Group #141697 — subgroups

Subgroups — #141697

Order 512.

336 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 122 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	#13290	C₂	1	4	yes	maximal
3	C₈×M₃₂	C₂	1	1	yes	maximal, characteristic
4	#8823	C₂	1	1	yes	maximal, characteristic
5	C₂×#500	C₂²	1	2	yes	—
6	C₂×#487	—	2	2	no	—
7	C₄×M₃₂	C₂²	1	2	yes	—
8	#1848	C₂²	1	2	yes	—
9	#1710	—	2	1	no	—
10	#1710	—	2	1	no	—
11	#1710	C₄	1	4	yes	—
12	#1567	—	2	2	no	—
13	C₂×C₈²	C₂²	1	1	yes	characteristic
14	#1387	—	2	8	no	—
15	#1311	—	2	1	no	—
16	C₈×C₁₆	—	2	1	no	—
17	C₂×C₄⋊Q₈	—	2	2	no	—
18	C₂×M₃₂	D₄	1	4	yes	—
19	#500	—	2	2	no	—
20	#500	—	2	4	no	—
21	#500	C₂×C₄	1	4	yes	—
22	#487	—	2	4	no	—
23	#487	—	2	4	no	—
24	#487	—	4	4	no	—
25	C₂×#108	—	2	4	no	—
26	C₂×#107	—	2	4	no	—
27	#423	—	2	2	no	—
28	C₂×#103	—	4	4	no	—
29	C₂×C₄×C₈	D₄	1	2	yes	—
30	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
31	#346	—	2	2	no	—
32	C₄×C₁₆	—	2	2	no	—
33	#335	—	4	4	no	—
34	C₈²	C₂×C₄	1	2	yes	—
35	C₈²	D₄	1	1	yes	characteristic
36	C₈²	D₄	1	1	yes	characteristic
37	C₂²×Q₈	—	4	2	no	—
38	C₂×M₁₆	—	4	2	no	—
39	C₂²×C₈	SD₁₆	1	4	yes	—
40	C₂²×C₈	C₂×D₄	1	2	yes	—
41	C₄⋊Q₈	—	2	4	no	—
42	C₄⋊Q₈	—	4	2	no	—
43	C₄⋊Q₈	—	4	2	no	—
44	C₄⋊Q₈	—	4	2	no	—
45	C₂×C₄⋊C₄	—	4	2	no	—
46	C₂×C₄⋊C₄	—	4	2	no	—
47	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
48	M₃₂	—	2	4	no	—
49	M₃₂	Q₁₆	1	8	yes	—
50	C₂×C₁₆	—	2	4	no	—
51	#108	—	4	8	no	—
52	#107	—	2	8	no	—
53	#107	—	4	4	no	—
54	C₄⋊C₈	—	4	4	no	—
55	#103	—	4	8	no	—
56	#103	—	4	8	no	—
57	C₄×C₈	C₂²⋊C₄	1	2	yes	—
58	C₄×C₈	C₂²⋊C₄	1	2	yes	—
59	C₄×C₈	D₈	1	4	yes	—
60	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
61	C₄×C₈	C₂×D₄	1	1	yes	characteristic
62	C₄×C₈	#41	1	2	yes	—
63	C₂×Q₈	—	4	4	no	—
64	C₂×Q₈	—	8	8	no	—
65	C₂×Q₈	—	8	2	no	—
66	C₂²×C₄	—	8	2	no	—
67	C₂²×C₄	#127	1	1	yes	characteristic
68	C₂²×C₄	C₂×SD₁₆	1	2	yes	—
69	M₁₆	—	8	4	no	—
70	C₂×C₈	—	4	4	no	—
71	C₂×C₈	—	8	2	no	—
72	C₂×C₈	#102	1	4	yes	—
73	C₂×C₈	#102	1	4	yes	—
74	C₂×C₈	C₄×D₄	1	2	yes	—
75	C₂×C₈	#121	1	2	yes	—
76	C₂×C₈	C₂×Q₁₆	1	4	yes	—
77	C₂×C₈	#135	1	4	yes	—
78	C₄⋊C₄	—	4	4	no	—
79	C₄⋊C₄	—	4	4	no	—
80	C₄⋊C₄	—	8	2	no	—
81	C₄⋊C₄	—	8	2	no	—
82	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₄²	#124	1	1	yes	characteristic
84	C₄²	C₂×D₈	1	2	yes	—
85	C₁₆	—	2	8	no	—
86	C₂³	#492	1	1	yes	characteristic
87	Q₈	—	8	8	no	—
88	Q₈	—	16	4	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	—	8	4	no	—
91	C₂×C₄	—	16	4	no	—
92	C₂×C₄	#390	1	1	yes	characteristic
93	C₂×C₄	C₂×#102	1	2	yes	—
94	C₂×C₄	#486	1	2	yes	—
95	C₂×C₄	#493	1	1	yes	characteristic
96	C₂×C₄	#494	1	2	yes	—
97	C₂×C₄	#495	1	2	yes	—
98	C₈	—	2	4	no	—
99	C₈	—	2	4	no	—
100	C₈	—	16	2	no	—
101	C₈	C₄×Q₁₆	1	4	yes	—
102	C₈	#467	1	4	yes	—
103	C₂²	—	2	2	no	—
104	C₂²	#1707	1	2	yes	—
105	C₂²	#1957	1	1	yes	characteristic
106	C₄	—	2	2	no	—
107	C₄	—	16	4	no	—
108	C₄	#1709	1	1	yes	characteristic
109	C₄	#1713	1	1	yes	characteristic
110	C₄	#1969	1	2	yes	—
111	C₄	#2148	1	4	yes	—
112	C₂	—	2	2	no	—
113	C₂	#9825	1	1	yes	characteristic
114	C₂	#13214	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#1710

C₂

yes

maximal, characteristic

#13290

C₂

yes

maximal

C₈×M₃₂

C₂

yes

maximal, characteristic

#8823

C₂

yes

maximal, characteristic

C₂×#500

C₂²

yes

—

C₂×#487

—

C₄×M₃₂

C₂²

yes

—

#1848

C₂²

yes

—

#1710

—

#1710

—

#1710

C₄

yes

—

#1567

—

C₂×C₈²

C₂²

yes

characteristic

#1387

—

#1311

—

C₈×C₁₆

—

C₂×C₄⋊Q₈

—

C₂×M₃₂

D₄

yes

—

#500

—

#500

—

#500

C₂×C₄

yes

—

—

—

—

—

—

—

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

C₂³

yes

characteristic

#346

—

C₄×C₁₆

—

#335

—

C₈²

C₂×C₄

yes

—

C₈²

D₄

yes

characteristic

C₈²

D₄

yes

characteristic

C₂²×Q₈

—

C₂×M₁₆

—

C₂²×C₈

SD₁₆

yes

—

C₂²×C₈

C₂×D₄

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

M₃₂

Q₁₆

yes

—

C₂×C₁₆

—

#108

—

#107

—

#107

—

C₄⋊C₈

—

#103

—

#103

—

C₄×C₈

C₂²⋊C₄

yes

—

C₄×C₈

C₂²⋊C₄

yes

—

C₄×C₈

D₈

yes

—

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

#41

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

#127

yes

characteristic

C₂²×C₄

C₂×SD₁₆

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#102

yes

—

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

—

C₄²

C₂×C₂²⋊C₄

yes

characteristic

C₄²

#124

yes

characteristic

C₄²

C₂×D₈

yes

—

C₁₆

—

C₂³

#492

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#390

yes

characteristic

C₂×C₄

C₂×#102

yes

—

C₂×C₄

#486

yes

—

C₂×C₄

#493

yes

characteristic

C₂×C₄

#494

yes

—

C₂×C₄

#495

yes

—

C₈

—

C₈

—

100

C₈

—

101

C₈

C₄×Q₁₆

yes

—

102

C₈

#467

yes

—

103

C₂²

—

104

C₂²

#1707

yes

—

105

C₂²

#1957

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

#1709

yes

characteristic

109

C₄

#1713

yes

characteristic

110

C₄

#1969

yes

—

111

C₄

#2148

yes

—

112

C₂

—

113

C₂

#9825

yes

characteristic

114

C₂

#13214

yes

—

	subgroup	G/H
center, socle	C₂²	#1957
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#492
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

#492

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

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

subgroup	index
C₂×#1710	2
#13290	2
C₈×M₃₂	2
#8823	2

subgroup

index

C₂×#1710

#13290

C₈×M₃₂

#8823

Subgroups — #141697

Conjugacy classes of subgroups

Maximal subgroups