Finite Group Databasebeta

← #16932076

Subgroups — #16932076

Order 4352.

726 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 107 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#14532438C211yesmaximal, characteristic
2#14531370C211yesmaximal, characteristic, complemented
3#14527314C212yesmaximal
4#14526638C212yesmaximal, complemented
5C17×#1784C211yesmaximal, characteristic, complemented
6C2×#10587633C412yescomplemented
7C2×#10587616C2211yescharacteristic
8C2×#10587612—21no—
9#11762720—22no—
10#11762716—24no—
11#11762716C412yescomplemented
12#11762688—21no—
13#11762688—21no—
14#11762672—21no—
15#11762672C2212yes—
16C4×C22⋊Dic17C2211yescharacteristic
17#11762544—24no—
18C23×Q8×C17C2211yescharacteristic
19C2×C17×#101—22no—
20C17×#407C2212yes—
21C22×Q8×C17—22no—
22C22×Q8×C17—22no—
23C22×Q8×C17—44no—
24C22×Q8×C17C2×C412yes—
25C22×Q8×C17D412yes—
26C23×C4×C17D411yescharacteristic
27C23×C4×C17D411yescharacteristic
28C23×C4×C17C2311yescharacteristic
29C22×C22⋊D17—21no—
30C22×D68C2×C411yescharacteristic
31C2×M16×C17—22no—
32C2×C22⋊Dic17—21no—
33C2×C22⋊Dic17—22no—
34#10587633—44no—
35#10587616—22no—
36#10587616—22no—
37C2×#63227—22no—
38C2×#63227—22no—
39C2×#63227—21no—
40#10587612—44no—
41C2×C4⋊Dic17—22no—
42C2×C4⋊Dic17C2×C411yescharacteristic
43C2×C4×Dic17—22no—
44C2×C4×Dic17C2×C412yes—
45C17×#101—44no—
46C17×C22⋊C8—22no—
47C17×C22⋊C8—22no—
48#10587502—24no—
49#10587502—22no—
50#10587502—22no—
51#10587502—22no—
52#10587502—21no—
53#10587502D414yes—
54C24×C17C2×D411yescharacteristic
55C23×D17—21no—
56C2×Q8×C17—24no—
57C2×Q8×C17—24no—
58C2×Q8×C17—24no—
59C2×Q8×C17—24no—
60C2×Q8×C17—42no—
61C2×Q8×C17—42no—
62C2×Q8×C17—48no—
63C2×Q8×C17—44no—
64C2×Q8×C17—44no—
65C22×C4×C17—21no—
66C22×C4×C17—21no—
67C22×C4×C17—24no—
68C22×C4×C17—21no—
69C22×C4×C17—21no—
70C22×C4×C17—42no—
71C22×C4×C17—42no—
72C22×C4×C17C22⋊C412yes—
73C22×C4×C17C22⋊C412yes—
74C22×C4×C17C22×C411yescharacteristic
75C22×C4×C17C2×D412yes—
76C22×C4×C17C2×D411yescharacteristic
77C22×C4×C17#4112yes—
78C2×C22⋊D17—44no—
79C2×C22⋊D17—41no—
80C2×C22⋊D17—41no—
81C22×Dic17—21no—
82C22×Dic17—22no—
83C22×Dic17—24no—
84C2×D68—22no—
85C2×D68—42no—
86M16×C17—44no—
87C2×C8×C17—42no—
88C22⋊Dic17—44no—
89C22⋊Dic17—42no—
90#63227—44no—
91#63227—44no—
92#63227—42no—
93C4⋊Dic17—22no—
94C4⋊Dic17—44no—
95C4×Dic17—44no—
96C4×Dic17—42no—
97#9466—171nomaximal, Sylow
98C23×C17—22no—
99C23×C17—22no—
100C23×C17—41no—
101C23×C17C2×C22⋊C411yescharacteristic
102C23×C17#12011yescharacteristic
103C23×C17#12311yescharacteristic
104C22×D17—42no—
105C22×D17—42no—
106C22×D17—41no—
107C22×D17—42no—
108Q8×C17—48no—
109Q8×C17—48no—
110C2×C4×C17—24no—
111C2×C4×C17—22no—
112C2×C4×C17—22no—
113C2×C4×C17—22no—
114C2×C4×C17—22no—
115C2×C4×C17—22no—
116C2×C4×C17—22no—
117C2×C4×C17—22no—
118C2×C4×C17—22no—
119C2×C4×C17—22no—
120C2×C4×C17—42no—
121C2×C4×C17—42no—
122C2×C4×C17—42no—
123C2×C4×C17—42no—
124C2×C4×C17#9914yes—
125C2×C4×C17C4×D412yes—
126C2×C4×C17#12112yes—
127C22⋊D17—82no—
128C22⋊D17—82no—
129C2×Dic17—42no—
130C2×Dic17—44no—
131C2×Dic17—44no—
132C2×Dic17—42no—
133C2×Dic17—42no—
134C2×Dic17—41no—
135C2×Dic17—42no—
136C2×Dic17—44no—
137D68—82no—
138C8×C17—82no—
139#2436—171no—
140#2310—171no—
141#1784D1711yescharacteristic, complemented
142#1510—172no—
143#1284—172no—
144C22×C17—21no—
145C22×C17—21no—
146C22×C17—21no—
147C22×C17—21no—
148C22×C17—21no—
149C22×C17—21no—
150C22×C17—21no—
151C22×C17—44no—
152C22×C17#38611yescharacteristic
153C22×C17C2×#9912yes—
154C22×C17#45412yes—
155C2×D17—84no—
156C2×D17—81no—
157C2×D17—81no—
158C2×D17—81no—
159C4×C17—22no—
160C4×C17—22no—
161C4×C17—24no—
162C4×C17—42no—
163Dic17—84no—
164Dic17—82no—
165Dic17—81no—
166C23×Q8C2×D1711yescharacteristic
167C2×#124—172no—
168C2×#121—171no—
169C2×#117—341no—
170C2×#101—22no—
171#407C2×D1712yes—
172#393—342no—
173#391—172no—
174#391—344no—
175#386—341no—
176#386—341no—
177#381—172no—
178#381—341no—
179C4×C22⋊C4—171no—
180#327—344no—
181C2×C17—22no—
182C2×C17—22no—
183C2×C17—41no—
184C2×C17#188811yescharacteristic
185C2×C17#189811yescharacteristic
186C2×C17#191211yescharacteristic
187D17—161no—
188C22×Q8—22no—
189C22×Q8—22no—
190C22×Q8—44no—
191C22×Q8C4×D1712yes—
192C22×Q8C22⋊D1712yes—
193C22×D4—171no—
194C22×D4—341no—
195C23×C4D6811yescharacteristic
196C23×C4D6811yescharacteristic
197C23×C4C22×D1711yescharacteristic
198C2×M16—22no—
199#124—684no—
200#121—342no—
201#121—342no—
202#117—684no—
203C2×C4⋊C4—171no—
204C2×C4⋊C4—342no—
205C2×C22⋊C4—342no—
206C2×C22⋊C4—341no—
207C2×C22⋊C4—341no—
208C2×C22⋊C4—342no—
209C2×C22⋊C4—342no—
210C2×C42—172no—
211C2×C42—342no—
212#101—44no—
213C22⋊C8—22no—
214C22⋊C8—22no—
215#95—174no—
216#95—344no—
217#95—342no—
218#95—341no—
219#95—342no—
220#95—342no—
221C17#946611yescharacteristic, complemented, Sylow
222C24—341no—
223C24C2×D6811yescharacteristic
224C2×Q8—24no—
225C2×Q8—24no—
226C2×Q8—24no—
227C2×Q8—24no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—44no—
231C2×Q8—44no—
232C2×Q8—48no—
233C2×D4—342no—
234C2×D4—682no—
235C2×D4—684no—
236C2×D4—681no—
237C2×D4—681no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—24no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—344no—
246C22×C4—341no—
247C22×C4—342no—
248C22×C4#6322712yes—
249C22×C4#6322712yes—
250C22×C4C2×C4×D1711yescharacteristic
251C22×C4D4×D1712yes—
252C22×C4#6325412yes—
253C22×C4C2×C22⋊D1711yescharacteristic
254M16—44no—
255C2×C8—42no—
256C4⋊C4—342no—
257C4⋊C4—684no—
258C22⋊C4—684no—
259C22⋊C4—682no—
260C22⋊C4—684no—
261C22⋊C4—684no—
262C22⋊C4—682no—
263C42—682no—
264C42—684no—
265C23—22no—
266C23—22no—
267C23—41no—
268C23—682no—
269C23—682no—
270C23—682no—
271C23—681no—
272C23#1058756811yescharacteristic
273C23#1058757211yescharacteristic
274C23C2×#6322711yescharacteristic
275Q8—48no—
276Q8—48no—
277D4—1362no—
278D4—1362no—
279D4—1362no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—24no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—684no—
295C2×C4—682no—
296C2×C4—682no—
297C2×C4—684no—
298C2×C4—682no—
299C2×C4—684no—
300C2×C4—682no—
301C2×C4—681no—
302C2×C4#1058747714yes—
303C2×C4#1058756712yes—
304C2×C4#1058762512yes—
305C8—82no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—44no—
314C22—1364no—
315C22—1361no—
316C22—1361no—
317C22—1361no—
318C22#1176268611yescharacteristic
319C22C2×#1058747712yes—
320C22#1176298412yes—
321C4—22no—
322C4—24no—
323C4—22no—
324C4—42no—
325C4—1364no—
326C4—1362no—
327C4—1361no—
328C2—22no—
329C2—22no—
330C2—41no—
331C2—2721no—
332C2#1452918511yescharacteristic
333C2#1452923511yescharacteristic
334C2#1452931911yescharacteristic
subgroupG/H
centerC22#11762686
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#1784C2
socleC22×C17#386
prime radicalC2×#10587616C22
cosocleC23×C4×C17C23
hypercenter#1784D17
hypocenterC17#9466
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1784

Complements

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

NH ≅ G/Ntype
2. #14531370C2semidirect
4. #14526638C2semidirect
5. C17×#1784C2semidirect
6. C2×#10587633C4semidirect
11. #11762716C4semidirect
141. #1784D17semidirect
221. C17#9466semidirect

Maximal subgroups

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

subgroupindex
#145324382
#145313702
#145273142
#145266382
C17×#17842
#946617