Finite Group Databasebeta

← C22×#11763819 #17979551

Subgroups — C22×#11763819 #17979551

Order 4352.

3006 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587677C212yesmaximal
2C2×#11763819C2124yesmaximal, direct product factor
3C22×#10587534C212yesmaximal, complemented
4C22×#10587533C212yesmaximal, complemented
5C23×C8×D17C211yesmaximal, characteristic
6C2×#10587677—212no—
7C2×#10587677C22124yes—
8C22×C4⋊#595C2214yes—
9C22×C22⋊#595—22no—
10C22×C4×#595—22no—
11#11763819C22164yesdirect product factor
12C2×#10587534C22124yescomplemented
13C2×#10587533C22124yescomplemented
14C23×C4×D17C2211yescharacteristic
15C23×#3622C411yescharacteristic, complemented
16C22×C8×D17C411yescharacteristic, complemented
17C22×C8×D17C411yescharacteristic, complemented
18C22×C8×D17C4112yescomplemented
19C22×C8×D17C2211yescharacteristic
20C22×C8×D17C2211yescharacteristic
21C22×C8×D17C22112yes—
22C23×C8×C17C411yescharacteristic, complemented
23C24×D17D411yescharacteristic
24C23×#595—24no—
25C23×C4×C17C2×C411yescharacteristic
26C23×Dic17Q811yescharacteristic
27C22×C4×D17C2×C416yes—
28C22×C4×D17C2×C411yescharacteristic
29C22×C4×D17D411yescharacteristic
30C22×C4×D17D416yes—
31C22×C4×D17Q816yes—
32C22×C4×D17Q811yescharacteristic
33C22×C4×D17C2311yescharacteristic
34C22×C4×D17C2316yes—
35C2×C22⋊#595—224no—
36#10587677—248no—
37#10587677C23132yes—
38C2×C4⋊#595—212no—
39C2×C4⋊#595C23124yes—
40C2×C4×#595—224no—
41C22×C8×C17C2×C411yescharacteristic
42C22×C8×C17C2×C411yescharacteristic
43C22×C8×C17C2×C4112yescomplemented
44C22×#3622C2×C411yescharacteristic
45C22×#3622C2×C411yescharacteristic
46C22×#3622C2×C4112yescomplemented
47C2×C8×D17C2×C416yes—
48C2×C8×D17C2×C416yes—
49C2×C8×D17C2×C4112yescomplemented
50C2×C8×D17C2×C4112yescomplemented
51C2×C8×D17C2×C4148yescomplemented
52C2×C8×D17C2316yes—
53C2×C8×D17C2316yes—
54C2×C8×D17C23116yes—
55#10587534C23132yescomplemented
56#10587533C23132yescomplemented
57C24×C17C4⋊C411yescharacteristic
58C23×D17—28no—
59C23×D17C4⋊C411yescharacteristic
60C23×D17C4⋊C416yes—
61C23×D17C2×D416yes—
62C23×D17C2×D411yescharacteristic
63C22×#595—224no—
64C22×#595—416no—
65C22×C4×C17C4⋊C416yes—
66C22×C4×C17C4⋊C411yescharacteristic
67C22×C4×C17C22×C411yescharacteristic
68C22×C4×C17C22×C416yes—
69C22×Dic17C4⋊C416yes—
70C22×Dic17C4⋊C411yescharacteristic
71C22×Dic17C2×Q816yes—
72C22×Dic17C2×Q811yescharacteristic
73C2×C4×D17C4⋊C416yes—
74C2×C4×D17C4⋊C4124yes—
75C2×C4×D17C4⋊C416yes—
76C2×C4×D17C4⋊C416yes—
77C2×C4×D17C22×C413yes—
78C2×C4×D17C22×C4112yes—
79C2×C4×D17C22×C416yes—
80C2×C4×D17C2×D416yes—
81C2×C4×D17C2×D413yes—
82C2×C4×D17C2×D4112yes—
83C2×C4×D17C2×Q8112yes—
84C2×C4×D17C2×Q816yes—
85C2×C4×D17C2×Q813yes—
86C2×C4×D17C2413yes—
87C2×C4×D17C2414yes—
88C22⋊#595—232no—
89C4⋊#595—224no—
90C4⋊#595C24116yes—
91C4×#595—232no—
92C2×C8×C17C22×C416yes—
93C2×C8×C17C22×C416yes—
94C2×C8×C17C22×C4116yescomplemented
95C2×#3622C22×C416yes—
96C2×#3622C22×C416yes—
97C2×#3622C22×C4116yescomplemented
98C8×D17C22×C4112yes—
99C8×D17C22×C4112yes—
100C8×D17C22×C4116yescomplemented
101C8×D17C22×C4116yescomplemented
102C8×D17C2414yes—
103C8×D17C2414yes—
104C22×#427—171nomaximal, Sylow
105C23×C17—24no—
106C23×C17C2×C4⋊C416yes—
107C23×C17C2×C4⋊C411yescharacteristic
108C22×D17—24no—
109C22×D17—224no—
110C22×D17—224no—
111C22×D17C2×C4⋊C416yes—
112C22×D17C2×C4⋊C4112yes—
113C22×D17C2×C4⋊C413yes—
114C22×D17#13518yes—
115C22×D17C22×D414yes—
116C22×D17C22×D413yes—
117C2×#595—216no—
118C2×#595—448no—
119C2×C4×C17C2×C4⋊C4112yes—
120C2×C4×C17C2×C4⋊C413yes—
121C2×C4×C17C2×C4⋊C416yes—
122C2×C4×C17C23×C413yes—
123C2×C4×C17C23×C414yes—
124C2×Dic17C2×C4⋊C4112yes—
125C2×Dic17C2×C4⋊C413yes—
126C2×Dic17C2×C4⋊C416yes—
127C2×Dic17C22×Q814yes—
128C2×Dic17C22×Q813yes—
129C4×D17C2×C4⋊C4112yes—
130C4×D17C2×C4⋊C4112yes—
131C4×D17C2×C4⋊C416yes—
132C4×D17C2×C4⋊C4112yes—
133C4×D17C23×C413yes—
134C4×D17C23×C414yes—
135C4×D17C22×D414yes—
136C4×D17C22×D413yes—
137C4×D17C22×Q814yes—
138C4×D17C22×Q813yes—
139C4×D17C2511yescharacteristic
140C8×C17C23×C414yes—
141C8×C17C23×C414yes—
142#3622C23×C414yes—
143#3622C23×C414yes—
144C24×C8—171no—
145C22×#117—172no—
146C2×#427—1724no—
147C22×#107—172no—
148C22×#106—172no—
149C22×C17—212no—
150C22×C17#42714yescomplemented
151C22×C17C22×C4⋊C414yes—
152C22×C17C22×C4⋊C413yes—
153C2×D17—212no—
154C2×D17—224no—
155C2×D17—28no—
156C2×D17#427112yescomplemented
157C2×D17C22×C4⋊C414yes—
158C2×D17C22×C4⋊C413yes—
159C2×D17C2×#135112yes—
160C2×D17C23×D411yescharacteristic
161#595—416no—
162C4×C17C22×C4⋊C414yes—
163C4×C17C22×C4⋊C413yes—
164C4×C17C24×C411yescharacteristic
165Dic17C22×C4⋊C413yes—
166Dic17C22×C4⋊C414yes—
167Dic17C23×Q811yescharacteristic
168C24×C4—171no—
169C23×C8—171no—
170C23×C8—171no—
171C23×C8—171no—
172C23×C8—171no—
173C23×C8—171no—
174C23×C8—1712no—
175C23×C8—1712no—
176C23×C8#59511yescharacteristic, complemented
177C2×#117—1724no—
178C2×#117—3412no—
179C22×C4⋊C4—174no—
180C22×C22⋊C4—342no—
181C22×C42—342no—
182#427—1764no—
183C2×#107—1724no—
184C2×#106—1724no—
185C2×C17—24no—
186C2×C17C2×#42716yescomplemented
187C2×C17C23×C4⋊C411yescharacteristic
188D17—24no—
189D17C2×#42716yescomplemented
190D17C22×#13512yes—
191C25—171no—
192C23×C4—171no—
193C23×C4—176no—
194C23×C4—176no—
195C23×C4—171no—
196C23×C4—171no—
197C23×C4—171no—
198C23×C4—176no—
199C23×C4—171no—
200C23×C4—176no—
201C23×C4—344no—
202C23×C4C2×#59511yescharacteristic
203C22×C8—171no—
204C22×C8—171no—
205C22×C8—1712no—
206C22×C8—176no—
207C22×C8—176no—
208C22×C8—176no—
209C22×C8—176no—
210C22×C8—1712no—
211C22×C8—1712no—
212C22×C8—1716no—
213C22×C8—1748no—
214C22×C8C2×#59511yescharacteristic
215C22×C8C2×#59511yescharacteristic
216C22×C8C2×#595112yescomplemented
217#117—1732no—
218#117—3448no—
219C2×C4⋊C4—1724no—
220C2×C4⋊C4—3412no—
221C2×C22⋊C4—3424no—
222C2×C42—3424no—
223#107—1732no—
224#106—1732no—
225C17C22×#42711yescharacteristic, complemented, Sylow
226C24—171no—
227C24—176no—
228C24—171no—
229C24—176no—
230C24—348no—
231C24C4⋊#59511yescharacteristic
232C22×C4—176no—
233C22×C4—171no—
234C22×C4—176no—
235C22×C4—171no—
236C22×C4—1724no—
237C22×C4—176no—
238C22×C4—176no—
239C22×C4—176no—
240C22×C4—176no—
241C22×C4—173no—
242C22×C4—1712no—
243C22×C4—173no—
244C22×C4—1712no—
245C22×C4—176no—
246C22×C4—173no—
247C22×C4—176no—
248C22×C4—1712no—
249C22×C4—173no—
250C22×C4—174no—
251C22×C4—3424no—
252C22×C4—6816no—
253C22×C4C4⋊#59516yes—
254C22×C4C4⋊#59511yescharacteristic
255C22×C4C22×#59511yescharacteristic
256C22×C4C22×#59516yes—
257C2×C8—176no—
258C2×C8—176no—
259C2×C8—1716no—
260C2×C8—174no—
261C2×C8—174no—
262C2×C8—1712no—
263C2×C8—1712no—
264C2×C8—1716no—
265C2×C8—1716no—
266C2×C8C22×#59516yes—
267C2×C8C22×#59516yes—
268C2×C8C22×#595116yescomplemented
269C4⋊C4—1716no—
270C4⋊C4—3424no—
271C22⋊C4—3432no—
272C42—3432no—
273C23—24no—
274C23—178no—
275C23—173no—
276C23—173no—
277C23—174no—
278C23—1712no—
279C23—176no—
280C23—3424no—
281C23—3424no—
282C23—344no—
283C23C2×C4⋊#59516yes—
284C23C2×C4⋊#59511yescharacteristic
285C2×C4—1712no—
286C2×C4—176no—
287C2×C4—173no—
288C2×C4—174no—
289C2×C4—173no—
290C2×C4—174no—
291C2×C4—173no—
292C2×C4—1712no—
293C2×C4—176no—
294C2×C4—1712no—
295C2×C4—1712no—
296C2×C4—173no—
297C2×C4—174no—
298C2×C4—171no—
299C2×C4—173no—
300C2×C4—174no—
301C2×C4—3416no—
302C2×C4—6848no—
303C2×C4C2×C4⋊#595112yes—
304C2×C4C2×C4⋊#59513yes—
305C2×C4C2×C4⋊#59516yes—
306C2×C4C23×#59513yes—
307C2×C4C23×#59514yes—
308C8—174no—
309C8—174no—
310C8C23×#59514yes—
311C8C23×#59514yes—
312C22—212no—
313C22—1712no—
314C22—1712no—
315C22—171no—
316C22—173no—
317C22—174no—
318C22—348no—
319C22—3424no—
320C22—3412no—
321C22#1176381914yesdirect product factor
322C22C22×C4⋊#59514yes—
323C22C22×C4⋊#59513yes—
324C4—174no—
325C4—173no—
326C4—171no—
327C4—6816no—
328C4C22×C4⋊#59514yes—
329C4C22×C4⋊#59513yes—
330C4C24×#59511yescharacteristic
331C2—24no—
332C2—172no—
333C2—176no—
334C2—344no—
335C2C2×#1176381916yesdirect product factor
336C2C23×C4⋊#59511yescharacteristic
subgroupG/H
centerC22×C4C4⋊#595
commutatorC4×C17C24×C4
FrattiniC4C24×#595
FittingC23×C8×C17C4
socleC23×C17C2×C4⋊C4
prime radicalC24×D17D4
cosocleC4×D17C25
hypercenterC23×C8#595
hypocenterC17C22×#427
LCSG ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C23×C8

Complements

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

NH ≅ G/Ntype
2. C2×#11763819C2direct (N × H)
3. C22×#10587534C2semidirect
4. C22×#10587533C2semidirect
11. #11763819C22direct (N × H)
12. C2×#10587534C22semidirect
13. C2×#10587533C22semidirect
15. C23×#3622C4semidirect
16. C22×C8×D17C4semidirect
17. C22×C8×D17C4semidirect
18. C22×C8×D17C4semidirect
22. C23×C8×C17C4semidirect
43. C22×C8×C17C2×C4semidirect
46. C22×#3622C2×C4semidirect
49. C2×C8×D17C2×C4semidirect
50. C2×C8×D17C2×C4semidirect
51. C2×C8×D17C2×C4semidirect
55. #10587534C23semidirect
56. #10587533C23semidirect
94. C2×C8×C17C22×C4semidirect
97. C2×#3622C22×C4semidirect
100. C8×D17C22×C4semidirect
101. C8×D17C22×C4semidirect
150. C22×C17#427semidirect
156. C2×D17#427semidirect
176. C23×C8#595semidirect
186. C2×C17C2×#427semidirect
189. D17C2×#427semidirect
216. C22×C8C2×#595semidirect
225. C17C22×#427semidirect
268. C2×C8C22×#595semidirect
321. C22#11763819direct (N × H)
335. C2C2×#11763819direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#105876772
C2×#117638192
C22×#105875342
C22×#105875332
C23×C8×D172
C22×#42717