Group #13444 — subgroups

Subgroups — #13444

Order 256.

196 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 53 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	#3414	C₂	1	1	yes	maximal, characteristic
2	#2150	C₂	1	1	yes	maximal, characteristic, complemented
3	#2148	C₂	1	2	yes	maximal, complemented
4	#1974	C₂	1	2	yes	maximal, complemented
5	#1385	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#135	C₂²	1	2	yes	—
7	#510	—	2	2	no	—
8	#509	—	2	2	no	—
9	#508	—	2	2	no	—
10	C₂×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×M₃₂	C₂²	1	1	yes	characteristic
12	C₂×#108	C₂²	1	2	yes	—
13	C₂×#104	—	2	2	no	—
14	#408	—	2	2	no	—
15	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₄⋊C₁₆	—	2	2	no	—
17	#362	—	2	2	no	—
18	#361	—	2	2	no	—
19	C₂×#41	—	2	2	no	—
20	#135	—	2	2	no	—
21	#135	—	2	2	no	—
22	#135	—	2	2	no	—
23	#135	—	2	2	no	—
24	C₂×Q₁₆	D₄	1	2	yes	—
25	C₂×SD₁₆	—	2	2	no	—
26	C₂×D₈	D₄	1	2	yes	—
27	C₂×M₁₆	—	2	2	no	—
28	C₂²×C₈	D₄	1	1	yes	characteristic
29	C₂²×C₈	C₂³	1	1	yes	characteristic
30	C₂×C₄²	D₄	1	1	yes	characteristic
31	Q₃₂	—	2	2	no	—
32	SD₃₂	—	2	4	no	—
33	D₁₆	—	2	2	no	—
34	M₃₂	—	2	2	no	—
35	M₃₂	—	2	2	no	—
36	C₂×C₁₆	—	2	1	no	—
37	C₂×C₁₆	D₄	1	2	yes	—
38	#108	—	2	2	no	—
39	#108	—	2	2	no	—
40	#104	—	4	2	no	—
41	#104	—	4	2	no	—
42	C₂²⋊C₈	—	4	4	no	—
43	C₄×C₈	—	2	1	no	—
44	C₄×C₈	—	2	1	no	—
45	#41	—	4	2	no	—
46	#41	—	4	2	no	—
47	#41	—	4	2	no	—
48	#41	—	4	2	no	—
49	C₂×Q₈	—	2	2	no	—
50	C₂×D₄	—	2	2	no	—
51	C₂×D₄	—	4	2	no	—
52	C₂²×C₄	—	2	1	no	—
53	C₂²×C₄	—	4	2	no	—
54	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
55	Q₁₆	—	2	2	no	—
56	Q₁₆	—	2	2	no	—
57	SD₁₆	—	4	2	no	—
58	SD₁₆	—	4	2	no	—
59	D₈	—	2	2	no	—
60	D₈	—	2	2	no	—
61	M₁₆	—	4	2	no	—
62	M₁₆	—	4	2	no	—
63	C₂×C₈	—	2	1	no	—
64	C₂×C₈	—	2	2	no	—
65	C₂×C₈	—	4	2	no	—
66	C₂×C₈	C₂×D₄	1	2	yes	—
67	C₂×C₈	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈	#41	1	1	yes	characteristic
69	C₂×C₈	#41	1	2	yes	—
70	C₄²	—	2	1	no	—
71	C₄²	D₈	1	1	yes	characteristic
72	C₄²	D₈	1	1	yes	characteristic
73	C₁₆	—	2	2	no	—
74	C₁₆	—	4	1	no	—
75	C₂³	—	4	2	no	—
76	C₂³	#135	1	1	yes	characteristic
77	Q₈	—	4	2	no	—
78	Q₈	—	4	2	no	—
79	D₄	—	4	2	no	—
80	D₄	—	4	2	no	—
81	D₄	—	8	2	no	—
82	D₄	—	8	2	no	—
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	4	2	no	—
86	C₂×C₄	—	4	2	no	—
87	C₂×C₄	—	8	2	no	—
88	C₂×C₄	—	8	2	no	—
89	C₂×C₄	#121	1	2	yes	—
90	C₂×C₄	#121	1	1	yes	characteristic
91	C₂×C₄	C₂×D₈	1	1	yes	characteristic
92	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
93	C₈	—	4	1	no	—
94	C₈	—	8	2	no	—
95	C₈	#120	1	1	yes	characteristic
96	C₈	#123	1	2	yes	—
97	C₈	#124	1	1	yes	characteristic
98	C₂²	—	4	1	no	—
99	C₂²	—	8	4	no	—
100	C₂²	—	8	2	no	—
101	C₂²	#463	1	2	yes	—
102	C₂²	#466	1	1	yes	characteristic
103	C₄	—	2	2	no	—
104	C₄	—	4	1	no	—
105	C₄	—	8	2	no	—
106	C₄	#394	1	1	yes	characteristic
107	C₄	#459	1	2	yes	—
108	C₄	#469	1	1	yes	characteristic
109	C₂	—	2	1	no	—
110	C₂	—	2	2	no	—
111	C₂	—	16	2	no	—
112	C₂	#2054	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#3414

C₂

yes

maximal, characteristic

#2150

C₂

yes

maximal, characteristic, complemented

#2148

C₂

yes

maximal, complemented

#1974

C₂

yes

maximal, complemented

#1385

C₂

yes

maximal, characteristic, complemented

C₂×#135

C₂²

yes

—

#510

—

#509

—

#508

—

C₂×M₃₂

C₂²

yes

characteristic

C₂×M₃₂

C₂²

yes

characteristic

C₂×#108

C₂²

yes

—

C₂×#104

—

#408

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₄⋊C₁₆

—

#362

—

#361

—

C₂×#41

—

—

—

—

—

C₂×Q₁₆

D₄

yes

—

C₂×SD₁₆

—

C₂×D₈

D₄

yes

—

C₂×M₁₆

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₂×C₄²

D₄

yes

characteristic

Q₃₂

—

SD₃₂

—

D₁₆

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

D₄

yes

—

—

—

—

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

—

—

—

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

Q₁₆

—

Q₁₆

—

SD₁₆

—

SD₁₆

—

D₈

—

D₈

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₄²

—

C₄²

D₈

yes

characteristic

C₄²

D₈

yes

characteristic

C₁₆

—

C₁₆

—

C₂³

—

C₂³

#135

yes

characteristic

Q₈

—

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#121

yes

—

C₂×C₄

#121

yes

characteristic

C₂×C₄

C₂×D₈

yes

characteristic

C₂×C₄

AGL(1,ℤ/8)

yes

—

C₈

—

C₈

—

C₈

#120

yes

characteristic

C₈

#123

yes

—

C₈

#124

yes

characteristic

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

#463

yes

—

102

C₂²

#466

yes

characteristic

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

#394

yes

characteristic

107

C₄

#459

yes

—

108

C₄

#469

yes

characteristic

109

C₂

—

110

C₂

—

111

C₂

—

112

C₂

#2054

yes

characteristic

	subgroup	G/H
center	C₄	#469
commutator, Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂	#2054
prime radical	#3414	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₄

#469

commutator, Frattini, cosocle

C₂²×C₈

C₂³

C₂

C₂

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. #2150	C₂	semidirect
3. #2148	C₂	semidirect
4. #1974	C₂	semidirect
5. #1385	C₂	semidirect

H ≅ G/N

type

2. #2150

C₂

semidirect

3. #2148

C₂

semidirect

4. #1974

C₂

semidirect

5. #1385

C₂

semidirect

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

subgroup	index
#3414	2
#2150	2
#2148	2
#1974	2
#1385	2

subgroup

index

Subgroups — #13444

Conjugacy classes of subgroups

Complements

Maximal subgroups