Group #141691 — subgroups

Subgroups — #141691

Order 512.

372 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 142 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×#1710	C₂	1	1	yes	maximal, characteristic
2	#13282	C₂	1	4	yes	maximal
3	C₂×#1311	C₂	1	1	yes	maximal, characteristic
4	#8818	C₂	1	1	yes	maximal, characteristic
5	C₂×#500	C₂²	1	2	yes	—
6	C₂×#487	—	2	2	no	—
7	C₂×C₄×C₁₆	C₂²	1	2	yes	—
8	#1845	—	2	2	no	—
9	#1845	C₂²	1	2	yes	—
10	#1710	—	2	2	no	—
11	#1710	C₄	1	4	yes	complemented
12	C₂×C₈²	C₂²	1	1	yes	characteristic
13	#1380	—	2	8	no	—
14	#1311	—	2	2	no	—
15	C₂×C₄⋊Q₈	—	2	2	no	—
16	C₂²×C₁₆	D₄	1	4	yes	—
17	#500	—	2	2	no	—
18	#500	—	2	4	no	—
19	#500	C₂×C₄	1	4	yes	—
20	#487	—	2	8	no	—
21	#487	—	4	4	no	—
22	C₂×#107	—	2	4	no	—
23	C₂×#107	—	2	4	no	—
24	C₂×#107	—	4	4	no	—
25	C₂×#103	—	4	4	no	—
26	C₂×C₄×C₈	D₄	1	2	yes	—
27	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
28	#384	—	2	2	no	—
29	C₄×C₁₆	—	2	2	no	—
30	C₄×C₁₆	—	2	2	no	—
31	C₈²	C₂×C₄	1	2	yes	—
32	C₈²	D₄	1	2	yes	—
33	C₂²×Q₈	—	4	2	no	—
34	C₂²×C₈	D₈	1	4	yes	—
35	C₂²×C₈	C₂×D₄	1	2	yes	—
36	C₄⋊Q₈	—	2	4	no	—
37	C₄⋊Q₈	—	4	2	no	—
38	C₄⋊Q₈	—	4	2	no	—
39	C₄⋊Q₈	—	4	2	no	—
40	C₂×C₄⋊C₄	—	4	2	no	—
41	C₂×C₄⋊C₄	—	4	2	no	—
42	C₂×C₄⋊C₄	—	4	2	no	—
43	C₂×C₄⋊C₄	—	4	4	no	—
44	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
45	C₂×C₁₆	—	2	4	no	—
46	C₂×C₁₆	—	2	4	no	—
47	C₂×C₁₆	Q₁₆	1	8	yes	—
48	#107	—	2	8	no	—
49	#107	—	4	4	no	—
50	#107	—	4	8	no	—
51	#107	—	8	8	no	—
52	#103	—	4	16	no	—
53	C₄×C₈	C₂²⋊C₄	1	4	yes	—
54	C₄×C₈	SD₁₆	1	4	yes	—
55	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
56	C₄×C₈	C₂×D₄	1	1	yes	characteristic
57	C₄×C₈	#41	1	2	yes	—
58	C₂×Q₈	—	4	4	no	—
59	C₂×Q₈	—	8	2	no	—
60	C₂×Q₈	—	8	4	no	—
61	C₂×Q₈	—	8	4	no	—
62	C₂²×C₄	—	8	2	no	—
63	C₂²×C₄	—	8	2	no	—
64	C₂²×C₄	#127	1	1	yes	characteristic
65	C₂²×C₄	C₂×D₈	1	2	yes	—
66	C₂×C₈	—	2	4	no	—
67	C₂×C₈	—	2	4	no	—
68	C₂×C₈	#102	1	8	yes	—
69	C₂×C₈	C₄×D₄	1	2	yes	—
70	C₂×C₈	#121	1	2	yes	—
71	C₂×C₈	C₂×Q₁₆	1	4	yes	—
72	C₂×C₈	#135	1	4	yes	—
73	C₄⋊C₄	—	4	4	no	—
74	C₄⋊C₄	—	4	4	no	—
75	C₄⋊C₄	—	8	2	no	—
76	C₄⋊C₄	—	8	2	no	—
77	C₄⋊C₄	—	8	4	no	—
78	C₄⋊C₄	—	8	8	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	—	2	8	no	—
83	C₂³	#493	1	1	yes	characteristic
84	Q₈	—	8	8	no	—
85	Q₈	—	16	4	no	—
86	C₂×C₄	—	8	4	no	—
87	C₂×C₄	—	16	2	no	—
88	C₂×C₄	—	16	2	no	—
89	C₂×C₄	—	16	2	no	—
90	C₂×C₄	—	16	4	no	—
91	C₂×C₄	#390	1	1	yes	characteristic
92	C₂×C₄	C₂×#102	1	2	yes	—
93	C₂×C₄	#486	1	2	yes	—
94	C₂×C₄	#492	1	1	yes	characteristic
95	C₂×C₄	#494	1	2	yes	—
96	C₂×C₄	#495	1	2	yes	—
97	C₂×C₄	#509	1	4	yes	—
98	C₂×C₄	#510	1	4	yes	—
99	C₈	—	2	8	no	—
100	C₈	C₄×Q₁₆	1	4	yes	—
101	C₈	#467	1	4	yes	—
102	C₂²	#1709	1	1	yes	characteristic
103	C₂²	#1713	1	1	yes	characteristic
104	C₂²	#1957	1	1	yes	characteristic
105	C₂²	#2250	1	2	yes	—
106	C₂²	#2251	1	2	yes	—
107	C₄	—	16	4	no	—
108	C₄	—	32	2	no	—
109	C₄	#1707	1	2	yes	—
110	C₄	#1969	1	2	yes	—
111	C₄	#2139	1	4	yes	—
112	C₄	#2243	1	4	yes	—
113	C₂	#9825	1	1	yes	characteristic
114	C₂	#12513	1	1	yes	characteristic
115	C₂	#12514	1	1	yes	characteristic
116	C₂	#12524	1	2	yes	—
117	C₂	#13205	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#1710

C₂

yes

maximal, characteristic

#13282

C₂

yes

maximal

C₂×#1311

C₂

yes

maximal, characteristic

#8818

C₂

yes

maximal, characteristic

C₂×#500

C₂²

yes

—

C₂×#487

—

C₂×C₄×C₁₆

C₂²

yes

—

#1845

—

#1845

C₂²

yes

—

#1710

—

#1710

C₄

yes

complemented

C₂×C₈²

C₂²

yes

characteristic

#1380

—

#1311

—

C₂×C₄⋊Q₈

—

C₂²×C₁₆

D₄

yes

—

#500

—

#500

—

#500

C₂×C₄

yes

—

—

—

—

—

—

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

C₂³

yes

characteristic

#384

—

C₄×C₁₆

—

C₄×C₁₆

—

C₈²

C₂×C₄

yes

—

C₈²

D₄

yes

—

C₂²×Q₈

—

C₂²×C₈

D₈

yes

—

C₂²×C₈

C₂×D₄

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

Q₁₆

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

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#127

yes

characteristic

C₂²×C₄

C₂×D₈

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#102

yes

—

C₂×C₈

C₄×D₄

yes

—

C₂×C₈

#121

yes

—

C₂×C₈

C₂×Q₁₆

yes

—

C₂×C₈

#135

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×C₂²⋊C₄

yes

characteristic

C₄²

#124

yes

characteristic

C₄²

C₂×SD₁₆

yes

—

C₁₆

—

C₂³

#493

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

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

#494

yes

—

C₂×C₄

#495

yes

—

C₂×C₄

#509

yes

—

C₂×C₄

#510

yes

—

C₈

—

100

C₈

C₄×Q₁₆

yes

—

101

C₈

#467

yes

—

102

C₂²

#1709

yes

characteristic

103

C₂²

#1713

yes

characteristic

104

C₂²

#1957

yes

characteristic

105

C₂²

#2250

yes

—

106

C₂²

#2251

yes

—

107

C₄

—

108

C₄

—

109

C₄

#1707

yes

—

110

C₄

#1969

yes

—

111

C₄

#2139

yes

—

112

C₄

#2243

yes

—

113

C₂

#9825

yes

characteristic

114

C₂

#12513

yes

characteristic

115

C₂

#12514

yes

characteristic

116

C₂

#12524

yes

—

117

C₂

#13205

yes

—

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

subgroup

G/H

center, socle, prime radical

C₂³

#493

commutator

C₄×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

H ≅ G/N

type

11. #1710

C₄

semidirect

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

subgroup	index
C₂×#1710	2
#13282	2
C₂×#1311	2
#8818	2

subgroup

index

Subgroups — #141691

Conjugacy classes of subgroups

Complements

Maximal subgroups