Finite Group Databasebeta

← C22×#11763154 #17903507

Subgroups — C22×#11763154 #17903507

Order 4352.

3086 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587574C211yesmaximal, characteristic, complemented
2C22×#10587626C211yesmaximal, characteristic
3C2×#11763154C2124yesmaximal, direct product factor
4C22×#10587492C211yesmaximal, characteristic, complemented
5C22×#10587503C211yesmaximal, characteristic, complemented
6C22×#10587480C211yesmaximal, characteristic, complemented
7C22×C8×Dic17C211yesmaximal, characteristic, complemented
8C22×C17×#102C211yesmaximal, characteristic, complemented
9C23×Dic34—21no—
10C23×D68C2211yescharacteristic
11C23×C22⋊D17—21no—
12C23×C22⋊D17—21no—
13C23×#3622C2211yescharacteristic
14C2×#10587574—26no—
15C2×#10587574C22112yescomplemented
16C2×#10587626—26no—
17C2×#10587626C22112yes—
18C22×C4⋊Dic17C2211yescharacteristic
19C22×#63225—21no—
20C22×C4×Dic17C2211yescharacteristic
21#11763154C22164yesdirect product factor
22C2×#10587492C22112yescomplemented
23C2×#10587503C22112yescomplemented
24C2×#10587480C22112yescomplemented
25C2×C8×Dic17C22112yescomplemented
26C23×D4×C17C2211yescharacteristic
27C23×C8×C17C2211yescharacteristic
28C22×C4⋊C4×C17C2211yescharacteristic
29C2×C17×#102C22112yescomplemented
30C25×C17—21no—
31C24×D17—21no—
32C22×D4×C17—23no—
33C22×D4×C17—24no—
34C22×D4×C17—24no—
35C22×D4×C17C2316yes—
36C23×C4×C17—21no—
37C23×C4×C17C2311yescharacteristic
38C22×C22⋊D17—212no—
39C22×C22⋊D17—212no—
40C22×C22⋊D17—48no—
41C22×C22⋊D17—48no—
42C23×Dic17—21no—
43C23×Dic17D411yescharacteristic
44C23×Dic17D411yescharacteristic
45C22×D68—24no—
46C22×D68—24no—
47C22×D68—23no—
48C22×D68C2316yes—
49C22×Dic34—212no—
50C22×Dic34—48no—
51C22×C8×C17—24no—
52C22×C8×C17C2316yes—
53C2×C4⋊C4×C17—23no—
54C2×C4⋊C4×C17C2316yes—
55#10587626—224no—
56#10587626C23116yes—
57C2×C4⋊Dic17—23no—
58C2×C4⋊Dic17C2316yes—
59C2×#63225—212no—
60C2×C4×Dic17D416yes—
61C2×C4×Dic17C2316yes—
62C22×#3622—24no—
63C22×#3622C2316yes—
64#10587574—224no—
65#10587574C23116yescomplemented
66C17×#102C23116yescomplemented
67#10587503C23116yescomplemented
68#10587492C23116yescomplemented
69C8×Dic17C23116yescomplemented
70#10587480C23116yescomplemented
71C24×C17—26no—
72C24×C17—44no—
73C24×C17—44no—
74C24×C17—44no—
75C24×C17C2×D411yescharacteristic
76C23×D17—26no—
77C23×D17—44no—
78C23×D17—44no—
79C23×D17—44no—
80C2×D4×C17—26no—
81C2×D4×C17—212no—
82C2×D4×C17—212no—
83C2×D4×C17—412no—
84C2×D4×C17C2414yes—
85C22×C4×C17—26no—
86C22×C4×C17—44no—
87C22×C4×C17C2×D413yes—
88C22×C4×C17#4114yes—
89C22×C4×C17#4114yes—
90C22×C4×C17C2413yes—
91C2×C22⋊D17—216no—
92C2×C22⋊D17—216no—
93C2×C22⋊D17—448no—
94C2×C22⋊D17—448no—
95C22×Dic17—26no—
96C22×Dic17—44no—
97C22×Dic17D818yes—
98C22×Dic17SD1618yes—
99C22×Dic17C2×D416yes—
100C22×Dic17C2×D416yes—
101C2×D68—212no—
102C2×D68—212no—
103C2×D68—26no—
104C2×D68—412no—
105C2×D68C2414yes—
106C2×Dic34—216no—
107C2×Dic34—448no—
108C2×C8×C17—212no—
109C2×C8×C17C2414yes—
110C4⋊C4×C17—26no—
111C4⋊C4×C17C2414yes—
112C4⋊Dic17—26no—
113C4⋊Dic17C2414yes—
114#63225—216no—
115C4×Dic17C2×D4112yes—
116C4×Dic17C2414yes—
117C2×#3622—212no—
118C2×#3622C2414yes—
119C22×#486—171nomaximal, Sylow
120C23×C17—24no—
121C23×C17—412no—
122C23×C17—412no—
123C23×C17—412no—
124C23×C17—816no—
125C23×C17#12414yes—
126C23×C17C2×D814yes—
127C23×C17C2×SD1614yes—
128C23×C17C22×D413yes—
129C22×D17—24no—
130C22×D17—412no—
131C22×D17—412no—
132C22×D17—412no—
133C22×D17—816no—
134D4×C17—24no—
135D4×C17—24no—
136D4×C17—412no—
137C2×C4×C17—24no—
138C2×C4×C17—412no—
139C2×C4×C17#124112yes—
140C2×C4×C17C22×D413yes—
141C2×C4×C17C2×#4116yes—
142C2×C4×C17C2×#4116yes—
143C2×C4×C17C2511yescharacteristic
144C22⋊D17—432no—
145C22⋊D17—432no—
146C2×Dic17—24no—
147C2×Dic17—412no—
148C2×Dic17C2×D8124yes—
149C2×Dic17C2×SD16124yes—
150C2×Dic17C22×D414yes—
151C2×Dic17C22×D414yes—
152D68—24no—
153D68—24no—
154D68—412no—
155Dic34—432no—
156C8×C17—24no—
157#3622—24no—
158C22×C4⋊Q8—171no—
159C22×#127—171no—
160C2×#486—1724no—
161C22×#102—171no—
162C22×#102—171no—
163C22×#102—171no—
164C22×#102D1711yescharacteristic, complemented
165C22×C4×C8—171no—
166C22×C17—44no—
167C22×C17—44no—
168C22×C17—44no—
169C22×C17—824no—
170C22×C17#486116yescomplemented
171C22×C17C2×#12416yes—
172C22×C17C22×D816yes—
173C22×C17C22×SD1616yes—
174C22×C17C23×D411yescharacteristic
175C2×D17—44no—
176C2×D17—44no—
177C2×D17—44no—
178C2×D17—824no—
179C4×C17—44no—
180C4×C17C2×#12416yes—
181C4×C17C22×#4111yescharacteristic
182C4×C17C22×#4111yescharacteristic
183Dic17—44no—
184Dic17C22×D818yes—
185Dic17C22×SD1618yes—
186C23×Q8—341no—
187C23×D4—171no—
188C23×D4—341no—
189C23×D4—341no—
190C23×D4C2×D1711yescharacteristic
191C23×C8—171no—
192C23×C8C2×D1711yescharacteristic
193C2×C4⋊Q8—1712no—
194C2×C4⋊Q8—346no—
195C2×#127—1712no—
196C2×#127—346no—
197C22×C4⋊C4—171no—
198C22×C4⋊C4—341no—
199C22×C4⋊C4C2×D1711yescharacteristic
200C22×C42—171no—
201#486—1764no—
202C2×#102—1712no—
203C2×#102—1712no—
204C2×#102—1712no—
205C2×#102C2×D17112yescomplemented
206C2×C4×C8—1712no—
207C2×C17—84no—
208C2×C17C2×#486112yescomplemented
209C2×C17C22×#12411yescharacteristic
210C2×C17C23×D811yescharacteristic
211C2×C17C23×SD1611yescharacteristic
212D17—84no—
213C25—21no—
214C25—341no—
215C22×Q8—3412no—
216C22×Q8—688no—
217C22×D4—24no—
218C22×D4—23no—
219C22×D4—24no—
220C22×D4—176no—
221C22×D4—3412no—
222C22×D4—3412no—
223C22×D4—344no—
224C22×D4—343no—
225C22×D4—344no—
226C22×D4—688no—
227C22×D4—688no—
228C22×D4C22×D1716yes—
229C23×C4—21no—
230C23×C4—171no—
231C23×C4—171no—
232C23×C4—341no—
233C23×C4C22×D1711yescharacteristic
234C22×C8—24no—
235C22×C8—176no—
236C22×C8—344no—
237C22×C8C22×D1716yes—
238C4⋊Q8—1716no—
239C4⋊Q8—3424no—
240#127—1716no—
241#127—3424no—
242C2×C4⋊C4—23no—
243C2×C4⋊C4—176no—
244C2×C4⋊C4—3412no—
245C2×C4⋊C4—343no—
246C2×C4⋊C4C22×D1716yes—
247C2×C42—176no—
248C2×C42—176no—
249#102—1716no—
250#102—1716no—
251#102—1716no—
252#102C22×D17116yescomplemented
253C4×C8—1716no—
254C17C22×#48611yescharacteristic, complemented, Sylow
255C24—26no—
256C24—44no—
257C24—44no—
258C24—44no—
259C24—346no—
260C24—684no—
261C24—684no—
262C24—684no—
263C24D4×D1711yescharacteristic
264C2×Q8—3416no—
265C2×Q8—6848no—
266C2×D4—212no—
267C2×D4—26no—
268C2×D4—212no—
269C2×D4—412no—
270C2×D4—174no—
271C2×D4—3416no—
272C2×D4—3416no—
273C2×D4—3412no—
274C2×D4—346no—
275C2×D4—3412no—
276C2×D4—6848no—
277C2×D4—6848no—
278C2×D4—6812no—
279C2×D4C23×D1714yes—
280C22×C4—26no—
281C22×C4—44no—
282C22×C4—176no—
283C22×C4—176no—
284C22×C4—178no—
285C22×C4—178no—
286C22×C4—346no—
287C22×C4—684no—
288C22×C4#6325214yes—
289C22×C4D4×D1713yes—
290C22×C4#6325414yes—
291C22×C4C23×D1713yes—
292C2×C8—212no—
293C2×C8—174no—
294C2×C8—3412no—
295C2×C8C23×D1714yes—
296C4⋊C4—26no—
297C4⋊C4—174no—
298C4⋊C4—3416no—
299C4⋊C4—346no—
300C4⋊C4C23×D1714yes—
301C42—1712no—
302C42—174no—
303C23—24no—
304C23—412no—
305C23—412no—
306C23—412no—
307C23—816no—
308C23—344no—
309C23—6812no—
310C23—6812no—
311C23—6812no—
312C23—13616no—
313C23#1058757114yes—
314C23D8×D1714yes—
315C23SD16×D1714yes—
316C23C2×D4×D1713yes—
317Q8—6832no—
318D4—24no—
319D4—24no—
320D4—412no—
321D4—344no—
322D4—344no—
323D4—6832no—
324D4—6832no—
325D4—6812no—
326C2×C4—24no—
327C2×C4—412no—
328C2×C4—1724no—
329C2×C4—1724no—
330C2×C4—174no—
331C2×C4—174no—
332C2×C4—344no—
333C2×C4—6812no—
334C2×C4#10587571112yes—
335C2×C4C2×#6325216yes—
336C2×C4C2×D4×D1713yes—
337C2×C4C2×#6325416yes—
338C2×C4C24×D1711yescharacteristic
339C8—24no—
340C8—344no—
341C22—44no—
342C22—44no—
343C22—44no—
344C22—824no—
345C22—684no—
346C22—684no—
347C22—684no—
348C22—13624no—
349C22#11763154116yesdirect product factor
350C22C2×#1058757116yes—
351C22C2×D8×D1716yes—
352C22C2×SD16×D1716yes—
353C22C22×D4×D1711yescharacteristic
354C4—44no—
355C4—178no—
356C4—178no—
357C4—684no—
358C4C2×#1058757116yes—
359C4C22×#6325411yescharacteristic
360C4C22×#6325211yescharacteristic
361C2—84no—
362C2—1364no—
363C2C2×#11763154112yesdirect product factor
364C2C22×#1058757111yescharacteristic
365C2C22×D8×D1711yescharacteristic
366C2C22×SD16×D1711yescharacteristic
subgroupG/H
centerC24D4×D17
commutator, cosocleC2×C4×C17C25
FrattiniC2×C4C24×D17
FittingC22×C17×#102C2
socleC24×C17C2×D4
prime radicalC22×#10587626C2
hypercenterC22×#102D17
hypocenterC17C22×#486
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C24 ◁ C23×C4 ◁ C22×#102

Complements

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

NH ≅ G/Ntype
1. C22×#10587574C2semidirect
3. C2×#11763154C2direct (N × H)
4. C22×#10587492C2semidirect
5. C22×#10587503C2semidirect
6. C22×#10587480C2semidirect
7. C22×C8×Dic17C2semidirect
8. C22×C17×#102C2semidirect
15. C2×#10587574C22semidirect
21. #11763154C22direct (N × H)
22. C2×#10587492C22semidirect
23. C2×#10587503C22semidirect
24. C2×#10587480C22semidirect
25. C2×C8×Dic17C22semidirect
29. C2×C17×#102C22semidirect
65. #10587574C23semidirect
66. C17×#102C23semidirect
67. #10587503C23semidirect
68. #10587492C23semidirect
69. C8×Dic17C23semidirect
70. #10587480C23semidirect
164. C22×#102D17semidirect
170. C22×C17#486semidirect
205. C2×#102C2×D17semidirect
208. C2×C17C2×#486semidirect
252. #102C22×D17semidirect
254. C17C22×#486semidirect
349. C22#11763154direct (N × H)
363. C2C2×#11763154direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#105875742
C22×#105876262
C2×#117631542
C22×#105874922
C22×#105875032
C22×#105874802
C22×C8×Dic172
C22×C17×#1022
C22×#48617