Finite Group Databasebeta

← #17009173

Subgroups — #17009173

Order 4352.

998 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 998 conjugacy classes of subgroups — too many to draw. Only its 207 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#14539722C212yesmaximal
2#14533783C211yesmaximal, characteristic
3#14527724C218yesmaximal, complemented
4#14527283C212yesmaximal, complemented
5#14527108C211yesmaximal, characteristic, complemented
6C17×#3502C211yesmaximal, characteristic, complemented
7C2×#10587560C2211yescharacteristic
8C2×#10587633—21no—
9#11763183—22no—
10#11763157—22no—
11#11763009—24no—
12#11762959C2218yes—
13#11762862C2212yes—
14#11762726C2214yes—
15#11762711—22no—
16#11762710—21no—
17C4×C4⋊Dic17C2211yescharacteristic
18#11762546C2218yes—
19#11762535C2214yes—
20Q82×C17—22no—
21C17×#557—24no—
22C17×#557—22no—
23C17×#544—21no—
24C17×#544C2214yes—
25C17×#533—22no—
26C17×#533C2212yes—
27C17×#519—21no—
28C2×C4×Q8×C17C2211yescharacteristic
29C22×Q8×C17D411yescharacteristic
30C22×D68—21no—
31C22×D68—21no—
32C22×D68C2312yes—
33C17×C4⋊Q8—22no—
34C17×C4⋊Q8—22no—
35C17×C4⋊Q8—24no—
36C17×C4⋊Q8—28no—
37C17×C4⋊Q8D412yes—
38C17×C4⋊Q8D412yes—
39C17×#125—24no—
40C17×#125—24no—
41C17×#125—28no—
42C17×#125—22no—
43C17×C22⋊Q8—22no—
44C17×C22⋊Q8—24no—
45C17×C22⋊Q8C2314yes—
46C4×Q8×C17—24no—
47C4×Q8×C17—24no—
48C4×Q8×C17—24no—
49C4×Q8×C17—24no—
50C4×Q8×C17—22no—
51C4×Q8×C17—22no—
52C4×Q8×C17—22no—
53C4×Q8×C17D414yes—
54C17×#117—24no—
55C17×#117—22no—
56C2×C4⋊C4×C17D411yescharacteristic
57C2×C4⋊C4×C17C2312yes—
58C2×C42×C17D412yes—
59C2×C42×C17C2311yescharacteristic
60#10587633—44no—
61C2×#63230—28no—
62C2×#63227—24no—
63C2×#63227—24no—
64C2×C4⋊Dic17—21no—
65C2×C4×Dic17—21no—
66#10587560—22no—
67#10587560—22no—
68#10587502C2312yes—
69C22⋊#3622C2314yes—
70#10587480—28no—
71C4⋊#3622—22no—
72C4×#3622—22no—
73C23×D17—22no—
74C2×Q8×C17—24no—
75C2×Q8×C17—22no—
76C2×Q8×C17—22no—
77C2×Q8×C17—22no—
78C2×Q8×C17SD1614yescomplemented
79C2×Q8×C17C2×D414yes—
80C22×C4×C17C2×D411yescharacteristic
81C22×C4×C17C2×D412yes—
82C22×C4×C17C2×D412yes—
83C22×C4×C17C2×D411yescharacteristic
84C22×C4×C17C2411yescharacteristic
85C22×Dic17—22no—
86C2×D68—24no—
87C2×D68—44no—
88C2×D68—41no—
89C2×D68—42no—
90C2×D68—41no—
91C2×D68—42no—
92C2×D68—42no—
93C2×D68—82no—
94C4⋊C4×C17—24no—
95C4⋊C4×C17—28no—
96C4⋊C4×C17—24no—
97C4⋊C4×C17—24no—
98C4⋊C4×C17—24no—
99C4⋊C4×C17—22no—
100C4⋊C4×C17—22no—
101C4⋊C4×C17—22no—
102C4⋊C4×C17—22no—
103C4⋊C4×C17—24no—
104C4⋊C4×C17—24no—
105C4⋊C4×C17C2×D414yes—
106C22⋊C4×C17—24no—
107C42×C17—28no—
108C42×C17—22no—
109C42×C17—24no—
110C42×C17—22no—
111C42×C17C2×D412yes—
112C42×C17C2×D412yes—
113#63230—88no—
114#63227—48no—
115#63227—84no—
116C4⋊Dic17—42no—
117C4×Dic17—42no—
118C2×#3622—24no—
119#18403—171nomaximal, Sylow
120C23×C17C22×D411yescharacteristic
121C22×D17—44no—
122C22×D17—44no—
123C22×D17—42no—
124C22×D17—42no—
125C22×D17—42no—
126Q8×C17—28no—
127Q8×C17—44no—
128Q8×C17—44no—
129Q8×C17—84no—
130C2×C4×C17—22no—
131C2×C4×C17—22no—
132C2×C4×C17—22no—
133C2×C4×C17—24no—
134C2×C4×C17—24no—
135C2×C4×C17—22no—
136C2×C4×C17—41no—
137C2×C4×C17—42no—
138C2×C4×C17#12011yescharacteristic
139C2×C4×C17#12012yes—
140C2×C4×C17#12011yescharacteristic
141C2×C4×C17C2×SD1614yes—
142C2×C4×C17C2×SD1612yes—
143C2×C4×C17#13712yes—
144C2×C4×C17C22×D412yes—
145C2×Dic17—42no—
146C2×Dic17—42no—
147C2×Dic17—42no—
148D68—48no—
149D68—82no—
150D68—84no—
151D68—82no—
152#3622—44no—
153#3502D1711yescharacteristic, complemented
154#3208—172no—
155#2548—171no—
156#1599—178no—
157#1503—172no—
158#1461—171no—
159C22×C17—21no—
160C22×C17—21no—
161C22×C17C2×#12011yescharacteristic
162C22×C17C22×SD1611yescharacteristic
163C22×C17C2×#13711yescharacteristic
164C2×D17—84no—
165C2×D17—84no—
166C2×D17—82no—
167C2×D17—82no—
168C2×D17—82no—
169C4×C17—24no—
170C4×C17—21no—
171C4×C17—22no—
172C4×C17—44no—
173C4×C17—42no—
174C4×C17—44no—
175C4×C17#44814yes—
176C4×C17#45412yes—
177Dic17—82no—
178Q82—22no—
179#557—22no—
180#557—24no—
181#544—21no—
182#544C2×D1714yes—
183#533—22no—
184#533C2×D1712yes—
185C2×#127—171no—
186C2×#124—341no—
187#519—21no—
188C2×C4×Q8C2×D1711yescharacteristic
189#492—342no—
190#486—342no—
191#460—344no—
192#450—178no—
193#431—172no—
194#394—174no—
195#390—341no—
196#390—342no—
197C4×C4⋊C4—171no—
198#328—178no—
199#324—174no—
200C2×C17—21no—
201C2×C17—21no—
202C2×C17C2×#44811yescharacteristic
203C2×C17C2×#45411yescharacteristic
204C2×C17#302711yescharacteristic
205D17—162no—
206C22×Q8C22⋊D1711yescharacteristic
207C22×D4—172no—
208C22×D4—341no—
209C22×D4—341no—
210C2×SD16—348no—
211C4⋊Q8—22no—
212C4⋊Q8—22no—
213C4⋊Q8—24no—
214C4⋊Q8—28no—
215C4⋊Q8C22⋊D1712yes—
216C4⋊Q8C22⋊D1712yes—
217#127—342no—
218#127—342no—
219#125—24no—
220#125—24no—
221#125—28no—
222#125—22no—
223#124—684no—
224C22⋊Q8—22no—
225C22⋊Q8—24no—
226C22⋊Q8C22×D1714yes—
227C4×Q8—24no—
228C4×Q8—24no—
229C4×Q8—24no—
230C4×Q8—24no—
231C4×Q8—22no—
232C4×Q8—22no—
233C4×Q8—22no—
234C4×Q8C22⋊D1714yes—
235#117—24no—
236#117—22no—
237C2×C4⋊C4—341no—
238C2×C4⋊C4C22⋊D1711yescharacteristic
239C2×C4⋊C4C22×D1712yes—
240C2×C22⋊C4—344no—
241C2×C22⋊C4—344no—
242C2×C42—341no—
243C2×C42C22⋊D1712yes—
244C2×C42C22×D1711yescharacteristic
245C4⋊C8—342no—
246#102—348no—
247C22⋊C8—174no—
248C4×C8—342no—
249#95—172no—
250C17#1840311yescharacteristic, complemented, Sylow
251C24—342no—
252C2×Q8—24no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—22no—
256C2×Q8#6323014yescomplemented
257C2×Q8C2×C22⋊D1714yes—
258C2×D4—344no—
259C2×D4—684no—
260C2×D4—682no—
261C2×D4—682no—
262C2×D4—681no—
263C2×D4—682no—
264C2×D4—681no—
265C2×D4—1362no—
266C22×C4—342no—
267C22×C4C2×C22⋊D1712yes—
268C22×C4C2×C22⋊D1711yescharacteristic
269C22×C4C2×C22⋊D1712yes—
270C22×C4C2×C22⋊D1711yescharacteristic
271C22×C4C23×D1711yescharacteristic
272SD16—1368no—
273C2×C8—344no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—28no—
278C4⋊C4—24no—
279C4⋊C4—24no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—24no—
285C4⋊C4—682no—
286C4⋊C4C2×C22⋊D1714yes—
287C22⋊C4—24no—
288C22⋊C4—688no—
289C22⋊C4—1364no—
290C42—28no—
291C42—22no—
292C42—22no—
293C42—24no—
294C42—682no—
295C42C2×C22⋊D1712yes—
296C42C2×C22⋊D1712yes—
297C23—684no—
298C23—684no—
299C23—682no—
300C23—682no—
301C23—682no—
302C23C22×C22⋊D1711yescharacteristic
303Q8—28no—
304Q8—44no—
305Q8—44no—
306Q8—84no—
307D4—688no—
308D4—1362no—
309D4—1364no—
310D4—1362no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—22no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—682no—
320C2×C4—682no—
321C2×C4—682no—
322C2×C4C2×#6323014yes—
323C2×C4C2×#6323012yes—
324C2×C4#1058762812yes—
325C2×C4#1058763911yescharacteristic
326C2×C4#1058763912yes—
327C2×C4#1058763911yescharacteristic
328C2×C4C22×C22⋊D1712yes—
329C8—684no—
330C22—21no—
331C22—21no—
332C22—1364no—
333C22—1364no—
334C22—1362no—
335C22—1362no—
336C22—1362no—
337C22C2×#1058763911yescharacteristic
338C22C22×#6323011yescharacteristic
339C22C2×#1058762811yescharacteristic
340C4—21no—
341C4—22no—
342C4—24no—
343C4—44no—
344C4—42no—
345C4—44no—
346C4—1362no—
347C4#1176294814yes—
348C4#1176298212yes—
349C2—21no—
350C2—21no—
351C2—2722no—
352C2C2×#1176294811yescharacteristic
353C2C2×#1176298211yescharacteristic
354C2#1453818611yescharacteristic
subgroupG/H
centerC22C2×#10587639
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#3502C2
socleC22×C17C2×#120
prime radicalC2×#10587560C22
hypercenter#3502D17
hypocenterC17#18403
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3502

Complements

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

NH ≅ G/Ntype
3. #14527724C2semidirect
4. #14527283C2semidirect
5. #14527108C2semidirect
6. C17×#3502C2semidirect
78. C2×Q8×C17SD16semidirect
153. #3502D17semidirect
250. C17#18403semidirect
256. C2×Q8#63230semidirect

Maximal subgroups

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

subgroupindex
#145397222
#145337832
#145277242
#145272832
#145271082
C17×#35022
#1840317