Group #141771 — subgroups

Subgroups — #141771

Order 512.

334 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 114 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	#19604	C₂	1	1	yes	maximal, characteristic
2	#13287	C₂	1	4	yes	maximal
3	#11402	C₂	1	1	yes	maximal, characteristic
4	#8815	C₂	1	1	yes	maximal, characteristic
5	#3170	—	2	2	no	—
6	#3160	C₂²	1	2	yes	—
7	C₄×M₃₂	C₂²	1	2	yes	—
8	#1844	C₂²	1	2	yes	—
9	#1834	—	2	2	no	—
10	#1713	C₄	1	2	yes	complemented
11	#1710	C₄	1	2	yes	complemented
12	#1707	—	2	2	no	—
13	C₂×C₈²	C₂²	1	1	yes	characteristic
14	#1386	—	2	8	no	—
15	#1312	—	2	2	no	—
16	#533	—	2	2	no	—
17	C₂×M₃₂	D₄	1	4	yes	—
18	#500	—	2	2	no	—
19	#500	C₂×C₄	1	4	yes	—
20	#499	—	2	4	no	—
21	#498	—	2	2	no	—
22	#494	—	2	2	no	—
23	#487	—	2	4	no	—
24	#467	—	4	4	no	—
25	C₄×Q₁₆	—	4	4	no	—
26	#427	—	2	4	no	—
27	C₂×#106	—	2	4	no	—
28	C₂×C₄×C₈	D₄	1	2	yes	—
29	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
30	#384	—	2	2	no	—
31	#346	—	2	2	no	—
32	C₄×C₁₆	—	2	2	no	—
33	#340	—	4	4	no	—
34	C₈²	C₂×C₄	1	1	yes	characteristic
35	C₈²	C₂×C₄	1	1	yes	characteristic
36	C₈²	D₄	1	2	yes	—
37	C₂×Q₁₆	—	4	4	no	—
38	C₂²×C₈	D₈	1	4	yes	—
39	C₂²×C₈	C₂×D₄	1	2	yes	—
40	C₄⋊Q₈	—	2	4	no	—
41	#125	—	4	2	no	—
42	C₂²⋊Q₈	—	4	4	no	—
43	C₄×Q₈	—	4	4	no	—
44	#117	—	4	2	no	—
45	C₂×C₄⋊C₄	—	4	2	no	—
46	C₂×C₄⋊C₄	—	4	4	no	—
47	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
48	M₃₂	—	2	8	no	—
49	C₂×C₁₆	—	2	4	no	—
50	#107	—	2	8	no	—
51	#107	—	4	4	no	—
52	#106	—	4	4	no	—
53	#106	—	4	8	no	—
54	#103	—	4	8	no	—
55	C₄×C₈	C₂²⋊C₄	1	2	yes	—
56	C₄×C₈	C₂²⋊C₄	1	2	yes	—
57	C₄×C₈	SD₁₆	1	4	yes	—
58	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
59	C₄×C₈	C₂×D₄	1	1	yes	characteristic
60	C₄×C₈	#41	1	2	yes	—
61	C₂×Q₈	—	4	4	no	—
62	C₂²×C₄	—	8	2	no	—
63	C₂²×C₄	#127	1	1	yes	characteristic
64	C₂²×C₄	C₂×D₈	1	2	yes	—
65	Q₁₆	—	8	8	no	—
66	C₂×C₈	—	4	4	no	—
67	C₂×C₈	#102	1	4	yes	—
68	C₂×C₈	#102	1	4	yes	—
69	C₂×C₈	C₄×D₄	1	2	yes	—
70	C₂×C₈	#121	1	2	yes	—
71	C₂×C₈	#137	1	8	yes	—
72	C₄⋊C₄	—	4	4	no	—
73	C₄⋊C₄	—	4	4	no	—
74	C₄⋊C₄	—	8	8	no	—
75	C₄⋊C₄	—	8	4	no	—
76	C₄⋊C₄	—	8	4	no	—
77	C₂²⋊C₄	—	8	2	no	—
78	C₄²	—	8	2	no	—
79	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₄²	#124	1	1	yes	characteristic
81	C₄²	C₂×SD₁₆	1	2	yes	—
82	C₁₆	—	4	4	no	—
83	C₂³	#493	1	1	yes	characteristic
84	Q₈	—	8	8	no	—
85	C₂×C₄	—	2	4	no	—
86	C₂×C₄	—	8	4	no	—
87	C₂×C₄	—	16	4	no	—
88	C₂×C₄	—	16	2	no	—
89	C₂×C₄	#390	1	1	yes	characteristic
90	C₂×C₄	C₂×#102	1	2	yes	—
91	C₂×C₄	#486	1	2	yes	—
92	C₂×C₄	#492	1	1	yes	characteristic
93	C₂×C₄	#497	1	4	yes	—
94	C₈	—	2	4	no	—
95	C₈	—	2	4	no	—
96	C₈	#441	1	4	yes	—
97	C₈	#470	1	4	yes	—
98	C₂²	—	2	2	no	—
99	C₂²	#1722	1	2	yes	—
100	C₂²	#1957	1	1	yes	characteristic
101	C₄	—	2	2	no	—
102	C₄	—	16	4	no	—
103	C₄	—	32	2	no	—
104	C₄	#1716	1	2	yes	—
105	C₄	#1971	1	2	yes	—
106	C₄	#2145	1	4	yes	—
107	C₂	—	2	2	no	—
108	C₂	#9839	1	1	yes	characteristic
109	C₂	#13211	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19604

C₂

yes

maximal, characteristic

#13287

C₂

yes

maximal

#11402

C₂

yes

maximal, characteristic

#8815

C₂

yes

maximal, characteristic

#3170

—

#3160

C₂²

yes

—

C₄×M₃₂

C₂²

yes

—

#1844

C₂²

yes

—

#1834

—

#1713

C₄

yes

complemented

#1710

C₄

yes

complemented

#1707

—

C₂×C₈²

C₂²

yes

—

—

—

C₂×M₃₂

D₄

yes

—

#500

—

#500

C₂×C₄

yes

—

—

—

—

—

—

C₄×Q₁₆

—

#427

—

C₂×#106

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

C₂³

yes

characteristic

#384

—

#346

—

C₄×C₁₆

—

#340

—

C₈²

C₂×C₄

yes

characteristic

C₈²

C₂×C₄

yes

characteristic

C₈²

D₄

yes

—

C₂×Q₁₆

—

C₂²×C₈

D₈

yes

—

C₂²×C₈

C₂×D₄

yes

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

C₂×C₁₆

—

—

—

—

—

—

C₄×C₈

C₂²⋊C₄

yes

—

C₄×C₈

C₂²⋊C₄

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

#41

yes

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

#127

yes

characteristic

C₂²×C₄

C₂×D₈

yes

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

#102

yes

—

C₂×C₈

#102

yes

—

C₂×C₈

C₄×D₄

yes

—

C₂×C₈

#121

yes

—

C₂×C₈

#137

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×C₂²⋊C₄

yes

characteristic

C₄²

#124

yes

characteristic

C₄²

C₂×SD₁₆

yes

—

C₁₆

—

C₂³

#493

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#390

yes

characteristic

C₂×C₄

C₂×#102

yes

—

C₂×C₄

#486

yes

—

C₂×C₄

#492

yes

characteristic

C₂×C₄

#497

yes

—

C₈

—

C₈

—

C₈

#441

yes

—

C₈

#470

yes

—

C₂²

—

C₂²

#1722

yes

—

100

C₂²

#1957

yes

characteristic

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

#1716

yes

—

105

C₄

#1971

yes

—

106

C₄

#2145

yes

—

107

C₂

—

108

C₂

#9839

yes

characteristic

109

C₂

#13211

yes

—

	subgroup	G/H
center, socle	C₂²	#1957
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#493
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1957

commutator

C₄×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#493

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
10. #1713	C₄	semidirect
11. #1710	C₄	semidirect

H ≅ G/N

type

10. #1713

C₄

semidirect

11. #1710

C₄

semidirect

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

subgroup	index
#19604	2
#13287	2
#11402	2
#8815	2

subgroup

index

Subgroups — #141771

Conjugacy classes of subgroups

Complements

Maximal subgroups