Finite Group Databasebeta

← #31720237

Subgroups — #31720237

Order 17152.

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#22573612C211yesmaximal, characteristic
2#22569157C218yesmaximal, complemented
3#22569154C218yesmaximal, complemented
4#22569149C211yesmaximal, characteristic, complemented
5#22569148C212yesmaximal, complemented
6#22569144C211yesmaximal, characteristic, complemented
7#22569141C212yesmaximal, complemented
8#22569139C211yesmaximal, characteristic, complemented
9#22568771C211yesmaximal, characteristic
10#424×D67C212yesmaximal
11#22568625C212yesmaximal
12C2×#16863353C211yesmaximal, characteristic
13C67×#2961C211yesmaximal, characteristic, complemented
14C22×#11761611C2211yescharacteristic
15C2×M16×D67C2212yes—
16C22×C8×D67C2211yescharacteristic
17#16864026C414yes—
18#16864025C414yes—
19#16864023C414yes—
20#16864021C414yes—
21#16863953C2211yescharacteristic
22#16863950C2218yes—
23#16863949C2212yes—
24C2×#14524000C2211yescharacteristic
25#117×D67C2212yes—
26#16863527C2214yescomplemented
27#16863525C2218yescomplemented
28#16863524C2218yescomplemented
29#16863524C2218yescomplemented
30#16863523C2214yescomplemented
31#16863523C2218yescomplemented
32Q8×C67⋊C8C2214yescomplemented
33C8×Dic134C2218yescomplemented
34#16863520C2214yescomplemented
35#16863424C2212yes—
36#16863423C2212yes—
37#16863422C2212yes—
38#16863421C2218yes—
39#16863421C2218yes—
40#16863420C2211yescharacteristic
41#16863419C2212yes—
42C4⋊C8×D67C2218yes—
43#16863354C2212yes—
44#16863353C2214yes—
45#16863353C2214yes—
46#16863353C2214yes—
47#16863352C2212yes—
48#16863351C2212yes—
49#16863350C2214yes—
50#16863348C2214yes—
51C4×#11761611C2214yes—
52C2×#14523953C2211yescharacteristic
53C2×C4×C67⋊C8C2211yescharacteristic
54C2×C8×Dic67C2211yescharacteristic
55#533×C67C2211yescharacteristic
56#446×C67C2218yescomplemented
57#424×C67C2212yes—
58#423×C67C2211yescharacteristic
59#405×C67C2212yes—
60C2×#97×C67C2211yescharacteristic
61C22×C4×D67C2311yescharacteristic
62C2×#11761612C2312yes—
63C2×#11761611Q818yescomplemented
64C2×#11761611C2314yes—
65C2×#11761611C2312yes—
66C2×#11761611C2312yes—
67M16×D67—28no—
68C22×C67⋊C8C2311yescharacteristic
69C22×C67⋊C8C2311yescharacteristic
70C2×C8×D67—24no—
71C2×C8×D67C2314yes—
72C2×C8×D67C2314yes—
73#14524043C2×C412yes—
74#14524042C2×C414yes—
75#14524042C2×C418yes—
76#14524041C2×C418yes—
77#14524035C2×C412yes—
78#14524034C2×C414yes—
79#14524033C2×C418yes—
80#14524033C2×C418yes—
81#14524023C2×C418yes—
82#14524022C2×C414yes—
83#14524021C2×C418yes—
84#14524021C2×C418yes—
85#14524020C2×C418yes—
86#14524019C2×C418yes—
87C4×Dic134C2314yes—
88C4×Dic134C2318yes—
89#14524002C2312yes—
90#14524001C2×C414yes—
91#14524000C2×C412yes—
92#14524000C2×C412yes—
93#14524000C2314yes—
94#14524000C2314yes—
95#14523999C2×C414yes—
96C2×Dic67⋊C4C2×C412yes—
97C4⋊C4×D67C2×C414yes—
98C2×(C2×D67)⋊C4C2×C412yes—
99C22⋊C4×D67C2×C414yes—
100C2×C4×Dic67C2311yescharacteristic
101C42×D67C2314yes—
102C4⋊(C67⋊C8)C2318yes—
103C4⋊(C67⋊C8)C2314yes—
104Dic67⋊C8C2318yes—
105Dic67⋊C8C2318yes—
106Dic67⋊C8C2318yes—
107#14523954C2314yes—
108#14523953C2312yes—
109#14523953C2314yes—
110#14523953C2314yes—
111#14523953C2312yes—
112C4×C67⋊C8C2312yes—
113C4×C67⋊C8C2312yes—
114C4×C67⋊C8C2314yes—
115C8×Dic67C2314yes—
116C8×Dic67C2314yes—
117C8×Dic67C2314yes—
118C2×M16×C67C2312yes—
119C22×C8×C67C2311yescharacteristic
120C22×C8×C67C2311yescharacteristic
121C4⋊Q8×C67C2×C412yes—
122#125×C67C2×C412yes—
123C22⋊Q8×C67C2×C414yes—
124C4×Q8×C67C2314yes—
125#117×C67C2312yes—
126C2×C42×C67C2311yescharacteristic
127C4⋊C8×C67C2314yes—
128C4⋊C8×C67C2318yes—
129#97×C67C2314yes—
130#97×C67C2314yes—
131C4×C8×C67C2314yes—
132C23×D67C22×C411yescharacteristic
133C2×Dic134C22×C418yes—
134C2×Dic134C22×C414yes—
135C22×Dic67C22×C412yes—
136C22×Dic67C22×C411yescharacteristic
137C2×C4×D67C22×C418yes—
138C2×C4×D67C22×C412yes—
139C2×C4×D67C2×Q814yes—
140C2×C4×D67#4114yes—
141C2×C4×D67C2412yes—
142#11761612—24no—
143#11761611—28no—
144#11761611C2×Q818yes—
145#11761611C2×Q818yes—
146C2×C67⋊C8—22no—
147C2×C67⋊C8C2×Q814yes—
148C2×C67⋊C8C2412yes—
149C2×C67⋊C8C2414yes—
150C2×C67⋊C8C2411yescharacteristic
151C2×C67⋊C8C2411yescharacteristic
152C8×D67—28no—
153C8×D67—28no—
154C4⋊Dic67C22×C414yes—
155C4⋊Dic67C22×C418yes—
156C4⋊Dic67C22×C414yes—
157Dic67⋊C4C22×C418yes—
158Dic67⋊C4C22×C414yes—
159Dic67⋊C4C22×C414yes—
160Dic67⋊C4C22×C418yes—
161Dic67⋊C4C22×C418yes—
162Dic67⋊C4C22×C418yes—
163C22⋊Dic67C22×C414yes—
164(C2×D67)⋊C4C22×C418yes—
165(C2×D67)⋊C4C22×C414yes—
166(C2×D67)⋊C4C22×C414yes—
167C4×Dic67C22×C418yes—
168C4×Dic67C22×C412yes—
169C4×Dic67C2414yes—
170C4×Dic67C2412yes—
171C2×Q8×C67C22×C414yes—
172C22×C4×C67C22×C412yes—
173C22×C4×C67C2411yescharacteristic
174M16×C67—24no—
175C2×C8×C67—22no—
176C2×C8×C67C2×Q814yes—
177C2×C8×C67C2412yes—
178C2×C8×C67C2414yes—
179C2×C8×C67C2412yes—
180C4⋊C4×C67C22×C414yes—
181C4⋊C4×C67C22×C418yes—
182C4⋊C4×C67C22×C414yes—
183C22⋊C4×C67C22×C414yes—
184C42×C67C22×C411yescharacteristic
185C42×C67C22×C411yescharacteristic
186C42×C67C2412yes—
187C42×C67C2414yes—
188C23×C67C23×C411yescharacteristic
189C22×D67—42no—
190C22×D67C4×Q814yes—
191C22×D67C23×C412yes—
192Q8×C67—44no—
193C2×C4×C67—42no—
194C2×C4×C67C23×C412yes—
195C2×C4×C67C23×C412yes—
196C2×C4×C67C23×C418yes—
197C2×C4×C67C23×C411yescharacteristic
198C2×C4×C67C22×Q812yes—
199C2×C4×C67C2×#4112yes—
200C2×C4×C67C2511yescharacteristic
201C2×Dic67—42no—
202C2×Dic67C4×Q814yes—
203C2×Dic67C23×C414yes—
204C2×Dic67C23×C418yes—
205C2×Dic67C23×C412yes—
206C4×D67—48no—
207C4×D67C4×Q818yes—
208C4×D67C22×Q814yes—
209C4×D67C2×#4114yes—
210Dic134—44no—
211Dic134—48no—
212C8×C67—22no—
213C8×C67—24no—
214C8×C67C22×Q814yes—
215C67⋊C8—22no—
216C67⋊C8—24no—
217C67⋊C8C22×Q812yes—
218C67⋊C8C22×Q812yes—
219C22×C67—22no—
220C22×C67C2×C4×Q812yes—
221C22×C67C24×C411yescharacteristic
222C2×D67—24no—
223C2×D67—44no—
224C2×D67C2×C4×Q814yes—
225C4×C67—22no—
226C4×C67—28no—
227C4×C67—22no—
228C4×C67C2×C4×Q812yes—
229C4×C67C23×Q811yescharacteristic
230C4×C67C22×#4111yescharacteristic
231Dic67—28no—
232Dic67—24no—
233Dic67C2×C4×Q814yes—
234#60193—671nomaximal, Sylow
235C2×C67—22no—
236C2×C67C22×C4×Q811yescharacteristic
237C2×C67#357312yes—
238D67—24no—
239#3458—671no—
240#2962—678no—
241#2962—678no—
242#2961—672no—
243#2961—671no—
244#2961D6711yescharacteristic, complemented
245#2960—671no—
246#2960—672no—
247#2960—671no—
248#2905—671no—
249C2×#424—672no—
250#2873—672no—
251C2×#405—671no—
252C67#6019311yescharacteristic, complemented, Sylow
253C22×M16—672no—
254C22×M16—671no—
255C23×C8—671no—
256#544—674no—
257#544—674no—
258#544—674no—
259#543—674no—
260#533—678no—
261#533—672no—
262#533—671no—
263#533C2×D6711yescharacteristic
264C2×#117—672no—
265C2×#117—671no—
266#446—674no—
267#446—674no—
268#446—678no—
269#446—678no—
270#446—678no—
271#446—678no—
272#446C2×D6718yescomplemented
273C8×Q8—674no—
274C8×Q8—674no—
275C8×Q8—678no—
276#424—672no—
277#424—672no—
278#424—672no—
279#424—678no—
280#424—678no—
281#424C2×D6712yes—
282#423—672no—
283#423—671no—
284#423C2×D6711yescharacteristic
285C2×C4⋊C8—678no—
286#405—672no—
287#405—672no—
288#405—672no—
289#405—674no—
290#405—674no—
291#405—674no—
292#405—674no—
293#405C2×D6712yes—
294C4×M16—674no—
295C4×M16—674no—
296C2×#97—671no—
297C2×#97C2×D6711yescharacteristic
298C2×C4×C8—671no—
299C2×C4×C8—671no—
300C23×C4—671no—
301C2×M16—672no—
302C2×M16—678no—
303C2×M16—672no—
304C2×M16—672no—
305C2×M16—674no—
306C2×M16—1348no—
307C2×M16C22×D6712yes—
308C22×C8—671no—
309C22×C8—671no—
310C22×C8—674no—
311C22×C8—674no—
312C22×C8—1344no—
313C22×C8C22×D6711yescharacteristic
314C22×C8C22×D6711yescharacteristic
315C4⋊Q8—678no—
316C4⋊Q8—678no—
317C4⋊Q8—674no—
318C4⋊Q8—672no—
319C4⋊Q8C4×D6712yes—
320#125—678no—
321#125—678no—
322#125—674no—
323#125—672no—
324#125C4×D6712yes—
325C22⋊Q8—678no—
326C22⋊Q8—674no—
327C22⋊Q8—678no—
328C22⋊Q8—678no—
329C22⋊Q8—678no—
330C22⋊Q8—678no—
331C22⋊Q8C4×D6714yes—
332C4×Q8—674no—
333C4×Q8—678no—
334C4×Q8C22×D6714yes—
335#117—672no—
336#117—674no—
337#117—674no—
338#117—672no—
339#117—672no—
340#117—674no—
341#117—674no—
342#117C22×D6712yes—
343C2×C4⋊C4—672no—
344C2×C4⋊C4—674no—
345C2×C22⋊C4—672no—
346C2×C22⋊C4—674no—
347C2×C42—671no—
348C2×C42—674no—
349C2×C42C22×D6711yescharacteristic
350C4⋊C8—678no—
351C4⋊C8—678no—
352C4⋊C8—678no—
353C4⋊C8—678no—
354C4⋊C8—674no—
355C4⋊C8C22×D6718yes—
356C4⋊C8C22×D6714yes—
357#97—672no—
358#97—672no—
359#97—674no—
360#97—674no—
361#97—674no—
362#97C22×D6714yes—
363#97C22×D6714yes—
364C4×C8—674no—
365C4×C8—674no—
366C4×C8—674no—
367C4×C8—672no—
368C4×C8—672no—
369C4×C8—674no—
370C4×C8C22×D6714yes—
371C24—671no—
372C2×Q8—678no—
373C2×Q8—674no—
374C2×Q8C2×C4×D6714yes—
375C22×C4—672no—
376C22×C4—678no—
377C22×C4—671no—
378C22×C4—672no—
379C22×C4—674no—
380C22×C4—674no—
381C22×C4—672no—
382C22×C4C2×C4×D6712yes—
383C22×C4C23×D6711yescharacteristic
384M16—24no—
385M16—678no—
386M16—678no—
387M16—1348no—
388M16—1344no—
389C2×C8—22no—
390C2×C8—674no—
391C2×C8—671no—
392C2×C8—671no—
393C2×C8—672no—
394C2×C8—674no—
395C2×C8—1348no—
396C2×C8—1348no—
397C2×C8—1342no—
398C2×C8Q8×D6714yes—
399C2×C8C23×D6714yes—
400C2×C8C23×D6712yes—
401C2×C8C23×D6712yes—
402C4⋊C4—678no—
403C4⋊C4—674no—
404C4⋊C4—674no—
405C4⋊C4—674no—
406C4⋊C4—678no—
407C4⋊C4—678no—
408C4⋊C4—678no—
409C4⋊C4—678no—
410C4⋊C4—674no—
411C4⋊C4C2×C4×D6714yes—
412C4⋊C4C2×C4×D6714yes—
413C4⋊C4C2×C4×D6718yes—
414C22⋊C4—674no—
415C22⋊C4—678no—
416C22⋊C4—674no—
417C22⋊C4—674no—
418C22⋊C4C2×C4×D6714yes—
419C42—678no—
420C42—672no—
421C42—672no—
422C42—674no—
423C42C2×C4×D6711yescharacteristic
424C42C2×C4×D6711yescharacteristic
425C42C23×D6712yes—
426C42C23×D6714yes—
427C23—2682no—
428C23—674no—
429C23—672no—
430C23C22×C4×D6711yescharacteristic
431Q8—44no—
432Q8—2684no—
433Q8—2688no—
434C2×C4—42no—
435C2×C4—2688no—
436C2×C4—2682no—
437C2×C4—674no—
438C2×C4—672no—
439C2×C4—678no—
440C2×C4—674no—
441C2×C4—674no—
442C2×C4—678no—
443C2×C4—674no—
444C2×C4C22×C4×D6718yes—
445C2×C4C22×C4×D6712yes—
446C2×C4C22×C4×D6712yes—
447C2×C4C22×C4×D6711yescharacteristic
448C2×C4C2×Q8×D6712yes—
449C2×C4#41×D6712yes—
450C2×C4C24×D6711yescharacteristic
451C8—22no—
452C8—24no—
453C8—672no—
454C8—672no—
455C8—1342no—
456C8—1344no—
457C8C2×Q8×D6714yes—
458C22—22no—
459C22—2684no—
460C22—674no—
461C22—1344no—
462C22C4×Q8×D6712yes—
463C22C23×C4×D6711yescharacteristic
464C4—22no—
465C4—22no—
466C4—28no—
467C4—674no—
468C4—1348no—
469C4—1344no—
470C4C4×Q8×D6712yes—
471C4C22×Q8×D6711yescharacteristic
472C4C2×#41×D6711yescharacteristic
473C2—22no—
474C2—1344no—
475C2C2×C4×Q8×D6711yescharacteristic
476C2#2257492212yes—
subgroupG/H
center, FrattiniC2×C4C24×D67
commutator, socleC22×C67C24×C4
FittingC67×#2961C2
prime radicalC23×D67C22×C4
cosocleC2×C4×C67C25
hypercenter#2961D67
hypocenterC67#60193
LCSG ▷ C22×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ #2961

Complements

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

NH ≅ G/Ntype
2. #22569157C2semidirect
3. #22569154C2semidirect
4. #22569149C2semidirect
5. #22569148C2semidirect
6. #22569144C2semidirect
7. #22569141C2semidirect
8. #22569139C2semidirect
13. C67×#2961C2semidirect
26. #16863527C22semidirect
27. #16863525C22semidirect
28. #16863524C22semidirect
29. #16863524C22semidirect
30. #16863523C22semidirect
31. #16863523C22semidirect
32. Q8×C67⋊C8C22semidirect
33. C8×Dic134C22semidirect
34. #16863520C22semidirect
56. #446×C67C22semidirect
63. C2×#11761611Q8semidirect
244. #2961D67semidirect
252. C67#60193semidirect
272. #446C2×D67semidirect

Maximal subgroups

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

subgroupindex
#225736122
#225691572
#225691542
#225691492
#225691482
#225691442
#225691412
#225691392
#225687712
#424×D672
#225686252
C2×#168633532
C67×#29612
#6019367