Group #10545 — subgroups

Subgroups — #10545

Order 256.

227 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 227 conjugacy classes of subgroups — too many to draw. Only its 75 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	#1981	C₂	1	2	yes	maximal
3	#1939	C₂	1	1	yes	maximal, characteristic
4	C₂×#332	C₂	1	2	yes	maximal
5	#1277	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Q₈	C₂²	1	2	yes	—
7	#500	—	2	4	no	—
8	C₂×#107	—	2	1	no	—
9	C₂×#107	—	2	1	no	—
10	C₂×C₄⋊C₈	—	2	1	no	—
11	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×#103	—	2	2	no	—
14	C₂×#103	—	2	2	no	—
15	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	#336	—	2	1	no	—
18	#332	—	2	4	no	—
19	C₂²×Q₈	—	2	2	no	—
20	C₂²×C₈	—	2	1	no	—
21	C₂²×C₈	—	2	2	no	—
22	C₂²×C₈	D₄	1	1	yes	characteristic
23	C₂²×C₈	D₄	1	1	yes	characteristic
24	C₄⋊Q₈	—	2	2	no	—
25	C₄⋊Q₈	—	2	2	no	—
26	C₄⋊Q₈	—	2	2	no	—
27	C₄⋊Q₈	D₄	1	4	yes	—
28	C₂×C₄⋊C₄	—	2	2	no	—
29	C₂×C₄⋊C₄	—	2	2	no	—
30	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
31	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
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	2	no	—
39	C₄⋊C₈	—	4	2	no	—
40	#103	—	4	4	no	—
41	#103	—	4	4	no	—
42	C₄×C₈	—	2	2	no	—
43	#95	—	2	1	no	—
44	C₂×Q₈	—	2	4	no	—
45	C₂×Q₈	—	4	2	no	—
46	C₂×Q₈	—	4	2	no	—
47	C₂×Q₈	—	4	2	no	—
48	C₂×Q₈	—	4	2	no	—
49	C₂×Q₈	—	4	2	no	—
50	C₂²×C₄	—	2	1	no	—
51	C₂²×C₄	—	2	1	no	—
52	C₂²×C₄	—	4	2	no	—
53	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄	#41	1	1	yes	characteristic
56	C₂×C₈	—	2	1	no	—
57	C₂×C₈	—	2	1	no	—
58	C₂×C₈	—	2	1	no	—
59	C₂×C₈	—	2	1	no	—
60	C₂×C₈	—	4	1	no	—
61	C₂×C₈	—	4	2	no	—
62	C₂×C₈	—	4	2	no	—
63	C₂×C₈	—	4	1	no	—
64	C₂×C₈	—	4	2	no	—
65	C₂×C₈	—	4	1	no	—
66	C₂×C₈	SD₁₆	1	2	yes	—
67	C₂×C₈	Q₁₆	1	2	yes	—
68	C₄⋊C₄	—	2	1	no	—
69	C₄⋊C₄	—	2	4	no	—
70	C₄⋊C₄	—	2	4	no	—
71	C₄⋊C₄	—	2	2	no	—
72	C₄⋊C₄	—	4	2	no	—
73	C₄⋊C₄	—	4	2	no	—
74	C₄⋊C₄	D₈	1	2	yes	—
75	C₄²	—	4	1	no	—
76	C₄²	—	4	1	no	—
77	C₄²	C₂×D₄	1	2	yes	—
78	C₄²	#41	1	2	yes	—
79	C₂³	#121	1	1	yes	characteristic
80	Q₈	—	4	4	no	—
81	Q₈	—	4	4	no	—
82	Q₈	—	8	2	no	—
83	Q₈	—	8	2	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	—	4	1	no	—
88	C₂×C₄	—	4	1	no	—
89	C₂×C₄	—	4	4	no	—
90	C₂×C₄	—	4	1	no	—
91	C₂×C₄	—	4	1	no	—
92	C₂×C₄	—	4	1	no	—
93	C₂×C₄	—	8	2	no	—
94	C₂×C₄	—	8	2	no	—
95	C₂×C₄	#120	1	1	yes	characteristic
96	C₂×C₄	#121	1	2	yes	—
97	C₂×C₄	#123	1	2	yes	—
98	C₂×C₄	#124	1	1	yes	characteristic
99	C₂×C₄	C₂×D₈	1	1	yes	characteristic
100	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
101	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
102	C₂×C₄	#135	1	1	yes	characteristic
103	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
104	C₂×C₄	#137	1	1	yes	characteristic
105	C₂×C₄	#137	1	1	yes	characteristic
106	C₈	—	2	2	no	—
107	C₈	—	2	2	no	—
108	C₈	—	8	1	no	—
109	C₈	—	8	2	no	—
110	C₂²	#394	1	1	yes	characteristic
111	C₂²	#461	1	1	yes	characteristic
112	C₂²	#462	1	1	yes	characteristic
113	C₂²	#466	1	1	yes	characteristic
114	C₂²	#469	1	1	yes	characteristic
115	C₂²	#473	1	1	yes	characteristic
116	C₂²	#473	1	1	yes	characteristic
117	C₄	—	2	2	no	—
118	C₄	—	4	2	no	—
119	C₄	—	8	4	no	—
120	C₄	—	8	1	no	—
121	C₄	#448	1	1	yes	characteristic
122	C₄	#451	1	1	yes	characteristic
123	C₄	#459	1	2	yes	—
124	C₄	#463	1	2	yes	—
125	C₄	#487	1	1	yes	characteristic
126	C₄	#491	1	1	yes	characteristic
127	C₂	#1673	1	1	yes	characteristic
128	C₂	#1674	1	1	yes	characteristic
129	C₂	#1693	1	1	yes	characteristic
130	C₂	#1694	1	1	yes	characteristic
131	C₂	#2043	1	1	yes	characteristic
132	C₂	#2047	1	1	yes	characteristic
133	C₂	#2054	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#500

C₂

yes

maximal, characteristic

#1981

C₂

yes

maximal

#1939

C₂

yes

maximal, characteristic

C₂×#332

C₂

yes

maximal

#1277

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈

C₂²

yes

—

#500

—

C₂×#107

—

C₂×#107

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

—

C₂×#103

—

C₂×#103

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₄×C₄⋊C₄

C₂²

yes

characteristic

#336

—

#332

—

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

D₄

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

D₄

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

D₄

yes

characteristic

C₂×C₄⋊C₄

D₄

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂³

yes

characteristic

#107

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

#103

—

C₄×C₈

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

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

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

SD₁₆

yes

—

C₂×C₈

Q₁₆

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₈

yes

—

C₄²

—

C₄²

—

C₄²

C₂×D₄

yes

—

C₄²

#41

yes

—

C₂³

#121

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#120

yes

characteristic

C₂×C₄

#121

yes

—

C₂×C₄

#123

yes

—

C₂×C₄

#124

yes

characteristic

C₂×C₄

C₂×D₈

yes

characteristic

100

C₂×C₄

C₂×SD₁₆

yes

characteristic

101

C₂×C₄

C₂×Q₁₆

yes

characteristic

102

C₂×C₄

#135

yes

characteristic

103

C₂×C₄

AGL(1,ℤ/8)

yes

—

104

C₂×C₄

#137

yes

characteristic

105

C₂×C₄

#137

yes

characteristic

106

C₈

—

107

C₈

—

108

C₈

—

109

C₈

—

110

C₂²

#394

yes

characteristic

111

C₂²

#461

yes

characteristic

112

C₂²

#462

yes

characteristic

113

C₂²

#466

yes

characteristic

114

C₂²

#469

yes

characteristic

115

C₂²

#473

yes

characteristic

116

C₂²

#473

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

#448

yes

characteristic

122

C₄

#451

yes

characteristic

123

C₄

#459

yes

—

124

C₄

#463

yes

—

125

C₄

#487

yes

characteristic

126

C₄

#491

yes

characteristic

127

C₂

#1673

yes

characteristic

128

C₂

#1674

yes

characteristic

129

C₂

#1693

yes

characteristic

130

C₂

#1694

yes

characteristic

131

C₂

#2043

yes

characteristic

132

C₂

#2047

yes

characteristic

133

C₂

#2054

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

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

subgroup	index
C₂×#500	2
#1981	2
#1939	2
C₂×#332	2
#1277	2

subgroup

index

Subgroups — #10545

Conjugacy classes of subgroups

Maximal subgroups