Group #17980 — subgroups

Subgroups — #17980

Order 256.

295 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 295 conjugacy classes of subgroups — too many to draw. Only its 155 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	#3170	C₂	1	1	yes	maximal, characteristic
2	#2960	C₂	1	2	yes	maximal
3	#1587	C₂	1	8	yes	maximal, complemented
4	#1566	C₂	1	1	yes	maximal, characteristic
5	#1493	C₂	1	2	yes	maximal
6	C₂×#322	C₂	1	1	yes	maximal, characteristic
7	#533	C₂²	1	2	yes	—
8	#500	C₄	1	1	yes	characteristic
9	#494	C₄	1	1	yes	characteristic
10	#487	C₄	1	2	yes	—
11	#467	C₄	1	4	yes	—
12	#446	—	2	2	no	—
13	#446	C₂²	1	4	yes	—
14	C₈×Q₈	—	2	2	no	—
15	C₈×Q₈	C₂²	1	4	yes	—
16	C₄×Q₁₆	C₂²	1	4	yes	—
17	#424	—	2	2	no	—
18	#423	C₂²	1	2	yes	—
19	#405	—	2	2	no	—
20	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
23	#334	C₂²	1	4	yes	—
24	Q₈⋊C₈	C₂²	1	8	yes	—
25	#322	C₂²	1	4	yes	—
26	C₂×Q₁₆	C₂×C₄	1	4	yes	—
27	C₂×M₁₆	—	2	2	no	—
28	C₂²×C₈	—	2	1	no	—
29	C₂²×C₈	C₂×C₄	1	2	yes	—
30	C₂²×C₈	D₄	1	2	yes	—
31	C₄⋊Q₈	C₂×C₄	1	2	yes	—
32	C₄⋊Q₈	C₂×C₄	1	2	yes	—
33	#125	—	2	2	no	—
34	C₂²⋊Q₈	—	2	2	no	—
35	C₂²⋊Q₈	C₂×C₄	1	4	yes	—
36	C₄×Q₈	—	2	2	no	—
37	C₄×Q₈	C₂³	1	4	yes	—
38	#117	—	2	2	no	—
39	C₂×C₄²	C₂³	1	1	yes	characteristic
40	#107	C₂×C₄	1	4	yes	—
41	C₄⋊C₈	—	2	4	no	—
42	C₄⋊C₈	—	2	4	no	—
43	C₄⋊C₈	—	2	2	no	—
44	C₄⋊C₈	C₂³	1	4	yes	—
45	#103	C₂×C₄	1	8	yes	—
46	#97	—	2	4	no	—
47	C₄×C₈	—	2	4	no	—
48	C₄×C₈	—	2	1	no	—
49	C₄×C₈	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈	D₄	1	1	yes	characteristic
52	C₄×C₈	D₄	1	1	yes	characteristic
53	C₄×C₈	C₂³	1	2	yes	—
54	C₄×C₈	C₂³	1	2	yes	—
55	C₄×C₈	C₂³	1	2	yes	—
56	C₂×Q₈	—	2	2	no	—
57	C₂×Q₈	C₂²×C₄	1	4	yes	—
58	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄	#41	1	1	yes	characteristic
61	Q₁₆	—	4	4	no	—
62	M₁₆	—	4	4	no	—
63	C₂×C₈	—	2	4	no	—
64	C₂×C₈	—	2	4	no	—
65	C₂×C₈	—	2	2	no	—
66	C₂×C₈	—	2	4	no	—
67	C₂×C₈	—	4	2	no	—
68	C₂×C₈	C₂²×C₄	1	2	yes	—
69	C₂×C₈	C₂²×C₄	1	2	yes	—
70	C₂×C₈	C₂×D₄	1	2	yes	—
71	C₂×C₈	C₂×D₄	1	2	yes	—
72	C₄⋊C₄	—	2	2	no	—
73	C₄⋊C₄	—	2	2	no	—
74	C₄⋊C₄	—	2	2	no	—
75	C₄⋊C₄	—	2	4	no	—
76	C₄⋊C₄	—	2	4	no	—
77	C₄⋊C₄	C₂²×C₄	1	4	yes	—
78	C₂²⋊C₄	—	2	4	no	—
79	C₄²	—	2	4	no	—
80	C₄²	C₂²×C₄	1	1	yes	characteristic
81	C₄²	C₂×D₄	1	1	yes	characteristic
82	C₄²	#41	1	1	yes	characteristic
83	C₄²	C₂⁴	1	1	yes	characteristic
84	C₂³	C₄×D₄	1	1	yes	characteristic
85	Q₈	—	2	8	no	—
86	Q₈	—	4	4	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	—	2	4	no	—
91	C₂×C₄	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄	C₄×D₄	1	1	yes	characteristic
93	C₂×C₄	C₄×D₄	1	1	yes	characteristic
94	C₂×C₄	#137	1	4	yes	—
95	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
96	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
97	C₂×C₄	C₂×#41	1	1	yes	characteristic
98	C₈	—	2	4	no	—
99	C₈	—	2	2	no	—
100	C₈	—	2	2	no	—
101	C₈	—	2	2	no	—
102	C₈	—	2	2	no	—
103	C₈	—	4	4	no	—
104	C₂²	—	2	1	no	—
105	C₂²	—	2	1	no	—
106	C₂²	#441	1	2	yes	—
107	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	—	4	4	no	—
111	C₄	—	4	4	no	—
112	C₄	#441	1	2	yes	—
113	C₄	#443	1	2	yes	—
114	C₄	#568	1	2	yes	—
115	C₄	C₂×#137	1	2	yes	—
116	C₂	—	2	2	no	—
117	C₂	#2932	1	1	yes	characteristic
118	C₂	C₂×#441	1	1	yes	characteristic
119	C₂	C₂×#443	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#3170

C₂

yes

maximal, characteristic

#2960

C₂

yes

maximal

#1587

C₂

yes

maximal, complemented

#1566

C₂

yes

maximal, characteristic

#1493

C₂

yes

maximal

C₂×#322

C₂

yes

maximal, characteristic

#533

C₂²

yes

—

#500

C₄

yes

characteristic

#494

C₄

yes

characteristic

#487

C₄

yes

—

#467

C₄

yes

—

#446

—

#446

C₂²

yes

—

C₈×Q₈

—

C₈×Q₈

C₂²

yes

—

C₄×Q₁₆

C₂²

yes

—

#424

—

#423

C₂²

yes

—

#405

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

#334

C₂²

yes

—

Q₈⋊C₈

C₂²

yes

—

#322

C₂²

yes

—

C₂×Q₁₆

C₂×C₄

yes

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×C₄

yes

—

C₂²×C₈

D₄

yes

—

C₄⋊Q₈

C₂×C₄

yes

—

C₄⋊Q₈

C₂×C₄

yes

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

C₂×C₄

yes

—

C₄×Q₈

—

C₄×Q₈

C₂³

yes

—

#117

—

C₂×C₄²

C₂³

yes

characteristic

#107

C₂×C₄

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂³

yes

—

#103

C₂×C₄

yes

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₄×C₈

C₂³

yes

—

C₄×C₈

C₂³

yes

—

C₄×C₈

C₂³

yes

—

C₂×Q₈

—

C₂×Q₈

C₂²×C₄

yes

—

C₂²×C₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

#41

yes

characteristic

Q₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×C₄

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

#41

yes

characteristic

C₄²

C₂⁴

yes

characteristic

C₂³

C₄×D₄

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×D₄

yes

characteristic

C₂×C₄

C₄×D₄

yes

characteristic

C₂×C₄

C₄×D₄

yes

characteristic

C₂×C₄

#137

yes

—

C₂×C₄

C₂³×C₄

yes

characteristic

C₂×C₄

C₂²×D₄

yes

characteristic

C₂×C₄

C₂×#41

yes

characteristic

C₈

—

C₈

—

100

C₈

—

101

C₈

—

102

C₈

—

103

C₈

—

104

C₂²

—

105

C₂²

—

106

C₂²

#441

yes

—

107

C₂²

C₂×C₄×D₄

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

#441

yes

—

113

C₄

#443

yes

—

114

C₄

#568

yes

—

115

C₄

C₂×#137

yes

—

116

C₂

—

117

C₂

#2932

yes

characteristic

118

C₂

C₂×#441

yes

characteristic

119

C₂

C₂×#443

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₄²	C₂⁴
socle	C₂²	C₂×C₄×D₄
prime radical	C₂³	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₂×C₄

C₂²×D₄

commutator

C₂×C₄

C₂³×C₄

Frattini, cosocle

C₄²

C₂⁴

socle

C₂²

C₂×C₄×D₄

prime radical

C₂³

C₄×D₄

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
3. #1587	C₂	semidirect

H ≅ G/N

type

3. #1587

C₂

semidirect

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

subgroup	index
#3170	2
#2960	2
#1587	2
#1566	2
#1493	2
C₂×#322	2

subgroup

index

Subgroups — #17980

Conjugacy classes of subgroups

Complements

Maximal subgroups