Group #9730 — subgroups

Subgroups — #9730

Order 256.

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

normal subgroups only

This group has 227 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	#1987	C₂	1	2	yes	maximal
3	C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₂×#335	C₂	1	1	yes	maximal, characteristic
5	C₂×#333	C₂	1	2	yes	maximal
6	C₂×C₄⋊Q₈	C₂²	1	2	yes	—
7	#500	—	2	2	no	—
8	#500	C₄	1	4	yes	complemented
9	C₂×#107	—	2	1	no	—
10	C₂×#107	—	2	1	no	—
11	C₂×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈	—	2	1	no	—
14	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	#340	—	2	4	no	—
17	#336	—	2	1	no	—
18	#335	—	2	2	no	—
19	#333	—	2	4	no	—
20	C₂²×Q₈	—	2	2	no	—
21	C₂²×C₈	—	2	1	no	—
22	C₂²×C₈	—	2	2	no	—
23	C₂²×C₈	D₄	1	1	yes	characteristic
24	C₂²×C₈	D₄	1	1	yes	characteristic
25	C₄⋊Q₈	—	2	2	no	—
26	C₄⋊Q₈	—	2	2	no	—
27	C₄⋊Q₈	—	2	2	no	—
28	C₄⋊Q₈	C₂×C₄	1	4	yes	—
29	C₂×C₄⋊C₄	—	2	2	no	—
30	C₂×C₄⋊C₄	—	2	2	no	—
31	C₂×C₄⋊C₄	D₄	1	2	yes	—
32	C₂×C₄²	—	2	1	no	—
33	C₂×C₄²	C₂³	1	1	yes	characteristic
34	#107	—	2	4	no	—
35	#107	—	2	4	no	—
36	C₄⋊C₈	—	2	2	no	—
37	C₄⋊C₈	—	2	2	no	—
38	C₄⋊C₈	—	2	1	no	—
39	C₄⋊C₈	D₄	1	2	yes	—
40	C₄×C₈	—	4	2	no	—
41	C₄×C₈	C₂×C₄	1	2	yes	—
42	C₄×C₈	D₄	1	2	yes	—
43	#95	—	2	1	no	—
44	C₂×Q₈	—	2	4	no	—
45	C₂×Q₈	—	4	2	no	—
46	C₂×Q₈	—	8	4	no	—
47	C₂²×C₄	—	2	2	no	—
48	C₂²×C₄	—	4	2	no	—
49	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄	#41	1	1	yes	characteristic
52	C₂×C₈	—	2	1	no	—
53	C₂×C₈	—	2	1	no	—
54	C₂×C₈	—	2	2	no	—
55	C₂×C₈	—	4	1	no	—
56	C₂×C₈	—	4	1	no	—
57	C₂×C₈	—	4	2	no	—
58	C₂×C₈	—	4	4	no	—
59	C₂×C₈	C₂²⋊C₄	1	2	yes	—
60	C₂×C₈	C₂²⋊C₄	1	2	yes	—
61	C₂×C₈	SD₁₆	1	2	yes	—
62	C₂×C₈	Q₁₆	1	2	yes	—
63	C₄⋊C₄	—	2	2	no	—
64	C₄⋊C₄	—	2	4	no	—
65	C₄⋊C₄	—	2	4	no	—
66	C₄⋊C₄	—	4	2	no	—
67	C₄⋊C₄	—	4	2	no	—
68	C₄⋊C₄	Q₁₆	1	4	yes	—
69	C₄²	—	4	1	no	—
70	C₄²	—	4	1	no	—
71	C₄²	C₂²×C₄	1	1	yes	characteristic
72	C₄²	C₂×D₄	1	1	yes	characteristic
73	C₄²	C₂×D₄	1	1	yes	characteristic
74	C₄²	#41	1	1	yes	characteristic
75	C₂³	#121	1	1	yes	characteristic
76	Q₈	—	8	4	no	—
77	Q₈	—	8	4	no	—
78	C₂×C₄	—	2	2	no	—
79	C₂×C₄	—	2	4	no	—
80	C₂×C₄	—	4	2	no	—
81	C₂×C₄	—	4	2	no	—
82	C₂×C₄	—	4	4	no	—
83	C₂×C₄	—	8	4	no	—
84	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄	C₄×D₄	1	1	yes	characteristic
87	C₂×C₄	#121	1	1	yes	characteristic
88	C₂×C₄	#124	1	1	yes	characteristic
89	C₂×C₄	#127	1	1	yes	characteristic
90	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
91	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
92	C₂×C₄	C₂×Q₁₆	1	2	yes	—
93	C₂×C₄	#135	1	2	yes	—
94	C₂×C₄	#137	1	1	yes	characteristic
95	C₂×C₄	#137	1	1	yes	characteristic
96	C₈	—	2	2	no	—
97	C₈	—	4	2	no	—
98	C₈	—	8	2	no	—
99	C₈	#103	1	4	yes	—
100	C₂²	#390	1	1	yes	characteristic
101	C₂²	#461	1	1	yes	characteristic
102	C₂²	#462	1	1	yes	characteristic
103	C₂²	#467	1	2	yes	—
104	C₂²	#472	1	2	yes	—
105	C₄	—	2	1	no	—
106	C₄	—	2	1	no	—
107	C₄	—	4	4	no	—
108	C₄	—	8	4	no	—
109	C₄	C₂×#103	1	1	yes	characteristic
110	C₄	#419	1	1	yes	characteristic
111	C₄	C₄×Q₁₆	1	2	yes	—
112	C₄	#461	1	1	yes	characteristic
113	C₄	#462	1	1	yes	characteristic
114	C₄	#494	1	1	yes	characteristic
115	C₄	#495	1	1	yes	characteristic
116	C₂	#1675	1	2	yes	—
117	C₂	#1676	1	2	yes	—
118	C₂	#1969	1	1	yes	characteristic
119	C₂	#1978	1	1	yes	characteristic
120	C₂	#1981	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#500

C₂

yes

maximal, characteristic

#1987

C₂

yes

maximal

C₈×C₄⋊C₄

C₂

yes

maximal, characteristic

C₂×#335

C₂

yes

maximal, characteristic

C₂×#333

C₂

yes

maximal

C₂×C₄⋊Q₈

C₂²

yes

—

#500

—

#500

C₄

yes

complemented

C₂×#107

—

C₂×#107

—

C₂×C₄⋊C₈

C₂²

yes

—

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₄×C₄⋊C₄

C₂²

yes

—

—

—

—

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

D₄

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×C₄

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

D₄

yes

—

C₂×C₄²

—

C₂×C₄²

C₂³

yes

characteristic

#107

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

D₄

yes

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

D₄

yes

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

#41

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄

yes

—

C₂×C₈

C₂²⋊C₄

yes

—

C₂×C₈

SD₁₆

yes

—

C₂×C₈

Q₁₆

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

Q₁₆

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

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×SD₁₆

yes

characteristic

C₂×C₄

C₂×Q₁₆

yes

characteristic

C₂×C₄

C₂×Q₁₆

yes

—

C₂×C₄

#135

yes

—

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

C₈

—

C₈

#103

yes

—

100

C₂²

#390

yes

characteristic

101

C₂²

#461

yes

characteristic

102

C₂²

#462

yes

characteristic

103

C₂²

#467

yes

—

104

C₂²

#472

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

C₂×#103

yes

characteristic

110

C₄

#419

yes

characteristic

111

C₄

C₄×Q₁₆

yes

—

112

C₄

#461

yes

characteristic

113

C₄

#462

yes

characteristic

114

C₄

#494

yes

characteristic

115

C₄

#495

yes

characteristic

116

C₂

#1675

yes

—

117

C₂

#1676

yes

—

118

C₂

#1969

yes

characteristic

119

C₂

#1978

yes

characteristic

120

C₂

#1981

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	#121
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

#121

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
#1987	2
C₈×C₄⋊C₄	2
C₂×#335	2
C₂×#333	2

subgroup

index

C₈×C₄⋊C₄

Subgroups — #9730

Conjugacy classes of subgroups

Complements

Maximal subgroups