Finite Group Databasebeta

← #17090474

Subgroups — #17090474

Order 4352.

1094 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14533263C212yesmaximal, complemented
2#14533101C211yesmaximal, characteristic
3#14530136C214yesmaximal, complemented
4#14529770C212yesmaximal, complemented
5#14529065C214yesmaximal
6C4×#10587505C211yesmaximal, characteristic
7C17×#3501C211yesmaximal, characteristic
8C22×C4⋊Dic17C2211yescharacteristic
9C2×#10587505C2212yes—
10#11762748—21no—
11#11762745—21no—
12#11762739—24no—
13#11762716—22no—
14#11762707—21no—
15#11762689C2212yes—
16#11762681C2218yes—
17#11762681C2212yes—
18#11762678—22no—
19#11762669—21no—
20#11762669C2211yescharacteristic
21C4×C22⋊Dic17C2212yes—
22#11762576C2218yes—
23#11762576C2212yes—
24C17×#554—22no—
25C17×#554—24no—
26D4×Q8×C17—22no—
27C17×#544—21no—
28C17×#543—21no—
29C17×#543C2212yes—
30C17×#533—21no—
31C2×C17×C22⋊Q8—21no—
32C2×C17×C22⋊Q8C2212yes—
33C17×#518C2212yes—
34C2×C4×D4×C17C2211yescharacteristic
35C22×Q8×C17—21no—
36C22×Q8×C17D412yes—
37C22×D4×C17C2311yescharacteristic
38C23×C4×C17D411yescharacteristic
39C23×C4×C17C2311yescharacteristic
40C23×Dic17C2312yes—
41C17×C4⋊Q8—21no—
42C17×C4⋊Q8—21no—
43C17×C4⋊Q8—24no—
44C17×#125—22no—
45C17×#125—24no—
46C17×C22⋊Q8—22no—
47C17×C22⋊Q8—22no—
48C17×C22⋊Q8—24no—
49C17×C22⋊Q8—24no—
50C17×C22⋊Q8—22no—
51C17×C22⋊Q8—22no—
52C17×C22⋊Q8—22no—
53C17×C22⋊Q8—22no—
54C17×C22⋊Q8—22no—
55C17×C22⋊Q8—22no—
56C17×C22⋊Q8—24no—
57C17×C22⋊Q8—24no—
58C4×Q8×C17—22no—
59C4×Q8×C17—22no—
60C4×D4×C17—22no—
61C4×D4×C17—22no—
62C4×D4×C17—24no—
63C4×D4×C17—24no—
64C17×#117—22no—
65C2×C4⋊C4×C17—21no—
66C2×C4⋊C4×C17—22no—
67C2×C4⋊C4×C17D411yescharacteristic
68C2×C4⋊C4×C17D414yes—
69C2×C4⋊C4×C17D412yes—
70C2×C22⋊C4×C17—22no—
71C2×C22⋊C4×C17D411yescharacteristic
72C2×C22⋊C4×C17C2311yescharacteristic
73C2×C22⋊C4×C17C2314yes—
74C2×C42×C17D411yescharacteristic
75C2×C22⋊Dic17—22no—
76C2×C22⋊Dic17C2312yes—
77C2×C22⋊Dic17C2314yes—
78C2×C4⋊Dic17—21no—
79C2×C4⋊Dic17—21no—
80C2×C4⋊Dic17—22no—
81C2×C4⋊Dic17—22no—
82C2×C4⋊Dic17—21no—
83C2×#63225—22no—
84C2×#63225—28no—
85C2×C4×Dic17—22no—
86#10587505Q818yescomplemented
87#10587502—22no—
88#10587502—28no—
89#10587502—22no—
90#10587502—21no—
91#10587502—22no—
92C24×C17C2×D411yescharacteristic
93C24×C17C2411yescharacteristic
94C2×Q8×C17—22no—
95C2×Q8×C17—22no—
96C2×Q8×C17—22no—
97C2×Q8×C17—22no—
98C2×Q8×C17—44no—
99C2×Q8×C17—82no—
100C2×D4×C17—22no—
101C2×D4×C17—22no—
102C2×D4×C17C2×Q814yes—
103C22×C4×C17—22no—
104C22×C4×C17—22no—
105C22×C4×C17—21no—
106C22×C4×C17—21no—
107C22×C4×C17—42no—
108C22×C4×C17—42no—
109C22×C4×C17C2×D411yescharacteristic
110C22×C4×C17C2×D412yes—
111C22×C4×C17C2×D411yescharacteristic
112C22×C4×C17C2×D411yescharacteristic
113C22×C4×C17C2×D411yescharacteristic
114C22×C4×C17C2×D414yes—
115C22×C4×C17C2×D414yes—
116C22×C4×C17C2×D411yescharacteristic
117C22×C4×C17C2×D412yes—
118C22×Dic17—24no—
119C22×Dic17—24no—
120C22×Dic17—22no—
121C22×Dic17—22no—
122C22×Dic17—82no—
123C4⋊C4×C17—24no—
124C4⋊C4×C17—22no—
125C4⋊C4×C17—22no—
126C4⋊C4×C17—24no—
127C4⋊C4×C17—22no—
128C4⋊C4×C17—22no—
129C4⋊C4×C17—22no—
130C4⋊C4×C17—24no—
131C4⋊C4×C17—24no—
132C4⋊C4×C17—22no—
133C4⋊C4×C17—22no—
134C22⋊C4×C17—24no—
135C22⋊C4×C17—24no—
136C22⋊C4×C17—22no—
137C22⋊C4×C17—24no—
138C22⋊C4×C17—22no—
139C42×C17—24no—
140C42×C17—21no—
141C42×C17—21no—
142C22⋊Dic17—24no—
143C22⋊Dic17—24no—
144C22⋊Dic17—82no—
145C22⋊Dic17C2×Q818yescomplemented
146C4⋊Dic17—42no—
147C4⋊Dic17—41no—
148C4⋊Dic17—42no—
149C4⋊Dic17—41no—
150#63225—88no—
151#63225—82no—
152C4×Dic17—44no—
153#25853—171nomaximal, Sylow
154C23×C17—21no—
155C23×C17—21no—
156C23×C17—41no—
157C23×C17—41no—
158C23×C17—41no—
159C23×C17—41no—
160C23×C17C4⋊Q812yes—
161C23×C17C22×D411yescharacteristic
162C23×C17C22×D412yes—
163C23×C17C22×Q812yes—
164C23×C172−1+412yes—
165Q8×C17—44no—
166Q8×C17—82no—
167Q8×C17—82no—
168Q8×C17—82no—
169D4×C17—28no—
170C2×C4×C17—21no—
171C2×C4×C17—21no—
172C2×C4×C17—22no—
173C2×C4×C17—24no—
174C2×C4×C17—21no—
175C2×C4×C17—24no—
176C2×C4×C17—24no—
177C2×C4×C17—21no—
178C2×C4×C17—42no—
179C2×C4×C17—41no—
180C2×C4×C17—42no—
181C2×C4×C17—41no—
182C2×C4×C17—44no—
183C2×C4×C17—44no—
184C2×C4×C17—41no—
185C2×C4×C17—81no—
186C2×C4×C17—82no—
187C2×C4×C17—82no—
188C2×C4×C17#12011yescharacteristic
189C2×C4×C17#12012yes—
190C2×C4×C17#12011yescharacteristic
191C2×C4×C17C4⋊Q812yes—
192C2×Dic17—28no—
193C2×Dic17—44no—
194C2×Dic17—44no—
195C2×Dic17—42no—
196C2×Dic17—42no—
197C2×Dic17—48no—
198C2×Dic17—42no—
199C2×Dic17—42no—
200C2×Dic17—42no—
201C2×Dic17—42no—
202#3501D1711yescharacteristic
203#2502—172no—
204#2487—171no—
205#2062—174no—
206#1996—172no—
207#1867—174no—
208C4×#99—171no—
209C22×C17—22no—
210C22×C17—21no—
211C22×C17—22no—
212C22×C17—21no—
213C22×C17—22no—
214C22×C17—42no—
215C22×C17—42no—
216C22×C17—42no—
217C22×C17—41no—
218C22×C17—41no—
219C22×C17—81no—
220C22×C17C2×#12011yescharacteristic
221C22×C17C2×C4⋊Q811yescharacteristic
222C22×C17#53611yescharacteristic
223C22×C17D4×Q814yes—
224C4×C17—22no—
225C4×C17—21no—
226C4×C17—22no—
227C4×C17—41no—
228C4×C17—44no—
229C4×C17—44no—
230C4×C17—41no—
231C4×C17#45812yes—
232Dic17—48no—
233Dic17—82no—
234Dic17—82no—
235#554—22no—
236#554—24no—
237D4×Q8—22no—
238#544—21no—
239#543—21no—
240#543C2×D1712yes—
241#533—21no—
242C2×C22⋊Q8—21no—
243C2×C22⋊Q8C2×D1712yes—
244#518C2×D1712yes—
245C2×C4×D4C2×D1711yescharacteristic
246C22×C4⋊C4—171no—
247C2×#99—172no—
248#400—341no—
249#399—341no—
250#398—344no—
251#391—342no—
252#389—341no—
253#386—172no—
254#385—172no—
255#385—178no—
256#384—342no—
257#380—171no—
258#380—341no—
259C4×C22⋊C4—172no—
260#342—172no—
261#342—178no—
262C2×C17—24no—
263C2×C17—21no—
264C2×C17—21no—
265C2×C17—42no—
266C2×C17#243411yescharacteristic
267C2×C17C2×#45811yescharacteristic
268C2×C17#302711yescharacteristic
269C22×Q8—21no—
270C22×Q8C22⋊D1712yes—
271C22×D4C22×D1711yescharacteristic
272C23×C4—172no—
273C23×C4C22⋊D1711yescharacteristic
274C23×C4C22×D1711yescharacteristic
275C4⋊Q8—21no—
276C4⋊Q8—21no—
277C4⋊Q8—24no—
278#125—22no—
279#125—24no—
280C22⋊Q8—22no—
281C22⋊Q8—22no—
282C22⋊Q8—24no—
283C22⋊Q8—24no—
284C22⋊Q8—22no—
285C22⋊Q8—22no—
286C22⋊Q8—22no—
287C22⋊Q8—22no—
288C22⋊Q8—22no—
289C22⋊Q8—22no—
290C22⋊Q8—24no—
291C22⋊Q8—24no—
292C4×Q8—22no—
293C4×Q8—22no—
294C4×D4—22no—
295C4×D4—22no—
296C4×D4—24no—
297C4×D4—24no—
298#117—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—341no—
302C2×C4⋊C4—342no—
303C2×C4⋊C4—342no—
304C2×C4⋊C4—341no—
305C2×C4⋊C4—341no—
306C2×C4⋊C4—348no—
307C2×C4⋊C4—342no—
308C2×C4⋊C4C22⋊D1711yescharacteristic
309C2×C4⋊C4C22⋊D1712yes—
310C2×C4⋊C4C22⋊D1714yes—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—174no—
313C2×C22⋊C4—172no—
314C2×C22⋊C4—342no—
315C2×C22⋊C4C22⋊D1711yescharacteristic
316C2×C22⋊C4C22×D1711yescharacteristic
317C2×C22⋊C4C22×D1714yes—
318C2×C42—342no—
319C2×C42C22⋊D1711yescharacteristic
320#99—178no—
321#95—342no—
322#95—341no—
323#95—342no—
324#95—348no—
325#95—342no—
326C17#2585311yescharacteristic, complemented, Sylow
327C24C2×C22⋊D1711yescharacteristic
328C24C23×D1711yescharacteristic
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—44no—
334C2×Q8—82no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4Q8×D1714yes—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—344no—
345C22×C4—342no—
346C22×C4—342no—
347C22×C4—344no—
348C22×C4—1362no—
349C22×C4C2×C22⋊D1711yescharacteristic
350C22×C4C2×C22⋊D1712yes—
351C22×C4C2×C22⋊D1711yescharacteristic
352C22×C4C2×C22⋊D1714yes—
353C22×C4C2×C22⋊D1711yescharacteristic
354C22×C4C2×C22⋊D1711yescharacteristic
355C22×C4C2×C22⋊D1714yes—
356C22×C4C2×C22⋊D1711yescharacteristic
357C22×C4C2×C22⋊D1712yes—
358C4⋊C4—22no—
359C4⋊C4—24no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—24no—
363C4⋊C4—22no—
364C4⋊C4—24no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—24no—
369C4⋊C4—682no—
370C4⋊C4—681no—
371C4⋊C4—682no—
372C4⋊C4—681no—
373C4⋊C4—1368no—
374C4⋊C4—1362no—
375C22⋊C4—24no—
376C22⋊C4—24no—
377C22⋊C4—22no—
378C22⋊C4—24no—
379C22⋊C4—22no—
380C22⋊C4—178no—
381C22⋊C4—344no—
382C22⋊C4—344no—
383C22⋊C4—1362no—
384C42—24no—
385C42—21no—
386C42—21no—
387C42—684no—
388C23—21no—
389C23—21no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23#1058763012yes—
395C23C2×Q8×D1712yes—
396C23#1058769412yes—
397C23C22×C22⋊D1712yes—
398C23C22×C22⋊D1711yescharacteristic
399Q8—44no—
400Q8—82no—
401Q8—82no—
402Q8—82no—
403D4—28no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—24no—
407C2×C4—22no—
408C2×C4—21no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—21no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—44no—
419C2×C4—81no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—348no—
423C2×C4—684no—
424C2×C4—684no—
425C2×C4—682no—
426C2×C4—682no—
427C2×C4—682no—
428C2×C4—688no—
429C2×C4—682no—
430C2×C4—682no—
431C2×C4—682no—
432C2×C4#1058763012yes—
433C2×C4#1058763911yescharacteristic
434C2×C4#1058763912yes—
435C2×C4#1058763911yescharacteristic
436C22—22no—
437C22—21no—
438C22—22no—
439C22—21no—
440C22—22no—
441C22—42no—
442C22—42no—
443C22—42no—
444C22—41no—
445C22—41no—
446C22—81no—
447C22C2×#1058763911yescharacteristic
448C22C2×#1058763011yescharacteristic
449C22#1176338311yescharacteristic
450C22Q8×C22⋊D1714yes—
451C4—22no—
452C4—21no—
453C4—22no—
454C4—41no—
455C4—44no—
456C4—44no—
457C4—41no—
458C4—688no—
459C4—1362no—
460C4—1362no—
461C4#1176299912yes—
462C2—24no—
463C2—21no—
464C2—21no—
465C2—42no—
466C2#1453242411yescharacteristic
467C2C2×#1176299911yescharacteristic
468C2#1453818611yescharacteristic
subgroupG/H
centerC22C2×#10587639
commutator, cosocleC24×C17C24
FrattiniC24C23×D17
FittingC17×#3501C2
socleC22×C17C2×#120
prime radicalC22×D4×C17C23
hypercenter#3501D17
hypocenterC17#25853
LCSG ▷ C24×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3501

Complements

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

NH ≅ G/Ntype
1. #14533263C2semidirect
3. #14530136C2semidirect
4. #14529770C2semidirect
86. #10587505Q8semidirect
145. C22⋊Dic17C2×Q8semidirect
326. C17#25853semidirect

Maximal subgroups

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

subgroupindex
#145332632
#145331012
#145301362
#145297702
#145290652
C4×#105875052
C17×#35012
#2585317