Finite Group Databasebeta

← #17951246

Subgroups — #17951246

Order 4352.

1226 conjugacy classes of subgroups; only the 172 normal subgroups are shown.

normal subgroups only

This group has 1226 conjugacy classes of subgroups — too many to draw. Only its 172 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
1C2×C4×C22⋊#595C212yesmaximal
2#14543664C214yesmaximal, complemented
3D17×#407C211yesmaximal, characteristic
4C22×C22⋊#595—22no—
5C22×C4×#595—22no—
6C4×C22⋊#595—22no—
7C4×C22⋊#595—22no—
8C4×C22⋊#595C414yescomplemented
9C4×C22⋊#595C2214yes—
10C2×(C2×#595)⋊C4C414yescomplemented
11C23×C4×D17C2211yescharacteristic
12C2×M16×D17C412yescomplemented
13#11762773C411yescharacteristic, complemented
14#11762772C2212yes—
15#11762771—22no—
16C22⋊C8×D17—22no—
17C17×#407C411yescharacteristic, complemented
18C24×D17D411yescharacteristic
19C23×#595—24no—
20C23×C4×C17C2×C411yescharacteristic
21C23×Dic17Q811yescharacteristic
22C22×C4×D17—44no—
23C22×C4×D17C2×C411yescharacteristic
24C22×C4×D17C2×C412yes—
25C22×C4×D17C2×C412yes—
26C22×C4×D17D412yes—
27C22×C4×D17D411yescharacteristic
28C22×C4×D17D412yes—
29C22×C4×D17Q811yescharacteristic
30C22×C4×D17C2311yescharacteristic
31C2×C22⋊#595—216no—
32C2×C22⋊#595—24no—
33C2×C22⋊#595—24no—
34C2×C4×#595—24no—
35C2×C4×#595—24no—
36C2×C4×#595—48no—
37C2×M16×C17C2×C412yescomplemented
38C2×#63223C2×C412yescomplemented
39M16×D17—44no—
40C2×#63218—22no—
41C2×C8×D17—22no—
42(C2×#595)⋊C4—24no—
43(C2×#595)⋊C4C2×C418yescomplemented
44C17×C22⋊C8—22no—
45C22⋊#3622—22no—
46#10587491—24no—
47C24×C17C4⋊C411yescharacteristic
48C23×D17—28no—
49C23×D17—42no—
50C23×D17C22⋊C412yes—
51C23×D17C22⋊C412yes—
52C23×D17C4⋊C411yescharacteristic
53C23×D17C2×D411yescharacteristic
54C22×#595—216no—
55C22×#595—28no—
56C22×#595—48no—
57C22×#595—48no—
58C22×C4×C17—42no—
59C22×C4×C17C4212yes—
60C22×C4×C17C22⋊C412yes—
61C22×C4×C17C4⋊C411yescharacteristic
62C22×C4×C17C22×C411yescharacteristic
63C22×Dic17—42no—
64C22×Dic17C4⋊C411yescharacteristic
65C22×Dic17C4⋊C412yes—
66C22×Dic17C4⋊C412yes—
67C22×Dic17C2×Q811yescharacteristic
68C2×C4×D17—28no—
69C2×C4×D17—24no—
70C2×C4×D17—44no—
71C2×C4×D17—42no—
72C2×C4×D17—44no—
73C2×C4×D17—42no—
74C2×C4×D17—44no—
75C2×C4×D17—42no—
76C2×C4×D17C4211yescharacteristic
77C2×C4×D17C4211yescharacteristic
78C2×C4×D17C22⋊C412yes—
79C2×C4×D17C22⋊C411yescharacteristic
80C2×C4×D17C22⋊C412yes—
81C2×C4×D17C22⋊C411yescharacteristic
82C2×C4×D17C4⋊C412yes—
83C2×C4×D17C4⋊C412yes—
84C2×C4×D17C22×C412yes—
85C2×C4×D17C2×D412yes—
86C22⋊#595—216no—
87C22⋊#595—216no—
88C4×#595—48no—
89C4×#595—48no—
90M16×C17—42no—
91C2×C8×C17—22no—
92#63223—42no—
93C2×#3622—22no—
94#63218—44no—
95C8×D17—44no—
96C2×#1297—171nomaximal, Sylow
97C23×C17—24no—
98C23×C17—41no—
99C23×C17#9512yes—
100C23×C17C2×C4⋊C411yescharacteristic
101C22×D17—24no—
102C22×D17—24no—
103C22×D17—24no—
104C22×D17—24no—
105C22×D17—22no—
106C22×D17—24no—
107C22×D17—24no—
108C22×D17—42no—
109C22×D17—41no—
110C22×D17—416no—
111C22×D17#9511yescharacteristic
112C22×D17#9511yescharacteristic
113C22×D17#10418yes—
114C22×D17C2×C22⋊C412yes—
115C2×#595—216no—
116C2×#595—48no—
117C2×#595—432no—
118C2×#595—48no—
119C2×C4×C17—22no—
120C2×C4×C17—24no—
121C2×C4×C17—42no—
122C2×C4×C17—41no—
123C2×C4×C17#9512yes—
124C2×C4×C17C2×C4211yescharacteristic
125C2×C4×C17C2×C22⋊C411yescharacteristic
126C2×Dic17—22no—
127C2×Dic17—24no—
128C2×Dic17—42no—
129C2×Dic17—41no—
130C2×Dic17#9511yescharacteristic
131C2×Dic17#9511yescharacteristic
132C2×Dic17C2×C4⋊C412yes—
133C4×D17—22no—
134C4×D17—24no—
135C4×D17—24no—
136C4×D17—22no—
137C4×D17—24no—
138C4×D17—22no—
139C4×D17—42no—
140C4×D17—42no—
141C4×D17—42no—
142C4×D17#9914yes—
143C8×C17—42no—
144#3622—42no—
145C2×#407—171no—
146C2×C4×C22⋊C4—172no—
147#1297—174no—
148C22×C17—22no—
149C22×C17—22no—
150C22×C17—22no—
151C22×C17—21no—
152C22×C17—44no—
153C22×C17#33914yes—
154C22×C17C2×#9511yescharacteristic
155C2×D17—22no—
156C2×D17—22no—
157C2×D17—216no—
158C2×D17—22no—
159C2×D17—21no—
160C2×D17—44no—
161C2×D17—48no—
162C2×D17—44no—
163C2×D17#33914yes—
164C2×D17#34212yes—
165C2×D17C2×#10414yes—
166#595—416no—
167C4×C17—22no—
168C4×C17—21no—
169C4×C17C2×#9912yes—
170Dic17—21no—
171Dic17—22no—
172Dic17#34212yes—
173C24×C4—171no—
174C22×M16—172no—
175C22×C22⋊C4—342no—
176C22×C42—342no—
177#407—171no—
178#407—172no—
179#407—342no—
180#407#59511yescharacteristic, complemented
181C2×C22⋊C8—342no—
182C4×C22⋊C4—174no—
183C4×C22⋊C4—174no—
184C4×C22⋊C4—342no—
185C4×C22⋊C4—342no—
186C2×#95—174no—
187C2×C17—24no—
188C2×C17—41no—
189C2×C17C2×#33912yes—
190C2×C17C2×#34211yescharacteristic
191D17—24no—
192D17—41no—
193D17#129714yescomplemented
194C25—171no—
195C23×C4—171no—
196C23×C4—172no—
197C23×C4—171no—
198C23×C4—172no—
199C23×C4—172no—
200C23×C4—171no—
201C23×C4—171no—
202C23×C4—172no—
203C23×C4—171no—
204C23×C4—344no—
205C23×C4—684no—
206C23×C4C2×#59511yescharacteristic
207C2×M16—172no—
208C2×M16—342no—
209C2×M16—684no—
210C2×M16C2×#59512yescomplemented
211C22×C8—342no—
212C2×C22⋊C4—344no—
213C2×C22⋊C4—344no—
214C2×C22⋊C4—3416no—
215C2×C42—344no—
216C2×C42—344no—
217C2×C42—688no—
218C22⋊C8—22no—
219C22⋊C8—344no—
220C22⋊C8—342no—
221#95—178no—
222#95—344no—
223C17C2×#129711yescharacteristic, complemented, Sylow
224C24—172no—
225C24—171no—
226C24—171no—
227C24—172no—
228C24—348no—
229C24—682no—
230C24C4⋊#59511yescharacteristic
231C22×C4—42no—
232C22×C4—171no—
233C22×C4—171no—
234C22×C4—172no—
235C22×C4—171no—
236C22×C4—172no—
237C22×C4—171no—
238C22×C4—171no—
239C22×C4—171no—
240C22×C4—172no—
241C22×C4—172no—
242C22×C4—172no—
243C22×C4—172no—
244C22×C4—172no—
245C22×C4—172no—
246C22×C4—348no—
247C22×C4—344no—
248C22×C4—348no—
249C22×C4—3416no—
250C22×C4—682no—
251C22×C4—684no—
252C22×C4—684no—
253C22×C4—682no—
254C22×C4—684no—
255C22×C4—682no—
256C22×C4—682no—
257C22×C4—688no—
258C22×C4—688no—
259C22×C4C4×#59512yes—
260C22×C4C4⋊#59511yescharacteristic
261C22×C4C22⋊#59512yes—
262C22×C4C22×#59511yescharacteristic
263M16—42no—
264M16—684no—
265M16—682no—
266C2×C8—22no—
267C2×C8—342no—
268C2×C8—684no—
269C22⋊C4—3416no—
270C22⋊C4—3416no—
271C42—688no—
272C42—688no—
273C23—24no—
274C23—41no—
275C23—171no—
276C23—172no—
277C23—178no—
278C23—171no—
279C23—344no—
280C23—344no—
281C23—344no—
282C23—342no—
283C23—344no—
284C23—344no—
285C23—344no—
286C23—682no—
287C23—681no—
288C23—6816no—
289C23(C2×#595)⋊C412yes—
290C23C2×C4⋊#59511yescharacteristic
291C2×C4—22no—
292C2×C4—24no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—171no—
296C2×C4—174no—
297C2×C4—171no—
298C2×C4—172no—
299C2×C4—344no—
300C2×C4—342no—
301C2×C4—342no—
302C2×C4—344no—
303C2×C4—344no—
304C2×C4—342no—
305C2×C4—344no—
306C2×C4—342no—
307C2×C4—3416no—
308C2×C4—682no—
309C2×C4—682no—
310C2×C4—682no—
311C2×C4—682no—
312C2×C4—681no—
313C2×C4—688no—
314C2×C4—688no—
315C2×C4—6832no—
316C2×C4(C2×#595)⋊C412yes—
317C2×C4C2×C4×#59511yescharacteristic
318C2×C4C2×C22⋊#59511yescharacteristic
319C8—42no—
320C8—682no—
321C22—22no—
322C22—22no—
323C22—22no—
324C22—21no—
325C22—44no—
326C22—174no—
327C22—172no—
328C22—174no—
329C22—3416no—
330C22—342no—
331C22—342no—
332C22—342no—
333C22—341no—
334C22—688no—
335C22—684no—
336C22—684no—
337C22#1176368114yes—
338C22C2×(C2×#595)⋊C411yescharacteristic
339C4—21no—
340C4—22no—
341C4—172no—
342C4—341no—
343C4—342no—
344C4—6816no—
345C4D17⋊#9912yes—
346C2—24no—
347C2—41no—
348C2—174no—
349C2—344no—
350C2—681no—
351C2C2×#1176368112yes—
352C2#1454448111yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#595
commutatorC2×C4×C17C2×C42
FrattiniC22×C4C22×#595
FittingC17×#407C4
socleC22×C17C2×#95
prime radicalC24×D17D4
cosocleC22×C4×D17C23
hypercenter#407#595
hypocenterC17C2×#1297
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. #14543664C2semidirect
8. C4×C22⋊#595C4semidirect
10. C2×(C2×#595)⋊C4C4semidirect
12. C2×M16×D17C4semidirect
13. #11762773C4semidirect
17. C17×#407C4semidirect
37. C2×M16×C17C2×C4semidirect
38. C2×#63223C2×C4semidirect
43. (C2×#595)⋊C4C2×C4semidirect
180. #407#595semidirect
193. D17#1297semidirect
210. C2×M16C2×#595semidirect
223. C17C2×#1297semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊#5952
#145436642
D17×#4072
C2×#129717