Finite Group Databasebeta

← #16931781

Subgroups — #16931781

Order 4352.

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

normal subgroups only

This group has 526 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#14534165C211yesmaximal, characteristic
2#14531384C211yesmaximal, characteristic
3#14527304C211yesmaximal, characteristic
4#14527113C211yesmaximal, characteristic
5#14526670C211yesmaximal, characteristic
6#14526669C211yesmaximal, characteristic
7C17×#1508C211yesmaximal, characteristic
8C22×#63225C2211yescharacteristic
9#11762773C2211yescharacteristic
10#11762740C412yescomplemented
11#11762728C412yescomplemented
12#11762704—22no—
13#11762693—22no—
14#11762691—22no—
15#11762681—21no—
16#11762681—21no—
17#11762670—21no—
18#11762667—21no—
19C4×C22⋊Dic17C2211yescharacteristic
20C4×C22⋊Dic17C2211yescharacteristic
21#11762546—22no—
22#11762534—22no—
23C2×C17×C22⋊Q8C2211yescharacteristic
24C17×#407C2211yescharacteristic
25C4×C22⋊C4×C17C2211yescharacteristic
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×C42×C17—21no—
41C2×C22⋊Dic17—21no—
42C2×C22⋊Dic17—21no—
43C2×#63225—22no—
44C2×#63225—21no—
45C2×#63225—21no—
46C2×#63225—21no—
47C2×#63225—21no—
48C2×#63225—22no—
49C2×#63225—22no—
50C2×#63225C2×C412yes—
51C2×#63225D412yes—
52C2×C4×Dic17—21no—
53C2×C4×Dic17—21no—
54C2×#63223—21no—
55C17×C22⋊C8—21no—
56C17×C22⋊C8—21no—
57C17×#95D412yes—
58#10587502—22no—
59#10587502—24no—
60#10587502—22no—
61#10587502—22no—
62#10587502C2×C412yes—
63#10587502D412yes—
64C22⋊#3622—21no—
65C22⋊#3622—21no—
66C24×C17C2×D411yescharacteristic
67C2×Q8×C17—21no—
68C2×Q8×C17—21no—
69C2×Q8×C17—81no—
70C22×C4×C17—21no—
71C22×C4×C17—21no—
72C22×C4×C17—22no—
73C22×C4×C17—81no—
74C22×C4×C17C22×C411yescharacteristic
75C22×C4×C17C2×D411yescharacteristic
76C22×C4×C17C2×D411yescharacteristic
77C22×C4×C17C2×D411yescharacteristic
78C22×C4×C17#4111yescharacteristic
79C22×C4×C17#4111yescharacteristic
80C22×Dic17—22no—
81C22×Dic17—21no—
82C22×Dic17—21no—
83C22×Dic17—21no—
84C22×Dic17—21no—
85C22×Dic17—22no—
86C22×Dic17—42no—
87C22×Dic17—81no—
88C22×Dic17C22⋊C412yes—
89C22×Dic17C22⋊C412yes—
90M16×C17—81no—
91C2×C8×C17—41no—
92C4⋊C4×C17—21no—
93C4⋊C4×C17—21no—
94C4⋊C4×C17—41no—
95C4⋊C4×C17—41no—
96C22⋊C4×C17—42no—
97C22⋊C4×C17—41no—
98C22⋊C4×C17—41no—
99C42×C17—81no—
100C22⋊Dic17—24no—
101C22⋊Dic17—42no—
102#63225—22no—
103#63225—22no—
104#63225—22no—
105#63225—22no—
106#63225—44no—
107#63225—82no—
108#63225—82no—
109C4×Dic17—81no—
110C4×Dic17—81no—
111#63223—81no—
112C2×#3622—41no—
113#9412—171nomaximal, Sylow
114C23×C17—22no—
115C23×C17—22no—
116C23×C17—41no—
117C23×C17C2×C22⋊C411yescharacteristic
118C23×C17#12011yescharacteristic
119C23×C17#12311yescharacteristic
120Q8×C17—81no—
121Q8×C17—81no—
122C2×C4×C17—41no—
123C2×C4×C17—41no—
124C2×C4×C17—41no—
125C2×C4×C17—41no—
126C2×C4×C17—41no—
127C2×C4×C17—41no—
128C2×C4×C17—42no—
129C2×C4×C17—42no—
130C2×C4×C17—42no—
131C2×C4×C17—41no—
132C2×C4×C17—41no—
133C2×C4×C17—42no—
134C2×C4×C17—42no—
135C2×C4×C17C4×D411yescharacteristic
136C2×C4×C17C4×D411yescharacteristic
137C2×C4×C17#12111yescharacteristic
138C2×C4×C17#12111yescharacteristic
139C2×Dic17—24no—
140C2×Dic17—22no—
141C2×Dic17—22no—
142C2×Dic17—22no—
143C2×Dic17—22no—
144C2×Dic17—42no—
145C2×Dic17—42no—
146C2×Dic17—42no—
147C2×Dic17—42no—
148C2×Dic17—42no—
149C2×Dic17—42no—
150C2×Dic17—42no—
151C2×Dic17—42no—
152C2×Dic17—42no—
153C2×Dic17—81no—
154C8×C17—81no—
155#3622—81no—
156#2583—171no—
157#2313—171no—
158#1508—171no—
159#1508D1711yescharacteristic
160#1463—171no—
161#1297—171no—
162#1297—171no—
163C22×C17—21no—
164C22×C17—21no—
165C22×C17—21no—
166C22×C17—21no—
167C22×C17—21no—
168C22×C17—21no—
169C22×C17—21no—
170C22×C17—44no—
171C22×C17#38611yescharacteristic
172C22×C17#42112yes—
173C22×C17#45612yes—
174C4×C17—21no—
175C4×C17—21no—
176C4×C17—21no—
177C4×C17—21no—
178C4×C17—82no—
179C4×C17—81no—
180C4×C17—81no—
181Dic17—44no—
182Dic17—42no—
183Dic17—42no—
184Dic17—82no—
185C2×C22⋊Q8C2×D1711yescharacteristic
186C22×C4⋊C4—171no—
187#407—171no—
188#407C2×D1711yescharacteristic
189#398—172no—
190#395—172no—
191#389—342no—
192#387—342no—
193#387—342no—
194#385—341no—
195#385—341no—
196#381—341no—
197#380—341no—
198C4×C22⋊C4—171no—
199C4×C22⋊C4—171no—
200C4×C22⋊C4C2×D1711yescharacteristic
201#328—22no—
202#328—342no—
203#324—342no—
204C2×C17—22no—
205C2×C17—22no—
206C2×C17—41no—
207C2×C17#188611yescharacteristic
208C2×C17#189711yescharacteristic
209C2×C17#191311yescharacteristic
210C22×Q8C4×D1711yescharacteristic
211C23×C4—171no—
212C23×C4—171no—
213C23×C4C22×D1711yescharacteristic
214C2×M16—21no—
215C2×M16—341no—
216C22⋊Q8—21no—
217C22⋊Q8—21no—
218C22⋊Q8—21no—
219C22⋊Q8—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—172no—
222C2×C4⋊C4—172no—
223C2×C4⋊C4—341no—
224C2×C4⋊C4—341no—
225C2×C4⋊C4—341no—
226C2×C4⋊C4—341no—
227C2×C4⋊C4—342no—
228C2×C4⋊C4—342no—
229C2×C4⋊C4—342no—
230C2×C4⋊C4C4×D1711yescharacteristic
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—341no—
234C2×C22⋊C4—341no—
235C2×C42—21no—
236C2×C42—341no—
237C2×C42—341no—
238C22⋊C8—21no—
239C22⋊C8—21no—
240C22⋊C8—341no—
241C22⋊C8—341no—
242#95—172no—
243#95—172no—
244#95—342no—
245#95—342no—
246#95—342no—
247#95—344no—
248#95C22⋊D1712yes—
249C17#941211yescharacteristic, complemented, Sylow
250C24D4×D1711yescharacteristic
251C2×Q8—21no—
252C2×Q8—21no—
253C2×Q8—81no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—81no—
258C22×C4—172no—
259C22×C4—172no—
260C22×C4—342no—
261C22×C4—341no—
262C22×C4—341no—
263C22×C4—341no—
264C22×C4—341no—
265C22×C4—342no—
266C22×C4—682no—
267C22×C4—1361no—
268C22×C4C2×C4×D1711yescharacteristic
269C22×C4#6325211yescharacteristic
270C22×C4D4×D1711yescharacteristic
271C22×C4D4×D1711yescharacteristic
272C22×C4#6325411yescharacteristic
273C22×C4C2×C22⋊D1711yescharacteristic
274M16—81no—
275M16—1361no—
276C2×C8—41no—
277C2×C8—681no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—342no—
283C4⋊C4—342no—
284C4⋊C4—342no—
285C4⋊C4—342no—
286C4⋊C4—684no—
287C4⋊C4—1362no—
288C4⋊C4—1362no—
289C22⋊C4—42no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—344no—
293C22⋊C4—682no—
294C42—81no—
295C42—1361no—
296C42—1361no—
297C23—22no—
298C23—22no—
299C23—41no—
300C23C22⋊C4×D1711yescharacteristic
301C23#1058756911yescharacteristic
302C23#1058762311yescharacteristic
303Q8—81no—
304Q8—81no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—344no—
319C2×C4—342no—
320C2×C4—342no—
321C2×C4—342no—
322C2×C4—342no—
323C2×C4—682no—
324C2×C4—682no—
325C2×C4—682no—
326C2×C4—682no—
327C2×C4—682no—
328C2×C4—682no—
329C2×C4—682no—
330C2×C4—682no—
331C2×C4—682no—
332C2×C4—1361no—
333C2×C4#1058756711yescharacteristic
334C2×C4#1058757011yescharacteristic
335C2×C4C4×C22⋊D1711yescharacteristic
336C2×C4#1058762411yescharacteristic
337C8—81no—
338C8—1361no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—44no—
347C22#1176268511yescharacteristic
348C22#1176282412yes—
349C22#1176299012yes—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—82no—
355C4—81no—
356C4—81no—
357C4—684no—
358C4—682no—
359C4—682no—
360C4—1362no—
361C2—22no—
362C2—22no—
363C2—41no—
364C2#1452917411yescharacteristic
365C2#1452922911yescharacteristic
366C2#1452932311yescharacteristic
subgroupG/H
centerC22#11762685
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#1508C2
socleC22×C17#386
prime radicalC24×C17C2×D4
cosocleC23×C4×C17C23
hypercenter#1508D17
hypocenterC17#9412
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1508

Complements

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

NH ≅ G/Ntype
10. #11762740C4semidirect
11. #11762728C4semidirect
249. C17#9412semidirect

Maximal subgroups

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

subgroupindex
#145341652
#145313842
#145273042
#145271132
#145266702
#145266692
C17×#15082
#941217