Group #10279 — subgroups

Subgroups — #10279

Order 256.

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

normal subgroups only

This group has 255 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×#477	C₂	1	2	yes	maximal
3	#1984	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	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
8	#500	—	2	4	no	—
9	#477	—	2	4	no	—
10	#477	—	2	4	no	—
11	C₂×#107	—	2	2	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	#333	—	2	4	no	—
19	C₂²×Q₈	—	2	2	no	—
20	C₂²×Q₈	D₄	1	1	yes	characteristic
21	C₂²×C₈	—	2	2	no	—
22	C₂²×C₈	—	2	1	no	—
23	C₂²×C₈	D₄	1	2	yes	—
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	C₄×Q₈	—	2	2	no	—
29	C₄×Q₈	—	2	2	no	—
30	C₂×C₄⋊C₄	—	2	2	no	—
31	C₂×C₄⋊C₄	—	2	2	no	—
32	C₂×C₄⋊C₄	—	2	1	no	—
33	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
34	C₂×C₄²	—	2	1	no	—
35	C₂×C₄²	C₂³	1	1	yes	characteristic
36	#107	—	2	8	no	—
37	#107	—	2	8	no	—
38	C₄⋊C₈	—	2	2	no	—
39	C₄⋊C₈	—	2	2	no	—
40	C₄⋊C₈	D₄	1	4	yes	—
41	#103	—	2	8	no	—
42	C₄×C₈	—	2	2	no	—
43	C₂×Q₈	—	2	1	no	—
44	C₂×Q₈	—	2	2	no	—
45	C₂×Q₈	—	2	4	no	—
46	C₂×Q₈	—	2	2	no	—
47	C₂×Q₈	—	4	2	no	—
48	C₂×Q₈	—	8	4	no	—
49	C₂×Q₈	Q₁₆	1	2	yes	—
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	2	no	—
57	C₂×C₈	—	2	2	no	—
58	C₂×C₈	—	2	4	no	—
59	C₂×C₈	—	4	2	no	—
60	C₂×C₈	—	4	2	no	—
61	C₂×C₈	—	4	1	no	—
62	C₂×C₈	—	4	2	no	—
63	C₂×C₈	Q₁₆	1	4	yes	—
64	C₄⋊C₄	—	2	4	no	—
65	C₄⋊C₄	—	2	4	no	—
66	C₄⋊C₄	—	2	2	no	—
67	C₄⋊C₄	—	4	2	no	—
68	C₄⋊C₄	—	4	2	no	—
69	C₄⋊C₄	—	4	2	no	—
70	C₄²	—	4	1	no	—
71	C₄²	—	4	1	no	—
72	C₄²	C₂×D₄	1	2	yes	—
73	C₄²	C₂×D₄	1	2	yes	—
74	C₂³	#121	1	1	yes	characteristic
75	Q₈	—	2	4	no	—
76	Q₈	—	4	2	no	—
77	Q₈	—	8	4	no	—
78	Q₈	—	8	4	no	—
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	4	1	no	—
82	C₂×C₄	—	4	1	no	—
83	C₂×C₄	—	4	1	no	—
84	C₂×C₄	—	4	4	no	—
85	C₂×C₄	—	4	2	no	—
86	C₂×C₄	—	8	4	no	—
87	C₂×C₄	#120	1	1	yes	characteristic
88	C₂×C₄	#120	1	2	yes	—
89	C₂×C₄	#121	1	2	yes	—
90	C₂×C₄	#127	1	1	yes	characteristic
91	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
92	C₂×C₄	C₂×Q₁₆	1	2	yes	—
93	C₂×C₄	#135	1	1	yes	characteristic
94	C₂×C₄	#137	1	2	yes	—
95	C₂×C₄	#137	1	2	yes	—
96	C₈	—	2	4	no	—
97	C₈	—	4	4	no	—
98	C₈	—	8	1	no	—
99	C₂²	#392	1	1	yes	characteristic
100	C₂²	#462	1	2	yes	—
101	C₂²	#467	1	1	yes	characteristic
102	C₂²	#470	1	1	yes	characteristic
103	C₂²	#472	1	2	yes	—
104	C₄	—	2	2	no	—
105	C₄	—	4	2	no	—
106	C₄	—	8	1	no	—
107	C₄	—	8	4	no	—
108	C₄	#451	1	2	yes	—
109	C₄	#451	1	2	yes	—
110	C₄	#452	1	2	yes	—
111	C₄	#494	1	1	yes	characteristic
112	C₄	#497	1	1	yes	characteristic
113	C₂	#1676	1	2	yes	—
114	C₂	#1696	1	2	yes	—
115	C₂	#2001	1	1	yes	characteristic
116	C₂	#2016	1	1	yes	characteristic
117	C₂	#2018	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#500

C₂

yes

maximal, characteristic

C₂×#477

C₂

yes

maximal

#1984

C₂

yes

maximal, characteristic

C₂×#333

C₂

yes

maximal

#1277

C₂

yes

maximal, characteristic

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

—

#333

—

C₂²×Q₈

—

C₂²×Q₈

D₄

yes

characteristic

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

—

#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

Q₁₆

yes

—

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

Q₁₆

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

—

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×Q₁₆

yes

characteristic

C₂×C₄

C₂×Q₁₆

yes

—

C₂×C₄

#135

yes

characteristic

C₂×C₄

#137

yes

—

C₂×C₄

#137

yes

—

C₈

—

C₈

—

C₈

—

C₂²

#392

yes

characteristic

100

C₂²

#462

yes

—

101

C₂²

#467

yes

characteristic

102

C₂²

#470

yes

characteristic

103

C₂²

#472

yes

—

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

#451

yes

—

109

C₄

#451

yes

—

110

C₄

#452

yes

—

111

C₄

#494

yes

characteristic

112

C₄

#497

yes

characteristic

113

C₂

#1676

yes

—

114

C₂

#1696

yes

—

115

C₂

#2001

yes

characteristic

116

C₂

#2016

yes

characteristic

117

C₂

#2018

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
C₂×#477	2
#1984	2
C₂×#333	2
#1277	2

subgroup

index

Subgroups — #10279

Conjugacy classes of subgroups

Maximal subgroups