Finite Group Databasebeta

← #17959698

Subgroups — #17959698

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#14544644C211yesmaximal, characteristic
2#14544023C211yesmaximal, characteristic
3#14543765C214yesmaximal, complemented
4#14533093C211yesmaximal, characteristic
5#11763826—21no—
6C2×#10587551—21no—
7#11763751—21no—
8#11763687C2212yes—
9#11763653C2212yes—
10#11763652—21no—
11C2×#10587572—21no—
12C2×C4×C22⋊D17C2211yescharacteristic
13C22×C4⋊Dic17C411yescharacteristic, complemented
14#11762741C412yescomplemented
15#11762736C2212yes—
16#11762708—21no—
17#11762670—21no—
18#11762666—21no—
19C17×#380C411yescharacteristic, complemented
20C23×C4×C17C2×C411yescharacteristic
21C22×C22⋊D17D411yescharacteristic
22C23×Dic17—21no—
23C22×C4×D17D411yescharacteristic
24C2×C22⋊#595—22no—
25C2×C4⋊#595—41no—
26C2×C4⋊C4×C17—21no—
27C2×C22⋊C4×C17—21no—
28C2×C22⋊Dic17D411yescharacteristic
29C4×C22⋊D17—41no—
30C4×C22⋊D17—41no—
31C2×#63227—22no—
32C2×#63227C2311yescharacteristic
33C2×C4⋊Dic17—21no—
34C2×C4⋊Dic17—22no—
35C2×C4⋊Dic17—41no—
36C2×C4⋊Dic17—41no—
37C2×C4⋊Dic17C2×C412yes—
38C2×#63225C2×C411yescharacteristic
39C2×C4×Dic17D411yescharacteristic
40#10587572—42no—
41#10587572—42no—
42C22⋊#3624—22no—
43#10587551—42no—
44(C2×#595)⋊C4—41no—
45#10587544—41no—
46C4×#3624—41no—
47C17×#95C2×C412yes—
48#10587502—22no—
49#10587502—22no—
50C24×C17C22⋊C411yescharacteristic
51C23×D17C2×D411yescharacteristic
52C22×#595—42no—
53C22×C4×C17—21no—
54C22×C4×C17—21no—
55C22×C4×C17—22no—
56C22×C4×C17—42no—
57C22×C4×C17C22⋊C411yescharacteristic
58C22×C4×C17C22×C411yescharacteristic
59C2×C22⋊D17—41no—
60C2×C22⋊D17—41no—
61C2×C22⋊D17D812yes—
62C2×C22⋊D17SD1612yes—
63C22×Dic17—22no—
64C22×Dic17—42no—
65C22×Dic17—42no—
66C22×Dic17—42no—
67C22×Dic17C22⋊C411yescharacteristic
68C22×Dic17C22⋊C411yescharacteristic
69C22×Dic17C2×D411yescharacteristic
70C2×C4×D17—21no—
71C2×C4×D17—21no—
72C2×C4×D17—42no—
73C22⋊#595—42no—
74C22⋊#595—42no—
75C2×#3624—42no—
76C4⋊#595—82no—
77C4⋊C4×C17—42no—
78C22⋊C4×C17—42no—
79C22⋊Dic17—41no—
80#63227—41no—
81#63227—42no—
82#63227—42no—
83C4⋊Dic17—41no—
84C4⋊Dic17—41no—
85C4⋊Dic17—44no—
86C4⋊Dic17—42no—
87#63225—41no—
88C4×Dic17—21no—
89C4×Dic17—21no—
90#11874—171nomaximal, Sylow
91C23×C17—21no—
92C23×C17—21no—
93C23×C17—41no—
94C23×C17—41no—
95C23×C17#10212yes—
96C23×C17C2×C22⋊C411yescharacteristic
97C22×D17—21no—
98C22×D17—21no—
99C22×D17—41no—
100C22×D17—41no—
101C22×D17C2×D811yescharacteristic
102C22×D17C2×SD1611yescharacteristic
103C2×#595—82no—
104C2×#595—82no—
105C2×#595—82no—
106C2×C4×C17—22no—
107C2×C4×C17—21no—
108C2×C4×C17—42no—
109C2×C4×C17—42no—
110C2×C4×C17—41no—
111C2×C4×C17—42no—
112C2×C4×C17—42no—
113C2×C4×C17—42no—
114C2×C4×C17#9911yescharacteristic
115C2×C4×C17#9911yescharacteristic
116C22⋊D17—42no—
117C22⋊D17—42no—
118C2×Dic17—21no—
119C2×Dic17—21no—
120C2×Dic17—21no—
121C2×Dic17—21no—
122C2×Dic17—41no—
123C2×Dic17—41no—
124C2×Dic17—42no—
125C2×Dic17—42no—
126C2×Dic17—42no—
127C2×Dic17—84no—
128C2×Dic17#10212yes—
129C4×D17—42no—
130C4×D17—81no—
131#3624—82no—
132#2486—171no—
133#1799—171no—
134#1460—171no—
135#1344—174no—
136C22×C17—21no—
137C22×C17—21no—
138C22×C17—22no—
139C22×C17—42no—
140C22×C17—42no—
141C22×C17—41no—
142C22×C17—41no—
143C22×C17C2×#9911yescharacteristic
144C22×C17C2×#10211yescharacteristic
145C22×C17#42111yescharacteristic
146C2×D17—21no—
147C2×D17—21no—
148C2×D17—41no—
149C2×D17—41no—
150C2×D17—41no—
151C2×D17—41no—
152C2×D17—81no—
153#595—162no—
154C4×C17—21no—
155C4×C17—41no—
156C4×C17—82no—
157C4×C17#35212yes—
158Dic17—22no—
159Dic17—41no—
160Dic17—41no—
161Dic17—82no—
162C2×#123—341no—
163C2×C4×D4—171no—
164C22×C4⋊C4—171no—
165#431—341no—
166C2×#99—341no—
167#398—172no—
168#397—172no—
169#389—341no—
170#381—341no—
171#380—341no—
172#380—341no—
173#380#59511yescharacteristic, complemented
174#342—172no—
175#323—172no—
176#323—341no—
177C2×C17—21no—
178C2×C17—21no—
179C2×C17—42no—
180C2×C17#151911yescharacteristic
181C2×C17C2×#35211yescharacteristic
182C2×C17#212911yescharacteristic
183D17—41no—
184D17—41no—
185C22×D4—171no—
186C23×C4—171no—
187C23×C4—341no—
188C23×C4C2×#59511yescharacteristic
189#123—682no—
190#123—682no—
191C4×D4—681no—
192C4×D4—681no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—171no—
195C2×C4⋊C4—172no—
196C2×C4⋊C4—342no—
197C2×C4⋊C4—341no—
198C2×C4⋊C4—681no—
199C2×C4⋊C4—681no—
200C2×C4⋊C4—681no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—171no—
203C2×C22⋊C4—171no—
204C2×C22⋊C4—342no—
205C2×C22⋊C4—342no—
206C2×C42—171no—
207C4⋊C8—681no—
208#99—682no—
209C22⋊C8—342no—
210C4×C8—681no—
211#95—342no—
212#95—342no—
213#95—681no—
214#95C2×#59512yes—
215C17#1187411yescharacteristic, complemented, Sylow
216C24—171no—
217C24C22⋊#59511yescharacteristic
218C2×D4—172no—
219C2×D4—172no—
220C2×D4—681no—
221C2×D4—681no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—42no—
226C22×C4—171no—
227C22×C4—171no—
228C22×C4—171no—
229C22×C4—341no—
230C22×C4—341no—
231C22×C4—342no—
232C22×C4—682no—
233C22×C4—682no—
234C22×C4—682no—
235C22×C4—682no—
236C22×C4—682no—
237C22×C4C22⋊#59511yescharacteristic
238C22×C4C22×#59511yescharacteristic
239C2×C8—682no—
240C4⋊C4—42no—
241C4⋊C4—681no—
242C4⋊C4—682no—
243C4⋊C4—684no—
244C4⋊C4—681no—
245C4⋊C4—681no—
246C4⋊C4—1362no—
247C22⋊C4—42no—
248C22⋊C4—681no—
249C22⋊C4—682no—
250C22⋊C4—682no—
251C22⋊C4—681no—
252C22⋊C4—682no—
253C22⋊C4—682no—
254C42—341no—
255C42—341no—
256C23—21no—
257C23—21no—
258C23—41no—
259C23—41no—
260C23—171no—
261C23—171no—
262C23—341no—
263C23—341no—
264C23—681no—
265C23—681no—
266C23#1058754712yes—
267C23C2×C22⋊#59511yescharacteristic
268D4—682no—
269D4—682no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—172no—
279C2×C4—341no—
280C2×C4—341no—
281C2×C4—341no—
282C2×C4—341no—
283C2×C4—682no—
284C2×C4—681no—
285C2×C4—681no—
286C2×C4—682no—
287C2×C4—682no—
288C2×C4—682no—
289C2×C4—1361no—
290C2×C4—1362no—
291C2×C4—1362no—
292C2×C4—1362no—
293C2×C4—1364no—
294C2×C4#1058753911yescharacteristic
295C2×C4#1058753911yescharacteristic
296C8—1362no—
297C22—21no—
298C22—21no—
299C22—22no—
300C22—42no—
301C22—42no—
302C22—41no—
303C22—41no—
304C22—341no—
305C22—341no—
306C22—681no—
307C22—681no—
308C22—681no—
309C22—681no—
310C22—1361no—
311C22C2×#1058753911yescharacteristic
312C22C2×#1058754711yescharacteristic
313C22#1176380811yescharacteristic
314C4—21no—
315C4—41no—
316C4—82no—
317C4—2722no—
318C4—342no—
319C4—681no—
320C4—681no—
321C4—1362no—
322C4#1176371212yes—
323C2—21no—
324C2—21no—
325C2—42no—
326C2—681no—
327C2—681no—
328C2#1454422111yescharacteristic
329C2C2×#1176371211yescharacteristic
330C2#1454506811yescharacteristic
subgroupG/H
centerC22C2×#10587539
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC17×#380C4
socleC22×C17C2×#99
prime radicalC22×C22⋊D17D4
cosocleC2×#63227C23
hypercenter#380#595
hypocenterC17#11874
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
3. #14543765C2semidirect
13. C22×C4⋊Dic17C4semidirect
14. #11762741C4semidirect
19. C17×#380C4semidirect
173. #380#595semidirect
215. C17#11874semidirect

Maximal subgroups

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

subgroupindex
#145446442
#145440232
#145437652
#145330932
#1187417