Group #10276 — subgroups

Subgroups — #10276

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×#494	C₂	1	1	yes	maximal, characteristic
2	C₂×#475	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	#494	—	2	4	no	—
10	#475	—	2	4	no	—
11	#475	—	2	4	no	—
12	C₂×#106	—	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	#106	—	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	4	no	—
51	C₂×Q₈	—	4	4	no	—
52	C₂×Q₈	Q₁₆	1	2	yes	complemented
53	C₂²×C₄	—	2	1	no	—
54	C₂²×C₄	—	2	1	no	—
55	C₂²×C₄	—	4	2	no	—
56	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄	#41	1	1	yes	characteristic
59	Q₁₆	—	4	16	no	—
60	C₂×C₈	—	2	2	no	—
61	C₂×C₈	—	2	2	no	—
62	C₂×C₈	—	2	4	no	—
63	C₂×C₈	—	4	2	no	—
64	C₂×C₈	—	4	2	no	—
65	C₂×C₈	—	4	1	no	—
66	C₂×C₈	—	4	2	no	—
67	C₂×C₈	SD₁₆	1	4	yes	—
68	C₄⋊C₄	—	2	4	no	—
69	C₄⋊C₄	—	2	2	no	—
70	C₄⋊C₄	—	2	4	no	—
71	C₄⋊C₄	—	4	2	no	—
72	C₄⋊C₄	—	4	2	no	—
73	C₄⋊C₄	—	4	2	no	—
74	C₄²	—	4	1	no	—
75	C₄²	—	4	1	no	—
76	C₄²	C₂×D₄	1	2	yes	—
77	C₄²	C₂×D₄	1	2	yes	—
78	C₂³	#121	1	1	yes	characteristic
79	Q₈	—	2	4	no	—
80	Q₈	—	4	2	no	—
81	Q₈	—	4	8	no	—
82	Q₈	—	8	2	no	—
83	Q₈	—	8	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	4	1	no	—
87	C₂×C₄	—	4	1	no	—
88	C₂×C₄	—	4	1	no	—
89	C₂×C₄	—	4	4	no	—
90	C₂×C₄	—	4	2	no	—
91	C₂×C₄	—	8	4	no	—
92	C₂×C₄	#120	1	1	yes	characteristic
93	C₂×C₄	#120	1	2	yes	—
94	C₂×C₄	#121	1	2	yes	—
95	C₂×C₄	#127	1	1	yes	characteristic
96	C₂×C₄	C₂×SD₁₆	1	2	yes	—
97	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
98	C₂×C₄	#135	1	1	yes	characteristic
99	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
100	C₂×C₄	#137	1	2	yes	—
101	C₈	—	2	4	no	—
102	C₈	—	4	4	no	—
103	C₈	—	8	1	no	—
104	C₂²	#392	1	1	yes	characteristic
105	C₂²	#460	1	2	yes	—
106	C₂²	#467	1	1	yes	characteristic
107	C₂²	#470	1	1	yes	characteristic
108	C₂²	#473	1	2	yes	—
109	C₄	—	2	2	no	—
110	C₄	—	4	2	no	—
111	C₄	—	8	4	no	—
112	C₄	—	8	1	no	—
113	C₄	#450	1	2	yes	—
114	C₄	#451	1	2	yes	—
115	C₄	#452	1	2	yes	—
116	C₄	#492	1	1	yes	characteristic
117	C₄	#496	1	1	yes	characteristic
118	C₂	#1670	1	2	yes	—
119	C₂	#1690	1	2	yes	—
120	C₂	#2001	1	1	yes	characteristic
121	C₂	#2014	1	1	yes	characteristic
122	C₂	#2020	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#494

C₂

yes

maximal, characteristic

C₂×#475

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

#106

—

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

Q₁₆

yes

complemented

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

SD₁₆

yes

—

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

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#120

yes

characteristic

C₂×C₄

#120

yes

—

C₂×C₄

#121

yes

—

C₂×C₄

#127

yes

characteristic

C₂×C₄

C₂×SD₁₆

yes

—

C₂×C₄

C₂×Q₁₆

yes

C₂×C₄

yes

C₂×C₄

yes

—

100

C₂×C₄

#137

yes

—

101

C₈

—

102

C₈

—

103

C₈

—

104

C₂²

#392

yes

characteristic

105

C₂²

#460

yes

—

106

C₂²

#467

yes

characteristic

107

C₂²

#470

yes

characteristic

108

C₂²

#473

yes

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

#450

yes

—

114

C₄

#451

yes

—

115

C₄

#452

yes

—

116

C₄

#492

yes

characteristic

117

C₄

#496

yes

characteristic

118

C₂

#1670

yes

—

119

C₂

#1690

yes

—

120

C₂

#2001

yes

characteristic

121

C₂

#2014

yes

characteristic

122

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
52. C₂×Q₈	Q₁₆	semidirect

H ≅ G/N

type

52. C₂×Q₈

Q₁₆

semidirect

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

subgroup	index
C₂×#494	2
C₂×#475	2
#1984	2
C₂×#332	2
#1277	2

subgroup

index

Subgroups — #10276

Conjugacy classes of subgroups

Complements

Maximal subgroups