Finite Group Databasebeta

← #16926825

Subgroups — #16926825

Order 4352.

818 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 818 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763414C211yesmaximal, characteristic
2#14528867C211yesmaximal, characteristic, complemented
3#14527312C214yesmaximal, complemented
4C17×#1820C211yesmaximal, characteristic, complemented
5#11763414—28no—
6C2×#10587560C411yescharacteristic, complemented
7C2×#10587556C411yescharacteristic, complemented
8C2×#10587580—22no—
9C2×#10587616C2212yes—
10C2×#10587570—22no—
11C2×C4×D68C412yescomplemented
12#11762773C2212yes—
13#11762578—22no—
14#11762542—28no—
15C2×C17×#117C2211yescharacteristic
16C17×#381C2212yes—
17C17×#342—22no—
18C23×C4×C17C2311yescharacteristic
19C22×C22⋊D17—22no—
20C22×D68—22no—
21C22×D68C2×C412yes—
22C22×C4×D17—22no—
23C17×#117—24no—
24C2×C4⋊C4×C17D412yes—
25C2×C22⋊C4×C17—22no—
26C2×C22⋊C4×C17D412yes—
27C2×C42×C17C2×C411yescharacteristic
28C2×C42×C17C2×C411yescharacteristic
29#10587616—24no—
30#10587616D418yes—
31C2×#63227—22no—
32C2×#63227—22no—
33C2×C4⋊Dic17C2×C412yes—
34C2×#63225—22no—
35C2×#63223—22no—
36#10587580—216no—
37#10587570—216no—
38#10587560—24no—
39C4×D68—24no—
40C4×D68—24no—
41#10587556—24no—
42C17×#95—24no—
43C22⋊#3622—24no—
44C24×C17C2×D411yescharacteristic
45C23×D17—22no—
46C22×C4×C17—42no—
47C22×C4×C17—42no—
48C22×C4×C17C22⋊C412yes—
49C22×C4×C17C22⋊C412yes—
50C22×C4×C17C22×C411yescharacteristic
51C22×C4×C17C2×D411yescharacteristic
52C22×C4×C17C2×D414yes—
53C2×C22⋊D17—28no—
54C2×C22⋊D17—84no—
55C22×Dic17—22no—
56C2×D68—22no—
57C2×D68—28no—
58C2×D68—44no—
59C2×D68—84no—
60C2×D68C22⋊C414yes—
61C2×C4×D17—28no—
62C2×C4×D17—84no—
63C4⋊C4×C17—24no—
64C22⋊C4×C17—24no—
65C22⋊C4×C17—44no—
66C42×C17—22no—
67C42×C17—22no—
68#63227—28no—
69#63227—28no—
70C4⋊Dic17—22no—
71C4⋊Dic17C22⋊C414yes—
72#63225—28no—
73#63223—44no—
74C2×#3622—42no—
75#8518—171nomaximal, Sylow
76C23×C17—22no—
77C23×C17—42no—
78C23×C17C2×C22⋊C411yescharacteristic
79C23×C17#12012yes—
80C22×D17—24no—
81C22×D17—44no—
82C22×D17—44no—
83C22×D17—44no—
84C2×C4×C17—22no—
85C2×C4×C17—22no—
86C2×C4×C17—22no—
87C2×C4×C17—22no—
88C2×C4×C17—44no—
89C2×C4×C17—42no—
90C2×C4×C17—42no—
91C2×C4×C17—82no—
92C2×C4×C17—84no—
93C2×C4×C17C2×C22⋊C412yes—
94C2×C4×C17#12012yes—
95C22⋊D17—88no—
96C2×Dic17—24no—
97C2×Dic17—44no—
98D68—44no—
99D68—44no—
100D68—88no—
101C4×D17—88no—
102#3622—82no—
103C2×#540—171no—
104#1822—171no—
105#1820D1711yescharacteristic, complemented
106#1510—174no—
107C22×C17—22no—
108C22×C17—24no—
109C22×C17—22no—
110C22×C17—82no—
111C22×C17#37911yescharacteristic
112C22×C17#41012yes—
113C2×D17—44no—
114C2×D17—44no—
115C2×D17—44no—
116C2×D17—88no—
117C4×C17—22no—
118C4×C17—22no—
119C4×C17—42no—
120C4×C17—42no—
121C4×C17—162no—
122Dic17—44no—
123#540—348no—
124C2×#127—171no—
125C2×#125—171no—
126C2×#123—342no—
127C2×#121—172no—
128C2×#121—342no—
129C2×C4×D4—172no—
130C2×#117C2×D1711yescharacteristic
131#407—172no—
132#381C2×D1712yes—
133#343—342no—
134#342—22no—
135#327—348no—
136C2×C17—22no—
137C2×C17—42no—
138C2×C17#178211yescharacteristic
139C2×C17#179512yes—
140D17—84no—
141C22×D4—172no—
142C22×D4—342no—
143C22×D4—342no—
144C23×C4—342no—
145C23×C4C22×D1711yescharacteristic
146C2×M16—342no—
147#127—344no—
148#125—344no—
149#123—3416no—
150#121—178no—
151#121—3416no—
152#121—344no—
153C4×D4—344no—
154C4×D4—344no—
155#117—24no—
156C2×C4⋊C4—172no—
157C2×C4⋊C4—342no—
158C2×C4⋊C4D6812yes—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—342no—
161C2×C22⋊C4—342no—
162C2×C22⋊C4C22⋊D1712yes—
163C2×C42C4×D1711yescharacteristic
164C2×C42C4×D1711yescharacteristic
165C22⋊C8—344no—
166#95—24no—
167C17#851811yescharacteristic, complemented, Sylow
168C24—342no—
169C24C2×C22⋊D1711yescharacteristic
170C2×D4—174no—
171C2×D4—348no—
172C2×D4—348no—
173C2×D4—342no—
174C2×D4—684no—
175C2×D4—1364no—
176C2×D4—1364no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—348no—
180C22×C4—342no—
181C22×C4—1364no—
182C22×C4#6322712yes—
183C22×C4#6322712yes—
184C22×C4C2×C4×D1711yescharacteristic
185C22×C4C2×D6811yescharacteristic
186C22×C4D4×D1714yes—
187M16—684no—
188C2×C8—682no—
189C4⋊C4—24no—
190C4⋊C4—174no—
191C4⋊C4—348no—
192C4⋊C4—342no—
193C22⋊C4—24no—
194C22⋊C4—44no—
195C22⋊C4—348no—
196C22⋊C4—348no—
197C42—22no—
198C42—22no—
199C23—22no—
200C23—42no—
201C23—344no—
202C23—684no—
203C23—684no—
204C23—684no—
205C23C2×#6322711yescharacteristic
206C23#1058762312yes—
207D4—684no—
208D4—684no—
209D4—1368no—
210D4—1368no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—44no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—82no—
219C2×C4—84no—
220C2×C4—344no—
221C2×C4—684no—
222C2×C4—1368no—
223C2×C4C22⋊C4×D1712yes—
224C2×C4#1058756812yes—
225C8—1362no—
226C22—22no—
227C22—22no—
228C22—24no—
229C22—82no—
230C22—684no—
231C22—684no—
232C22—684no—
233C22—1368no—
234C22#1176266411yescharacteristic
235C22#1176278212yes—
236C4—22no—
237C4—22no—
238C4—42no—
239C4—42no—
240C4—162no—
241C4—684no—
242C2—22no—
243C2—42no—
244C2—1364no—
245C2#1452870111yescharacteristic
246C2#1452875212yes—
subgroupG/H
centerC22#11762664
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#1820C2
socleC22×C17#379
prime radicalC2×#11763414C2
cosocleC23×C4×C17C23
hypercenter#1820D17
hypocenterC17#8518
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1820

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #14528867C2semidirect
3. #14527312C2semidirect
4. C17×#1820C2semidirect
6. C2×#10587560C4semidirect
7. C2×#10587556C4semidirect
11. C2×C4×D68C4semidirect
105. #1820D17semidirect
167. C17#8518semidirect

Maximal subgroups

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

subgroupindex
C2×#117634142
#145288672
#145273122
C17×#18202
#851817