Group #13277 — subgroups

Subgroups — #13277

Order 256.

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

normal subgroups only

This group has 183 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	C₂×#499	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₄×#107	C₂	1	1	yes	maximal, characteristic
4	#1380	C₂	1	2	yes	maximal
5	#1380	C₂	1	2	yes	maximal
6	C₂×#125	—	2	1	no	—
7	C₂²×C₁₆	C₂²	1	2	yes	—
8	#499	—	2	2	no	—
9	#499	C₄	1	4	yes	complemented
10	C₂×#107	C₂²	1	1	yes	characteristic
11	C₂×#107	C₂²	1	1	yes	characteristic
12	C₂×#107	C₂²	1	2	yes	—
13	C₂×#106	—	2	1	no	—
14	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄	—	2	1	no	—
16	C₄⋊C₁₆	—	2	2	no	—
17	#340	—	2	1	no	—
18	C₂²×C₈	D₄	1	1	yes	characteristic
19	C₂²×C₈	C₂³	1	1	yes	characteristic
20	#125	—	2	4	no	—
21	#125	—	4	2	no	—
22	C₂×C₄⋊C₄	—	2	1	no	—
23	C₂×C₄⋊C₄	—	2	1	no	—
24	C₂×C₄⋊C₄	—	2	2	no	—
25	C₂×C₄⋊C₄	—	4	1	no	—
26	C₂×C₄⋊C₄	—	4	1	no	—
27	C₂×C₄²	—	4	1	no	—
28	C₂×C₄²	D₄	1	1	yes	characteristic
29	C₂×C₁₆	—	2	2	no	—
30	C₂×C₁₆	—	2	4	no	—
31	#107	—	2	1	no	—
32	#107	—	2	1	no	—
33	#107	—	2	2	no	—
34	#107	C₂×C₄	1	2	yes	—
35	#107	C₂×C₄	1	2	yes	—
36	#107	Q₈	1	4	yes	—
37	#106	—	2	4	no	—
38	C₄×C₈	C₂×C₄	1	2	yes	—
39	C₄×C₈	D₄	1	2	yes	—
40	#95	—	4	1	no	—
41	C₂²×C₄	—	4	2	no	—
42	C₂²×C₄	—	4	1	no	—
43	C₂²×C₄	—	4	1	no	—
44	C₂²×C₄	D₈	1	1	yes	characteristic
45	C₂²×C₄	D₈	1	1	yes	characteristic
46	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈	—	2	2	no	—
48	C₂×C₈	C₂²⋊C₄	1	2	yes	—
49	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈	C₂×Q₈	1	2	yes	—
52	C₂×C₈	#41	1	2	yes	—
53	C₄⋊C₄	—	2	2	no	—
54	C₄⋊C₄	—	2	2	no	—
55	C₄⋊C₄	—	2	4	no	—
56	C₄⋊C₄	—	4	4	no	—
57	C₄⋊C₄	—	4	4	no	—
58	C₄⋊C₄	—	4	1	no	—
59	C₄⋊C₄	—	4	1	no	—
60	C₄⋊C₄	—	4	2	no	—
61	C₄²	—	8	2	no	—
62	C₄²	C₂²⋊C₄	1	2	yes	—
63	C₄²	SD₁₆	1	2	yes	—
64	C₁₆	—	4	2	no	—
65	C₂³	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	—	4	2	no	—
70	C₂×C₄	—	4	2	no	—
71	C₂×C₄	—	4	4	no	—
72	C₂×C₄	—	8	2	no	—
73	C₂×C₄	—	8	2	no	—
74	C₂×C₄	—	8	2	no	—
75	C₂×C₄	—	8	1	no	—
76	C₂×C₄	—	8	1	no	—
77	C₂×C₄	#102	1	2	yes	—
78	C₂×C₄	#102	1	2	yes	—
79	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄	C₂²⋊Q₈	1	2	yes	—
81	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
82	C₂×C₄	#137	1	2	yes	—
83	C₈	—	2	2	no	—
84	C₈	C₄×Q₈	1	2	yes	—
85	C₈	#124	1	1	yes	characteristic
86	C₈	C₄⋊Q₈	1	1	yes	characteristic
87	C₂²	C₂×#102	1	1	yes	characteristic
88	C₂²	#474	1	2	yes	—
89	C₂²	#508	1	2	yes	—
90	C₂²	#509	1	1	yes	characteristic
91	C₂²	#510	1	1	yes	characteristic
92	C₄	—	2	2	no	—
93	C₄	—	2	2	no	—
94	C₄	—	8	2	no	—
95	C₄	—	8	2	no	—
96	C₄	—	8	4	no	—
97	C₄	#391	1	1	yes	characteristic
98	C₄	#419	1	1	yes	characteristic
99	C₄	#475	1	2	yes	—
100	C₂	#1988	1	1	yes	characteristic
101	C₂	#2138	1	1	yes	characteristic
102	C₂	#2139	1	1	yes	characteristic
103	C₂	#2228	1	2	yes	—
104	C₂	#2229	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#499

C₂

yes

maximal, characteristic

C₂×C₄⋊C₁₆

C₂

yes

maximal, characteristic

C₄×#107

C₂

yes

maximal, characteristic

#1380

C₂

yes

maximal

#1380

C₂

yes

maximal

C₂×#125

—

C₂²×C₁₆

C₂²

yes

—

#499

—

#499

C₄

yes

complemented

C₂×#107

C₂²

yes

characteristic

C₂×#107

C₂²

yes

characteristic

C₂×#107

C₂²

yes

—

C₂×#106

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₄×C₄⋊C₄

—

C₄⋊C₁₆

—

#340

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

#125

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

C₂×C₄

yes

—

#107

C₂×C₄

yes

—

#107

Q₈

yes

—

#106

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

D₄

yes

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₈

yes

characteristic

C₂²×C₄

D₈

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂×C₈

—

C₂×C₈

C₂²⋊C₄

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

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₂²⋊C₄

yes

—

C₄²

SD₁₆

yes

—

C₁₆

—

C₂³

C₂×D₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

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

C₂²⋊Q₈

yes

—

C₂×C₄

C₂×SD₁₆

yes

characteristic

C₂×C₄

#137

yes

—

C₈

—

C₈

C₄×Q₈

yes

—

C₈

#124

yes

characteristic

C₈

C₄⋊Q₈

yes

C₂²

yes

C₂²

yes

—

C₂²

#508

yes

—

C₂²

yes

C₂²

yes

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

yes

C₄

yes

C₄

yes

—

100

C₂

#1988

yes

characteristic

101

C₂

#2138

yes

characteristic

102

C₂

#2139

yes

characteristic

103

C₂

#2228

yes

—

104

C₂

#2229

yes

—

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

subgroup

G/H

center, socle, prime radical

C₂³

C₂×D₈

commutator

C₂×C₈

C₂²×C₄

Frattini, cosocle

C₂²×C₈

C₂³

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
9. #499	C₄	semidirect

H ≅ G/N

type

9. #499

C₄

semidirect

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

subgroup	index
C₂×#499	2
C₂×C₄⋊C₁₆	2
C₄×#107	2
#1380	2

subgroup

index

C₂×#499

C₂×C₄⋊C₁₆

C₄×#107

#1380

Subgroups — #13277

Conjugacy classes of subgroups

Complements

Maximal subgroups