Group #13203 — subgroups

Subgroups — #13203

Order 256.

255 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 255 conjugacy classes of subgroups — too many to draw. Only its 107 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×#494	C₂	1	1	yes	maximal, characteristic
2	C₂×#358	C₂	1	4	yes	maximal
3	C₂×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
4	#1845	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₁₆	—	2	1	no	—
6	C₂²×Q₁₆	C₂²	1	2	yes	—
7	C₂×C₄⋊Q₈	—	2	1	no	—
8	C₂²×C₁₆	C₂²	1	2	yes	—
9	#494	—	2	2	no	—
10	#494	C₄	1	4	yes	complemented
11	C₂×#107	—	2	1	no	—
12	C₂×#107	C₂²	1	2	yes	—
13	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
14	#384	—	2	1	no	—
15	#358	—	2	8	no	—
16	C₄×C₁₆	—	2	2	no	—
17	C₂²×Q₈	—	2	2	no	—
18	C₂×Q₁₆	—	2	4	no	—
19	C₂×Q₁₆	—	2	4	no	—
20	C₂×Q₁₆	—	2	2	no	—
21	C₂×Q₁₆	—	2	4	no	—
22	C₂×Q₁₆	—	4	2	no	—
23	C₂×Q₁₆	—	4	2	no	—
24	C₂×Q₁₆	C₂×C₄	1	4	yes	—
25	C₂²×C₈	D₄	1	1	yes	characteristic
26	C₂²×C₈	C₂³	1	1	yes	characteristic
27	C₄⋊Q₈	—	2	4	no	—
28	C₄⋊Q₈	—	4	2	no	—
29	C₂×C₄⋊C₄	—	2	2	no	—
30	C₂×C₄⋊C₄	—	4	2	no	—
31	C₂×C₄⋊C₄	—	4	2	no	—
32	C₂×C₄²	D₄	1	1	yes	characteristic
33	C₂×C₁₆	—	2	2	no	—
34	C₂×C₁₆	—	2	2	no	—
35	C₂×C₁₆	D₄	1	4	yes	—
36	#107	—	2	4	no	—
37	#107	—	4	2	no	—
38	C₄×C₈	C₂×C₄	1	2	yes	—
39	C₄×C₈	D₄	1	2	yes	—
40	C₂×Q₈	—	2	4	no	—
41	C₂×Q₈	—	4	2	no	—
42	C₂×Q₈	—	4	2	no	—
43	C₂×Q₈	—	4	2	no	—
44	C₂×Q₈	—	4	2	no	—
45	C₂×Q₈	—	4	2	no	—
46	C₂²×C₄	—	4	2	no	—
47	C₂²×C₄	—	4	2	no	—
48	C₂²×C₄	D₈	1	2	yes	—
49	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
50	Q₁₆	—	2	8	no	—
51	Q₁₆	—	4	8	no	—
52	Q₁₆	—	4	4	no	—
53	C₂×C₈	C₂²⋊C₄	1	2	yes	—
54	C₂×C₈	D₈	1	2	yes	—
55	C₂×C₈	SD₁₆	1	2	yes	—
56	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈	C₂×D₄	1	2	yes	—
58	C₂×C₈	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈	#41	1	2	yes	—
60	C₄⋊C₄	—	4	8	no	—
61	C₄⋊C₄	—	4	2	no	—
62	C₄⋊C₄	—	4	2	no	—
63	C₄⋊C₄	—	8	4	no	—
64	C₄²	C₂²⋊C₄	1	2	yes	—
65	C₄²	SD₁₆	1	2	yes	—
66	C₁₆	—	2	4	no	—
67	C₂³	C₂×D₈	1	1	yes	characteristic
68	Q₈	—	4	4	no	—
69	Q₈	—	4	4	no	—
70	Q₈	—	8	2	no	—
71	Q₈	—	8	2	no	—
72	C₂×C₄	—	4	4	no	—
73	C₂×C₄	—	8	2	no	—
74	C₂×C₄	—	8	2	no	—
75	C₂×C₄	—	8	2	no	—
76	C₂×C₄	—	8	4	no	—
77	C₂×C₄	#102	1	4	yes	—
78	C₂×C₄	SD₃₂	1	4	yes	—
79	C₂×C₄	Q₃₂	1	4	yes	—
80	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄	#124	1	1	yes	characteristic
82	C₂×C₄	#127	1	1	yes	characteristic
83	C₂×C₄	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
85	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
86	C₈	#102	1	4	yes	—
87	C₈	C₄×D₄	1	2	yes	—
88	C₈	#121	1	2	yes	—
89	C₂²	C₂×#102	1	1	yes	characteristic
90	C₂²	#486	1	1	yes	characteristic
91	C₂²	#493	1	1	yes	characteristic
92	C₂²	C₂×SD₃₂	1	2	yes	—
93	C₂²	C₂×Q₃₂	1	2	yes	—
94	C₄	—	8	4	no	—
95	C₄	—	16	2	no	—
96	C₄	#358	1	8	yes	complemented
97	C₄	#390	1	1	yes	characteristic
98	C₄	C₂×#102	1	1	yes	characteristic
99	C₄	#486	1	1	yes	characteristic
100	C₄	#492	1	1	yes	characteristic
101	C₂	#1957	1	1	yes	characteristic
102	C₂	C₂×#358	1	2	yes	—
103	C₂	#2241	1	2	yes	—
104	C₂	#2247	1	1	yes	characteristic
105	C₂	#2248	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#494

C₂

yes

maximal, characteristic

C₂×#358

C₂

yes

maximal

C₂×C₄×C₁₆

C₂

yes

maximal, characteristic

#1845

C₂

yes

maximal, characteristic

C₂²×Q₁₆

—

C₂²×Q₁₆

C₂²

yes

—

C₂×C₄⋊Q₈

—

C₂²×C₁₆

C₂²

yes

—

#494

—

#494

C₄

yes

complemented

C₂×#107

—

C₂×#107

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

#384

—

#358

—

C₄×C₁₆

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×C₄

yes

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

D₄

yes

—

#107

—

#107

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

D₄

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₈

yes

—

C₂²×C₄

C₂×D₄

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

—

C₂×C₈

C₂²⋊C₄

yes

—

C₂×C₈

D₈

yes

—

C₂×C₈

SD₁₆

yes

—

C₂×C₈

C₂²×C₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

#41

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²⋊C₄

yes

—

C₄²

SD₁₆

yes

—

C₁₆

—

C₂³

C₂×D₈

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#102

yes

—

C₂×C₄

SD₃₂

yes

—

C₂×C₄

Q₃₂

yes

—

C₂×C₄

C₂×C₂²⋊C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂×D₈

yes

characteristic

C₂×C₄

C₂×SD₁₆

yes

characteristic

C₂×C₄

C₂×SD₁₆

yes

characteristic

C₈

#102

yes

—

C₈

C₄×D₄

yes

—

C₈

#121

yes

—

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

C₂×SD₃₂

yes

—

C₂²

C₂×Q₃₂

yes

—

C₄

—

C₄

—

C₄

yes

C₄

yes

C₄

yes

C₄

yes

100

C₄

#492

yes

characteristic

101

C₂

#1957

yes

characteristic

102

C₂

C₂×#358

yes

—

103

C₂

#2241

yes

—

104

C₂

#2247

yes

characteristic

105

C₂

#2248

yes

characteristic

	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
10. #494	C₄	semidirect
96. C₄	#358	semidirect

H ≅ G/N

type

10. #494

C₄

semidirect

96. C₄

#358

semidirect

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

subgroup	index
C₂×#494	2
C₂×#358	2
C₂×C₄×C₁₆	2
#1845	2

subgroup

index

C₂×#494

C₂×#358

C₂×C₄×C₁₆

#1845

Subgroups — #13203

Conjugacy classes of subgroups

Complements

Maximal subgroups