Finite Group Databasebeta

← #16943816

Subgroups — #16943816

Order 4352.

440 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 95 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
1#14527122C212yesmaximal
2#14526730C212yesmaximal, complemented
3#14526718C212yesmaximal, complemented
4C17×#2326C211yesmaximal, characteristic
5#11762773—22no—
6#11762535C2214yes—
7#11762533—22no—
8C2×C17×#124C411yescharacteristic
9C2×C4×Q8×C17C411yescharacteristic
10C2×C4×D4×C17C2211yescharacteristic
11C17×#391C2211yescharacteristic
12C17×#390C411yescharacteristic
13C17×#388—22no—
14C17×#385—22no—
15C17×#384C2211yescharacteristic
16C17×#376C411yescharacteristic
17C22×Q8×C17C2×C411yescharacteristic
18C22×D4×C17C2×C411yescharacteristic
19C23×C4×C17D412yes—
20C17×#124—22no—
21C17×#124—22no—
22C4×Q8×C17—42no—
23C4×D4×C17—42no—
24C2×C4⋊C4×C17—21no—
25C2×C4⋊C4×C17—22no—
26C2×C4⋊C4×C17C2×C411yescharacteristic
27C2×C4⋊C4×C17C2311yescharacteristic
28C2×C22⋊C4×C17—22no—
29C2×C22⋊C4×C17—22no—
30C2×C22⋊C4×C17D412yes—
31C2×C42×C17—21no—
32C2×C42×C17C2×C411yescharacteristic
33C2×C42×C17C2×C411yescharacteristic
34C2×C42×C17C2×C411yescharacteristic
35C2×#63223—42no—
36C17×#95—21no—
37C17×#95—21no—
38C17×#95—22no—
39C22⋊#3622—24no—
40C22⋊#3622—42no—
41C24×C17C22⋊C412yes—
42C2×Q8×C17—21no—
43C2×Q8×C17—41no—
44C2×Q8×C17—41no—
45C2×Q8×C17M1612yescomplemented
46C2×D4×C17—21no—
47C2×D4×C17—21no—
48C2×D4×C17—41no—
49C2×D4×C17—41no—
50C22×C4×C17—21no—
51C22×C4×C17—21no—
52C22×C4×C17—22no—
53C22×C4×C17—22no—
54C22×C4×C17—21no—
55C22×C4×C17—21no—
56C22×C4×C17—82no—
57C22×C4×C17C22⋊C411yescharacteristic
58C22×C4×C17C22⋊C411yescharacteristic
59C22×C4×C17C22×C411yescharacteristic
60C22×C4×C17C2×D412yes—
61C4⋊C4×C17—41no—
62C4⋊C4×C17—41no—
63C4⋊C4×C17—42no—
64C4⋊C4×C17—82no—
65C22⋊C4×C17—28no—
66C22⋊C4×C17—42no—
67C22⋊C4×C17—82no—
68C42×C17—22no—
69C42×C17—21no—
70C42×C17—41no—
71C42×C17—42no—
72C42×C17M1612yescomplemented
73#63223—84no—
74C2×#3622—44no—
75#11535—171nomaximal, Sylow
76C23×C17—22no—
77C23×C17—22no—
78C23×C17—22no—
79C23×C17—42no—
80C23×C17—42no—
81C23×C17C2×C22⋊C411yescharacteristic
82Q8×C17—42no—
83Q8×C17—42no—
84D4×C17—42no—
85D4×C17—42no—
86C2×C4×C17—21no—
87C2×C4×C17—21no—
88C2×C4×C17—22no—
89C2×C4×C17—22no—
90C2×C4×C17—21no—
91C2×C4×C17—22no—
92C2×C4×C17—41no—
93C2×C4×C17—44no—
94C2×C4×C17—41no—
95C2×C4×C17—41no—
96C2×C4×C17—41no—
97C2×C4×C17—41no—
98C2×C4×C17—42no—
99C2×C4×C17—41no—
100C2×C4×C17—41no—
101C2×C4×C17—42no—
102C2×C4×C17—41no—
103C2×C4×C17—41no—
104C2×C4×C17—41no—
105C2×C4×C17—82no—
106C2×C4×C17#9912yes—
107C2×C4×C17C2×M1611yescharacteristic
108C2×C4×C17C2×M1611yescharacteristic
109#3622—84no—
110#2326D1711yescharacteristic
111#1465—172no—
112#1325—172no—
113#1319—172no—
114C22×C17—21no—
115C22×C17—21no—
116C22×C17—44no—
117C22×C17—42no—
118C22×C17—42no—
119C22×C17—42no—
120C22×C17—82no—
121C22×C17#40711yescharacteristic
122C22×C17C2×#9911yescharacteristic
123C22×C17#41311yescharacteristic
124C4×C17—22no—
125C4×C17—21no—
126C4×C17—21no—
127C4×C17—42no—
128C4×C17—42no—
129C4×C17—41no—
130C4×C17—42no—
131C4×C17—81no—
132C4×C17—81no—
133C2×#124Dic1711yescharacteristic
134C2×C4×Q8Dic1711yescharacteristic
135C2×C4×D4C2×D1711yescharacteristic
136#407—342no—
137#391C2×D1711yescharacteristic
138#390Dic1711yescharacteristic
139#388—22no—
140#385—22no—
141#384C2×D1711yescharacteristic
142#376Dic1711yescharacteristic
143#324—174no—
144#323—342no—
145C2×C17—21no—
146C2×C17—21no—
147C2×C17—82no—
148C2×C17#146511yescharacteristic
149C2×C17#212711yescharacteristic
150C2×C17#213511yescharacteristic
151C22×Q8C2×Dic1711yescharacteristic
152C22×D4C2×Dic1711yescharacteristic
153C23×C4C22⋊D1712yes—
154C2×M16—682no—
155#124—22no—
156#124—22no—
157C4×Q8—42no—
158C4×D4—42no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4C2×Dic1711yescharacteristic
162C2×C4⋊C4C22×D1711yescharacteristic
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4C22⋊D1712yes—
166C2×C42—21no—
167C2×C42C2×Dic1711yescharacteristic
168C2×C42C2×Dic1711yescharacteristic
169C2×C42C2×Dic1711yescharacteristic
170C22⋊C8—344no—
171C22⋊C8—682no—
172#95—21no—
173#95—21no—
174#95—22no—
175C17#1153511yescharacteristic, complemented, Sylow
176C24C22⋊Dic1712yes—
177C2×Q8—21no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8#6322312yescomplemented
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—41no—
184C2×D4—41no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—82no—
192C22×C4C22⋊Dic1711yescharacteristic
193C22×C4C22⋊Dic1711yescharacteristic
194C22×C4C22×Dic1711yescharacteristic
195C22×C4C2×C22⋊D1712yes—
196M16—1364no—
197C2×C8—684no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—42no—
201C4⋊C4—82no—
202C22⋊C4—28no—
203C22⋊C4—42no—
204C22⋊C4—82no—
205C42—22no—
206C42—21no—
207C42—41no—
208C42—42no—
209C42#6322312yescomplemented
210C23—22no—
211C23—22no—
212C23—22no—
213C23—42no—
214C23—42no—
215C23C2×C22⋊Dic1711yescharacteristic
216Q8—42no—
217Q8—42no—
218D4—42no—
219D4—42no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—41no—
227C2×C4—44no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—82no—
240C2×C4#1058750512yes—
241C2×C4C2×#6322311yescharacteristic
242C2×C4C2×#6322311yescharacteristic
243C8—1364no—
244C22—21no—
245C22—21no—
246C22—44no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—82no—
251C22#1176277311yescharacteristic
252C22C2×#1058750511yescharacteristic
253C22#1176279311yescharacteristic
254C4—21no—
255C4—21no—
256C4—22no—
257C4—42no—
258C4—42no—
259C4—42no—
260C4—41no—
261C4—81no—
262C4—81no—
263C2—21no—
264C2—21no—
265C2—82no—
266C2#1452712211yescharacteristic
267C2#1453045311yescharacteristic
268C2#1453047911yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#2326C2
socleC22×C17C2×#99
prime radicalC22×D4×C17C2×C4
cosocleC2×C4⋊C4×C17C23
hypercenter#2326D17
hypocenterC17#11535
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2326

Complements

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

NH ≅ G/Ntype
2. #14526730C2semidirect
3. #14526718C2semidirect
45. C2×Q8×C17M16semidirect
72. C42×C17M16semidirect
175. C17#11535semidirect
180. C2×Q8#63223semidirect
209. C42#63223semidirect

Maximal subgroups

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

subgroupindex
#145271222
#145267302
#145267182
C17×#23262
#1153517