Finite Group Databasebeta

← #17960851

Subgroups — #17960851

Order 4352.

564 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 100 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×#11763716C212yesmaximal
2#14544529C212yesmaximal
3#14543879C212yesmaximal
4#14531492C211yesmaximal, characteristic
5C2×#10587539C2214yes—
6C4×C22⋊#595—22no—
7#11763716C418yescomplemented
8#11763687—22no—
9C2×#10587561C2211yescharacteristic
10C2×#10587573C411yescharacteristic, complemented
11C2×#10587579C2211yescharacteristic
12#11762714C411yescharacteristic, complemented
13#11762710C2211yescharacteristic
14#11762699—22no—
15#11762680—22no—
16#11762655C411yescharacteristic, complemented
17C17×#384C411yescharacteristic, complemented
18C22×D68C2311yescharacteristic
19C22×C4×D17D412yes—
20C22×Dic34C2×C411yescharacteristic
21C2×C22⋊#595—24no—
22C2×C4×#595—42no—
23C2×C4⋊C4×C17—22no—
24C2×C4⋊C4×C17C2×C411yescharacteristic
25C2×C4⋊C4×C17C2×C411yescharacteristic
26C2×C42×C17C2×C411yescharacteristic
27C2×#63227—22no—
28C2×#63227—22no—
29C2×#63227D412yes—
30C2×#63225—21no—
31C2×C4×Dic17—21no—
32C2×C4×Dic17C2×C411yescharacteristic
33C2×C4×Dic17C2×C411yescharacteristic
34#10587579—42no—
35#10587573—42no—
36#10587561—22no—
37#10587561C2×C414yes—
38(C2×#595)⋊C4—42no—
39#10587539—24no—
40#10587539C2×C418yes—
41#10587502—22no—
42#10587502—21no—
43#10587502—21no—
44C23×D17C2×D412yes—
45C22×#595—44no—
46C22×C4×C17—21no—
47C22×C4×C17—21no—
48C22×C4×C17C22⋊C412yes—
49C22×C4×C17C22×C411yescharacteristic
50C22×Dic17—21no—
51C22×Dic17—21no—
52C22×Dic17C22⋊C411yescharacteristic
53C22×Dic17C22⋊C411yescharacteristic
54C2×D68—41no—
55C2×D68—41no—
56C2×D68C22×C412yes—
57C2×D68#4112yes—
58C2×C4×D17—22no—
59C2×C4×D17—22no—
60C2×C4×D17—82no—
61C2×Dic34—21no—
62C2×Dic34—41no—
63C2×Dic34—41no—
64C2×Dic34C4212yescomplemented
65C22⋊#595—28no—
66C22⋊#595—44no—
67C4×#595—84no—
68C4⋊C4×C17—41no—
69C4⋊C4×C17—41no—
70C4⋊C4×C17—82no—
71C42×C17—21no—
72C42×C17C4212yescomplemented
73#63227—28no—
74#63227—42no—
75#63227—82no—
76#63225—42no—
77C4×Dic17—22no—
78C4×Dic17—42no—
79C4×Dic17—41no—
80#12627—171nomaximal, Sylow
81C23×C17C2×C22⋊C411yescharacteristic
82C22×D17—22no—
83C22×D17—22no—
84C22×D17—42no—
85C22×D17—42no—
86C22×D17C4×D414yes—
87C2×#595—48no—
88C2×#595—84no—
89C2×#595—84no—
90C2×C4×C17—21no—
91C2×C4×C17—21no—
92C2×C4×C17—22no—
93C2×C4×C17—22no—
94C2×C4×C17—41no—
95C2×C4×C17—41no—
96C2×C4×C17—42no—
97C2×C4×C17—42no—
98C2×C4×C17—41no—
99C2×C4×C17C2×C4211yescharacteristic
100C2×C4×C17#11711yescharacteristic
101C2×Dic17—22no—
102C2×Dic17—21no—
103C2×Dic17—41no—
104C2×Dic17—41no—
105C2×Dic17—41no—
106C2×Dic17—41no—
107C2×Dic17—41no—
108C2×Dic17—41no—
109C2×Dic17—41no—
110C2×Dic17#9912yes—
111D68—42no—
112D68—42no—
113C4×D17—44no—
114C4×D17—82no—
115Dic34—42no—
116Dic34—42no—
117#2326—171no—
118C2×#354—172no—
119C4×#99—172no—
120#1393—172no—
121C22×C17—21no—
122C22×C17—21no—
123C22×C17C4×C22⋊C411yescharacteristic
124C22×C17C2×#9911yescharacteristic
125C22×C17#41011yescharacteristic
126C2×D17—24no—
127C2×D17—42no—
128C2×D17—42no—
129C2×D17—44no—
130C2×D17—82no—
131#595—88no—
132C4×C17—21no—
133C4×C17—21no—
134C4×C17—42no—
135C4×C17—42no—
136C4×C17—81no—
137Dic17—22no—
138Dic17—41no—
139Dic17—42no—
140Dic17—81no—
141C2×#124—171no—
142C2×C4×Q8—171no—
143C2×C4×D4—171no—
144C2×#99—174no—
145#391—171no—
146#390—171no—
147#388—342no—
148#385—342no—
149#384#59511yescharacteristic, complemented
150C4×C22⋊C4—342no—
151#376—171no—
152#354—178no—
153#342—342no—
154C2×C17—21no—
155C2×C17—21no—
156C2×C17C4×#9911yescharacteristic
157C2×C17#212711yescharacteristic
158C2×C17#212911yescharacteristic
159D17—44no—
160C22×Q8—171no—
161C22×D4—171no—
162C23×C4—172no—
163#124—174no—
164#124—342no—
165C4×Q8—682no—
166C4×D4—682no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—341no—
169C2×C4⋊C4C2×#59511yescharacteristic
170C2×C4⋊C4C2×#59511yescharacteristic
171C2×C22⋊C4—172no—
172C2×C22⋊C4—342no—
173C2×C22⋊C4—344no—
174C2×C22⋊C4—342no—
175C2×C42—171no—
176C2×C42—171no—
177C2×C42—341no—
178C2×C42—682no—
179C2×C42C2×#59511yescharacteristic
180#99—178no—
181#99—344no—
182#95—341no—
183#95—341no—
184#95—342no—
185#95—682no—
186C17#1262711yescharacteristic, complemented, Sylow
187C24—172no—
188C2×Q8—172no—
189C2×Q8—341no—
190C2×Q8—681no—
191C2×Q8—681no—
192C2×D4—172no—
193C2×D4—172no—
194C2×D4—681no—
195C2×D4—681no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—171no—
199C22×C4—171no—
200C22×C4—342no—
201C22×C4—342no—
202C22×C4—341no—
203C22×C4—341no—
204C22×C4—684no—
205C22×C4—1362no—
206C22×C4C22⋊#59512yes—
207C22×C4C22×#59511yescharacteristic
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—82no—
211C4⋊C4—682no—
212C22⋊C4—348no—
213C22⋊C4—348no—
214C22⋊C4—682no—
215C22⋊C4—684no—
216C22⋊C4—1362no—
217C42—21no—
218C42—342no—
219C42—681no—
220C42—682no—
221C42—1364no—
222C42C4×#59512yescomplemented
223C23—174no—
224C23—342no—
225C23—342no—
226C23—682no—
227C23—682no—
228C23C2×C22⋊#59511yescharacteristic
229Q8—682no—
230Q8—682no—
231D4—682no—
232D4—682no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—41no—
242C2×C4—172no—
243C2×C4—342no—
244C2×C4—341no—
245C2×C4—684no—
246C2×C4—681no—
247C2×C4—681no—
248C2×C4—681no—
249C2×C4—681no—
250C2×C4—681no—
251C2×C4—681no—
252C2×C4—688no—
253C2×C4—681no—
254C2×C4—1362no—
255C2×C4—1364no—
256C2×C4—1364no—
257C2×C4C2×C4×#59511yescharacteristic
258C2×C4#1058767711yescharacteristic
259C22—21no—
260C22—21no—
261C22—344no—
262C22—684no—
263C22—682no—
264C22—682no—
265C22—1362no—
266C22C4×C22⋊#59511yescharacteristic
267C22D17⋊#9911yescharacteristic
268C22#1176379011yescharacteristic
269C4—21no—
270C4—21no—
271C4—42no—
272C4—42no—
273C4—81no—
274C4—342no—
275C4—682no—
276C4—681no—
277C4—1361no—
278C4—1368no—
279C2—21no—
280C2—21no—
281C2—684no—
282C2#1454452911yescharacteristic
283C2#1454506011yescharacteristic
284C2#1454506711yescharacteristic
subgroupG/H
centerC22D17⋊#99
commutatorC2×C4×C17C2×C42
FrattiniC22×C4C22×#595
FittingC17×#384C4
socleC22×C17C2×#99
prime radical, cosocleC22×D68C23
hypercenter#384#595
hypocenterC17#12627
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #384

Complements

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

NH ≅ G/Ntype
7. #11763716C4semidirect
10. C2×#10587573C4semidirect
12. #11762714C4semidirect
16. #11762655C4semidirect
17. C17×#384C4semidirect
64. C2×Dic34C42semidirect
72. C42×C17C42semidirect
149. #384#595semidirect
186. C17#12627semidirect
222. C42C4×#595semidirect

Maximal subgroups

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

subgroupindex
C2×#117637162
#145445292
#145438792
#145314922
#1262717