Finite Group Databasebeta

← #17960843

Subgroups — #17960843

Order 4352.

720 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 102 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×#11763713C212yesmaximal
2#14544529C212yesmaximal, complemented
3#14543878C212yesmaximal, complemented
4#14531348C211yesmaximal, characteristic
5C2×#10587539C2214yes—
6C4×C22⋊#595—22no—
7#11763713C418yescomplemented
8#11763687—22no—
9C2×#10587626C2211yescharacteristic
10C2×D4×Dic17C411yescharacteristic, complemented
11C2×#10587579C2211yescharacteristic
12C2×#10587567—22no—
13#11762711C2211yescharacteristic
14#11762709—22no—
15#11762678C411yescharacteristic, complemented
16C42×Dic17C411yescharacteristic, complemented
17C17×#390C411yescharacteristic, complemented
18C22×D4×C17C2×C411yescharacteristic
19C22×C22⋊D17—22no—
20C23×Dic17—21no—
21C22×D68C2311yescharacteristic
22C22×C4×D17D412yes—
23C2×C22⋊#595—24no—
24C2×C4×#595—42no—
25C2×C4⋊C4×C17C2×C411yescharacteristic
26C2×C22⋊C4×C17—22no—
27C2×C42×C17C2×C411yescharacteristic
28C2×C22⋊Dic17—21no—
29#10587626—22no—
30#10587626C2×C414yes—
31D4×Dic17—41no—
32D4×Dic17—41no—
33C2×#63227—22no—
34C2×#63227D412yes—
35C2×C4⋊Dic17C2×C411yescharacteristic
36C2×#63225—22no—
37C2×C4×Dic17—21no—
38C2×C4×Dic17—21no—
39C2×C4×Dic17C2×C411yescharacteristic
40C2×C4×Dic17C2×C411yescharacteristic
41#10587579—41no—
42#10587579—41no—
43#10587567—48no—
44(C2×#595)⋊C4—42no—
45#10587539—24no—
46#10587539C2×C418yescomplemented
47C24×C17—21no—
48C23×D17C2×D412yes—
49C22×#595—44no—
50C2×D4×C17—21no—
51C2×D4×C17—41no—
52C2×D4×C17—41no—
53C2×D4×C17C4212yescomplemented
54C22×C4×C17—21no—
55C22×C4×C17C22⋊C412yes—
56C22×C4×C17C22×C411yescharacteristic
57C2×C22⋊D17—28no—
58C2×C22⋊D17—44no—
59C2×C22⋊D17—44no—
60C22×Dic17—21no—
61C22×Dic17—21no—
62C22×Dic17—44no—
63C22×Dic17—41no—
64C22×Dic17—41no—
65C22×Dic17C22⋊C411yescharacteristic
66C22×Dic17C22⋊C411yescharacteristic
67C2×D68—41no—
68C2×D68—41no—
69C2×D68C22×C412yes—
70C2×D68#4112yes—
71C2×C4×D17—22no—
72C2×C4×D17—22no—
73C2×C4×D17—44no—
74C22⋊#595—28no—
75C22⋊#595—44no—
76C4×#595—84no—
77C4⋊C4×C17—41no—
78C22⋊C4×C17—44no—
79C42×C17—41no—
80C22⋊Dic17—42no—
81#63227—42no—
82#63227—44no—
83C4⋊Dic17—41no—
84#63225—44no—
85C4×Dic17—24no—
86C4×Dic17—21no—
87C4×Dic17—21no—
88C4×Dic17—21no—
89C4×Dic17—42no—
90C4×Dic17C4212yescomplemented
91C4×#353—171nomaximal, Sylow
92C23×C17—22no—
93C23×C17—42no—
94C23×C17—42no—
95C23×C17—41no—
96C23×C17—41no—
97C23×C17C2×C22⋊C411yescharacteristic
98C22×D17—22no—
99C22×D17—22no—
100C22×D17—42no—
101C22×D17—42no—
102C22×D17C4×D414yes—
103C2×#595—48no—
104C2×#595—84no—
105C2×#595—84no—
106D4×C17—42no—
107D4×C17—42no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—22no—
111C2×C4×C17—22no—
112C2×C4×C17—41no—
113C2×C4×C17—41no—
114C2×C4×C17—42no—
115C2×C4×C17C2×C4211yescharacteristic
116C2×C4×C17#11711yescharacteristic
117C22⋊D17—416no—
118C2×Dic17—22no—
119C2×Dic17—22no—
120C2×Dic17—22no—
121C2×Dic17—22no—
122C2×Dic17—21no—
123C2×Dic17—41no—
124C2×Dic17—42no—
125C2×Dic17—41no—
126C2×Dic17—82no—
127C2×Dic17—82no—
128C2×Dic17#9911yescharacteristic
129C2×Dic17#9911yescharacteristic
130D68—42no—
131D68—42no—
132C4×D17—44no—
133C4×D17—82no—
134C4×#127—171no—
135C2×#353—172no—
136C4×#99—172no—
137#1393—172no—
138C22×C17—21no—
139C22×C17—21no—
140C22×C17—44no—
141C22×C17—42no—
142C22×C17—82no—
143C22×C17—81no—
144C22×C17—81no—
145C22×C17C4×C22⋊C411yescharacteristic
146C22×C17C2×#9911yescharacteristic
147C22×C17#41011yescharacteristic
148C2×D17—24no—
149C2×D17—42no—
150C2×D17—42no—
151C2×D17—44no—
152C2×D17—82no—
153#595—88no—
154C4×C17—21no—
155C4×C17—21no—
156C4×C17—42no—
157C4×C17—42no—
158Dic17—24no—
159Dic17—21no—
160Dic17—41no—
161Dic17—81no—
162Dic17#35312yescomplemented
163C2×#127—171no—
164C2×C4×D4—171no—
165C2×C4×D4—171no—
166C2×C4×D4—342no—
167C2×#99—174no—
168#390—171no—
169#390—342no—
170#390#59511yescharacteristic, complemented
171#384—171no—
172C4×C22⋊C4—342no—
173C43—171no—
174#353—178no—
175#342—342no—
176C2×C17—21no—
177C2×C17—21no—
178C2×C17—82no—
179C2×C17C4×#9911yescharacteristic
180C2×C17C2×#35311yescharacteristic
181C2×C17#212611yescharacteristic
182D17—44no—
183C22×D4—171no—
184C22×D4—342no—
185C22×D4C2×#59511yescharacteristic
186C23×C4—172no—
187C23×C4—341no—
188#127—174no—
189#127—342no—
190C4×D4—688no—
191C4×D4—681no—
192C4×D4—681no—
193C4×D4—681no—
194C4×D4—681no—
195C2×C4⋊C4—171no—
196C2×C4⋊C4—342no—
197C2×C4⋊C4C2×#59511yescharacteristic
198C2×C22⋊C4—22no—
199C2×C22⋊C4—172no—
200C2×C22⋊C4—344no—
201C2×C22⋊C4—342no—
202C2×C22⋊C4—341no—
203C2×C42—171no—
204C2×C42—171no—
205C2×C42—341no—
206C2×C42—341no—
207C2×C42—682no—
208C2×C42C2×#59511yescharacteristic
209#99—178no—
210#99—344no—
211#95—682no—
212C17C4×#35311yescharacteristic, complemented, Sylow
213C24—21no—
214C24—172no—
215C2×D4—21no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—172no—
219C2×D4—172no—
220C2×D4—348no—
221C2×D4—681no—
222C2×D4—681no—
223C2×D4—684no—
224C2×D4—684no—
225C2×D4C4×#59512yescomplemented
226C22×C4—21no—
227C22×C4—171no—
228C22×C4—171no—
229C22×C4—342no—
230C22×C4—342no—
231C22×C4—341no—
232C22×C4—341no—
233C22×C4—684no—
234C22×C4—681no—
235C22×C4—681no—
236C22×C4—684no—
237C22×C4—684no—
238C22×C4C22⋊#59512yes—
239C22×C4C22×#59511yescharacteristic
240C4⋊C4—41no—
241C4⋊C4—681no—
242C4⋊C4—684no—
243C22⋊C4—44no—
244C22⋊C4—348no—
245C22⋊C4—682no—
246C22⋊C4—684no—
247C22⋊C4—682no—
248C22⋊C4—684no—
249C42—41no—
250C42—172no—
251C42—341no—
252C42—341no—
253C42—344no—
254C42—341no—
255C42—682no—
256C42—1364no—
257C23—22no—
258C23—42no—
259C23—42no—
260C23—41no—
261C23—41no—
262C23—174no—
263C23—342no—
264C23—342no—
265C23—682no—
266C23—682no—
267C23C2×C22⋊#59511yescharacteristic
268D4—42no—
269D4—42no—
270D4—682no—
271D4—682no—
272D4—6816no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—171no—
281C2×C4—171no—
282C2×C4—342no—
283C2×C4—342no—
284C2×C4—342no—
285C2×C4—342no—
286C2×C4—341no—
287C2×C4—684no—
288C2×C4—681no—
289C2×C4—682no—
290C2×C4—688no—
291C2×C4—681no—
292C2×C4—1362no—
293C2×C4—1362no—
294C2×C4—1362no—
295C2×C4—1364no—
296C2×C4—1364no—
297C2×C4C2×C4×#59511yescharacteristic
298C2×C4#1058767711yescharacteristic
299C22—21no—
300C22—21no—
301C22—44no—
302C22—42no—
303C22—82no—
304C22—81no—
305C22—81no—
306C22—344no—
307C22—684no—
308C22—682no—
309C22—682no—
310C22—1362no—
311C22C4×C22⋊#59511yescharacteristic
312C22D17⋊#9911yescharacteristic
313C22#1176378911yescharacteristic
314C4—21no—
315C4—21no—
316C4—42no—
317C4—42no—
318C4—172no—
319C4—344no—
320C4—341no—
321C4—681no—
322C4—1368no—
323C4—1361no—
324C2—21no—
325C2—21no—
326C2—82no—
327C2—684no—
328C2#1454453111yescharacteristic
329C2D17⋊#35311yescharacteristic
330C2#1454505611yescharacteristic
subgroupG/H
centerC22D17⋊#99
commutatorC2×C4×C17C2×C42
FrattiniC22×C4C22×#595
FittingC17×#390C4
socleC22×C17C2×#99
prime radicalC2×#10587626C22
cosocleC22×D68C23
hypercenter#390#595
hypocenterC17C4×#353
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #390

Complements

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

NH ≅ G/Ntype
2. #14544529C2semidirect
3. #14543878C2semidirect
7. #11763713C4semidirect
10. C2×D4×Dic17C4semidirect
15. #11762678C4semidirect
16. C42×Dic17C4semidirect
17. C17×#390C4semidirect
46. #10587539C2×C4semidirect
53. C2×D4×C17C42semidirect
90. C4×Dic17C42semidirect
162. Dic17#353semidirect
170. #390#595semidirect
212. C17C4×#353semidirect
225. C2×D4C4×#595semidirect

Maximal subgroups

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

subgroupindex
C2×#117637132
#145445292
#145438782
#145313482
C4×#35317