Group #14530360 — subgroups

Subgroups — #14530360

Order 2176.

158 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 59 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	#11762858	C₂	1	1	yes	maximal, characteristic
2	#11762832	C₂	1	1	yes	maximal, characteristic
3	#11762579	C₂	1	2	yes	maximal
4	#11762568	C₂	1	2	yes	maximal
5	C₁₇×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic
6	C₂×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
7	C₁₇×#117	C₂²	1	2	yes	—
8	C₂×C₄×Dic₁₇	C₂²	1	1	yes	characteristic
9	C₂×#63223	C₂²	1	2	yes	—
10	C₂×#63223	C₂²	1	1	yes	characteristic
11	#10587490	—	2	2	no	—
12	#10587485	—	2	2	no	—
13	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
14	C₂²×Dic₁₇	—	2	1	no	—
15	M₁₆×C₁₇	—	2	2	no	—
16	C₂×C₈×C₁₇	D₄	1	2	yes	—
17	C₄⋊C₄×C₁₇	—	2	2	no	—
18	C₂²⋊C₄×C₁₇	—	2	2	no	—
19	C₄²×C₁₇	—	2	2	no	—
20	C₄×Dic₁₇	—	2	1	no	—
21	C₄×Dic₁₇	Q₈	1	1	yes	characteristic
22	C₄×Dic₁₇	Q₈	1	1	yes	characteristic
23	#63223	—	2	2	no	—
24	#63223	—	2	2	no	—
25	#63223	—	2	2	no	—
26	C₂×#3622	—	2	1	no	—
27	C₂×#3622	—	2	2	no	—
28	C₂³×C₁₇	#41	1	1	yes	characteristic
29	C₂×C₄×C₁₇	—	4	2	no	—
30	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₄×C₁₇	#41	1	2	yes	—
33	C₂×C₄×C₁₇	#41	1	2	yes	—
34	C₂×Dic₁₇	—	2	1	no	—
35	C₂×Dic₁₇	—	2	1	no	—
36	C₂×Dic₁₇	—	4	2	no	—
37	C₈×C₁₇	—	2	2	no	—
38	#3622	—	4	1	no	—
39	#3622	—	4	2	no	—
40	#2102	—	17	1	no	maximal, Sylow
41	C₂²×C₁₇	—	4	1	no	—
42	C₂²×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
43	C₂²×C₁₇	#126	1	2	yes	—
44	C₄×C₁₇	—	8	2	no	—
45	C₄×C₁₇	#123	1	1	yes	characteristic
46	C₄×C₁₇	#124	1	1	yes	characteristic
47	C₄×C₁₇	#125	1	2	yes	—
48	Dic₁₇	—	2	2	no	—
49	Dic₁₇	—	4	1	no	—
50	#430	—	17	1	no	—
51	#423	—	17	1	no	—
52	#344	—	17	2	no	—
53	#339	—	17	2	no	—
54	H(3,ℤ/4)	D₁₇	1	1	yes	characteristic
55	C₂×C₁₇	—	2	1	no	—
56	C₂×C₁₇	—	2	2	no	—
57	C₂×C₁₇	#400	1	1	yes	characteristic
58	C₂×M₁₆	—	17	2	no	—
59	C₂×M₁₆	—	17	1	no	—
60	C₂×M₁₆	C₂×D₁₇	1	1	yes	characteristic
61	#117	C₂×D₁₇	1	2	yes	—
62	C₂×C₄²	—	17	1	no	—
63	#108	—	34	2	no	—
64	C₄⋊C₈	—	34	2	no	—
65	C₁₇	#2102	1	1	yes	characteristic, complemented, Sylow
66	C₂²×C₄	—	34	1	no	—
67	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
68	M₁₆	—	2	2	no	—
69	M₁₆	—	34	2	no	—
70	M₁₆	—	34	2	no	—
71	M₁₆	—	34	2	no	—
72	C₂×C₈	—	34	1	no	—
73	C₂×C₈	—	34	2	no	—
74	C₂×C₈	D₆₈	1	2	yes	—
75	C₄⋊C₄	—	2	2	no	—
76	C₂²⋊C₄	—	2	2	no	—
77	C₄²	—	2	2	no	—
78	C₄²	—	17	1	no	—
79	C₄²	—	17	1	no	—
80	C₄²	—	34	1	no	—
81	C₂³	#63254	1	1	yes	characteristic
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	34	1	no	—
84	C₂×C₄	—	34	1	no	—
85	C₂×C₄	—	68	2	no	—
86	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
87	C₂×C₄	#63252	1	2	yes	—
88	C₂×C₄	#63254	1	2	yes	—
89	C₂×C₄	Q₈×D₁₇	1	1	yes	characteristic
90	C₈	—	2	2	no	—
91	C₈	—	68	2	no	—
92	C₈	—	68	1	no	—
93	C₂²	—	4	1	no	—
94	C₂²	#10587565	1	2	yes	—
95	C₂²	#10587583	1	1	yes	characteristic
96	C₄	—	8	2	no	—
97	C₄	—	34	2	no	—
98	C₄	—	68	1	no	—
99	C₄	#10587561	1	1	yes	characteristic
100	C₄	#10587572	1	1	yes	characteristic
101	C₄	#10587575	1	2	yes	—
102	C₂	—	2	2	no	—
103	C₂	—	2	1	no	—
104	C₂	#11762749	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11762858

C₂

yes

maximal, characteristic

#11762832

C₂

yes

maximal, characteristic

#11762579

C₂

yes

maximal

#11762568

C₂

yes

maximal

C₁₇×H(3,ℤ/4)

C₂

yes

maximal, characteristic

C₂×M₁₆×C₁₇

C₂²

yes

characteristic

C₁₇×#117

C₂²

yes

—

C₂×C₄×Dic₁₇

C₂²

yes

characteristic

C₂×#63223

C₂²

yes

—

C₂×#63223

C₂²

yes

characteristic

#10587490

—

#10587485

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂²×Dic₁₇

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

D₄

yes

—

C₄⋊C₄×C₁₇

—

C₂²⋊C₄×C₁₇

—

C₄²×C₁₇

—

C₄×Dic₁₇

—

C₄×Dic₁₇

Q₈

yes

characteristic

C₄×Dic₁₇

Q₈

yes

—

—

—

—

—

C₂³×C₁₇

#41

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₇

#41

yes

—

C₂×C₄×C₁₇

#41

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₈×C₁₇

—

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₇

#126

yes

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

—

Dic₁₇

—

Dic₁₇

—

—

—

—

—

D₁₇

yes

characteristic

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

#400

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₁₇

yes

characteristic

#117

C₂×D₁₇

yes

—

C₂×C₄²

—

#108

—

C₄⋊C₈

—

C₁₇

#2102

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₆₈

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₂³

#63254

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₆₈

yes

characteristic

C₂×C₄

#63252

yes

—

C₂×C₄

#63254

yes

—

C₂×C₄

Q₈×D₁₇

yes

characteristic

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

#10587565

yes

—

C₂²

#10587583

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

#10587561

yes

characteristic

100

C₄

#10587572

yes

characteristic

101

C₄

#10587575

yes

—

102

C₂

—

103

C₂

—

104

C₂

#11762749

yes

characteristic

	subgroup	G/H
center	C₄	#10587561
commutator, cosocle	C₂²×C₄×C₁₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₁₇×H(3,ℤ/4)	C₂
socle	C₂×C₁₇	#400
prime radical	C₂³×C₁₇	#41
hypercenter	H(3,ℤ/4)	D₁₇
hypocenter	C₁₇	#2102
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#10587561

commutator, cosocle

C₂²×C₄×C₁₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₁₇×H(3,ℤ/4)

C₂

C₂×C₁₇

C₂³×C₁₇

D₁₇

C₁₇

G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
65. C₁₇	#2102	semidirect

H ≅ G/N

type

65. C₁₇

#2102

semidirect

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

subgroup	index
#11762858	2
#11762832	2
#11762579	2
#11762568	2
C₁₇×H(3,ℤ/4)	2
#2102	17

subgroup

index

C₁₇×H(3,ℤ/4)

Subgroups — #14530360

Conjugacy classes of subgroups

Complements

Maximal subgroups