Finite Group Databasebeta

← #16932052

Subgroups — #16932052

Order 4352.

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

normal subgroups only

This group has 528 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#14533262C211yesmaximal, 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#11762741—22no—
12#11762715C412yescomplemented
13#11762694—22no—
14#11762692—22no—
15#11762684—21no—
16#11762684—21no—
17#11762679C412yescomplemented
18#11762672C2211yescharacteristic
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⋊Dic17—22no—
42C2×C4⋊Dic17—22no—
43C2×C4⋊Dic17C2×C411yescharacteristic
44C2×C4⋊Dic17C2×C412yes—
45C2×C4⋊Dic17D412yes—
46#10587610—21no—
47#10587610—21no—
48#10587610—21no—
49#10587610—21no—
50C2×#63225—22no—
51C2×#63225—21no—
52C2×C4×Dic17—22no—
53C2×#63223—21no—
54C17×C22⋊C8—21no—
55C17×C22⋊C8—21no—
56C17×#95—21no—
57C17×#95—21no—
58C17×#95D412yes—
59#10587502—22no—
60#10587502—22no—
61#10587502—21no—
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—81no—
85C22×Dic17—81no—
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—24no—
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—
105C4⋊Dic17—82no—
106C4⋊Dic17—82no—
107#63225—41no—
108#63225—41no—
109#63225—82no—
110C4×Dic17—82no—
111#63223—42no—
112C2×#3622—41no—
113#9461—171nomaximal, Sylow
114C23×C17—22no—
115C23×C17—22no—
116C23×C17—41no—
117C23×C17C2×C22⋊C411yescharacteristic
118C23×C17#12011yescharacteristic
119C23×C17#12311yescharacteristic
120C2×C4×C17—22no—
121C2×C4×C17—22no—
122C2×C4×C17—24no—
123C2×C4×C17—41no—
124C2×C4×C17—41no—
125C2×C4×C17—41no—
126C2×C4×C17—41no—
127C2×C4×C17—42no—
128C2×C4×C17—42no—
129C2×C4×C17—42no—
130C2×C4×C17—42no—
131C2×C4×C17—42no—
132C2×C4×C17C4×D411yescharacteristic
133C2×C4×C17C4×D411yescharacteristic
134C2×C4×C17#12111yescharacteristic
135C2×C4×C17#12111yescharacteristic
136C2×Dic17—22no—
137C2×Dic17—22no—
138C2×Dic17—42no—
139C2×Dic17—42no—
140C2×Dic17—42no—
141C2×Dic17—42no—
142C2×Dic17—42no—
143C2×Dic17—42no—
144C2×Dic17—42no—
145C2×Dic17—41no—
146C2×Dic17—41no—
147C2×Dic17—42no—
148C2×Dic17—42no—
149C2×Dic17—81no—
150C2×Dic17—81no—
151Dic34—81no—
152Dic34—81no—
153C8×C17—81no—
154#3622—81no—
155#2502—171no—
156#2313—171no—
157#1821—171no—
158#1511—171no—
159#1511—171no—
160#1284—171no—
161#1284D1711yescharacteristic
162C22×C17—21no—
163C22×C17—21no—
164C22×C17—21no—
165C22×C17—21no—
166C22×C17—21no—
167C22×C17—21no—
168C22×C17—21no—
169C22×C17—44no—
170C22×C17#38611yescharacteristic
171C22×C17#41012yes—
172C22×C17#45612yes—
173C4×C17—21no—
174C4×C17—21no—
175C4×C17—21no—
176C4×C17—21no—
177C4×C17—44no—
178C4×C17—82no—
179Dic17—42no—
180Dic17—42no—
181Dic17—82no—
182Dic17—81no—
183Dic17—81no—
184C2×C22⋊Q8—171no—
185C22×C4⋊C4—171no—
186#407—171no—
187#407C2×D1711yescharacteristic
188#398—342no—
189#391—172no—
190#387—342no—
191#387—342no—
192#385—341no—
193#385—341no—
194#384—172no—
195#381—171no—
196#381C2×D1711yescharacteristic
197#380—341no—
198C4×C22⋊C4—341no—
199C4×C22⋊C4C2×D1711yescharacteristic
200#343—342no—
201#328—342no—
202#328—342no—
203C2×C17—22no—
204C2×C17—22no—
205C2×C17—41no—
206C2×C17#188611yescharacteristic
207C2×C17#189911yescharacteristic
208C2×C17#191111yescharacteristic
209C22×Q8—171no—
210C23×C4—171no—
211C23×C4—171no—
212C23×C4C22×D1711yescharacteristic
213C2×M16—21no—
214C2×M16—341no—
215C22⋊Q8—341no—
216C22⋊Q8—341no—
217C22⋊Q8—341no—
218C22⋊Q8—341no—
219C2×C4⋊C4—172no—
220C2×C4⋊C4—172no—
221C2×C4⋊C4—171no—
222C2×C4⋊C4—341no—
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×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—341no—
233C2×C22⋊C4—341no—
234C2×C42—342no—
235C2×C42C4×D1712yes—
236C22⋊C8—21no—
237C22⋊C8—21no—
238C22⋊C8—341no—
239C22⋊C8—341no—
240#95—21no—
241#95—21no—
242#95—172no—
243#95—342no—
244#95—342no—
245#95—342no—
246#95—342no—
247#95—341no—
248#95D6812yes—
249C17#946111yescharacteristic, complemented, Sylow
250C24D4×D1711yescharacteristic
251C2×Q8—341no—
252C2×Q8—341no—
253C2×Q8—682no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—42no—
257C22×C4—172no—
258C22×C4—172no—
259C22×C4—341no—
260C22×C4—341no—
261C22×C4—341no—
262C22×C4—341no—
263C22×C4—341no—
264C22×C4—341no—
265C22×C4—342no—
266C22×C4—1361no—
267C22×C4—1361no—
268C22×C4C2×C4×D1711yescharacteristic
269C22×C4C2×D6811yescharacteristic
270C22×C4#6325211yescharacteristic
271C22×C4D4×D1711yescharacteristic
272C22×C4D4×D1711yescharacteristic
273C22×C4#6325611yescharacteristic
274M16—42no—
275M16—682no—
276C2×C8—41no—
277C2×C8—681no—
278C4⋊C4—342no—
279C4⋊C4—342no—
280C4⋊C4—342no—
281C4⋊C4—342no—
282C4⋊C4—341no—
283C4⋊C4—341no—
284C4⋊C4—681no—
285C4⋊C4—681no—
286C4⋊C4—1362no—
287C4⋊C4—1362no—
288C4⋊C4—1362no—
289C22⋊C4—24no—
290C22⋊C4—42no—
291C22⋊C4—682no—
292C22⋊C4—681no—
293C22⋊C4—681no—
294C42—42no—
295C42—1362no—
296C23—22no—
297C23—22no—
298C23—41no—
299C23C22⋊C4×D1711yescharacteristic
300C23#1058756811yescharacteristic
301C23#1058758011yescharacteristic
302Q8—1361no—
303Q8—1361no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—24no—
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—342no—
317C2×C4—342no—
318C2×C4—682no—
319C2×C4—682no—
320C2×C4—682no—
321C2×C4—682no—
322C2×C4—682no—
323C2×C4—682no—
324C2×C4—682no—
325C2×C4—682no—
326C2×C4—682no—
327C2×C4—681no—
328C2×C4—681no—
329C2×C4—1361no—
330C2×C4—1361no—
331C2×C4C4×D6811yescharacteristic
332C2×C4#1058757011yescharacteristic
333C2×C4#1058757911yescharacteristic
334C2×C4#1058758111yescharacteristic
335C8—81no—
336C8—1361no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—44no—
345C22#1176268711yescharacteristic
346C22#1176278112yes—
347C22#1176299312yes—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—44no—
353C4—82no—
354C4—682no—
355C4—682no—
356C4—1362no—
357C4—1361no—
358C4—1361no—
359C2—22no—
360C2—22no—
361C2—41no—
362C2#1452917611yescharacteristic
363C2#1452924111yescharacteristic
364C2#1452931511yescharacteristic
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#9461
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
17. #11762679C4semidirect
249. C17#9461semidirect

Maximal subgroups

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

subgroupindex
#145332622
#145313852
#145288622
#145273182
#145273162
#145266372
C17×#12842
#946117