Finite Group Databasebeta

← #33154191

Subgroups — #33154191

Order 18688.

726 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22612956C211yesmaximal, characteristic
2#22611253C211yesmaximal, characteristic, complemented
3#22611134C211yesmaximal, characteristic
4#22611079C211yesmaximal, characteristic, complemented
5#22610939C211yesmaximal, characteristic
6#22610889C211yesmaximal, characteristic, complemented
7#22610610C211yesmaximal, characteristic, complemented
8#22610569C211yesmaximal, characteristic, complemented
9#22610539C211yesmaximal, characteristic
10#22610502C211yesmaximal, characteristic, complemented
11#22610275C211yesmaximal, characteristic, complemented
12#22610244C211yesmaximal, characteristic
13#22610023C211yesmaximal, characteristic
14C4×#14556514C211yesmaximal, characteristic, complemented
15C73×#2489C211yesmaximal, characteristic
16C22×#11924018C2211yescharacteristic
17C2×#14556570C2211yescharacteristic
18C2×C4×Dic146C2211yescharacteristic
19C2×#14556550C2211yescharacteristic
20C2×#14556524C2211yescharacteristic
21C2×C4⋊#10596831C2211yescharacteristic
22C2×#14556515C2211yescharacteristic
23C2×#14556515C2211yescharacteristic
24C2×#14556515C2211yescharacteristic
25C2×#14556514C2211yescharacteristic
26C2×#14556514C2211yescharacteristic
27C2×#14556514C2211yescharacteristic
28C2×C22⋊#10596831C2211yescharacteristic
29C2×C22⋊#10596831C2211yescharacteristic
30C2×C4×#10596831C2211yescharacteristic
31#17988493C2211yescharacteristic
32#17988485C2211yescharacteristic
33#17988481—21no—
34#17988475—21no—
35#17988469—21no—
36C4×C4⋊Dic73C2211yescharacteristic
37#17988438—21no—
38#17988349C2211yescharacteristic
39#17988349C2211yescharacteristic
40#17988349C2211yescharacteristic
41#17988349C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988342C2211yescharacteristic
47#17988342C2211yescharacteristic
48C2×#125×C73C2211yescharacteristic
49C2×C22⋊Q8×C73C2211yescharacteristic
50#400×C73—21no—
51#397×C73—21no—
52#391×C73C2211yescharacteristic
53#391×C73C2211yescharacteristic
54#388×C73—21no—
55#382×C73—21no—
56#380×C73C2211yescharacteristic
57C4×C4⋊C4×C73C2211yescharacteristic
58C4×C22⋊C4×C73C2211yescharacteristic
59C22×Dic146C2311yescharacteristic
60C2×#11924018—22no—
61C2×#11924018—41no—
62C2×#11924018—41no—
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65C22×#10596831C2311yescharacteristic
66C22×#10596831C2311yescharacteristic
67#14556570—22no—
68#14556570—22no—
69C4×Dic146—22no—
70C4×Dic146—41no—
71#14556550—22no—
72#14556550—41no—
73C2×C4⋊Dic73C2311yescharacteristic
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×#11924007—21no—
77C2×#11924007—21no—
78C2×#11924007—21no—
79C2×C22⋊Dic73—21no—
80C2×C22⋊Dic73—21no—
81C2×C4×Dic73—21no—
82C2×C4×Dic73—21no—
83C2×C4×Dic73—21no—
84#14556524—22no—
85C4⋊#10596831—41no—
86#14556515—22no—
87#14556515—22no—
88#14556515—41no—
89#14556514—22no—
90#14556514—22no—
91#14556514—41no—
92C22⋊#10596831—41no—
93C22⋊#10596831—41no—
94C4×#10596831—41no—
95#14556498—21no—
96#14556498—21no—
97#14556498—21no—
98#14556498—21no—
99C22×Q8×C73C2311yescharacteristic
100C23×C4×C73C2311yescharacteristic
101#125×C73—22no—
102#125×C73—41no—
103C22⋊Q8×C73—22no—
104C22⋊Q8×C73—41no—
105C2×C4⋊C4×C73—21no—
106C2×C4⋊C4×C73—21no—
107C2×C4⋊C4×C73—21no—
108C2×C4⋊C4×C73C2311yescharacteristic
109C2×C4⋊C4×C73C2311yescharacteristic
110C2×C4⋊C4×C73C2311yescharacteristic
111C2×C22⋊C4×C73—21no—
112C2×C22⋊C4×C73—21no—
113C2×C22⋊C4×C73D411yescharacteristic
114C2×C22⋊C4×C73D411yescharacteristic
115C2×C42×C73—21no—
116C2×C42×C73C2311yescharacteristic
117C2×C42×C73C2311yescharacteristic
118#95×C73—21no—
119#95×C73—21no—
120#95×C73—21no—
121#95×C73—21no—
122#95×C73—21no—
123#95×C73—21no—
124#95×C73D411yescharacteristic
125#95×C73D411yescharacteristic
126C2×Dic146—21no—
127C2×Dic146—21no—
128C2×Dic146—21no—
129C2×Dic146—21no—
130C2×Dic146—21no—
131C2×Dic146#4112yes—
132C22×Dic73—21no—
133C22×Dic73—21no—
134C22×Dic73—21no—
135C22×Dic73—21no—
136#11924018—44no—
137C2×#10596831—21no—
138C2×#10596831—21no—
139C2×#10596831—21no—
140C2×#10596831—21no—
141C2×#10596831—21no—
142C2×#10596831—21no—
143C2×#10596831—21no—
144C2×#10596831—21no—
145C2×#10596831—21no—
146C2×#10596831—21no—
147C2×#10596831—21no—
148C2×#10596831—21no—
149C4⋊Dic73—21no—
150C4⋊Dic73—21no—
151C4⋊Dic73—21no—
152C4⋊Dic73—21no—
153C4⋊Dic73#4112yes—
154C4⋊Dic73#4112yes—
155#11924007—24no—
156#11924007—42no—
157#11924007—42no—
158C22⋊Dic73—24no—
159C22⋊Dic73—42no—
160C4×Dic73—42no—
161C4×Dic73—42no—
162C4×Dic73—42no—
163C24×C73C2×D411yescharacteristic
164C2×Q8×C73—21no—
165C2×Q8×C73—21no—
166C2×Q8×C73—21no—
167C2×Q8×C73—21no—
168C2×Q8×C73—21no—
169C2×Q8×C73—21no—
170C22×C4×C73—21no—
171C22×C4×C73—21no—
172C22×C4×C73—21no—
173C22×C4×C73—21no—
174C22×C4×C73—21no—
175C22×C4×C73—21no—
176C22×C4×C73—21no—
177C22×C4×C73—21no—
178C22×C4×C73—21no—
179C22×C4×C73—21no—
180C22×C4×C73C2×D411yescharacteristic
181C22×C4×C73C2×D411yescharacteristic
182C22×C4×C73C2×D411yescharacteristic
183C22×C4×C73C2×D411yescharacteristic
184C22×C4×C73C2×D411yescharacteristic
185C22×C4×C73C2411yescharacteristic
186C4⋊C4×C73—21no—
187C4⋊C4×C73—21no—
188C4⋊C4×C73—21no—
189C4⋊C4×C73—21no—
190C4⋊C4×C73—21no—
191C4⋊C4×C73—42no—
192C4⋊C4×C73—42no—
193C4⋊C4×C73—42no—
194C4⋊C4×C73#4112yes—
195C22⋊C4×C73—22no—
196C22⋊C4×C73—41no—
197C22⋊C4×C73—42no—
198C22⋊C4×C73—42no—
199C42×C73—21no—
200C42×C73—21no—
201C42×C73—21no—
202C42×C73—21no—
203C42×C73—42no—
204C23×C73—21no—
205C23×C73—21no—
206C23×C73—21no—
207C23×C73—21no—
208C23×C73—21no—
209C23×C73—21no—
210C23×C73—21no—
211C23×C73C22×D411yescharacteristic
212Q8×C73—41no—
213Q8×C73—41no—
214Q8×C73—41no—
215Q8×C73—41no—
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—22no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—21no—
228C2×C4×C73—21no—
229C2×C4×C73—21no—
230C2×C4×C73—21no—
231C2×C4×C73—41no—
232C2×C4×C73—41no—
233C2×C4×C73—41no—
234C2×C4×C73—41no—
235C2×C4×C73—41no—
236C2×C4×C73—41no—
237C2×C4×C73—41no—
238C2×C4×C73—41no—
239C2×C4×C73—41no—
240C2×C4×C73—41no—
241C2×C4×C73—41no—
242C2×C4×C73—41no—
243C2×C4×C73—41no—
244C2×C4×C73—41no—
245C2×C4×C73—41no—
246C2×C4×C73#13512yes—
247C2×C4×C73C2×#4111yescharacteristic
248C2×C4×C73C2×#4111yescharacteristic
249C2×C4×C73C2×#4111yescharacteristic
250C2×C4×C73C2×#4111yescharacteristic
251C2×C4×C73H(5,𝔽2)11yescharacteristic
252C2×C4×C73H(5,𝔽2)11yescharacteristic
253C2×Dic73—22no—
254C2×Dic73—22no—
255C2×Dic73—22no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260C2×Dic73—41no—
261C2×Dic73—41no—
262C2×Dic73—41no—
263C2×Dic73—41no—
264C2×Dic73—41no—
265Dic146—22no—
266Dic146—22no—
267Dic146—41no—
268Dic146—41no—
269#10596831—41no—
270#10596831—41no—
271#10596831—41no—
272#10596831—41no—
273C22×C73—41no—
274C22×C73—41no—
275C22×C73—41no—
276C22×C73—41no—
277C22×C73—41no—
278C22×C73—41no—
279C22×C73—41no—
280C22×C73#52511yescharacteristic
281C22×C73#54611yescharacteristic
282C22×C73#54611yescharacteristic
283C22×C73C2×#13511yescharacteristic
284C22×C73#57511yescharacteristic
285C22×C73#57811yescharacteristic
286C22×C73#57811yescharacteristic
287C4×C73—22no—
288C4×C73—41no—
289C4×C73—41no—
290C4×C73—42no—
291C4×C73—41no—
292C4×C73—41no—
293C4×C73—81no—
294C4×C73—81no—
295C4×C73—81no—
296C4×C73#52911yescharacteristic
297C4×C73#53811yescharacteristic
298C4×C73#55511yescharacteristic
299C4×C73#55911yescharacteristic
300Dic73—42no—
301Dic73—42no—
302Dic73—42no—
303Dic73—81no—
304#26732—731nomaximal, Sylow
305C2×C73—81no—
306C2×C73#245911yescharacteristic
307C2×C73#310211yescharacteristic
308C2×C73#312711yescharacteristic
309C2×C73#330011yescharacteristic
310C2×C73#330211yescharacteristic
311C2×C73#331511yescharacteristic
312C2×C73#332711yescharacteristic
313#2489D7311yescharacteristic
314#2362—731no—
315#2056—731no—
316#2036—731no—
317#2026—731no—
318#2001—731no—
319#1989—731no—
320#1933—731no—
321#1928—731no—
322#1923—731no—
323#1918—731no—
324#1878—731no—
325#1873—731no—
326#1825—731no—
327C4×#103—731no—
328C73#2673211yescharacteristic, complemented, Sylow
329C22×Q16—731no—
330C2×#125C2×D7311yescharacteristic
331C2×C22⋊Q8—731no—
332C2×C22⋊Q8C2×D7311yescharacteristic
333C2×C4×Q8—731no—
334C2×#117—731no—
335C2×#107—731no—
336C2×C4⋊C8—731no—
337C2×#103—731no—
338C2×#103—731no—
339C2×#103—731no—
340C2×#103—731no—
341C2×#103—731no—
342C2×#103—731no—
343C2×C22⋊C8—731no—
344C2×C22⋊C8—731no—
345C2×C4×C8—731no—
346#400—21no—
347#397—21no—
348#391—731no—
349#391C2×D7311yescharacteristic
350#391C2×D7311yescharacteristic
351#389—731no—
352#388—21no—
353#388—1461no—
354#388—1461no—
355#387—1461no—
356#382—21no—
357#380C2×D7311yescharacteristic
358C4×C4⋊C4—731no—
359C4×C4⋊C4—1461no—
360C4×C4⋊C4C2×D7311yescharacteristic
361C4×C22⋊C4C2×D7311yescharacteristic
362#340—731no—
363#340—731no—
364#340—731no—
365#340—731no—
366#340—731no—
367#340—731no—
368#340—731no—
369#340—731no—
370#336—731no—
371#336—731no—
372C22×Q8—731no—
373C22×Q8C22×D7311yescharacteristic
374C23×C4C22×D7311yescharacteristic
375C2×Q16—2921no—
376C2×Q16—2921no—
377C2×Q16—1462no—
378C22×C8—731no—
379C22×C8—731no—
380C22×C8—731no—
381C22×C8—731no—
382#125—22no—
383#125—41no—
384C22⋊Q8—22no—
385C22⋊Q8—41no—
386C22⋊Q8—1462no—
387C22⋊Q8—1462no—
388C4×Q8—2921no—
389C4×Q8—1462no—
390#117—2921no—
391#117—1462no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—731no—
396C2×C4⋊C4—731no—
397C2×C4⋊C4—731no—
398C2×C4⋊C4—1461no—
399C2×C4⋊C4—1461no—
400C2×C4⋊C4—1461no—
401C2×C4⋊C4C22×D7311yescharacteristic
402C2×C4⋊C4C22×D7311yescharacteristic
403C2×C4⋊C4C22×D7311yescharacteristic
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—1461no—
407C2×C22⋊C4—1461no—
408C2×C22⋊C4C22⋊D7311yescharacteristic
409C2×C22⋊C4C22⋊D7311yescharacteristic
410C2×C42—21no—
411C2×C42—1461no—
412C2×C42—1461no—
413C2×C42—1461no—
414C2×C42C22×D7311yescharacteristic
415C2×C42C22×D7311yescharacteristic
416#107—1462no—
417C4⋊C8—2921no—
418#103—2921no—
419#103—2921no—
420#103—1462no—
421#103—1462no—
422#103—1462no—
423#103—1462no—
424C22⋊C8—2921no—
425C22⋊C8—2921no—
426C4×C8—2921no—
427#95—21no—
428#95—21no—
429#95—21no—
430#95—21no—
431#95—21no—
432#95—21no—
433#95—1461no—
434#95—1461no—
435#95—1461no—
436#95—1461no—
437#95C22⋊D7311yescharacteristic
438#95C22⋊D7311yescharacteristic
439C24C2×C22⋊D7311yescharacteristic
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—732no—
447C2×Q8—1461no—
448C2×Q8—1461no—
449C2×Q8—1461no—
450C2×Q8—1461no—
451C2×Q8—1461no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—1461no—
463C22×C4—1461no—
464C22×C4—1461no—
465C22×C4—1461no—
466C22×C4C2×C22⋊D7311yescharacteristic
467C22×C4C2×C22⋊D7311yescharacteristic
468C22×C4C2×C22⋊D7311yescharacteristic
469C22×C4C2×C22⋊D7311yescharacteristic
470C22×C4C2×C22⋊D7311yescharacteristic
471C22×C4C23×D7311yescharacteristic
472Q16—2924no—
473C2×C8—1461no—
474C2×C8—1461no—
475C2×C8—1461no—
476C2×C8—1461no—
477C2×C8—1461no—
478C2×C8—1461no—
479C2×C8—1461no—
480C2×C8—1461no—
481C2×C8—1461no—
482C2×C8—1461no—
483C2×C8—1461no—
484C2×C8—1461no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—2922no—
494C4⋊C4—2922no—
495C4⋊C4—732no—
496C4⋊C4—732no—
497C4⋊C4—1464no—
498C4⋊C4—1461no—
499C4⋊C4—1461no—
500C4⋊C4—1461no—
501C4⋊C4—1461no—
502C4⋊C4#1192402712yes—
503C22⋊C4—22no—
504C22⋊C4—41no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—2922no—
508C22⋊C4—1464no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—42no—
514C42—2922no—
515C42—2922no—
516C42—2922no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23C22×C22⋊D7311yescharacteristic
525Q8—41no—
526Q8—41no—
527Q8—41no—
528Q8—41no—
529Q8—2921no—
530Q8—2921no—
531Q8—1462no—
532Q8—1462no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—2921no—
564C2×C4—2921no—
565C2×C4—2921no—
566C2×C4—2921no—
567C2×C4—2921no—
568C2×C4—2921no—
569C2×C4—2921no—
570C2×C4—2921no—
571C2×C4—2921no—
572C2×C4—1462no—
573C2×C4—1462no—
574C2×C4—1462no—
575C2×C4#1455661112yes—
576C2×C4#41×D7311yescharacteristic
577C2×C4#41×D7311yescharacteristic
578C2×C4#41×D7311yescharacteristic
579C2×C4C2×#1192402711yescharacteristic
580C2×C4#1455663611yescharacteristic
581C2×C4#1455663711yescharacteristic
582C8—2921no—
583C8—2921no—
584C8—2921no—
585C8—2921no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22#1798909411yescharacteristic
594C22#1798922511yescharacteristic
595C22#1798923111yescharacteristic
596C22C2×#1455661111yescharacteristic
597C22#1798938511yescharacteristic
598C22#1798940211yescharacteristic
599C22#1798940211yescharacteristic
600C4—22no—
601C4—41no—
602C4—41no—
603C4—41no—
604C4—42no—
605C4—41no—
606C4—81no—
607C4—81no—
608C4—81no—
609C4—2922no—
610C4—2922no—
611C4—2922no—
612C4—5841no—
613C4#1798911611yescharacteristic
614C4#1798917111yescharacteristic
615C4#1798929911yescharacteristic
616C4#1798931911yescharacteristic
617C2—81no—
618C2#2261386411yescharacteristic
619C2#2261994311yescharacteristic
620C2#2262022611yescharacteristic
621C2#2262181211yescharacteristic
622C2#2262184811yescharacteristic
623C2#2262202111yescharacteristic
624C2#2262219911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2489C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
hypercenter#2489D73
hypocenterC73#26732
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2489

Complements

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

NH ≅ G/Ntype
2. #22611253C2semidirect
4. #22611079C2semidirect
6. #22610889C2semidirect
7. #22610610C2semidirect
8. #22610569C2semidirect
10. #22610502C2semidirect
11. #22610275C2semidirect
14. C4×#14556514C2semidirect
328. C73#26732semidirect

Maximal subgroups

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

subgroupindex
#226129562
#226112532
#226111342
#226110792
#226109392
#226108892
#226106102
#226105692
#226105392
#226105022
#226102752
#226102442
#226100232
C4×#145565142
C73×#24892
#2673273