Group #17961638 — subgroups

Subgroups — #17961638

Order 4352.

282 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 104 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	#14544724	C₂	1	2	yes	maximal
2	#14543844	C₂	1	4	yes	maximal
3	C₄×C₁₆×D₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#10587536	C₂²	1	4	yes	—
5	C₂×#10587535	—	2	2	no	—
6	#11763814	—	2	2	no	—
7	C₂×C₁₆×D₁₇	C₂²	1	2	yes	—
8	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
9	C₄×#63214	C₄	1	1	yes	characteristic
10	C₁₆×Dic₁₇	C₄	1	2	yes	—
11	C₄×C₁₆×C₁₇	C₄	1	1	yes	characteristic
12	C₂×#63243	—	2	4	no	—
13	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
14	C₂×C₈×D₁₇	C₂³	1	1	yes	characteristic
15	C₄²×D₁₇	D₄	1	1	yes	characteristic
16	#10587544	—	4	4	no	—
17	#10587536	—	2	8	no	—
18	#10587535	—	4	4	no	—
19	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
20	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
21	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
22	C₈×Dic₁₇	Q₈	1	1	yes	characteristic
23	C₂×#63214	C₂×C₄	1	2	yes	—
24	C₄×#3622	Q₈	1	1	yes	characteristic
25	C₁₆×D₁₇	—	2	4	no	—
26	C₂×C₄×D₁₇	Q₁₆	1	2	yes	—
27	C₂×C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
28	C₂×#3624	—	4	4	no	—
29	#63243	—	4	4	no	—
30	#63243	—	4	4	no	—
31	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
33	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
34	C₄×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
35	C₄×Dic₁₇	D₈	1	2	yes	—
36	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
37	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
38	C₈×D₁₇	SD₁₆	1	4	yes	—
39	C₈×D₁₇	#41	1	4	yes	—
40	C₁₆×C₁₇	—	2	2	no	—
41	#63214	—	2	2	no	—
42	#12891	—	17	1	no	maximal, Sylow
43	C₂²×D₁₇	C₂×Q₁₆	1	1	yes	characteristic
44	C₂×C₄×C₁₇	#107	1	2	yes	—
45	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂×Dic₁₇	#107	1	2	yes	—
47	C₂×Dic₁₇	C₂×D₈	1	1	yes	characteristic
48	C₄×D₁₇	—	2	4	no	—
49	C₄×D₁₇	#124	1	2	yes	—
50	C₄×D₁₇	C₂×SD₁₆	1	2	yes	—
51	#3624	—	8	4	no	—
52	C₈×C₁₇	#106	1	2	yes	—
53	C₈×C₁₇	#117	1	2	yes	—
54	#3622	#106	1	2	yes	—
55	#3622	#125	1	2	yes	—
56	C₂×C₄×C₁₆	—	17	1	no	—
57	#1848	—	17	2	no	—
58	#1381	—	17	4	no	—
59	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
60	C₂×D₁₇	—	2	2	no	—
61	C₂×D₁₇	#487	1	2	yes	—
62	C₄×C₁₇	—	2	2	no	—
63	C₄×C₁₇	#382	1	1	yes	characteristic
64	C₄×C₁₇	C₂×#106	1	1	yes	characteristic
65	Dic₁₇	#368	1	4	yes	—
66	Dic₁₇	#486	1	2	yes	—
67	C₂²×C₁₆	—	17	2	no	—
68	C₂×#108	—	17	4	no	—
69	C₂×#108	—	34	2	no	—
70	#423	—	34	2	no	—
71	C₂×C₄×C₈	—	17	1	no	—
72	C₄×C₁₆	—	17	2	no	—
73	C₄×C₁₆	—	17	1	no	—
74	C₄×C₁₆	#595	1	1	yes	characteristic
75	C₂×C₁₇	#1834	1	1	yes	characteristic
76	C₂×C₁₇	C₂×#368	1	2	yes	—
77	D₁₇	—	2	2	no	—
78	C₂×M₁₆	—	34	4	no	—
79	C₂²×C₈	—	17	1	no	—
80	C₂²×C₈	—	17	1	no	—
81	C₂×C₄²	—	17	1	no	—
82	C₂×C₁₆	—	17	2	no	—
83	C₂×C₁₆	—	34	4	no	—
84	C₂×C₁₆	C₂×#595	1	2	yes	—
85	#108	—	34	8	no	—
86	#108	—	68	4	no	—
87	C₄⋊C₈	—	68	4	no	—
88	C₄×C₈	—	17	1	no	—
89	C₄×C₈	—	17	1	no	—
90	C₄×C₈	—	17	1	no	—
91	C₄×C₈	C₂×#595	1	1	yes	characteristic
92	C₁₇	#12891	1	1	yes	characteristic, complemented, Sylow
93	C₂²×C₄	—	17	1	no	—
94	C₂²×C₄	—	17	2	no	—
95	M₁₆	—	68	4	no	—
96	M₁₆	—	68	4	no	—
97	C₂×C₈	—	17	1	no	—
98	C₂×C₈	—	17	1	no	—
99	C₂×C₈	—	17	4	no	—
100	C₂×C₈	—	17	4	no	—
101	C₂×C₈	—	68	4	no	—
102	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
103	C₂×C₈	C₂²×#595	1	1	yes	characteristic
104	C₄²	—	17	1	no	—
105	C₄²	—	17	2	no	—
106	C₄²	C₄⋊#595	1	1	yes	characteristic
107	C₁₆	—	2	2	no	—
108	C₁₆	—	34	2	no	—
109	C₂³	—	17	1	no	—
110	C₂×C₄	—	17	1	no	—
111	C₂×C₄	—	17	2	no	—
112	C₂×C₄	—	17	2	no	—
113	C₂×C₄	—	17	2	no	—
114	C₂×C₄	—	34	4	no	—
115	C₂×C₄	#10587534	1	2	yes	—
116	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
117	C₈	—	17	2	no	—
118	C₈	—	17	2	no	—
119	C₈	—	136	4	no	—
120	C₈	#10587533	1	2	yes	—
121	C₈	#10587677	1	2	yes	—
122	C₂²	—	17	2	no	—
123	C₂²	—	34	2	no	—
124	C₂²	C₂×#10587534	1	1	yes	characteristic
125	C₄	—	2	2	no	—
126	C₄	—	17	2	no	—
127	C₄	—	17	4	no	—
128	C₄	#11763753	1	1	yes	characteristic
129	C₄	C₂×#10587533	1	1	yes	characteristic
130	C₂	—	34	2	no	—
131	C₂	#14544705	1	1	yes	characteristic
132	C₂	D₁₇⋊#368	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14544724

C₂

yes

maximal

#14543844

C₂

yes

maximal

C₄×C₁₆×D₁₇

C₂

yes

maximal, characteristic

C₂×#10587536

C₂²

yes

—

C₂×#10587535

—

#11763814

—

C₂×C₁₆×D₁₇

C₂²

yes

—

C₄×C₈×D₁₇

C₂²

yes

characteristic

C₄×#63214

C₄

yes

characteristic

C₁₆×Dic₁₇

C₄

yes

—

C₄×C₁₆×C₁₇

C₄

yes

characteristic

C₂×#63243

—

C₂×C₈×D₁₇

D₄

yes

characteristic

C₂×C₈×D₁₇

C₂³

yes

characteristic

C₄²×D₁₇

D₄

yes

—

—

—

C₂×C₁₆×C₁₇

C₂×C₄

yes

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

Q₈

yes

characteristic

C₂×#63214

C₂×C₄

yes

—

C₄×#3622

Q₈

yes

characteristic

C₁₆×D₁₇

—

C₂×C₄×D₁₇

Q₁₆

yes

—

C₂×C₄×D₁₇

C₂×D₄

yes

—

—

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₇

C₂²×C₄

yes

characteristic

C₄²×C₁₇

C₄⋊C₄

yes

characteristic

C₄×Dic₁₇

C₄⋊C₄

yes

characteristic

C₄×Dic₁₇

D₈

yes

—

C₂×#3622

C₄⋊C₄

yes

characteristic

C₂×#3622

C₂×Q₈

yes

characteristic

C₈×D₁₇

SD₁₆

yes

—

C₈×D₁₇

#41

yes

—

C₁₆×C₁₇

—

#63214

—

#12891

—

maximal, Sylow

C₂²×D₁₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₁₇

#107

yes

—

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₁₇

#107

yes

—

C₂×Dic₁₇

C₂×D₈

yes

characteristic

C₄×D₁₇

—

C₄×D₁₇

#124

yes

—

C₄×D₁₇

C₂×SD₁₆

yes

—

#3624

—

C₈×C₁₇

#106

yes

—

C₈×C₁₇

#117

yes

—

#3622

#106

yes

—

#3622

#125

yes

—

C₂×C₄×C₁₆

—

#1848

—

#1381

—

C₂²×C₁₇

C₂×#107

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

#487

yes

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

yes

—

Dic₁₇

#486

yes

—

C₂²×C₁₆

—

C₂×#108

—

C₂×#108

—

#423

—

C₂×C₄×C₈

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

—

D₁₇

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×#595

yes

—

#108

—

#108

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×#595

yes

characteristic

C₁₇

#12891

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

C₄⋊#595

yes

characteristic

103

C₂×C₈

C₂²×#595

yes

characteristic

104

C₄²

—

105

C₄²

—

106

C₄²

C₄⋊#595

yes

characteristic

107

C₁₆

—

108

C₁₆

—

109

C₂³

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

#10587534

yes

—

116

C₂×C₄

C₂×C₄⋊#595

yes

characteristic

117

C₈

—

118

C₈

—

119

C₈

—

136

—

120

C₈

#10587533

yes

—

121

C₈

#10587677

yes

—

122

C₂²

—

123

C₂²

—

124

C₂²

C₂×#10587534

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

#11763753

yes

characteristic

129

C₄

C₂×#10587533

yes

characteristic

130

C₂

—

131

C₂

#14544705

yes

characteristic

132

C₂

D₁₇⋊#368

yes

—

	subgroup	G/H
center	C₂²	C₂×#10587534
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×#595
Fitting	C₄×C₁₆×C₁₇	C₄
socle	C₂²×C₁₇	C₂×#107
prime radical	C₂²×D₁₇	C₂×Q₁₆
cosocle	C₂×C₈×D₁₇	C₂³
hypercenter	C₄×C₁₆	#595
hypocenter	C₁₇	#12891
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

subgroup

G/H

center

C₂²

C₂×#10587534

commutator

C₂×C₈×C₁₇

C₂²×C₄

Frattini

C₂×C₈

C₂²×#595

Fitting

C₄×C₁₆×C₁₇

C₄

socle

C₂²×C₁₇

C₂×#107

prime radical

C₂²×D₁₇

C₂×Q₁₆

cosocle

C₂×C₈×D₁₇

C₂³

C₄×C₁₆

C₁₇

G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

N	H ≅ G/N	type
92. C₁₇	#12891	semidirect

H ≅ G/N

type

92. C₁₇

#12891

semidirect

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

subgroup	index
#14544724	2
#14543844	2
C₄×C₁₆×D₁₇	2
#12891	17

subgroup

index

#14544724

#14543844

C₄×C₁₆×D₁₇

#12891

Subgroups — #17961638

Conjugacy classes of subgroups

Complements

Maximal subgroups