Finite Group Databasebeta

← #17898715

Subgroups — #17898715

Order 4352.

1810 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14542187C211yesmaximal, characteristic
2#14537724C211yesmaximal, characteristic, complemented
3#14537722C212yesmaximal, complemented
4#14537720C218yesmaximal, complemented
5#14537719C211yesmaximal, characteristic, complemented
6#14537717C212yesmaximal, complemented
7#14537715C218yesmaximal, complemented
8#14537714C211yesmaximal, characteristic, complemented
9#14537346C211yesmaximal, characteristic
10#14537345C212yesmaximal
11C2×#11762766C211yesmaximal, characteristic
12D17×#405C212yesmaximal
13C17×#2961C211yesmaximal, characteristic, complemented
14C22×#63218C2211yescharacteristic
15C2×M16×D17C2212yes—
16C22×C8×D17C2211yescharacteristic
17#11763439C414yes—
18#11763438C414yes—
19#11763436C414yes—
20#11763434C414yes—
21#11763366C2211yescharacteristic
22#11763363C2218yes—
23#11763362C2212yes—
24C2×#10587558C2211yescharacteristic
25#117×D17C2212yes—
26#11762940C2214yescomplemented
27#11762939C2218yescomplemented
28#11762938C2218yescomplemented
29#11762937C2218yescomplemented
30#11762937C2218yescomplemented
31#11762936C2214yescomplemented
32Q8×#3622C2214yescomplemented
33#11762933C2214yescomplemented
34#11762933C2218yescomplemented
35#11762837C2212yes—
36#11762836C2212yes—
37#11762835C2212yes—
38#11762834C2218yes—
39#11762834C2218yes—
40#11762833C2211yescharacteristic
41#11762832C2212yes—
42#11762831C2218yes—
43#11762767C2212yes—
44#11762766C2214yes—
45#11762766C2214yes—
46#11762766C2214yes—
47#11762765C2212yes—
48#11762764C2212yes—
49#11762763C2214yes—
50C2×#10587486C2211yescharacteristic
51#97×D17C2214yes—
52C2×C4×#3622C2211yescharacteristic
53C2×C8×Dic17C2211yescharacteristic
54C4×C8×D17C2214yes—
55C17×#533C2211yescharacteristic
56C17×#446C2218yescomplemented
57C17×#424C2212yes—
58C17×#423C2211yescharacteristic
59C17×#405C2212yes—
60C2×C17×#97C2211yescharacteristic
61C22×C4×D17C2311yescharacteristic
62C2×M16×C17C2312yes—
63C22×C8×C17C2311yescharacteristic
64C22×C8×C17C2311yescharacteristic
65C17×C4⋊Q8C2×C412yes—
66C17×#125C2×C412yes—
67C17×C22⋊Q8C2×C414yes—
68C4×Q8×C17C2314yes—
69C17×#117C2312yes—
70C2×C42×C17C2311yescharacteristic
71#10587632C2×C418yes—
72#10587630C2×C418yes—
73C2×#63227C2×C412yes—
74#10587612C2312yes—
75#10587610C2×C414yes—
76C2×#63225C2×C412yes—
77C2×C4×Dic17C2311yescharacteristic
78C2×#63223C2312yes—
79C22×#3622C2311yescharacteristic
80C22×#3622C2311yescharacteristic
81M16×D17—28no—
82C2×#63218—24no—
83C2×#63218C2314yes—
84C2×#63218C2314yes—
85C2×C8×D17Q818yescomplemented
86C2×C8×D17C2314yes—
87C2×C8×D17C2312yes—
88C2×C8×D17C2312yes—
89#10587583C2×C418yes—
90#10587582C2×C418yes—
91#10587582C2×C418yes—
92#10587578C2×C414yes—
93C4⋊C4×D17C2×C414yes—
94#10587576C2×C414yes—
95#10587575C2×C418yes—
96#10587575C2×C418yes—
97#10587574C2×C414yes—
98#10587574C2×C418yes—
99C22⋊C4×D17C2×C414yes—
100#10587564C2×C418yes—
101#10587563C2×C414yes—
102#10587558C2×C412yes—
103#10587558C2×C412yes—
104#10587558C2314yes—
105#10587558C2314yes—
106C42×D17C2314yes—
107#10587556C2×C412yes—
108#10587555C2×C412yes—
109C4×Dic34C2318yes—
110C4×Dic34C2314yes—
111C17×C4⋊C8C2318yes—
112C17×C4⋊C8C2314yes—
113C17×#97C2314yes—
114C17×#97C2314yes—
115C4×C8×C17C2314yes—
116#10587486C2314yes—
117#10587486C2314yes—
118#10587486C2314yes—
119#10587485C2318yes—
120#10587485C2318yes—
121#10587485C2318yes—
122C8×Dic17C2314yes—
123C8×Dic17C2314yes—
124C8×Dic17C2312yes—
125C8×Dic17C2312yes—
126C4⋊#3622C2318yes—
127C4⋊#3622C2314yes—
128#10587474C2314yes—
129C4×#3622C2314yes—
130C4×#3622C2312yes—
131C4×#3622C2312yes—
132C23×D17C22×C411yescharacteristic
133C2×Q8×C17C22×C414yes—
134C22×C4×C17C22×C412yes—
135C22×C4×C17C2411yescharacteristic
136C22×Dic17C22×C412yes—
137C22×Dic17C22×C411yescharacteristic
138C2×C4×D17C22×C418yes—
139C2×C4×D17C22×C412yes—
140C2×C4×D17C2×Q814yes—
141C2×C4×D17#4114yes—
142C2×C4×D17C2412yes—
143C2×Dic34C22×C414yes—
144C2×Dic34C22×C418yes—
145M16×C17—24no—
146C2×C8×C17—22no—
147C2×C8×C17C2×Q814yes—
148C2×C8×C17C2412yes—
149C2×C8×C17C2414yes—
150C2×C8×C17C2412yes—
151C4⋊C4×C17C22×C418yes—
152C4⋊C4×C17C22×C414yes—
153C4⋊C4×C17C22×C414yes—
154C22⋊C4×C17C22×C414yes—
155C42×C17C22×C411yescharacteristic
156C42×C17C22×C411yescharacteristic
157C42×C17C2412yes—
158C42×C17C2414yes—
159C22⋊Dic17C22×C414yes—
160#63227C22×C414yes—
161#63227C22×C414yes—
162#63227C22×C418yes—
163C4⋊Dic17C22×C414yes—
164C4⋊Dic17C22×C418yes—
165C4⋊Dic17C22×C414yes—
166#63225C22×C418yes—
167#63225C22×C418yes—
168#63225C22×C418yes—
169#63225C22×C414yes—
170#63225C22×C414yes—
171#63225C22×C418yes—
172C4×Dic17C22×C418yes—
173C4×Dic17C22×C412yes—
174C4×Dic17C2414yes—
175C4×Dic17C2412yes—
176#63223—24no—
177C2×#3622—22no—
178C2×#3622C2×Q814yes—
179C2×#3622C2412yes—
180C2×#3622C2414yes—
181C2×#3622C2411yescharacteristic
182C2×#3622C2411yescharacteristic
183#63218—28no—
184#63218—28no—
185C8×D17—28no—
186C8×D17C2×Q818yes—
187C8×D17C2×Q818yes—
188#60191—171nomaximal, Sylow
189C23×C17C23×C411yescharacteristic
190C22×D17—42no—
191C22×D17C4×Q814yes—
192C22×D17C23×C412yes—
193Q8×C17—44no—
194C2×C4×C17—42no—
195C2×C4×C17C23×C412yes—
196C2×C4×C17C23×C412yes—
197C2×C4×C17C23×C418yes—
198C2×C4×C17C23×C411yescharacteristic
199C2×C4×C17C22×Q812yes—
200C2×C4×C17C2×#4112yes—
201C2×C4×C17C2511yescharacteristic
202C2×Dic17—42no—
203C2×Dic17C4×Q814yes—
204C2×Dic17C23×C418yes—
205C2×Dic17C23×C414yes—
206C2×Dic17C23×C412yes—
207C4×D17—48no—
208C4×D17C4×Q818yes—
209C4×D17C22×Q814yes—
210C4×D17C2×#4114yes—
211Dic34—44no—
212Dic34—48no—
213C8×C17—22no—
214C8×C17—24no—
215C8×C17C22×Q814yes—
216#3622—24no—
217#3622—22no—
218#3622C22×Q812yes—
219#3622C22×Q812yes—
220#3458—171no—
221#2961—178no—
222#2961—172no—
223#2961—171no—
224#2961D1711yescharacteristic, complemented
225#2960—178no—
226#2960—172no—
227#2960—171no—
228#2960—171no—
229#2905—172no—
230#2905—171no—
231C2×#405—172no—
232C2×#405—171no—
233C22×C17—22no—
234C22×C17C2×C4×Q812yes—
235C22×C17C24×C411yescharacteristic
236C2×D17—24no—
237C2×D17—44no—
238C2×D17C2×C4×Q814yes—
239C4×C17—22no—
240C4×C17—28no—
241C4×C17—22no—
242C4×C17C2×C4×Q812yes—
243C4×C17C23×Q811yescharacteristic
244C4×C17C22×#4111yescharacteristic
245Dic17—28no—
246Dic17—24no—
247Dic17C2×C4×Q814yes—
248C22×M16—172no—
249C22×M16—171no—
250C23×C8—171no—
251#544—174no—
252#544—174no—
253#544—174no—
254#543—174no—
255#533—178no—
256#533—172no—
257#533—171no—
258#533C2×D1711yescharacteristic
259C2×#117—172no—
260C2×#117—171no—
261#446—174no—
262#446—174no—
263#446—178no—
264#446—178no—
265#446—178no—
266#446—178no—
267#446C2×D1718yescomplemented
268C8×Q8—174no—
269C8×Q8—178no—
270C8×Q8—174no—
271#424—172no—
272#424—172no—
273#424—172no—
274#424—178no—
275#424—178no—
276#424C2×D1712yes—
277#423—172no—
278#423—171no—
279#423—178no—
280#423C2×D1711yescharacteristic
281#405—172no—
282#405—172no—
283#405—172no—
284#405—174no—
285#405—174no—
286#405—174no—
287#405—174no—
288#405C2×D1712yes—
289C2×#97—171no—
290C2×#97—174no—
291C2×#97C2×D1711yescharacteristic
292C2×C4×C8—171no—
293C2×C4×C8—171no—
294C2×C4×C8—174no—
295C2×C17—22no—
296C2×C17C22×C4×Q811yescharacteristic
297C2×C17#357312yes—
298D17—24no—
299C23×C4—171no—
300C2×M16—172no—
301C2×M16—174no—
302C2×M16—174no—
303C2×M16—344no—
304C2×M16—348no—
305C2×M16C22×D1712yes—
306C22×C8—171no—
307C22×C8—178no—
308C22×C8—172no—
309C22×C8—172no—
310C22×C8—171no—
311C22×C8—174no—
312C22×C8C22×D1711yescharacteristic
313C22×C8C22×D1711yescharacteristic
314C4⋊Q8—178no—
315C4⋊Q8—178no—
316C4⋊Q8—174no—
317C4⋊Q8—172no—
318C4⋊Q8C4×D1712yes—
319#125—178no—
320#125—178no—
321#125—174no—
322#125—172no—
323#125C4×D1712yes—
324C22⋊Q8—178no—
325C22⋊Q8—174no—
326C22⋊Q8—178no—
327C22⋊Q8—178no—
328C22⋊Q8—178no—
329C22⋊Q8—178no—
330C22⋊Q8C4×D1714yes—
331C4×Q8—174no—
332C4×Q8—178no—
333C4×Q8C22×D1714yes—
334#117—172no—
335#117—174no—
336#117—174no—
337#117—172no—
338#117—172no—
339#117—174no—
340#117—174no—
341#117C22×D1712yes—
342C2×C4⋊C4—172no—
343C2×C4⋊C4—174no—
344C2×C22⋊C4—172no—
345C2×C22⋊C4—174no—
346C2×C42—171no—
347C2×C42—174no—
348C2×C42C22×D1711yescharacteristic
349C4⋊C8—178no—
350C4⋊C8—178no—
351C4⋊C8—178no—
352C4⋊C8—174no—
353C4⋊C8—178no—
354C4⋊C8C22×D1718yes—
355C4⋊C8C22×D1714yes—
356#97—174no—
357#97—174no—
358#97—174no—
359#97—174no—
360#97C22×D1714yes—
361#97C22×D1714yes—
362C4×C8—172no—
363C4×C8—174no—
364C4×C8—172no—
365C4×C8—174no—
366C4×C8—172no—
367C4×C8—174no—
368C4×C8—172no—
369C4×C8C22×D1714yes—
370C17#6019111yescharacteristic, complemented, Sylow
371C24—171no—
372C2×Q8—178no—
373C2×Q8—174no—
374C2×Q8C2×C4×D1714yes—
375C22×C4—171no—
376C22×C4—172no—
377C22×C4—174no—
378C22×C4—174no—
379C22×C4—172no—
380C22×C4—172no—
381C22×C4—178no—
382C22×C4C2×C4×D1712yes—
383C22×C4C23×D1711yescharacteristic
384M16—24no—
385M16—348no—
386M16—348no—
387M16—344no—
388C2×C8—22no—
389C2×C8—174no—
390C2×C8—171no—
391C2×C8—171no—
392C2×C8—178no—
393C2×C8—178no—
394C2×C8—174no—
395C2×C8—172no—
396C2×C8—348no—
397C2×C8—342no—
398C2×C8Q8×D1714yes—
399C2×C8C23×D1712yes—
400C2×C8C23×D1714yes—
401C2×C8C23×D1712yes—
402C4⋊C4—178no—
403C4⋊C4—174no—
404C4⋊C4—174no—
405C4⋊C4—174no—
406C4⋊C4—178no—
407C4⋊C4—178no—
408C4⋊C4—178no—
409C4⋊C4—178no—
410C4⋊C4—174no—
411C4⋊C4C2×C4×D1714yes—
412C4⋊C4C2×C4×D1714yes—
413C4⋊C4C2×C4×D1718yes—
414C22⋊C4—174no—
415C22⋊C4—178no—
416C22⋊C4—174no—
417C22⋊C4—174no—
418C22⋊C4C2×C4×D1714yes—
419C42—178no—
420C42—172no—
421C42—172no—
422C42—174no—
423C42C2×C4×D1711yescharacteristic
424C42C2×C4×D1711yescharacteristic
425C42C23×D1712yes—
426C42C23×D1714yes—
427C23—172no—
428C23—174no—
429C23—682no—
430C23C22×C4×D1711yescharacteristic
431Q8—44no—
432Q8—684no—
433Q8—688no—
434C2×C4—42no—
435C2×C4—174no—
436C2×C4—172no—
437C2×C4—178no—
438C2×C4—174no—
439C2×C4—174no—
440C2×C4—178no—
441C2×C4—174no—
442C2×C4—688no—
443C2×C4—682no—
444C2×C4C22×C4×D1718yes—
445C2×C4C22×C4×D1712yes—
446C2×C4C22×C4×D1712yes—
447C2×C4C22×C4×D1711yescharacteristic
448C2×C4C2×Q8×D1712yes—
449C2×C4#41×D1712yes—
450C2×C4C24×D1711yescharacteristic
451C8—22no—
452C8—24no—
453C8—172no—
454C8—172no—
455C8—344no—
456C8—342no—
457C8C2×Q8×D1714yes—
458C22—22no—
459C22—174no—
460C22—344no—
461C22—684no—
462C22C4×Q8×D1712yes—
463C22C23×C4×D1711yescharacteristic
464C4—22no—
465C4—22no—
466C4—28no—
467C4—174no—
468C4—348no—
469C4—344no—
470C4C4×Q8×D1712yes—
471C4C22×Q8×D1711yescharacteristic
472C4C2×#41×D1711yescharacteristic
473C2—22no—
474C2—344no—
475C2C2×C4×Q8×D1711yescharacteristic
476C2#1454349712yes—
subgroupG/H
center, FrattiniC2×C4C24×D17
commutator, socleC22×C17C24×C4
FittingC17×#2961C2
prime radicalC23×D17C22×C4
cosocleC2×C4×C17C25
hypercenter#2961D17
hypocenterC17#60191
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ #2961

Complements

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

NH ≅ G/Ntype
2. #14537724C2semidirect
3. #14537722C2semidirect
4. #14537720C2semidirect
5. #14537719C2semidirect
6. #14537717C2semidirect
7. #14537715C2semidirect
8. #14537714C2semidirect
13. C17×#2961C2semidirect
26. #11762940C22semidirect
27. #11762939C22semidirect
28. #11762938C22semidirect
29. #11762937C22semidirect
30. #11762937C22semidirect
31. #11762936C22semidirect
32. Q8×#3622C22semidirect
33. #11762933C22semidirect
34. #11762933C22semidirect
56. C17×#446C22semidirect
85. C2×C8×D17Q8semidirect
224. #2961D17semidirect
267. #446C2×D17semidirect
370. C17#60191semidirect

Maximal subgroups

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

subgroupindex
#145421872
#145377242
#145377222
#145377202
#145377192
#145377172
#145377152
#145377142
#145373462
#145373452
C2×#117627662
D17×#4052
C17×#29612
#6019117