Group #7263 — subgroups

Subgroups — #7263

Order 256.

168 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 53 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	#2112	C₂	1	1	yes	maximal, characteristic
2	#1807	C₂	1	1	yes	maximal, characteristic
3	C₂×#341	C₂	1	1	yes	maximal, characteristic
4	C₂²×M₁₆	—	2	1	no	—
5	C₂³×C₈	C₂²	1	1	yes	characteristic
6	C₂×M₃₂	—	2	1	no	—
7	C₂×#106	C₄	1	2	yes	—
8	#380	—	2	1	no	—
9	C₂²⋊C₁₆	—	2	1	no	—
10	C₂²⋊C₁₆	C₄	1	2	yes	complemented
11	#341	—	2	1	no	—
12	#341	C₄	1	2	yes	complemented
13	#340	—	2	1	no	—
14	C₂³×C₄	D₄	1	1	yes	characteristic
15	C₂×M₁₆	—	2	4	no	—
16	C₂×M₁₆	—	2	2	no	—
17	C₂×M₁₆	—	4	2	no	—
18	C₂×M₁₆	—	4	1	no	—
19	C₂²×C₈	—	4	2	no	—
20	C₂²×C₈	—	4	1	no	—
21	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈	D₄	1	1	yes	characteristic
25	C₂²×C₈	D₄	1	1	yes	characteristic
26	C₂²×C₈	Q₈	1	1	yes	characteristic
27	C₂×C₄⋊C₄	—	2	2	no	—
28	C₂×C₂²⋊C₄	—	4	1	no	—
29	M₃₂	—	4	2	no	—
30	C₂×C₁₆	—	2	2	no	—
31	#106	—	4	2	no	—
32	#95	—	4	1	no	—
33	C₂⁴	SD₁₆	1	1	yes	characteristic
34	C₂²×C₄	—	2	2	no	—
35	C₂²×C₄	—	2	2	no	—
36	C₂²×C₄	—	4	2	no	—
37	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₄	D₈	1	1	yes	characteristic
41	C₂²×C₄	SD₁₆	1	1	yes	characteristic
42	C₂²×C₄	Q₁₆	1	1	yes	characteristic
43	M₁₆	—	2	4	no	—
44	M₁₆	—	4	4	no	—
45	M₁₆	—	4	2	no	—
46	C₂×C₈	—	2	2	no	—
47	C₂×C₈	—	2	2	no	—
48	C₂×C₈	—	4	1	no	—
49	C₂×C₈	—	4	1	no	—
50	C₂×C₈	—	4	1	no	—
51	C₂×C₈	—	4	1	no	—
52	C₂×C₈	—	4	4	no	—
53	C₂×C₈	—	4	2	no	—
54	C₂×C₈	C₄²	1	1	yes	characteristic
55	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
58	C₄⋊C₄	—	4	2	no	—
59	C₄⋊C₄	—	4	2	no	—
60	C₂²⋊C₄	—	8	2	no	—
61	C₁₆	—	4	2	no	—
62	C₂³	—	2	2	no	—
63	C₂³	—	2	2	no	—
64	C₂³	—	4	1	no	—
65	C₂³	#102	1	1	yes	characteristic
66	C₂³	#103	1	1	yes	characteristic
67	C₂³	#106	1	1	yes	characteristic
68	C₂×C₄	—	2	2	no	—
69	C₂×C₄	—	2	2	no	—
70	C₂×C₄	—	2	2	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	8	2	no	—
76	C₂×C₄	—	8	2	no	—
77	C₂×C₄	—	8	2	no	—
78	C₂×C₄	#95	1	1	yes	characteristic
79	C₂×C₄	#99	1	2	yes	—
80	C₂×C₄	#102	1	1	yes	characteristic
81	C₂×C₄	#103	1	1	yes	characteristic
82	C₂×C₄	#104	1	2	yes	—
83	C₂×C₄	#107	1	1	yes	characteristic
84	C₈	—	2	1	no	—
85	C₈	—	2	1	no	—
86	C₈	—	2	1	no	—
87	C₈	—	2	1	no	—
88	C₈	—	4	4	no	—
89	C₂²	—	2	1	no	—
90	C₂²	—	2	1	no	—
91	C₂²	—	2	1	no	—
92	C₂²	—	2	1	no	—
93	C₂²	—	2	1	no	—
94	C₂²	—	2	1	no	—
95	C₂²	—	2	1	no	—
96	C₂²	—	4	4	no	—
97	C₂²	#327	1	2	yes	—
98	C₂²	#340	1	1	yes	characteristic
99	C₂²	#365	1	2	yes	—
100	C₄	—	2	1	no	—
101	C₄	—	2	1	no	—
102	C₄	—	16	2	no	—
103	C₄	#328	1	2	yes	—
104	C₄	#339	1	1	yes	characteristic
105	C₄	#342	1	1	yes	characteristic
106	C₂	—	2	2	no	—
107	C₂	—	2	2	no	—
108	C₂	—	4	1	no	—
109	C₂	#1294	1	1	yes	characteristic
110	C₂	#1383	1	1	yes	characteristic
111	C₂	#1389	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#2112

C₂

yes

maximal, characteristic

#1807

C₂

yes

maximal, characteristic

C₂×#341

C₂

yes

maximal, characteristic

C₂²×M₁₆

—

C₂³×C₈

C₂²

yes

characteristic

C₂×M₃₂

—

C₂×#106

C₄

yes

—

#380

—

C₂²⋊C₁₆

—

C₂²⋊C₁₆

C₄

yes

complemented

#341

—

#341

C₄

yes

complemented

#340

—

C₂³×C₄

D₄

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

Q₈

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

M₃₂

—

C₂×C₁₆

—

#106

—

#95

—

C₂⁴

SD₁₆

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊C₄

yes

characteristic

C₂²×C₄

C₂²⋊C₄

yes

characteristic

C₂²×C₄

C₄⋊C₄

yes

characteristic

C₂²×C₄

D₈

yes

characteristic

C₂²×C₄

SD₁₆

yes

characteristic

C₂²×C₄

Q₁₆

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄²

yes

characteristic

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₈

C₄⋊C₄

yes

characteristic

C₂×C₈

C₄⋊C₄

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₁₆

—

C₂³

—

C₂³

—

C₂³

—

C₂³

yes

C₂³

yes

C₂³

yes

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#95

yes

characteristic

C₂×C₄

#99

yes

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₂×C₄

#107

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#327

yes

—

C₂²

#340

yes

characteristic

C₂²

#365

yes

—

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

#328

yes

—

104

C₄

#339

yes

characteristic

105

C₄

#342

yes

characteristic

106

C₂

—

107

C₂

—

108

C₂

—

109

C₂

#1294

yes

characteristic

110

C₂

#1383

yes

characteristic

111

C₂

#1389

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#340
commutator	C₂×C₈	C₄²
Frattini, cosocle	C₂³×C₈	C₂²
prime radical	C₂⁴	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#340

commutator

C₂×C₈

C₄²

Frattini, cosocle

C₂³×C₈

C₂²

prime radical

C₂⁴

SD₁₆

LCS

G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ G

LECS

G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
10. C₂²⋊C₁₆	C₄	semidirect
12. #341	C₄	semidirect

H ≅ G/N

type

10. C₂²⋊C₁₆

C₄

semidirect

12. #341

C₄

semidirect

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

subgroup	index
#2112	2
#1807	2
C₂×#341	2

subgroup

index

#2112

#1807

C₂×#341

Subgroups — #7263

Conjugacy classes of subgroups

Complements

Maximal subgroups