Group #10689 — subgroups

Subgroups — #10689

Order 256.

124 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 49 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	#1940	C₂	1	1	yes	maximal, characteristic
2	#1838	C₂	1	1	yes	maximal, characteristic
3	#1303	C₂	1	1	yes	maximal, characteristic
4	#1303	C₂	1	1	yes	maximal, characteristic
5	#1301	C₂	1	1	yes	maximal, characteristic
6	#1301	C₂	1	1	yes	maximal, characteristic
7	#1275	C₂	1	1	yes	maximal, characteristic
8	#431	C₂²	1	1	yes	characteristic
9	#431	C₂²	1	1	yes	characteristic
10	#423	C₂²	1	1	yes	characteristic
11	#423	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₈	—	2	1	no	—
13	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×#97	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	#341	—	2	1	no	—
17	#341	—	2	1	no	—
18	#336	—	2	1	no	—
19	C₂×M₁₆	—	2	1	no	—
20	C₂×M₁₆	—	2	1	no	—
21	C₂²×C₈	—	2	1	no	—
22	C₂²×C₈	D₄	1	1	yes	characteristic
23	C₂²×C₈	D₄	1	1	yes	characteristic
24	C₂×C₄⋊C₄	—	2	1	no	—
25	C₂×C₄²	—	2	1	no	—
26	C₂×C₄²	C₂³	1	1	yes	characteristic
27	C₄⋊C₈	—	2	1	no	—
28	C₄⋊C₈	—	2	1	no	—
29	C₄⋊C₈	—	2	1	no	—
30	C₄⋊C₈	—	2	1	no	—
31	C₄⋊C₈	—	2	1	no	—
32	C₄⋊C₈	—	2	1	no	—
33	C₄⋊C₈	—	2	1	no	—
34	C₄⋊C₈	—	2	1	no	—
35	C₄⋊C₈	—	4	2	no	—
36	C₂²⋊C₈	—	2	1	no	—
37	C₂²⋊C₈	—	2	1	no	—
38	#97	—	2	1	no	—
39	#97	—	2	1	no	—
40	C₄×C₈	—	2	1	no	—
41	C₄×C₈	—	2	1	no	—
42	#95	Q₈	1	1	yes	characteristic
43	#95	Q₈	1	1	yes	characteristic
44	C₂²×C₄	—	2	1	no	—
45	C₂²×C₄	—	2	1	no	—
46	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
48	C₂²×C₄	#41	1	1	yes	characteristic
49	M₁₆	—	4	1	no	—
50	M₁₆	—	4	1	no	—
51	M₁₆	—	4	1	no	—
52	M₁₆	—	4	1	no	—
53	C₂×C₈	—	2	1	no	—
54	C₂×C₈	—	2	1	no	—
55	C₂×C₈	—	2	1	no	—
56	C₂×C₈	—	2	1	no	—
57	C₂×C₈	—	2	1	no	—
58	C₂×C₈	—	2	1	no	—
59	C₂×C₈	—	4	1	no	—
60	C₂×C₈	—	4	1	no	—
61	C₂×C₈	—	4	1	no	—
62	C₂×C₈	—	4	1	no	—
63	C₂×C₈	—	4	1	no	—
64	C₂×C₈	—	4	1	no	—
65	C₂×C₈	—	4	1	no	—
66	C₄⋊C₄	—	4	2	no	—
67	C₄²	—	4	1	no	—
68	C₄²	—	4	1	no	—
69	C₄²	#41	1	1	yes	characteristic
70	C₄²	#41	1	1	yes	characteristic
71	C₄²	#41	1	1	yes	characteristic
72	C₄²	#41	1	1	yes	characteristic
73	C₂³	C₂²⋊Q₈	1	1	yes	characteristic
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	—	4	1	no	—
76	C₂×C₄	—	4	1	no	—
77	C₂×C₄	—	4	1	no	—
78	C₂×C₄	—	4	1	no	—
79	C₂×C₄	—	4	1	no	—
80	C₂×C₄	—	4	1	no	—
81	C₂×C₄	—	4	1	no	—
82	C₂×C₄	#123	1	1	yes	characteristic
83	C₂×C₄	#124	1	1	yes	characteristic
84	C₂×C₄	#125	1	1	yes	characteristic
85	C₂×C₄	#125	1	1	yes	characteristic
86	C₂×C₄	#126	1	1	yes	characteristic
87	C₂×C₄	#126	1	1	yes	characteristic
88	C₂×C₄	#135	1	1	yes	characteristic
89	C₂×C₄	#135	1	1	yes	characteristic
90	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
91	C₂×C₄	#137	1	1	yes	characteristic
92	C₈	—	4	1	no	—
93	C₈	—	4	1	no	—
94	C₈	—	8	1	no	—
95	C₈	—	8	1	no	—
96	C₈	—	8	1	no	—
97	C₈	—	8	1	no	—
98	C₈	—	8	1	no	—
99	C₂²	—	2	1	no	—
100	C₂²	—	2	1	no	—
101	C₂²	#400	1	1	yes	characteristic
102	C₂²	#478	1	1	yes	characteristic
103	C₂²	#479	1	1	yes	characteristic
104	C₄	—	2	1	no	—
105	C₄	—	2	1	no	—
106	C₄	—	2	1	no	—
107	C₄	—	2	1	no	—
108	C₄	—	4	2	no	—
109	C₄	—	8	1	no	—
110	C₄	#482	1	1	yes	characteristic
111	C₄	#485	1	1	yes	characteristic
112	C₄	#488	1	1	yes	characteristic
113	C₄	#489	1	1	yes	characteristic
114	C₂	—	2	1	no	—
115	C₂	—	2	1	no	—
116	C₂	#2101	1	1	yes	characteristic
117	C₂	#2102	1	1	yes	characteristic
118	C₂	#2103	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#1940

C₂

yes

maximal, characteristic

#1838

C₂

yes

maximal, characteristic

#1303

C₂

yes

maximal, characteristic

#1303

C₂

yes

maximal, characteristic

#1301

C₂

yes

maximal, characteristic

#1301

C₂

yes

maximal, characteristic

#1275

C₂

yes

maximal, characteristic

#431

C₂²

yes

characteristic

#431

C₂²

yes

characteristic

#423

C₂²

yes

characteristic

#423

C₂²

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×#97

C₂²

yes

characteristic

C₄×C₄⋊C₄

C₂²

yes

—

—

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

D₄

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

#95

Q₈

yes

characteristic

#95

Q₈

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×Q₈

yes

characteristic

C₂²×C₄

#41

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

yes

C₄²

yes

C₄²

yes

C₄²

yes

C₂³

C₂²⋊Q₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₂²

—

100

C₂²

—

101

C₂²

#400

yes

characteristic

102

C₂²

#478

yes

characteristic

103

C₂²

#479

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

#482

yes

characteristic

111

C₄

#485

yes

characteristic

112

C₄

#488

yes

characteristic

113

C₄

#489

yes

characteristic

114

C₂

—

115

C₂

—

116

C₂

#2101

yes

characteristic

117

C₂

#2102

yes

characteristic

118

C₂

#2103

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#124
commutator, Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	#400
prime radical	C₂³	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₂×C₄

#124

commutator, Frattini, cosocle

C₂×C₄²

C₂³

socle

C₂²

#400

prime radical

C₂³

C₂²⋊Q₈

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ 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
#1940	2
#1838	2
#1303	2
#1301	2
#1275	2

subgroup

index

Subgroups — #10689

Conjugacy classes of subgroups

Maximal subgroups