Finite Group Databasebeta

← #17008316

Subgroups — #17008316

Order 4352.

912 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14539729C212yesmaximal, complemented
2#14534605C211yesmaximal, characteristic
3#14527758C214yesmaximal, complemented
4#14527741C214yesmaximal
5#14527348C212yesmaximal, complemented
6#14527098C211yesmaximal, characteristic
7C17×#3458C211yesmaximal, characteristic
8C2×#10587555C2211yescharacteristic
9#11763188—22no—
10#11763161—22no—
11#11763019—24no—
12#11762964C2218yes—
13#11762862C2212yes—
14#11762748—21no—
15#11762745—21no—
16#11762729C2212yes—
17#11762721C2212yes—
18#11762707—21no—
19#11762675C2211yescharacteristic
20#11762669—21no—
21#11762546C2218yes—
22#11762535C2212yes—
23#11762533C2212yes—
24C17×#544—21no—
25C17×#544—22no—
26C17×#544—22no—
27C17×#544C2212yes—
28C17×#543—21no—
29C17×#543C2212yes—
30C17×#533—24no—
31C17×#533—21no—
32C17×#533C2212yes—
33C2×C17×#117—21no—
34C2×C17×#117C2211yescharacteristic
35C23×C4×C17D411yescharacteristic
36C22×Dic34C2312yes—
37C17×C4⋊Q8—24no—
38C17×C4⋊Q8—21no—
39C17×C4⋊Q8—21no—
40C17×C4⋊Q8—24no—
41C17×C4⋊Q8D412yes—
42C17×C4⋊Q8D412yes—
43C17×#125—24no—
44C17×#125—22no—
45C17×#125—24no—
46C17×#125—22no—
47C17×C22⋊Q8—24no—
48C17×C22⋊Q8—24no—
49C17×C22⋊Q8—24no—
50C17×C22⋊Q8—22no—
51C17×C22⋊Q8—22no—
52C17×C22⋊Q8—22no—
53C17×C22⋊Q8—24no—
54C17×C22⋊Q8C2314yes—
55C4×Q8×C17—22no—
56C4×Q8×C17—22no—
57C4×Q8×C17—22no—
58C4×Q8×C17D414yes—
59C17×#117—22no—
60C17×#117—22no—
61C17×#117—22no—
62C17×#117—22no—
63C17×#117—22no—
64C17×#117—22no—
65C2×C4⋊C4×C17—21no—
66C2×C4⋊C4×C17—21no—
67C2×C4⋊C4×C17D411yescharacteristic
68C2×C4⋊C4×C17C2311yescharacteristic
69C2×C22⋊C4×C17—22no—
70C2×C22⋊C4×C17D411yescharacteristic
71C2×C22⋊C4×C17C2311yescharacteristic
72C2×C42×C17—22no—
73C2×C42×C17D411yescharacteristic
74C2×C42×C17C2311yescharacteristic
75C2×C22⋊Dic17—22no—
76C2×#63231—28no—
77C2×C4⋊Dic17—21no—
78C2×C4⋊Dic17—21no—
79C2×C4⋊Dic17—21no—
80C2×#63225—22no—
81#10587555—22no—
82#10587555—22no—
83#10587502—22no—
84#10587502—22no—
85#10587502—21no—
86#10587502C2312yes—
87C22⋊#3622C2314yes—
88#10587481—28no—
89C4⋊#3622—22no—
90C4×#3622—22no—
91C24×C17C2×D411yescharacteristic
92C2×Q8×C17—22no—
93C2×Q8×C17—22no—
94C2×Q8×C17—22no—
95C2×Q8×C17C2×D414yes—
96C22×C4×C17—21no—
97C22×C4×C17—21no—
98C22×C4×C17—22no—
99C22×C4×C17—22no—
100C22×C4×C17—22no—
101C22×C4×C17Q1614yescomplemented
102C22×C4×C17C2×D411yescharacteristic
103C22×C4×C17C2×D411yescharacteristic
104C22×C4×C17C2×D411yescharacteristic
105C22×C4×C17C2×D411yescharacteristic
106C22×C4×C17C2×D411yescharacteristic
107C22×C4×C17C2411yescharacteristic
108C22×Dic17—22no—
109C22×Dic17—22no—
110C2×Dic34—24no—
111C2×Dic34—82no—
112C4⋊C4×C17—22no—
113C4⋊C4×C17—24no—
114C4⋊C4×C17—24no—
115C4⋊C4×C17—22no—
116C4⋊C4×C17—22no—
117C4⋊C4×C17—22no—
118C4⋊C4×C17—24no—
119C4⋊C4×C17—22no—
120C4⋊C4×C17—22no—
121C4⋊C4×C17—22no—
122C4⋊C4×C17—22no—
123C4⋊C4×C17—24no—
124C4⋊C4×C17—22no—
125C4⋊C4×C17C2×D414yes—
126C22⋊C4×C17—24no—
127C22⋊C4×C17—24no—
128C22⋊C4×C17—22no—
129C22⋊C4×C17—22no—
130C42×C17—24no—
131C42×C17—21no—
132C42×C17—24no—
133C42×C17—21no—
134C42×C17C2×D412yes—
135C42×C17C2×D412yes—
136C22⋊Dic17—82no—
137#63231—88no—
138C4⋊Dic17—42no—
139C4⋊Dic17—41no—
140C4⋊Dic17—41no—
141C4⋊Dic17—42no—
142#63225—82no—
143C2×#3622—24no—
144#18303—171nomaximal, Sylow
145C23×C17—21no—
146C23×C17—21no—
147C23×C17—41no—
148C23×C17—41no—
149C23×C17C2×Q1612yes—
150C23×C17C22×D411yescharacteristic
151Q8×C17—44no—
152Q8×C17—82no—
153Q8×C17—82no—
154Q8×C17—82no—
155C2×C4×C17—21no—
156C2×C4×C17—21no—
157C2×C4×C17—24no—
158C2×C4×C17—24no—
159C2×C4×C17—24no—
160C2×C4×C17—21no—
161C2×C4×C17—21no—
162C2×C4×C17—24no—
163C2×C4×C17—41no—
164C2×C4×C17—41no—
165C2×C4×C17—41no—
166C2×C4×C17—41no—
167C2×C4×C17—82no—
168C2×C4×C17—82no—
169C2×C4×C17#12012yes—
170C2×C4×C17#12011yescharacteristic
171C2×C4×C17#12011yescharacteristic
172C2×C4×C17C2×Q1612yes—
173C2×C4×C17C2×Q1612yes—
174C2×C4×C17#13712yes—
175C2×C4×C17C22×D412yes—
176C2×Dic17—42no—
177C2×Dic17—42no—
178C2×Dic17—42no—
179C2×Dic17—42no—
180C2×Dic17—42no—
181C2×Dic17—42no—
182Dic34—48no—
183#3622—44no—
184#3458D1711yescharacteristic
185#3209—172no—
186#2628—171no—
187#1605—174no—
188#1602—174no—
189#1518—172no—
190#1458—171no—
191C22×C17—21no—
192C22×C17—21no—
193C22×C17—22no—
194C22×C17—42no—
195C22×C17—41no—
196C22×C17—41no—
197C22×C17—81no—
198C22×C17C2×#12011yescharacteristic
199C22×C17C22×Q1611yescharacteristic
200C22×C17C2×#13711yescharacteristic
201C4×C17—22no—
202C4×C17—22no—
203C4×C17—22no—
204C4×C17—41no—
205C4×C17—44no—
206C4×C17—41no—
207C4×C17—44no—
208C4×C17#45112yes—
209C4×C17#45112yes—
210Dic17—82no—
211Dic17—82no—
212#544—22no—
213#544—22no—
214#544—21no—
215#544C2×D1712yes—
216#543—21no—
217#543C2×D1712yes—
218#533—24no—
219#533—21no—
220#533C2×D1712yes—
221C2×C4⋊Q8—171no—
222C2×#117—21no—
223C2×#117C2×D1711yescharacteristic
224#494—342no—
225#487—342no—
226#462—344no—
227#451—178no—
228#431—172no—
229#400—341no—
230#399—341no—
231#395—172no—
232#393—172no—
233#389—341no—
234#382—171no—
235#380—341no—
236#328—178no—
237#324—172no—
238#323—172no—
239C2×C17—21no—
240C2×C17—21no—
241C2×C17—42no—
242C2×C17C2×#45111yescharacteristic
243C2×C17#302211yescharacteristic
244C2×C17#302711yescharacteristic
245C22×Q8—172no—
246C23×C4C22⋊D1711yescharacteristic
247C2×Q16—348no—
248C4⋊Q8—24no—
249C4⋊Q8—21no—
250C4⋊Q8—21no—
251C4⋊Q8—24no—
252C4⋊Q8—342no—
253C4⋊Q8—342no—
254C4⋊Q8C22⋊D1712yes—
255C4⋊Q8C22⋊D1712yes—
256#125—24no—
257#125—22no—
258#125—24no—
259#125—22no—
260C22⋊Q8—24no—
261C22⋊Q8—24no—
262C22⋊Q8—24no—
263C22⋊Q8—22no—
264C22⋊Q8—22no—
265C22⋊Q8—22no—
266C22⋊Q8—24no—
267C22⋊Q8C22×D1714yes—
268C4×Q8—22no—
269C4×Q8—22no—
270C4×Q8—22no—
271C4×Q8C22⋊D1714yes—
272#117—22no—
273#117—22no—
274#117—22no—
275#117—22no—
276#117—22no—
277#117—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—341no—
281C2×C4⋊C4—341no—
282C2×C4⋊C4—341no—
283C2×C4⋊C4—342no—
284C2×C4⋊C4C22⋊D1711yescharacteristic
285C2×C4⋊C4C22×D1711yescharacteristic
286C2×C22⋊C4—22no—
287C2×C22⋊C4—342no—
288C2×C22⋊C4C22⋊D1711yescharacteristic
289C2×C22⋊C4C22×D1711yescharacteristic
290C2×C42—22no—
291C2×C42C22⋊D1711yescharacteristic
292C2×C42C22×D1711yescharacteristic
293C4⋊C8—342no—
294#103—348no—
295C22⋊C8—174no—
296C4×C8—342no—
297#95—172no—
298#95—342no—
299#95—341no—
300#95—342no—
301C17#1830311yescharacteristic, complemented, Sylow
302C24C2×C22⋊D1711yescharacteristic
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—344no—
307C2×Q8—1362no—
308C2×Q8C2×C22⋊D1714yes—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—342no—
315C22×C4—342no—
316C22×C4#6323114yescomplemented
317C22×C4C2×C22⋊D1711yescharacteristic
318C22×C4C2×C22⋊D1711yescharacteristic
319C22×C4C2×C22⋊D1711yescharacteristic
320C22×C4C2×C22⋊D1711yescharacteristic
321C22×C4C2×C22⋊D1711yescharacteristic
322C22×C4C23×D1711yescharacteristic
323Q16—1368no—
324C2×C8—344no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—24no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—24no—
338C4⋊C4—682no—
339C4⋊C4—682no—
340C4⋊C4—681no—
341C4⋊C4—681no—
342C4⋊C4—1362no—
343C4⋊C4C2×C22⋊D1714yes—
344C22⋊C4—24no—
345C22⋊C4—24no—
346C22⋊C4—22no—
347C22⋊C4—22no—
348C22⋊C4—1362no—
349C42—24no—
350C42—24no—
351C42—21no—
352C42—21no—
353C42C2×C22⋊D1712yes—
354C42C2×C22⋊D1712yes—
355C23—21no—
356C23—21no—
357C23—41no—
358C23—41no—
359C23C2×#6323112yes—
360C23C22×C22⋊D1711yescharacteristic
361Q8—44no—
362Q8—82no—
363Q8—82no—
364Q8—82no—
365Q8—688no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—24no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—682no—
381C2×C4—682no—
382C2×C4—682no—
383C2×C4—682no—
384C2×C4—682no—
385C2×C4—682no—
386C2×C4#1058762812yes—
387C2×C4C2×#6323112yes—
388C2×C4C2×#6323112yes—
389C2×C4#1058763912yes—
390C2×C4#1058763911yescharacteristic
391C2×C4#1058763911yescharacteristic
392C2×C4C22×C22⋊D1712yes—
393C8—684no—
394C22—21no—
395C22—21no—
396C22—22no—
397C22—42no—
398C22—41no—
399C22—41no—
400C22—81no—
401C22C2×#1058763911yescharacteristic
402C22C22×#6323111yescharacteristic
403C22C2×#1058762811yescharacteristic
404C4—22no—
405C4—22no—
406C4—22no—
407C4—41no—
408C4—44no—
409C4—41no—
410C4—44no—
411C4—1362no—
412C4—1362no—
413C4#1176296512yes—
414C4#1176296512yes—
415C2—21no—
416C2—21no—
417C2—42no—
418C2C2×#1176296511yescharacteristic
419C2#1453816311yescharacteristic
420C2#1453818611yescharacteristic
subgroupG/H
centerC22C2×#10587639
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#3458C2
socleC22×C17C2×#120
prime radicalC24×C17C2×D4
hypercenter#3458D17
hypocenterC17#18303
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3458

Complements

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

NH ≅ G/Ntype
1. #14539729C2semidirect
3. #14527758C2semidirect
5. #14527348C2semidirect
101. C22×C4×C17Q16semidirect
301. C17#18303semidirect
316. C22×C4#63231semidirect

Maximal subgroups

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

subgroupindex
#145397292
#145346052
#145277582
#145277412
#145273482
#145270982
C17×#34582
#1830317