Group #13191 — subgroups

Subgroups — #13191

Order 256.

225 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 225 conjugacy classes of subgroups — too many to draw. Only its 87 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	#2931	C₂	1	1	yes	maximal, characteristic
2	#2150	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#350	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#131	C₂²	1	2	yes	—
6	#532	C₄	1	1	yes	characteristic
7	C₂×M₃₂	C₂²	1	2	yes	—
8	C₂²×C₁₆	C₂²	1	2	yes	—
9	#436	—	2	2	no	—
10	C₈×D₄	—	2	2	no	—
11	#423	C₄	1	1	yes	characteristic
12	#408	C₄	1	2	yes	—
13	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₄⋊C₁₆	—	2	2	no	—
15	#350	—	2	8	no	—
16	C₄×C₁₆	—	2	2	no	—
17	C₂×#41	C₂×C₄	1	2	yes	—
18	#131	—	2	4	no	—
19	#131	—	2	4	no	—
20	C₂×M₁₆	—	2	2	no	—
21	C₂×M₁₆	C₂×C₄	1	2	yes	—
22	C₂²×C₈	—	2	2	no	—
23	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈	C₂³	1	1	yes	characteristic
25	C₄⋊Q₈	C₈	1	1	yes	characteristic
26	#127	C₈	1	1	yes	characteristic
27	#124	C₈	1	2	yes	—
28	#121	—	2	2	no	—
29	C₄×D₄	—	2	2	no	—
30	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
31	M₃₂	—	2	4	no	—
32	C₂×C₁₆	—	2	2	no	—
33	C₂×C₁₆	—	2	2	no	—
34	C₂×C₁₆	—	2	2	no	—
35	C₂×C₁₆	D₄	1	4	yes	complemented
36	C₄⋊C₈	—	2	2	no	—
37	C₂²⋊C₈	—	2	4	no	—
38	C₄×C₈	D₄	1	2	yes	—
39	C₄×C₈	D₄	1	2	yes	—
40	#41	—	2	4	no	—
41	#41	—	2	4	no	—
42	C₂×Q₈	C₂×C₈	1	2	yes	—
43	C₂×D₄	—	2	2	no	—
44	C₂×D₄	—	2	2	no	—
45	C₂×D₄	C₂×C₈	1	2	yes	—
46	C₂²×C₄	—	2	2	no	—
47	C₂²×C₄	M₁₆	1	2	yes	—
48	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
49	M₁₆	—	2	4	no	—
50	M₁₆	—	4	4	no	—
51	C₂×C₈	—	2	2	no	—
52	C₂×C₈	—	2	2	no	—
53	C₂×C₈	—	4	2	no	—
54	C₂×C₈	—	4	2	no	—
55	C₂×C₈	C₂²⋊C₄	1	2	yes	—
56	C₂×C₈	C₂×D₄	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	—	2	2	no	—
61	C₂²⋊C₄	—	2	4	no	—
62	C₄²	C₂²⋊C₄	1	2	yes	—
63	C₄²	C₂×C₈	1	1	yes	characteristic
64	C₄²	C₂×C₈	1	1	yes	characteristic
65	C₁₆	—	2	4	no	—
66	C₁₆	—	4	2	no	—
67	C₂³	—	2	2	no	—
68	C₂³	C₂×M₁₆	1	1	yes	characteristic
69	Q₈	—	2	4	no	—
70	D₄	—	2	4	no	—
71	D₄	—	4	4	no	—
72	D₄	—	4	4	no	—
73	C₂×C₄	—	2	4	no	—
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	—	4	2	no	—
76	C₂×C₄	—	4	2	no	—
77	C₂×C₄	C₂²⋊C₈	1	2	yes	—
78	C₂×C₄	C₂²⋊C₈	1	2	yes	—
79	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄	C₄×D₄	1	2	yes	—
81	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄	#131	1	2	yes	—
83	C₈	—	2	2	no	—
84	C₈	—	4	2	no	—
85	C₈	#121	1	2	yes	—
86	C₈	#124	1	1	yes	characteristic
87	C₈	#127	1	1	yes	characteristic
88	C₂²	—	2	2	no	—
89	C₂²	—	4	4	no	—
90	C₂²	—	4	2	no	—
91	C₂²	C₂×C₂²⋊C₈	1	1	yes	characteristic
92	C₂²	#436	1	2	yes	—
93	C₄	—	2	2	no	—
94	C₄	—	4	2	no	—
95	C₄	#350	1	4	yes	complemented
96	C₄	#390	1	1	yes	characteristic
97	C₄	#408	1	1	yes	characteristic
98	C₄	C₈×D₄	1	2	yes	—
99	C₂	—	2	2	no	—
100	C₂	—	8	2	no	—
101	C₂	#1954	1	1	yes	characteristic
102	C₂	C₂×#350	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#2931

C₂

yes

maximal, characteristic

#2150

C₂

yes

maximal, characteristic, complemented

C₂×#350

C₂

yes

maximal, complemented

C₂×C₄×C₁₆

C₂

yes

maximal, characteristic, complemented

C₂×#131

C₂²

yes

—

#532

C₄

yes

characteristic

C₂×M₃₂

C₂²

yes

—

C₂²×C₁₆

C₂²

yes

—

#436

—

C₈×D₄

—

#423

C₄

yes

characteristic

#408

C₄

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₄⋊C₁₆

—

#350

—

C₄×C₁₆

—

C₂×#41

C₂×C₄

yes

—

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

C₂×C₄

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₄⋊Q₈

C₈

yes

characteristic

#127

C₈

yes

characteristic

#124

C₈

yes

—

#121

—

C₄×D₄

—

C₂×C₄²

C₂×C₄

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

D₄

yes

complemented

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

D₄

yes

—

C₄×C₈

D₄

yes

—

#41

—

#41

—

C₂×Q₈

C₂×C₈

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×C₈

yes

—

C₂²×C₄

—

C₂²×C₄

M₁₆

yes

—

C₂²×C₄

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄

yes

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

#41

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂²⋊C₄

yes

—

C₄²

C₂×C₈

yes

characteristic

C₄²

C₂×C₈

yes

characteristic

C₁₆

—

C₁₆

—

C₂³

—

C₂³

C₂×M₁₆

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊C₈

yes

—

C₂×C₄

C₂²⋊C₈

yes

—

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄

C₄×D₄

yes

—

C₂×C₄

C₂²×C₈

yes

characteristic

C₂×C₄

#131

yes

—

C₈

—

C₈

—

C₈

#121

yes

—

C₈

yes

C₈

yes

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂×C₂²⋊C₈

yes

characteristic

C₂²

#436

yes

—

C₄

—

C₄

—

C₄

yes

C₄

yes

C₄

yes

C₄

C₈×D₄

yes

—

C₂

—

100

C₂

—

101

C₂

#1954

yes

characteristic

102

C₂

C₂×#350

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₈
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂²	C₂×C₂²⋊C₈
prime radical	#532	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₈

C₂×D₄

commutator

C₂×C₄

C₂²×C₈

Frattini, cosocle

C₂²×C₈

C₂³

socle

C₂²

C₂×C₂²⋊C₈

C₄

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. #2150	C₂	semidirect
3. C₂×#350	C₂	semidirect
4. C₂×C₄×C₁₆	C₂	semidirect
35. C₂×C₁₆	D₄	semidirect
95. C₄	#350	semidirect

H ≅ G/N

type

2. #2150

C₂

semidirect

3. C₂×#350

C₂

semidirect

4. C₂×C₄×C₁₆

C₂

semidirect

35. C₂×C₁₆

D₄

semidirect

95. C₄

#350

semidirect

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

subgroup	index
#2931	2
#2150	2
C₂×#350	2
C₂×C₄×C₁₆	2

subgroup

index

#2931

#2150

C₂×#350

C₂×C₄×C₁₆

Subgroups — #13191

Conjugacy classes of subgroups

Complements

Maximal subgroups