Finite Group Databasebeta

← #33189183

Subgroups — #33189183

Order 18688.

674 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22615896C211yesmaximal, characteristic, complemented
2#22611433C211yesmaximal, characteristic, complemented
3#22611309C211yesmaximal, characteristic, complemented
4#22611305C211yesmaximal, characteristic
5#22611248C211yesmaximal, characteristic
6#22610784C211yesmaximal, characteristic
7#22610656C211yesmaximal, characteristic, complemented
8#22610631C211yesmaximal, characteristic
9#22610620C211yesmaximal, characteristic, complemented
10#22610565C211yesmaximal, characteristic, complemented
11#22610509C211yesmaximal, characteristic, complemented
12#22610265C211yesmaximal, characteristic
13#22610192C211yesmaximal, characteristic
14#22609788C211yesmaximal, characteristic, complemented
15C73×#2319C211yesmaximal, characteristic
16C22×#11924016C2211yescharacteristic
17C2×#14556522C2211yescharacteristic
18C2×#14556522C2211yescharacteristic
19C2×#14556522C2211yescharacteristic
20C2×C4⋊#10596831C2211yescharacteristic
21C2×C4⋊#10596831C2211yescharacteristic
22C2×#14556514C2211yescharacteristic
23C2×#14556514C2211yescharacteristic
24C2×#14556514C2211yescharacteristic
25C2×#14556512C2211yescharacteristic
26C2×#14556512C2211yescharacteristic
27C2×#14556512C2211yescharacteristic
28C2×#14556502C2211yescharacteristic
29C2×C4×#10596831C2211yescharacteristic
30C73⋊#401—21no—
31#17988528—21no—
32#17988528—21no—
33#17988528—21no—
34#17988493C2211yescharacteristic
35#17988493C2211yescharacteristic
36#17988493C2211yescharacteristic
37#17988485C2211yescharacteristic
38#17988485C2211yescharacteristic
39#17988485C2211yescharacteristic
40#17988349C2211yescharacteristic
41#17988349C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988342C2211yescharacteristic
47#17988342C2211yescharacteristic
48C2×C4×D4×C73C2211yescharacteristic
49#390×C73C2211yescharacteristic
50#390×C73C2211yescharacteristic
51#389×C73C2211yescharacteristic
52#389×C73C2211yescharacteristic
53#388×C73—21no—
54#388×C73—21no—
55#381×C73—21no—
56#380×C73—21no—
57C4×C4⋊C4×C73C2211yescharacteristic
58#376×C73C2211yescharacteristic
59C22×Dic146C2311yescharacteristic
60C2×#11924016—22no—
61C2×#11924016—22no—
62C2×#11924016—41no—
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65C22×#10596831C2311yescharacteristic
66C22×#10596831C2311yescharacteristic
67C2×C4⋊Dic73C2311yescharacteristic
68C2×C4⋊Dic73C2311yescharacteristic
69C2×C4⋊Dic73C2311yescharacteristic
70C2×#11924007—21no—
71C2×#11924007—21no—
72C2×#11924007—21no—
73#14556522—22no—
74#14556522—22no—
75#14556522—41no—
76C4⋊#10596831—22no—
77C4⋊#10596831—22no—
78#14556514—22no—
79#14556514—22no—
80#14556514—41no—
81#14556512—22no—
82#14556512—22no—
83#14556512—41no—
84#14556502—41no—
85C4×#10596831—41no—
86#14556498—21no—
87#14556498—21no—
88#14556498—21no—
89#14556498—21no—
90#14556498—21no—
91#14556498—21no—
92#14556498—21no—
93#14556498—21no—
94#14556498—21no—
95C22×D4×C73C2311yescharacteristic
96C23×C4×C73—21no—
97C4×D4×C73—22no—
98C4×D4×C73—22no—
99C2×C4⋊C4×C73—21no—
100C2×C4⋊C4×C73—21no—
101C2×C4⋊C4×C73C2311yescharacteristic
102C2×C4⋊C4×C73C2311yescharacteristic
103C2×C4⋊C4×C73C2311yescharacteristic
104C2×C22⋊C4×C73—21no—
105C2×C22⋊C4×C73—21no—
106C2×C22⋊C4×C73—21no—
107C2×C22⋊C4×C73—21no—
108C2×C22⋊C4×C73—21no—
109C2×C42×C73—21no—
110C2×C42×C73C2311yescharacteristic
111C2×C42×C73C2311yescharacteristic
112C2×C42×C73C2311yescharacteristic
113#95×C73—21no—
114#95×C73—21no—
115#95×C73—21no—
116#95×C73D411yescharacteristic
117#95×C73D411yescharacteristic
118#95×C73D411yescharacteristic
119#95×C73D411yescharacteristic
120C2×Dic146—21no—
121C2×Dic146—21no—
122C2×Dic146—21no—
123C2×Dic146—21no—
124C2×Dic146—21no—
125C2×Dic146—21no—
126C22×Dic73—21no—
127C22×Dic73—21no—
128C22×Dic73—21no—
129C22×Dic73—21no—
130#11924016—44no—
131C2×#10596831—21no—
132C2×#10596831—21no—
133C2×#10596831—21no—
134C2×#10596831—21no—
135C2×#10596831—21no—
136C2×#10596831—21no—
137C2×#10596831—21no—
138C2×#10596831—21no—
139C2×#10596831—21no—
140C2×#10596831—21no—
141C2×#10596831#4112yes—
142C2×#10596831#4112yes—
143C4⋊Dic73—21no—
144C4⋊Dic73—21no—
145C4⋊Dic73—21no—
146C4⋊Dic73—21no—
147C4⋊Dic73—21no—
148C4⋊Dic73—21no—
149#11924007—81no—
150#11924007—81no—
151#11924007—81no—
152C24×C73—21no—
153C2×D4×C73—21no—
154C2×D4×C73—21no—
155C2×D4×C73—21no—
156C2×D4×C73—21no—
157C2×D4×C73—21no—
158C2×D4×C73#4112yes—
159C22×C4×C73—21no—
160C22×C4×C73—24no—
161C22×C4×C73—21no—
162C22×C4×C73—21no—
163C22×C4×C73—81no—
164C22×C4×C73—81no—
165C22×C4×C73C2×D411yescharacteristic
166C22×C4×C73C2×D411yescharacteristic
167C22×C4×C73C2×D411yescharacteristic
168C22×C4×C73C2×D411yescharacteristic
169C22×C4×C73C2×D411yescharacteristic
170C22×C4×C73C2×D411yescharacteristic
171C22×C4×C73C2411yescharacteristic
172C4⋊C4×C73—21no—
173C4⋊C4×C73—21no—
174C4⋊C4×C73—21no—
175C4⋊C4×C73—21no—
176C4⋊C4×C73—21no—
177C4⋊C4×C73—81no—
178C4⋊C4×C73—81no—
179C4⋊C4×C73#4112yes—
180C22⋊C4×C73—24no—
181C22⋊C4×C73—81no—
182C22⋊C4×C73—81no—
183C22⋊C4×C73—81no—
184C22⋊C4×C73—81no—
185C42×C73—21no—
186C42×C73—21no—
187C42×C73—21no—
188C42×C73—21no—
189C42×C73—21no—
190C42×C73—21no—
191C42×C73—81no—
192C23×C73—22no—
193C23×C73—41no—
194C23×C73—41no—
195C23×C73—41no—
196C23×C73—41no—
197C23×C73—41no—
198C23×C73—41no—
199C23×C73C22×D411yescharacteristic
200D4×C73—22no—
201D4×C73—22no—
202D4×C73—41no—
203D4×C73—41no—
204C2×C4×C73—21no—
205C2×C4×C73—21no—
206C2×C4×C73—22no—
207C2×C4×C73—21no—
208C2×C4×C73—21no—
209C2×C4×C73—21no—
210C2×C4×C73—21no—
211C2×C4×C73—21no—
212C2×C4×C73—21no—
213C2×C4×C73—21no—
214C2×C4×C73—21no—
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—41no—
224C2×C4×C73—41no—
225C2×C4×C73—41no—
226C2×C4×C73—41no—
227C2×C4×C73—41no—
228C2×C4×C73—41no—
229C2×C4×C73—41no—
230C2×C4×C73—41no—
231C2×C4×C73—84no—
232C2×C4×C73C2×#4111yescharacteristic
233C2×C4×C73C2×#4111yescharacteristic
234C2×C4×C73C2×#4111yescharacteristic
235C2×C4×C73C2×#4111yescharacteristic
236C2×C4×C73H(5,𝔽2)11yescharacteristic
237C2×C4×C732−1+411yescharacteristic
238C2×Dic73—41no—
239C2×Dic73—41no—
240C2×Dic73—41no—
241C2×Dic73—41no—
242C2×Dic73—41no—
243C2×Dic73—41no—
244C2×Dic73—41no—
245C2×Dic73—41no—
246C2×Dic73—41no—
247C2×Dic73—41no—
248C2×Dic73—41no—
249C2×Dic73—41no—
250Dic146—41no—
251Dic146—41no—
252Dic146—41no—
253Dic146—41no—
254#10596831—22no—
255#10596831—22no—
256#10596831—41no—
257#10596831—41no—
258C22×C73—42no—
259C22×C73—42no—
260C22×C73—42no—
261C22×C73—81no—
262C22×C73—81no—
263C22×C73—81no—
264C22×C73—81no—
265C22×C73#52511yescharacteristic
266C22×C73#53211yescharacteristic
267C22×C73#53711yescharacteristic
268C22×C73#57711yescharacteristic
269C22×C73#57711yescharacteristic
270C22×C73#57711yescharacteristic
271C22×C73#57711yescharacteristic
272C4×C73—41no—
273C4×C73—42no—
274C4×C73—41no—
275C4×C73—41no—
276C4×C73—41no—
277C4×C73—41no—
278C4×C73—41no—
279C4×C73—81no—
280C4×C73—81no—
281C4×C73#55111yescharacteristic
282C4×C73#55211yescharacteristic
283C4×C73#55311yescharacteristic
284C4×C73#55611yescharacteristic
285Dic73—81no—
286Dic73—81no—
287Dic73—81no—
288Dic73—81no—
289#29440—731nomaximal, Sylow
290C2×C73—82no—
291C2×C73#260111yescharacteristic
292C2×C73#312311yescharacteristic
293C2×C73#312511yescharacteristic
294C2×C73#317311yescharacteristic
295C2×C73#317411yescharacteristic
296C2×C73#326011yescharacteristic
297C2×C73#326311yescharacteristic
298#2645—731no—
299#2319D7311yescharacteristic
300#2088—731no—
301#2066—731no—
302#2065—731no—
303#2055—731no—
304#1968—731no—
305#1941—731no—
306#1936—731no—
307#1935—731no—
308#1927—731no—
309#1919—731no—
310#1877—731no—
311#1864—731no—
312#1768—731no—
313C73#2944011yescharacteristic, complemented, Sylow
314C22×SD16—731no—
315C2×C4×D4C2×D7311yescharacteristic
316C2×#106—731no—
317C2×#106—731no—
318C2×#106—731no—
319C2×C4⋊C8—731no—
320C2×C4⋊C8—731no—
321C2×#103—731no—
322C2×#103—731no—
323C2×#103—731no—
324C2×#102—731no—
325C2×#102—731no—
326C2×#102—731no—
327C2×#97—731no—
328C2×C4×C8—731no—
329#401—1461no—
330#400—1461no—
331#400—1461no—
332#400—1461no—
333#391—731no—
334#391—731no—
335#391—731no—
336#390C2×D7311yescharacteristic
337#390C2×D7311yescharacteristic
338#389—731no—
339#389—731no—
340#389—731no—
341#389C2×D7311yescharacteristic
342#389C2×D7311yescharacteristic
343#388—21no—
344#388—21no—
345#381—21no—
346#380—21no—
347C4×C4⋊C4C2×D7311yescharacteristic
348#376C2×D7311yescharacteristic
349#340—731no—
350#340—731no—
351#340—731no—
352#340—731no—
353#340—731no—
354#340—731no—
355#336—731no—
356#336—731no—
357C22×Q8—731no—
358C22×D4C22×D7311yescharacteristic
359C23×C4—21no—
360C2×SD16—2921no—
361C2×SD16—1462no—
362C2×SD16—1462no—
363C22×C8—731no—
364C22×C8—731no—
365C22×C8—731no—
366C22×C8—731no—
367C4×D4—22no—
368C4×D4—22no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—731no—
372C2×C4⋊C4—731no—
373C2×C4⋊C4—731no—
374C2×C4⋊C4—1461no—
375C2×C4⋊C4—1461no—
376C2×C4⋊C4—1461no—
377C2×C4⋊C4C22×D7311yescharacteristic
378C2×C4⋊C4C22×D7311yescharacteristic
379C2×C4⋊C4C22×D7311yescharacteristic
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C42—21no—
386C2×C42C22×D7311yescharacteristic
387C2×C42C22×D7311yescharacteristic
388C2×C42C22×D7311yescharacteristic
389#106—2921no—
390#106—1462no—
391#106—1462no—
392C4⋊C8—1462no—
393C4⋊C8—1462no—
394#103—2921no—
395#103—1462no—
396#103—1462no—
397#102—2921no—
398#102—1462no—
399#102—1462no—
400#97—2921no—
401C4×C8—2921no—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—1461no—
406#95—1461no—
407#95—1461no—
408#95—1461no—
409#95—1461no—
410#95—1461no—
411#95—1461no—
412#95—1461no—
413#95—1461no—
414#95C22⋊D7311yescharacteristic
415#95C22⋊D7311yescharacteristic
416#95C22⋊D7311yescharacteristic
417#95C22⋊D7311yescharacteristic
418C24—21no—
419C2×Q8—1461no—
420C2×Q8—1461no—
421C2×Q8—1461no—
422C2×Q8—1461no—
423C2×Q8—1461no—
424C2×Q8—1461no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4#1192402812yes—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—81no—
436C22×C4—81no—
437C22×C4—1461no—
438C22×C4—1461no—
439C22×C4—1461no—
440C22×C4—1461no—
441C22×C4C2×C22⋊D7311yescharacteristic
442C22×C4C2×C22⋊D7311yescharacteristic
443C22×C4C2×C22⋊D7311yescharacteristic
444C22×C4C2×C22⋊D7311yescharacteristic
445C22×C4C2×C22⋊D7311yescharacteristic
446C22×C4C2×C22⋊D7311yescharacteristic
447C22×C4C23×D7311yescharacteristic
448SD16—2924no—
449C2×C8—732no—
450C2×C8—732no—
451C2×C8—1461no—
452C2×C8—1461no—
453C2×C8—1461no—
454C2×C8—1461no—
455C2×C8—1461no—
456C2×C8—1461no—
457C2×C8—1461no—
458C2×C8—1461no—
459C2×C8—1461no—
460C2×C8—1461no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—81no—
467C4⋊C4—81no—
468C4⋊C4—5841no—
469C4⋊C4—5841no—
470C4⋊C4—5841no—
471C4⋊C4—1461no—
472C4⋊C4—1461no—
473C4⋊C4—1461no—
474C4⋊C4—1461no—
475C4⋊C4—1461no—
476C4⋊C4—1461no—
477C4⋊C4#1192402812yes—
478C22⋊C4—24no—
479C22⋊C4—81no—
480C22⋊C4—81no—
481C22⋊C4—81no—
482C22⋊C4—81no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—81no—
490C23—22no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23C22×C22⋊D7311yescharacteristic
498Q8—2921no—
499Q8—2921no—
500Q8—2921no—
501Q8—2921no—
502D4—22no—
503D4—22no—
504D4—41no—
505D4—41no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—84no—
534C2×C4—2921no—
535C2×C4—2921no—
536C2×C4—2921no—
537C2×C4—2921no—
538C2×C4—2921no—
539C2×C4—2921no—
540C2×C4—2921no—
541C2×C4—2921no—
542C2×C4—2921no—
543C2×C4—2921no—
544C2×C4—2921no—
545C2×C4—2921no—
546C2×C4#41×D7311yescharacteristic
547C2×C4#41×D7311yescharacteristic
548C2×C4C2×#1192402811yescharacteristic
549C2×C4C2×#1192402811yescharacteristic
550C2×C4#1455663711yescharacteristic
551C2×C4#1455663811yescharacteristic
552C8—2921no—
553C8—2921no—
554C8—1462no—
555C8—1462no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—81no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22#1798909811yescharacteristic
564C22#1798913311yescharacteristic
565C22#1798916511yescharacteristic
566C22#1798939511yescharacteristic
567C22#1798939811yescharacteristic
568C22#1798939811yescharacteristic
569C22#1798939811yescharacteristic
570C4—41no—
571C4—42no—
572C4—41no—
573C4—41no—
574C4—41no—
575C4—41no—
576C4—41no—
577C4—81no—
578C4—81no—
579C4—5841no—
580C4—5841no—
581C4—5841no—
582C4—5841no—
583C4#1798926111yescharacteristic
584C4#1798927011yescharacteristic
585C4#1798928111yescharacteristic
586C4#1798930511yescharacteristic
587C2—82no—
588C2#2261552811yescharacteristic
589C2#2262018311yescharacteristic
590C2#2262020811yescharacteristic
591C2#2262054211yescharacteristic
592C2#2262055311yescharacteristic
593C2#2262137711yescharacteristic
594C2#2262139611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2319C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
hypercenter#2319D73
hypocenterC73#29440
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ #376 ◁ #2319

Complements

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

NH ≅ G/Ntype
1. #22615896C2semidirect
2. #22611433C2semidirect
3. #22611309C2semidirect
7. #22610656C2semidirect
9. #22610620C2semidirect
10. #22610565C2semidirect
11. #22610509C2semidirect
14. #22609788C2semidirect
313. C73#29440semidirect

Maximal subgroups

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

subgroupindex
#226158962
#226114332
#226113092
#226113052
#226112482
#226107842
#226106562
#226106312
#226106202
#226105652
#226105092
#226102652
#226101922
#226097882
C73×#23192
#2944073