Finite Group Databasebeta

← #16927965

Subgroups — #16927965

Order 4352.

466 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 103 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
1C4×#10587505C211yesmaximal, characteristic
2#14527122C211yesmaximal, characteristic
3#14526670C212yesmaximal, complemented
4#14526636C212yesmaximal, complemented
5C17×#2357C211yesmaximal, characteristic
6C2×#10587505—21no—
7#11762773C2212yes—
8C4×C22⋊Dic17C2212yes—
9#11762576—21no—
10#11762535—21no—
11#11762533—21no—
12C2×C4×D4×C17C2211yescharacteristic
13C17×#390—21no—
14C17×#388—21no—
15C17×#388—21no—
16C17×#382C411yescharacteristic, complemented
17C17×#381C411yescharacteristic, complemented
18C17×#381C2211yescharacteristic
19C4×C4⋊C4×C17—21no—
20C4×C4⋊C4×C17C411yescharacteristic, complemented
21C4×C22⋊C4×C17C411yescharacteristic, complemented
22C4×C22⋊C4×C17C2211yescharacteristic
23C22×D4×C17D411yescharacteristic
24C23×C4×C17C2×C411yescharacteristic
25C23×C4×C17C2311yescharacteristic
26C4×D4×C17—22no—
27C4×D4×C17—22no—
28C2×C4⋊C4×C17—21no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17D411yescharacteristic
31C2×C22⋊C4×C17—21no—
32C2×C22⋊C4×C17—21no—
33C2×C22⋊C4×C17—21no—
34C2×C22⋊C4×C17—21no—
35C2×C22⋊C4×C17D411yescharacteristic
36C2×C22⋊C4×C17D411yescharacteristic
37C2×C42×C17—21no—
38C2×C42×C17—21no—
39C2×C42×C17—21no—
40C2×C42×C17C2×C411yescharacteristic
41C2×C22⋊Dic17—22no—
42C2×C4×Dic17—22no—
43C2×#63223—22no—
44C17×#95—21no—
45C17×#95—21no—
46C17×#95—21no—
47C17×#95C2×C411yescharacteristic
48C17×#95C2×C411yescharacteristic
49C17×#95C2×C411yescharacteristic
50C17×#95C2×C411yescharacteristic
51#10587505—24no—
52#10587502—22no—
53C22⋊#3622—22no—
54C22⋊#3622—22no—
55C24×C17C22⋊C411yescharacteristic
56C24×C17C2×D411yescharacteristic
57C2×D4×C17—22no—
58C2×D4×C17—22no—
59C2×D4×C17—41no—
60C22×C4×C17—21no—
61C22×C4×C17—21no—
62C22×C4×C17—21no—
63C22×C4×C17—21no—
64C22×C4×C17—21no—
65C22×C4×C17—21no—
66C22×C4×C17—81no—
67C22×C4×C17—81no—
68C22×C4×C17C22⋊C411yescharacteristic
69C22×C4×C17C22⋊C411yescharacteristic
70C22×C4×C17C22⋊C411yescharacteristic
71C22×C4×C17C22×C411yescharacteristic
72C22×C4×C17C2×D411yescharacteristic
73C22×C4×C17#4112yes—
74C22×C4×C17#4112yes—
75C22×Dic17—42no—
76C4⋊C4×C17—22no—
77C4⋊C4×C17—42no—
78C4⋊C4×C17—42no—
79C22⋊C4×C17—22no—
80C22⋊C4×C17—22no—
81C22⋊C4×C17—42no—
82C22⋊C4×C17—42no—
83C22⋊C4×C17—81no—
84C22⋊C4×C17—81no—
85C42×C17—21no—
86C42×C17—21no—
87C42×C17—42no—
88C42×C17—81no—
89C42×C17—81no—
90C22⋊Dic17—24no—
91C22⋊Dic17—42no—
92C4×Dic17—82no—
93#63223—82no—
94C2×#3622—42no—
95#8779—171nomaximal, Sylow
96C23×C17—21no—
97C23×C17—21no—
98C23×C17—21no—
99C23×C17—21no—
100C23×C17—21no—
101C23×C17—41no—
102C23×C17—41no—
103C23×C17—41no—
104C23×C17—41no—
105C23×C17C2×C22⋊C411yescharacteristic
106C23×C17#12312yes—
107D4×C17—44no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—22no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—41no—
114C2×C4×C17—41no—
115C2×C4×C17—42no—
116C2×C4×C17—42no—
117C2×C4×C17—41no—
118C2×C4×C17—41no—
119C2×C4×C17—41no—
120C2×C4×C17—41no—
121C2×C4×C17—41no—
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—41no—
129C2×C4×C17—41no—
130C2×C4×C17—81no—
131C2×C4×C17—81no—
132C2×C4×C17#10412yes—
133C2×C4×C17#11711yescharacteristic
134C2×C4×C17#11711yescharacteristic
135C2×C4×C17#12312yes—
136C2×Dic17—44no—
137C2×Dic17—82no—
138C2×Dic17—82no—
139#3622—82no—
140#2357D1711yescharacteristic
141C4×#99—171no—
142#1465—171no—
143#1297—172no—
144#1284—172no—
145C22×C17—21no—
146C22×C17—21no—
147C22×C17—22no—
148C22×C17—22no—
149C22×C17—22no—
150C22×C17—41no—
151C22×C17—41no—
152C22×C17—41no—
153C22×C17—81no—
154C22×C17—81no—
155C22×C17—81no—
156C22×C17—81no—
157C22×C17#38111yescharacteristic
158C22×C17C2×#10411yescharacteristic
159C22×C17#42111yescharacteristic
160C4×C17—22no—
161C4×C17—22no—
162C4×C17—21no—
163C4×C17—21no—
164C4×C17—42no—
165C4×C17—41no—
166C4×C17—41no—
167C4×C17—81no—
168C4×C17—81no—
169C4×C17—81no—
170Dic17—84no—
171C2×C4×D4C2×D1711yescharacteristic
172C2×#99—341no—
173#407—172no—
174#390—21no—
175#388—21no—
176#388—21no—
177#382Dic1711yescharacteristic, complemented
178#381Dic1711yescharacteristic, complemented
179#381C2×D1711yescharacteristic
180C4×C4⋊C4—21no—
181C4×C4⋊C4Dic1711yescharacteristic, complemented
182C4×C22⋊C4—172no—
183C4×C22⋊C4Dic1711yescharacteristic, complemented
184C4×C22⋊C4C2×D1711yescharacteristic
185#342—341no—
186#324—341no—
187#323—341no—
188C2×C17—21no—
189C2×C17—21no—
190C2×C17—42no—
191C2×C17—81no—
192C2×C17#182611yescharacteristic
193C2×C17#182711yescharacteristic
194C2×C17#182811yescharacteristic
195C22×D4C22⋊D1711yescharacteristic
196C23×C4C2×Dic1711yescharacteristic
197C23×C4C22×D1711yescharacteristic
198C2×M16—342no—
199C4×D4—22no—
200C4×D4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4C22⋊D1711yescharacteristic
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—342no—
209C2×C22⋊C4C22⋊D1711yescharacteristic
210C2×C22⋊C4C22⋊D1711yescharacteristic
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—342no—
215C2×C42C2×Dic1711yescharacteristic
216#99—344no—
217C22⋊C8—342no—
218C22⋊C8—342no—
219#95—21no—
220#95—21no—
221#95—21no—
222#95—342no—
223#95C2×Dic1711yescharacteristic
224#95C2×Dic1711yescharacteristic
225#95C2×Dic1711yescharacteristic
226#95C2×Dic1711yescharacteristic
227C17#877911yescharacteristic, complemented, Sylow
228C24C22⋊Dic1711yescharacteristic
229C24C2×C22⋊D1711yescharacteristic
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—41no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—81no—
240C22×C4—81no—
241C22×C4—682no—
242C22×C4C22⋊Dic1711yescharacteristic
243C22×C4C22⋊Dic1711yescharacteristic
244C22×C4C22⋊Dic1711yescharacteristic
245C22×C4#6325212yes—
246C22×C4#6325212yes—
247C22×C4C22×Dic1711yescharacteristic
248C22×C4C2×C22⋊D1711yescharacteristic
249M16—1362no—
250C2×C8—682no—
251C4⋊C4—22no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C22⋊C4—22no—
255C22⋊C4—22no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—81no—
259C22⋊C4—81no—
260C22⋊C4—344no—
261C22⋊C4—682no—
262C42—21no—
263C42—21no—
264C42—42no—
265C42—81no—
266C42—81no—
267C42—1362no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23#1058761512yes—
278C23C2×C22⋊Dic1711yescharacteristic
279D4—44no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—684no—
305C2×C4—1362no—
306C2×C4—1362no—
307C2×C4#1058750812yes—
308C2×C4#1058761211yescharacteristic
309C2×C4#1058761211yescharacteristic
310C2×C4#1058761512yes—
311C8—1362no—
312C22—21no—
313C22—21no—
314C22—22no—
315C22—22no—
316C22—22no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22#1176267211yescharacteristic
325C22C2×#1058750811yescharacteristic
326C22#1176282611yescharacteristic
327C4—22no—
328C4—22no—
329C4—21no—
330C4—21no—
331C4—42no—
332C4—41no—
333C4—41no—
334C4—81no—
335C4—81no—
336C4—81no—
337C4—1364no—
338C2—21no—
339C2—21no—
340C2—42no—
341C2—81no—
342C2#1452888311yescharacteristic
343C2#1452888811yescharacteristic
344C2#1452889211yescharacteristic
subgroupG/H
centerC22#11762672
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#2357C2
socleC22×C17#381
prime radicalC22×D4×C17D4
cosocleC23×C4×C17C23
hypercenter#2357D17
hypocenterC17#8779
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2357

Complements

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

NH ≅ G/Ntype
3. #14526670C2semidirect
4. #14526636C2semidirect
16. C17×#382C4semidirect
17. C17×#381C4semidirect
20. C4×C4⋊C4×C17C4semidirect
21. C4×C22⋊C4×C17C4semidirect
177. #382Dic17semidirect
178. #381Dic17semidirect
181. C4×C4⋊C4Dic17semidirect
183. C4×C22⋊C4Dic17semidirect
227. C17#8779semidirect

Maximal subgroups

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

subgroupindex
C4×#105875052
#145271222
#145266702
#145266362
C17×#23572
#877917