Group #13244 — subgroups

Subgroups — #13244

Order 256.

201 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 201 conjugacy classes of subgroups — too many to draw. Only its 71 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	C₂²×Q₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#368	C₂	1	1	yes	maximal, characteristic
3	C₂×#362	C₂	1	2	yes	maximal
4	#2108	C₂	1	1	yes	maximal, characteristic
5	#1979	C₂	1	2	yes	maximal
6	C₂²×Q₁₆	C₂²	1	2	yes	—
7	C₂×Q₃₂	—	2	1	no	—
8	C₂×Q₃₂	—	2	1	no	—
9	C₂×Q₃₂	—	2	2	no	—
10	C₂×Q₃₂	C₄	1	4	yes	complemented
11	C₂×M₃₂	C₂²	1	1	yes	characteristic
12	C₂²×C₁₆	C₂²	1	1	yes	characteristic
13	C₂×#108	C₂²	1	2	yes	—
14	#419	—	2	2	no	—
15	C₂×#101	—	2	2	no	—
16	#405	C₂²	1	1	yes	characteristic
17	#368	—	2	2	no	—
18	#362	—	2	4	no	—
19	#346	—	2	1	no	—
20	C₄×C₁₆	—	2	1	no	—
21	C₂²×Q₈	—	2	2	no	—
22	C₂×Q₁₆	—	2	2	no	—
23	C₂×Q₁₆	—	2	2	no	—
24	C₂×Q₁₆	—	2	2	no	—
25	C₂×Q₁₆	—	2	2	no	—
26	C₂×Q₁₆	—	2	2	no	—
27	C₂×Q₁₆	C₂×C₄	1	4	yes	—
28	C₂×M₁₆	—	2	2	no	—
29	C₂×M₁₆	D₄	1	1	yes	characteristic
30	C₂²×C₈	C₂³	1	1	yes	characteristic
31	#117	D₄	1	1	yes	characteristic
32	Q₃₂	—	2	4	no	—
33	Q₃₂	—	2	4	no	—
34	M₃₂	—	2	1	no	—
35	M₃₂	D₄	1	2	yes	—
36	C₂×C₁₆	—	2	1	no	—
37	C₂×C₁₆	C₂×C₄	1	2	yes	—
38	C₂×C₁₆	D₄	1	1	yes	characteristic
39	C₂×C₁₆	D₄	1	1	yes	characteristic
40	C₂×C₁₆	D₄	1	2	yes	—
41	#108	—	2	2	no	—
42	#108	—	2	2	no	—
43	#103	—	4	4	no	—
44	#101	—	4	4	no	—
45	#97	—	2	1	no	—
46	C₄×C₈	—	2	1	no	—
47	C₂×Q₈	—	2	4	no	—
48	C₂×Q₈	—	4	2	no	—
49	C₂×Q₈	—	4	2	no	—
50	C₂×Q₈	—	4	2	no	—
51	C₂×Q₈	—	4	2	no	—
52	C₂×Q₈	—	4	2	no	—
53	C₂²×C₄	—	4	2	no	—
54	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
55	Q₁₆	—	2	4	no	—
56	Q₁₆	—	2	4	no	—
57	Q₁₆	—	4	2	no	—
58	Q₁₆	—	4	2	no	—
59	M₁₆	—	2	1	no	—
60	M₁₆	—	4	2	no	—
61	M₁₆	—	4	2	no	—
62	M₁₆	D₈	1	2	yes	—
63	C₂×C₈	—	2	1	no	—
64	C₂×C₈	—	4	2	no	—
65	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₈	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈	#41	1	1	yes	characteristic
70	C₂×C₈	#41	1	1	yes	characteristic
71	C₄⋊C₄	D₈	1	2	yes	—
72	C₂²⋊C₄	—	2	1	no	—
73	C₄²	—	2	1	no	—
74	C₁₆	—	2	2	no	—
75	C₁₆	C₂²⋊C₄	1	2	yes	—
76	C₁₆	C₂²⋊C₄	1	2	yes	—
77	C₂³	#135	1	1	yes	characteristic
78	Q₈	—	4	4	no	—
79	Q₈	—	4	4	no	—
80	Q₈	—	8	2	no	—
81	Q₈	—	8	2	no	—
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	4	4	no	—
84	C₂×C₄	—	8	2	no	—
85	C₂×C₄	—	8	2	no	—
86	C₂×C₄	C₄×D₄	1	1	yes	characteristic
87	C₂×C₄	#121	1	1	yes	characteristic
88	C₂×C₄	#127	1	1	yes	characteristic
89	C₂×C₄	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄	C₂×D₈	1	1	yes	characteristic
91	C₂×C₄	#135	1	1	yes	characteristic
92	C₈	—	2	2	no	—
93	C₈	—	8	2	no	—
94	C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₈	C₄×D₄	1	1	yes	characteristic
96	C₈	#121	1	1	yes	characteristic
97	C₈	#124	1	1	yes	characteristic
98	C₂²	—	2	2	no	—
99	C₂²	C₄×D₈	1	1	yes	characteristic
100	C₂²	#466	1	1	yes	characteristic
101	C₂²	#495	1	1	yes	characteristic
102	C₄	—	4	2	no	—
103	C₄	—	8	4	no	—
104	C₄	#390	1	1	yes	characteristic
105	C₄	C₄×D₈	1	1	yes	characteristic
106	C₄	#466	1	1	yes	characteristic
107	C₄	#493	1	1	yes	characteristic
108	C₂	—	2	1	no	—
109	C₂	—	2	1	no	—
110	C₂	#1970	1	1	yes	characteristic
111	C₂	#2222	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Q₃₂

C₂

yes

maximal, characteristic

C₂×#368

C₂

yes

maximal, characteristic

C₂×#362

C₂

yes

maximal

#2108

C₂

yes

maximal, characteristic

#1979

C₂

yes

maximal

C₂²×Q₁₆

C₂²

yes

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

C₄

yes

complemented

C₂×M₃₂

C₂²

yes

characteristic

C₂²×C₁₆

C₂²

yes

characteristic

C₂×#108

C₂²

yes

—

#419

—

C₂×#101

—

#405

C₂²

yes

—

—

—

C₄×C₁₆

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×C₄

yes

—

C₂×M₁₆

—

C₂×M₁₆

D₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

#117

D₄

yes

characteristic

Q₃₂

—

Q₃₂

—

M₃₂

—

M₃₂

D₄

yes

—

C₂×C₁₆

—

C₂×C₁₆

C₂×C₄

yes

—

C₂×C₁₆

D₄

yes

characteristic

C₂×C₁₆

D₄

yes

characteristic

C₂×C₁₆

D₄

yes

—

—

—

—

—

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

D₈

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²×C₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

D₈

yes

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

C₁₆

C₂²⋊C₄

yes

—

C₁₆

C₂²⋊C₄

yes

—

C₂³

#135

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×D₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂×D₈

yes

characteristic

C₂×C₄

C₂×D₈

yes

characteristic

C₂×C₄

#135

yes

characteristic

C₈

—

C₈

—

C₈

C₂×C₂²⋊C₄

yes

characteristic

C₈

C₄×D₄

yes

C₈

yes

C₈

yes

C₂²

—

C₂²

C₄×D₈

yes

characteristic

100

C₂²

#466

yes

characteristic

101

C₂²

#495

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

#390

yes

characteristic

105

C₄

C₄×D₈

yes

characteristic

106

C₄

#466

yes

characteristic

107

C₄

#493

yes

characteristic

108

C₂

—

109

C₂

—

110

C₂

#1970

yes

characteristic

111

C₂

#2222

yes

—

	subgroup	G/H
center, socle	C₂²	#466
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂³	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#466

commutator

C₂×C₈

C₂²×C₄

Frattini, cosocle

C₂²×C₈

C₂³

prime radical

C₂³

#135

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
10. C₂×Q₃₂	C₄	semidirect

H ≅ G/N

type

10. C₂×Q₃₂

C₄

semidirect

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

subgroup	index
C₂²×Q₃₂	2
C₂×#368	2
C₂×#362	2
#2108	2
#1979	2

subgroup

index

C₂²×Q₃₂

Subgroups — #13244

Conjugacy classes of subgroups

Complements

Maximal subgroups