Finite Group Databasebeta

← #33241086

Subgroups — #33241086

Order 18688.

568 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618784C211yesmaximal, characteristic, complemented
2#22612136C211yesmaximal, characteristic, complemented
3#22612119C211yesmaximal, characteristic, complemented
4#22612075C211yesmaximal, characteristic, complemented
5#22612065C211yesmaximal, characteristic, complemented
6#22611983C211yesmaximal, characteristic, complemented
7#22611983C211yesmaximal, characteristic, complemented
8#22611859C211yesmaximal, characteristic, complemented
9#22611842C211yesmaximal, characteristic
10#22611805C211yesmaximal, characteristic, complemented
11#22611793C211yesmaximal, characteristic, complemented
12C4×#14556534C211yesmaximal, characteristic, complemented
13C4×#14556532C211yesmaximal, characteristic, complemented
14#22611738C211yesmaximal, characteristic, complemented
15C73×#3175C211yesmaximal, characteristic, complemented
16C22×D584C2211yescharacteristic
17#17989061—21no—
18C2×#14556534C2211yescharacteristic
19C2×#14556534C2211yescharacteristic
20C2×#14556532C2211yescharacteristic
21C2×#14556532C2211yescharacteristic
22#17988700—21no—
23#17988700—21no—
24C4×D584C2211yescharacteristic
25C4×D584C2211yescharacteristic
26C4×D584C2211yescharacteristic
27C4×D584C2211yescharacteristic
28#17988625C2211yescharacteristic
29#17988593—21no—
30#17988414C2211yescharacteristic, complemented
31#17988414C2211yescharacteristic, complemented
32#17988412C2211yescharacteristic, complemented
33#17988412C2211yescharacteristic, complemented
34C4⋊#11924003C2211yescharacteristic, complemented
35C4⋊#11924003C2211yescharacteristic, complemented
36#17988391C2211yescharacteristic, complemented
37#17988391C2211yescharacteristic, complemented
38#17988388C2211yescharacteristic, complemented
39#17988388C2211yescharacteristic, complemented
40#17988387C2211yescharacteristic
41#17988387C2211yescharacteristic
42#17988387C2211yescharacteristic
43#17988387C2211yescharacteristic
44C22⋊#11924003C2211yescharacteristic
45C22⋊#11924003C2211yescharacteristic
46#17988363C2211yescharacteristic, complemented
47C4×#11924003C2211yescharacteristic, complemented
48#542×C73—21no—
49#500×C73C2211yescharacteristic, complemented
50#495×C73C2211yescharacteristic, complemented
51#489×C73—21no—
52#468×C73—21no—
53#467×C73C2211yescharacteristic
54#466×C73C2211yescharacteristic
55#440×C73—21no—
56C4×Q16×C73C2211yescharacteristic, complemented
57C4×D8×C73C2211yescharacteristic, complemented
58#405×C73C2211yescharacteristic
59C22×D292—21no—
60C2×D584—22no—
61C2×D584—22no—
62C2×D584—21no—
63C2×D584C2311yescharacteristic
64C2×D584C2311yescharacteristic
65C4×D292—21no—
66C4×D292—21no—
67C4×D292—21no—
68C4×D292—21no—
69#14556553—41no—
70#14556553—41no—
71#14556550—21no—
72C22⋊C4×D73—41no—
73C22⋊C4×D73—41no—
74#14556534—22no—
75#14556534—22no—
76#14556532—22no—
77#14556532—22no—
78C2×#11924003C2311yescharacteristic
79C2×#11924003C2311yescharacteristic
80C2×#11924003C2311yescharacteristic
81C2×#11924003C2311yescharacteristic
82#14556525C2311yescharacteristic
83#14556525C2311yescharacteristic
84#14556523—21no—
85#14556513—21no—
86#14556513—21no—
87#14556513—21no—
88#14556513—21no—
89C2×Q16×C73C2311yescharacteristic
90C2×SD16×C73—21no—
91C2×D8×C73C2311yescharacteristic
92C2×M16×C73D411yescharacteristic
93C22×C8×C73C2311yescharacteristic
94C4⋊Q8×C73—21no—
95#126×C73—41no—
96#124×C73—21no—
97#124×C73—41no—
98#123×C73—41no—
99C22⋊Q8×C73—21no—
100C22⋊Q8×C73—41no—
101#121×C73—21no—
102C4×Q8×C73—21no—
103C4×D4×C73—21no—
104#117×C73D411yescharacteristic
105#107×C73—21no—
106#107×C73C2311yescharacteristic
107#107×C73C2311yescharacteristic
108#103×C73—21no—
109#103×C73—21no—
110#102×C73—21no—
111#102×C73—21no—
112#97×C73D411yescharacteristic
113#97×C73D411yescharacteristic
114C4×C8×C73C2311yescharacteristic
115C4×C8×C73C2311yescharacteristic
116C23×D73—41no—
117C2×D292—24no—
118C2×D292—21no—
119C2×D292—21no—
120C2×D292—42no—
121C2×D292—41no—
122C2×C4×D73—41no—
123C2×C4×D73—41no—
124C2×C4×D73—41no—
125C2×C4×D73—41no—
126D584—24no—
127D584—21no—
128D584—21no—
129D584#4112yes—
130D584#4112yes—
131C4⋊Dic73—21no—
132C4⋊Dic73—21no—
133C22⋊Dic73—41no—
134#11924005—41no—
135#11924005—41no—
136#11924005—41no—
137#11924005—41no—
138C4×Dic73—41no—
139#11924003—21no—
140#11924003—21no—
141#11924003—21no—
142#11924003—21no—
143C2×Q8×C73—21no—
144C2×Q8×C73—41no—
145C2×D4×C73—21no—
146C2×D4×C73—41no—
147C22×C4×C73—41no—
148C22×C4×C73C2×D411yescharacteristic
149Q16×C73—21no—
150Q16×C73—21no—
151SD16×C73—42no—
152D8×C73—21no—
153D8×C73—21no—
154M16×C73—41no—
155C2×C8×C73—21no—
156C2×C8×C73—21no—
157C2×C8×C73C2×D411yescharacteristic
158C2×C8×C73C2×D411yescharacteristic
159C2×C8×C73C2×D411yescharacteristic
160C2×C8×C73C2411yescharacteristic
161C4⋊C4×C73—21no—
162C4⋊C4×C73—21no—
163C4⋊C4×C73—41no—
164C4⋊C4×C73—41no—
165C4⋊C4×C73—41no—
166C4⋊C4×C73D811yescharacteristic
167C4⋊C4×C73D811yescharacteristic
168C22⋊C4×C73—41no—
169C22⋊C4×C73—41no—
170C22⋊C4×C73—41no—
171C22⋊C4×C73—41no—
172C22⋊C4×C73—41no—
173C22⋊C4×C73D811yescharacteristic
174C22⋊C4×C73D811yescharacteristic
175C42×C73—41no—
176C42×C73C2×D411yescharacteristic
177C42×C73C2×D411yescharacteristic
178C23×C73—41no—
179C23×C73C2×D811yescharacteristic
180C22×D73—41no—
181C22×D73—41no—
182C22×D73—44no—
183C22×D73—161no—
184C22×D73—161no—
185Q8×C73—41no—
186Q8×C73—41no—
187Q8×C73—161no—
188D4×C73—41no—
189D4×C73—41no—
190D4×C73—161no—
191C2×C4×C73—21no—
192C2×C4×C73—21no—
193C2×C4×C73—41no—
194C2×C4×C73—41no—
195C2×C4×C73—41no—
196C2×C4×C73—161no—
197C2×C4×C73C2×D811yescharacteristic
198C2×C4×C73C2×D811yescharacteristic
199C2×C4×C73C2×D811yescharacteristic
200C2×C4×C73C2×D811yescharacteristic
201C2×C4×C73C2×D811yescharacteristic
202C2×C4×C73C22×D411yescharacteristic
203C2×Dic73—41no—
204C2×Dic73—41no—
205D292—22no—
206D292—22no—
207D292—44no—
208D292—41no—
209D292—41no—
210C4×D73—161no—
211C4×D73—161no—
212C4×D73—161no—
213C4×D73—161no—
214C8×C73—21no—
215C8×C73—21no—
216C8×C73—21no—
217C8×C73C2×#4111yescharacteristic
218C8×C73C2×#4111yescharacteristic
219C22×C73—21no—
220C22×C73—21no—
221C22×C73—21no—
222C22×C73—81no—
223C22×C73—81no—
224C22×C73—81no—
225C22×C73C22×D811yescharacteristic
226C2×D73—42no—
227C2×D73—42no—
228C2×D73—81no—
229C2×D73—81no—
230C2×D73—81no—
231C2×D73—81no—
232C2×D73—164no—
233C4×C73—21no—
234C4×C73—21no—
235C4×C73—21no—
236C4×C73—21no—
237C4×C73—21no—
238C4×C73—81no—
239C4×C73—81no—
240C4×C73—81no—
241C4×C73#52511yescharacteristic
242C4×C73#57511yescharacteristic
243Dic73—81no—
244Dic73—81no—
245#33678—731nomaximal, Sylow
246C2×C73—41no—
247C2×C73—161no—
248C2×C73#310011yescharacteristic
249C2×C73#341511yescharacteristic
250C2×C73#341611yescharacteristic
251D73—82no—
252D73—82no—
253#3175D7311yescharacteristic, complemented
254#2948—731no—
255#2237—731no—
256#2234—731no—
257#2226—731no—
258#2224—731no—
259#2210—731no—
260#2210—731no—
261#2187—731no—
262#2184—731no—
263#2177—731no—
264#2175—731no—
265C4×SD32—731no—
266C4×D16—731no—
267#2164—731no—
268C73#3367811yescharacteristic, complemented, Sylow
269C22×D8—731no—
270#542—21no—
271#518—1461no—
272C2×SD32—731no—
273C2×SD32—731no—
274C2×D16—731no—
275C2×D16—731no—
276#500C2×D7311yescharacteristic, complemented
277#495C2×D7311yescharacteristic, complemented
278#489—21no—
279#468—21no—
280#467C2×D7311yescharacteristic
281#466C2×D7311yescharacteristic
282#442—1461no—
283#442—1461no—
284#440—21no—
285C4×Q16C2×D7311yescharacteristic, complemented
286C4×D8—731no—
287C4×D8—731no—
288C4×D8—731no—
289C4×D8—731no—
290C4×D8C2×D7311yescharacteristic, complemented
291#427—731no—
292#418—1461no—
293#405C2×D7311yescharacteristic
294#367—731no—
295#367—731no—
296#366—731no—
297#366—731no—
298C4⋊C16—731no—
299C4⋊C16—731no—
300#358—731no—
301#358—731no—
302#357—731no—
303#357—731no—
304#357—731no—
305#357—731no—
306#357—731no—
307#357—731no—
308C22⋊C16—731no—
309C22⋊C16—731no—
310#346—731no—
311C4×C16—731no—
312C22×D4—1461no—
313C2×Q16C22×D7311yescharacteristic
314C2×SD16—21no—
315C2×D8—731no—
316C2×D8—731no—
317C2×D8—1462no—
318C2×D8—1462no—
319C2×D8—1461no—
320C2×D8C22×D7311yescharacteristic
321C2×M16C22⋊D7311yescharacteristic
322C22×C8C22×D7311yescharacteristic
323C4⋊Q8—21no—
324#126—41no—
325#124—21no—
326#124—41no—
327#123—41no—
328C22⋊Q8—21no—
329C22⋊Q8—41no—
330#121—21no—
331C4×Q8—21no—
332C4×D4—21no—
333C4×D4—2921no—
334C4×D4—2921no—
335C4×D4—1461no—
336C4×D4—1461no—
337C4×D4—1461no—
338C4×D4—1461no—
339#117—1461no—
340#117C22⋊D7311yescharacteristic
341C2×C22⋊C4—2921no—
342C2×C22⋊C4—2921no—
343SD32—1462no—
344SD32—1462no—
345D16—1462no—
346D16—1462no—
347C2×C16—731no—
348C2×C16—731no—
349C2×C16—731no—
350C2×C16—731no—
351#107—21no—
352#107—731no—
353#107—731no—
354#107C22×D7311yescharacteristic
355#107C22×D7311yescharacteristic
356#106—1461no—
357#103—21no—
358#103—21no—
359#102—21no—
360#102—21no—
361#102—1461no—
362#102—1461no—
363#102—1461no—
364#102—1461no—
365#97C22⋊D7311yescharacteristic
366#97C22⋊D7311yescharacteristic
367C4×C8C22×D7311yescharacteristic
368C4×C8C22×D7311yescharacteristic
369C24—2921no—
370C2×Q8—21no—
371C2×Q8—41no—
372C2×D4—21no—
373C2×D4—41no—
374C2×D4—2922no—
375C2×D4—2921no—
376C2×D4—1464no—
377C2×D4—1461no—
378C2×D4—1461no—
379C22×C4—41no—
380C22×C4—2921no—
381C22×C4—2921no—
382C22×C4—2921no—
383C22×C4—2921no—
384C22×C4C2×C22⋊D7311yescharacteristic
385Q16—21no—
386Q16—21no—
387SD16—42no—
388D8—21no—
389D8—21no—
390D8—732no—
391D8—732no—
392D8—1464no—
393D8—1461no—
394D8—1461no—
395M16—41no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8C2×C22⋊D7311yescharacteristic
399C2×C8C2×C22⋊D7311yescharacteristic
400C2×C8C2×C22⋊D7311yescharacteristic
401C2×C8C23×D7311yescharacteristic
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—1461no—
408C4⋊C4—1461no—
409C4⋊C4#1192401311yescharacteristic
410C4⋊C4#1192401311yescharacteristic
411C22⋊C4—41no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C22⋊C4—41no—
415C22⋊C4—41no—
416C22⋊C4—2921no—
417C22⋊C4—2921no—
418C22⋊C4—2921no—
419C22⋊C4—2921no—
420C22⋊C4—2921no—
421C22⋊C4#1192401311yescharacteristic
422C22⋊C4#1192401311yescharacteristic
423C42—41no—
424C42—2921no—
425C42C2×C22⋊D7311yescharacteristic
426C42C2×C22⋊D7311yescharacteristic
427C16—1461no—
428C16—1461no—
429C16—1461no—
430C16—1461no—
431C23—41no—
432C23—2924no—
433C23—2921no—
434C23—2921no—
435C23—11681no—
436C23—11681no—
437C23C2×#1192401311yescharacteristic
438Q8—41no—
439Q8—41no—
440Q8—161no—
441D4—41no—
442D4—41no—
443D4—161no—
444D4—2924no—
445D4—2921no—
446D4—2921no—
447D4—1462no—
448D4—1462no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—161no—
455C2×C4—2921no—
456C2×C4—2921no—
457C2×C4—11681no—
458C2×C4—11681no—
459C2×C4—11681no—
460C2×C4—11681no—
461C2×C4C2×#1192401311yescharacteristic
462C2×C4C2×#1192401311yescharacteristic
463C2×C4C2×#1192401311yescharacteristic
464C2×C4C2×#1192401311yescharacteristic
465C2×C4C2×#1192401311yescharacteristic
466C2×C4C22×C22⋊D7311yescharacteristic
467C8—21no—
468C8—21no—
469C8—21no—
470C8#41×D7311yescharacteristic
471C8#41×D7311yescharacteristic
472C22—21no—
473C22—21no—
474C22—21no—
475C22—81no—
476C22—81no—
477C22—81no—
478C22—2922no—
479C22—2922no—
480C22—5841no—
481C22—5841no—
482C22—5841no—
483C22—5841no—
484C22—11684no—
485C22C22×#1192401311yescharacteristic
486C4—21no—
487C4—21no—
488C4—21no—
489C4—21no—
490C4—21no—
491C4—81no—
492C4—81no—
493C4—81no—
494C4—5841no—
495C4—5841no—
496C4#1798909411yescharacteristic
497C4#1798938511yescharacteristic
498C2—41no—
499C2—161no—
500C2—5842no—
501C2—5842no—
502C2#2261992711yescharacteristic
503C2#2262307011yescharacteristic
504C2#2262307511yescharacteristic
subgroupG/H
centerC22C22×#11924013
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#3175C2
socleC22×C73C22×D8
prime radical#22611842C2
hypercenter#3175D73
hypocenterC73#33678
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #3175

Complements

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

NH ≅ G/Ntype
1. #22618784C2semidirect
2. #22612136C2semidirect
3. #22612119C2semidirect
4. #22612075C2semidirect
5. #22612065C2semidirect
6. #22611983C2semidirect
7. #22611983C2semidirect
8. #22611859C2semidirect
10. #22611805C2semidirect
11. #22611793C2semidirect
12. C4×#14556534C2semidirect
13. C4×#14556532C2semidirect
14. #22611738C2semidirect
15. C73×#3175C2semidirect
30. #17988414C22semidirect
31. #17988414C22semidirect
32. #17988412C22semidirect
33. #17988412C22semidirect
34. C4⋊#11924003C22semidirect
35. C4⋊#11924003C22semidirect
36. #17988391C22semidirect
37. #17988391C22semidirect
38. #17988388C22semidirect
39. #17988388C22semidirect
46. #17988363C22semidirect
47. C4×#11924003C22semidirect
49. #500×C73C22semidirect
50. #495×C73C22semidirect
56. C4×Q16×C73C22semidirect
57. C4×D8×C73C22semidirect
253. #3175D73semidirect
268. C73#33678semidirect
276. #500C2×D73semidirect
277. #495C2×D73semidirect
285. C4×Q16C2×D73semidirect
290. C4×D8C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226187842
#226121362
#226121192
#226120752
#226120652
#226119832
#226118592
#226118422
#226118052
#226117932
C4×#145565342
C4×#145565322
#226117382
C73×#31752
#3367873