Finite Group Databasebeta

← #17033432

Subgroups — #17033432

Order 4352.

5990 conjugacy classes of subgroups; only the 479 normal subgroups are shown.

normal subgroups only

This group has 5990 conjugacy classes of subgroups — too many to draw. Only its 479 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
1C24×C4×D17C211yesmaximal, characteristic
2C2×#11762771C216yesmaximal, complemented
3#14528694C211yesmaximal, characteristic, complemented
4#14528693C216yesmaximal, complemented
5C17×#1779C211yesmaximal, characteristic, complemented
6C25×D17C411yescharacteristic
7C24×Dic17C411yescharacteristic
8C23×C4×D17—224no—
9C23×C4×D17C413yes—
10C23×C4×D17C413yes—
11C23×C4×D17C2213yes—
12C23×C4×D17C2213yes—
13C22×#63218C2214yescomplemented
14#11762771—224no—
15C2×C22⋊#3622C2216yescomplemented
16C2×#10587491C22112yescomplemented
17C24×C4×C17C2211yescharacteristic
18C2×C17×C22⋊C8C2216yescomplemented
19C25×C17C2×C411yescharacteristic
20C24×D17—24no—
21C24×D17—212no—
22C24×D17—212no—
23C24×D17C2×C413yes—
24C24×D17C2×C413yes—
25C23×C4×C17—212no—
26C23×C4×C17C2×C413yes—
27C23×C4×C17C2313yes—
28C23×Dic17—212no—
29C23×Dic17C2×C413yes—
30C23×Dic17C2×C413yes—
31C22×C4×D17—2128no—
32C22×C4×D17—212no—
33C22×C4×D17—212no—
34C22×C4×D17—212no—
35C22×C4×D17—224no—
36C22×C4×D17—224no—
37C22×C4×D17—224no—
38C22×C4×D17—224no—
39C22×C4×D17C2×C416yes—
40C22×C4×D17C2×C413yes—
41C22×C4×D17C2×C413yes—
42C22×C4×D17D4124yes—
43C22×C4×D17C2313yes—
44C22×C4×D17C2311yescharacteristic
45C22×C8×C17C2314yescomplemented
46C22×#3622C2314yescomplemented
47C2×#63218—224no—
48C17×C22⋊C8—212no—
49C22⋊#3622—212no—
50#10587491—224no—
51C24×C17—26no—
52C24×C17—26no—
53C24×C17M1614yescomplemented
54C24×C17C22×C413yes—
55C23×D17—232no—
56C23×D17—296no—
57C23×D17—296no—
58C23×D17—26no—
59C23×D17—26no—
60C23×D17—26no—
61C23×D17—26no—
62C23×D17—212no—
63C23×D17—212no—
64C23×D17—212no—
65C23×D17—212no—
66C23×D17C22⋊C4112yes—
67C23×D17M16112yescomplemented
68C23×D17C22×C411yescharacteristic
69C23×D17C22×C413yes—
70C22×C4×C17—232no—
71C22×C4×C17—26no—
72C22×C4×C17—212no—
73C22×C4×C17—212no—
74C22×C4×C17C22×C413yes—
75C22×C4×C17C2×D4112yes—
76C22×C4×C17C2411yescharacteristic
77C22×Dic17—232no—
78C22×Dic17—26no—
79C22×Dic17—212no—
80C22×Dic17—212no—
81C22×Dic17C22⋊C4112yes—
82C22×Dic17C22×C413yes—
83C22×Dic17C22×C411yescharacteristic
84C2×C4×D17—296no—
85C2×C4×D17—296no—
86C2×C4×D17—296no—
87C2×C4×D17—212no—
88C2×C4×D17—212no—
89C2×C4×D17—212no—
90C2×C4×D17—212no—
91C2×C4×D17—212no—
92C2×C4×D17—212no—
93C2×C4×D17—296no—
94C2×C4×D17C22⋊C4112yes—
95C2×C4×D17C22⋊C4112yes—
96C2×C4×D17C2×D4112yes—
97C2×C4×D17C2×D4112yes—
98C2×C8×C17—212no—
99C2×#3622—212no—
100#63218—416no—
101#20579—171nomaximal, Sylow
102C23×C17—28no—
103C23×C17—224no—
104C23×C17—26no—
105C23×C17—26no—
106C23×C17—224no—
107C23×C17C2×C22⋊C416yes—
108C23×C17C2×M1616yes—
109C23×C17C2×M1616yes—
110C23×C17C23×C411yescharacteristic
111C22×D17—224no—
112C22×D17—2256no—
113C22×D17—248no—
114C22×D17—248no—
115C22×D17—248no—
116C22×D17—224no—
117C22×D17—26no—
118C22×D17—23no—
119C22×D17—23no—
120C22×D17—21no—
121C22×D17—23no—
122C22×D17—224no—
123C22×D17—224no—
124C22×D17—224no—
125C22×D17—224no—
126C22×D17—248no—
127C22×D17C2×C22⋊C416yes—
128C22×D17C2×C22⋊C416yes—
129C22×D17C2×M16112yes—
130C2×C4×C17—224no—
131C2×C4×C17—26no—
132C2×C4×C17—224no—
133C2×C4×C17C2×C22⋊C416yes—
134C2×C4×C17C22×D416yes—
135C2×Dic17—224no—
136C2×Dic17—26no—
137C2×Dic17—224no—
138C2×Dic17C2×C22⋊C416yes—
139C2×Dic17C2×C22⋊C416yes—
140C4×D17—248no—
141C4×D17—224no—
142C4×D17—224no—
143C4×D17—224no—
144C4×D17#120116yes—
145C8×C17—44no—
146#3622—44no—
147C25×C4—171no—
148C2×#407—176no—
149#1779—176no—
150#1779—171no—
151#1779D1711yescharacteristic, complemented
152C22×C17—232no—
153C22×C17—26no—
154C22×C17—212no—
155C22×C17—212no—
156C22×C17—26no—
157C22×C17#407112yescomplemented
158C22×C17C22×C22⋊C413yes—
159C22×C17C22×M1613yes—
160C22×C17C22×M1611yescharacteristic
161C2×D17—296no—
162C2×D17—296no—
163C2×D17—26no—
164C2×D17—26no—
165C2×D17—26no—
166C2×D17—212no—
167C2×D17—212no—
168C2×D17—26no—
169C2×D17#37914yes—
170C2×D17#407112yescomplemented
171C4×C17—26no—
172C4×C17C2×#12014yes—
173Dic17—26no—
174Dic17#37914yes—
175C26—171no—
176C24×C4—173no—
177C24×C4—171no—
178C24×C4—173no—
179C24×C4—173no—
180C24×C4—173no—
181C24×C4—3424no—
182C24×C4C2×D1711yescharacteristic
183C22×M16—174no—
184#407—3424no—
185C2×C22⋊C8—176no—
186C2×C22⋊C8—1712no—
187C2×C22⋊C8C2×D1716yescomplemented
188C2×C17—212no—
189C2×C17C2×#37911yescharacteristic
190C2×C17C2×#40713yes—
191C2×C17C2×#40713yes—
192D17—24no—
193D17—212no—
194C25—173no—
195C25—173no—
196C25—3412no—
197C25—3412no—
198C25—344no—
199C25C4×D1711yescharacteristic
200C23×C4—212no—
201C23×C4—171no—
202C23×C4—173no—
203C23×C4—173no—
204C23×C4—173no—
205C23×C4—1724no—
206C23×C4—173no—
207C23×C4—173no—
208C23×C4—176no—
209C23×C4—3412no—
210C23×C4—3424no—
211C23×C4—3424no—
212C23×C4—3424no—
213C23×C4—3424no—
214C23×C4—3412no—
215C23×C4—34128no—
216C23×C4—3412no—
217C23×C4—3412no—
218C23×C4C4×D1713yes—
219C23×C4C22×D1713yes—
220C2×M16—3424no—
221C22×C8—174no—
222C22×C8C22×D1714yescomplemented
223C22⋊C8—212no—
224C22⋊C8—3424no—
225C22⋊C8—3412no—
226C17#2057911yescharacteristic, complemented, Sylow
227C24—26no—
228C24—26no—
229C24—171no—
230C24—173no—
231C24—1712no—
232C24—1712no—
233C24—3412no—
234C24—3412no—
235C24—3412no—
236C24—3412no—
237C24—3432no—
238C24—3496no—
239C24—3496no—
240C24—346no—
241C24—346no—
242C24—346no—
243C24—346no—
244C24#6321814yescomplemented
245C24C2×C4×D1713yes—
246C22×C4—232no—
247C22×C4—26no—
248C22×C4—212no—
249C22×C4—212no—
250C22×C4—171no—
251C22×C4—1712no—
252C22×C4—173no—
253C22×C4—1712no—
254C22×C4—1712no—
255C22×C4—1712no—
256C22×C4—1712no—
257C22×C4—3412no—
258C22×C4—3412no—
259C22×C4—3496no—
260C22×C4—3412no—
261C22×C4—3412no—
262C22×C4—3496no—
263C22×C4—3496no—
264C22×C4—3496no—
265C22×C4—3412no—
266C22×C4—3412no—
267C22×C4—3432no—
268C22×C4—346no—
269C22×C4—3412no—
270C22×C4—3412no—
271C22×C4C2×C4×D1713yes—
272C22×C4D4×D17112yes—
273C22×C4C23×D1711yescharacteristic
274M16—6816no—
275C2×C8—212no—
276C2×C8—3412no—
277C23—28no—
278C23—224no—
279C23—26no—
280C23—224no—
281C23—26no—
282C23—176no—
283C23—176no—
284C23—1712no—
285C23—3424no—
286C23—3424no—
287C23—3448no—
288C23—3424no—
289C23—34256no—
290C23—3424no—
291C23—3448no—
292C23—3448no—
293C23—3448no—
294C23—346no—
295C23—3424no—
296C23—3424no—
297C23—343no—
298C23—343no—
299C23—341no—
300C23—343no—
301C23C22⋊C4×D1716yes—
302C23C2×#6321816yes—
303C23M16×D1716yes—
304C23C22×C4×D1711yescharacteristic
305C2×C4—224no—
306C2×C4—26no—
307C2×C4—224no—
308C2×C4—176no—
309C2×C4—1716no—
310C2×C4—176no—
311C2×C4—3424no—
312C2×C4—3424no—
313C2×C4—3448no—
314C2×C4—3424no—
315C2×C4—3424no—
316C2×C4—346no—
317C2×C4—3424no—
318C2×C4C22⋊C4×D1716yes—
319C2×C4C2×D4×D1716yes—
320C8—44no—
321C8—684no—
322C22—232no—
323C22—26no—
324C22—212no—
325C22—212no—
326C22—26no—
327C22—1712no—
328C22—174no—
329C22—3496no—
330C22—3496no—
331C22—346no—
332C22—3412no—
333C22—346no—
334C22—346no—
335C22—3412no—
336C22—346no—
337C22#11762771112yescomplemented
338C22C2×C22⋊C4×D1713yes—
339C22C2×M16×D1713yes—
340C22C22×#6321811yescharacteristic
341C4—26no—
342C4—174no—
343C4—346no—
344C4#120×D1714yes—
345C2—212no—
346C2—3412no—
347C2—344no—
348C2D17×#37911yescharacteristic
349C2D17×#40713yes—
350C2C2×#1176277113yes—
subgroupG/H
center, FrattiniC22×C4C23×D17
commutator, socleC23×C17C23×C4
FittingC17×#1779C2
prime radicalC25×D17C4
cosocleC22×C4×C17C24
hypercenter#1779D17
hypocenterC17#20579
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C22×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
2. C2×#11762771C2semidirect
3. #14528694C2semidirect
4. #14528693C2semidirect
5. C17×#1779C2semidirect
13. C22×#63218C22semidirect
15. C2×C22⋊#3622C22semidirect
16. C2×#10587491C22semidirect
18. C2×C17×C22⋊C8C22semidirect
45. C22×C8×C17C23semidirect
46. C22×#3622C23semidirect
53. C24×C17M16semidirect
67. C23×D17M16semidirect
151. #1779D17semidirect
157. C22×C17#407semidirect
170. C2×D17#407semidirect
187. C2×C22⋊C8C2×D17semidirect
222. C22×C8C22×D17semidirect
226. C17#20579semidirect
244. C24#63218semidirect
337. C22#11762771semidirect

Maximal subgroups

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

subgroupindex
C24×C4×D172
C2×#117627712
#145286942
#145286932
C17×#17792
#2057917