Finite Group Databasebeta

← #33090360

Subgroups — #33090360

Order 18688.

814 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988689C211yesmaximal, characteristic
2#22612583C211yesmaximal, characteristic, complemented
3#22612575C211yesmaximal, characteristic
4#22610812C211yesmaximal, characteristic
5#22610785C211yesmaximal, characteristic
6#22610669C211yesmaximal, characteristic, complemented
7#22610664C211yesmaximal, characteristic, complemented
8#22610628C211yesmaximal, characteristic
9#22610621C211yesmaximal, characteristic, complemented
10#22610614C211yesmaximal, characteristic
11#22610607C211yesmaximal, characteristic, complemented
12#22609828C211yesmaximal, characteristic, complemented
13C4×#14556516C211yesmaximal, characteristic, complemented
14C4×#14556513C211yesmaximal, characteristic
15C73×#1726C211yesmaximal, characteristic, complemented
16C22×#11924017C2211yescharacteristic
17C2×C4×Dic146C2211yescharacteristic
18C2×C4×D292C2211yescharacteristic
19#17988689C418yescomplemented
20C2×#14556525C2211yescharacteristic
21C2×#14556525C2211yescharacteristic
22C2×#14556523C2211yescharacteristic
23C2×#14556516C2211yescharacteristic
24C2×#14556516C2211yescharacteristic
25C2×#14556516C2211yescharacteristic
26C2×#14556513C2211yescharacteristic
27C2×#14556513C2211yescharacteristic
28C2×#14556513C2211yescharacteristic
29#17988493C2211yescharacteristic
30#17988493C2211yescharacteristic
31#17988489C2211yescharacteristic
32#17988489C2211yescharacteristic
33#17988485C2211yescharacteristic
34#17988485C2211yescharacteristic
35#17988485C2211yescharacteristic
36#17988485C2211yescharacteristic
37#17988476—21no—
38#17988476—21no—
39#17988469—21no—
40#17988469—21no—
41#17988454—21no—
42#17988451—21no—
43#17988449—21no—
44#17988446—21no—
45C4×C4⋊Dic73C2211yescharacteristic
46C4×C4⋊Dic73C2211yescharacteristic
47#17988350C2211yescharacteristic
48#17988350C2211yescharacteristic
49#17988350C2211yescharacteristic
50#17988350C2211yescharacteristic
51#17988350C2211yescharacteristic
52#17988350C2211yescharacteristic
53C2×#97×C73C2211yescharacteristic
54C2×#97×C73C2211yescharacteristic
55C2×#97×C73C2211yescharacteristic
56C2×C4×C8×C73C2211yescharacteristic
57C2×C4×C8×C73C2211yescharacteristic
58C2×C4×C8×C73C2211yescharacteristic
59#376×C73C2211yescharacteristic
60C22×Dic146C2311yescharacteristic
61C22×D292C2311yescharacteristic
62C22×C4×D73—21no—
63C2×#11924017—22no—
64C2×#11924017—22no—
65C2×#11924017C2×C414yes—
66C4×Dic146—22no—
67C4×Dic146C2×C414yes—
68C4×D292—22no—
69C4×D292C2×C414yes—
70C2×C4⋊Dic73C2311yescharacteristic
71C2×C4⋊Dic73C2311yescharacteristic
72C2×C4⋊Dic73C2311yescharacteristic
73C2×C4⋊Dic73C2311yescharacteristic
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×#11924007—21no—
77C2×#11924007—21no—
78C2×#11924007—21no—
79C2×#11924007—21no—
80C2×#11924007—21no—
81C2×#11924005—21no—
82C2×#11924005—21no—
83C2×#11924005—21no—
84C2×#11924005—21no—
85C2×#11924005—21no—
86C2×C4×Dic73—21no—
87C2×C4×Dic73—21no—
88C2×C4×Dic73—21no—
89#14556525—22no—
90#14556525C2×C414yes—
91#14556523—22no—
92#14556516—22no—
93#14556516—22no—
94#14556516C2×C414yes—
95#14556513—22no—
96#14556513—22no—
97#14556513C2×C414yes—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102#14556498—21no—
103#14556498—21no—
104#14556498—21no—
105#14556498—21no—
106#14556498—21no—
107#14556498—21no—
108C22×C8×C73C2311yescharacteristic
109C22×C8×C73C2311yescharacteristic
110C22×C8×C73C2311yescharacteristic
111C22×C8×C73C2311yescharacteristic
112C2×C42×C73C2311yescharacteristic
113C2×C42×C73C2311yescharacteristic
114C2×C42×C73C2311yescharacteristic
115#97×C73—22no—
116#97×C73—22no—
117#97×C73C2×C414yes—
118C4×C8×C73—22no—
119C4×C8×C73—22no—
120C4×C8×C73—22no—
121#95×C73D411yescharacteristic
122#95×C73D411yescharacteristic
123#95×C73D411yescharacteristic
124#95×C73D411yescharacteristic
125C23×D73—21no—
126C2×Dic146—21no—
127C2×Dic146—21no—
128C2×Dic146—21no—
129C2×Dic146—21no—
130C2×Dic146—21no—
131C2×Dic146C22×C412yes—
132C2×D292—21no—
133C2×D292—21no—
134C2×D292—21no—
135C2×D292—21no—
136C2×D292—21no—
137C2×D292C22×C412yes—
138C22×Dic73—21no—
139C22×Dic73—21no—
140C22×Dic73—21no—
141C22×Dic73—21no—
142C22×Dic73—21no—
143C22×Dic73—21no—
144C22×Dic73—21no—
145C2×C4×D73—24no—
146C2×C4×D73—81no—
147C2×C4×D73—81no—
148#11924017—28no—
149C4⋊Dic73—21no—
150C4⋊Dic73—21no—
151C4⋊Dic73—21no—
152C4⋊Dic73—21no—
153C4⋊Dic73—21no—
154C4⋊Dic73—21no—
155C4⋊Dic73—21no—
156C4⋊Dic73—21no—
157C4⋊Dic73—21no—
158C4⋊Dic73—21no—
159C4⋊Dic73C22×C412yes—
160C4⋊Dic73C22×C412yes—
161#11924007—24no—
162#11924007—81no—
163#11924007—81no—
164#11924007—81no—
165#11924007—81no—
166#11924005—24no—
167#11924005—81no—
168#11924005—81no—
169#11924005—81no—
170#11924005—81no—
171C4×Dic73—24no—
172C4×Dic73—81no—
173C4×Dic73—81no—
174C22×C4×C73C2×D411yescharacteristic
175C22×C4×C73C2×D411yescharacteristic
176C22×C4×C73C2×D411yescharacteristic
177C22×C4×C73C2×D411yescharacteristic
178C22×C4×C73C2×D411yescharacteristic
179C22×C4×C73C2×D411yescharacteristic
180C22×C4×C73C2411yescharacteristic
181C2×C8×C73—21no—
182C2×C8×C73—21no—
183C2×C8×C73—21no—
184C2×C8×C73—21no—
185C2×C8×C73—21no—
186C2×C8×C73—21no—
187C2×C8×C73C22×C412yes—
188C2×C8×C73C22×C412yes—
189C2×C8×C73#4112yes—
190C2×C8×C73#4112yes—
191C2×C8×C73#4112yes—
192C2×C8×C73#4112yes—
193C42×C73—21no—
194C42×C73—21no—
195C42×C73—21no—
196C42×C73—21no—
197C42×C73C22×C412yes—
198C42×C73#4112yes—
199C23×C73C22×D411yescharacteristic
200C22×D73—22no—
201C22×D73—41no—
202C22×D73—41no—
203C22×D73—41no—
204C22×D73—41no—
205C22×D73—41no—
206C22×D73—41no—
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×C73C4×D412yes—
224C2×C4×C73C4×D412yes—
225C2×C4×C73C23×C411yescharacteristic
226C2×C4×C73C2×#4111yescharacteristic
227C2×C4×C73C2×#4111yescharacteristic
228C2×C4×C73C2×#4111yescharacteristic
229C2×C4×C73C2×#4111yescharacteristic
230C2×C4×C73C2×#4111yescharacteristic
231C2×Dic73—22no—
232C2×Dic73—22no—
233C2×Dic73—22no—
234C2×Dic73—41no—
235C2×Dic73—41no—
236C2×Dic73—41no—
237C2×Dic73—41no—
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—
250C2×Dic73—41no—
251C2×Dic73—41no—
252D292—22no—
253D292—22no—
254D292—41no—
255D292—41no—
256C4×D73—84no—
257Dic146—22no—
258Dic146—22no—
259Dic146—41no—
260Dic146—41no—
261C8×C73—22no—
262C8×C73—22no—
263C8×C73—22no—
264C8×C73#11714yes—
265C22×C73C2×C4×D411yescharacteristic
266C22×C73#53211yescharacteristic
267C22×C73#53211yescharacteristic
268C22×C73#57611yescharacteristic
269C22×C73#57711yescharacteristic
270C22×C73#57711yescharacteristic
271C22×C73#57811yescharacteristic
272C2×D73—42no—
273C2×D73—42no—
274C2×D73—42no—
275C2×D73—81no—
276C2×D73—81no—
277C2×D73—81no—
278C2×D73—81no—
279C4×C73—22no—
280C4×C73—22no—
281C4×C73—41no—
282C4×C73—41no—
283C4×C73—41no—
284C4×C73—41no—
285C4×C73C2×#11711yescharacteristic
286C4×C73C4×#4111yescharacteristic
287C4×C73#52911yescharacteristic
288C4×C73#52911yescharacteristic
289Dic73—42no—
290Dic73—42no—
291Dic73—42no—
292Dic73—81no—
293Dic73—81no—
294Dic73—81no—
295Dic73—81no—
296#21291—731nomaximal, Sylow
297C2×C73#229311yescharacteristic
298C2×C73#294611yescharacteristic
299C2×C73#294711yescharacteristic
300C2×C73#317411yescharacteristic
301C2×C73#317511yescharacteristic
302C2×C73#317811yescharacteristic
303C2×C73#317911yescharacteristic
304D73—82no—
305C2×#440—731no—
306#2320—731no—
307#2319—731no—
308#1973—731no—
309#1968—731no—
310#1943—731no—
311#1942—731no—
312#1936—731no—
313#1935—731no—
314#1934—731no—
315#1933—731no—
316#1776—731no—
317C4×#103—731no—
318C4×#102—731no—
319#1726D7311yescharacteristic, complemented
320C73#2129111yescharacteristic, complemented, Sylow
321C22×SD16—731no—
322C2×C4×Q8—731no—
323C2×C4×D4—731no—
324#440—738no—
325C2×#107—731no—
326C2×#107—731no—
327C2×#106—731no—
328C2×#103—731no—
329C2×#103—731no—
330C2×#103—731no—
331C2×#102—731no—
332C2×#102—731no—
333C2×#102—731no—
334C2×#97C2×D7311yescharacteristic
335C2×#97C2×D7311yescharacteristic
336C2×#97C2×D7311yescharacteristic
337C2×C4×C8C2×D7311yescharacteristic
338C2×C4×C8C2×D7311yescharacteristic
339C2×C4×C8C2×D7311yescharacteristic
340#391—731no—
341#391—731no—
342#390—731no—
343#390—731no—
344#389—731no—
345#389—731no—
346#389—731no—
347#389—731no—
348#388—1461no—
349#388—1461no—
350#387—1461no—
351#387—1461no—
352#383—1461no—
353#382—1461no—
354#381—1461no—
355#380—1461no—
356C4×C4⋊C4—731no—
357C4×C4⋊C4—731no—
358#376C2×D7311yescharacteristic
359#340—731no—
360#340—731no—
361#340—731no—
362#340—731no—
363#340—731no—
364#340—731no—
365C22×Q8—731no—
366C22×D4—731no—
367C23×C4—1461no—
368C2×SD16—734no—
369C2×SD16—1462no—
370C2×SD16—1462no—
371C22×C8C22×D7311yescharacteristic
372C22×C8C22×D7311yescharacteristic
373C22×C8C22×D7311yescharacteristic
374C22×C8C22×D7311yescharacteristic
375C4×Q8—734no—
376C4×Q8—1462no—
377C4×D4—734no—
378C4×D4—1462no—
379C2×C4⋊C4—731no—
380C2×C4⋊C4—731no—
381C2×C4⋊C4—731no—
382C2×C4⋊C4—731no—
383C2×C4⋊C4—731no—
384C2×C4⋊C4—731no—
385C2×C4⋊C4—1461no—
386C2×C4⋊C4—1461no—
387C2×C4⋊C4—1461no—
388C2×C4⋊C4—1461no—
389C2×C4⋊C4—1461no—
390C2×C22⋊C4—1461no—
391C2×C22⋊C4—1461no—
392C2×C22⋊C4—1461no—
393C2×C22⋊C4—1461no—
394C2×C22⋊C4—1461no—
395C2×C42—1461no—
396C2×C42—1461no—
397C2×C42—1461no—
398C2×C42C22×D7311yescharacteristic
399C2×C42C22×D7311yescharacteristic
400C2×C42C22×D7311yescharacteristic
401#107—734no—
402#107—1462no—
403#106—1462no—
404#103—734no—
405#103—1462no—
406#103—1462no—
407#102—734no—
408#102—1462no—
409#102—1462no—
410#97—22no—
411#97—22no—
412#97C4×D7314yes—
413C4×C8—22no—
414C4×C8—22no—
415C4×C8—22no—
416#95—1461no—
417#95—1461no—
418#95—1461no—
419#95—1461no—
420#95—1461no—
421#95—1461no—
422#95—1461no—
423#95—1461no—
424#95—1461no—
425#95—1461no—
426#95D29211yescharacteristic
427#95D29211yescharacteristic
428#95D29211yescharacteristic
429#95D29211yescharacteristic
430C24—1461no—
431C2×Q8—732no—
432C2×Q8—1461no—
433C2×Q8—1461no—
434C2×Q8—1461no—
435C2×Q8—1461no—
436C2×Q8—1461no—
437C2×D4—732no—
438C2×D4—1461no—
439C2×D4—1461no—
440C2×D4—1461no—
441C2×D4—1461no—
442C2×D4—1461no—
443C22×C4—5841no—
444C22×C4—5841no—
445C22×C4—1461no—
446C22×C4—1461no—
447C22×C4—1461no—
448C22×C4—1461no—
449C22×C4—1461no—
450C22×C4—1461no—
451C22×C4—1461no—
452C22×C4—1464no—
453C22×C4C2×D29211yescharacteristic
454C22×C4C2×D29211yescharacteristic
455C22×C4C2×D29211yescharacteristic
456C22×C4C2×D29211yescharacteristic
457C22×C4C2×D29211yescharacteristic
458C22×C4C2×D29211yescharacteristic
459C22×C4C23×D7311yescharacteristic
460SD16—1468no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8C2×C4×D7312yes—
468C2×C8C2×C4×D7312yes—
469C2×C8#1192402812yes—
470C2×C8#1192402812yes—
471C2×C8#1192402812yes—
472C2×C8#1192402812yes—
473C4⋊C4—5841no—
474C4⋊C4—5841no—
475C4⋊C4—5841no—
476C4⋊C4—5841no—
477C4⋊C4—732no—
478C4⋊C4—732no—
479C4⋊C4—1461no—
480C4⋊C4—1461no—
481C4⋊C4—1461no—
482C4⋊C4—1461no—
483C4⋊C4—1461no—
484C4⋊C4—1461no—
485C4⋊C4—1461no—
486C4⋊C4—1461no—
487C4⋊C4—1464no—
488C4⋊C4—1461no—
489C4⋊C4—1461no—
490C22⋊C4—5841no—
491C22⋊C4—5841no—
492C22⋊C4—5841no—
493C22⋊C4—5841no—
494C22⋊C4—1464no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—5841no—
500C42—5841no—
501C42—1464no—
502C42C2×C4×D7312yes—
503C42#1192402812yes—
504C23—2921no—
505C23—2921no—
506C23—2921no—
507C23—2921no—
508C23—2921no—
509C23—2921no—
510C23—1462no—
511C23C22×D29211yescharacteristic
512Q8—2921no—
513Q8—2921no—
514Q8—1462no—
515Q8—1462no—
516D4—2921no—
517D4—2921no—
518D4—1462no—
519D4—1462no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
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—2921no—
547C2×C4—2921no—
548C2×C4—2921no—
549C2×C4—2921no—
550C2×C4—2921no—
551C2×C4—2921no—
552C2×C4—2921no—
553C2×C4—2921no—
554C2×C4—5844no—
555C2×C4—1462no—
556C2×C4—1462no—
557C2×C4—1462no—
558C2×C4C4×D29212yes—
559C2×C4C4×D29212yes—
560C2×C4C22×C4×D7311yescharacteristic
561C2×C4C2×#1192402811yescharacteristic
562C2×C4C2×#1192402811yescharacteristic
563C2×C4C2×#1192402811yescharacteristic
564C2×C4C2×#1192402811yescharacteristic
565C2×C4C2×#1192402811yescharacteristic
566C8—22no—
567C8—22no—
568C8—22no—
569C8#1455654814yes—
570C22—2922no—
571C22—2922no—
572C22—2922no—
573C22—5841no—
574C22—5841no—
575C22—5841no—
576C22—5841no—
577C22C2×C4×D29211yescharacteristic
578C22#1798913711yescharacteristic
579C22#1798913711yescharacteristic
580C22#1798939211yescharacteristic
581C22#1798939911yescharacteristic
582C22#1798939911yescharacteristic
583C22#1798940411yescharacteristic
584C4—22no—
585C4—22no—
586C4—41no—
587C4—41no—
588C4—41no—
589C4—41no—
590C4—2922no—
591C4—2922no—
592C4—2922no—
593C4—5841no—
594C4—5841no—
595C4—5841no—
596C4—5841no—
597C4C2×#1455654811yescharacteristic
598C4C4×#1192402811yescharacteristic
599C4#1798912111yescharacteristic
600C4#1798912111yescharacteristic
601C2—5842no—
602C2#2261239511yescharacteristic
603C2#2261876511yescharacteristic
604C2#2261877611yescharacteristic
605C2#2262055911yescharacteristic
606C2#2262057011yescharacteristic
607C2#2262059911yescharacteristic
608C2#2262061411yescharacteristic
subgroupG/H
centerC23C22×D292
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1726C2
socleC23×C73C22×D4
prime radicalC22×D292C23
cosocleC22×C4×C73C24
hypercenter#1726D73
hypocenterC73#21291
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ #376 ◁ #1726

Complements

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

NH ≅ G/Ntype
2. #22612583C2semidirect
6. #22610669C2semidirect
7. #22610664C2semidirect
9. #22610621C2semidirect
11. #22610607C2semidirect
12. #22609828C2semidirect
13. C4×#14556516C2semidirect
15. C73×#1726C2semidirect
19. #17988689C4semidirect
319. #1726D73semidirect
320. C73#21291semidirect

Maximal subgroups

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

subgroupindex
C2×#179886892
#226125832
#226125752
#226108122
#226107852
#226106692
#226106642
#226106282
#226106212
#226106142
#226106072
#226098282
C4×#145565162
C4×#145565132
C73×#17262
#2129173