Group #14544724 — subgroups

Subgroups — #14544724

Order 2176.

220 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 90 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×#10587536	C₂	1	2	yes	maximal
2	C₂×#10587535	C₂	1	2	yes	maximal
3	#11763814	C₂	1	2	yes	maximal
4	C₄×C₈×D₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#63243	C₂²	1	4	yes	—
6	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
7	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
8	C₄²×D₁₇	C₂²	1	1	yes	characteristic
9	#10587544	—	2	4	no	—
10	#10587536	—	2	4	no	—
11	#10587535	—	2	4	no	—
12	C₄×C₈×C₁₇	C₄	1	1	yes	characteristic
13	C₈×Dic₁₇	C₄	1	1	yes	characteristic
14	C₈×Dic₁₇	C₄	1	1	yes	characteristic
15	C₄×#3622	C₄	1	1	yes	characteristic
16	C₂×C₄×D₁₇	Q₈	1	2	yes	—
17	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
18	C₂×#3624	—	2	4	no	—
19	#63243	—	2	4	no	—
20	#63243	—	2	4	no	—
21	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
25	C₄×Dic₁₇	D₄	1	2	yes	—
26	C₂×#3622	C₂×C₄	1	1	yes	characteristic
27	C₂×#3622	C₂×C₄	1	1	yes	characteristic
28	C₈×D₁₇	D₄	1	2	yes	—
29	C₈×D₁₇	D₄	1	2	yes	—
30	C₈×D₁₇	Q₈	1	2	yes	—
31	C₈×D₁₇	Q₈	1	2	yes	—
32	C₂²×D₁₇	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
34	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
35	C₂×Dic₁₇	C₄⋊C₄	1	2	yes	—
36	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
37	C₄×D₁₇	—	2	4	no	—
38	C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
39	C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
40	C₄×D₁₇	C₂×Q₈	1	1	yes	characteristic
41	C₄×D₁₇	C₂×Q₈	1	1	yes	characteristic
42	#3624	—	4	4	no	—
43	C₈×C₁₇	C₄⋊C₄	1	2	yes	—
44	C₈×C₁₇	C₄⋊C₄	1	2	yes	—
45	#3622	C₄⋊C₄	1	2	yes	—
46	#3622	C₄⋊C₄	1	2	yes	—
47	#1848	—	17	1	no	maximal, Sylow
48	C₂²×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂×D₁₇	—	2	2	no	—
50	C₂×D₁₇	C₄⋊Q₈	1	1	yes	characteristic
51	C₂×D₁₇	C₄⋊Q₈	1	1	yes	characteristic
52	C₄×C₁₇	—	2	2	no	—
53	C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
55	Dic₁₇	#108	1	4	yes	complemented
56	Dic₁₇	#127	1	1	yes	characteristic
57	Dic₁₇	C₄⋊Q₈	1	1	yes	characteristic
58	C₂×#108	—	17	2	no	—
59	C₂×#108	—	17	2	no	—
60	#423	—	17	2	no	—
61	C₂×C₄×C₈	—	17	1	no	—
62	C₂×C₁₇	#384	1	1	yes	characteristic
63	C₂×C₁₇	C₂×#108	1	2	yes	—
64	D₁₇	—	2	1	no	—
65	D₁₇	—	2	1	no	—
66	C₂×M₁₆	—	17	4	no	—
67	C₂²×C₈	—	17	1	no	—
68	C₂²×C₈	—	17	1	no	—
69	C₂×C₄²	—	17	1	no	—
70	#108	—	34	4	no	—
71	#108	—	34	4	no	—
72	C₄⋊C₈	—	34	4	no	—
73	C₄×C₈	—	17	1	no	—
74	C₄×C₈	—	17	1	no	—
75	C₄×C₈	—	17	1	no	—
76	C₄×C₈	#595	1	1	yes	characteristic
77	C₁₇	#1848	1	1	yes	characteristic, complemented, Sylow
78	C₂²×C₄	—	17	1	no	—
79	C₂²×C₄	—	17	2	no	—
80	M₁₆	—	34	4	no	—
81	M₁₆	—	34	4	no	—
82	C₂×C₈	—	17	1	no	—
83	C₂×C₈	—	17	1	no	—
84	C₂×C₈	—	17	2	no	—
85	C₂×C₈	—	17	2	no	—
86	C₂×C₈	—	17	2	no	—
87	C₂×C₈	—	17	2	no	—
88	C₂×C₈	—	34	4	no	—
89	C₂×C₈	C₂×#595	1	1	yes	characteristic
90	C₂×C₈	C₂×#595	1	1	yes	characteristic
91	C₄²	—	17	2	no	—
92	C₄²	—	17	1	no	—
93	C₄²	C₂×#595	1	1	yes	characteristic
94	C₂³	—	17	1	no	—
95	C₂×C₄	—	17	1	no	—
96	C₂×C₄	—	17	2	no	—
97	C₂×C₄	—	17	1	no	—
98	C₂×C₄	—	17	1	no	—
99	C₂×C₄	—	17	1	no	—
100	C₂×C₄	—	17	1	no	—
101	C₂×C₄	—	34	4	no	—
102	C₂×C₄	C₄⋊#595	1	2	yes	—
103	C₂×C₄	C₂²×#595	1	1	yes	characteristic
104	C₈	—	17	2	no	—
105	C₈	—	17	2	no	—
106	C₈	—	68	4	no	—
107	C₈	C₄⋊#595	1	2	yes	—
108	C₈	C₄⋊#595	1	2	yes	—
109	C₂²	—	17	1	no	—
110	C₂²	—	17	1	no	—
111	C₂²	—	34	2	no	—
112	C₂²	C₂×C₄⋊#595	1	1	yes	characteristic
113	C₄	—	2	2	no	—
114	C₄	—	17	1	no	—
115	C₄	—	17	1	no	—
116	C₄	—	17	4	no	—
117	C₄	C₂×C₄⋊#595	1	1	yes	characteristic
118	C₄	C₂×C₄⋊#595	1	1	yes	characteristic
119	C₂	—	34	1	no	—
120	C₂	—	34	1	no	—
121	C₂	#11763755	1	1	yes	characteristic
122	C₂	D₁₇⋊#108	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10587536

C₂

yes

maximal

C₂×#10587535

C₂

yes

maximal

#11763814

C₂

yes

maximal

C₄×C₈×D₁₇

C₂

yes

maximal, characteristic

C₂×#63243

C₂²

yes

—

C₂×C₈×D₁₇

C₂²

yes

characteristic

C₂×C₈×D₁₇

C₂²

yes

characteristic

C₄²×D₁₇

C₂²

yes

—

—

—

C₄×C₈×C₁₇

C₄

yes

characteristic

C₈×Dic₁₇

C₄

yes

characteristic

C₈×Dic₁₇

C₄

yes

characteristic

C₄×#3622

C₄

yes

characteristic

C₂×C₄×D₁₇

Q₈

yes

—

C₂×C₄×D₁₇

C₂³

yes

—

—

—

C₂×C₈×C₁₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄²×C₁₇

C₂×C₄

yes

characteristic

C₄×Dic₁₇

C₂×C₄

yes

characteristic

C₄×Dic₁₇

D₄

yes

—

C₂×#3622

C₂×C₄

yes

characteristic

C₂×#3622

C₂×C₄

yes

characteristic

C₈×D₁₇

D₄

yes

—

C₈×D₁₇

D₄

yes

—

C₈×D₁₇

Q₈

yes

—

C₈×D₁₇

Q₈

yes

—

C₂²×D₁₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×Dic₁₇

C₄⋊C₄

yes

—

C₂×Dic₁₇

C₂×D₄

yes

characteristic

C₄×D₁₇

—

C₄×D₁₇

C₂×D₄

yes

characteristic

C₄×D₁₇

C₂×D₄

yes

characteristic

C₄×D₁₇

C₂×Q₈

yes

characteristic

C₄×D₁₇

C₂×Q₈

yes

characteristic

#3624

—

C₈×C₁₇

C₄⋊C₄

yes

—

C₈×C₁₇

C₄⋊C₄

yes

—

#3622

C₄⋊C₄

yes

—

#3622

C₄⋊C₄

yes

—

#1848

—

maximal, Sylow

C₂²×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

C₄⋊Q₈

yes

characteristic

C₂×D₁₇

C₄⋊Q₈

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₇

C₂×C₄⋊C₄

yes

Dic₁₇

yes

Dic₁₇

yes

Dic₁₇

C₄⋊Q₈

yes

—

—

—

C₂×C₄×C₈

—

C₂×C₁₇

#384

yes

characteristic

C₂×C₁₇

C₂×#108

yes

—

D₁₇

—

D₁₇

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

#108

—

#108

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

#595

yes

characteristic

C₁₇

#1848

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₄²

—

C₄²

—

C₄²

C₂×#595

yes

characteristic

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

C₄⋊#595

yes

—

103

C₂×C₄

C₂²×#595

yes

characteristic

104

C₈

—

105

C₈

—

106

C₈

—

107

C₈

C₄⋊#595

yes

—

108

C₈

C₄⋊#595

yes

—

109

C₂²

—

110

C₂²

—

111

C₂²

—

112

C₂²

C₂×C₄⋊#595

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

C₂×C₄⋊#595

yes

characteristic

118

C₄

C₂×C₄⋊#595

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

#11763755

yes

characteristic

122

C₂

D₁₇⋊#108

yes

—

	subgroup	G/H
center	C₂²	C₂×C₄⋊#595
commutator	C₂×C₄×C₁₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#595
Fitting	C₄×C₈×C₁₇	C₄
socle	C₂²×C₁₇	C₂×C₄⋊C₄
prime radical	C₂²×D₁₇	C₂×Q₈
cosocle	C₂×C₄×D₁₇	C₂³
hypercenter	C₄×C₈	#595
hypocenter	C₁₇	#1848
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₂²

C₂×C₄⋊#595

commutator

C₂×C₄×C₁₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×#595

Fitting

C₄×C₈×C₁₇

C₄

socle

C₂²×C₁₇

C₂×C₄⋊C₄

prime radical

C₂²×D₁₇

C₂×Q₈

cosocle

C₂×C₄×D₁₇

C₂³

C₄×C₈

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
55. Dic₁₇	#108	semidirect
77. C₁₇	#1848	semidirect

H ≅ G/N

type

55. Dic₁₇

#108

semidirect

77. C₁₇

#1848

semidirect

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

subgroup	index
C₂×#10587536	2
C₂×#10587535	2
#11763814	2
C₄×C₈×D₁₇	2
#1848	17

subgroup

index

C₄×C₈×D₁₇

Subgroups — #14544724

Conjugacy classes of subgroups

Complements

Maximal subgroups