Finite Group Databasebeta

← #16932045

Subgroups — #16932045

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#14533270C211yesmaximal, characteristic
2#14531385C211yesmaximal, characteristic
3#14528862C211yesmaximal, characteristic
4#14527317C211yesmaximal, characteristic
5#14526638C211yesmaximal, characteristic
6#14526636C211yesmaximal, characteristic
7C17×#1511C211yesmaximal, characteristic
8C22×C4⋊Dic17C2211yescharacteristic
9#11762773C2211yescharacteristic
10#11762748—22no—
11#11762716C412yescomplemented
12#11762694—22no—
13#11762692—22no—
14#11762674C412yescomplemented
15#11762672C2211yescharacteristic
16#11762670—21no—
17#11762670—21no—
18#11762667—21no—
19C22⋊C4×Dic17—21no—
20C4×C22⋊Dic17C2211yescharacteristic
21#11762577—22no—
22#11762546—22no—
23C2×C17×C22⋊Q8C2211yescharacteristic
24C17×#407C2211yescharacteristic
25C17×#381C2211yescharacteristic
26C17×#328—22no—
27C22×Q8×C17C2×C411yescharacteristic
28C23×C4×C17C2311yescharacteristic
29C23×Dic17D411yescharacteristic
30C23×Dic17D411yescharacteristic
31C2×M16×C17—21no—
32C17×C22⋊Q8—21no—
33C17×C22⋊Q8—21no—
34C17×C22⋊Q8—21no—
35C17×C22⋊Q8—21no—
36C2×C4⋊C4×C17—21no—
37C2×C4⋊C4×C17C2×C411yescharacteristic
38C2×C22⋊C4×C17—21no—
39C2×C22⋊C4×C17—21no—
40C2×C22⋊Dic17—21no—
41C2×C22⋊Dic17—21no—
42C2×C4⋊Dic17—21no—
43C2×C4⋊Dic17—21no—
44C2×C4⋊Dic17—21no—
45C2×C4⋊Dic17—21no—
46C2×C4⋊Dic17C2×C412yes—
47C2×C4⋊Dic17D412yes—
48C2×#63225—22no—
49C2×C4×Dic17—22no—
50C2×C4×Dic17C2×C412yes—
51C2×#63223—21no—
52C17×C22⋊C8—21no—
53C17×C22⋊C8—21no—
54C17×#95—21no—
55C17×#95D412yes—
56#10587502—22no—
57#10587502—22no—
58#10587502—21no—
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
68C2×Q8×C17—21no—
69C2×Q8×C17—21no—
70C2×Q8×C17—42no—
71C22×C4×C17—21no—
72C22×C4×C17—21no—
73C22×C4×C17—22no—
74C22×C4×C17—42no—
75C22×C4×C17C22×C411yescharacteristic
76C22×C4×C17C2×D411yescharacteristic
77C22×C4×C17C2×D411yescharacteristic
78C22×C4×C17C2×D411yescharacteristic
79C22×C4×C17#4111yescharacteristic
80C22×C4×C17#4111yescharacteristic
81C22×Dic17—21no—
82C22×Dic17—21no—
83C22×Dic17—21no—
84C22×Dic17—21no—
85C22×Dic17—22no—
86C22×Dic17—22no—
87C22×Dic17—42no—
88C22×Dic17—42no—
89C22×Dic17C22⋊C412yes—
90C22×Dic17C22⋊C412yes—
91M16×C17—42no—
92C2×C8×C17—41no—
93C4⋊C4×C17—21no—
94C4⋊C4×C17—21no—
95C4⋊C4×C17—41no—
96C4⋊C4×C17—41no—
97C22⋊C4×C17—42no—
98C22⋊C4×C17—41no—
99C22⋊C4×C17—41no—
100C22⋊Dic17—42no—
101C22⋊Dic17—42no—
102C4⋊Dic17—22no—
103C4⋊Dic17—22no—
104C4⋊Dic17—22no—
105C4⋊Dic17—22no—
106#63225—82no—
107C4×Dic17—42no—
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
118Q8×C17—81no—
119Q8×C17—81no—
120C2×C4×C17—22no—
121C2×C4×C17—22no—
122C2×C4×C17—41no—
123C2×C4×C17—41no—
124C2×C4×C17—41no—
125C2×C4×C17—41no—
126C2×C4×C17—42no—
127C2×C4×C17—42no—
128C2×C4×C17—42no—
129C2×C4×C17—41no—
130C2×C4×C17—41no—
131C2×C4×C17—42no—
132C2×C4×C17—42no—
133C2×C4×C17C4×D411yescharacteristic
134C2×C4×C17C4×D411yescharacteristic
135C2×C4×C17#12111yescharacteristic
136C2×C4×C17#12111yescharacteristic
137C2×Dic17—22no—
138C2×Dic17—22no—
139C2×Dic17—42no—
140C2×Dic17—42no—
141C2×Dic17—42no—
142C2×Dic17—42no—
143C2×Dic17—42no—
144C2×Dic17—42no—
145C2×Dic17—42no—
146C2×Dic17—42no—
147C2×Dic17—42no—
148C2×Dic17—42no—
149C2×Dic17—81no—
150C2×Dic17—81no—
151C8×C17—81no—
152#3622—81no—
153#2503—171no—
154#2313—171no—
155#1821—171no—
156#1511—171no—
157#1511D1711yescharacteristic
158#1284—171no—
159#1284—171no—
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—81no—
177C4×C17—81no—
178Dic17—42no—
179Dic17—42no—
180Dic17—82no—
181Dic17—82no—
182C2×C22⋊Q8C2×D1711yescharacteristic
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—171no—
197C4×C22⋊C4—341no—
198#343—342no—
199#328—22no—
200#328—342no—
201C2×C17—22no—
202C2×C17—22no—
203C2×C17—41no—
204C2×C17#188511yescharacteristic
205C2×C17#189911yescharacteristic
206C2×C17#191211yescharacteristic
207C22×Q8C4×D1711yescharacteristic
208C23×C4—171no—
209C23×C4—171no—
210C23×C4C22×D1711yescharacteristic
211C2×M16—21no—
212C2×M16—341no—
213C22⋊Q8—21no—
214C22⋊Q8—21no—
215C22⋊Q8—21no—
216C22⋊Q8—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—172no—
219C2×C4⋊C4—172no—
220C2×C4⋊C4—341no—
221C2×C4⋊C4—341no—
222C2×C4⋊C4—341no—
223C2×C4⋊C4—341no—
224C2×C4⋊C4—342no—
225C2×C4⋊C4C4×D1711yescharacteristic
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—341no—
229C2×C22⋊C4—341no—
230C2×C42—172no—
231C2×C42—342no—
232C22⋊C8—21no—
233C22⋊C8—21no—
234C22⋊C8—341no—
235C22⋊C8—341no—
236#95—21no—
237#95—172no—
238#95—342no—
239#95—342no—
240#95—342no—
241#95—342no—
242#95—342no—
243#95—342no—
244#95—341no—
245#95—341no—
246#95C22⋊D1712yes—
247C17#946011yescharacteristic, complemented, Sylow
248C24D4×D1711yescharacteristic
249C2×Q8—21no—
250C2×Q8—21no—
251C2×Q8—42no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—42no—
256C22×C4—172no—
257C22×C4—172no—
258C22×C4—341no—
259C22×C4—341no—
260C22×C4—341no—
261C22×C4—341no—
262C22×C4—342no—
263C22×C4—342no—
264C22×C4—682no—
265C22×C4—682no—
266C22×C4C2×C4×D1711yescharacteristic
267C22×C4#6325211yescharacteristic
268C22×C4D4×D1711yescharacteristic
269C22×C4D4×D1711yescharacteristic
270C22×C4#6325411yescharacteristic
271C22×C4C2×C22⋊D1711yescharacteristic
272M16—42no—
273M16—682no—
274C2×C8—41no—
275C2×C8—681no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—342no—
281C4⋊C4—342no—
282C4⋊C4—342no—
283C4⋊C4—342no—
284C4⋊C4—1362no—
285C22⋊C4—42no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—682no—
289C22⋊C4—682no—
290C42—682no—
291C42—1362no—
292C23—22no—
293C23—22no—
294C23—41no—
295C23C22⋊C4×D1711yescharacteristic
296C23#1058756911yescharacteristic
297C23#1058762311yescharacteristic
298Q8—81no—
299Q8—81no—
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—41no—
312C2×C4—41no—
313C2×C4—342no—
314C2×C4—342no—
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—682no—
324C2×C4—682no—
325C2×C4—1361no—
326C2×C4—1361no—
327C2×C4#1058756711yescharacteristic
328C2×C4#1058757011yescharacteristic
329C2×C4C4×C22⋊D1711yescharacteristic
330C2×C4#1058762411yescharacteristic
331C8—81no—
332C8—1361no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—44no—
341C22#1176268511yescharacteristic
342C22#1176278112yes—
343C22#1176298512yes—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—82no—
349C4—81no—
350C4—81no—
351C4—682no—
352C4—682no—
353C4—1362no—
354C4—1362no—
355C2—22no—
356C2—22no—
357C2—41no—
358C2#1452916911yescharacteristic
359C2#1452923911yescharacteristic
360C2#1452931811yescharacteristic
subgroupG/H
centerC22#11762685
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#1511C2
socleC22×C17#386
prime radicalC24×C17C2×D4
cosocleC23×C4×C17C23
hypercenter#1511D17
hypocenterC17#9460
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1511

Complements

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

NH ≅ G/Ntype
11. #11762716C4semidirect
14. #11762674C4semidirect
247. C17#9460semidirect

Maximal subgroups

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

subgroupindex
#145332702
#145313852
#145288622
#145273172
#145266382
#145266362
C17×#15112
#946017