×C₈	—	2	4	no	—
50	C₄×C₈	—	2	2	no	—
51	C₄×C₈	—	2	2	no	—
52	C₄×C₈	—	2	2	no	—
53	C₄×C₈	—	2	2	no	—
54	C₄×C₈	M₁₆	1	4	yes	—
55	C₄×C₈	#41	1	4	yes	—
56	C₂²×C₄	C₄×C₈	1	2	yes	—
57	C₂²×C₄	C₄×C₈	1	2	yes	—
58	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
59	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
60	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
61	C₂×C₈	—	2	2	no	—
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	—	2	4	no	—
64	C₂×C₈	—	2	4	no	—
65	C₂×C₈	—	2	4	no	—
66	C₂×C₈	—	2	8	no	—
67	C₂×C₈	#97	1	4	yes	—
68	C₂×C₈	C₂×C₁₆	1	8	yes	—
69	C₂×C₈	#117	1	2	yes	—
70	C₂×C₈	#117	1	2	yes	—
71	C₂×C₈	C₂×M₁₆	1	4	yes	—
72	C₂×C₈	#131	1	4	yes	—
73	C₄²	—	2	2	no	—
74	C₄²	—	2	2	no	—
75	C₄²	—	2	4	no	—
76	C₄²	C₂×C₁₆	1	4	yes	—
77	C₄²	M₃₂	1	4	yes	—
78	C₄²	#117	1	2	yes	—
79	C₄²	C₂×M₁₆	1	2	yes	—
80	C₁₆	—	2	8	no	—
81	C₁₆	—	4	4	no	—
82	C₂³	C₂×C₄×C₈	1	1	yes	characteristic
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	C₄×C₁₆	1	4	yes	—
88	C₂×C₄	#346	1	4	yes	—
89	C₂×C₄	#376	1	1	yes	characteristic
90	C₂×C₄	C₂×#97	1	1	yes	characteristic
91	C₂×C₄	C₄×M₁₆	1	2	yes	—
92	C₂×C₄	#405	1	2	yes	—
93	C₂×C₄	#431	1	2	yes	—
94	C₂×C₄	#431	1	2	yes	—
95	C₂×C₄	C₂²×C₁₆	1	2	yes	—
96	C₂×C₄	C₂×M₃₂	1	2	yes	—
97	C₂×C₄	#504	1	4	yes	—
98	C₈	—	2	8	no	—
99	C₈	—	2	4	no	—
100	C₈	—	4	4	no	—
101	C₈	#432	1	4	yes	—
102	C₈	#433	1	4	yes	—
103	C₂²	C₈×M₁₆	1	2	yes	—
104	C₂²	#1744	1	1	yes	characteristic
105	C₂²	C₂×C₄×C₁₆	1	1	yes	characteristic
106	C₂²	C₂×#346	1	1	yes	characteristic
107	C₂²	#2108	1	2	yes	—
108	C₄	—	2	4	no	—
109	C₄	—	2	4	no	—
110	C₄	#1451	1	2	yes	—
111	C₄	#1745	1	1	yes	characteristic
112	C₄	#1746	1	1	yes	characteristic
113	C₄	#2162	1	4	yes	—
114	C₄	#2164	1	4	yes	—
115	C₂	#8132	1	1	yes	characteristic
116	C₂	C₁₆×M₁₆	1	2	yes	—
117	C₂	#11408	1	2	yes	—
118	C₂	#12590	1	1	yes	characteristic
119	C₂	#12592	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12590

C₂

yes

maximal, characteristic

C₄²×C₁₆

C₂

yes

maximal, characteristic

C₈×#97

C₂

yes

maximal, characteristic

#7136

C₂

yes

maximal

C₂×C₄×C₁₆

C₄

yes

—

C₂×C₄×C₁₆

C₂²

yes

—

C₂×C₄×C₁₆

C₂²

yes

—

#1744

C₄

yes

characteristic

C₄×#97

C₄

yes

characteristic

C₄²×C₈

C₂²

yes

characteristic

C₂×#322

C₄

yes

—

C₂×C₈²

C₂²

yes

—

#1374

C₄

yes

—

C₂²×C₁₆

C₂×C₄

yes

—

C₂²×C₁₆

C₂×C₄

yes

—

C₂×#97

C₈

yes

—

C₂×#97

C₈

yes

—

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄³

C₂×C₄

yes

characteristic

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

#336

C₈

yes

—

#322

—

C₈²

—

C₂²×C₈

C₄²

yes

—

C₂²×C₈

C₂×C₈

yes

—

C₂²×C₈

C₂×C₈

yes

—

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂×C₄²

C₄²

yes

—

C₂×C₄²

C₂×C₈

yes

—

C₂×C₄²

C₂×C₈

yes

characteristic

C₂×C₄²

C₂×C₈

yes

characteristic

C₂×C₄²

C₂²×C₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

M₁₆

yes

—

#97

—

#97

C₁₆

yes

complemented

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

#41

yes

—

C₂²×C₄

C₄×C₈

yes

—

C₂²×C₄

C₄×C₈

yes

—

C₂²×C₄

C₂×C₄²

yes

characteristic

C₂²×C₄

C₂²×C₈

yes

characteristic

C₂²×C₄

C₂²×C₈

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#97

yes

—

C₂×C₈

C₂×C₁₆

yes

—

C₂×C₈

#117

yes

—

C₂×C₈

#117

yes

—

C₂×C₈

C₂×M₁₆

yes

—

C₂×C₈

#131

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×C₁₆

yes

—

C₄²

M₃₂

yes

—

C₄²

#117

yes

—

C₄²

C₂×M₁₆

yes

—

C₁₆

—

C₁₆

—

C₂³

C₂×C₄×C₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×C₁₆

yes

—

C₂×C₄

#346

yes

—

C₂×C₄

#376

yes

characteristic

C₂×C₄

C₂×#97

yes

characteristic

C₂×C₄

C₄×M₁₆

yes

—

C₂×C₄

#405

yes

—

C₂×C₄

#431

yes

—

C₂×C₄

#431

yes

—

C₂×C₄

C₂²×C₁₆

yes

—

C₂×C₄

C₂×M₃₂

yes

—

C₂×C₄

#504

yes

—

C₈

—

C₈

—

100

C₈

—

101

C₈

#432

yes

—

102

C₈

#433

yes

—

103

C₂²

C₈×M₁₆

yes

—

104

C₂²

#1744

yes

characteristic

105

C₂²

C₂×C₄×C₁₆

yes

characteristic

106

C₂²

C₂×#346

yes

characteristic

107

C₂²

#2108

yes

—

108

C₄

—

109

C₄

—

110

C₄

#1451

yes

—

111

C₄

#1745

yes

characteristic

112

C₄

#1746

yes

characteristic

113

C₄

#2162

yes

—

114

C₄

#2164

yes

—

115

C₂

#8132

yes

characteristic

116

C₂

C₁₆×M₁₆

yes

—

117

C₂

#11408

yes

—

118

C₂

#12590

yes

characteristic

119

C₂

#12592

yes

characteristic

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

subgroup

G/H

center, Frattini, cosocle

C₂×C₄×C₈

C₂³

commutator

C₂²

C₂×C₄×C₁₆

socle, prime radical

C₂³

C₂×C₄×C₈

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
47. #97	C₁₆	semidirect

H ≅ G/N

type

47. #97

C₁₆

semidirect

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

subgroup	index
#12590	2
C₄²×C₁₆	2
C₈×#97	2
#7136	2

subgroup

index

#12590

C₄²×C₁₆

C₈×#97

#7136

Subgroups — #128209

Conjugacy classes of subgroups

Complements

Maximal subgroups