Group #13164 — subgroups

Subgroups — #13164

Order 256.

142 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 81 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	#2153	C₂	1	2	yes	maximal
2	C₂×#364	C₂	1	1	yes	maximal, characteristic
3	C₄×M₃₂	C₂	1	1	yes	maximal, characteristic
4	#1772	C₂	1	1	yes	maximal, characteristic
5	#1378	C₂	1	2	yes	maximal
6	C₂×M₃₂	C₂²	1	2	yes	—
7	C₂×M₃₂	C₂²	1	1	yes	characteristic
8	C₂×M₃₂	C₂²	1	1	yes	characteristic
9	C₂×#108	C₄	1	1	yes	characteristic
10	#427	C₄	1	1	yes	characteristic
11	#405	C₂²	1	2	yes	—
12	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
13	#364	—	2	4	no	—
14	#364	—	2	1	no	—
15	#364	—	2	1	no	—
16	#346	—	2	1	no	—
17	C₄×C₁₆	—	2	1	no	—
18	#339	C₄	1	2	yes	—
19	C₂×M₁₆	C₂×C₄	1	2	yes	—
20	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈	C₂³	1	1	yes	characteristic
22	#117	C₂×C₄	1	2	yes	—
23	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
24	M₃₂	—	2	1	no	—
25	M₃₂	—	2	1	no	—
26	M₃₂	—	2	2	no	—
27	M₃₂	—	2	2	no	—
28	M₃₂	D₄	1	2	yes	—
29	M₃₂	Q₈	1	2	yes	—
30	C₂×C₁₆	—	2	1	no	—
31	C₂×C₁₆	—	2	1	no	—
32	C₂×C₁₆	—	2	2	no	—
33	#108	C₈	1	2	yes	—
34	#108	C₈	1	2	yes	—
35	#107	—	2	1	no	—
36	#106	—	2	1	no	—
37	#97	—	2	2	no	—
38	C₄×C₈	—	2	2	no	—
39	C₄×C₈	D₄	1	1	yes	characteristic
40	C₄×C₈	D₄	1	1	yes	characteristic
41	C₄×C₈	Q₈	1	1	yes	characteristic
42	C₄×C₈	Q₈	1	1	yes	characteristic
43	C₂²×C₄	—	2	1	no	—
44	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
45	M₁₆	—	2	2	no	—
46	M₁₆	C₂×C₈	1	4	yes	—
47	C₂×C₈	—	2	2	no	—
48	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈	C₂×C₈	1	1	yes	characteristic
51	C₂×C₈	C₂×C₈	1	1	yes	characteristic
52	C₂×C₈	M₁₆	1	1	yes	characteristic
53	C₂×C₈	M₁₆	1	1	yes	characteristic
54	C₂×C₈	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₈	#41	1	1	yes	characteristic
59	C₂×C₈	#41	1	1	yes	characteristic
60	C₄⋊C₄	—	2	2	no	—
61	C₂²⋊C₄	—	2	2	no	—
62	C₄²	—	2	1	no	—
63	C₄²	—	2	2	no	—
64	C₄²	C₄⋊C₄	1	1	yes	characteristic
65	C₄²	C₄⋊C₄	1	1	yes	characteristic
66	C₁₆	—	2	2	no	—
67	C₁₆	—	2	2	no	—
68	C₁₆	—	4	2	no	—
69	C₂³	#131	1	1	yes	characteristic
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	4	2	no	—
73	C₂×C₄	—	4	2	no	—
74	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄	C₄×D₄	1	1	yes	characteristic
76	C₂×C₄	C₄×Q₈	1	1	yes	characteristic
77	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
78	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
79	C₂×C₄	#131	1	1	yes	characteristic
80	C₈	—	2	4	no	—
81	C₈	C₄⋊C₈	1	1	yes	characteristic
82	C₈	C₄⋊C₈	1	1	yes	characteristic
83	C₈	C₄⋊C₈	1	1	yes	characteristic
84	C₈	C₄⋊C₈	1	1	yes	characteristic
85	C₈	#121	1	1	yes	characteristic
86	C₈	C₂²⋊Q₈	1	1	yes	characteristic
87	C₈	#125	1	1	yes	characteristic
88	C₈	C₄⋊Q₈	1	1	yes	characteristic
89	C₂²	—	4	1	no	—
90	C₂²	#424	1	1	yes	characteristic
91	C₂²	#436	1	1	yes	characteristic
92	C₂²	C₈×Q₈	1	1	yes	characteristic
93	C₄	—	2	2	no	—
94	C₄	—	4	1	no	—
95	C₄	—	8	2	no	—
96	C₄	#389	1	1	yes	characteristic
97	C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
98	C₄	C₈×D₄	1	1	yes	characteristic
99	C₄	#446	1	1	yes	characteristic
100	C₂	—	2	1	no	—
101	C₂	—	2	1	no	—
102	C₂	—	2	1	no	—
103	C₂	#1939	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#2153

C₂

yes

maximal

C₂×#364

C₂

yes

maximal, characteristic

C₄×M₃₂

C₂

yes

maximal, characteristic

#1772

C₂

yes

maximal, characteristic

#1378

C₂

yes

maximal

C₂×M₃₂

C₂²

yes

—

C₂×M₃₂

C₂²

yes

characteristic

C₂×M₃₂

C₂²

yes

characteristic

C₂×#108

C₄

yes

characteristic

#427

C₄

yes

characteristic

#405

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

—

—

—

—

C₄×C₁₆

—

#339

C₄

yes

—

C₂×M₁₆

C₂×C₄

yes

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

#117

C₂×C₄

yes

—

C₂×C₄²

C₂×C₄

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

D₄

yes

—

M₃₂

Q₈

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#108

C₈

yes

—

#108

C₈

yes

—

#107

—

#106

—

#97

—

C₄×C₈

—

C₄×C₈

D₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₄×C₈

Q₈

yes

characteristic

C₄×C₈

Q₈

yes

characteristic

C₂²×C₄

—

C₂²×C₄

yes

characteristic

M₁₆

—

M₁₆

C₂×C₈

yes

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄

yes

characteristic

C₂×C₈

C₄⋊C₄

yes

characteristic

C₂×C₈

yes

characteristic

C₂×C₈

yes

characteristic

C₂×C₈

M₁₆

yes

characteristic

C₂×C₈

M₁₆

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×Q₈

yes

characteristic

C₂×C₈

C₂×Q₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₄⋊C₄

yes

characteristic

C₄²

C₄⋊C₄

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

—

C₂³

#131

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄

C₄×D₄

yes

characteristic

C₂×C₄

C₄×Q₈

yes

characteristic

C₂×C₄

C₂²×C₈

yes

characteristic

C₂×C₄

C₂×M₁₆

yes

characteristic

C₂×C₄

#131

yes

characteristic

C₈

—

C₈

C₄⋊C₈

yes

characteristic

C₈

C₄⋊C₈

yes

characteristic

C₈

C₄⋊C₈

yes

characteristic

C₈

C₄⋊C₈

yes

characteristic

C₈

#121

yes

characteristic

C₈

C₂²⋊Q₈

yes

characteristic

C₈

#125

yes

characteristic

C₈

C₄⋊Q₈

yes

characteristic

C₂²

—

C₂²

yes

C₂²

yes

C₂²

C₈×Q₈

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

#389

yes

characteristic

C₄

C₂×C₄⋊C₈

yes

characteristic

C₄

C₈×D₄

yes

characteristic

C₄

#446

yes

characteristic

100

C₂

—

101

C₂

—

102

C₂

—

103

C₂

#1939

yes

characteristic

	subgroup	G/H
center	C₈	#121
commutator	C₂×C₄	C₂²×C₈
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂	#1939
prime radical	C₂³	#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₈

#121

commutator

C₂×C₄

C₂²×C₈

Frattini, cosocle

C₂²×C₈

C₂³

C₂

C₂³

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ G

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

subgroup	index
#2153	2
C₂×#364	2
C₄×M₃₂	2
#1772	2
#1378	2

subgroup

index

C₄×M₃₂

Subgroups — #13164

Conjugacy classes of subgroups

Maximal subgroups