Group #13368 — subgroups

Subgroups — #13368

Order 256.

185 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 185 conjugacy classes of subgroups — too many to draw. Only its 57 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	#3162	C₂	1	1	yes	maximal, characteristic
2	#2141	C₂	1	2	yes	maximal
3	#1971	C₂	1	1	yes	maximal, characteristic
4	#1386	C₂	1	2	yes	maximal
5	#1377	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆	C₂²	1	1	yes	characteristic
7	#533	—	2	1	no	—
8	C₂×M₃₂	C₂²	1	2	yes	—
9	#501	—	2	4	no	—
10	#427	—	2	1	no	—
11	#427	C₂²	1	2	yes	—
12	C₂×#106	C₂²	1	1	yes	characteristic
13	C₂×#103	—	2	1	no	—
14	C₂×#97	C₂²	1	1	yes	characteristic
15	#391	—	2	1	no	—
16	#358	—	2	4	no	—
17	C₂²×Q₈	—	2	1	no	—
18	C₂×Q₁₆	—	2	2	no	—
19	C₂×Q₁₆	—	2	2	no	—
20	C₂×Q₁₆	—	2	1	no	—
21	C₂×Q₁₆	Q₈	1	2	yes	complemented
22	C₂²×C₈	D₄	1	1	yes	characteristic
23	C₂²×C₈	C₂³	1	1	yes	characteristic
24	C₄⋊Q₈	—	2	2	no	—
25	#125	—	2	2	no	—
26	C₂²⋊Q₈	—	4	2	no	—
27	C₄×Q₈	—	4	2	no	—
28	#117	—	2	2	no	—
29	C₂×C₄⋊C₄	—	2	1	no	—
30	C₂×C₄²	D₄	1	1	yes	characteristic
31	M₃₂	—	2	2	no	—
32	M₃₂	—	2	2	no	—
33	C₂×C₁₆	—	2	2	no	—
34	#107	—	2	2	no	—
35	#107	D₄	1	4	yes	—
36	#106	—	2	2	no	—
37	#106	—	2	2	no	—
38	#106	—	2	2	no	—
39	#103	—	4	2	no	—
40	#97	—	2	2	no	—
41	#95	—	4	2	no	—
42	C₂×Q₈	—	2	2	no	—
43	C₂×Q₈	—	4	2	no	—
44	C₂×Q₈	—	4	1	no	—
45	C₂×Q₈	—	4	1	no	—
46	C₂×Q₈	—	4	1	no	—
47	C₂×Q₈	—	4	1	no	—
48	C₂×Q₈	—	4	1	no	—
49	C₂²×C₄	—	4	1	no	—
50	C₂²×C₄	—	4	1	no	—
51	C₂²×C₄	D₈	1	2	yes	—
52	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
53	Q₁₆	—	2	4	no	—
54	Q₁₆	—	4	2	no	—
55	C₂×C₈	—	2	1	no	—
56	C₂×C₈	—	2	1	no	—
57	C₂×C₈	SD₁₆	1	2	yes	—
58	C₂×C₈	C₂×D₄	1	2	yes	—
59	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₈	#41	1	2	yes	—
61	C₂×C₈	#41	1	1	yes	characteristic
62	C₄⋊C₄	—	2	4	no	—
63	C₄⋊C₄	—	4	4	no	—
64	C₄⋊C₄	—	4	2	no	—
65	C₄⋊C₄	—	4	1	no	—
66	C₄⋊C₄	—	4	1	no	—
67	C₂²⋊C₄	—	4	2	no	—
68	C₄²	—	2	1	no	—
69	C₄²	—	2	1	no	—
70	C₄²	—	4	2	no	—
71	C₁₆	—	4	2	no	—
72	C₂³	C₂×D₈	1	1	yes	characteristic
73	Q₈	—	4	2	no	—
74	Q₈	—	4	2	no	—
75	Q₈	—	8	4	no	—
76	Q₈	—	8	1	no	—
77	Q₈	—	8	1	no	—
78	C₂×C₄	—	2	2	no	—
79	C₂×C₄	—	4	2	no	—
80	C₂×C₄	—	4	4	no	—
81	C₂×C₄	—	4	2	no	—
82	C₂×C₄	—	8	1	no	—
83	C₂×C₄	—	8	1	no	—
84	C₂×C₄	—	8	2	no	—
85	C₂×C₄	—	8	1	no	—
86	C₂×C₄	#120	1	1	yes	characteristic
87	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
88	C₂×C₄	#124	1	1	yes	characteristic
89	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
90	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
91	C₂×C₄	#137	1	1	yes	characteristic
92	C₈	—	2	2	no	—
93	C₈	C₂²⋊Q₈	1	2	yes	—
94	C₈	#124	1	2	yes	—
95	C₂²	—	2	2	no	—
96	C₂²	#447	1	1	yes	characteristic
97	C₂²	#474	1	1	yes	characteristic
98	C₂²	#486	1	1	yes	characteristic
99	C₄	—	4	2	no	—
100	C₄	—	8	4	no	—
101	C₄	—	8	2	no	—
102	C₄	—	16	1	no	—
103	C₄	#393	1	1	yes	characteristic
104	C₄	#448	1	1	yes	characteristic
105	C₄	#476	1	1	yes	characteristic
106	C₄	#489	1	1	yes	characteristic
107	C₂	—	2	1	no	—
108	C₂	—	2	1	no	—
109	C₂	#2023	1	1	yes	characteristic
110	C₂	#2191	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#3162

C₂

yes

maximal, characteristic

#2141

C₂

yes

maximal

#1971

C₂

yes

maximal, characteristic

#1386

C₂

yes

maximal

#1377

C₂

yes

maximal, characteristic

C₂²×Q₁₆

C₂²

yes

characteristic

#533

—

C₂×M₃₂

C₂²

yes

—

#501

—

#427

—

#427

C₂²

yes

—

C₂×#106

C₂²

yes

characteristic

C₂×#103

—

C₂×#97

C₂²

yes

characteristic

#391

—

#358

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

Q₈

yes

complemented

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

—

C₂×C₁₆

—

#107

—

#107

D₄

yes

—

—

—

—

—

—

—

C₂×Q₈

—

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

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

SD₁₆

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×Q₈

yes

characteristic

C₂×C₈

#41

yes

—

C₂×C₈

#41

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₁₆

—

C₂³

C₂×D₈

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#120

yes

characteristic

C₂×C₄

C₂²⋊Q₈

yes

characteristic

C₂×C₄

#124

yes

characteristic

C₂×C₄

C₂×SD₁₆

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

C₂²⋊Q₈

yes

—

C₈

#124

yes

—

C₂²

—

C₂²

yes

C₂²

yes

C₂²

yes

C₄

—

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

#393

yes

characteristic

104

C₄

#448

yes

characteristic

105

C₄

#476

yes

characteristic

106

C₄

#489

yes

characteristic

107

C₂

—

108

C₂

—

109

C₂

#2023

yes

characteristic

110

C₂

#2191

yes

—

	subgroup	G/H
center, socle	C₂²	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂³	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#447

commutator, Frattini, cosocle

C₂²×C₈

C₂³

prime radical

C₂³

C₂×D₈

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
21. C₂×Q₁₆	Q₈	semidirect

H ≅ G/N

type

21. C₂×Q₁₆

Q₈

semidirect

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

subgroup	index
#3162	2
#2141	2
#1971	2
#1386	2
#1377	2

subgroup

index

Subgroups — #13368

Conjugacy classes of subgroups

Complements

Maximal subgroups