Group #9831 — subgroups

Subgroups — #9831

Order 256.

247 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 247 conjugacy classes of subgroups — too many to draw. Only its 95 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×#500	C₂	1	1	yes	maximal, characteristic
2	#1990	C₂	1	2	yes	maximal
3	#1956	C₂	1	2	yes	maximal
4	#1834	C₂	1	1	yes	maximal, characteristic
5	C₂×#322	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Q₈	C₂²	1	2	yes	—
7	#500	—	2	2	no	—
8	#500	C₄	1	4	yes	complemented
9	C₂×#107	—	2	2	no	—
10	C₂×#103	—	2	4	no	—
11	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
14	#384	C₂²	1	2	yes	—
15	#340	—	2	2	no	—
16	#340	—	2	4	no	—
17	#322	—	2	2	no	—
18	C₂²×Q₈	—	2	2	no	—
19	C₂²×C₈	D₄	1	2	yes	—
20	C₂²×C₈	D₄	1	2	yes	—
21	C₂²×C₈	D₄	1	2	yes	—
22	C₄⋊Q₈	—	2	2	no	—
23	C₄⋊Q₈	—	2	2	no	—
24	C₄⋊Q₈	—	2	2	no	—
25	C₄⋊Q₈	C₂×C₄	1	4	yes	—
26	C₂×C₄⋊C₄	—	2	2	no	—
27	C₂×C₄⋊C₄	—	2	2	no	—
28	C₂×C₄⋊C₄	—	2	2	no	—
29	C₂×C₄⋊C₄	—	2	2	no	—
30	C₂×C₄⋊C₄	—	2	2	no	—
31	C₂×C₄²	C₂³	1	1	yes	characteristic
32	#107	—	2	8	no	—
33	#103	—	4	8	no	—
34	C₄×C₈	—	2	1	no	—
35	C₄×C₈	—	2	1	no	—
36	C₄×C₈	—	2	1	no	—
37	C₄×C₈	—	2	1	no	—
38	C₄×C₈	C₂×C₄	1	2	yes	—
39	C₄×C₈	D₄	1	2	yes	—
40	C₂×Q₈	—	2	4	no	—
41	C₂×Q₈	—	4	2	no	—
42	C₂×Q₈	—	4	4	no	—
43	C₂×Q₈	—	4	4	no	—
44	C₂²×C₄	—	4	2	no	—
45	C₂²×C₄	—	4	2	no	—
46	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈	—	2	2	no	—
50	C₂×C₈	—	2	2	no	—
51	C₂×C₈	—	2	4	no	—
52	C₂×C₈	—	2	2	no	—
53	C₂×C₈	—	2	4	no	—
54	C₂×C₈	C₂²⋊C₄	1	4	yes	—
55	C₂×C₈	D₈	1	4	yes	—
56	C₄⋊C₄	—	2	4	no	—
57	C₄⋊C₄	—	2	4	no	—
58	C₄⋊C₄	—	4	2	no	—
59	C₄⋊C₄	—	4	2	no	—
60	C₄⋊C₄	—	4	4	no	—
61	C₄⋊C₄	—	4	4	no	—
62	C₄⋊C₄	—	4	4	no	—
63	C₄²	C₂²×C₄	1	1	yes	characteristic
64	C₄²	C₂×D₄	1	1	yes	characteristic
65	C₄²	#41	1	1	yes	characteristic
66	C₄²	#41	1	1	yes	characteristic
67	C₂³	#127	1	1	yes	characteristic
68	Q₈	—	4	8	no	—
69	Q₈	—	8	4	no	—
70	C₂×C₄	—	4	4	no	—
71	C₂×C₄	—	8	2	no	—
72	C₂×C₄	—	8	2	no	—
73	C₂×C₄	—	8	2	no	—
74	C₂×C₄	—	8	2	no	—
75	C₂×C₄	—	8	2	no	—
76	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄	C₄×D₄	1	1	yes	characteristic
78	C₂×C₄	C₄×D₄	1	1	yes	characteristic
79	C₂×C₄	#121	1	1	yes	characteristic
80	C₂×C₄	#121	1	1	yes	characteristic
81	C₂×C₄	#124	1	1	yes	characteristic
82	C₂×C₄	C₂×D₈	1	2	yes	—
83	C₂×C₄	#135	1	2	yes	—
84	C₂×C₄	#137	1	4	yes	—
85	C₂×C₄	#137	1	4	yes	—
86	C₈	—	2	4	no	—
87	C₈	—	2	4	no	—
88	C₈	—	4	2	no	—
89	C₂²	#390	1	1	yes	characteristic
90	C₂²	#493	1	1	yes	characteristic
91	C₂²	#495	1	1	yes	characteristic
92	C₂²	#497	1	2	yes	—
93	C₂²	#497	1	2	yes	—
94	C₄	—	8	4	no	—
95	C₄	—	16	2	no	—
96	C₄	#419	1	2	yes	—
97	C₄	C₄×D₈	1	2	yes	—
98	C₄	#441	1	2	yes	—
99	C₄	#466	1	2	yes	—
100	C₄	#470	1	2	yes	—
101	C₄	#491	1	2	yes	—
102	C₂	#1718	1	2	yes	—
103	C₂	#1722	1	2	yes	—
104	C₂	#1960	1	1	yes	characteristic
105	C₂	#1970	1	1	yes	characteristic
106	C₂	#1971	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#500

C₂

yes

maximal, characteristic

#1990

C₂

yes

maximal

#1956

C₂

yes

maximal

#1834

C₂

yes

maximal, characteristic

C₂×#322

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈

C₂²

yes

—

#500

—

#500

C₄

yes

complemented

C₂×#107

—

C₂×#103

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

#384

C₂²

yes

—

#340

—

#340

—

#322

—

C₂²×Q₈

—

C₂²×C₈

D₄

yes

—

C₂²×C₈

D₄

yes

—

C₂²×C₈

D₄

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×C₄

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂³

yes

characteristic

#107

—

#103

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

D₄

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄

yes

—

C₂×C₈

D₈

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂×D₄

yes

C₄²

yes

C₄²

yes

C₂³

yes

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄

C₄×D₄

yes

characteristic

C₂×C₄

C₄×D₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂×D₈

yes

—

C₂×C₄

#135

yes

—

C₂×C₄

#137

yes

—

C₂×C₄

#137

yes

—

C₈

—

C₈

—

C₈

—

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#497

yes

—

C₄

—

C₄

—

C₄

#419

yes

—

C₄

C₄×D₈

yes

—

C₄

#441

yes

—

C₄

#466

yes

—

100

C₄

#470

yes

—

101

C₄

#491

yes

—

102

C₂

#1718

yes

—

103

C₂

#1722

yes

—

104

C₂

#1960

yes

characteristic

105

C₂

#1970

yes

characteristic

106

C₂

#1971

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	#127
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

#127

commutator

C₄²

C₂²×C₄

Frattini, cosocle

C₂×C₄²

C₂³

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
8. #500	C₄	semidirect

H ≅ G/N

type

8. #500

C₄

semidirect

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

subgroup	index
C₂×#500	2
#1990	2
#1956	2
#1834	2
C₂×#322	2

subgroup

index

Subgroups — #9831

Conjugacy classes of subgroups

Complements

Maximal subgroups