Finite Group Databasebeta

← C2×#14536845 #17240885

Subgroups — C2×#14536845 #17240885

Order 4352.

2370 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587555C211yesmaximal, characteristic, complemented
2C22×#10587562C211yesmaximal, characteristic
3#14536845C2116yesmaximal, direct product factor
4C2×#11762749C212yesmaximal, complemented
5C2×#11762746C212yesmaximal
6C2×#11762741C212yesmaximal, complemented
7C2×#11762736C212yesmaximal
8C2×#11762730C212yesmaximal, complemented
9C2×#11762722C212yesmaximal
10C2×C17×#383C211yesmaximal, characteristic, complemented
11C23×Dic34C2212yes—
12C2×#10587555—28no—
13C2×#10587555C2218yescomplemented
14C2×#10587562—28no—
15C2×#10587562C2218yes—
16C22×C4⋊Dic17C2212yes—
17C22×C4⋊Dic17C2212yes—
18C22×C4⋊Dic17C2214yes—
19C22×#63225C2211yescharacteristic
20C22×#63225C2211yescharacteristic
21C22×#63225C2211yescharacteristic
22C22×#63227C2211yescharacteristic
23C22×#63227C2211yescharacteristic
24C22×#63227C2211yescharacteristic
25C22×#63227C2214yes—
26#11762749C22116yescomplemented
27#11762746C22116yes—
28#11762741C22116yescomplemented
29#11762736C22116yes—
30#11762730C22116yescomplemented
31#11762722C22116yes—
32C2×#10587502C2214yes—
33C2×#10587502C2214yes—
34C22×C42×C17C2211yescharacteristic
35C17×#383C2218yescomplemented
36C2×C17×#95C2212yes—
37C2×C17×#95C2212yes—
38C2×C17×#95C2212yes—
39C24×D17C2311yescharacteristic
40C23×C4×C17C2314yes—
41C23×C4×C17C2311yescharacteristic
42C23×C4×C17C2311yescharacteristic
43C23×C4×C17C2311yescharacteristic
44C23×Dic17C2311yescharacteristic
45C23×Dic17C2311yescharacteristic
46C23×Dic17C2311yescharacteristic
47C23×Dic17C2314yes—
48C22×Dic34—28no—
49C22×Dic34—48no—
50C22×Dic34C2318yes—
51C2×C42×C17D418yes—
52C2×C42×C17C2314yes—
53C2×#63227—24no—
54C2×#63227—24no—
55C2×#63227—24no—
56C2×#63227—48no—
57C2×#63227C2314yes—
58C2×#63227C2314yes—
59C2×#63227C2314yes—
60C2×#63227C23116yes—
61C2×C4⋊Dic17—28no—
62C2×C4⋊Dic17—28no—
63C2×C4⋊Dic17—48no—
64C2×C4⋊Dic17C2318yes—
65C2×C4⋊Dic17C2318yes—
66C2×C4⋊Dic17C23116yes—
67C2×#63225—24no—
68C2×#63225—24no—
69C2×#63225—24no—
70C2×#63225C2314yes—
71C2×#63225C2314yes—
72C2×#63225C2314yes—
73#10587562—232no—
74#10587555—232no—
75C17×#95C2318yes—
76C17×#95C2318yes—
77C17×#95C2318yes—
78#10587502C23116yes—
79#10587502C23116yes—
80C24×C17C2411yescharacteristic
81C23×D17—22no—
82C23×D17—24no—
83C23×D17—24no—
84C23×D17—24no—
85C23×D17C2412yes—
86C22×C4×C17—24no—
87C22×C4×C17—24no—
88C22×C4×C17—24no—
89C22×C4×C17—28no—
90C22×C4×C17—28no—
91C22×C4×C17—28no—
92C22×C4×C17C2×D414yes—
93C22×C4×C17C2×D414yes—
94C22×C4×C17C2×D414yes—
95C22×C4×C17C2418yes—
96C22×C4×C17C2412yes—
97C22×C4×C17C2412yes—
98C22×C4×C17C2412yes—
99C22×Dic17—28no—
100C22×Dic17—22no—
101C22×Dic17—22no—
102C22×Dic17—22no—
103C22×Dic17—28no—
104C22×Dic17—28no—
105C22×Dic17—24no—
106C22×Dic17—24no—
107C22×Dic17—24no—
108C22×Dic17C2418yes—
109C22×Dic17C2412yes—
110C22×Dic17C2412yes—
111C22×Dic17C2412yes—
112C2×Dic34—216no—
113C2×Dic34—416no—
114C2×Dic34—432no—
115C42×C17C2×D4116yes—
116#63227—28no—
117#63227—28no—
118#63227—28no—
119#63227—416no—
120C4⋊Dic17—216no—
121C4⋊Dic17—216no—
122C4⋊Dic17—416no—
123#63225—28no—
124#63225—28no—
125#63225—28no—
126C2×#2821—171nomaximal, Sylow
127C23×C17C22×D412yes—
128C23×C17H(5,𝔽2)14yes—
129C23×C172−1+414yes—
130C23×C172−1+414yes—
131C23×C17C2511yescharacteristic
132C22×D17—22no—
133C22×D17—24no—
134C22×D17—24no—
135C22×D17—24no—
136C22×D17—44no—
137C22×D17—44no—
138C22×D17—44no—
139C22×D17—416no—
140C2×C4×C17—28no—
141C2×C4×C17—28no—
142C2×C4×C17—28no—
143C2×C4×C17—24no—
144C2×C4×C17—24no—
145C2×C4×C17—24no—
146C2×C4×C17—28no—
147C2×C4×C17—28no—
148C2×C4×C17—28no—
149C2×C4×C17—416no—
150C2×C4×C17C22×D414yes—
151C2×C4×C17C22×D414yes—
152C2×C4×C17C22×D414yes—
153C2×Dic17—28no—
154C2×Dic17—22no—
155C2×Dic17—22no—
156C2×Dic17—22no—
157C2×Dic17—28no—
158C2×Dic17—28no—
159C2×Dic17—24no—
160C2×Dic17—24no—
161C2×Dic17—24no—
162C2×Dic17—416no—
163C2×Dic17—44no—
164C2×Dic17—44no—
165C2×Dic17—44no—
166Dic34—432no—
167C22×C4⋊Q8—171no—
168C22×#126—171no—
169#2821—1716no—
170C2×#400—172no—
171C2×#399—172no—
172C2×#398—172no—
173C2×#397—172no—
174C2×#395—172no—
175C2×#393—172no—
176C2×#383D1711yescharacteristic, complemented
177C22×C17#53514yes—
178C22×C17#53614yes—
179C22×C17#53614yes—
180C22×C17#563116yes—
181C22×C17C23×D411yescharacteristic
182C22×C17C2×H(5,𝔽2)12yes—
183C22×C17C2×2−1+412yes—
184C22×C17C2×2−1+412yes—
185C2×D17—42no—
186C2×D17—42no—
187C2×D17—42no—
188C2×D17—48no—
189C2×D17—88no—
190C4×C17—24no—
191C4×C17—24no—
192C4×C17—24no—
193C4×C17—48no—
194Dic17—48no—
195Dic17—42no—
196Dic17—42no—
197Dic17—42no—
198C23×Q8—172no—
199C2×C4⋊Q8—178no—
200C2×C4⋊Q8—348no—
201C2×#126—178no—
202C2×#126—348no—
203C22×C4⋊C4—171no—
204C22×C4⋊C4—171no—
205C22×C4⋊C4—171no—
206C22×C4⋊C4—172no—
207C22×C4⋊C4—172no—
208C22×C4⋊C4—174no—
209C22×C22⋊C4—171no—
210C22×C22⋊C4—171no—
211C22×C22⋊C4—171no—
212C22×C22⋊C4—174no—
213C22×C42C2×D1711yescharacteristic
214#400—1716no—
215#399—1716no—
216#398—1716no—
217#397—1716no—
218#395—1716no—
219#393—1716no—
220#383C2×D1718yescomplemented
221C2×#95—174no—
222C2×#95—174no—
223C2×#95C2×D1712yes—
224C2×#95C2×D1712yes—
225C2×#95C2×D1712yes—
226C2×C17#282118yescomplemented
227C2×C17C2×#53511yescharacteristic
228C2×C17C2×#53611yescharacteristic
229C2×C17C2×#53611yescharacteristic
230C2×C17C2×#56314yes—
231D17—82no—
232C25—171no—
233C22×Q8—178no—
234C22×Q8—348no—
235C22×Q8—688no—
236C23×C4—171no—
237C23×C4—174no—
238C23×C4—171no—
239C23×C4—171no—
240C23×C4C22×D1714yes—
241C23×C4C22×D1711yescharacteristic
242C23×C4C22×D1711yescharacteristic
243C23×C4C22×D1711yescharacteristic
244C4⋊Q8—3432no—
245#126—3432no—
246C2×C4⋊C4—174no—
247C2×C4⋊C4—174no—
248C2×C4⋊C4—174no—
249C2×C4⋊C4—178no—
250C2×C4⋊C4—178no—
251C2×C4⋊C4—1716no—
252C2×C4⋊C4—344no—
253C2×C4⋊C4—344no—
254C2×C4⋊C4—344no—
255C2×C4⋊C4—348no—
256C2×C4⋊C4—348no—
257C2×C4⋊C4—688no—
258C2×C22⋊C4—174no—
259C2×C22⋊C4—174no—
260C2×C22⋊C4—174no—
261C2×C22⋊C4—1716no—
262C2×C22⋊C4—344no—
263C2×C22⋊C4—344no—
264C2×C22⋊C4—344no—
265C2×C22⋊C4—688no—
266C2×C42D6818yes—
267C2×C42C22×D1714yes—
268#95—1716no—
269#95—1716no—
270#95C22×D1718yes—
271#95C22×D1718yes—
272#95C22×D1718yes—
273C17C2×#282111yescharacteristic, complemented, Sylow
274C24—172no—
275C24—342no—
276C24—344no—
277C24—344no—
278C24—344no—
279C24C23×D1711yescharacteristic
280C2×Q8—3416no—
281C2×Q8—6816no—
282C2×Q8—6832no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—28no—
287C22×C4—28no—
288C22×C4—28no—
289C22×C4—172no—
290C22×C4—172no—
291C22×C4—178no—
292C22×C4—172no—
293C22×C4—342no—
294C22×C4—342no—
295C22×C4—342no—
296C22×C4—348no—
297C22×C4—348no—
298C22×C4—344no—
299C22×C4—348no—
300C22×C4—344no—
301C22×C4—344no—
302C22×C4C2×D6814yes—
303C22×C4C2×D6814yes—
304C22×C4C2×D6814yes—
305C22×C4C23×D1718yes—
306C22×C4C23×D1712yes—
307C22×C4C23×D1712yes—
308C22×C4C23×D1712yes—
309C4⋊C4—348no—
310C4⋊C4—348no—
311C4⋊C4—348no—
312C4⋊C4—3416no—
313C4⋊C4—3416no—
314C4⋊C4—6816no—
315C22⋊C4—348no—
316C22⋊C4—348no—
317C22⋊C4—348no—
318C22⋊C4—6816no—
319C42C2×D68116yes—
320C23—342no—
321C23—344no—
322C23—344no—
323C23—344no—
324C23—6816no—
325C23—684no—
326C23—684no—
327C23—684no—
328C23C22×D6812yes—
329C23#1058769614yes—
330C23#1058769714yes—
331C23#1058769714yes—
332C23C24×D1711yescharacteristic
333Q8—6832no—
334C2×C4—28no—
335C2×C4—24no—
336C2×C4—28no—
337C2×C4—28no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—28no—
341C2×C4—28no—
342C2×C4—28no—
343C2×C4—416no—
344C2×C4—342no—
345C2×C4—342no—
346C2×C4—342no—
347C2×C4—348no—
348C2×C4—348no—
349C2×C4—344no—
350C2×C4—348no—
351C2×C4—344no—
352C2×C4—344no—
353C2×C4—6816no—
354C2×C4—684no—
355C2×C4—684no—
356C2×C4—684no—
357C2×C4C22×D6814yes—
358C2×C4C22×D6814yes—
359C2×C4C22×D6814yes—
360C22—682no—
361C22—682no—
362C22—682no—
363C22—688no—
364C22—1368no—
365C22#1176337814yes—
366C22#1176338514yes—
367C22#1176338514yes—
368C22#11763561116yes—
369C22C23×D6811yescharacteristic
370C22C2×#1058769612yes—
371C22C2×#1058769712yes—
372C22C2×#1058769712yes—
373C4—24no—
374C4—24no—
375C4—24no—
376C4—48no—
377C4—688no—
378C4—682no—
379C4—682no—
380C4—682no—
381C2—1362no—
382C2#1453684518yesdirect product factor
383C2C2×#1176337811yescharacteristic
384C2C2×#1176338511yescharacteristic
385C2C2×#1176338511yescharacteristic
386C2C2×#1176356114yes—
subgroupG/H
centerC24C23×D17
commutator, cosocleC23×C17C25
FrattiniC23C24×D17
FittingC2×C17×#383C2
socleC24×C17C24
prime radicalC24×D17C23
hypercenterC2×#383D17
hypocenterC17C2×#2821
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C2×#383

Complements

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

NH ≅ G/Ntype
1. C22×#10587555C2semidirect
3. #14536845C2direct (N × H)
4. C2×#11762749C2semidirect
6. C2×#11762741C2semidirect
8. C2×#11762730C2semidirect
10. C2×C17×#383C2semidirect
13. C2×#10587555C22semidirect
26. #11762749C22semidirect
28. #11762741C22semidirect
30. #11762730C22semidirect
35. C17×#383C22semidirect
176. C2×#383D17semidirect
220. #383C2×D17semidirect
226. C2×C17#2821semidirect
273. C17C2×#2821semidirect
382. C2#14536845direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#105875552
C22×#105875622
#145368452
C2×#117627492
C2×#117627462
C2×#117627412
C2×#117627362
C2×#117627302
C2×#117627222
C2×C17×#3832
C2×#282117