Group #9853 — subgroups

Subgroups — #9853

Order 256.

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

normal subgroups only

This group has 211 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	#3170	C₂	1	1	yes	maximal, characteristic
2	#1994	C₂	1	2	yes	maximal
3	#1744	C₂	1	1	yes	maximal, characteristic
4	#1566	C₂	1	1	yes	maximal, characteristic
5	#1493	C₂	1	2	yes	maximal
6	#533	C₂²	1	2	yes	—
7	#500	C₄	1	1	yes	characteristic, complemented
8	#494	C₄	1	1	yes	characteristic, complemented
9	#487	C₄	1	2	yes	complemented
10	#467	—	2	2	no	—
11	C₄×Q₁₆	—	2	2	no	—
12	#423	C₂²	1	2	yes	—
13	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₄³	C₂²	1	1	yes	characteristic
16	#339	—	2	4	no	—
17	#336	—	2	2	no	—
18	#334	—	2	2	no	—
19	Q₈⋊C₈	—	2	4	no	—
20	C₂×Q₁₆	—	2	2	no	—
21	C₂×M₁₆	—	2	2	no	—
22	C₂²×C₈	—	2	1	no	—
23	C₂²×C₈	D₄	1	2	yes	—
24	C₄⋊Q₈	C₂×C₄	1	2	yes	—
25	C₄⋊Q₈	C₂×C₄	1	2	yes	—
26	#125	—	2	2	no	—
27	C₂²⋊Q₈	—	2	2	no	—
28	C₂²⋊Q₈	—	2	2	no	—
29	C₄×Q₈	—	2	2	no	—
30	C₄×Q₈	—	2	2	no	—
31	#117	—	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	#103	—	2	4	no	—
40	C₄×C₈	—	2	1	no	—
41	C₄×C₈	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈	D₄	1	2	yes	—
44	C₄×C₈	D₄	1	2	yes	—
45	C₂×Q₈	—	2	2	no	—
46	C₂×Q₈	—	2	2	no	—
47	C₂²×C₄	—	2	2	no	—
48	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄	#41	1	1	yes	characteristic
51	Q₁₆	—	4	4	no	—
52	M₁₆	—	4	4	no	—
53	C₂×C₈	—	2	2	no	—
54	C₂×C₈	—	4	2	no	—
55	C₂×C₈	—	4	2	no	—
56	C₂×C₈	—	4	2	no	—
57	C₂×C₈	C₂²⋊C₄	1	2	yes	—
58	C₂×C₈	C₂²⋊C₄	1	2	yes	—
59	C₄⋊C₄	—	2	2	no	—
60	C₄⋊C₄	—	2	2	no	—
61	C₄⋊C₄	—	2	2	no	—
62	C₄⋊C₄	—	2	2	no	—
63	C₄⋊C₄	—	4	2	no	—
64	C₄⋊C₄	—	4	2	no	—
65	C₂²⋊C₄	—	4	2	no	—
66	C₄²	—	2	4	no	—
67	C₄²	—	2	4	no	—
68	C₄²	—	4	2	no	—
69	C₄²	—	4	2	no	—
70	C₄²	C₂²×C₄	1	1	yes	characteristic
71	C₄²	C₂×D₄	1	1	yes	characteristic
72	C₄²	C₂×D₄	1	1	yes	characteristic
73	C₄²	#41	1	1	yes	characteristic
74	C₂³	#121	1	1	yes	characteristic
75	Q₈	—	4	4	no	—
76	Q₈	—	4	4	no	—
77	C₂×C₄	—	2	2	no	—
78	C₂×C₄	—	2	2	no	—
79	C₂×C₄	—	2	4	no	—
80	C₂×C₄	—	4	2	no	—
81	C₂×C₄	—	4	2	no	—
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	4	2	no	—
84	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄	C₄×D₄	1	1	yes	characteristic
87	C₂×C₄	#121	1	1	yes	characteristic
88	C₂×C₄	#124	1	1	yes	characteristic
89	C₂×C₄	#127	1	1	yes	characteristic
90	C₂×C₄	#137	1	4	yes	—
91	C₈	—	2	2	no	—
92	C₈	—	2	2	no	—
93	C₈	—	4	2	no	—
94	C₈	—	8	2	no	—
95	C₂²	—	2	1	no	—
96	C₂²	—	2	1	no	—
97	C₂²	#390	1	1	yes	characteristic
98	C₂²	#470	1	2	yes	—
99	C₄	—	2	1	no	—
100	C₄	—	2	1	no	—
101	C₄	—	4	4	no	—
102	C₄	—	8	2	no	—
103	C₄	—	8	2	no	—
104	C₄	#421	1	2	yes	—
105	C₄	#441	1	2	yes	—
106	C₄	#443	1	2	yes	—
107	C₄	#497	1	2	yes	—
108	C₂	—	2	2	no	—
109	C₂	#1963	1	1	yes	characteristic
110	C₂	#1971	1	1	yes	characteristic
111	C₂	#1974	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#3170

C₂

yes

maximal, characteristic

#1994

C₂

yes

maximal

#1744

C₂

yes

maximal, characteristic

#1566

C₂

yes

maximal, characteristic

#1493

C₂

yes

maximal

#533

C₂²

yes

—

#500

C₄

yes

characteristic, complemented

#494

C₄

yes

characteristic, complemented

#487

C₄

yes

complemented

#467

—

C₄×Q₁₆

—

#423

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₄³

C₂²

yes

—

—

—

Q₈⋊C₈

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

D₄

yes

—

C₄⋊Q₈

C₂×C₄

yes

—

C₄⋊Q₈

C₂×C₄

yes

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

D₄

yes

—

C₂×C₄²

C₂³

yes

characteristic

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

D₄

yes

—

C₄×C₈

D₄

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

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

C₂²⋊C₄

yes

—

C₂×C₈

C₂²⋊C₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

C₄²

yes

C₂³

yes

Q₈

—

Q₈

—

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

—

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

#390

yes

characteristic

C₂²

#470

yes

—

C₄

—

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

#421

yes

—

105

C₄

#441

yes

—

106

C₄

#443

yes

—

107

C₄

#497

yes

—

108

C₂

—

109

C₂

#1963

yes

characteristic

110

C₂

#1971

yes

characteristic

111

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	C₂³	#121
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

C₂³

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
7. #500	C₄	semidirect
8. #494	C₄	semidirect
9. #487	C₄	semidirect

H ≅ G/N

type

7. #500

C₄

semidirect

8. #494

C₄

semidirect

9. #487

C₄

semidirect

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

subgroup	index
#3170	2
#1994	2
#1744	2
#1566	2
#1493	2

subgroup

index

Subgroups — #9853

Conjugacy classes of subgroups

Complements

Maximal subgroups