Finite Group Databasebeta

← #16932047

Subgroups — #16932047

Order 4352.

512 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 93 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#14533269C211yesmaximal, characteristic
2#14531385C211yesmaximal, characteristic
3#14528862C211yesmaximal, characteristic
4#14527318C211yesmaximal, characteristic
5#14527316C211yesmaximal, characteristic
6#14526637C211yesmaximal, characteristic
7C17×#1284C211yesmaximal, characteristic
8C2×#10587610C2211yescharacteristic
9C22×C4⋊Dic17C2211yescharacteristic
10#11762773C2211yescharacteristic
11#11762749—22no—
12#11762715C412yescomplemented
13#11762694—22no—
14#11762692—22no—
15#11762675C412yescomplemented
16#11762672C2211yescharacteristic
17#11762670—21no—
18#11762670—21no—
19#11762667—21no—
20C22⋊C4×Dic17—21no—
21#11762577—22no—
22#11762547—22no—
23#11762545—22no—
24C17×#407C2211yescharacteristic
25C17×#381C2211yescharacteristic
26C4×C22⋊C4×C17C2211yescharacteristic
27C23×C4×C17C2311yescharacteristic
28C23×Dic17D411yescharacteristic
29C23×Dic17D411yescharacteristic
30C22×Dic34C2×C411yescharacteristic
31C2×M16×C17—21no—
32C2×C22⋊C4×C17—21no—
33C2×C22⋊C4×C17—21no—
34C2×C42×C17C2×C412yes—
35C2×C22⋊Dic17—21no—
36C2×C22⋊Dic17—21no—
37C2×C4⋊Dic17—21no—
38C2×C4⋊Dic17—21no—
39C2×C4⋊Dic17—21no—
40C2×C4⋊Dic17—21no—
41C2×C4⋊Dic17C2×C411yescharacteristic
42C2×C4⋊Dic17C2×C412yes—
43C2×C4⋊Dic17D412yes—
44#10587610—21no—
45#10587610—21no—
46#10587610—21no—
47#10587610—21no—
48C2×#63225—22no—
49C2×#63225—21no—
50C2×C4×Dic17—22no—
51C2×#63223—21no—
52C17×C22⋊C8—21no—
53C17×C22⋊C8—21no—
54C17×#95—21no—
55C17×#95—21no—
56C17×#95D412yes—
57#10587502—22no—
58#10587502—22no—
59#10587502—21no—
60#10587502—22no—
61#10587502—22no—
62#10587502—22no—
63#10587502—22no—
64#10587502D412yes—
65C22⋊#3622—21no—
66C22⋊#3622—21no—
67C24×C17C2×D411yescharacteristic
68C22×C4×C17—22no—
69C22×C4×C17—22no—
70C22×C4×C17—42no—
71C22×C4×C17C22×C411yescharacteristic
72C22×C4×C17C2×D411yescharacteristic
73C22×C4×C17C2×D411yescharacteristic
74C22×C4×C17C2×D411yescharacteristic
75C22×C4×C17#4111yescharacteristic
76C22×C4×C17#4111yescharacteristic
77C22×Dic17—21no—
78C22×Dic17—21no—
79C22×Dic17—21no—
80C22×Dic17—21no—
81C22×Dic17—21no—
82C22×Dic17—21no—
83C22×Dic17—22no—
84C22×Dic17—42no—
85C22×Dic17—42no—
86C22×Dic17C22⋊C412yes—
87C22×Dic17C22⋊C412yes—
88C2×Dic34—21no—
89C2×Dic34—21no—
90C2×Dic34—42no—
91M16×C17—42no—
92C2×C8×C17—41no—
93C22⋊C4×C17—42no—
94C22⋊C4×C17—42no—
95C42×C17—42no—
96C22⋊Dic17—42no—
97C22⋊Dic17—41no—
98C22⋊Dic17—41no—
99C4⋊Dic17—22no—
100C4⋊Dic17—22no—
101C4⋊Dic17—22no—
102C4⋊Dic17—22no—
103C4⋊Dic17—21no—
104C4⋊Dic17—21no—
105#63225—41no—
106#63225—41no—
107#63225—82no—
108C4×Dic17—82no—
109#63223—42no—
110C2×#3622—41no—
111#9460—171nomaximal, Sylow
112C23×C17—22no—
113C23×C17—22no—
114C23×C17—41no—
115C23×C17C2×C22⋊C411yescharacteristic
116C23×C17#12011yescharacteristic
117C23×C17#12311yescharacteristic
118C2×C4×C17—22no—
119C2×C4×C17—22no—
120C2×C4×C17—41no—
121C2×C4×C17—41no—
122C2×C4×C17—41no—
123C2×C4×C17—41no—
124C2×C4×C17—42no—
125C2×C4×C17—42no—
126C2×C4×C17—42no—
127C2×C4×C17—42no—
128C2×C4×C17—42no—
129C2×C4×C17—42no—
130C2×C4×C17C4×D411yescharacteristic
131C2×C4×C17C4×D411yescharacteristic
132C2×C4×C17#12111yescharacteristic
133C2×C4×C17#12111yescharacteristic
134C2×Dic17—22no—
135C2×Dic17—22no—
136C2×Dic17—42no—
137C2×Dic17—42no—
138C2×Dic17—42no—
139C2×Dic17—42no—
140C2×Dic17—42no—
141C2×Dic17—42no—
142C2×Dic17—42no—
143C2×Dic17—41no—
144C2×Dic17—41no—
145C2×Dic17—42no—
146C2×Dic17—42no—
147C2×Dic17—81no—
148C2×Dic17—81no—
149Dic34—81no—
150Dic34—81no—
151C8×C17—81no—
152#3622—81no—
153#2503—171no—
154#2313—171no—
155#1821—171no—
156#1511—171no—
157#1511—171no—
158#1284—171no—
159#1284D1711yescharacteristic
160C22×C17—21no—
161C22×C17—21no—
162C22×C17—21no—
163C22×C17—21no—
164C22×C17—21no—
165C22×C17—21no—
166C22×C17—21no—
167C22×C17—44no—
168C22×C17#38611yescharacteristic
169C22×C17#41012yes—
170C22×C17#45512yes—
171C4×C17—21no—
172C4×C17—21no—
173C4×C17—21no—
174C4×C17—21no—
175C4×C17—82no—
176C4×C17—82no—
177Dic17—42no—
178Dic17—42no—
179Dic17—82no—
180Dic17—81no—
181Dic17—81no—
182C2×C22⋊Q8—171no—
183C22×C4⋊C4—171no—
184#407—171no—
185#407C2×D1711yescharacteristic
186#400—342no—
187#391—172no—
188#387—342no—
189#387—342no—
190#382—172no—
191#381—171no—
192#381—341no—
193#381—341no—
194#381C2×D1711yescharacteristic
195#380—341no—
196C4×C22⋊C4—341no—
197C4×C22⋊C4C2×D1711yescharacteristic
198#343—342no—
199#328—342no—
200#328—342no—
201C2×C17—22no—
202C2×C17—22no—
203C2×C17—41no—
204C2×C17#188511yescharacteristic
205C2×C17#189911yescharacteristic
206C2×C17#191211yescharacteristic
207C22×Q8—171no—
208C23×C4—171no—
209C23×C4—171no—
210C23×C4C22×D1711yescharacteristic
211C2×M16—21no—
212C2×M16—341no—
213C22⋊Q8—341no—
214C22⋊Q8—341no—
215C22⋊Q8—341no—
216C22⋊Q8—341no—
217C2×C4⋊C4—172no—
218C2×C4⋊C4—172no—
219C2×C4⋊C4—171no—
220C2×C4⋊C4—341no—
221C2×C4⋊C4—341no—
222C2×C4⋊C4—341no—
223C2×C4⋊C4—341no—
224C2×C4⋊C4—341no—
225C2×C4⋊C4—342no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—341no—
229C2×C22⋊C4—341no—
230C2×C42—342no—
231C2×C42C4×D1712yes—
232C22⋊C8—21no—
233C22⋊C8—21no—
234C22⋊C8—341no—
235C22⋊C8—341no—
236#95—21no—
237#95—21no—
238#95—172no—
239#95—342no—
240#95—342no—
241#95—342no—
242#95—342no—
243#95—342no—
244#95—342no—
245#95—341no—
246#95D6812yes—
247C17#946011yescharacteristic, complemented, Sylow
248C24D4×D1711yescharacteristic
249C2×Q8—341no—
250C2×Q8—341no—
251C2×Q8—682no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—42no—
255C22×C4—172no—
256C22×C4—172no—
257C22×C4—341no—
258C22×C4—341no—
259C22×C4—341no—
260C22×C4—341no—
261C22×C4—341no—
262C22×C4—341no—
263C22×C4—342no—
264C22×C4—682no—
265C22×C4—682no—
266C22×C4C2×C4×D1711yescharacteristic
267C22×C4C2×D6811yescharacteristic
268C22×C4#6325211yescharacteristic
269C22×C4D4×D1711yescharacteristic
270C22×C4D4×D1711yescharacteristic
271C22×C4#6325611yescharacteristic
272M16—42no—
273M16—682no—
274C2×C8—41no—
275C2×C8—681no—
276C4⋊C4—342no—
277C4⋊C4—342no—
278C4⋊C4—342no—
279C4⋊C4—342no—
280C4⋊C4—341no—
281C4⋊C4—341no—
282C4⋊C4—681no—
283C4⋊C4—681no—
284C4⋊C4—1362no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C22⋊C4—682no—
288C22⋊C4—681no—
289C22⋊C4—681no—
290C42—42no—
291C42—1362no—
292C23—22no—
293C23—22no—
294C23—41no—
295C23C22⋊C4×D1711yescharacteristic
296C23#1058756811yescharacteristic
297C23#1058758011yescharacteristic
298Q8—1361no—
299Q8—1361no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—342no—
313C2×C4—342no—
314C2×C4—682no—
315C2×C4—682no—
316C2×C4—682no—
317C2×C4—682no—
318C2×C4—682no—
319C2×C4—682no—
320C2×C4—682no—
321C2×C4—682no—
322C2×C4—682no—
323C2×C4—681no—
324C2×C4—681no—
325C2×C4—1361no—
326C2×C4—1361no—
327C2×C4C4×D6811yescharacteristic
328C2×C4#1058757011yescharacteristic
329C2×C4#1058757911yescharacteristic
330C2×C4#1058758111yescharacteristic
331C8—81no—
332C8—1361no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—44no—
341C22#1176268711yescharacteristic
342C22#1176278112yes—
343C22#1176298812yes—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—82no—
349C4—82no—
350C4—682no—
351C4—682no—
352C4—1362no—
353C4—1361no—
354C4—1361no—
355C2—22no—
356C2—22no—
357C2—41no—
358C2#1452917111yescharacteristic
359C2#1452924111yescharacteristic
360C2#1452932011yescharacteristic
subgroupG/H
centerC22#11762687
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#1284C2
socleC22×C17#386
prime radicalC24×C17C2×D4
cosocleC23×C4×C17C23
hypercenter#1284D17
hypocenterC17#9460
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1284

Complements

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

NH ≅ G/Ntype
12. #11762715C4semidirect
15. #11762675C4semidirect
247. C17#9460semidirect

Maximal subgroups

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

subgroupindex
#145332692
#145313852
#145288622
#145273182
#145273162
#145266372
C17×#12842
#946017