Group #10265 — subgroups

Subgroups — #10265

Order 256.

287 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 287 conjugacy classes of subgroups — too many to draw. Only its 79 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	C₂×#462	C₂	1	2	yes	maximal
3	#1984	C₂	1	1	yes	maximal, characteristic
4	C₂×#332	C₂	1	2	yes	maximal
5	#1277	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆	—	2	2	no	—
7	C₂×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
9	#500	—	2	4	no	—
10	#462	—	2	4	no	—
11	#462	—	2	4	no	—
12	C₂×#107	—	2	2	no	—
13	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×#103	—	2	2	no	—
16	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
17	#336	—	2	2	no	—
18	#332	—	2	4	no	—
19	C₂²×Q₈	—	2	2	no	—
20	C₂²×Q₈	D₄	1	1	yes	characteristic
21	C₂×Q₁₆	—	2	8	no	—
22	C₂×Q₁₆	—	4	4	no	—
23	C₂×Q₁₆	—	4	4	no	—
24	C₂²×C₈	—	2	2	no	—
25	C₂²×C₈	—	2	1	no	—
26	C₂²×C₈	D₄	1	2	yes	—
27	C₄⋊Q₈	—	2	2	no	—
28	C₄⋊Q₈	—	2	2	no	—
29	C₄⋊Q₈	—	2	2	no	—
30	C₄⋊Q₈	D₄	1	4	yes	—
31	C₄×Q₈	—	2	2	no	—
32	C₄×Q₈	—	2	2	no	—
33	C₂×C₄⋊C₄	—	2	2	no	—
34	C₂×C₄⋊C₄	—	2	2	no	—
35	C₂×C₄⋊C₄	—	2	1	no	—
36	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
37	C₂×C₄²	—	2	1	no	—
38	C₂×C₄²	C₂³	1	1	yes	characteristic
39	#107	—	2	8	no	—
40	C₄⋊C₈	—	2	2	no	—
41	C₄⋊C₈	—	2	2	no	—
42	C₄⋊C₈	D₄	1	4	yes	—
43	#103	—	2	8	no	—
44	C₄×C₈	—	2	2	no	—
45	C₂×Q₈	—	2	1	no	—
46	C₂×Q₈	—	2	2	no	—
47	C₂×Q₈	—	2	4	no	—
48	C₂×Q₈	—	2	2	no	—
49	C₂×Q₈	—	4	2	no	—
50	C₂×Q₈	—	4	2	no	—
51	C₂×Q₈	—	4	2	no	—
52	C₂×Q₈	—	4	2	no	—
53	C₂×Q₈	—	4	2	no	—
54	C₂×Q₈	D₈	1	2	yes	—
55	C₂²×C₄	—	2	1	no	—
56	C₂²×C₄	—	2	1	no	—
57	C₂²×C₄	—	4	2	no	—
58	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄	#41	1	1	yes	characteristic
61	Q₁₆	—	4	16	no	—
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	—	2	2	no	—
64	C₂×C₈	—	2	4	no	—
65	C₂×C₈	—	4	2	no	—
66	C₂×C₈	—	4	2	no	—
67	C₂×C₈	—	4	1	no	—
68	C₂×C₈	—	4	2	no	—
69	C₂×C₈	Q₁₆	1	4	yes	complemented
70	C₄⋊C₄	—	2	2	no	—
71	C₄⋊C₄	—	2	4	no	—
72	C₄⋊C₄	—	2	4	no	—
73	C₄⋊C₄	—	4	2	no	—
74	C₄⋊C₄	—	4	2	no	—
75	C₄⋊C₄	—	4	2	no	—
76	C₄²	—	4	1	no	—
77	C₄²	—	4	1	no	—
78	C₄²	C₂×D₄	1	2	yes	—
79	C₄²	C₂×D₄	1	2	yes	—
80	C₂³	#121	1	1	yes	characteristic
81	Q₈	—	2	4	no	—
82	Q₈	—	4	2	no	—
83	Q₈	—	4	4	no	—
84	Q₈	—	4	4	no	—
85	Q₈	—	8	2	no	—
86	Q₈	—	8	2	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	4	1	no	—
90	C₂×C₄	—	4	1	no	—
91	C₂×C₄	—	4	1	no	—
92	C₂×C₄	—	4	4	no	—
93	C₂×C₄	—	4	2	no	—
94	C₂×C₄	—	8	2	no	—
95	C₂×C₄	—	8	2	no	—
96	C₂×C₄	#120	1	2	yes	—
97	C₂×C₄	#120	1	1	yes	characteristic
98	C₂×C₄	#121	1	2	yes	—
99	C₂×C₄	#127	1	1	yes	characteristic
100	C₂×C₄	C₂×D₈	1	1	yes	characteristic
101	C₂×C₄	C₂×Q₁₆	1	2	yes	—
102	C₂×C₄	#135	1	1	yes	characteristic
103	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
104	C₂×C₄	#137	1	2	yes	—
105	C₈	—	2	4	no	—
106	C₈	—	4	4	no	—
107	C₈	—	8	1	no	—
108	C₂²	#392	1	1	yes	characteristic
109	C₂²	#462	1	2	yes	—
110	C₂²	#466	1	1	yes	characteristic
111	C₂²	#469	1	1	yes	characteristic
112	C₂²	#473	1	2	yes	—
113	C₄	—	2	2	no	—
114	C₄	—	4	2	no	—
115	C₄	—	8	4	no	—
116	C₄	—	8	1	no	—
117	C₄	#447	1	2	yes	—
118	C₄	#449	1	2	yes	—
119	C₄	#451	1	2	yes	—
120	C₄	#494	1	1	yes	characteristic
121	C₄	#497	1	1	yes	characteristic
122	C₂	#1674	1	2	yes	—
123	C₂	#1694	1	2	yes	—
124	C₂	#1999	1	1	yes	characteristic
125	C₂	#2016	1	1	yes	characteristic
126	C₂	#2020	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#500

C₂

yes

maximal, characteristic

C₂×#462

C₂

yes

maximal

#1984

C₂

yes

maximal, characteristic

C₂×#332

C₂

yes

maximal

#1277

C₂

yes

maximal, characteristic

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

C₂²

yes

—

C₂×C₄×Q₈

C₂²

yes

—

—

—

—

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

—

C₂×#103

—

C₂×C₄×C₈

C₂²

yes

characteristic

#336

—

#332

—

C₂²×Q₈

—

C₂²×Q₈

D₄

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

D₄

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

D₄

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

D₄

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂³

yes

characteristic

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

D₄

yes

—

#103

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₈

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

#41

yes

characteristic

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Q₁₆

yes

complemented

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂×D₄

yes

—

C₄²

C₂×D₄

yes

—

C₂³

#121

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#120

yes

—

C₂×C₄

#120

yes

characteristic

C₂×C₄

#121

yes

—

C₂×C₄

#127

yes

characteristic

100

C₂×C₄

C₂×D₈

yes

characteristic

101

C₂×C₄

C₂×Q₁₆

yes

—

102

C₂×C₄

#135

yes

characteristic

103

C₂×C₄

AGL(1,ℤ/8)

yes

—

104

C₂×C₄

#137

yes

—

105

C₈

—

106

C₈

—

107

C₈

—

108

C₂²

#392

yes

characteristic

109

C₂²

#462

yes

—

110

C₂²

#466

yes

characteristic

111

C₂²

#469

yes

characteristic

112

C₂²

#473

yes

—

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

#447

yes

—

118

C₄

#449

yes

—

119

C₄

#451

yes

—

120

C₄

#494

yes

characteristic

121

C₄

#497

yes

characteristic

122

C₂

#1674

yes

—

123

C₂

#1694

yes

—

124

C₂

#1999

yes

characteristic

125

C₂

#2016

yes

characteristic

126

C₂

#2020

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂³
LCS	G ▷ C₂×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, Frattini, cosocle

C₂×C₄²

C₂³

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
69. C₂×C₈	Q₁₆	semidirect

H ≅ G/N

type

69. C₂×C₈

Q₁₆

semidirect

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

subgroup	index
C₂×#500	2
C₂×#462	2
#1984	2
C₂×#332	2
#1277	2

subgroup

index

Subgroups — #10265

Conjugacy classes of subgroups

Complements

Maximal subgroups