Group #17948998 — subgroups

Subgroups — #17948998

Order 4352.

188 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 62 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	#14545147	C₂	1	1	yes	maximal, characteristic
2	#14544332	C₂	1	1	yes	maximal, characteristic
3	#14527538	C₂	1	1	yes	maximal, characteristic
4	#11763828	—	2	1	no	—
5	#11763730	—	2	1	no	—
6	#11763704	C₄	1	2	yes	—
7	#11763697	—	2	1	no	—
8	#11763673	C₄	1	2	yes	—
9	#11763670	—	2	1	no	—
10	#11762831	—	2	1	no	—
11	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
12	#11762560	C₄	1	1	yes	characteristic
13	#11762559	—	2	1	no	—
14	C₁₇×#334	C₄	1	1	yes	characteristic
15	C₂×#63218	—	4	1	no	—
16	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×D₁₇	Q₈	1	1	yes	characteristic
18	C₄²×D₁₇	D₄	1	1	yes	characteristic
19	#10587543	—	4	1	no	—
20	C₄⋊#3624	—	2	2	no	—
21	C₄×#3624	—	4	1	no	—
22	C₂×#63216	—	2	2	no	—
23	C₁₇×C₄⋊C₈	—	2	1	no	—
24	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
25	#10587485	—	4	1	no	—
26	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
27	C₈×Dic₁₇	D₄	1	1	yes	characteristic
28	C₄⋊#3622	—	2	1	no	—
29	C₄×#3622	D₄	1	1	yes	characteristic
30	C₂×C₄×D₁₇	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×D₁₇	D₈	1	1	yes	characteristic
32	C₂×C₄×D₁₇	Q₁₆	1	1	yes	characteristic
33	C₂×#3624	—	4	2	no	—
34	C₂×C₈×C₁₇	—	4	1	no	—
35	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
36	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
37	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
38	C₄×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
39	C₄×Dic₁₇	SD₁₆	1	1	yes	characteristic
40	C₄×Dic₁₇	SD₁₆	1	1	yes	characteristic
41	C₂×#3622	—	4	1	no	—
42	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
44	#63218	—	8	2	no	—
45	C₈×D₁₇	—	2	2	no	—
46	C₈×D₁₇	—	4	1	no	—
47	#63216	—	4	2	no	—
48	#7295	—	17	1	no	maximal, Sylow
49	C₂²×D₁₇	#107	1	1	yes	characteristic
50	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
51	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
52	C₂×C₄×C₁₇	#103	1	1	yes	characteristic
53	C₂×Dic₁₇	#102	1	1	yes	characteristic
54	C₂×Dic₁₇	#103	1	1	yes	characteristic
55	C₂×Dic₁₇	#106	1	1	yes	characteristic
56	C₄×D₁₇	—	2	1	no	—
57	C₄×D₁₇	—	2	1	no	—
58	C₄×D₁₇	—	2	2	no	—
59	C₄×D₁₇	—	4	1	no	—
60	#3624	—	8	2	no	—
61	C₈×C₁₇	—	2	1	no	—
62	C₈×C₁₇	—	2	1	no	—
63	C₈×C₁₇	—	8	1	no	—
64	#3622	—	2	1	no	—
65	#3622	—	8	1	no	—
66	#3622	#108	1	2	yes	—
67	#2163	—	17	1	no	—
68	#1566	—	17	1	no	—
69	#1564	—	17	1	no	—
70	C₂²×C₁₇	#340	1	1	yes	characteristic
71	C₂×D₁₇	—	2	1	no	—
72	C₂×D₁₇	—	2	1	no	—
73	C₂×D₁₇	—	2	1	no	—
74	C₄×C₁₇	—	2	1	no	—
75	C₄×C₁₇	—	2	1	no	—
76	C₄×C₁₇	#341	1	1	yes	characteristic
77	C₄×C₁₇	#344	1	1	yes	characteristic
78	Dic₁₇	—	2	1	no	—
79	Dic₁₇	—	2	1	no	—
80	Dic₁₇	#360	1	2	yes	—
81	#431	—	34	1	no	—
82	#423	—	34	1	no	—
83	C₂×C₄×C₈	—	17	1	no	—
84	C₄⋊C₁₆	—	34	1	no	—
85	C₂²⋊C₁₆	—	17	2	no	—
86	#346	—	34	1	no	—
87	#335	—	17	2	no	—
88	#334	—	17	1	no	—
89	#334	—	34	1	no	—
90	#334	—	34	1	no	—
91	#334	#595	1	1	yes	characteristic
92	C₂×C₁₇	#1301	1	1	yes	characteristic
93	C₂×C₁₇	#1385	1	1	yes	characteristic
94	C₂×C₁₇	#1386	1	1	yes	characteristic
95	D₁₇	—	4	1	no	—
96	C₂×M₁₆	—	68	1	no	—
97	C₂²×C₈	—	17	1	no	—
98	C₂²×C₈	—	17	1	no	—
99	C₂×C₄²	—	17	1	no	—
100	C₂×C₁₆	—	34	2	no	—
101	C₄⋊C₈	—	2	1	no	—
102	C₄⋊C₈	—	34	2	no	—
103	C₄⋊C₈	—	34	1	no	—
104	C₄⋊C₈	—	68	1	no	—
105	C₂²⋊C₈	—	68	1	no	—
106	C₄×C₈	—	17	1	no	—
107	C₄×C₈	—	17	1	no	—
108	C₄×C₈	—	17	1	no	—
109	C₄×C₈	—	68	1	no	—
110	C₄×C₈	C₂×#595	1	1	yes	characteristic
111	C₁₇	#7295	1	1	yes	characteristic, complemented, Sylow
112	C₂²×C₄	—	17	1	no	—
113	C₂²×C₄	—	17	1	no	—
114	C₂²×C₄	—	17	1	no	—
115	M₁₆	—	136	2	no	—
116	C₂×C₈	—	4	1	no	—
117	C₂×C₈	—	17	1	no	—
118	C₂×C₈	—	17	1	no	—
119	C₂×C₈	—	34	2	no	—
120	C₂×C₈	—	68	1	no	—
121	C₂×C₈	—	68	2	no	—
122	C₂×C₈	—	68	1	no	—
123	C₂×C₈	C₄×#595	1	1	yes	characteristic
124	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
125	C₄²	—	17	1	no	—
126	C₄²	—	17	1	no	—
127	C₄²	—	17	1	no	—
128	C₄²	C₂²⋊#595	1	1	yes	characteristic
129	C₁₆	—	68	2	no	—
130	C₂³	—	17	1	no	—
131	C₂×C₄	—	17	1	no	—
132	C₂×C₄	—	17	1	no	—
133	C₂×C₄	—	17	1	no	—
134	C₂×C₄	—	34	2	no	—
135	C₂×C₄	—	34	1	no	—
136	C₂×C₄	—	34	1	no	—
137	C₂×C₄	—	68	1	no	—
138	C₂×C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
139	C₂×C₄	#10587547	1	1	yes	characteristic
140	C₂×C₄	#10587549	1	1	yes	characteristic
141	C₈	—	2	1	no	—
142	C₈	—	2	1	no	—
143	C₈	—	8	1	no	—
144	C₈	—	17	2	no	—
145	C₈	—	34	1	no	—
146	C₈	—	136	1	no	—
147	C₈	—	136	2	no	—
148	C₂²	—	34	1	no	—
149	C₂²	—	34	1	no	—
150	C₂²	—	34	1	no	—
151	C₂²	#11763682	1	1	yes	characteristic
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	17	2	no	—
155	C₄	—	34	1	no	—
156	C₄	—	34	1	no	—
157	C₄	#11763684	1	1	yes	characteristic
158	C₄	#11763691	1	1	yes	characteristic
159	C₂	—	68	1	no	—
160	C₂	#14543674	1	1	yes	characteristic
161	C₂	#14543853	1	1	yes	characteristic
162	C₂	#14543857	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14545147

C₂

yes

maximal, characteristic

#14544332

C₂

yes

maximal, characteristic

#14527538

C₂

yes

maximal, characteristic

#11763828

—

#11763730

—

#11763704

C₄

yes

—

#11763697

—

#11763673

C₄

yes

—

#11763670

—

#11762831

—

C₄×C₈×D₁₇

C₂²

yes

characteristic

#11762560

C₄

yes

characteristic

#11762559

—

C₁₇×#334

C₄

yes

characteristic

C₂×#63218

—

C₂×C₈×D₁₇

C₂×C₄

yes

characteristic

C₂×C₈×D₁₇

Q₈

yes

characteristic

C₄²×D₁₇

D₄

yes

—

—

—

—

C₁₇×C₄⋊C₈

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

#10587485

—

C₈×Dic₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

D₄

yes

characteristic

C₄⋊#3622

—

C₄×#3622

D₄

yes

characteristic

C₂×C₄×D₁₇

C₄⋊C₄

yes

characteristic

C₂×C₄×D₁₇

D₈

yes

characteristic

C₂×C₄×D₁₇

Q₁₆

yes

characteristic

C₂×#3624

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₄²

yes

characteristic

C₂×C₈×C₁₇

C₄⋊C₄

yes

characteristic

C₄²×C₁₇

C₂²⋊C₄

yes

characteristic

C₄×Dic₁₇

C₂²⋊C₄

yes

characteristic

C₄×Dic₁₇

SD₁₆

yes

characteristic

C₄×Dic₁₇

SD₁₆

yes

characteristic

C₂×#3622

—

C₂×#3622

C₂²⋊C₄

yes

characteristic

C₂×#3622

C₄⋊C₄

yes

characteristic

#63218

—

C₈×D₁₇

—

C₈×D₁₇

—

—

—

C₂²×D₁₇

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×Dic₁₇

yes

C₂×Dic₁₇

yes

C₂×Dic₁₇

yes

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

—

#3624

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

—

—

yes

—

#2163

—

#1566

—

#1564

—

C₂²×C₁₇

#340

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

—

Dic₁₇

#360

yes

—

#431

—

#423

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

—

—

—

—

—

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

D₁₇

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

100

C₂×C₁₆

—

101

C₄⋊C₈

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

C₂²⋊C₈

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₄×C₈

C₂×#595

yes

characteristic

111

C₁₇

#7295

yes

characteristic, complemented, Sylow

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

M₁₆

—

136

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

C₄×#595

yes

characteristic

124

C₂×C₈

C₄⋊#595

yes

characteristic

125

C₄²

—

126

C₄²

—

127

C₄²

—

128

C₄²

C₂²⋊#595

yes

characteristic

129

C₁₆

—

130

C₂³

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

(C₂×#595)⋊C₄

yes

characteristic

139

C₂×C₄

#10587547

yes

characteristic

140

C₂×C₄

#10587549

yes

characteristic

141

C₈

—

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

—

136

—

147

C₈

—

136

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

#11763682

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

#11763684

yes

characteristic

158

C₄

#11763691

yes

characteristic

159

C₂

—

160

C₂

#14543674

yes

characteristic

161

C₂

#14543853

yes

characteristic

162

C₂

#14543857

yes

characteristic

	subgroup	G/H
center	C₂²	#11763682
commutator	C₂×C₈×C₁₇	C₄²
Frattini	C₄×C₈	C₂×#595
Fitting	C₁₇×#334	C₄
socle	C₂²×C₁₇	#340
prime radical	C₂²×D₁₇	#107
cosocle	C₄×C₈×D₁₇	C₂²
hypercenter	#334	#595
hypocenter	C₁₇	#7295
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #334

subgroup

G/H

center

C₂²

#11763682

commutator

C₂×C₈×C₁₇

C₄²

Frattini

C₄×C₈

C₂×#595

Fitting

C₁₇×#334

C₄

C₂²×C₁₇

C₂²×D₁₇

C₄×C₈×D₁₇

C₂²

C₁₇

G ▷ C₂×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #334

N	H ≅ G/N	type
111. C₁₇	#7295	semidirect

H ≅ G/N

type

111. C₁₇

#7295

semidirect

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

subgroup	index
#14545147	2
#14544332	2
#14527538	2
#7295	17

subgroup

index

Subgroups — #17948998

Conjugacy classes of subgroups

Complements

Maximal subgroups