Group #13288 — subgroups

Subgroups — #13288

Order 256.

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

normal subgroups only

This group has 177 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	C₄×M₃₂	C₂	1	1	yes	maximal, characteristic
3	#1846	C₂	1	1	yes	maximal, characteristic
4	#1386	C₂	1	4	yes	maximal
5	#533	—	2	1	no	—
6	C₂×M₃₂	C₂²	1	2	yes	—
7	#500	C₄	1	2	yes	complemented
8	#499	—	2	1	no	—
9	#499	C₄	1	2	yes	complemented
10	#498	—	2	1	no	—
11	#427	—	2	1	no	—
12	#427	C₂²	1	2	yes	—
13	C₂×#107	—	2	1	no	—
14	C₂×#106	C₂²	1	2	yes	—
15	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
16	#382	—	2	1	no	—
17	#346	—	2	1	no	—
18	C₄×C₁₆	—	2	1	no	—
19	C₂²×C₈	D₄	1	1	yes	characteristic
20	C₂²×C₈	C₂³	1	1	yes	characteristic
21	C₄⋊Q₈	—	2	2	no	—
22	#125	—	2	2	no	—
23	C₂²⋊Q₈	—	4	2	no	—
24	C₄×Q₈	—	4	2	no	—
25	#117	—	2	2	no	—
26	C₂×C₄⋊C₄	—	2	2	no	—
27	C₂×C₄²	D₄	1	1	yes	characteristic
28	M₃₂	—	2	2	no	—
29	M₃₂	Q₈	1	4	yes	complemented
30	C₂×C₁₆	—	2	2	no	—
31	#107	—	2	2	no	—
32	#107	—	4	2	no	—
33	#107	C₂×C₄	1	4	yes	—
34	#106	—	2	2	no	—
35	#106	—	2	2	no	—
36	#106	—	2	4	no	—
37	C₄×C₈	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈	D₄	1	1	yes	characteristic
40	C₄×C₈	D₄	1	1	yes	characteristic
41	#95	—	4	2	no	—
42	C₂×Q₈	—	4	2	no	—
43	C₂²×C₄	—	4	2	no	—
44	C₂²×C₄	D₈	1	2	yes	—
45	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₈	SD₁₆	1	2	yes	—
49	C₂×C₈	Q₁₆	1	2	yes	—
50	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈	C₂×Q₈	1	2	yes	—
53	C₂×C₈	#41	1	2	yes	—
54	C₄⋊C₄	—	2	4	no	—
55	C₄⋊C₄	—	4	4	no	—
56	C₄⋊C₄	—	4	2	no	—
57	C₄⋊C₄	—	4	2	no	—
58	C₄⋊C₄	—	4	2	no	—
59	C₂²⋊C₄	—	4	2	no	—
60	C₄²	—	4	2	no	—
61	C₄²	C₂²⋊C₄	1	1	yes	characteristic
62	C₄²	C₂²⋊C₄	1	1	yes	characteristic
63	C₄²	SD₁₆	1	2	yes	—
64	C₁₆	—	2	4	no	—
65	C₂³	C₂×D₈	1	1	yes	characteristic
66	Q₈	—	8	4	no	—
67	C₂×C₄	—	4	2	no	—
68	C₂×C₄	—	4	4	no	—
69	C₂×C₄	—	8	4	no	—
70	C₂×C₄	—	8	2	no	—
71	C₂×C₄	#102	1	2	yes	—
72	C₂×C₄	#102	1	2	yes	—
73	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄	#124	1	1	yes	characteristic
75	C₂×C₄	C₄⋊Q₈	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	C₈	#103	1	2	yes	—
80	C₈	#103	1	2	yes	—
81	C₈	C₄×Q₈	1	2	yes	—
82	C₈	C₂²⋊Q₈	1	2	yes	—
83	C₂²	—	2	2	no	—
84	C₂²	C₂×#102	1	1	yes	characteristic
85	C₂²	#486	1	1	yes	characteristic
86	C₂²	#500	1	1	yes	characteristic
87	C₄	—	2	2	no	—
88	C₄	—	2	2	no	—
89	C₄	—	8	4	no	—
90	C₄	—	16	2	no	—
91	C₄	#391	1	1	yes	characteristic
92	C₄	C₂×#103	1	1	yes	characteristic
93	C₄	#487	1	1	yes	characteristic
94	C₄	#498	1	1	yes	characteristic
95	C₂	—	2	1	no	—
96	C₂	—	2	1	no	—
97	C₂	#1990	1	1	yes	characteristic
98	C₂	#2145	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#3160

C₂

yes

maximal, characteristic

C₄×M₃₂

C₂

yes

maximal, characteristic

#1846

C₂

yes

maximal, characteristic

#1386

C₂

yes

maximal

#533

—

C₂×M₃₂

C₂²

yes

—

#500

C₄

yes

complemented

#499

—

#499

C₄

yes

—

—

C₂²

yes

—

C₂×#107

—

C₂×#106

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

#382

—

#346

—

C₄×C₁₆

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄²

D₄

yes

characteristic

M₃₂

—

M₃₂

Q₈

yes

complemented

C₂×C₁₆

—

#107

—

#107

—

#107

C₂×C₄

yes

—

#106

—

#106

—

#106

—

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

D₈

yes

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₈

SD₁₆

yes

—

C₂×C₈

Q₁₆

yes

—

C₂×C₈

C₂²×C₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×Q₈

yes

—

C₂×C₈

#41

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²⋊C₄

yes

characteristic

C₄²

C₂²⋊C₄

yes

characteristic

C₄²

SD₁₆

yes

—

C₁₆

—

C₂³

C₂×D₈

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#102

yes

—

C₂×C₄

#102

yes

—

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄

#124

yes

characteristic

C₂×C₄

C₄⋊Q₈

yes

characteristic

C₂×C₄

C₂×SD₁₆

yes

characteristic

C₂×C₄

C₂×SD₁₆

yes

characteristic

C₂×C₄

C₂×Q₁₆

yes

characteristic

C₈

#103

yes

—

C₈

#103

yes

—

C₈

C₄×Q₈

yes

—

C₈

C₂²⋊Q₈

yes

—

C₂²

—

C₂²

yes

C₂²

yes

C₂²

yes

C₄

—

C₄

—

C₄

—

C₄

—

C₄

yes

C₄

yes

C₄

yes

C₄

yes

C₂

—

C₂

—

C₂

#1990

yes

characteristic

C₂

#2145

yes

—

	subgroup	G/H
center, socle	C₂²	C₂×#102
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂³	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂×#102

commutator

C₂×C₈

C₂²×C₄

Frattini, cosocle

C₂²×C₈

C₂³

prime radical

C₂³

C₂×D₈

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
7. #500	C₄	semidirect
9. #499	C₄	semidirect
29. M₃₂	Q₈	semidirect

H ≅ G/N

type

7. #500

C₄

semidirect

9. #499

C₄

semidirect

29. M₃₂

Q₈

semidirect

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

subgroup	index
#3160	2
C₄×M₃₂	2
#1846	2
#1386	2

subgroup

index

#3160

C₄×M₃₂

#1846

#1386

Subgroups — #13288

Conjugacy classes of subgroups

Complements

Maximal subgroups