Group #10650 — subgroups

Subgroups — #10650

Order 256.

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

normal subgroups only

This group has 175 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	#1947	C₂	1	2	yes	maximal
2	#1939	C₂	1	1	yes	maximal, characteristic
3	#1845	C₂	1	1	yes	maximal, characteristic
4	#1296	C₂	1	2	yes	maximal
5	#1277	C₂	1	1	yes	maximal, characteristic
6	C₂×#107	—	2	1	no	—
7	C₂×#107	—	2	1	no	—
8	C₂×#107	—	2	2	no	—
9	C₂×C₄⋊C₈	—	2	1	no	—
10	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
13	#384	C₂²	1	2	yes	—
14	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	#340	—	2	2	no	—
16	#336	—	2	1	no	—
17	C₂²×C₈	—	2	2	no	—
18	C₂²×C₈	—	2	1	no	—
19	C₂²×C₈	D₄	1	1	yes	characteristic
20	C₂²×C₈	D₄	1	1	yes	characteristic
21	C₂×C₄⋊C₄	—	2	2	no	—
22	C₂×C₄⋊C₄	—	2	2	no	—
23	C₂×C₄⋊C₄	—	2	2	no	—
24	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
25	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
26	C₂×C₄²	—	2	1	no	—
27	C₂×C₄²	C₂³	1	1	yes	characteristic
28	#107	—	4	4	no	—
29	#107	—	4	2	no	—
30	#107	—	4	2	no	—
31	C₄⋊C₈	—	2	2	no	—
32	C₄⋊C₈	—	2	2	no	—
33	C₄⋊C₈	—	4	2	no	—
34	C₄⋊C₈	Q₈	1	4	yes	—
35	C₄×C₈	—	2	2	no	—
36	#95	—	2	1	no	—
37	C₂²×C₄	—	2	1	no	—
38	C₂²×C₄	—	2	1	no	—
39	C₂²×C₄	—	4	2	no	—
40	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄	#41	1	1	yes	characteristic
43	C₂×C₈	—	2	1	no	—
44	C₂×C₈	—	2	1	no	—
45	C₂×C₈	—	2	1	no	—
46	C₂×C₈	—	2	1	no	—
47	C₂×C₈	—	2	4	no	—
48	C₂×C₈	—	4	2	no	—
49	C₂×C₈	—	4	2	no	—
50	C₂×C₈	—	4	1	no	—
51	C₂×C₈	—	4	1	no	—
52	C₂×C₈	—	4	1	no	—
53	C₂×C₈	D₈	1	2	yes	—
54	C₂×C₈	Q₁₆	1	2	yes	—
55	C₄⋊C₄	—	2	1	no	—
56	C₄⋊C₄	—	2	2	no	—
57	C₄⋊C₄	—	4	2	no	—
58	C₄⋊C₄	—	4	2	no	—
59	C₄⋊C₄	—	4	2	no	—
60	C₄⋊C₄	—	4	2	no	—
61	C₄⋊C₄	—	8	2	no	—
62	C₄⋊C₄	Q₁₆	1	2	yes	—
63	C₄²	—	4	1	no	—
64	C₄²	—	4	1	no	—
65	C₄²	C₂×Q₈	1	2	yes	—
66	C₄²	#41	1	2	yes	—
67	C₂³	#121	1	1	yes	characteristic
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	2	2	no	—
71	C₂×C₄	—	4	1	no	—
72	C₂×C₄	—	4	1	no	—
73	C₂×C₄	—	4	1	no	—
74	C₂×C₄	—	4	1	no	—
75	C₂×C₄	—	4	1	no	—
76	C₂×C₄	—	8	2	no	—
77	C₂×C₄	—	8	2	no	—
78	C₂×C₄	—	8	2	no	—
79	C₂×C₄	C₂²⋊Q₈	1	2	yes	—
80	C₂×C₄	#123	1	1	yes	characteristic
81	C₂×C₄	#125	1	2	yes	—
82	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
83	C₂×C₄	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
85	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
86	C₂×C₄	#135	1	1	yes	characteristic
87	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
88	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
89	C₂×C₄	#137	1	1	yes	characteristic
90	C₈	—	2	2	no	—
91	C₈	—	2	2	no	—
92	C₈	—	4	4	no	—
93	C₈	—	8	1	no	—
94	C₂²	#398	1	1	yes	characteristic
95	C₂²	#459	1	1	yes	characteristic
96	C₂²	#462	1	1	yes	characteristic
97	C₂²	#467	1	1	yes	characteristic
98	C₂²	#469	1	1	yes	characteristic
99	C₂²	#472	1	1	yes	characteristic
100	C₂²	#473	1	1	yes	characteristic
101	C₄	—	2	2	no	—
102	C₄	—	4	2	no	—
103	C₄	—	8	1	no	—
104	C₄	—	16	2	no	—
105	C₄	#477	1	2	yes	—
106	C₄	#478	1	2	yes	—
107	C₄	#480	1	1	yes	characteristic
108	C₄	#484	1	1	yes	characteristic
109	C₄	#500	1	1	yes	characteristic
110	C₄	#501	1	1	yes	characteristic
111	C₂	#1669	1	1	yes	characteristic
112	C₂	#1676	1	1	yes	characteristic
113	C₂	#1687	1	1	yes	characteristic
114	C₂	#1694	1	1	yes	characteristic
115	C₂	#2083	1	1	yes	characteristic
116	C₂	#2087	1	1	yes	characteristic
117	C₂	#2089	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#1947

C₂

yes

maximal

#1939

C₂

yes

maximal, characteristic

#1845

C₂

yes

maximal, characteristic

#1296

C₂

yes

maximal

#1277

C₂

yes

maximal, characteristic

C₂×#107

—

C₂×#107

—

C₂×#107

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

#384

C₂²

yes

—

C₄×C₄⋊C₄

C₂²

yes

characteristic

#340

—

#336

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

D₄

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

D₄

yes

characteristic

C₂×C₄⋊C₄

D₄

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂³

yes

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

Q₈

yes

—

C₄×C₈

—

#95

—

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

D₈

yes

—

C₂×C₈

Q₁₆

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

Q₁₆

yes

—

C₄²

—

C₄²

—

C₄²

C₂×Q₈

yes

—

C₄²

#41

yes

—

C₂³

#121

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Q₈

yes

—

C₂×C₄

#123

yes

characteristic

C₂×C₄

#125

yes

—

C₂×C₄

C₄⋊Q₈

yes

characteristic

C₂×C₄

C₂×D₈

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

C₈

—

C₈

—

C₈

—

C₈

—

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

100

C₂²

#473

yes

characteristic

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

#477

yes

—

106

C₄

#478

yes

—

107

C₄

#480

yes

characteristic

108

C₄

#484

yes

characteristic

109

C₄

#500

yes

characteristic

110

C₄

#501

yes

characteristic

111

C₂

#1669

yes

characteristic

112

C₂

#1676

yes

characteristic

113

C₂

#1687

yes

characteristic

114

C₂

#1694

yes

characteristic

115

C₂

#2083

yes

characteristic

116

C₂

#2087

yes

characteristic

117

C₂

#2089

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
#1947	2
#1939	2
#1845	2
#1296	2
#1277	2

subgroup

index

Subgroups — #10650

Conjugacy classes of subgroups

Maximal subgroups