Group #9852 — subgroups

Subgroups — #9852

Order 256.

249 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 249 conjugacy classes of subgroups — too many to draw. Only its 71 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	#3169	C₂	1	1	yes	maximal, characteristic
2	#1962	C₂	1	2	yes	maximal, complemented
3	#1744	C₂	1	1	yes	maximal, characteristic, complemented
4	#1566	C₂	1	1	yes	maximal, characteristic, complemented
5	#1492	C₂	1	2	yes	maximal, complemented
6	#532	C₂²	1	2	yes	—
7	#500	C₄	1	1	yes	characteristic, complemented
8	#493	C₄	1	1	yes	characteristic, complemented
9	#486	C₄	1	2	yes	complemented
10	#466	—	2	2	no	—
11	C₄×D₈	—	2	2	no	—
12	#423	C₂²	1	2	yes	—
13	C₂×#104	—	2	4	no	—
14	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₄³	C₂²	1	1	yes	characteristic
17	#336	—	2	2	no	—
18	#334	—	2	2	no	—
19	D₄⋊C₈	—	2	4	no	—
20	C₂×#41	—	2	2	no	—
21	C₂×D₈	—	2	2	no	—
22	C₂×M₁₆	—	2	2	no	—
23	C₂²×C₈	—	2	1	no	—
24	C₂²×C₈	D₄	1	2	yes	—
25	C₄⋊Q₈	C₂×C₄	1	2	yes	—
26	#127	C₂×C₄	1	2	yes	—
27	#124	—	2	2	no	—
28	#121	—	2	2	no	—
29	#121	—	2	2	no	—
30	C₄×D₄	—	2	2	no	—
31	C₄×D₄	—	2	2	no	—
32	C₂×C₄²	—	2	1	no	—
33	C₂×C₄²	—	2	1	no	—
34	C₂×C₄²	D₄	1	2	yes	—
35	C₂×C₄²	C₂³	1	1	yes	characteristic
36	#107	—	2	2	no	—
37	C₄⋊C₈	—	2	2	no	—
38	C₄⋊C₈	—	2	2	no	—
39	#104	—	4	8	no	—
40	#102	—	2	4	no	—
41	C₄×C₈	—	2	1	no	—
42	C₄×C₈	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈	D₄	1	2	yes	—
45	C₄×C₈	D₄	1	2	yes	—
46	#41	—	4	4	no	—
47	#41	—	4	4	no	—
48	C₂×Q₈	—	2	2	no	—
49	C₂×D₄	—	2	2	no	—
50	C₂×D₄	—	2	2	no	—
51	C₂×D₄	—	2	2	no	—
52	C₂×D₄	—	4	2	no	—
53	C₂²×C₄	—	2	2	no	—
54	C₂²×C₄	—	4	2	no	—
55	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄	#41	1	1	yes	characteristic
58	D₈	—	4	4	no	—
59	M₁₆	—	4	4	no	—
60	C₂×C₈	—	2	2	no	—
61	C₂×C₈	—	4	2	no	—
62	C₂×C₈	—	4	2	no	—
63	C₂×C₈	—	4	2	no	—
64	C₂×C₈	C₂²⋊C₄	1	2	yes	—
65	C₂×C₈	C₂²⋊C₄	1	2	yes	—
66	C₄⋊C₄	—	2	2	no	—
67	C₄⋊C₄	—	2	2	no	—
68	C₂²⋊C₄	—	4	2	no	—
69	C₂²⋊C₄	—	4	2	no	—
70	C₄²	—	2	4	no	—
71	C₄²	—	2	4	no	—
72	C₄²	—	4	2	no	—
73	C₄²	C₂²×C₄	1	1	yes	characteristic
74	C₄²	C₂×D₄	1	1	yes	characteristic
75	C₄²	C₂×D₄	1	1	yes	characteristic
76	C₄²	#41	1	1	yes	characteristic
77	C₂³	—	4	2	no	—
78	C₂³	#121	1	1	yes	characteristic
79	Q₈	—	4	4	no	—
80	D₄	—	4	4	no	—
81	D₄	—	4	4	no	—
82	D₄	—	4	4	no	—
83	D₄	—	8	4	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	2	4	no	—
87	C₂×C₄	—	4	2	no	—
88	C₂×C₄	—	4	2	no	—
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	—	8	2	no	—
91	C₂×C₄	—	8	2	no	—
92	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄	C₄×D₄	1	1	yes	characteristic
94	C₂×C₄	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄	#121	1	1	yes	characteristic
96	C₂×C₄	#124	1	1	yes	characteristic
97	C₂×C₄	#127	1	1	yes	characteristic
98	C₂×C₄	AGL(1,ℤ/8)	1	4	yes	—
99	C₈	—	2	2	no	—
100	C₈	—	2	2	no	—
101	C₈	—	4	2	no	—
102	C₈	—	8	2	no	—
103	C₂²	—	2	1	no	—
104	C₂²	—	2	1	no	—
105	C₂²	—	8	2	no	—
106	C₂²	—	8	2	no	—
107	C₂²	—	8	2	no	—
108	C₂²	#390	1	1	yes	characteristic
109	C₂²	#469	1	2	yes	—
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	—	4	4	no	—
113	C₄	—	8	2	no	—
114	C₄	#421	1	2	yes	—
115	C₄	#442	1	2	yes	—
116	C₄	#443	1	2	yes	—
117	C₄	#496	1	2	yes	—
118	C₂	—	2	2	no	—
119	C₂	—	16	2	no	—
120	C₂	#1963	1	1	yes	characteristic
121	C₂	#1972	1	1	yes	characteristic
122	C₂	#1974	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#3169

C₂

yes

maximal, characteristic

#1962

C₂

yes

maximal, complemented

#1744

C₂

yes

maximal, characteristic, complemented

#1566

C₂

yes

maximal, characteristic, complemented

#1492

C₂

yes

maximal, complemented

#532

C₂²

yes

—

#500

C₄

yes

characteristic, complemented

#493

C₄

yes

characteristic, complemented

#486

C₄

yes

complemented

#466

—

C₄×D₈

—

#423

C₂²

yes

—

C₂×#104

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₄³

C₂²

yes

characteristic

#336

—

#334

—

D₄⋊C₈

—

C₂×#41

—

C₂×D₈

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

D₄

yes

—

C₄⋊Q₈

C₂×C₄

yes

—

#127

C₂×C₄

yes

—

#124

—

#121

—

#121

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

D₄

yes

—

C₂×C₄²

C₂³

yes

characteristic

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

#104

—

#102

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

D₄

yes

—

C₄×C₈

D₄

yes

—

#41

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

#41

yes

characteristic

D₈

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄

yes

—

C₂×C₈

C₂²⋊C₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

#41

yes

characteristic

C₂³

—

C₂³

#121

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄

C₄×D₄

yes

characteristic

C₂×C₄

C₄×D₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₈

—

100

C₈

—

101

C₈

—

102

C₈

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

#390

yes

characteristic

109

C₂²

#469

yes

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

#421

yes

—

115

C₄

#442

yes

—

116

C₄

#443

yes

—

117

C₄

#496

yes

—

118

C₂

—

119

C₂

—

120

C₂

#1963

yes

characteristic

121

C₂

#1972

yes

characteristic

122

C₂

#1974

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#127
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	#390
prime radical	#3169	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₂×C₄

#127

commutator

C₄²

C₂²×C₄

Frattini, cosocle

C₂×C₄²

C₂³

C₂²

C₂

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
2. #1962	C₂	semidirect
3. #1744	C₂	semidirect
4. #1566	C₂	semidirect
5. #1492	C₂	semidirect
7. #500	C₄	semidirect
8. #493	C₄	semidirect
9. #486	C₄	semidirect

H ≅ G/N

type

2. #1962

C₂

semidirect

3. #1744

C₂

semidirect

4. #1566

C₂

semidirect

5. #1492

C₂

semidirect

7. #500

C₄

semidirect

8. #493

C₄

semidirect

9. #486

C₄

semidirect

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

subgroup	index
#3169	2
#1962	2
#1744	2
#1566	2
#1492	2

subgroup

index

Subgroups — #9852

Conjugacy classes of subgroups

Complements

Maximal subgroups