Group #17961637 — subgroups

Subgroups — #17961637

Order 4352.

282 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 108 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	#14544725	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	—	2	2	no	—
5	C₂×#10587536	C₂²	1	4	yes	—
6	#11763813	—	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	C₄⋊#3624	—	4	2	no	—
17	C₄⋊#3624	—	4	2	no	—
18	#10587536	—	2	8	no	—
19	#10587536	—	4	4	no	—
20	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
21	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
22	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
23	C₈×Dic₁₇	Q₈	1	1	yes	characteristic
24	C₂×#63214	C₂×C₄	1	2	yes	—
25	C₄×#3622	Q₈	1	1	yes	characteristic
26	C₁₆×D₁₇	—	2	4	no	—
27	C₂×C₄×D₁₇	Q₁₆	1	1	yes	characteristic
28	C₂×C₄×D₁₇	Q₁₆	1	1	yes	characteristic
29	C₂×C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
30	C₂×#3624	—	4	4	no	—
31	#63243	—	4	4	no	—
32	#63243	—	4	4	no	—
33	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
35	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
36	C₄×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
37	C₄×Dic₁₇	D₈	1	1	yes	characteristic
38	C₄×Dic₁₇	D₈	1	1	yes	characteristic
39	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
40	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
41	C₈×D₁₇	SD₁₆	1	4	yes	—
42	C₈×D₁₇	#41	1	4	yes	—
43	C₁₆×C₁₇	—	2	2	no	—
44	#63214	—	2	2	no	—
45	#12891	—	17	1	no	maximal, Sylow
46	C₂²×D₁₇	C₂×Q₁₆	1	1	yes	characteristic
47	C₂×C₄×C₁₇	#107	1	1	yes	characteristic
48	C₂×C₄×C₁₇	#107	1	1	yes	characteristic
49	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂×Dic₁₇	#107	1	1	yes	characteristic
51	C₂×Dic₁₇	#107	1	1	yes	characteristic
52	C₂×Dic₁₇	C₂×D₈	1	1	yes	characteristic
53	C₄×D₁₇	—	2	2	no	—
54	C₄×D₁₇	—	2	2	no	—
55	C₄×D₁₇	#124	1	2	yes	—
56	C₄×D₁₇	C₂×SD₁₆	1	2	yes	—
57	#3624	—	8	4	no	—
58	C₈×C₁₇	#106	1	2	yes	—
59	C₈×C₁₇	#117	1	2	yes	—
60	#3622	#106	1	2	yes	—
61	#3622	#125	1	2	yes	—
62	C₂×C₄×C₁₆	—	17	1	no	—
63	#1848	—	17	2	no	—
64	#1381	—	17	4	no	—
65	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
66	C₂×D₁₇	—	2	1	no	—
67	C₂×D₁₇	—	2	1	no	—
68	C₂×D₁₇	#487	1	2	yes	—
69	C₄×C₁₇	#368	1	2	yes	—
70	C₄×C₁₇	#368	1	2	yes	—
71	C₄×C₁₇	#382	1	1	yes	characteristic
72	C₄×C₁₇	C₂×#106	1	1	yes	characteristic
73	Dic₁₇	—	2	1	no	—
74	Dic₁₇	—	2	1	no	—
75	Dic₁₇	#486	1	2	yes	—
76	C₂²×C₁₆	—	17	2	no	—
77	C₂×#108	—	17	4	no	—
78	C₂×#108	—	34	2	no	—
79	#423	—	34	2	no	—
80	C₂×C₄×C₈	—	17	1	no	—
81	C₄×C₁₆	—	17	2	no	—
82	C₄×C₁₆	—	17	1	no	—
83	C₄×C₁₆	#595	1	1	yes	characteristic
84	C₂×C₁₇	#1834	1	1	yes	characteristic
85	C₂×C₁₇	C₂×#368	1	1	yes	characteristic
86	C₂×C₁₇	C₂×#368	1	1	yes	characteristic
87	D₁₇	—	2	2	no	—
88	C₂×M₁₆	—	34	4	no	—
89	C₂²×C₈	—	17	1	no	—
90	C₂²×C₈	—	17	1	no	—
91	C₂×C₄²	—	17	1	no	—
92	C₂×C₁₆	—	17	2	no	—
93	C₂×C₁₆	—	34	4	no	—
94	C₂×C₁₆	C₂×#595	1	2	yes	—
95	#108	—	34	8	no	—
96	#108	—	68	4	no	—
97	C₄⋊C₈	—	68	2	no	—
98	C₄⋊C₈	—	68	2	no	—
99	C₄×C₈	—	17	1	no	—
100	C₄×C₈	—	17	1	no	—
101	C₄×C₈	—	17	1	no	—
102	C₄×C₈	C₂×#595	1	1	yes	characteristic
103	C₁₇	#12891	1	1	yes	characteristic, complemented, Sylow
104	C₂²×C₄	—	17	1	no	—
105	C₂²×C₄	—	17	1	no	—
106	C₂²×C₄	—	17	1	no	—
107	M₁₆	—	68	4	no	—
108	M₁₆	—	68	4	no	—
109	C₂×C₈	—	17	1	no	—
110	C₂×C₈	—	17	1	no	—
111	C₂×C₈	—	17	4	no	—
112	C₂×C₈	—	17	4	no	—
113	C₂×C₈	—	68	4	no	—
114	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
115	C₂×C₈	C₂²×#595	1	1	yes	characteristic
116	C₄²	—	17	1	no	—
117	C₄²	—	17	1	no	—
118	C₄²	—	17	1	no	—
119	C₄²	C₄⋊#595	1	1	yes	characteristic
120	C₁₆	—	2	2	no	—
121	C₁₆	—	34	2	no	—
122	C₂³	—	17	1	no	—
123	C₂×C₄	—	17	1	no	—
124	C₂×C₄	—	17	2	no	—
125	C₂×C₄	—	17	2	no	—
126	C₂×C₄	—	17	1	no	—
127	C₂×C₄	—	17	1	no	—
128	C₂×C₄	—	34	2	no	—
129	C₂×C₄	—	34	2	no	—
130	C₂×C₄	#10587534	1	1	yes	characteristic
131	C₂×C₄	#10587534	1	1	yes	characteristic
132	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
133	C₈	—	17	2	no	—
134	C₈	—	17	2	no	—
135	C₈	—	136	4	no	—
136	C₈	#10587533	1	2	yes	—
137	C₈	#10587677	1	2	yes	—
138	C₂²	—	17	2	no	—
139	C₂²	—	34	1	no	—
140	C₂²	—	34	1	no	—
141	C₂²	C₂×#10587534	1	1	yes	characteristic
142	C₄	—	17	2	no	—
143	C₄	—	34	1	no	—
144	C₄	—	34	1	no	—
145	C₄	#11763737	1	2	yes	—
146	C₄	#11763738	1	2	yes	—
147	C₄	#11763753	1	1	yes	characteristic
148	C₄	C₂×#10587533	1	1	yes	characteristic
149	C₂	—	34	2	no	—
150	C₂	#14544705	1	1	yes	characteristic
151	C₂	C₂×#11763737	1	1	yes	characteristic
152	C₂	C₂×#11763738	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14544725

C₂

yes

maximal

#14543844

C₂

yes

maximal

C₄×C₁₆×D₁₇

C₂

yes

maximal, characteristic

C₂×#10587536

—

C₂×#10587536

C₂²

yes

—

#11763813

—

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

characteristic

C₂×C₄×D₁₇

Q₁₆

yes

characteristic

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

characteristic

C₄×Dic₁₇

D₈

yes

characteristic

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

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

C₂×Dic₁₇

yes

C₂×Dic₁₇

yes

C₂×Dic₁₇

C₂×D₈

yes

characteristic

C₄×D₁₇

—

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

—

C₂×D₁₇

#487

yes

—

C₄×C₁₇

#368

yes

—

C₄×C₁₇

#368

yes

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

—

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

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

—

100

C₄×C₈

—

101

C₄×C₈

—

102

C₄×C₈

C₂×#595

yes

characteristic

103

C₁₇

#12891

yes

characteristic, complemented, Sylow

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

M₁₆

—

108

M₁₆

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

C₄⋊#595

yes

characteristic

115

C₂×C₈

C₂²×#595

yes

characteristic

116

C₄²

—

117

C₄²

—

118

C₄²

—

119

C₄²

C₄⋊#595

yes

characteristic

120

C₁₆

—

121

C₁₆

—

122

C₂³

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

#10587534

yes

characteristic

131

C₂×C₄

#10587534

yes

characteristic

132

C₂×C₄

C₂×C₄⋊#595

yes

characteristic

133

C₈

—

134

C₈

—

135

C₈

—

136

—

136

C₈

#10587533

yes

—

137

C₈

#10587677

yes

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

C₂×#10587534

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

#11763737

yes

—

146

C₄

#11763738

yes

—

147

C₄

#11763753

yes

characteristic

148

C₄

C₂×#10587533

yes

characteristic

149

C₂

—

150

C₂

#14544705

yes

characteristic

151

C₂

C₂×#11763737

yes

characteristic

152

C₂

C₂×#11763738

yes

characteristic

	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
103. C₁₇	#12891	semidirect

H ≅ G/N

type

103. C₁₇

#12891

semidirect

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

subgroup	index
#14544725	2
#14543844	2
C₄×C₁₆×D₁₇	2
#12891	17

subgroup

index

#14544725

#14543844

C₄×C₁₆×D₁₇

#12891

Subgroups — #17961637

Conjugacy classes of subgroups

Complements

Maximal subgroups