Group #14543857 — subgroups

Subgroups — #14543857

Order 2176.

140 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 50 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	#11763865	C₂	1	1	yes	maximal, characteristic
2	#11763819	C₂	1	1	yes	maximal, characteristic
3	#106×D₁₇	C₂	1	1	yes	maximal, characteristic
4	#10587677	—	2	1	no	—
5	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
6	C₄⋊C₄×D₁₇	—	2	1	no	—
7	#10587538	—	2	1	no	—
8	#10587534	C₄	1	2	yes	complemented
9	#10587533	—	2	1	no	—
10	#10587530	C₄	1	2	yes	complemented
11	#10587529	—	2	1	no	—
12	C₁₇×#106	C₄	1	1	yes	characteristic, complemented
13	#10587488	C₄	1	1	yes	characteristic, complemented
14	#10587479	—	2	1	no	—
15	C₂×C₄×D₁₇	—	4	1	no	—
16	C₂×C₄×D₁₇	D₄	1	1	yes	characteristic
17	C₂²⋊#595	—	4	1	no	—
18	C₄⋊#595	—	2	2	no	—
19	C₄×#595	—	4	1	no	—
20	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
21	C₄⋊C₄×C₁₇	—	2	1	no	—
22	C₄⋊Dic₁₇	—	2	1	no	—
23	#63225	—	4	1	no	—
24	C₂×#3622	D₄	1	1	yes	characteristic
25	C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
26	C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
27	C₈×D₁₇	D₄	1	1	yes	characteristic
28	C₈×D₁₇	Q₈	1	1	yes	characteristic
29	#63216	—	2	2	no	—
30	C₂²×D₁₇	D₈	1	1	yes	characteristic
31	C₂×#595	—	4	2	no	—
32	C₂×C₄×C₁₇	—	4	1	no	—
33	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×Dic₁₇	—	4	1	no	—
35	C₂×Dic₁₇	SD₁₆	1	1	yes	characteristic
36	C₄×D₁₇	—	8	2	no	—
37	C₄×D₁₇	C₂²⋊C₄	1	1	yes	characteristic
38	C₄×D₁₇	C₄⋊C₄	1	1	yes	characteristic
39	C₄×D₁₇	SD₁₆	1	1	yes	characteristic
40	C₄×D₁₇	Q₁₆	1	1	yes	characteristic
41	C₈×C₁₇	C₄²	1	1	yes	characteristic
42	C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
43	#3622	C₂²⋊C₄	1	1	yes	characteristic
44	#3622	C₄⋊C₄	1	1	yes	characteristic
45	#1386	—	17	1	no	maximal, Sylow
46	C₂²×C₁₇	#102	1	1	yes	characteristic
47	C₂×D₁₇	—	2	2	no	—
48	C₂×D₁₇	#102	1	1	yes	characteristic
49	C₂×D₁₇	#107	1	1	yes	characteristic
50	#595	—	8	2	no	—
51	C₄×C₁₇	—	8	1	no	—
52	C₄×C₁₇	#95	1	1	yes	characteristic
53	C₄×C₁₇	#103	1	1	yes	characteristic
54	Dic₁₇	—	8	1	no	—
55	Dic₁₇	#103	1	1	yes	characteristic
56	Dic₁₇	#106	1	1	yes	characteristic
57	C₂×M₃₂	—	17	1	no	—
58	#427	—	17	1	no	—
59	C₂×#106	—	17	1	no	—
60	C₂×C₁₇	—	2	1	no	—
61	C₂×C₁₇	#340	1	1	yes	characteristic
62	D₁₇	—	2	1	no	—
63	D₁₇	#360	1	2	yes	—
64	C₂²×C₈	—	17	1	no	—
65	#117	—	34	1	no	—
66	C₂×C₄⋊C₄	—	34	1	no	—
67	M₃₂	—	17	2	no	—
68	M₃₂	—	34	1	no	—
69	C₂×C₁₆	—	34	1	no	—
70	#107	—	17	2	no	—
71	#106	—	17	1	no	—
72	#106	—	34	1	no	—
73	#106	—	34	1	no	—
74	#106	#595	1	1	yes	characteristic, complemented
75	C₁₇	#1386	1	1	yes	characteristic, complemented, Sylow
76	C₂²×C₄	—	17	1	no	—
77	C₂²×C₄	—	68	1	no	—
78	C₂×C₈	—	17	1	no	—
79	C₂×C₈	—	17	1	no	—
80	C₂×C₈	—	17	1	no	—
81	C₂×C₈	—	17	1	no	—
82	C₂×C₈	—	17	1	no	—
83	C₂×C₈	C₂×#595	1	1	yes	characteristic
84	C₄⋊C₄	—	2	1	no	—
85	C₄⋊C₄	—	34	2	no	—
86	C₄⋊C₄	—	34	1	no	—
87	C₄⋊C₄	—	68	1	no	—
88	C₂²⋊C₄	—	68	1	no	—
89	C₄²	—	68	1	no	—
90	C₁₆	—	34	2	no	—
91	C₂³	—	17	1	no	—
92	C₂×C₄	—	4	1	no	—
93	C₂×C₄	—	17	1	no	—
94	C₂×C₄	—	17	1	no	—
95	C₂×C₄	—	17	1	no	—
96	C₂×C₄	—	17	1	no	—
97	C₂×C₄	—	17	1	no	—
98	C₂×C₄	—	68	2	no	—
99	C₂×C₄	—	68	1	no	—
100	C₂×C₄	—	136	2	no	—
101	C₂×C₄	C₂²⋊#595	1	1	yes	characteristic
102	C₈	—	17	1	no	—
103	C₈	—	17	1	no	—
104	C₈	C₄×#595	1	1	yes	characteristic
105	C₈	C₄⋊#595	1	1	yes	characteristic
106	C₂²	—	17	1	no	—
107	C₂²	—	17	1	no	—
108	C₂²	—	34	2	no	—
109	C₂²	#10587547	1	1	yes	characteristic
110	C₄	—	8	1	no	—
111	C₄	—	17	1	no	—
112	C₄	—	17	1	no	—
113	C₄	—	136	1	no	—
114	C₄	—	136	2	no	—
115	C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
116	C₄	#10587549	1	1	yes	characteristic
117	C₂	—	2	1	no	—
118	C₂	—	17	2	no	—
119	C₂	—	34	1	no	—
120	C₂	#11763682	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763865

C₂

yes

maximal, characteristic

#11763819

C₂

yes

maximal, characteristic

#106×D₁₇

C₂

yes

maximal, characteristic

#10587677

—

C₂×C₈×D₁₇

C₂²

yes

characteristic

C₄⋊C₄×D₁₇

—

#10587538

—

#10587534

C₄

yes

complemented

#10587533

—

#10587530

C₄

yes

complemented

#10587529

—

C₁₇×#106

C₄

yes

characteristic, complemented

#10587488

C₄

yes

characteristic, complemented

#10587479

—

C₂×C₄×D₁₇

—

C₂×C₄×D₁₇

D₄

yes

characteristic

C₂²⋊#595

—

C₄⋊#595

—

C₄×#595

—

C₂×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄⋊C₄×C₁₇

—

C₄⋊Dic₁₇

—

#63225

—

C₂×#3622

D₄

yes

characteristic

C₈×D₁₇

C₂×C₄

yes

characteristic

C₈×D₁₇

C₂×C₄

yes

characteristic

C₈×D₁₇

D₄

yes

characteristic

C₈×D₁₇

Q₈

yes

characteristic

#63216

—

C₂²×D₁₇

D₈

yes

characteristic

C₂×#595

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

SD₁₆

yes

characteristic

C₄×D₁₇

—

C₄×D₁₇

C₂²⋊C₄

yes

characteristic

C₄×D₁₇

C₄⋊C₄

yes

characteristic

C₄×D₁₇

SD₁₆

yes

characteristic

C₄×D₁₇

Q₁₆

yes

characteristic

C₈×C₁₇

C₄²

yes

characteristic

C₈×C₁₇

C₄⋊C₄

yes

characteristic

#3622

C₂²⋊C₄

yes

characteristic

#3622

C₄⋊C₄

yes

—

C₂²×C₁₇

yes

C₂×D₁₇

—

C₂×D₁₇

yes

C₂×D₁₇

yes

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

yes

Dic₁₇

yes

C₂×M₃₂

—

#427

—

C₂×#106

—

C₂×C₁₇

—

C₂×C₁₇

#340

yes

characteristic

D₁₇

—

D₁₇

#360

yes

—

C₂²×C₈

—

#117

—

C₂×C₄⋊C₄

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

—

—

—

—

yes

characteristic, complemented

C₁₇

#1386

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×#595

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

136

—

101

C₂×C₄

C₂²⋊#595

yes

characteristic

102

C₈

—

103

C₈

—

104

C₈

C₄×#595

yes

characteristic

105

C₈

C₄⋊#595

yes

characteristic

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

#10587547

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

136

—

114

C₄

—

136

—

115

C₄

(C₂×#595)⋊C₄

yes

characteristic

116

C₄

#10587549

yes

characteristic

117

C₂

—

118

C₂

—

119

C₂

—

120

C₂

#11763682

yes

characteristic

	subgroup	G/H
center	C₂	#11763682
commutator	C₈×C₁₇	C₄²
Frattini	C₂×C₈	C₂×#595
Fitting	C₁₇×#106	C₄
socle	C₂×C₁₇	#340
prime radical	C₂²×D₁₇	D₈
cosocle	C₂×C₈×D₁₇	C₂²
hypercenter	#106	#595
hypocenter	C₁₇	#1386
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #106

subgroup

G/H

center

C₂

#11763682

commutator

C₈×C₁₇

C₄²

Frattini

C₂×C₈

C₂×#595

Fitting

C₁₇×#106

C₄

socle

C₂×C₁₇

#340

prime radical

C₂²×D₁₇

D₈

cosocle

C₂×C₈×D₁₇

C₂²

C₁₇

G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #106

N	H ≅ G/N	type
8. #10587534	C₄	semidirect
10. #10587530	C₄	semidirect
12. C₁₇×#106	C₄	semidirect
13. #10587488	C₄	semidirect
74. #106	#595	semidirect
75. C₁₇	#1386	semidirect

H ≅ G/N

type

8. #10587534

C₄

semidirect

10. #10587530

C₄

semidirect

12. C₁₇×#106

C₄

semidirect

13. #10587488

C₄

semidirect

74. #106

#595

semidirect

75. C₁₇

#1386

semidirect

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

subgroup	index
#11763865	2
#11763819	2
#106×D₁₇	2
#1386	17

subgroup

index

#11763865

#11763819

#106×D₁₇

#1386

Subgroups — #14543857

Conjugacy classes of subgroups

Complements

Maximal subgroups