Group #9823 — subgroups

Subgroups — #9823

Order 256.

263 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 263 conjugacy classes of subgroups — too many to draw. Only its 115 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	#1845	C₂	1	1	yes	maximal, characteristic
5	C₂×C₈²	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×#107	—	2	2	no	—
11	C₂×#103	—	2	4	no	—
12	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
15	#384	C₂²	1	2	yes	—
16	#340	—	2	4	no	—
17	C₈²	—	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	#107	—	4	4	no	—
34	#103	—	4	8	no	—
35	C₄×C₈	—	2	1	no	—
36	C₄×C₈	—	2	1	no	—
37	C₄×C₈	—	2	1	no	—
38	C₄×C₈	—	2	1	no	—
39	C₄×C₈	C₂×C₄	1	2	yes	—
40	C₄×C₈	D₄	1	2	yes	—
41	C₂×Q₈	—	2	4	no	—
42	C₂×Q₈	—	4	2	no	—
43	C₂×Q₈	—	4	4	no	—
44	C₂×Q₈	—	4	4	no	—
45	C₂²×C₄	—	4	2	no	—
46	C₂²×C₄	—	4	2	no	—
47	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈	—	2	2	no	—
51	C₂×C₈	—	2	2	no	—
52	C₂×C₈	—	2	2	no	—
53	C₂×C₈	—	2	2	no	—
54	C₂×C₈	C₂²⋊C₄	1	4	yes	—
55	C₂×C₈	D₈	1	4	yes	—
56	C₂×C₈	SD₁₆	1	4	yes	—
57	C₂×C₈	Q₁₆	1	4	yes	—
58	C₂×C₈	Q₁₆	1	4	yes	—
59	C₄⋊C₄	—	2	4	no	—
60	C₄⋊C₄	—	2	4	no	—
61	C₄⋊C₄	—	4	2	no	—
62	C₄⋊C₄	—	4	2	no	—
63	C₄⋊C₄	—	4	4	no	—
64	C₄⋊C₄	—	4	4	no	—
65	C₄⋊C₄	—	4	4	no	—
66	C₄²	C₂²×C₄	1	1	yes	characteristic
67	C₄²	C₂×D₄	1	1	yes	characteristic
68	C₄²	#41	1	1	yes	characteristic
69	C₄²	#41	1	1	yes	characteristic
70	C₂³	#127	1	1	yes	characteristic
71	Q₈	—	4	8	no	—
72	Q₈	—	8	4	no	—
73	C₂×C₄	—	4	4	no	—
74	C₂×C₄	—	8	2	no	—
75	C₂×C₄	—	8	2	no	—
76	C₂×C₄	—	8	2	no	—
77	C₂×C₄	—	8	2	no	—
78	C₂×C₄	—	8	2	no	—
79	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄	C₄×D₄	1	1	yes	characteristic
81	C₂×C₄	C₄×D₄	1	1	yes	characteristic
82	C₂×C₄	#121	1	1	yes	characteristic
83	C₂×C₄	#121	1	1	yes	characteristic
84	C₂×C₄	#124	1	1	yes	characteristic
85	C₂×C₄	C₂×D₈	1	2	yes	—
86	C₂×C₄	C₂×SD₁₆	1	2	yes	—
87	C₂×C₄	C₂×Q₁₆	1	2	yes	—
88	C₂×C₄	C₂×Q₁₆	1	2	yes	—
89	C₂×C₄	#135	1	2	yes	—
90	C₂×C₄	#135	1	2	yes	—
91	C₈	—	2	4	no	—
92	C₈	—	2	4	no	—
93	C₈	#103	1	8	yes	complemented
94	C₂²	#390	1	1	yes	characteristic
95	C₂²	#492	1	1	yes	characteristic
96	C₂²	#493	1	1	yes	characteristic
97	C₂²	#494	1	1	yes	characteristic
98	C₂²	#494	1	1	yes	characteristic
99	C₂²	#495	1	1	yes	characteristic
100	C₂²	#495	1	1	yes	characteristic
101	C₄	—	8	4	no	—
102	C₄	—	16	2	no	—
103	C₄	C₂×#103	1	2	yes	—
104	C₄	C₄×D₈	1	2	yes	—
105	C₄	C₄×Q₁₆	1	2	yes	—
106	C₄	#466	1	2	yes	—
107	C₄	#467	1	2	yes	—
108	C₄	#487	1	2	yes	—
109	C₂	#1706	1	1	yes	characteristic
110	C₂	#1707	1	1	yes	characteristic
111	C₂	#1710	1	1	yes	characteristic
112	C₂	#1713	1	1	yes	characteristic
113	C₂	#1956	1	1	yes	characteristic
114	C₂	#1969	1	1	yes	characteristic
115	C₂	#1970	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

#1845

C₂

yes

maximal, characteristic

C₂×C₈²

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈

C₂²

yes

—

#500

—

#500

C₄

yes

—

—

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

#384

C₂²

yes

—

#340

—

C₈²

—

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

—

—

—

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

C₂²⋊C₄

yes

—

C₂×C₈

D₈

yes

—

C₂×C₈

SD₁₆

yes

—

C₂×C₈

Q₁₆

yes

—

C₂×C₈

Q₁₆

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

C₂×SD₁₆

yes

—

C₂×C₄

C₂×Q₁₆

yes

—

C₂×C₄

C₂×Q₁₆

yes

—

C₂×C₄

#135

yes

—

C₂×C₄

#135

yes

—

C₈

—

C₈

—

C₈

yes

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

100

C₂²

#495

yes

characteristic

101

C₄

—

102

C₄

—

103

C₄

C₂×#103

yes

—

104

C₄

C₄×D₈

yes

—

105

C₄

C₄×Q₁₆

yes

—

106

C₄

#466

yes

—

107

C₄

#467

yes

—

108

C₄

#487

yes

—

109

C₂

#1706

yes

characteristic

110

C₂

#1707

yes

characteristic

111

C₂

#1710

yes

characteristic

112

C₂

#1713

yes

characteristic

113

C₂

#1956

yes

characteristic

114

C₂

#1969

yes

characteristic

115

C₂

#1970

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
93. C₈	#103	semidirect

H ≅ G/N

type

8. #500

C₄

semidirect

93. C₈

#103

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
#1845	2
C₂×C₈²	2

subgroup

index

C₂×C₈²

Subgroups — #9823

Conjugacy classes of subgroups

Complements

Maximal subgroups