Group #14528501 — subgroups

Subgroups — #14528501

Order 2176.

424 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 205 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	C₂×#10587612	C₂	1	2	yes	maximal
2	#11762572	C₂	1	4	yes	maximal, complemented
3	C₂³×C₈×C₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₄×C₁₇	C₂²	1	1	yes	characteristic
5	C₂²×C₈×C₁₇	—	2	4	no	—
6	C₂²×C₈×C₁₇	C₄	1	4	yes	complemented
7	C₂²×C₈×C₁₇	C₂²	1	2	yes	—
8	C₂×C₂²⋊Dic₁₇	—	2	2	no	—
9	#10587612	—	2	4	no	—
10	#10587612	C₄	1	8	yes	complemented
11	C₂×C₄⋊Dic₁₇	C₂²	1	4	yes	—
12	C₂×C₄×Dic₁₇	—	2	2	no	—
13	C₂⁴×C₁₇	D₄	1	1	yes	characteristic
14	C₂²×C₄×C₁₇	C₂×C₄	1	4	yes	—
15	C₂²×C₄×C₁₇	C₂×C₄	1	2	yes	—
16	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₁₇	D₄	1	4	yes	—
18	C₂²×C₄×C₁₇	Q₈	1	2	yes	—
19	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
20	C₂²×Dic₁₇	—	2	4	no	—
21	C₂×C₈×C₁₇	—	2	8	no	—
22	C₂×C₈×C₁₇	—	2	4	no	—
23	C₂×C₈×C₁₇	C₂×C₄	1	4	yes	—
24	C₂²⋊Dic₁₇	—	2	8	no	—
25	C₄⋊Dic₁₇	—	2	4	no	—
26	C₄⋊Dic₁₇	C₂×C₄	1	8	yes	—
27	C₄×Dic₁₇	—	2	8	no	—
28	C₂³×C₁₇	—	2	4	no	—
29	C₂³×C₁₇	C₂²⋊C₄	1	4	yes	—
30	C₂³×C₁₇	C₄⋊C₄	1	2	yes	—
31	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₁₇	C₄²	1	4	yes	—
33	C₂×C₄×C₁₇	C₂²⋊C₄	1	4	yes	—
34	C₂×C₄×C₁₇	C₂²⋊C₄	1	4	yes	—
35	C₂×C₄×C₁₇	C₄⋊C₄	1	8	yes	—
36	C₂×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
37	C₂×C₄×C₁₇	C₂²×C₄	1	2	yes	—
38	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
41	C₂×Dic₁₇	—	2	8	no	—
42	C₂×Dic₁₇	—	4	8	no	—
43	C₈×C₁₇	—	2	4	no	—
44	#1735	—	17	1	no	maximal, Sylow
45	C₂²×C₁₇	—	2	8	no	—
46	C₂²×C₁₇	—	2	4	no	—
47	C₂²×C₁₇	#95	1	4	yes	—
48	C₂²×C₁₇	C₂×C₂²⋊C₄	1	2	yes	—
49	C₂²×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₁₇	#135	1	4	yes	—
51	C₄×C₁₇	#95	1	4	yes	—
52	C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
53	C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₄×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
55	Dic₁₇	—	4	8	no	—
56	C₂³×C₈	D₁₇	1	1	yes	characteristic
57	C₂×#117	—	17	2	no	—
58	#340	—	17	4	no	—
59	C₂×C₁₇	—	2	4	no	—
60	C₂×C₁₇	C₂×#95	1	1	yes	characteristic
61	C₂×C₁₇	#416	1	4	yes	—
62	C₂×C₁₇	#427	1	2	yes	—
63	C₂³×C₄	C₂×D₁₇	1	1	yes	characteristic
64	C₂²×C₈	—	2	4	no	—
65	C₂²×C₈	Dic₁₇	1	4	yes	complemented
66	C₂²×C₈	C₂×D₁₇	1	2	yes	—
67	#117	—	17	8	no	—
68	#117	—	34	4	no	—
69	C₂×C₄⋊C₄	—	17	4	no	—
70	C₂×C₂²⋊C₄	—	34	2	no	—
71	C₂×C₄²	—	34	2	no	—
72	C₁₇	#1735	1	1	yes	characteristic, complemented, Sylow
73	C₂⁴	D₆₈	1	1	yes	characteristic
74	C₂²×C₄	—	34	4	no	—
75	C₂²×C₄	Dic₃₄	1	2	yes	—
76	C₂²×C₄	C₄×D₁₇	1	4	yes	—
77	C₂²×C₄	D₆₈	1	1	yes	characteristic
78	C₂²×C₄	C₂×Dic₁₇	1	2	yes	—
79	C₂²×C₄	C₂²⋊D₁₇	1	4	yes	—
80	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
81	C₂×C₈	—	2	8	no	—
82	C₂×C₈	—	2	4	no	—
83	C₂×C₈	C₂×Dic₁₇	1	4	yes	—
84	C₄⋊C₄	—	17	8	no	—
85	C₄⋊C₄	—	34	4	no	—
86	C₂²⋊C₄	—	34	8	no	—
87	C₄²	—	34	8	no	—
88	C₂³	—	2	4	no	—
89	C₂³	C₄⋊Dic₁₇	1	2	yes	—
90	C₂³	#63227	1	4	yes	—
91	C₂³	C₂×D₆₈	1	1	yes	characteristic
92	C₂×C₄	—	34	8	no	—
93	C₂×C₄	—	68	8	no	—
94	C₂×C₄	C₄×Dic₁₇	1	4	yes	—
95	C₂×C₄	#63225	1	8	yes	—
96	C₂×C₄	C₄⋊Dic₁₇	1	2	yes	—
97	C₂×C₄	#63227	1	4	yes	—
98	C₂×C₄	C₂²⋊Dic₁₇	1	4	yes	—
99	C₂×C₄	C₂×Dic₃₄	1	1	yes	characteristic
100	C₂×C₄	C₂×C₄×D₁₇	1	2	yes	—
101	C₂×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
102	C₂×C₄	C₂×C₂²⋊D₁₇	1	2	yes	—
103	C₈	—	2	4	no	—
104	C₂²	—	2	8	no	—
105	C₂²	—	2	4	no	—
106	C₂²	#10587502	1	4	yes	—
107	C₂²	#10587590	1	4	yes	—
108	C₂²	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
109	C₂²	C₂×#63227	1	2	yes	—
110	C₄	—	68	8	no	—
111	C₄	#10587502	1	4	yes	—
112	C₄	C₂×C₄×Dic₁₇	1	1	yes	characteristic
113	C₄	C₂×#63225	1	2	yes	—
114	C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
115	C₂	—	2	4	no	—
116	C₂	C₂×#10587502	1	1	yes	characteristic
117	C₂	#11762806	1	4	yes	—
118	C₂	#11762847	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10587612

C₂

yes

maximal

#11762572

C₂

yes

maximal, complemented

C₂³×C₈×C₁₇

C₂

yes

maximal, characteristic

C₂³×C₄×C₁₇

C₂²

yes

characteristic

C₂²×C₈×C₁₇

—

C₂²×C₈×C₁₇

C₄

yes

complemented

C₂²×C₈×C₁₇

C₂²

yes

—

C₂×C₂²⋊Dic₁₇

—

#10587612

—

#10587612

C₄

yes

complemented

C₂×C₄⋊Dic₁₇

C₂²

yes

—

C₂×C₄×Dic₁₇

—

C₂⁴×C₁₇

D₄

yes

characteristic

C₂²×C₄×C₁₇

C₂×C₄

yes

—

C₂²×C₄×C₁₇

C₂×C₄

yes

—

C₂²×C₄×C₁₇

D₄

yes

characteristic

C₂²×C₄×C₁₇

D₄

yes

—

C₂²×C₄×C₁₇

Q₈

yes

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂²×Dic₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₂×C₄

yes

—

C₂²⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

C₂×C₄

yes

—

C₄×Dic₁₇

—

C₂³×C₁₇

—

C₂³×C₁₇

C₂²⋊C₄

yes

—

C₂³×C₁₇

C₄⋊C₄

yes

—

C₂³×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₄²

yes

—

C₂×C₄×C₁₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₇

C₂²×C₄

yes

—

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

C₂×Q₈

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₈×C₁₇

—

#1735

—

maximal, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#95

yes

—

C₂²×C₁₇

C₂×C₂²⋊C₄

yes

—

C₂²×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₇

#135

yes

—

C₄×C₁₇

#95

yes

—

C₄×C₁₇

C₂×C₄²

yes

characteristic

C₄×C₁₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₇

C₂×C₄⋊C₄

yes

—

Dic₁₇

—

C₂³×C₈

D₁₇

yes

characteristic

C₂×#117

—

#340

—

C₂×C₁₇

—

C₂×C₁₇

C₂×#95

yes

characteristic

C₂×C₁₇

#416

yes

—

C₂×C₁₇

#427

yes

—

C₂³×C₄

C₂×D₁₇

yes

characteristic

C₂²×C₈

—

C₂²×C₈

Dic₁₇

yes

complemented

C₂²×C₈

C₂×D₁₇

yes

—

#117

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₁₇

#1735

yes

characteristic, complemented, Sylow

C₂⁴

D₆₈

yes

characteristic

C₂²×C₄

—

C₂²×C₄

Dic₃₄

yes

—

C₂²×C₄

C₄×D₁₇

yes

—

C₂²×C₄

D₆₈

yes

characteristic

C₂²×C₄

C₂×Dic₁₇

yes

—

C₂²×C₄

C₂²⋊D₁₇

yes

—

C₂²×C₄

C₂²×D₁₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₁₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂³

—

C₂³

C₄⋊Dic₁₇

yes

—

C₂³

#63227

yes

—

C₂³

C₂×D₆₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₁₇

yes

—

C₂×C₄

#63225

yes

—

C₂×C₄

C₄⋊Dic₁₇

yes

—

C₂×C₄

#63227

yes

—

C₂×C₄

C₂²⋊Dic₁₇

yes

—

C₂×C₄

C₂×Dic₃₄

yes

characteristic

100

C₂×C₄

C₂×C₄×D₁₇

yes

—

101

C₂×C₄

C₂²×Dic₁₇

yes

characteristic

102

C₂×C₄

C₂×C₂²⋊D₁₇

yes

—

103

C₈

—

104

C₂²

—

105

C₂²

—

106

C₂²

#10587502

yes

—

107

C₂²

#10587590

yes

—

108

C₂²

C₂×C₄⋊Dic₁₇

yes

characteristic

109

C₂²

C₂×#63227

yes

—

110

C₄

—

111

C₄

#10587502

yes

—

112

C₄

C₂×C₄×Dic₁₇

yes

characteristic

113

C₄

C₂×#63225

yes

—

114

C₄

C₂×C₂²⋊Dic₁₇

yes

characteristic

115

C₂

—

116

C₂

C₂×#10587502

yes

characteristic

117

C₂

#11762806

yes

—

118

C₂

#11762847

yes

—

	subgroup	G/H
center	C₂²×C₄	D₆₈
commutator	C₄×C₁₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₂³×C₈×C₁₇	C₂
socle	C₂³×C₁₇	C₂×D₄
prime radical	C₂⁴×C₁₇	D₄
cosocle	C₂²×C₄×C₁₇	C₂³
hypercenter	C₂³×C₈	D₁₇
hypocenter	C₁₇	#1735
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈

subgroup

G/H

center

C₂²×C₄

D₆₈

commutator

C₄×C₁₇

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₂³×C₈×C₁₇

C₂

socle

C₂³×C₁₇

C₂×D₄

prime radical

C₂⁴×C₁₇

D₄

cosocle

C₂²×C₄×C₁₇

C₂³

hypercenter

C₂³×C₈

D₁₇

hypocenter

C₁₇

#1735

LCS

G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈

N	H ≅ G/N	type
2. #11762572	C₂	semidirect
6. C₂²×C₈×C₁₇	C₄	semidirect
10. #10587612	C₄	semidirect
65. C₂²×C₈	Dic₁₇	semidirect
72. C₁₇	#1735	semidirect

H ≅ G/N

type

2. #11762572

C₂

semidirect

6. C₂²×C₈×C₁₇

C₄

semidirect

10. #10587612

C₄

semidirect

65. C₂²×C₈

Dic₁₇

semidirect

72. C₁₇

#1735

semidirect

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

subgroup	index
C₂×#10587612	2
#11762572	2
C₂³×C₈×C₁₇	2
#1735	17

subgroup

index

C₂×#10587612

#11762572

C₂³×C₈×C₁₇

#1735

Subgroups — #14528501

Conjugacy classes of subgroups

Complements

Maximal subgroups