Finite Group Databasebeta

← C22×#11763775 #17979412

Subgroups — C22×#11763775 #17979412

Order 4352.

3518 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#63242C211yesmaximal, characteristic
2C22×#10587677C211yesmaximal, characteristic
3C2×#11763775C2124yesmaximal, direct product factor
4C22×#10587532C212yesmaximal, complemented
5C22×C8×#595C212yesmaximal, complemented
6C22×M16×D17C211yesmaximal, characteristic
7C22×#63242C4112yescomplemented
8C22×#63242C412yescomplemented
9C22×#63242C2211yescharacteristic
10C22×#63242C2211yescharacteristic
11C22×#63242C22112yes—
12C23×#3624C412yescomplemented
13C2×#10587677C4112yes—
14C2×#10587677C22112yes—
15C22×C4⋊#595C412yes—
16C22×C22⋊#595C412yes—
17C22×C4×#595C2212yes—
18#11763775C22164yesdirect product factor
19C2×#10587532C22124yescomplemented
20C2×C8×#595C22124yescomplemented
21C23×C4×D17C2211yescharacteristic
22C22×#63223C411yescharacteristic, complemented
23C22×#63218C412yescomplemented
24C2×M16×D17C4112yescomplemented
25C2×M16×D17C22112yes—
26C22×C8×D17C2212yes—
27C22×M16×C17C411yescharacteristic, complemented
28C24×D17C2×C411yescharacteristic
29C23×#595C2×C414yes—
30C23×C4×C17C2×C411yescharacteristic
31C23×Dic17C2×C411yescharacteristic
32C22×C4×D17C2×C411yescharacteristic
33C22×C4×D17C2×C416yes—
34C22×C4×D17C2×C416yes—
35C22×C4×D17C2×C411yescharacteristic
36C22×C4×D17C2×C416yes—
37C22×C4×D17C2×C411yescharacteristic
38C22×C4×D17C2311yescharacteristic
39C22×C4×D17C2316yes—
40C2×C22⋊#595C2×C4124yes—
41C22×#3624C2×C412yes—
42C22×#3624C2×C412yes—
43C22×#3624C2×C4124yescomplemented
44#10587677C2×C4148yes—
45#10587677C23116yes—
46C2×C4⋊#595C2×C4124yes—
47C2×C4×#595C2×C4112yes—
48C2×C4×#595C23112yes—
49C2×#63242C2×C416yes—
50C2×#63242C2×C416yes—
51C2×#63242C2×C4148yescomplemented
52C2×#63242C2×C4124yescomplemented
53C2×#63242C2316yes—
54C2×#63242C2316yes—
55C2×#63242C23116yes—
56C2×M16×C17C2×C4112yescomplemented
57C22×C8×C17C2×C412yes—
58C2×#63223C2×C4112yescomplemented
59C22×#3622C2×C412yes—
60M16×D17C2×C4148yescomplemented
61M16×D17C23116yes—
62C2×#63218C2×C4124yescomplemented
63C2×C8×D17C2×C4112yes—
64C2×C8×D17C23112yes—
65#10587532C23132yescomplemented
66C8×#595C23132yescomplemented
67C24×C17C4211yescharacteristic
68C23×D17—28no—
69C23×D17C4211yescharacteristic
70C23×D17C4216yes—
71C23×D17C22×C416yes—
72C23×D17C22×C411yescharacteristic
73C22×#595—216no—
74C22×#595C22×C4124yes—
75C22×C4×C17C4216yes—
76C22×C4×C17C4211yescharacteristic
77C22×C4×C17C22×C416yes—
78C22×C4×C17C22×C411yescharacteristic
79C22×Dic17C4216yes—
80C22×Dic17C4211yescharacteristic
81C22×Dic17C22×C411yescharacteristic
82C22×Dic17C22×C416yes—
83C2×C4×D17C42124yes—
84C2×C4×D17C4216yes—
85C2×C4×D17C4216yes—
86C2×C4×D17C4216yes—
87C2×C4×D17C22×C413yes—
88C2×C4×D17C22×C416yes—
89C2×C4×D17C22×C4112yes—
90C2×C4×D17C22×C413yes—
91C2×C4×D17C22×C416yes—
92C2×C4×D17C22×C4112yes—
93C2×C4×D17C22×C413yes—
94C2×C4×D17C22×C4112yes—
95C2×C4×D17C22×C416yes—
96C2×C4×D17C2413yes—
97C2×C4×D17C2414yes—
98C22⋊#595C22×C4132yes—
99C2×#3624C22×C4112yes—
100C2×#3624C22×C4112yes—
101C2×#3624C22×C4132yescomplemented
102C4⋊#595C22×C4132yes—
103C4×#595C22×C4124yes—
104C4×#595C2418yes—
105#63242C22×C4112yes—
106#63242C22×C4112yes—
107#63242C22×C4132yescomplemented
108#63242C2414yes—
109#63242C2414yes—
110M16×C17C22×C4116yescomplemented
111C2×C8×C17C22×C4112yes—
112#63223C22×C4116yescomplemented
113C2×#3622C22×C4112yes—
114#63218C22×C4132yescomplemented
115C8×D17C22×C4124yes—
116C8×D17C2418yes—
117C22×#405—171nomaximal, Sylow
118C23×C17—24no—
119C23×C17C2×C4216yes—
120C23×C17C2×C4211yescharacteristic
121C22×D17—24no—
122C22×D17—224no—
123C22×D17—224no—
124C22×D17C2×C4216yes—
125C22×D17C2×C42112yes—
126C22×D17C2×C4213yes—
127C22×D17#13118yes—
128C22×D17C23×C414yes—
129C22×D17C23×C413yes—
130C2×#595—248no—
131C2×#595C23×C4116yes—
132C2×C4×C17C2×C42112yes—
133C2×C4×C17C2×C4216yes—
134C2×C4×C17C2×C4213yes—
135C2×C4×C17C23×C413yes—
136C2×C4×C17C23×C414yes—
137C2×Dic17C2×C42112yes—
138C2×Dic17C2×C4213yes—
139C2×Dic17C2×C4216yes—
140C2×Dic17C23×C413yes—
141C2×Dic17C23×C414yes—
142C4×D17C2×C42112yes—
143C4×D17C2×C42112yes—
144C4×D17C2×C4216yes—
145C4×D17C2×C42112yes—
146C4×D17C23×C413yes—
147C4×D17C23×C414yes—
148C4×D17C23×C413yes—
149C4×D17C23×C414yes—
150C4×D17C23×C413yes—
151C4×D17C23×C414yes—
152C4×D17C2511yescharacteristic
153#3624C23×C418yes—
154#3624C23×C418yes—
155C8×C17C23×C418yes—
156#3622C23×C418yes—
157C23×M16—171no—
158C24×C8—171no—
159C22×#117—171no—
160C2×#405—1724no—
161C22×#97—172no—
162C22×C4×C8—172no—
163C22×C17—212no—
164C22×C17#40514yescomplemented
165C22×C17C22×C4214yes—
166C22×C17C22×C4213yes—
167C2×D17—212no—
168C2×D17—224no—
169C2×D17—28no—
170C2×D17#405112yescomplemented
171C2×D17C22×C4214yes—
172C2×D17C22×C4213yes—
173C2×D17C2×#131112yes—
174C2×D17C24×C411yescharacteristic
175#595—216no—
176C4×C17C22×C4213yes—
177C4×C17C22×C4214yes—
178C4×C17C24×C411yescharacteristic
179Dic17C22×C4213yes—
180Dic17C22×C4214yes—
181Dic17C24×C411yescharacteristic
182C24×C4—171no—
183C22×M16—171no—
184C22×M16—1712no—
185C22×M16—1712no—
186C22×M16—172no—
187C22×M16#59511yescharacteristic, complemented
188C23×C8—171no—
189C23×C8—171no—
190C23×C8—1712no—
191C23×C8—172no—
192C23×C8—1712no—
193C23×C8—172no—
194C23×C8—172no—
195C2×#117—1712no—
196C2×#117—1712no—
197C22×C4⋊C4—172no—
198C22×C22⋊C4—172no—
199C22×C42—172no—
200#405—1764no—
201C2×#97—1724no—
202C2×C4×C8—1724no—
203C2×C17—24no—
204C2×C17C2×#40516yescomplemented
205C2×C17C23×C4211yescharacteristic
206D17—24no—
207D17C2×#40516yescomplemented
208D17C22×#13112yes—
209C25—171no—
210C23×C4—171no—
211C23×C4—171no—
212C23×C4—176no—
213C23×C4—176no—
214C23×C4—171no—
215C23×C4—176no—
216C23×C4—176no—
217C23×C4—171no—
218C23×C4—171no—
219C23×C4—174no—
220C23×C4C2×#59511yescharacteristic
221C2×M16—1712no—
222C2×M16—1716no—
223C2×M16—1748no—
224C2×M16—1724no—
225C2×M16C2×#595112yescomplemented
226C22×C8—176no—
227C22×C8—176no—
228C22×C8—1748no—
229C22×C8—1724no—
230C22×C8—176no—
231C22×C8—176no—
232C22×C8—1716no—
233C22×C8—172no—
234C22×C8—172no—
235C22×C8—1724no—
236C22×C8—172no—
237C22×C8—1712no—
238C22×C8—1712no—
239C22×C8C2×#59512yes—
240#117—1748no—
241#117—1716no—
242C2×C4⋊C4—1724no—
243C2×C22⋊C4—1724no—
244C2×C42—1712no—
245C2×C42—1712no—
246#97—1732no—
247C4×C8—1732no—
248C17C22×#40511yescharacteristic, complemented, Sylow
249C24—171no—
250C24—176no—
251C24—171no—
252C24—176no—
253C24—348no—
254C24C4×#59511yescharacteristic
255C22×C4—176no—
256C22×C4—171no—
257C22×C4—176no—
258C22×C4—171no—
259C22×C4—176no—
260C22×C4—1724no—
261C22×C4—176no—
262C22×C4—176no—
263C22×C4—173no—
264C22×C4—176no—
265C22×C4—1712no—
266C22×C4—1712no—
267C22×C4—176no—
268C22×C4—173no—
269C22×C4—173no—
270C22×C4—1712no—
271C22×C4—176no—
272C22×C4—173no—
273C22×C4—174no—
274C22×C4—1724no—
275C22×C4—3416no—
276C22×C4C4×#59511yescharacteristic
277C22×C4C4×#59516yes—
278C22×C4C22×#59516yes—
279C22×C4C22×#59511yescharacteristic
280M16—1716no—
281M16—1732no—
282M16C22×#595116yescomplemented
283C2×C8—1712no—
284C2×C8—1712no—
285C2×C8—1732no—
286C2×C8—174no—
287C2×C8—174no—
288C2×C8—1712no—
289C2×C8—1712no—
290C2×C8—1732no—
291C2×C8—1712no—
292C2×C8—178no—
293C2×C8—1724no—
294C2×C8C22×#595112yes—
295C4⋊C4—1732no—
296C22⋊C4—1732no—
297C42—1724no—
298C42—178no—
299C23—24no—
300C23—178no—
301C23—173no—
302C23—173no—
303C23—174no—
304C23—1712no—
305C23—176no—
306C23—3424no—
307C23—3424no—
308C23—344no—
309C23C2×C4×#59516yes—
310C23C2×C4×#59511yescharacteristic
311C2×C4—174no—
312C2×C4—173no—
313C2×C4—1712no—
314C2×C4—173no—
315C2×C4—176no—
316C2×C4—173no—
317C2×C4—174no—
318C2×C4—173no—
319C2×C4—174no—
320C2×C4—176no—
321C2×C4—1712no—
322C2×C4—1712no—
323C2×C4—1712no—
324C2×C4—171no—
325C2×C4—173no—
326C2×C4—174no—
327C2×C4—1716no—
328C2×C4—3448no—
329C2×C4C2×C4×#595112yes—
330C2×C4C2×C4×#59516yes—
331C2×C4C2×C4×#59513yes—
332C2×C4C23×#59513yes—
333C2×C4C23×#59514yes—
334C8—178no—
335C8—178no—
336C8—178no—
337C8C23×#59518yes—
338C22—212no—
339C22—171no—
340C22—173no—
341C22—1712no—
342C22—1712no—
343C22—174no—
344C22—348no—
345C22—3424no—
346C22—3412no—
347C22#1176377514yesdirect product factor
348C22C22×C4×#59514yes—
349C22C22×C4×#59513yes—
350C4—171no—
351C4—173no—
352C4—174no—
353C4—3416no—
354C4C22×C4×#59513yes—
355C4C22×C4×#59514yes—
356C4C24×#59511yescharacteristic
357C2—24no—
358C2—172no—
359C2—176no—
360C2—344no—
361C2C2×#1176377516yesdirect product factor
362C2C23×C4×#59511yescharacteristic
subgroupG/H
centerC22×C4C22×#595
commutatorC2×C17C23×C42
FrattiniC4C24×#595
FittingC22×M16×C17C4
socleC23×C17C2×C42
prime radicalC24×D17C2×C4
cosocleC4×D17C25
hypercenterC22×M16#595
hypocenterC17C22×#405
LCSG ▷ C2×C17 ▷ C17
UCSC1 ◁ C22×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
3. C2×#11763775C2direct (N × H)
4. C22×#10587532C2semidirect
5. C22×C8×#595C2semidirect
7. C22×#63242C4semidirect
8. C22×#63242C4semidirect
12. C23×#3624C4semidirect
18. #11763775C22direct (N × H)
19. C2×#10587532C22semidirect
20. C2×C8×#595C22semidirect
22. C22×#63223C4semidirect
23. C22×#63218C4semidirect
24. C2×M16×D17C4semidirect
27. C22×M16×C17C4semidirect
43. C22×#3624C2×C4semidirect
51. C2×#63242C2×C4semidirect
52. C2×#63242C2×C4semidirect
56. C2×M16×C17C2×C4semidirect
58. C2×#63223C2×C4semidirect
60. M16×D17C2×C4semidirect
62. C2×#63218C2×C4semidirect
65. #10587532C23semidirect
66. C8×#595C23semidirect
101. C2×#3624C22×C4semidirect
107. #63242C22×C4semidirect
110. M16×C17C22×C4semidirect
112. #63223C22×C4semidirect
114. #63218C22×C4semidirect
164. C22×C17#405semidirect
170. C2×D17#405semidirect
187. C22×M16#595semidirect
204. C2×C17C2×#405semidirect
207. D17C2×#405semidirect
225. C2×M16C2×#595semidirect
248. C17C22×#405semidirect
282. M16C22×#595semidirect
347. C22#11763775direct (N × H)
361. C2C2×#11763775direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#632422
C22×#105876772
C2×#117637752
C22×#105875322
C22×C8×#5952
C22×M16×D172
C22×#40517