Group #17948768 — subgroups

Subgroups — #17948768

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	#14545144	C₂	1	1	yes	maximal, characteristic
2	#14544708	C₂	1	1	yes	maximal, characteristic
3	#14527546	C₂	1	1	yes	maximal, characteristic
4	#11763813	—	2	1	no	—
5	C₄⋊#63216	—	2	1	no	—
6	#11763704	—	2	1	no	—
7	#11763695	C₄	1	2	yes	—
8	#11763685	—	2	1	no	—
9	#11763685	C₄	1	2	yes	—
10	#11762865	—	2	1	no	—
11	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
12	#11762562	C₄	1	1	yes	characteristic
13	#11762559	—	2	1	no	—
14	C₁₇×#335	C₄	1	1	yes	characteristic
15	C₂×#63243	—	2	2	no	—
16	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
18	C₄²×D₁₇	D₄	1	1	yes	characteristic
19	C₄⋊#3624	—	4	1	no	—
20	C₄⋊#3624	—	4	1	no	—
21	C₂×#63216	—	2	2	no	—
22	C₁₇×C₄⋊C₈	—	2	1	no	—
23	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
24	#10587491	—	4	1	no	—
25	#10587486	—	4	1	no	—
26	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
27	C₈×Dic₁₇	Q₈	1	1	yes	characteristic
28	C₄⋊#3622	—	2	1	no	—
29	C₄×#3622	D₄	1	1	yes	characteristic
30	C₂×C₄×D₁₇	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₄×D₁₇	SD₁₆	1	1	yes	characteristic
32	C₂×C₄×D₁₇	Q₁₆	1	1	yes	characteristic
33	C₂×#3624	—	4	2	no	—
34	#63243	—	4	2	no	—
35	#63243	—	4	2	no	—
36	C₂×C₈×C₁₇	—	4	1	no	—
37	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
38	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
39	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
40	C₄×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
41	C₄×Dic₁₇	D₈	1	1	yes	characteristic
42	C₄×Dic₁₇	SD₁₆	1	1	yes	characteristic
43	C₂×#3622	—	4	1	no	—
44	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
46	C₈×D₁₇	—	4	1	no	—
47	C₈×D₁₇	—	4	1	no	—
48	#63216	—	4	2	no	—
49	#7220	—	17	1	no	maximal, Sylow
50	C₂²×D₁₇	#103	1	1	yes	characteristic
51	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
52	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
53	C₂×C₄×C₁₇	#103	1	1	yes	characteristic
54	C₂×Dic₁₇	#102	1	1	yes	characteristic
55	C₂×Dic₁₇	#106	1	1	yes	characteristic
56	C₂×Dic₁₇	#107	1	1	yes	characteristic
57	C₄×D₁₇	—	2	1	no	—
58	C₄×D₁₇	—	2	1	no	—
59	C₄×D₁₇	—	2	1	no	—
60	C₄×D₁₇	—	2	1	no	—
61	C₄×D₁₇	—	4	1	no	—
62	#3624	—	8	2	no	—
63	C₈×C₁₇	—	2	1	no	—
64	C₈×C₁₇	—	2	1	no	—
65	C₈×C₁₇	—	8	1	no	—
66	#3622	—	2	1	no	—
67	#3622	—	2	1	no	—
68	#3622	—	8	1	no	—
69	#2162	—	17	1	no	—
70	#1837	—	17	1	no	—
71	#1568	—	17	1	no	—
72	C₂²×C₁₇	#340	1	1	yes	characteristic
73	C₂×D₁₇	—	2	1	no	—
74	C₂×D₁₇	—	2	1	no	—
75	C₂×D₁₇	—	2	1	no	—
76	C₄×C₁₇	—	2	1	no	—
77	C₄×C₁₇	H(3,ℤ/4)	1	1	yes	characteristic
78	C₄×C₁₇	#344	1	1	yes	characteristic
79	C₄×C₁₇	#359	1	1	yes	characteristic
80	C₄×C₁₇	#359	1	1	yes	characteristic
81	Dic₁₇	—	2	1	no	—
82	Dic₁₇	—	2	1	no	—
83	Dic₁₇	—	2	1	no	—
84	#432	—	34	1	no	—
85	#423	—	34	1	no	—
86	C₂×C₄×C₈	—	17	1	no	—
87	C₄⋊C₁₆	—	34	1	no	—
88	C₂²⋊C₁₆	—	34	1	no	—
89	C₄×C₁₆	—	17	2	no	—
90	#341	—	17	2	no	—
91	#341	—	34	1	no	—
92	#335	—	17	1	no	—
93	#335	#595	1	1	yes	characteristic
94	#334	—	34	1	no	—
95	C₂×C₁₇	#1286	1	1	yes	characteristic
96	C₂×C₁₇	#1382	1	1	yes	characteristic
97	C₂×C₁₇	#1385	1	1	yes	characteristic
98	D₁₇	—	4	1	no	—
99	C₂×M₁₆	—	34	2	no	—
100	C₂²×C₈	—	17	1	no	—
101	C₂²×C₈	—	17	1	no	—
102	C₂×C₄²	—	17	1	no	—
103	C₂×C₁₆	—	34	2	no	—
104	C₄⋊C₈	—	2	1	no	—
105	C₄⋊C₈	—	34	1	no	—
106	C₄⋊C₈	—	68	1	no	—
107	C₄⋊C₈	—	68	1	no	—
108	C₂²⋊C₈	—	68	1	no	—
109	#97	—	68	1	no	—
110	C₄×C₈	—	17	1	no	—
111	C₄×C₈	—	17	1	no	—
112	C₄×C₈	—	17	1	no	—
113	C₄×C₈	C₂×#595	1	1	yes	characteristic
114	C₁₇	#7220	1	1	yes	characteristic, complemented, Sylow
115	C₂²×C₄	—	17	1	no	—
116	C₂²×C₄	—	17	1	no	—
117	C₂²×C₄	—	17	1	no	—
118	M₁₆	—	68	2	no	—
119	M₁₆	—	68	2	no	—
120	C₂×C₈	—	4	1	no	—
121	C₂×C₈	—	17	1	no	—
122	C₂×C₈	—	17	1	no	—
123	C₂×C₈	—	68	1	no	—
124	C₂×C₈	—	68	1	no	—
125	C₂×C₈	—	68	2	no	—
126	C₂×C₈	—	68	1	no	—
127	C₂×C₈	C₄×#595	1	1	yes	characteristic
128	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
129	C₄²	—	17	1	no	—
130	C₄²	—	17	1	no	—
131	C₄²	—	17	1	no	—
132	C₄²	C₂²⋊#595	1	1	yes	characteristic
133	C₁₆	—	68	2	no	—
134	C₂³	—	17	1	no	—
135	C₂×C₄	—	17	1	no	—
136	C₂×C₄	—	17	1	no	—
137	C₂×C₄	—	17	1	no	—
138	C₂×C₄	—	34	1	no	—
139	C₂×C₄	—	34	1	no	—
140	C₂×C₄	—	34	1	no	—
141	C₂×C₄	—	34	1	no	—
142	C₂×C₄	—	68	1	no	—
143	C₂×C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
144	C₂×C₄	#10587547	1	1	yes	characteristic
145	C₂×C₄	#10587549	1	1	yes	characteristic
146	C₈	—	2	1	no	—
147	C₈	—	2	1	no	—
148	C₈	—	8	1	no	—
149	C₈	—	34	1	no	—
150	C₈	—	34	1	no	—
151	C₈	—	136	2	no	—
152	C₈	—	136	1	no	—
153	C₂²	—	34	1	no	—
154	C₂²	—	34	1	no	—
155	C₂²	—	34	1	no	—
156	C₂²	#11763682	1	1	yes	characteristic
157	C₄	—	2	1	no	—
158	C₄	—	34	1	no	—
159	C₄	—	34	1	no	—
160	C₄	—	34	1	no	—
161	C₄	#11763677	1	1	yes	characteristic
162	C₄	#11763692	1	1	yes	characteristic
163	C₄	#11763723	1	1	yes	characteristic
164	C₄	#11763724	1	1	yes	characteristic
165	C₂	—	68	1	no	—
166	C₂	#14543639	1	1	yes	characteristic
167	C₂	#14543845	1	1	yes	characteristic
168	C₂	#14543853	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14545144

C₂

yes

maximal, characteristic

#14544708

C₂

yes

maximal, characteristic

#14527546

C₂

yes

maximal, characteristic

—

—

—

C₄

yes

—

#11763685

—

#11763685

C₄

yes

—

#11762865

—

C₄×C₈×D₁₇

C₂²

yes

characteristic

#11762562

C₄

yes

characteristic

#11762559

—

C₁₇×#335

C₄

yes

characteristic

C₂×#63243

—

C₂×C₈×D₁₇

C₂×C₄

yes

characteristic

C₂×C₈×D₁₇

D₄

yes

characteristic

C₄²×D₁₇

D₄

yes

—

—

—

C₁₇×C₄⋊C₈

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

#10587491

—

#10587486

—

C₈×Dic₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

Q₈

yes

characteristic

C₄⋊#3622

—

C₄×#3622

D₄

yes

characteristic

C₂×C₄×D₁₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×D₁₇

SD₁₆

yes

characteristic

C₂×C₄×D₁₇

Q₁₆

yes

—

—

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₄²

yes

characteristic

C₂×C₈×C₁₇

C₄⋊C₄

yes

characteristic

C₄²×C₁₇

C₂²⋊C₄

yes

characteristic

C₄×Dic₁₇

C₄⋊C₄

yes

characteristic

C₄×Dic₁₇

D₈

yes

characteristic

C₄×Dic₁₇

SD₁₆

yes

characteristic

C₂×#3622

—

C₂×#3622

C₂²⋊C₄

yes

characteristic

C₂×#3622

C₄⋊C₄

yes

characteristic

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

—

C₄×D₁₇

—

#3624

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

—

—

—

—

—

—

C₂²×C₁₇

#340

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

#432

—

#423

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

—

—

—

yes

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

D₁₇

—

C₂×M₁₆

—

100

C₂²×C₈

—

101

C₂²×C₈

—

102

C₂×C₄²

—

103

C₂×C₁₆

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

C₂²⋊C₈

—

109

#97

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

C₂×#595

yes

characteristic

114

C₁₇

#7220

yes

characteristic, complemented, Sylow

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

M₁₆

—

119

M₁₆

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

C₄×#595

yes

characteristic

128

C₂×C₈

C₄⋊#595

yes

characteristic

129

C₄²

—

130

C₄²

—

131

C₄²

—

132

C₄²

C₂²⋊#595

yes

characteristic

133

C₁₆

—

134

C₂³

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

(C₂×#595)⋊C₄

yes

characteristic

144

C₂×C₄

#10587547

yes

characteristic

145

C₂×C₄

#10587549

yes

characteristic

146

C₈

—

147

C₈

—

148

C₈

—

149

C₈

—

150

C₈

—

151

C₈

—

136

—

152

C₈

—

136

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

#11763682

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

#11763677

yes

characteristic

162

C₄

#11763692

yes

characteristic

163

C₄

#11763723

yes

characteristic

164

C₄

#11763724

yes

characteristic

165

C₂

—

166

C₂

#14543639

yes

characteristic

167

C₂

#14543845

yes

characteristic

168

C₂

#14543853

yes

characteristic

	subgroup	G/H
center	C₂²	#11763682
commutator	C₂×C₈×C₁₇	C₄²
Frattini	C₄×C₈	C₂×#595
Fitting	C₁₇×#335	C₄
socle	C₂²×C₁₇	#340
prime radical	C₂²×D₁₇	#103
cosocle	C₄×C₈×D₁₇	C₂²
hypercenter	#335	#595
hypocenter	C₁₇	#7220
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #335

subgroup

G/H

center

C₂²

#11763682

commutator

C₂×C₈×C₁₇

C₄²

Frattini

C₄×C₈

C₂×#595

Fitting

C₁₇×#335

C₄

C₂²×C₁₇

C₂²×D₁₇

C₄×C₈×D₁₇

C₂²

C₁₇

G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #335

N	H ≅ G/N	type
114. C₁₇	#7220	semidirect

H ≅ G/N

type

114. C₁₇

#7220

semidirect

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

subgroup	index
#14545144	2
#14544708	2
#14527546	2
#7220	17

subgroup

index

Subgroups — #17948768

Conjugacy classes of subgroups

Complements

Maximal subgroups