Group #14529535 — subgroups

Subgroups — #14529535

Order 2176.

200 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 89 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	#11762847	C₂	1	1	yes	maximal, characteristic
2	#11762837	C₂	1	2	yes	maximal
3	C₂×#10587486	C₂	1	1	yes	maximal, characteristic
4	#11762570	C₂	1	2	yes	maximal
5	C₂×C₁₇×#108	C₂	1	1	yes	maximal, characteristic
6	C₂×M₁₆×C₁₇	C₂²	1	2	yes	—
7	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
8	#10587612	C₂²	1	2	yes	—
9	C₂×C₄×Dic₁₇	C₂²	1	1	yes	characteristic
10	C₂²×#3622	C₂²	1	1	yes	characteristic
11	C₁₇×#108	—	2	2	no	—
12	#10587489	C₄	1	2	yes	complemented
13	#10587488	C₄	1	2	yes	complemented
14	#10587486	—	2	2	no	—
15	#10587485	—	2	4	no	—
16	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
17	C₂²×Dic₁₇	—	2	1	no	—
18	M₁₆×C₁₇	—	2	2	no	—
19	M₁₆×C₁₇	—	2	2	no	—
20	C₂×C₈×C₁₇	—	2	2	no	—
21	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₁₇	D₄	1	2	yes	—
24	C₂×C₈×C₁₇	Q₈	1	2	yes	—
25	C₂²⋊Dic₁₇	—	2	2	no	—
26	C₄⋊Dic₁₇	C₂×C₄	1	4	yes	—
27	C₄×Dic₁₇	—	2	2	no	—
28	C₄×Dic₁₇	—	2	1	no	—
29	C₄×Dic₁₇	Q₈	1	2	yes	—
30	C₂×#3622	—	2	1	no	—
31	C₂×#3622	—	2	1	no	—
32	C₂×#3622	D₄	1	2	yes	—
33	C₂³×C₁₇	#41	1	1	yes	characteristic
34	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
40	C₂×Dic₁₇	—	2	4	no	—
41	C₂×Dic₁₇	—	2	2	no	—
42	C₂×Dic₁₇	—	4	2	no	—
43	C₈×C₁₇	—	4	2	no	—
44	C₈×C₁₇	C₄⋊C₄	1	2	yes	—
45	C₈×C₁₇	C₄⋊C₄	1	2	yes	—
46	#3622	—	2	2	no	—
47	#1946	—	17	1	no	maximal, Sylow
48	C₂²×C₁₇	—	2	2	no	—
49	C₂²×C₁₇	C₄×D₄	1	1	yes	characteristic
50	C₂²×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
51	C₂²×C₁₇	#125	1	1	yes	characteristic
52	C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
54	C₄×C₁₇	#121	1	1	yes	characteristic
55	C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
56	Dic₁₇	—	4	2	no	—
57	Dic₁₇	—	4	4	no	—
58	C₂×#108	D₁₇	1	1	yes	characteristic
59	#427	—	17	1	no	—
60	#424	—	17	2	no	—
61	C₂×#97	—	17	1	no	—
62	#339	—	17	2	no	—
63	C₂×C₁₇	—	2	1	no	—
64	C₂×C₁₇	—	2	1	no	—
65	C₂×C₁₇	#389	1	1	yes	characteristic
66	C₂×C₁₇	#444	1	2	yes	—
67	C₂×M₁₆	C₂×D₁₇	1	2	yes	—
68	C₂²×C₈	—	17	1	no	—
69	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
70	#117	—	17	2	no	—
71	C₂×C₄²	—	17	1	no	—
72	#108	—	2	2	no	—
73	#107	—	17	2	no	—
74	#106	—	17	2	no	—
75	C₄⋊C₈	—	34	4	no	—
76	#97	—	34	2	no	—
77	C₁₇	#1946	1	1	yes	characteristic, complemented, Sylow
78	C₂²×C₄	—	34	1	no	—
79	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
80	M₁₆	—	2	2	no	—
81	M₁₆	—	2	2	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	17	2	no	—
84	C₂×C₈	—	34	1	no	—
85	C₂×C₈	—	34	1	no	—
86	C₂×C₈	Dic₃₄	1	2	yes	—
87	C₂×C₈	C₄×D₁₇	1	1	yes	characteristic
88	C₂×C₈	C₄×D₁₇	1	1	yes	characteristic
89	C₂×C₈	C₂²⋊D₁₇	1	2	yes	—
90	C₄⋊C₄	—	17	4	no	—
91	C₂²⋊C₄	—	34	2	no	—
92	C₄²	—	17	2	no	—
93	C₄²	—	34	2	no	—
94	C₄²	—	34	1	no	—
95	C₂³	#63256	1	1	yes	characteristic
96	C₂×C₄	—	34	4	no	—
97	C₂×C₄	—	34	2	no	—
98	C₂×C₄	—	68	2	no	—
99	C₂×C₄	C₂×Dic₃₄	1	1	yes	characteristic
100	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
101	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
102	C₂×C₄	#63254	1	1	yes	characteristic
103	C₂×C₄	Q₈×D₁₇	1	1	yes	characteristic
104	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
105	C₈	—	4	2	no	—
106	C₈	—	34	2	no	—
107	C₈	#63225	1	2	yes	—
108	C₈	#63225	1	2	yes	—
109	C₂²	—	2	2	no	—
110	C₂²	#10587576	1	1	yes	characteristic
111	C₂²	#10587579	1	1	yes	characteristic
112	C₂²	#10587632	1	1	yes	characteristic
113	C₄	—	68	2	no	—
114	C₄	—	68	4	no	—
115	C₄	#10587573	1	1	yes	characteristic
116	C₄	#10587574	1	1	yes	characteristic
117	C₄	C₂×#63225	1	1	yes	characteristic
118	C₄	#10587624	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	2	1	no	—
121	C₂	#11762705	1	1	yes	characteristic
122	C₂	#11762930	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11762847

C₂

yes

maximal, characteristic

#11762837

C₂

yes

maximal

C₂×#10587486

C₂

yes

maximal, characteristic

#11762570

C₂

yes

maximal

C₂×C₁₇×#108

C₂

yes

maximal, characteristic

C₂×M₁₆×C₁₇

C₂²

yes

—

C₂²×C₈×C₁₇

C₂²

yes

characteristic

#10587612

C₂²

yes

—

C₂×C₄×Dic₁₇

C₂²

yes

characteristic

C₂²×#3622

C₂²

yes

characteristic

C₁₇×#108

—

#10587489

C₄

yes

complemented

#10587488

C₄

yes

complemented

#10587486

—

#10587485

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂²×Dic₁₇

—

M₁₆×C₁₇

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₇

D₄

yes

—

C₂×C₈×C₁₇

Q₈

yes

—

C₂²⋊Dic₁₇

—

C₄⋊Dic₁₇

C₂×C₄

yes

—

C₄×Dic₁₇

—

C₄×Dic₁₇

—

C₄×Dic₁₇

Q₈

yes

—

C₂×#3622

—

C₂×#3622

—

C₂×#3622

D₄

yes

—

C₂³×C₁₇

#41

yes

characteristic

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₇

#41

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₈×C₁₇

—

C₈×C₁₇

C₄⋊C₄

yes

—

C₈×C₁₇

C₄⋊C₄

yes

—

#3622

—

#1946

—

maximal, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

C₄×D₄

yes

characteristic

C₂²×C₁₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₇

#125

yes

characteristic

C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₇

C₄×Q₈

yes

characteristic

C₄×C₁₇

#121

yes

characteristic

C₄×C₁₇

C₄⋊Q₈

yes

characteristic

Dic₁₇

—

Dic₁₇

—

C₂×#108

D₁₇

yes

characteristic

#427

—

#424

—

C₂×#97

—

#339

—

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

#389

yes

characteristic

C₂×C₁₇

#444

yes

—

C₂×M₁₆

C₂×D₁₇

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₁₇

yes

characteristic

#117

—

C₂×C₄²

—

#108

—

#107

—

#106

—

C₄⋊C₈

—

#97

—

C₁₇

#1946

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₇

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₃₄

yes

—

C₂×C₈

C₄×D₁₇

yes

characteristic

C₂×C₈

C₄×D₁₇

yes

characteristic

C₂×C₈

C₂²⋊D₁₇

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₂³

#63256

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×Dic₃₄

yes

characteristic

100

C₂×C₄

C₂×C₄×D₁₇

yes

characteristic

101

C₂×C₄

D₄×D₁₇

yes

characteristic

102

C₂×C₄

#63254

yes

characteristic

103

C₂×C₄

Q₈×D₁₇

yes

characteristic

104

C₂×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

105

C₈

—

106

C₈

—

107

C₈

#63225

yes

—

108

C₈

#63225

yes

—

109

C₂²

—

110

C₂²

#10587576

yes

characteristic

111

C₂²

#10587579

yes

characteristic

112

C₂²

#10587632

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

#10587573

yes

characteristic

116

C₄

#10587574

yes

characteristic

117

C₄

C₂×#63225

yes

characteristic

118

C₄

#10587624

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

#11762705

yes

characteristic

122

C₂

#11762930

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₇
commutator	C₂×C₄×C₁₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₂×C₁₇×#108	C₂
socle	C₂²×C₁₇	C₄×D₄
prime radical	C₂³×C₁₇	#41
cosocle	C₂²×C₄×C₁₇	C₂³
hypercenter	C₂×#108	D₁₇
hypocenter	C₁₇	#1946
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂×C₄

D₄×D₁₇

commutator

C₂×C₄×C₁₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₂×C₁₇×#108

C₂

socle

C₂²×C₁₇

C₄×D₄

prime radical

C₂³×C₁₇

#41

cosocle

C₂²×C₄×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
12. #10587489	C₄	semidirect
13. #10587488	C₄	semidirect
77. C₁₇	#1946	semidirect

H ≅ G/N

type

12. #10587489

C₄

semidirect

13. #10587488

C₄

semidirect

77. C₁₇

#1946

semidirect

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

subgroup	index
#11762847	2
#11762837	2
C₂×#10587486	2
#11762570	2
C₂×C₁₇×#108	2
#1946	17

subgroup

index

C₂×C₁₇×#108

Subgroups — #14529535

Conjugacy classes of subgroups

Complements

Maximal subgroups