Group #126539 — subgroups

Subgroups — #126539

Order 512.

292 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 186 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	#12593	C₂	1	1	yes	maximal, characteristic
2	#12539	C₂	1	1	yes	maximal, characteristic
3	#11406	C₂	1	2	yes	maximal
4	#11401	C₂	1	2	yes	maximal
5	#8137	C₂	1	1	yes	maximal, characteristic
6	C₄×M₃₂	C₂²	1	2	yes	—
7	C₂×#346	C₄	1	2	yes	—
8	C₂×#346	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₁₆	C₄	1	2	yes	—
10	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
11	#1726	C₂²	1	1	yes	characteristic
12	#1450	C₂²	1	2	yes	—
13	#1375	C₄	1	4	yes	—
14	#1313	C₄	1	8	yes	complemented
15	#1312	C₄	1	8	yes	complemented
16	#1275	C₄	1	2	yes	—
17	#1274	C₄	1	2	yes	—
18	C₂×M₃₂	C₂×C₄	1	4	yes	—
19	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₁₆	C₂×C₄	1	2	yes	—
21	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
22	#431	C₂×C₄	1	4	yes	—
23	C₂×#97	C₂×C₄	1	2	yes	—
24	C₂×#97	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
26	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
27	#376	C₂×C₄	1	1	yes	characteristic
28	#346	—	2	4	no	—
29	#346	C₂×C₄	1	4	yes	—
30	#346	C₂×C₄	1	4	yes	—
31	C₄×C₁₆	—	2	4	no	—
32	C₄×C₁₆	C₂×C₄	1	4	yes	—
33	C₄×C₁₆	C₂×C₄	1	4	yes	—
34	#322	C₂×C₄	1	4	yes	—
35	C₈²	C₂×C₄	1	4	yes	—
36	C₂²×C₈	C₄²	1	2	yes	—
37	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄²	C₄²	1	2	yes	—
40	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
41	M₃₂	—	2	8	no	—
42	C₂×C₁₆	—	2	2	no	—
43	C₂×C₁₆	—	2	2	no	—
44	C₂×C₁₆	—	2	2	no	—
45	C₂×C₁₆	—	2	2	no	—
46	C₂×C₁₆	C₄²	1	8	yes	—
47	C₂×C₁₆	C₄²	1	4	yes	—
48	C₂×C₁₆	C₄²	1	2	yes	—
49	C₂×C₁₆	C₄²	1	2	yes	—
50	C₄⋊C₈	—	2	4	no	—
51	C₂²⋊C₈	—	2	4	no	—
52	#97	—	2	4	no	—
53	#97	—	2	2	no	—
54	C₄×C₈	C₄²	1	8	yes	—
55	C₄×C₈	M₁₆	1	4	yes	—
56	C₄×C₈	M₁₆	1	4	yes	—
57	C₄×C₈	C₂²×C₄	1	2	yes	—
58	C₄×C₈	C₂²×C₄	1	2	yes	—
59	#95	—	2	2	no	—
60	C₂²×C₄	—	2	2	no	—
61	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
62	C₂²×C₄	#131	1	2	yes	—
63	C₂×C₈	—	2	2	no	—
64	C₂×C₈	—	2	2	no	—
65	C₂×C₈	—	2	8	no	—
66	C₂×C₈	#97	1	4	yes	—
67	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
68	C₂×C₈	C₂×C₄²	1	2	yes	—
69	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
70	C₂×C₈	C₂×M₁₆	1	2	yes	—
71	C₂×C₈	C₂×M₁₆	1	2	yes	—
72	C₂×C₈	#131	1	4	yes	—
73	C₄²	—	2	2	no	—
74	C₄²	#97	1	4	yes	—
75	C₄²	C₂×C₄²	1	2	yes	—
76	C₄²	#131	1	2	yes	—
77	C₁₆	—	2	4	no	—
78	C₁₆	—	2	2	no	—
79	C₁₆	—	2	2	no	—
80	C₁₆	—	2	8	no	—
81	C₂³	#405	1	1	yes	characteristic
82	C₂×C₄	—	2	4	no	—
83	C₂×C₄	—	4	2	no	—
84	C₂×C₄	—	4	4	no	—
85	C₂×C₄	C₄³	1	1	yes	characteristic
86	C₂×C₄	C₂×#97	1	2	yes	—
87	C₂×C₄	#405	1	4	yes	—
88	C₂×C₄	#405	1	1	yes	characteristic
89	C₂×C₄	#405	1	2	yes	—
90	C₈	—	2	4	no	—
91	C₈	—	4	8	no	—
92	C₈	C₄×M₁₆	1	2	yes	—
93	C₈	C₄×M₁₆	1	1	yes	characteristic
94	C₈	C₄×M₁₆	1	1	yes	characteristic
95	C₈	#405	1	4	yes	—
96	C₂²	—	2	2	no	—
97	C₂²	#1452	1	2	yes	—
98	C₂²	#1726	1	1	yes	characteristic
99	C₄	—	2	4	no	—
100	C₄	—	4	4	no	—
101	C₄	#1452	1	2	yes	—
102	C₄	C₄×#97	1	1	yes	characteristic
103	C₄	#1726	1	1	yes	characteristic
104	C₂	—	2	2	no	—
105	C₂	#7843	1	1	yes	characteristic
106	C₂	#12542	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#12593

C₂

yes

maximal, characteristic

#12539

C₂

yes

maximal, characteristic

#11406

C₂

yes

maximal

#11401

C₂

yes

maximal

#8137

C₂

yes

maximal, characteristic

C₄×M₃₂

C₂²

yes

—

C₂×#346

C₄

yes

—

C₂×#346

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₄

yes

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1726

C₂²

yes

characteristic

#1450

C₂²

yes

—

#1375

C₄

yes

—

#1313

C₄

yes

complemented

#1312

C₄

yes

complemented

#1275

C₄

yes

—

#1274

C₄

yes

—

C₂×M₃₂

C₂×C₄

yes

—

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

—

C₂²×C₁₆

C₂×C₄

yes

characteristic

#431

C₂×C₄

yes

—

C₂×#97

C₂×C₄

yes

—

C₂×#97

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

C₂³

yes

characteristic

#376

C₂×C₄

yes

characteristic

#346

—

#346

C₂×C₄

yes

—

#346

C₂×C₄

yes

—

C₄×C₁₆

—

C₄×C₁₆

C₂×C₄

yes

—

C₄×C₁₆

C₂×C₄

yes

—

#322

C₂×C₄

yes

—

C₈²

C₂×C₄

yes

—

C₂²×C₈

C₄²

yes

—

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂×C₄²

C₄²

yes

—

C₂×C₄²

C₂²×C₄

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄²

yes

—

C₂×C₁₆

C₄²

yes

—

C₂×C₁₆

C₄²

yes

—

C₂×C₁₆

C₄²

yes

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

C₄×C₈

C₄²

yes

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

C₂²×C₄

yes

—

C₄×C₈

C₂²×C₄

yes

—

#95

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄²

yes

characteristic

C₂²×C₄

#131

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#97

yes

—

C₂×C₈

C₂×C₄²

yes

characteristic

C₂×C₈

C₂×C₄²

yes

—

C₂×C₈

C₂×C₄²

yes

characteristic

C₂×C₈

C₂×M₁₆

yes

—

C₂×C₈

C₂×M₁₆

yes

—

C₂×C₈

#131

yes

—

C₄²

—

C₄²

#97

yes

—

C₄²

C₂×C₄²

yes

—

C₄²

#131

yes

—

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

—

C₂³

#405

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄³

yes

characteristic

C₂×C₄

C₂×#97

yes

—

C₂×C₄

#405

yes

—

C₂×C₄

#405

yes

characteristic

C₂×C₄

#405

yes

—

C₈

—

C₈

—

C₈

C₄×M₁₆

yes

—

C₈

C₄×M₁₆

yes

characteristic

C₈

C₄×M₁₆

yes

characteristic

C₈

#405

yes

—

C₂²

—

C₂²

#1452

yes

—

C₂²

#1726

yes

characteristic

C₄

—

100

C₄

—

101

C₄

#1452

yes

—

102

C₄

C₄×#97

yes

characteristic

103

C₄

#1726

yes

characteristic

104

C₂

—

105

C₂

#7843

yes

characteristic

106

C₂

#12542

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×C₄²
commutator	C₂×C₄	C₄³
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1726
prime radical	C₂³	#405
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₈

C₂×C₄²

commutator

C₂×C₄

C₄³

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂³

G ▷ C₂×C₄ ▷ C₁

C₁ ◁ C₂×C₈ ◁ G

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
14. #1313	C₄	semidirect
15. #1312	C₄	semidirect

H ≅ G/N

type

14. #1313

C₄

semidirect

15. #1312

C₄

semidirect

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

subgroup	index
#12593	2
#12539	2
#11406	2
#11401	2
#8137	2

subgroup

index

Subgroups — #126539

Conjugacy classes of subgroups

Complements

Maximal subgroups