Finite Group Databasebeta

← C2×#14534733 #17201558

Subgroups — C2×#14534733 #17201558

Order 4352.

2286 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587630C211yesmaximal, characteristic
2C22×#10587576C211yesmaximal, characteristic
3#14534733C2116yesmaximal, direct product factor
4C2×#11762751C214yesmaximal
5C2×#11762750C214yesmaximal
6C2×#11762716C211yesmaximal, characteristic
7C2×#11762715C211yesmaximal, characteristic
8C2×#11762714C211yesmaximal, characteristic
9C2×#11762655C211yesmaximal, characteristic
10C2×C17×#391C211yesmaximal, characteristic
11C23×Dic34C2211yescharacteristic
12C2×#10587630—28no—
13C2×#10587630C2218yes—
14C2×#10587576—28no—
15C2×#10587576C2218yes—
16C22×C4⋊Dic17C2211yescharacteristic
17C22×C4⋊Dic17C2212yes—
18C22×#63225C2212yes—
19C22×#63225C2212yes—
20C22×#63225C2212yes—
21C22×C4×Dic17C2211yescharacteristic
22C22×C4×Dic17C2211yescharacteristic
23#11762751C22132yes—
24#11762750C22132yes—
25#11762716C2218yes—
26#11762715C2218yes—
27#11762714C2218yes—
28#11762655C2218yes—
29C2×#10587502C2212yes—
30C2×#10587502C2212yes—
31C2×#10587502C2214yes—
32C2×#10587502C2214yes—
33C2×#10587502C2214yes—
34C23×Q8×C17C2211yescharacteristic
35C22×C4⋊C4×C17C2211yescharacteristic
36C22×C42×C17C2211yescharacteristic
37C17×#391C2218yes—
38C2×C17×#95C2214yes—
39C22×Q8×C17—24no—
40C22×Q8×C17—24no—
41C22×Q8×C17—24no—
42C22×Q8×C17C2314yes—
43C23×C4×C17C2312yes—
44C23×C4×C17C2312yes—
45C23×C4×C17C2312yes—
46C23×C4×C17C2311yescharacteristic
47C23×Dic17C2312yes—
48C23×Dic17C2312yes—
49C23×Dic17C2312yes—
50C23×Dic17C2311yescharacteristic
51C23×Dic17C2311yescharacteristic
52C22×Dic34—24no—
53C22×Dic34—24no—
54C22×Dic34—24no—
55C22×Dic34C2314yes—
56C2×C4⋊C4×C17—24no—
57C2×C4⋊C4×C17C2314yes—
58C2×C42×C17—24no—
59C2×C42×C17C2314yes—
60#10587630—232no—
61C2×C4⋊Dic17—24no—
62C2×C4⋊Dic17—28no—
63C2×C4⋊Dic17C2314yes—
64C2×C4⋊Dic17C2318yes—
65C2×#63225—28no—
66C2×#63225—28no—
67C2×#63225—28no—
68C2×#63225C2318yes—
69C2×#63225C2318yes—
70C2×#63225C2318yes—
71C2×C4×Dic17—24no—
72C2×C4×Dic17D418yes—
73C2×C4×Dic17C2314yes—
74C2×C4×Dic17C2314yes—
75#10587576—232no—
76C17×#95C23116yes—
77#10587502C23116yes—
78#10587502C23116yes—
79#10587502C23116yes—
80#10587502C2318yes—
81#10587502C2318yes—
82C24×C17C2411yescharacteristic
83C2×Q8×C17—28no—
84C2×Q8×C17—28no—
85C2×Q8×C17—28no—
86C2×Q8×C17—416no—
87C22×C4×C17—24no—
88C22×C4×C17—24no—
89C22×C4×C17—24no—
90C22×C4×C17—24no—
91C22×C4×C17—24no—
92C22×C4×C17—24no—
93C22×C4×C17—28no—
94C22×C4×C17—24no—
95C22×C4×C17C2×D414yes—
96C22×C4×C17#4114yes—
97C22×C4×C17#4114yes—
98C22×C4×C17C2414yes—
99C22×C4×C17C2414yes—
100C22×C4×C17C2414yes—
101C22×C4×C17C2412yes—
102C22×Dic17—24no—
103C22×Dic17—24no—
104C22×Dic17—24no—
105C22×Dic17—24no—
106C22×Dic17—24no—
107C22×Dic17—24no—
108C22×Dic17—24no—
109C22×Dic17—28no—
110C22×Dic17—24no—
111C22×Dic17—24no—
112C22×Dic17C2×D414yes—
113C22×Dic17C2×D414yes—
114C22×Dic17C2414yes—
115C22×Dic17C2414yes—
116C22×Dic17C2414yes—
117C22×Dic17C2412yes—
118C22×Dic17C2412yes—
119C2×Dic34—28no—
120C2×Dic34—28no—
121C2×Dic34—28no—
122C2×Dic34—416no—
123C4⋊C4×C17—28no—
124C42×C17—28no—
125C4⋊Dic17—28no—
126C4⋊Dic17—216no—
127#63225—216no—
128#63225—216no—
129#63225—216no—
130C4×Dic17—28no—
131C4×Dic17C2×D4116yes—
132C2×#2642—171nomaximal, Sylow
133C23×C17C22×D412yes—
134C23×C17C2×#4112yes—
135C23×C17C2×#4112yes—
136C23×C172−1+414yes—
137C23×C172−1+414yes—
138C23×C17C2511yescharacteristic
139Q8×C17—416no—
140C2×C4×C17—24no—
141C2×C4×C17—24no—
142C2×C4×C17—24no—
143C2×C4×C17—24no—
144C2×C4×C17—24no—
145C2×C4×C17—28no—
146C2×C4×C17—24no—
147C2×C4×C17—24no—
148C2×C4×C17—48no—
149C2×C4×C17—48no—
150C2×C4×C17—48no—
151C2×C4×C17#124116yes—
152C2×C4×C17C22×D414yes—
153C2×C4×C17C2×#4114yes—
154C2×C4×C17C2×#4114yes—
155C2×Dic17—28no—
156C2×Dic17—28no—
157C2×Dic17—24no—
158C2×Dic17—24no—
159C2×Dic17—24no—
160C2×Dic17—24no—
161C2×Dic17—24no—
162C2×Dic17—24no—
163C2×Dic17—24no—
164C2×Dic17—24no—
165C2×Dic17—28no—
166C2×Dic17—24no—
167C2×Dic17—48no—
168C2×Dic17—48no—
169C2×Dic17—48no—
170C2×Dic17C22×D414yes—
171C2×Dic17C22×D414yes—
172Dic34—416no—
173C22×C4⋊Q8—171no—
174C22×#125—171no—
175#2642—1716no—
176C2×#400—174no—
177C2×#400—174no—
178C2×#391—171no—
179C2×#391—171no—
180C2×#391—171no—
181C2×#391D1711yescharacteristic
182C2×#376—171no—
183C22×C17C2×#12414yes—
184C22×C17#53614yes—
185C22×C17#53614yes—
186C22×C17#54118yes—
187C22×C17#54118yes—
188C22×C17C23×D411yescharacteristic
189C22×C17C22×#4111yescharacteristic
190C22×C17C22×#4111yescharacteristic
191C22×C17C2×2−1+412yes—
192C22×C17C2×2−1+412yes—
193C4×C17—44no—
194C4×C17—44no—
195C4×C17—44no—
196C4×C17C2×#12418yes—
197Dic17—24no—
198Dic17—24no—
199Dic17—44no—
200Dic17—44no—
201Dic17—44no—
202C23×Q8—171no—
203C23×Q8C2×D1711yescharacteristic
204C2×C4⋊Q8—178no—
205C2×C4⋊Q8—348no—
206C2×#125—178no—
207C2×#125—348no—
208C22×C4⋊C4—172no—
209C22×C4⋊C4—172no—
210C22×C4⋊C4—172no—
211C22×C4⋊C4—172no—
212C22×C4⋊C4—171no—
213C22×C4⋊C4C2×D1711yescharacteristic
214C22×C42—171no—
215C22×C42—171no—
216C22×C42C2×D1711yescharacteristic
217#400—1732no—
218#400—1732no—
219#391—178no—
220#391—178no—
221#391—178no—
222#391C2×D1718yes—
223#376—178no—
224C2×#95—172no—
225C2×#95—172no—
226C2×#95—174no—
227C2×#95—174no—
228C2×#95—174no—
229C2×#95C2×D1714yes—
230C2×C17#264218yescomplemented
231C2×C17C22×#12411yescharacteristic
232C2×C17C2×#53611yescharacteristic
233C2×C17C2×#53611yescharacteristic
234C2×C17C2×#54112yes—
235C2×C17C2×#54112yes—
236C22×Q8—24no—
237C22×Q8—24no—
238C22×Q8—24no—
239C22×Q8—174no—
240C22×Q8—344no—
241C22×Q8—344no—
242C22×Q8—344no—
243C22×Q8C22×D1714yes—
244C23×C4—171no—
245C23×C4—171no—
246C23×C4—172no—
247C23×C4—172no—
248C23×C4—172no—
249C23×C4C22×D1712yes—
250C23×C4C22×D1712yes—
251C23×C4C22×D1712yes—
252C23×C4C22×D1711yescharacteristic
253C4⋊Q8—3432no—
254#125—3432no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—174no—
257C2×C4⋊C4—178no—
258C2×C4⋊C4—178no—
259C2×C4⋊C4—178no—
260C2×C4⋊C4—178no—
261C2×C4⋊C4—348no—
262C2×C4⋊C4—348no—
263C2×C4⋊C4—348no—
264C2×C4⋊C4—348no—
265C2×C4⋊C4—344no—
266C2×C4⋊C4C22×D1714yes—
267C2×C42—24no—
268C2×C42—178no—
269C2×C42—174no—
270C2×C42—174no—
271C2×C42—344no—
272C2×C42C22×D1714yes—
273#95—178no—
274#95—178no—
275#95—1716no—
276#95—1716no—
277#95—1716no—
278#95C22×D17116yes—
279C17C2×#264211yescharacteristic, complemented, Sylow
280C24C23×D1711yescharacteristic
281C2×Q8—28no—
282C2×Q8—28no—
283C2×Q8—28no—
284C2×Q8—416no—
285C2×Q8—348no—
286C2×Q8—348no—
287C2×Q8—348no—
288C2×Q8—6816no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—28no—
296C22×C4—24no—
297C22×C4—174no—
298C22×C4—174no—
299C22×C4—172no—
300C22×C4—172no—
301C22×C4—174no—
302C22×C4—174no—
303C22×C4—174no—
304C22×C4—344no—
305C22×C4—344no—
306C22×C4—344no—
307C22×C4—344no—
308C22×C4—344no—
309C22×C4—344no—
310C22×C4—344no—
311C22×C4—348no—
312C22×C4—344no—
313C22×C4—344no—
314C22×C4#6325214yes—
315C22×C4D4×D1714yes—
316C22×C4#6325414yes—
317C22×C4C23×D1714yes—
318C22×C4C23×D1714yes—
319C22×C4C23×D1714yes—
320C22×C4C23×D1712yes—
321C4⋊C4—28no—
322C4⋊C4—3416no—
323C4⋊C4—3416no—
324C4⋊C4—3416no—
325C4⋊C4—3416no—
326C4⋊C4—348no—
327C42—28no—
328C42—1716no—
329C42—348no—
330C23C2×#6325212yes—
331C23C2×D4×D1712yes—
332C23C2×#6325412yes—
333C23#1058769414yes—
334C23#1058769714yes—
335C23C24×D1711yescharacteristic
336Q8—416no—
337Q8—6816no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—28no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—48no—
347C2×C4—48no—
348C2×C4—48no—
349C2×C4—174no—
350C2×C4—174no—
351C2×C4—348no—
352C2×C4—348no—
353C2×C4—344no—
354C2×C4—344no—
355C2×C4—344no—
356C2×C4—344no—
357C2×C4—344no—
358C2×C4—344no—
359C2×C4—344no—
360C2×C4—344no—
361C2×C4—348no—
362C2×C4—344no—
363C2×C4—688no—
364C2×C4—688no—
365C2×C4—688no—
366C2×C4#10587571116yes—
367C2×C4C2×#6325214yes—
368C2×C4C2×D4×D1714yes—
369C2×C4C2×#6325414yes—
370C22C2×#1058757114yes—
371C22#1176337914yes—
372C22#1176338414yes—
373C22#1176341718yes—
374C22#1176341918yes—
375C22C22×D4×D1711yescharacteristic
376C22C22×#6325411yescharacteristic
377C22C22×#6325211yescharacteristic
378C22C2×#1058769712yes—
379C22C2×#1058769412yes—
380C4—44no—
381C4—44no—
382C4—44no—
383C4—344no—
384C4—344no—
385C4—684no—
386C4—684no—
387C4—684no—
388C4C2×#1058757118yes—
389C2#1453473318yesdirect product factor
390C2C22×#1058757111yescharacteristic
391C2C2×#1176337911yescharacteristic
392C2C2×#1176338411yescharacteristic
393C2C2×#1176341712yes—
394C2C2×#1176341912yes—
subgroupG/H
centerC24C23×D17
commutator, cosocleC23×C17C25
FrattiniC23C24×D17
FittingC2×C17×#391C2
socle, prime radicalC24×C17C24
hypercenterC2×#391D17
hypocenterC17C2×#2642
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
3. #14534733C2direct (N × H)
230. C2×C17#2642semidirect
279. C17C2×#2642semidirect
389. C2#14534733direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#105876302
C22×#105875762
#145347332
C2×#117627512
C2×#117627502
C2×#117627162
C2×#117627152
C2×#117627142
C2×#117626552
C2×C17×#3912
C2×#264217