Group #9755 — subgroups

Subgroups — #9755

Order 256.

197 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 197 conjugacy classes of subgroups — too many to draw. Only its 83 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	#3160	C₂	1	1	yes	maximal, characteristic
2	#1846	C₂	1	1	yes	maximal, characteristic
3	#1518	C₂	1	4	yes	maximal
4	#1450	C₂	1	1	yes	maximal, characteristic
5	#533	C₂²	1	2	yes	—
6	#500	C₄	1	2	yes	complemented
7	#499	—	2	1	no	—
8	#499	—	2	1	no	—
9	#498	C₄	1	2	yes	complemented
10	#431	C₂²	1	2	yes	—
11	#427	—	2	1	no	—
12	#427	—	2	1	no	—
13	C₂×#107	—	2	1	no	—
14	C₂×#106	—	2	1	no	—
15	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
16	#382	C₂²	1	2	yes	—
17	#328	—	2	8	no	—
18	#322	—	2	1	no	—
19	C₈²	—	2	1	no	—
20	C₂²×C₈	D₄	1	1	yes	characteristic
21	C₂²×C₈	D₄	1	1	yes	characteristic
22	C₄⋊Q₈	C₂×C₄	1	4	yes	—
23	#125	—	2	2	no	—
24	C₂²⋊Q₈	—	2	4	no	—
25	C₄×Q₈	—	2	4	no	—
26	#117	—	2	2	no	—
27	C₂×C₄⋊C₄	—	2	2	no	—
28	C₂×C₄²	C₂³	1	1	yes	characteristic
29	#107	—	2	2	no	—
30	#107	—	2	2	no	—
31	#107	—	4	2	no	—
32	#106	—	2	2	no	—
33	#106	—	2	2	no	—
34	#106	—	4	2	no	—
35	C₄⋊C₈	—	2	2	no	—
36	C₂²⋊C₈	D₄	1	4	yes	—
37	C₄×C₈	—	2	2	no	—
38	C₄×C₈	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈	D₄	1	1	yes	characteristic
41	C₄×C₈	D₄	1	1	yes	characteristic
42	#95	—	2	4	no	—
43	C₂×Q₈	—	2	4	no	—
44	C₂²×C₄	—	4	2	no	—
45	C₂²×C₄	C₂×D₄	1	2	yes	—
46	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈	—	2	4	no	—
48	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₈	SD₁₆	1	2	yes	—
53	C₂×C₈	SD₁₆	1	2	yes	—
54	C₂×C₈	Q₁₆	1	2	yes	—
55	C₂×C₈	Q₁₆	1	2	yes	—
56	C₄⋊C₄	—	2	4	no	—
57	C₄⋊C₄	—	2	4	no	—
58	C₄⋊C₄	—	4	4	no	—
59	C₄⋊C₄	—	4	2	no	—
60	C₄⋊C₄	—	4	2	no	—
61	C₂²⋊C₄	—	4	2	no	—
62	C₄²	—	4	2	no	—
63	C₄²	C₂²×C₄	1	1	yes	characteristic
64	C₄²	C₂×D₄	1	1	yes	characteristic
65	C₄²	#41	1	2	yes	—
66	C₂³	#127	1	1	yes	characteristic
67	Q₈	—	8	4	no	—
68	C₂×C₄	—	4	2	no	—
69	C₂×C₄	—	4	4	no	—
70	C₂×C₄	—	8	4	no	—
71	C₂×C₄	—	8	2	no	—
72	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₄	C₄×D₄	1	2	yes	—
74	C₂×C₄	#121	1	2	yes	—
75	C₂×C₄	#124	1	1	yes	characteristic
76	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
79	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
80	C₈	—	4	4	no	—
81	C₈	#103	1	2	yes	—
82	C₈	#103	1	2	yes	—
83	C₈	#103	1	2	yes	—
84	C₈	#103	1	2	yes	—
85	C₂²	—	2	2	no	—
86	C₂²	#390	1	1	yes	characteristic
87	C₂²	#492	1	1	yes	characteristic
88	C₂²	#494	1	1	yes	characteristic
89	C₄	—	2	2	no	—
90	C₄	—	2	2	no	—
91	C₄	—	8	4	no	—
92	C₄	—	16	2	no	—
93	C₄	C₂×#103	1	1	yes	characteristic
94	C₄	C₂×#103	1	1	yes	characteristic
95	C₄	#487	1	1	yes	characteristic
96	C₄	#487	1	1	yes	characteristic
97	C₂	—	2	1	no	—
98	C₂	—	2	1	no	—
99	C₂	#1956	1	1	yes	characteristic
100	C₂	#1976	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#3160

C₂

yes

maximal, characteristic

#1846

C₂

yes

maximal, characteristic

#1518

C₂

yes

maximal

#1450

C₂

yes

maximal, characteristic

#533

C₂²

yes

—

#500

C₄

yes

—

—

C₄

yes

complemented

#431

C₂²

yes

—

—

—

—

—

C₂×C₄×C₈

C₂²

yes

characteristic

#382

C₂²

yes

—

#328

—

#322

—

C₈²

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

D₄

yes

characteristic

C₄⋊Q₈

C₂×C₄

yes

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂³

yes

—

—

—

—

—

—

C₄⋊C₈

—

C₂²⋊C₈

D₄

yes

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

#95

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂×C₈

—

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₈

SD₁₆

yes

—

C₂×C₈

SD₁₆

yes

—

C₂×C₈

Q₁₆

yes

—

C₂×C₈

Q₁₆

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

#41

yes

—

C₂³

#127

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄

C₄×D₄

yes

—

C₂×C₄

#121

yes

—

C₂×C₄

#124

yes

characteristic

C₂×C₄

C₂×SD₁₆

yes

characteristic

C₂×C₄

C₂×SD₁₆

yes

characteristic

C₂×C₄

C₂×Q₁₆

yes

characteristic

C₂×C₄

C₂×Q₁₆

yes

characteristic

C₈

—

C₈

#103

yes

—

C₈

#103

yes

—

C₈

#103

yes

—

C₈

#103

yes

—

C₂²

—

C₂²

yes

C₂²

yes

C₂²

yes

C₄

—

C₄

—

C₄

—

C₄

—

C₄

yes

C₄

yes

C₄

yes

C₄

yes

C₂

—

C₂

—

C₂

#1956

yes

characteristic

100

C₂

#1976

yes

—

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

subgroup

G/H

center, socle

C₂²

#390

commutator

C₄²

C₂²×C₄

Frattini, cosocle

C₂×C₄²

C₂³

C₂³

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂² ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
6. #500	C₄	semidirect
9. #498	C₄	semidirect

H ≅ G/N

type

6. #500

C₄

semidirect

9. #498

C₄

semidirect

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

subgroup	index
#3160	2
#1846	2
#1518	2
#1450	2

subgroup

index

Subgroups — #9755

Conjugacy classes of subgroups

Complements

Maximal subgroups