Finite Group Databasebeta

← #17959711

Subgroups — #17959711

Order 4352.

472 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 80 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#14544645C211yesmaximal, characteristic
2#14544021C211yesmaximal, characteristic
3#14543764C214yesmaximal, complemented
4#14533780C211yesmaximal, characteristic
5C2×#10587539—21no—
6#11763777—21no—
7#11763751—21no—
8#11763687C2212yes—
9#11763654—21no—
10#11763653C2212yes—
11C2×#10587633—21no—
12C2×C4×D68C2211yescharacteristic
13#11762735C412yescomplemented
14#11762735C2212yes—
15#11762716—21no—
16#11762666—21no—
17#11762666—21no—
18C4×C4⋊Dic17C411yescharacteristic, complemented
19C2×C17×C4⋊Q8C411yescharacteristic, complemented
20C22×Q8×C17—21no—
21C22×D68D411yescharacteristic
22C22×C4×D17D411yescharacteristic
23C22×C4×D17D411yescharacteristic
24C2×C22⋊#595—22no—
25C2×C4⋊#595—41no—
26C2×#63242—41no—
27C17×C4⋊Q8—41no—
28C17×C4⋊Q8—41no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17C2×C412yes—
31C2×C42×C17C2×C411yescharacteristic
32#10587633—42no—
33#10587633—42no—
34C2×#63227—22no—
35C2×#63227—22no—
36C2×#63227C2×C411yescharacteristic
37C2×#63227C2311yescharacteristic
38C2×C4⋊Dic17—21no—
39C2×C4⋊Dic17D411yescharacteristic
40C2×C4×Dic17—21no—
41C4×D68—41no—
42C4×D68—41no—
43C22⋊#3624—22no—
44(C2×#595)⋊C4—41no—
45#10587543—41no—
46#10587539—42no—
47#10587502—22no—
48#10587502—22no—
49#10587502C2×C412yes—
50C23×D17C22⋊C411yescharacteristic
51C23×D17C2×D411yescharacteristic
52C22×#595—42no—
53C2×Q8×C17—42no—
54C2×Q8×C17—42no—
55C2×Q8×C17—42no—
56C22×C4×C17—22no—
57C22×C4×C17C22⋊C411yescharacteristic
58C22×C4×C17C22⋊C411yescharacteristic
59C22×C4×C17C22×C411yescharacteristic
60C22×Dic17—22no—
61C22×Dic17C22⋊C411yescharacteristic
62C22×Dic17C2×D411yescharacteristic
63C2×D68—41no—
64C2×D68—41no—
65C2×D68D812yes—
66C2×D68SD1612yes—
67C2×C4×D17—21no—
68C2×C4×D17—21no—
69C2×C4×D17—21no—
70C2×C4×D17—21no—
71C2×C4×D17—42no—
72C2×C4×D17—42no—
73C22⋊#595—42no—
74C22⋊#595—42no—
75C2×#3624—42no—
76C4⋊#595—82no—
77#63242—82no—
78C4⋊C4×C17—42no—
79C4⋊C4×C17—41no—
80C4⋊C4×C17—41no—
81C42×C17—21no—
82C42×C17—21no—
83#63227—41no—
84#63227—41no—
85#63227—44no—
86#63227—42no—
87#63227—42no—
88C4⋊Dic17—41no—
89C4⋊Dic17—42no—
90C4×Dic17—42no—
91#11878—171nomaximal, Sylow
92C23×C17C2×C22⋊C411yescharacteristic
93C22×D17—21no—
94C22×D17—21no—
95C22×D17—21no—
96C22×D17—21no—
97C22×D17—41no—
98C22×D17—41no—
99C22×D17—41no—
100C22×D17—41no—
101C22×D17#10212yes—
102C22×D17C2×D811yescharacteristic
103C22×D17C2×SD1611yescharacteristic
104C2×#595—82no—
105C2×#595—82no—
106C2×#595—82no—
107Q8×C17—84no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—22no—
111C2×C4×C17—21no—
112C2×C4×C17—41no—
113C2×C4×C17—42no—
114C2×C4×C17—42no—
115C2×C4×C17—42no—
116C2×C4×C17#9911yescharacteristic
117C2×C4×C17#9911yescharacteristic
118C2×C4×C17#10212yes—
119C2×Dic17—21no—
120C2×Dic17—21no—
121C2×Dic17—41no—
122C2×Dic17—41no—
123C2×Dic17—42no—
124C2×Dic17—42no—
125C2×Dic17—42no—
126D68—42no—
127D68—42no—
128C4×D17—42no—
129C4×D17—42no—
130C4×D17—81no—
131C4×D17—81no—
132#3624—82no—
133#2547—171no—
134#1799—171no—
135#1459—171no—
136#1344—174no—
137C22×C17—21no—
138C22×C17—21no—
139C22×C17C2×#9911yescharacteristic
140C22×C17C2×#10211yescharacteristic
141C22×C17#42111yescharacteristic
142C2×D17—21no—
143C2×D17—22no—
144C2×D17—21no—
145C2×D17—41no—
146C2×D17—41no—
147C2×D17—41no—
148C2×D17—41no—
149C2×D17—42no—
150C2×D17—41no—
151C2×D17—41no—
152C2×D17—81no—
153C2×D17—81no—
154#595—162no—
155C4×C17—21no—
156C4×C17—22no—
157C4×C17—41no—
158C4×C17—82no—
159C4×C17#35412yes—
160Dic17—41no—
161Dic17—41no—
162Dic17—82no—
163C2×C4⋊Q8#59511yescharacteristic, complemented
164C2×#124—341no—
165C2×C4×D4—171no—
166C2×#99—341no—
167C2×C22⋊C8—341no—
168#397—172no—
169#397—172no—
170#391—341no—
171#380—341no—
172#380—341no—
173#380—341no—
174C4×C4⋊C4—171no—
175#342—172no—
176#324—341no—
177#323—172no—
178C2×C17—21no—
179C2×C17—21no—
180C2×C17#151911yescharacteristic
181C2×C17#212311yescharacteristic
182C2×C17C2×#35411yescharacteristic
183D17—41no—
184D17—42no—
185D17—41no—
186C22×Q8—21no—
187C22×D4—171no—
188C23×C4—171no—
189C23×C4—171no—
190C22×C8—681no—
191C4⋊Q8—41no—
192C4⋊Q8—41no—
193#124—682no—
194#124—682no—
195C4×D4—681no—
196C4×D4—681no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—171no—
199C2×C4⋊C4—341no—
200C2×C4⋊C4—681no—
201C2×C4⋊C4C2×#59512yes—
202C2×C22⋊C4—171no—
203C2×C22⋊C4—171no—
204C2×C22⋊C4—342no—
205C2×C22⋊C4—342no—
206C2×C22⋊C4—342no—
207C2×C42—341no—
208C2×C42C2×#59511yescharacteristic
209#99—682no—
210C22⋊C8—342no—
211C22⋊C8—681no—
212#95—172no—
213#95—342no—
214#95—342no—
215#95—681no—
216C17#1187811yescharacteristic, complemented, Sylow
217C24—171no—
218C24—171no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×D4—172no—
223C2×D4—172no—
224C2×D4—681no—
225C2×D4—681no—
226C22×C4—22no—
227C22×C4—171no—
228C22×C4—171no—
229C22×C4—341no—
230C22×C4—341no—
231C22×C4—341no—
232C22×C4—341no—
233C22×C4—342no—
234C22×C4—682no—
235C22×C4—682no—
236C22×C4—682no—
237C22×C4C22⋊#59511yescharacteristic
238C22×C4C22⋊#59511yescharacteristic
239C22×C4C22×#59511yescharacteristic
240C2×C8—682no—
241C2×C8—1362no—
242C4⋊C4—42no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—681no—
246C4⋊C4—682no—
247C4⋊C4—1362no—
248C22⋊C4—681no—
249C22⋊C4—681no—
250C22⋊C4—682no—
251C22⋊C4—684no—
252C22⋊C4—682no—
253C22⋊C4—682no—
254C22⋊C4—682no—
255C42—21no—
256C42—21no—
257C42—682no—
258C23—171no—
259C23—172no—
260C23—171no—
261C23—341no—
262C23—341no—
263C23—341no—
264C23—341no—
265C23—681no—
266C23—681no—
267C23—681no—
268C23—681no—
269C23C2×C22⋊#59511yescharacteristic
270Q8—84no—
271D4—682no—
272D4—682no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—341no—
282C2×C4—341no—
283C2×C4—682no—
284C2×C4—682no—
285C2×C4—681no—
286C2×C4—681no—
287C2×C4—682no—
288C2×C4—682no—
289C2×C4—682no—
290C2×C4—1361no—
291C2×C4—1361no—
292C2×C4—1362no—
293C2×C4—1362no—
294C2×C4—1362no—
295C2×C4#1058754712yes—
296C2×C4#1058755111yescharacteristic
297C2×C4#1058755111yescharacteristic
298C8—1362no—
299C22—21no—
300C22—21no—
301C22—341no—
302C22—342no—
303C22—341no—
304C22—681no—
305C22—682no—
306C22—681no—
307C22—681no—
308C22—681no—
309C22—681no—
310C22—681no—
311C22—1361no—
312C22—1361no—
313C22C2×#1058755111yescharacteristic
314C22C2×#1058754711yescharacteristic
315C22#1176380711yescharacteristic
316C4—22no—
317C4—21no—
318C4—41no—
319C4—82no—
320C4—2722no—
321C4—681no—
322C4—681no—
323C4—1362no—
324C4#1176371512yes—
325C2—21no—
326C2—21no—
327C2—681no—
328C2—681no—
329C2—682no—
330C2#1454422011yescharacteristic
331C2#1454504611yescharacteristic
332C2C2×#1176371511yescharacteristic
subgroupG/H
centerC22C2×#10587551
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC2×C17×C4⋊Q8C4
socleC22×C17C2×#99
prime radicalC22×D68D4
cosocleC2×#63227C23
hypercenterC2×C4⋊Q8#595
hypocenterC17#11878
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
3. #14543764C2semidirect
13. #11762735C4semidirect
18. C4×C4⋊Dic17C4semidirect
19. C2×C17×C4⋊Q8C4semidirect
163. C2×C4⋊Q8#595semidirect
216. C17#11878semidirect

Maximal subgroups

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

subgroupindex
#145446452
#145440212
#145437642
#145337802
#1187817