Finite Group Databasebeta

← #33242875

Subgroups — #33242875

Order 18688.

546 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22621674C211yesmaximal, characteristic
2#22612165C211yesmaximal, characteristic, complemented
3#22612139C211yesmaximal, characteristic, complemented
4#22612111C211yesmaximal, characteristic, complemented
5#22612092C211yesmaximal, characteristic, complemented
6#22612029C211yesmaximal, characteristic, complemented
7#22612020C211yesmaximal, characteristic, complemented
8#22611993C211yesmaximal, characteristic, complemented
9#22611869C211yesmaximal, characteristic, complemented
10#22611864C211yesmaximal, characteristic, complemented
11C4×#14556537C211yesmaximal, characteristic, complemented
12D4×#11924003C211yesmaximal, characteristic, complemented
13#22611636C211yesmaximal, characteristic, complemented
14C2×#17988390C211yesmaximal, characteristic, complemented
15C73×#3336C211yesmaximal, characteristic, complemented
16C2×#14556613C2211yescharacteristic
17C22×Dic292C2211yescharacteristic
18#17989246—21no—
19C2×#14556537C2211yescharacteristic
20C2×#14556537C2211yescharacteristic
21C2×#14556534C2211yescharacteristic
22C22×#11924003C2211yescharacteristic
23C22×#11924003C2211yescharacteristic
24#17988971C2211yescharacteristic
25#17988828C2211yescharacteristic
26#17988823C2211yescharacteristic
27#17988810—21no—
28#17988751—21no—
29#17988728—21no—
30C4×Dic292C2211yescharacteristic
31#17988414C2211yescharacteristic
32#17988412C2211yescharacteristic
33#17988412C2211yescharacteristic
34C4⋊#11924003C2211yescharacteristic
35#17988391C2211yescharacteristic
36#17988390C2211yescharacteristic
37#17988390C2211yescharacteristic
38#17988390C2211yescharacteristic
39#17988390C2211yescharacteristic
40#17988390C2211yescharacteristic
41#17988390C2211yescharacteristic
42#17988387C2211yescharacteristic
43#17988387C2211yescharacteristic
44#17988387C2211yescharacteristic
45C22⋊#11924003C2211yescharacteristic
46C22⋊#11924003C2211yescharacteristic
47C4×#11924003C2211yescharacteristic
48#552×C73—21no—
49#499×C73C2211yescharacteristic
50#485×C73—21no—
51#483×C73—21no—
52#479×C73—21no—
53#467×C73C2211yescharacteristic
54#467×C73C2211yescharacteristic
55C4×Q16×C73C2211yescharacteristic
56C8×D4×C73C2211yescharacteristic
57#427×C73C2211yescharacteristic
58C2×#107×C73C2211yescharacteristic
59C2×#11924028—21no—
60C22×Dic146—21no—
61#14556613—22no—
62#14556613—41no—
63C2×Dic292—22no—
64C2×Dic292—21no—
65C2×Dic292—41no—
66C2×Dic292C2311yescharacteristic
67C2×Dic292C2311yescharacteristic
68C2×#11924017—21no—
69C2×#11924017—21no—
70C2×D584C2311yescharacteristic
71#14556579—21no—
72#14556577—41no—
73#14556570—21no—
74#14556569—41no—
75#14556565—21no—
76#14556563—41no—
77C4×Dic146—21no—
78#14556551—41no—
79#14556537—22no—
80#14556537—41no—
81#14556534—41no—
82C2×#11924003—41no—
83C2×#11924003—41no—
84C2×#11924003C2311yescharacteristic
85C2×#11924003C2311yescharacteristic
86C2×#11924003C2311yescharacteristic
87C2×#11924003C2311yescharacteristic
88#14556525C2311yescharacteristic
89#14556516—21no—
90#14556516—21no—
91#14556513—21no—
92C2×Q16×C73C2311yescharacteristic
93C22×C8×C73C2311yescharacteristic
94C22×C8×C73C2311yescharacteristic
95#126×C73—41no—
96#125×C73—21no—
97#125×C73—41no—
98#123×C73—41no—
99C22⋊Q8×C73—21no—
100C22⋊Q8×C73—21no—
101C4×Q8×C73—21no—
102C4×D4×C73D411yescharacteristic
103#117×C73—21no—
104#117×C73—41no—
105C2×C4⋊C4×C73—21no—
106#107×C73—21no—
107#107×C73C2311yescharacteristic
108#107×C73C2311yescharacteristic
109#107×C73C2311yescharacteristic
110#106×C73—21no—
111#106×C73—21no—
112C4⋊C8×C73D411yescharacteristic
113#103×C73—21no—
114#103×C73—21no—
115#103×C73—21no—
116C22⋊C8×C73D411yescharacteristic
117C22⋊C8×C73D411yescharacteristic
118C4×C8×C73C2311yescharacteristic
119#11924028—42no—
120#11924028—81no—
121C2×Dic146—21no—
122C2×Dic146—21no—
123C2×Dic146—42no—
124C2×Dic146—41no—
125C2×Dic146—81no—
126C2×D292—21no—
127C2×C22⋊D73—41no—
128C2×C22⋊D73—41no—
129C22×Dic73—41no—
130C2×C4×D73—41no—
131Dic292—21no—
132Dic292—21no—
133Dic292—21no—
134Dic292—42no—
135Dic292#4112yes—
136#11924017—42no—
137#11924017—81no—
138D584—21no—
139D584—21no—
140C4⋊Dic73—21no—
141#11924007—41no—
142#11924007—41no—
143C22⋊Dic73—41no—
144#11924005—41no—
145#11924005—41no—
146#11924005—41no—
147C4×Dic73—41no—
148#11924003—21no—
149#11924003—21no—
150#11924003—21no—
151#11924003—21no—
152C2×Q8×C73—21no—
153C2×D4×C73D811yescharacteristic
154C22×C4×C73—41no—
155C22×C4×C73C2×D411yescharacteristic
156C22×C4×C73C2×D411yescharacteristic
157Q16×C73—21no—
158Q16×C73—21no—
159C2×C8×C73—21no—
160C2×C8×C73—21no—
161C2×C8×C73—21no—
162C2×C8×C73—21no—
163C2×C8×C73C2×D411yescharacteristic
164C2×C8×C73C2×D411yescharacteristic
165C2×C8×C73C2×D411yescharacteristic
166C2×C8×C73C2411yescharacteristic
167C4⋊C4×C73—21no—
168C4⋊C4×C73—21no—
169C4⋊C4×C73—21no—
170C4⋊C4×C73—41no—
171C4⋊C4×C73—41no—
172C4⋊C4×C73—41no—
173C4⋊C4×C73—41no—
174C4⋊C4×C73—41no—
175C4⋊C4×C73—41no—
176C4⋊C4×C73D811yescharacteristic
177C22⋊C4×C73—41no—
178C22⋊C4×C73—41no—
179C22⋊C4×C73—41no—
180C22⋊C4×C73D811yescharacteristic
181C22⋊C4×C73D811yescharacteristic
182C42×C73—41no—
183C42×C73—41no—
184C42×C73C2×D411yescharacteristic
185C23×C73C2×D811yescharacteristic
186C23×C73C2×D811yescharacteristic
187C22×D73—41no—
188Q8×C73—41no—
189Q8×C73—41no—
190D4×C73—22no—
191C2×C4×C73—21no—
192C2×C4×C73—21no—
193C2×C4×C73—21no—
194C2×C4×C73—21no—
195C2×C4×C73—41no—
196C2×C4×C73—41no—
197C2×C4×C73—41no—
198C2×C4×C73—41no—
199C2×C4×C73—82no—
200C2×C4×C73C2×D811yescharacteristic
201C2×C4×C73C2×D811yescharacteristic
202C2×C4×C73C2×D811yescharacteristic
203C2×C4×C73C2×D811yescharacteristic
204C2×C4×C73C22×D411yescharacteristic
205C22⋊D73—82no—
206C22⋊D73—82no—
207C2×Dic73—41no—
208C2×Dic73—41no—
209C2×Dic73—41no—
210C2×Dic73—82no—
211D292—41no—
212D292—41no—
213C4×D73—82no—
214Dic146—22no—
215Dic146—41no—
216Dic146—41no—
217Dic146—41no—
218Dic146—82no—
219C8×C73—21no—
220C8×C73—21no—
221C8×C73—21no—
222C8×C73C2×#4111yescharacteristic
223C8×C73H(5,𝔽2)11yescharacteristic
224C22×C73—21no—
225C22×C73—21no—
226C22×C73—21no—
227C22×C73—21no—
228C22×C73—21no—
229C22×C73#50812yes—
230C22×C73C22×D811yescharacteristic
231C2×D73—81no—
232C2×D73—81no—
233C2×D73—81no—
234C4×C73—21no—
235C4×C73—21no—
236C4×C73—21no—
237C4×C73—21no—
238C4×C73—81no—
239C4×C73—81no—
240C4×C73—81no—
241C4×C73—81no—
242C4×C73#54611yescharacteristic
243C4×C73#57711yescharacteristic
244Dic73—42no—
245Dic73—81no—
246Dic73—81no—
247#33867—731nomaximal, Sylow
248C2×C73—22no—
249C2×C73—41no—
250C2×C73#329411yescharacteristic
251C2×C73C2×#50811yescharacteristic
252C2×C73#341711yescharacteristic
253D73—161no—
254#3336D7311yescharacteristic, complemented
255#3285—731no—
256#2242—731no—
257#2237—731no—
258#2232—731no—
259#2229—731no—
260#2217—731no—
261#2216—731no—
262#2211—731no—
263#2189—731no—
264#2188—731no—
265C4×Q32—731no—
266D4×C16—731no—
267#2138—731no—
268C2×#358—731no—
269C73#3386711yescharacteristic, complemented, Sylow
270C2×#135—731no—
271C22×Q16—731no—
272#552—21no—
273#548—1461no—
274C2×Q32—731no—
275C2×Q32—731no—
276C2×SD32—731no—
277C22×C16—731no—
278C22×C16—731no—
279#499C2×D7311yescharacteristic
280#495—731no—
281#485—21no—
282#483—21no—
283#479—21no—
284#467—731no—
285#467C2×D7311yescharacteristic
286#467C2×D7311yescharacteristic
287#466—731no—
288#464—1461no—
289#452—1461no—
290#448—1461no—
291C4×Q16—731no—
292C4×Q16C2×D7311yescharacteristic
293C8×D4C2×D7311yescharacteristic
294#427C2×D7311yescharacteristic
295C2×#107C2×D7311yescharacteristic
296#367—731no—
297#366—731no—
298#366—731no—
299C4⋊C16—731no—
300#358—731no—
301#358—731no—
302#358—731no—
303#358—731no—
304#358—731no—
305#358—731no—
306#358—731no—
307#357—731no—
308#357—731no—
309#357—731no—
310C22⋊C16—731no—
311C22⋊C16—731no—
312C4×C16—731no—
313C2×#41—1461no—
314C22×Q8—1461no—
315#135—2921no—
316#135—1462no—
317C2×Q16—2921no—
318C2×Q16—731no—
319C2×Q16—731no—
320C2×Q16—1462no—
321C2×Q16—1461no—
322C2×Q16C22×D7311yescharacteristic
323C2×SD16—1461no—
324C2×SD16—1461no—
325C2×D8—731no—
326C22×C8C22×D7311yescharacteristic
327C22×C8C22×D7311yescharacteristic
328#126—41no—
329#125—21no—
330#125—41no—
331#124—2921no—
332#124—1461no—
333#123—41no—
334C22⋊Q8—21no—
335C22⋊Q8—21no—
336C22⋊Q8—2921no—
337C22⋊Q8—1461no—
338#121—2921no—
339#121—1461no—
340C4×Q8—21no—
341C4×Q8—1461no—
342C4×D4—2921no—
343C4×D4C22⋊D7311yescharacteristic
344#117—21no—
345#117—41no—
346C2×C4⋊C4—21no—
347Q32—2921no—
348Q32—1462no—
349SD32—2921no—
350C2×C16—2921no—
351C2×C16—2921no—
352C2×C16—731no—
353C2×C16—731no—
354C2×C16—731no—
355C2×C16—731no—
356#107—21no—
357#107—731no—
358#107C22×D7311yescharacteristic
359#107C22×D7311yescharacteristic
360#107C22×D7311yescharacteristic
361#106—21no—
362#106—21no—
363C4⋊C8C22⋊D7311yescharacteristic
364#103—21no—
365#103—21no—
366#103—21no—
367#103—1461no—
368#103—1461no—
369#102—1461no—
370C22⋊C8C22⋊D7311yescharacteristic
371C22⋊C8C22⋊D7311yescharacteristic
372C4×C8C22×D7311yescharacteristic
373#41—2922no—
374#41—5841no—
375C2×Q8—21no—
376C2×Q8—2922no—
377C2×Q8—2921no—
378C2×Q8—5841no—
379C2×Q8—1461no—
380C2×Q8—1461no—
381C2×D4—2921no—
382C2×D4—2921no—
383C2×D4—1461no—
384C2×D4#1192401311yescharacteristic
385C22×C4—41no—
386C22×C4—2921no—
387C22×C4—2921no—
388C22×C4C2×C22⋊D7311yescharacteristic
389C22×C4C2×C22⋊D7311yescharacteristic
390Q16—21no—
391Q16—21no—
392Q16—2922no—
393Q16—732no—
394Q16—1461no—
395Q16—1461no—
396Q16—1461no—
397SD16—2922no—
398SD16—5841no—
399D8—1461no—
400D8—1461no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8C2×C22⋊D7311yescharacteristic
406C2×C8C2×C22⋊D7311yescharacteristic
407C2×C8C2×C22⋊D7311yescharacteristic
408C2×C8C23×D7311yescharacteristic
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—2921no—
419C4⋊C4—2921no—
420C4⋊C4—1461no—
421C4⋊C4#1192401311yescharacteristic
422C22⋊C4—41no—
423C22⋊C4—41no—
424C22⋊C4—41no—
425C22⋊C4—2921no—
426C22⋊C4—2921no—
427C22⋊C4—2921no—
428C22⋊C4—2921no—
429C22⋊C4#1192401311yescharacteristic
430C22⋊C4#1192401311yescharacteristic
431C42—41no—
432C42—41no—
433C42—2921no—
434C42C2×C22⋊D7311yescharacteristic
435C16—1461no—
436C16—1461no—
437C16—1461no—
438C16—1461no—
439C23—2921no—
440C23C2×#1192401311yescharacteristic
441C23C2×#1192401311yescharacteristic
442Q8—41no—
443Q8—41no—
444Q8—2921no—
445Q8—2921no—
446Q8—2921no—
447Q8—5842no—
448Q8—1462no—
449D4—22no—
450D4—2921no—
451D4—2921no—
452D4—5842no—
453D4—5842no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—82no—
463C2×C4—2921no—
464C2×C4—2921no—
465C2×C4—2921no—
466C2×C4—5842no—
467C2×C4—5842no—
468C2×C4C2×#1192401311yescharacteristic
469C2×C4C2×#1192401311yescharacteristic
470C2×C4C2×#1192401311yescharacteristic
471C2×C4C2×#1192401311yescharacteristic
472C2×C4C22×C22⋊D7311yescharacteristic
473C8—21no—
474C8—21no—
475C8—21no—
476C8#41×D7311yescharacteristic
477C8#1455663711yescharacteristic
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—5841no—
484C22—5841no—
485C22—5841no—
486C22#1798901712yes—
487C22C22×#1192401311yescharacteristic
488C4—21no—
489C4—21no—
490C4—21no—
491C4—21no—
492C4—81no—
493C4—81no—
494C4—81no—
495C4—81no—
496C4—2922no—
497C4—5841no—
498C4—5841no—
499C4#1798922511yescharacteristic
500C4#1798939511yescharacteristic
501C2—22no—
502C2—41no—
503C2—11681no—
504C2#2262173811yescharacteristic
505C2C2×#1798901711yescharacteristic
506C2#2262308211yescharacteristic
subgroupG/H
centerC22C22×#11924013
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#3336C2
socleC22×C73C22×D8
prime radical#22621674C2
hypercenter#3336D73
hypocenterC73#33867
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3336

Complements

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

NH ≅ G/Ntype
2. #22612165C2semidirect
3. #22612139C2semidirect
4. #22612111C2semidirect
5. #22612092C2semidirect
6. #22612029C2semidirect
7. #22612020C2semidirect
8. #22611993C2semidirect
9. #22611869C2semidirect
10. #22611864C2semidirect
11. C4×#14556537C2semidirect
12. D4×#11924003C2semidirect
13. #22611636C2semidirect
14. C2×#17988390C2semidirect
15. C73×#3336C2semidirect
254. #3336D73semidirect
269. C73#33867semidirect

Maximal subgroups

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

subgroupindex
#226216742
#226121652
#226121392
#226121112
#226120922
#226120292
#226120202
#226119932
#226118692
#226118642
C4×#145565372
D4×#119240032
#226116362
C2×#179883902
C73×#33362
#3386773