Finite Group Databasebeta

← #17008179

Subgroups — #17008179

Order 4352.

1064 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14539556C212yesmaximal, complemented
2#14534589C211yesmaximal, characteristic
3#14527809C214yesmaximal, complemented
4#14527741C214yesmaximal, complemented
5#14527356C212yesmaximal, complemented
6C2×#11762533C211yesmaximal, characteristic
7C2×C17×#543C211yesmaximal, characteristic
8C2×#10587610C2211yescharacteristic
9#11763144—22no—
10#11763142C2214yescomplemented
11#11763048—24no—
12#11762965—22no—
13#11762964C2214yescomplemented
14C2×C22⋊#3622C2212yes—
15#11762745C2212yes—
16#11762732—22no—
17#11762721C2212yes—
18#11762682—21no—
19#11762669C2211yescharacteristic
20#11762665—21no—
21#11762546C2218yescomplemented
22#11762533C2214yes—
23C17×#543—22no—
24C17×#543—24no—
25C17×#543C2214yes—
26C2×C17×C22⋊Q8—22no—
27C2×C17×C22⋊Q8—22no—
28C2×C17×C22⋊Q8—21no—
29C2×C17×C22⋊Q8C2212yes—
30C22×C22⋊C4×C17—21no—
31C22×C22⋊C4×C17C2211yescharacteristic
32C25×C17D411yescharacteristic
33C22×Q8×C17—21no—
34C22×Q8×C17D412yes—
35C23×C4×C17—21no—
36C23×C4×C17—21no—
37C23×C4×C17D411yescharacteristic
38C23×C4×C17C2311yescharacteristic
39C22×Dic34C2311yescharacteristic
40C17×C22⋊Q8—28no—
41C17×C22⋊Q8—28no—
42C17×C22⋊Q8—24no—
43C17×C22⋊Q8—28no—
44C17×C22⋊Q8—24no—
45C17×C22⋊Q8—22no—
46C17×C22⋊Q8—22no—
47C17×C22⋊Q8—28no—
48C17×C22⋊Q8D414yes—
49C17×C22⋊Q8C2314yes—
50C2×C4⋊C4×C17—22no—
51C2×C4⋊C4×C17—21no—
52C2×C4⋊C4×C17—22no—
53C2×C4⋊C4×C17D412yes—
54C2×C22⋊C4×C17—22no—
55C2×C22⋊C4×C17—24no—
56C2×C22⋊C4×C17—22no—
57C2×C22⋊C4×C17—24no—
58C2×C22⋊C4×C17—22no—
59C2×C22⋊C4×C17—22no—
60C2×C22⋊C4×C17D412yes—
61C2×C22⋊C4×C17C2312yes—
62C2×C22⋊Dic17—21no—
63C2×C22⋊Dic17—21no—
64C2×C22⋊Dic17—22no—
65C2×C22⋊Dic17—22no—
66C2×#63231—24no—
67C2×C4⋊Dic17—21no—
68C2×C4⋊Dic17C2311yescharacteristic
69#10587610—22no—
70#10587610—22no—
71C2×#63225—21no—
72C2×#63225—22no—
73C22×#3622—22no—
74#10587506—24no—
75#10587502—22no—
76#10587502C2312yes—
77C22⋊#3622—22no—
78C22⋊#3622C2314yes—
79#10587481—24no—
80#10587478—24no—
81C24×C17—21no—
82C24×C17—21no—
83C24×C17—22no—
84C24×C17—81no—
85C24×C17—81no—
86C24×C17Q1614yescomplemented
87C24×C17C2×D412yes—
88C24×C17C2×D411yescharacteristic
89C2×Q8×C17—24no—
90C2×Q8×C17—22no—
91C2×Q8×C17—82no—
92C2×Q8×C17—82no—
93C2×Q8×C17C2×D414yes—
94C22×C4×C17—22no—
95C22×C4×C17—22no—
96C22×C4×C17—21no—
97C22×C4×C17—24no—
98C22×C4×C17—24no—
99C22×C4×C17—21no—
100C22×C4×C17—81no—
101C22×C4×C17—81no—
102C22×C4×C17—82no—
103C22×C4×C17—82no—
104C22×C4×C17C2×D412yes—
105C22×C4×C17C2×D411yescharacteristic
106C22×C4×C17C2×D412yes—
107C22×C4×C17C2×D412yes—
108C22×C4×C17C2411yescharacteristic
109C22×Dic17—21no—
110C22×Dic17—22no—
111C22×Dic17—21no—
112C2×Dic34—22no—
113C2×Dic34—81no—
114C4⋊C4×C17—24no—
115C4⋊C4×C17—28no—
116C4⋊C4×C17—22no—
117C4⋊C4×C17—24no—
118C4⋊C4×C17—24no—
119C4⋊C4×C17C2×D414yes—
120C22⋊C4×C17—28no—
121C22⋊C4×C17—24no—
122C22⋊C4×C17—24no—
123C22⋊C4×C17—24no—
124C22⋊C4×C17—24no—
125C22⋊Dic17—41no—
126C22⋊Dic17—41no—
127C22⋊Dic17—42no—
128C22⋊Dic17—82no—
129C22⋊Dic17—82no—
130#63231—84no—
131C4⋊Dic17—22no—
132C4⋊Dic17—81no—
133#63225—42no—
134#63225—82no—
135C2×#3622—24no—
136C2×#3622—82no—
137#18287—171nomaximal, Sylow
138C23×C17—22no—
139C23×C17—24no—
140C23×C17—22no—
141C23×C17—24no—
142C23×C17—41no—
143C23×C17—41no—
144C23×C17—42no—
145C23×C17—42no—
146C23×C17—42no—
147C23×C17—42no—
148C23×C17—81no—
149C23×C17—81no—
150C23×C17—84no—
151C23×C17—84no—
152C23×C17#12011yescharacteristic
153C23×C17#12011yescharacteristic
154C23×C17C2×Q1614yes—
155C23×C17C2×Q1612yes—
156C23×C17#13712yes—
157C23×C17C22×D411yescharacteristic
158Q8×C17—44no—
159Q8×C17—82no—
160Q8×C17—84no—
161C2×C4×C17—24no—
162C2×C4×C17—22no—
163C2×C4×C17—24no—
164C2×C4×C17—41no—
165C2×C4×C17—42no—
166C2×C4×C17—42no—
167C2×C4×C17—41no—
168C2×C4×C17—42no—
169C2×C4×C17—44no—
170C2×C4×C17—44no—
171C2×C4×C17—42no—
172C2×C4×C17—81no—
173C2×C4×C17—84no—
174C2×C4×C17—84no—
175C2×C4×C17#12012yes—
176C2×C4×C17C22×D412yes—
177C2×Dic17—42no—
178C2×Dic17—42no—
179C2×Dic17—42no—
180C2×Dic17—42no—
181C2×Dic17—41no—
182C2×Dic17—41no—
183C2×Dic17—42no—
184Dic34—44no—
185#3622—44no—
186C2×#543D1711yescharacteristic
187#3189—172no—
188#2626—171no—
189#1614—174no—
190#1602—174no—
191#1520—172no—
192C2×#323—171no—
193C22×C17—21no—
194C22×C17—21no—
195C22×C17—21no—
196C22×C17—21no—
197C22×C17—21no—
198C22×C17—21no—
199C22×C17—24no—
200C22×C17—42no—
201C22×C17—42no—
202C22×C17—44no—
203C22×C17—48no—
204C22×C17—42no—
205C22×C17—42no—
206C22×C17—82no—
207C22×C17—84no—
208C22×C17#45112yes—
209C22×C17#45112yes—
210C22×C17C2×#12011yescharacteristic
211C22×C17C22×Q1611yescharacteristic
212C22×C17C2×#13711yescharacteristic
213C4×C17—22no—
214C4×C17—22no—
215C4×C17—42no—
216C4×C17—44no—
217C4×C17—44no—
218Dic17—82no—
219Dic17—81no—
220Dic17—81no—
221#543—24no—
222#543—22no—
223#543C2×D1714yes—
224C2×C22⋊Q8—22no—
225C2×C22⋊Q8—21no—
226C2×C22⋊Q8—22no—
227C2×C22⋊Q8—171no—
228C2×C22⋊Q8C2×D1712yes—
229C22×C22⋊C4—21no—
230C22×C22⋊C4C2×D1711yescharacteristic
231#484—174no—
232#484—342no—
233#467—344no—
234#451—174no—
235#451—342no—
236C2×C22⋊C8—172no—
237#399—172no—
238#396—342no—
239#393—172no—
240#385—341no—
241#380—171no—
242#379—341no—
243#328—178no—
244#323—174no—
245C2×C17—21no—
246C2×C17—21no—
247C2×C17—22no—
248C2×C17—22no—
249C2×C17—44no—
250C2×C17C2×#45111yescharacteristic
251C2×C17#301911yescharacteristic
252C2×C17#302611yescharacteristic
253C25C22⋊D1711yescharacteristic
254C22×Q8—21no—
255C22×Q8—171no—
256C22×Q8C22⋊D1712yes—
257C23×C4—21no—
258C23×C4—21no—
259C23×C4C22⋊D1711yescharacteristic
260C23×C4C22×D1711yescharacteristic
261C2×Q16—344no—
262C22×C8—342no—
263C22⋊Q8—28no—
264C22⋊Q8—28no—
265C22⋊Q8—24no—
266C22⋊Q8—28no—
267C22⋊Q8—24no—
268C22⋊Q8—22no—
269C22⋊Q8—22no—
270C22⋊Q8—28no—
271C22⋊Q8—342no—
272C22⋊Q8—342no—
273C22⋊Q8C22⋊D1714yes—
274C22⋊Q8C22×D1714yes—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—171no—
279C2×C4⋊C4—341no—
280C2×C4⋊C4—341no—
281C2×C4⋊C4—342no—
282C2×C4⋊C4C22⋊D1712yes—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—341no—
290C2×C22⋊C4—341no—
291C2×C22⋊C4—342no—
292C2×C22⋊C4—342no—
293C2×C22⋊C4C22⋊D1712yes—
294C2×C22⋊C4C22×D1712yes—
295#107—344no—
296#103—344no—
297#103—344no—
298C22⋊C8—174no—
299C22⋊C8—342no—
300#95—172no—
301#95—342no—
302C17#1828711yescharacteristic, complemented, Sylow
303C24—21no—
304C24—22no—
305C24—21no—
306C24—81no—
307C24—81no—
308C24#6323114yescomplemented
309C24C2×C22⋊D1712yes—
310C24C2×C22⋊D1711yescharacteristic
311C2×Q8—24no—
312C2×Q8—22no—
313C2×Q8—82no—
314C2×Q8—82no—
315C2×Q8—342no—
316C2×Q8—1361no—
317C2×Q8C2×C22⋊D1714yes—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—21no—
324C22×C4—81no—
325C22×C4—81no—
326C22×C4—82no—
327C22×C4—82no—
328C22×C4—342no—
329C22×C4—341no—
330C22×C4—341no—
331C22×C4C2×C22⋊D1712yes—
332C22×C4C2×C22⋊D1712yes—
333C22×C4C2×C22⋊D1712yes—
334C22×C4C2×C22⋊D1711yescharacteristic
335C22×C4C23×D1711yescharacteristic
336Q16—1364no—
337C2×C8—344no—
338C2×C8—1362no—
339C4⋊C4—24no—
340C4⋊C4—28no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—22no—
344C4⋊C4—342no—
345C4⋊C4—682no—
346C4⋊C4—1362no—
347C4⋊C4—1361no—
348C4⋊C4C2×C22⋊D1714yes—
349C22⋊C4—28no—
350C22⋊C4—24no—
351C22⋊C4—24no—
352C22⋊C4—24no—
353C22⋊C4—24no—
354C22⋊C4—682no—
355C22⋊C4—681no—
356C22⋊C4—681no—
357C22⋊C4—1362no—
358C22⋊C4—1362no—
359C23—24no—
360C23—22no—
361C23—24no—
362C23—22no—
363C23—41no—
364C23—41no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—81no—
370C23—81no—
371C23—84no—
372C23—84no—
373C23#1058762812yes—
374C23C2×#6323114yes—
375C23C2×#6323112yes—
376C23#1058763911yescharacteristic
377C23#1058763911yescharacteristic
378C23C22×C22⋊D1711yescharacteristic
379Q8—44no—
380Q8—82no—
381Q8—84no—
382Q8—684no—
383C2×C4—24no—
384C2×C4—22no—
385C2×C4—24no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—42no—
394C2×C4—81no—
395C2×C4—84no—
396C2×C4—84no—
397C2×C4—682no—
398C2×C4—682no—
399C2×C4—682no—
400C2×C4—682no—
401C2×C4—681no—
402C2×C4—681no—
403C2×C4—682no—
404C2×C4#1058763912yes—
405C2×C4C22×C22⋊D1712yes—
406C8—684no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—24no—
414C22—42no—
415C22—42no—
416C22—44no—
417C22—48no—
418C22—42no—
419C22—42no—
420C22—82no—
421C22—84no—
422C22#1176296512yes—
423C22#1176296512yes—
424C22C2×#1058763911yescharacteristic
425C22C22×#6323111yescharacteristic
426C22C2×#1058762811yescharacteristic
427C4—22no—
428C4—22no—
429C4—42no—
430C4—44no—
431C4—44no—
432C4—1362no—
433C4—1361no—
434C4—1361no—
435C2—21no—
436C2—22no—
437C2—21no—
438C2—22no—
439C2—44no—
440C2C2×#1176296511yescharacteristic
441C2#1453814111yescharacteristic
442C2#1453818111yescharacteristic
subgroupG/H
centerC22C2×#10587639
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×C17×#543C2
socleC22×C17C2×#120
prime radicalC25×C17D4
hypercenterC2×#543D17
hypocenterC17#18287
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#543

Complements

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

NH ≅ G/Ntype
1. #14539556C2semidirect
3. #14527809C2semidirect
4. #14527741C2semidirect
5. #14527356C2semidirect
10. #11763142C22semidirect
13. #11762964C22semidirect
21. #11762546C22semidirect
86. C24×C17Q16semidirect
302. C17#18287semidirect
308. C24#63231semidirect

Maximal subgroups

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

subgroupindex
#145395562
#145345892
#145278092
#145277412
#145273562
C2×#117625332
C2×C17×#5432
#1828717