Group #10543 — subgroups

Subgroups — #10543

Order 256.

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

normal subgroups only

This group has 211 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	#1987	C₂	1	2	yes	maximal
3	#1939	C₂	1	1	yes	maximal, characteristic
4	C₂×#333	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×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	#340	—	2	2	no	—
16	#340	—	2	2	no	—
17	#336	—	2	1	no	—
18	#333	—	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	C₄×C₈	—	2	2	no	—
41	#95	—	2	1	no	—
42	C₂×Q₈	—	2	4	no	—
43	C₂×Q₈	—	4	2	no	—
44	C₂×Q₈	—	8	2	no	—
45	C₂×Q₈	—	8	2	no	—
46	C₂²×C₄	—	2	1	no	—
47	C₂²×C₄	—	2	1	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	1	no	—
55	C₂×C₈	—	2	1	no	—
56	C₂×C₈	—	4	1	no	—
57	C₂×C₈	—	4	2	no	—
58	C₂×C₈	—	4	1	no	—
59	C₂×C₈	—	4	2	no	—
60	C₂×C₈	—	4	1	no	—
61	C₂×C₈	—	4	2	no	—
62	C₂×C₈	SD₁₆	1	2	yes	—
63	C₂×C₈	Q₁₆	1	2	yes	—
64	C₄⋊C₄	—	2	1	no	—
65	C₄⋊C₄	—	2	4	no	—
66	C₄⋊C₄	—	2	4	no	—
67	C₄⋊C₄	—	2	2	no	—
68	C₄⋊C₄	—	4	2	no	—
69	C₄⋊C₄	—	4	2	no	—
70	C₄⋊C₄	Q₁₆	1	2	yes	—
71	C₄²	—	4	1	no	—
72	C₄²	—	4	1	no	—
73	C₄²	C₂×D₄	1	2	yes	—
74	C₄²	#41	1	2	yes	—
75	C₂³	#121	1	1	yes	characteristic
76	Q₈	—	8	4	no	—
77	Q₈	—	8	4	no	—
78	C₂×C₄	—	2	1	no	—
79	C₂×C₄	—	2	1	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	4	1	no	—
82	C₂×C₄	—	4	1	no	—
83	C₂×C₄	—	4	4	no	—
84	C₂×C₄	—	4	1	no	—
85	C₂×C₄	—	4	1	no	—
86	C₂×C₄	—	4	1	no	—
87	C₂×C₄	—	8	2	no	—
88	C₂×C₄	—	8	2	no	—
89	C₂×C₄	#120	1	1	yes	characteristic
90	C₂×C₄	#121	1	2	yes	—
91	C₂×C₄	#123	1	2	yes	—
92	C₂×C₄	#124	1	1	yes	characteristic
93	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
94	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
95	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
96	C₂×C₄	#135	1	1	yes	characteristic
97	C₂×C₄	#137	1	1	yes	characteristic
98	C₂×C₄	#137	1	1	yes	characteristic
99	C₂×C₄	#137	1	2	yes	—
100	C₈	—	2	2	no	—
101	C₈	—	2	2	no	—
102	C₈	—	8	1	no	—
103	C₈	—	8	2	no	—
104	C₂²	#394	1	1	yes	characteristic
105	C₂²	#461	1	1	yes	characteristic
106	C₂²	#462	1	1	yes	characteristic
107	C₂²	#467	1	1	yes	characteristic
108	C₂²	#470	1	1	yes	characteristic
109	C₂²	#472	1	1	yes	characteristic
110	C₂²	#472	1	1	yes	characteristic
111	C₄	—	2	2	no	—
112	C₄	—	4	2	no	—
113	C₄	—	8	1	no	—
114	C₄	—	8	4	no	—
115	C₄	#448	1	1	yes	characteristic
116	C₄	#451	1	1	yes	characteristic
117	C₄	#462	1	2	yes	—
118	C₄	#464	1	2	yes	—
119	C₄	#487	1	1	yes	characteristic
120	C₄	#491	1	1	yes	characteristic
121	C₂	#1675	1	1	yes	characteristic
122	C₂	#1676	1	1	yes	characteristic
123	C₂	#1695	1	1	yes	characteristic
124	C₂	#1696	1	1	yes	characteristic
125	C₂	#2043	1	1	yes	characteristic
126	C₂	#2045	1	1	yes	characteristic
127	C₂	#2056	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#500

C₂

yes

maximal, characteristic

#1987

C₂

yes

maximal

#1939

C₂

yes

maximal, characteristic

C₂×#333

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×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

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

—

C₄×C₈

—

#95

—

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

Q₁₆

yes

—

C₄²

—

C₄²

—

C₄²

C₂×D₄

yes

—

C₄²

#41

yes

—

C₂³

#121

yes

characteristic

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×SD₁₆

yes

characteristic

C₂×C₄

C₂×Q₁₆

yes

characteristic

C₂×C₄

C₂×Q₁₆

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

—

100

C₈

—

101

C₈

—

102

C₈

—

103

C₈

—

104

C₂²

#394

yes

characteristic

105

C₂²

#461

yes

characteristic

106

C₂²

#462

yes

characteristic

107

C₂²

#467

yes

characteristic

108

C₂²

#470

yes

characteristic

109

C₂²

#472

yes

characteristic

110

C₂²

#472

yes

characteristic

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

#448

yes

characteristic

116

C₄

#451

yes

characteristic

117

C₄

#462

yes

—

118

C₄

#464

yes

—

119

C₄

#487

yes

characteristic

120

C₄

#491

yes

characteristic

121

C₂

#1675

yes

characteristic

122

C₂

#1676

yes

characteristic

123

C₂

#1695

yes

characteristic

124

C₂

#1696

yes

characteristic

125

C₂

#2043

yes

characteristic

126

C₂

#2045

yes

characteristic

127

C₂

#2056

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
#1987	2
#1939	2
C₂×#333	2
#1277	2

subgroup

index

Subgroups — #10543

Conjugacy classes of subgroups

Maximal subgroups