Group #14527608 — subgroups

Subgroups — #14527608

Order 2176.

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

normal subgroups only

This group has 192 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	D₄×#3622	C₂	1	2	yes	maximal, complemented
2	#11762560	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄⋊#3622	C₂	1	2	yes	maximal, complemented
4	#11762530	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×D₈×C₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂×D₈×C₁₇	C₄	1	1	yes	characteristic
7	C₄×D₄×C₁₇	C₂²	1	2	yes	—
8	C₂²×#3622	—	2	2	no	—
9	C₁₇×#107	C₄	1	1	yes	characteristic
10	C₁₇×#102	C₄	1	2	yes	—
11	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
12	C₂²⋊#3622	—	2	2	no	—
13	C₄⋊#3622	C₂²	1	2	yes	—
14	C₄×#3622	C₂²	1	1	yes	characteristic
15	C₄×#3622	C₂²	1	1	yes	characteristic
16	C₂×D₄×C₁₇	C₂×C₄	1	2	yes	—
17	C₂²×C₄×C₁₇	—	2	2	no	—
18	D₈×C₁₇	C₈	1	4	yes	complemented
19	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
21	C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
22	C₂²⋊C₄×C₁₇	—	2	2	no	—
23	C₄²×C₁₇	C₂³	1	1	yes	characteristic
24	C₂×#3622	—	2	2	no	—
25	C₂×#3622	—	2	1	no	—
26	C₂×#3622	—	4	4	no	—
27	C₂×#3622	D₄	1	2	yes	—
28	C₂³×C₁₇	—	2	2	no	—
29	D₄×C₁₇	—	2	2	no	—
30	D₄×C₁₇	C₂×C₈	1	4	yes	complemented
31	C₂×C₄×C₁₇	—	2	2	no	—
32	C₂×C₄×C₁₇	—	4	2	no	—
33	C₂×C₄×C₁₇	—	4	2	no	—
34	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
37	C₈×C₁₇	—	2	1	no	—
38	C₈×C₁₇	C₂×C₈	1	2	yes	complemented
39	#3622	—	2	2	no	—
40	#3622	—	4	1	no	—
41	#3622	—	4	2	no	—
42	#1580	—	17	1	no	maximal, Sylow
43	C₂²×C₁₇	—	2	4	no	—
44	C₂²×C₁₇	—	4	2	no	—
45	C₂²×C₁₇	—	4	2	no	—
46	C₂²×C₁₇	C₄×D₄	1	1	yes	characteristic
47	C₄×C₁₇	—	2	1	no	—
48	C₄×C₁₇	—	4	2	no	—
49	C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
50	C₄×C₁₇	#131	1	1	yes	characteristic
51	C₄×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
52	C₄×D₈	D₁₇	1	1	yes	characteristic
53	C₈×D₄	—	17	2	no	—
54	#334	—	17	1	no	—
55	D₄⋊C₈	—	17	2	no	—
56	#322	—	17	1	no	—
57	C₂×C₁₇	—	4	4	no	—
58	C₂×C₁₇	C₈×D₄	1	1	yes	characteristic
59	C₂×C₁₇	#442	1	1	yes	characteristic
60	C₂×C₁₇	#444	1	1	yes	characteristic
61	C₂×D₈	Dic₁₇	1	1	yes	characteristic
62	C₂²×C₈	—	34	2	no	—
63	C₄×D₄	C₂×D₁₇	1	2	yes	—
64	#107	Dic₁₇	1	1	yes	characteristic
65	C₄⋊C₈	—	17	2	no	—
66	#102	Dic₁₇	1	2	yes	—
67	C₂²⋊C₈	—	34	2	no	—
68	C₄×C₈	—	17	1	no	—
69	C₄×C₈	—	17	1	no	—
70	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
71	C₁₇	#1580	1	1	yes	characteristic, complemented, Sylow
72	C₂×D₄	C₂×Dic₁₇	1	2	yes	—
73	C₂²×C₄	—	2	2	no	—
74	D₈	#3622	1	4	yes	complemented
75	C₂×C₈	—	17	2	no	—
76	C₂×C₈	—	34	2	no	—
77	C₂×C₈	—	34	1	no	—
78	C₂×C₈	—	68	4	no	—
79	C₂×C₈	C₂×Dic₁₇	1	1	yes	characteristic
80	C₂×C₈	C₂×Dic₁₇	1	1	yes	characteristic
81	C₄⋊C₄	C₂×Dic₁₇	1	2	yes	—
82	C₂²⋊C₄	—	2	2	no	—
83	C₄²	C₂²×D₁₇	1	1	yes	characteristic
84	C₂³	—	2	2	no	—
85	D₄	—	2	2	no	—
86	D₄	C₂×#3622	1	4	yes	complemented
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	4	2	no	—
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
91	C₂×C₄	#63254	1	1	yes	characteristic
92	C₂×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
93	C₈	—	2	1	no	—
94	C₈	—	34	2	no	—
95	C₈	—	68	1	no	—
96	C₈	—	68	2	no	—
97	C₈	C₂×#3622	1	2	yes	complemented
98	C₂²	—	2	4	no	—
99	C₂²	—	4	2	no	—
100	C₂²	—	4	2	no	—
101	C₂²	D₄×Dic₁₇	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	4	2	no	—
104	C₄	#10587597	1	2	yes	—
105	C₄	C₂²×#3622	1	1	yes	characteristic
106	C₄	#10587634	1	1	yes	characteristic
107	C₂	—	4	4	no	—
108	C₂	D₄×#3622	1	1	yes	characteristic
109	C₂	#11762921	1	1	yes	characteristic
110	C₂	#11762931	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₄×#3622

C₂

yes

maximal, complemented

#11762560

C₂

yes

maximal, characteristic, complemented

D₄⋊#3622

C₂

yes

maximal, complemented

#11762530

C₂

yes

maximal, characteristic, complemented

C₄×D₈×C₁₇

C₂

yes

maximal, characteristic

C₂×D₈×C₁₇

C₄

yes

characteristic

C₄×D₄×C₁₇

C₂²

yes

—

C₂²×#3622

—

C₁₇×#107

C₄

yes

characteristic

C₁₇×#102

C₄

yes

—

C₄×C₈×C₁₇

C₂²

yes

characteristic

C₂²⋊#3622

—

C₄⋊#3622

C₂²

yes

—

C₄×#3622

C₂²

yes

characteristic

C₄×#3622

C₂²

yes

characteristic

C₂×D₄×C₁₇

C₂×C₄

yes

—

C₂²×C₄×C₁₇

—

D₈×C₁₇

C₈

yes

complemented

C₂×C₈×C₁₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄⋊C₄×C₁₇

C₂×C₄

yes

—

C₂²⋊C₄×C₁₇

—

C₄²×C₁₇

C₂³

yes

—

—

—

D₄

yes

—

C₂³×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

C₂×C₈

yes

complemented

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

#41

yes

characteristic

C₈×C₁₇

—

C₈×C₁₇

C₂×C₈

yes

—

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

C₄×D₄

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₂²×C₈

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

—

C₄×D₈

D₁₇

yes

characteristic

C₈×D₄

—

#334

—

D₄⋊C₈

—

#322

—

C₂×C₁₇

—

C₂×C₁₇

C₈×D₄

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×D₈

Dic₁₇

yes

characteristic

C₂²×C₈

—

C₄×D₄

C₂×D₁₇

yes

—

#107

Dic₁₇

yes

characteristic

C₄⋊C₈

—

#102

Dic₁₇

yes

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₁₇

yes

characteristic

C₁₇

#1580

yes

characteristic, complemented, Sylow

C₂×D₄

C₂×Dic₁₇

yes

—

C₂²×C₄

—

D₈

#3622

yes

complemented

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₁₇

yes

characteristic

C₂×C₈

C₂×Dic₁₇

yes

characteristic

C₄⋊C₄

C₂×Dic₁₇

yes

—

C₂²⋊C₄

—

C₄²

C₂²×D₁₇

yes

characteristic

C₂³

—

D₄

—

D₄

C₂×#3622

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₄×D₁₇

yes

characteristic

C₂×C₄

#63254

yes

characteristic

C₂×C₄

C₂²×Dic₁₇

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

C₈

C₂×#3622

yes

complemented

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

D₄×Dic₁₇

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

#10587597

yes

—

105

C₄

C₂²×#3622

yes

characteristic

106

C₄

#10587634

yes

characteristic

107

C₂

—

108

C₂

D₄×#3622

yes

characteristic

109

C₂

#11762921

yes

characteristic

110

C₂

#11762931

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₁₇
commutator	C₄×C₁₇	C₂²×C₈
Frattini	C₄²	C₂²×D₁₇
Fitting	C₄×D₈×C₁₇	C₂
socle	C₂²×C₁₇	C₄×D₄
prime radical	C₂×D₈×C₁₇	C₄
cosocle	C₄²×C₁₇	C₂³
hypercenter	C₄×D₈	D₁₇
hypocenter	C₁₇	#1580
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₈

subgroup

G/H

center

C₂×C₄

D₄×D₁₇

commutator

C₄×C₁₇

C₂²×C₈

Frattini

C₄²

C₂²×D₁₇

Fitting

C₄×D₈×C₁₇

C₂

socle

C₂²×C₁₇

C₄×D₄

prime radical

C₂×D₈×C₁₇

C₄

cosocle

C₄²×C₁₇

C₂³

hypercenter

C₄×D₈

D₁₇

hypocenter

C₁₇

#1580

LCS

G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₈

N	H ≅ G/N	type
1. D₄×#3622	C₂	semidirect
2. #11762560	C₂	semidirect
3. D₄⋊#3622	C₂	semidirect
4. #11762530	C₂	semidirect
18. D₈×C₁₇	C₈	semidirect
30. D₄×C₁₇	C₂×C₈	semidirect
38. C₈×C₁₇	C₂×C₈	semidirect
71. C₁₇	#1580	semidirect
74. D₈	#3622	semidirect
86. D₄	C₂×#3622	semidirect
97. C₈	C₂×#3622	semidirect

H ≅ G/N

type

1. D₄×#3622

C₂

semidirect

2. #11762560

C₂

semidirect

3. D₄⋊#3622

C₂

semidirect

4. #11762530

C₂

semidirect

18. D₈×C₁₇

C₈

semidirect

30. D₄×C₁₇

C₂×C₈

semidirect

38. C₈×C₁₇

C₂×C₈

semidirect

71. C₁₇

#1580

semidirect

74. D₈

#3622

semidirect

86. D₄

C₂×#3622

semidirect

97. C₈

C₂×#3622

semidirect

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

subgroup	index
D₄×#3622	2
#11762560	2
D₄⋊#3622	2
#11762530	2
C₄×D₈×C₁₇	2
#1580	17

subgroup

index

C₄×D₈×C₁₇

Subgroups — #14527608

Conjugacy classes of subgroups

Complements

Maximal subgroups