Group #14531010 — subgroups

Subgroups — #14531010

Order 2176.

140 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 51 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	#11763213	C₂	1	1	yes	maximal, characteristic
2	#11763187	C₂	1	1	yes	maximal, characteristic
3	#11762614	C₂	1	1	yes	maximal, characteristic
4	#11762613	C₂	1	1	yes	maximal, characteristic
5	#11762612	C₂	1	1	yes	maximal, characteristic
6	C₁₆×Dic₁₇	C₂	1	1	yes	maximal, characteristic
7	C₁₇×#358	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₁₆×C₁₇	C₂²	1	1	yes	characteristic
9	#10587630	—	2	1	no	—
10	C₂×#63231	—	2	1	no	—
11	C₂×Dic₆₈	C₂²	1	1	yes	characteristic
12	#10587574	—	2	1	no	—
13	C₂×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
14	C₁₇×#107	C₂²	1	1	yes	characteristic
15	#10587489	C₂²	1	1	yes	characteristic
16	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
17	C₂×#63214	C₂²	1	1	yes	characteristic
18	#10587478	—	2	1	no	—
19	C₂×Q₈×C₁₇	—	2	1	no	—
20	C₂×Dic₃₄	—	2	1	no	—
21	C₂×Dic₃₄	—	4	1	no	—
22	Q₁₆×C₁₇	—	2	1	no	—
23	Q₁₆×C₁₇	—	2	1	no	—
24	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₁₇	—	2	1	no	—
26	#63231	—	4	2	no	—
27	C₄⋊Dic₁₇	—	2	1	no	—
28	#63225	—	4	1	no	—
29	#63225	—	4	1	no	—
30	#63225	—	4	1	no	—
31	C₄×Dic₁₇	D₄	1	1	yes	characteristic
32	C₂×#3622	D₄	1	1	yes	characteristic
33	Dic₆₈	—	2	1	no	—
34	Dic₆₈	—	2	1	no	—
35	C₁₆×C₁₇	—	2	1	no	—
36	#63214	—	2	1	no	—
37	Q₈×C₁₇	—	4	1	no	—
38	Q₈×C₁₇	—	4	1	no	—
39	C₂×C₄×C₁₇	—	4	1	no	—
40	C₂×C₄×C₁₇	—	4	1	no	—
41	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
42	C₂×Dic₁₇	—	4	1	no	—
43	C₂×Dic₁₇	—	4	1	no	—
44	C₂×Dic₁₇	D₈	1	1	yes	characteristic
45	C₂×Dic₁₇	D₈	1	1	yes	characteristic
46	Dic₃₄	—	4	1	no	—
47	Dic₃₄	—	4	1	no	—
48	Dic₃₄	—	8	2	no	—
49	C₈×C₁₇	#41	1	1	yes	characteristic
50	C₈×C₁₇	#41	1	1	yes	characteristic
51	#3622	SD₁₆	1	2	yes	—
52	#2241	—	17	1	no	maximal, Sylow
53	C₂²×C₁₇	C₂×D₈	1	1	yes	characteristic
54	C₄×C₁₇	—	8	1	no	—
55	C₄×C₁₇	—	8	1	no	—
56	C₄×C₁₇	#124	1	1	yes	characteristic
57	C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
58	Dic₁₇	—	8	1	no	—
59	Dic₁₇	—	8	1	no	—
60	Dic₁₇	SD₃₂	1	2	yes	—
61	Dic₁₇	Q₃₂	1	2	yes	—
62	#500	—	17	1	no	—
63	#494	—	17	1	no	—
64	#358	—	17	1	no	—
65	#358	—	17	1	no	—
66	#358	—	17	1	no	—
67	#358	D₁₇	1	1	yes	characteristic
68	C₄×C₁₆	—	17	1	no	—
69	C₂×C₁₇	#486	1	1	yes	characteristic
70	C₂×C₁₇	C₂×SD₃₂	1	1	yes	characteristic
71	C₂×C₁₇	C₂×Q₃₂	1	1	yes	characteristic
72	C₂×Q₁₆	—	17	1	no	—
73	C₂×Q₁₆	—	34	1	no	—
74	C₂×Q₁₆	C₂×D₁₇	1	1	yes	characteristic
75	C₄⋊Q₈	—	34	1	no	—
76	C₄⋊Q₈	—	34	1	no	—
77	C₂×C₁₆	—	17	1	no	—
78	C₂×C₁₆	C₂×D₁₇	1	1	yes	characteristic
79	#107	—	17	1	no	—
80	#107	—	34	1	no	—
81	#107	C₂×D₁₇	1	1	yes	characteristic
82	C₄×C₈	—	17	1	no	—
83	C₁₇	#2241	1	1	yes	characteristic, complemented, Sylow
84	C₂×Q₈	—	2	1	no	—
85	C₂×Q₈	—	34	1	no	—
86	C₂×Q₈	—	68	1	no	—
87	Q₁₆	—	2	1	no	—
88	Q₁₆	—	2	1	no	—
89	Q₁₆	—	34	1	no	—
90	Q₁₆	—	34	1	no	—
91	Q₁₆	—	68	2	no	—
92	C₂×C₈	—	17	1	no	—
93	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
94	C₄⋊C₄	—	2	1	no	—
95	C₄⋊C₄	—	34	1	no	—
96	C₄⋊C₄	—	68	1	no	—
97	C₄⋊C₄	—	68	1	no	—
98	C₄⋊C₄	—	68	1	no	—
99	C₄²	—	17	1	no	—
100	C₁₆	—	2	1	no	—
101	C₁₆	—	34	1	no	—
102	Q₈	—	4	1	no	—
103	Q₈	—	4	1	no	—
104	Q₈	—	68	1	no	—
105	Q₈	—	68	1	no	—
106	Q₈	—	136	2	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	4	1	no	—
109	C₂×C₄	—	17	1	no	—
110	C₂×C₄	—	17	1	no	—
111	C₂×C₄	—	68	1	no	—
112	C₂×C₄	—	68	1	no	—
113	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
114	C₈	—	17	2	no	—
115	C₈	#63252	1	1	yes	characteristic
116	C₈	#63254	1	1	yes	characteristic
117	C₂²	D₈×D₁₇	1	1	yes	characteristic
118	C₄	—	8	1	no	—
119	C₄	—	8	1	no	—
120	C₄	—	17	2	no	—
121	C₄	—	17	2	no	—
122	C₄	—	136	1	no	—
123	C₄	—	136	1	no	—
124	C₄	#10587571	1	1	yes	characteristic
125	C₄	SD₁₆×D₁₇	1	1	yes	characteristic
126	C₂	#11763154	1	1	yes	characteristic
127	C₂	SD₃₂×D₁₇	1	1	yes	characteristic
128	C₂	Q₃₂×D₁₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763213

C₂

yes

maximal, characteristic

#11763187

C₂

yes

maximal, characteristic

#11762614

C₂

yes

maximal, characteristic

#11762613

C₂

yes

maximal, characteristic

#11762612

C₂

yes

maximal, characteristic

C₁₆×Dic₁₇

C₂

yes

maximal, characteristic

C₁₇×#358

C₂

yes

maximal, characteristic

C₂×Q₁₆×C₁₇

C₂²

yes

characteristic

#10587630

—

C₂×#63231

—

C₂×Dic₆₈

C₂²

yes

characteristic

#10587574

—

C₂×C₁₆×C₁₇

C₂²

yes

characteristic

C₁₇×#107

C₂²

yes

characteristic

#10587489

C₂²

yes

characteristic

C₈×Dic₁₇

C₂²

yes

characteristic

C₂×#63214

C₂²

yes

characteristic

#10587478

—

C₂×Q₈×C₁₇

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

Q₁₆×C₁₇

—

Q₁₆×C₁₇

—

C₂×C₈×C₁₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₇

—

#63231

—

C₄⋊Dic₁₇

—

#63225

—

#63225

—

#63225

—

C₄×Dic₁₇

D₄

yes

characteristic

C₂×#3622

D₄

yes

characteristic

Dic₆₈

—

Dic₆₈

—

C₁₆×C₁₇

—

#63214

—

Q₈×C₁₇

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

D₈

yes

characteristic

C₂×Dic₁₇

D₈

yes

characteristic

Dic₃₄

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

yes

C₈×C₁₇

yes

SD₁₆

yes

—

#2241

—

maximal, Sylow

C₂²×C₁₇

C₂×D₈

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#124

yes

characteristic

C₄×C₁₇

C₂×SD₁₆

yes

characteristic

Dic₁₇

—

Dic₁₇

—

Dic₁₇

SD₃₂

yes

—

Dic₁₇

Q₃₂

yes

—

—

—

—

—

—

D₁₇

yes

characteristic

C₄×C₁₆

—

C₂×C₁₇

#486

yes

characteristic

C₂×C₁₇

C₂×SD₃₂

yes

characteristic

C₂×C₁₇

C₂×Q₃₂

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×D₁₇

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₁₇

yes

—

—

C₂×D₁₇

yes

characteristic

C₄×C₈

—

C₁₇

#2241

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

C₂²×D₁₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

100

C₁₆

—

101

C₁₆

—

102

Q₈

—

103

Q₈

—

104

Q₈

—

105

Q₈

—

106

Q₈

—

136

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

D₄×D₁₇

yes

characteristic

114

C₈

—

115

C₈

#63252

yes

characteristic

116

C₈

#63254

yes

characteristic

117

C₂²

D₈×D₁₇

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

136

—

123

C₄

—

136

—

124

C₄

#10587571

yes

characteristic

125

C₄

SD₁₆×D₁₇

yes

characteristic

126

C₂

#11763154

yes

characteristic

127

C₂

SD₃₂×D₁₇

yes

characteristic

128

C₂

Q₃₂×D₁₇

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₁₇
commutator, cosocle	C₂×C₈×C₁₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₇
Fitting	C₁₇×#358	C₂
socle, prime radical	C₂²×C₁₇	C₂×D₈
hypercenter	#358	D₁₇
hypocenter	C₁₇	#2241
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

subgroup

G/H

center

C₂²

D₈×D₁₇

commutator, cosocle

C₂×C₈×C₁₇

C₂³

Frattini

C₂×C₈

C₂²×D₁₇

Fitting

C₁₇×#358

C₂

socle, prime radical

C₂²×C₁₇

C₂×D₈

D₁₇

C₁₇

G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

N	H ≅ G/N	type
83. C₁₇	#2241	semidirect

H ≅ G/N

type

83. C₁₇

#2241

semidirect

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

subgroup	index
#11763213	2
#11763187	2
#11762614	2
#11762613	2
#11762612	2
C₁₆×Dic₁₇	2
C₁₇×#358	2
#2241	17

subgroup

index

C₁₆×Dic₁₇

C₁₇×#358

#2241

Subgroups — #14531010

Conjugacy classes of subgroups

Complements

Maximal subgroups