Finite Group Databasebeta

← #33090323

Subgroups — #33090323

Order 18688.

874 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988689C211yesmaximal, characteristic
2#22612408C211yesmaximal, characteristic, complemented
3#22612395C211yesmaximal, 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
13#22609762C211yesmaximal, characteristic, complemented
14#22609757C211yesmaximal, characteristic
15C4×#97×C73C211yesmaximal, characteristic, complemented
16C22×#11924017C2211yescharacteristic
17C2×C4×Dic146C2211yescharacteristic
18C2×C4×D292C2211yescharacteristic
19C2×#14556548—21no—
20#17988689C418yescomplemented
21C2×#14556525C2211yescharacteristic
22C2×#14556525C2211yescharacteristic
23C2×#14556523C2211yescharacteristic
24C2×#14556516C2211yescharacteristic
25C2×#14556516C2211yescharacteristic
26C2×#14556516C2211yescharacteristic
27C2×#14556513C2211yescharacteristic
28C2×#14556513C2211yescharacteristic
29C2×#14556513C2211yescharacteristic
30#17988493C2211yescharacteristic
31#17988493C2211yescharacteristic
32#17988489C2211yescharacteristic
33#17988489C2211yescharacteristic
34#17988485C2211yescharacteristic
35#17988485C2211yescharacteristic
36#17988485C2211yescharacteristic
37#17988485C2211yescharacteristic
38#17988478—21no—
39#17988478—21no—
40#17988471—21no—
41#17988471—21no—
42C4×#11924007—21no—
43C4×C4⋊Dic73C2211yescharacteristic
44C4×C4⋊Dic73C2211yescharacteristic
45C4×#11924005—21no—
46#17988433—21no—
47#17988350C2211yescharacteristic
48#17988350C2211yescharacteristic
49#17988350C2211yescharacteristic
50#17988350C2211yescharacteristic
51#17988350C2211yescharacteristic
52#17988350C2211yescharacteristic
53C2×#97×C73C2211yescharacteristic
54C2×#97×C73C2211yescharacteristic
55C2×#97×C73C2211yescharacteristic
56C2×#97×C73C2211yescharacteristic
57C2×C4×C8×C73C2211yescharacteristic
58C2×C4×C8×C73C2211yescharacteristic
59C43×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—
70#14556548—44no—
71C2×C4⋊Dic73C2311yescharacteristic
72C2×C4⋊Dic73C2311yescharacteristic
73C2×C4⋊Dic73C2311yescharacteristic
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2311yescharacteristic
77C2×#11924007—21no—
78C2×#11924007—21no—
79C2×#11924007—21no—
80C2×#11924007—21no—
81C2×#11924007—21no—
82C2×#11924005—21no—
83C2×#11924005—21no—
84C2×#11924005—21no—
85C2×#11924005—21no—
86C2×#11924005—21no—
87C2×C4×Dic73—21no—
88C2×C4×Dic73—21no—
89C2×C4×Dic73—21no—
90#14556525—22no—
91#14556525C2×C414yes—
92#14556523—22no—
93#14556516—22no—
94#14556516—22no—
95#14556516C2×C414yes—
96#14556513—22no—
97#14556513—22no—
98#14556513C2×C414yes—
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—
108#14556498—21no—
109C22×C8×C73C2311yescharacteristic
110C22×C8×C73C2311yescharacteristic
111C22×C8×C73C2311yescharacteristic
112C22×C8×C73C2311yescharacteristic
113C2×C42×C73D411yescharacteristic
114C2×C42×C73D411yescharacteristic
115C2×C42×C73D411yescharacteristic
116C2×C42×C73D411yescharacteristic
117C2×C42×C73C2311yescharacteristic
118C2×C42×C73C2311yescharacteristic
119C2×C42×C73C2311yescharacteristic
120#97×C73—22no—
121#97×C73—22no—
122#97×C73—22no—
123#97×C73C2×C414yes—
124C4×C8×C73—22no—
125C4×C8×C73—22no—
126C23×D73—21no—
127C2×Dic146—21no—
128C2×Dic146—21no—
129C2×Dic146—21no—
130C2×Dic146—21no—
131C2×Dic146—21no—
132C2×Dic146C22×C412yes—
133C2×D292—21no—
134C2×D292—21no—
135C2×D292—21no—
136C2×D292—21no—
137C2×D292—21no—
138C2×D292C22×C412yes—
139C22×Dic73—21no—
140C22×Dic73—21no—
141C22×Dic73—21no—
142C22×Dic73—21no—
143C22×Dic73—21no—
144C22×Dic73—21no—
145C22×Dic73—21no—
146C2×C4×D73—24no—
147C2×C4×D73—42no—
148C2×C4×D73—42no—
149#11924017—28no—
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⋊Dic73—21no—
160C4⋊Dic73C22×C412yes—
161C4⋊Dic73C22×C412yes—
162#11924007—24no—
163#11924007—42no—
164#11924007—42no—
165#11924007—42no—
166#11924007—42no—
167#11924005—24no—
168#11924005—42no—
169#11924005—42no—
170#11924005—42no—
171#11924005—42no—
172C4×Dic73—24no—
173C4×Dic73—42no—
174C4×Dic73—42no—
175C22×C4×C73C2×D411yescharacteristic
176C22×C4×C73C2×D411yescharacteristic
177C22×C4×C73C2×D411yescharacteristic
178C22×C4×C73C2×D411yescharacteristic
179C22×C4×C73C2×D411yescharacteristic
180C22×C4×C73C2×D411yescharacteristic
181C22×C4×C73C2411yescharacteristic
182C2×C8×C73—21no—
183C2×C8×C73—21no—
184C2×C8×C73—21no—
185C2×C8×C73—21no—
186C2×C8×C73—21no—
187C2×C8×C73—21no—
188C2×C8×C73C22×C412yes—
189C2×C8×C73C22×C412yes—
190C2×C8×C73#4112yes—
191C2×C8×C73#4112yes—
192C2×C8×C73#4112yes—
193C2×C8×C73#4112yes—
194C42×C73—22no—
195C42×C73—22no—
196C42×C73—21no—
197C42×C73—21no—
198C42×C73—21no—
199C42×C73—21no—
200C42×C73—41no—
201C42×C73—41no—
202C42×C73C22×C412yes—
203C42×C73#4112yes—
204C23×C73C22×D411yescharacteristic
205C22×D73—22no—
206C22×D73—41no—
207C22×D73—41no—
208C22×D73—41no—
209C22×D73—41no—
210C22×D73—41no—
211C22×D73—41no—
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—21no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73C4×D412yes—
227C2×C4×C73C4×D412yes—
228C2×C4×C73AGL(1,ℤ/8)12yes—
229C2×C4×C73#13712yes—
230C2×C4×C73C23×C411yescharacteristic
231C2×C4×C73C2×#4111yescharacteristic
232C2×C4×C73C2×#4111yescharacteristic
233C2×C4×C73C2×#4111yescharacteristic
234C2×C4×C73C2×#4111yescharacteristic
235C2×C4×C73C2×#4111yescharacteristic
236C2×Dic73—22no—
237C2×Dic73—22no—
238C2×Dic73—22no—
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—
252C2×Dic73—41no—
253C2×Dic73—41no—
254C2×Dic73—41no—
255C2×Dic73—41no—
256C2×Dic73—41no—
257D292—22no—
258D292—22no—
259D292—41no—
260D292—41no—
261C4×D73—48no—
262Dic146—22no—
263Dic146—22no—
264Dic146—41no—
265Dic146—41no—
266C8×C73—22no—
267C8×C73—22no—
268C8×C73—22no—
269C8×C73#11714yescomplemented
270C22×C73C2×C4×D411yescharacteristic
271C22×C73#53211yescharacteristic
272C22×C73#53211yescharacteristic
273C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
274C22×C73C2×#13711yescharacteristic
275C22×C73#57511yescharacteristic
276C22×C73#57511yescharacteristic
277C2×D73—42no—
278C2×D73—42no—
279C2×D73—42no—
280C2×D73—81no—
281C2×D73—81no—
282C2×D73—81no—
283C2×D73—81no—
284C4×C73—22no—
285C4×C73—22no—
286C4×C73—22no—
287C4×C73—22no—
288C4×C73—41no—
289C4×C73—41no—
290C4×C73C2×#11711yescharacteristic
291C4×C73C4×#4111yescharacteristic
292C4×C73#52911yescharacteristic
293C4×C73#52911yescharacteristic
294Dic73—42no—
295Dic73—42no—
296Dic73—42no—
297Dic73—81no—
298Dic73—81no—
299Dic73—81no—
300Dic73—81no—
301#21288—731nomaximal, Sylow
302C2×C73#229311yescharacteristic
303C2×C73C4×AGL(1,ℤ/8)11yescharacteristic
304C2×C73C4×#13711yescharacteristic
305C2×C73#318011yescharacteristic
306C2×C73#318111yescharacteristic
307C2×C73#318211yescharacteristic
308C2×C73#318311yescharacteristic
309D73—82no—
310C2×#440—731no—
311#2295—731no—
312#2293—731no—
313#1973—731no—
314#1968—731no—
315#1943—731no—
316#1942—731no—
317#1936—731no—
318#1935—731no—
319#1934—731no—
320#1933—731no—
321#1776—731no—
322#1763—731no—
323#1762—731no—
324C4×#97D7311yescharacteristic, complemented
325C73#2128811yescharacteristic, complemented, Sylow
326C22×SD16—731no—
327C2×C4×Q8—731no—
328C2×C4×D4—731no—
329C2×#117—1461no—
330#440—738no—
331C2×#107—731no—
332C2×#107—731no—
333C2×#106—731no—
334C2×#103—731no—
335C2×#103—731no—
336C2×#103—731no—
337C2×#102—731no—
338C2×#102—731no—
339C2×#102—731no—
340C2×#97C2×D7311yescharacteristic
341C2×#97C2×D7311yescharacteristic
342C2×#97C2×D7311yescharacteristic
343C2×#97C2×D7311yescharacteristic
344C2×C4×C8C2×D7311yescharacteristic
345C2×C4×C8C2×D7311yescharacteristic
346#391—731no—
347#391—731no—
348#390—731no—
349#390—731no—
350#389—731no—
351#389—731no—
352#389—731no—
353#389—731no—
354#388—1461no—
355#388—1461no—
356#387—1461no—
357#387—1461no—
358C4×C4⋊C4—731no—
359C4×C4⋊C4—731no—
360C4×C4⋊C4—1461no—
361C4×C22⋊C4—1461no—
362#376—1461no—
363C43C2×D7311yescharacteristic
364#340—731no—
365#340—731no—
366#340—731no—
367#340—731no—
368#340—731no—
369#340—731no—
370C22×Q8—731no—
371C22×D4—731no—
372C23×C4—1461no—
373C2×SD16—734no—
374C2×SD16—1462no—
375C2×SD16—1462no—
376C22×C8C22×D7311yescharacteristic
377C22×C8C22×D7311yescharacteristic
378C22×C8C22×D7311yescharacteristic
379C22×C8C22×D7311yescharacteristic
380C4×Q8—734no—
381C4×Q8—1462no—
382C4×D4—734no—
383C4×D4—1462no—
384#117—2924no—
385C2×C4⋊C4—731no—
386C2×C4⋊C4—731no—
387C2×C4⋊C4—731no—
388C2×C4⋊C4—731no—
389C2×C4⋊C4—731no—
390C2×C4⋊C4—731no—
391C2×C4⋊C4—1461no—
392C2×C4⋊C4—1461no—
393C2×C4⋊C4—1461no—
394C2×C4⋊C4—1461no—
395C2×C4⋊C4—1461no—
396C2×C22⋊C4—1461no—
397C2×C22⋊C4—1461no—
398C2×C22⋊C4—1461no—
399C2×C22⋊C4—1461no—
400C2×C22⋊C4—1461no—
401C2×C42—1461no—
402C2×C42—1461no—
403C2×C42—1461no—
404C2×C42D29211yescharacteristic
405C2×C42D29211yescharacteristic
406C2×C42D29211yescharacteristic
407C2×C42D29211yescharacteristic
408C2×C42C22×D7311yescharacteristic
409C2×C42C22×D7311yescharacteristic
410C2×C42C22×D7311yescharacteristic
411#107—734no—
412#107—1462no—
413#106—1462no—
414#103—734no—
415#103—1462no—
416#103—1462no—
417#102—734no—
418#102—1462no—
419#102—1462no—
420#97—22no—
421#97—22no—
422#97—22no—
423#97C4×D7314yes—
424C4×C8—22no—
425C4×C8—22no—
426#95—1461no—
427#95—1461no—
428#95—1461no—
429#95—1461no—
430#95—1461no—
431#95—1461no—
432#95—1461no—
433#95—1461no—
434#95—1461no—
435#95—1461no—
436C24—1461no—
437C2×Q8—732no—
438C2×Q8—1461no—
439C2×Q8—1461no—
440C2×Q8—1461no—
441C2×Q8—1461no—
442C2×Q8—1461no—
443C2×D4—732no—
444C2×D4—1461no—
445C2×D4—1461no—
446C2×D4—1461no—
447C2×D4—1461no—
448C2×D4—1461no—
449C22×C4—2922no—
450C22×C4—2922no—
451C22×C4—1461no—
452C22×C4—1461no—
453C22×C4—1461no—
454C22×C4—1461no—
455C22×C4—1461no—
456C22×C4—1461no—
457C22×C4—1461no—
458C22×C4—1464no—
459C22×C4C2×D29211yescharacteristic
460C22×C4C2×D29211yescharacteristic
461C22×C4C2×D29211yescharacteristic
462C22×C4C2×D29211yescharacteristic
463C22×C4C2×D29211yescharacteristic
464C22×C4C2×D29211yescharacteristic
465C22×C4C23×D7311yescharacteristic
466SD16—1468no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8C2×C4×D7312yes—
474C2×C8C2×C4×D7312yes—
475C2×C8#1192402812yes—
476C2×C8#1192402812yes—
477C2×C8#1192402812yes—
478C2×C8#1192402812yes—
479C4⋊C4—2922no—
480C4⋊C4—2922no—
481C4⋊C4—2922no—
482C4⋊C4—2922no—
483C4⋊C4—732no—
484C4⋊C4—732no—
485C4⋊C4—1461no—
486C4⋊C4—1461no—
487C4⋊C4—1461no—
488C4⋊C4—1461no—
489C4⋊C4—1461no—
490C4⋊C4—1461no—
491C4⋊C4—1461no—
492C4⋊C4—1461no—
493C4⋊C4—1464no—
494C4⋊C4—1461no—
495C4⋊C4—1461no—
496C22⋊C4—2922no—
497C22⋊C4—2922no—
498C22⋊C4—2922no—
499C22⋊C4—2922no—
500C22⋊C4—1464no—
501C42—22no—
502C42—22no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—41no—
508C42—41no—
509C42—2922no—
510C42—2922no—
511C42—1464no—
512C42C2×C4×D7312yes—
513C42#1192402812yes—
514C23—2921no—
515C23—2921no—
516C23—2921no—
517C23—2921no—
518C23—2921no—
519C23—2921no—
520C23—1462no—
521C23C22×D29211yescharacteristic
522Q8—2921no—
523Q8—2921no—
524Q8—1462no—
525Q8—1462no—
526D4—2921no—
527D4—2921no—
528D4—1462no—
529D4—1462no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
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—2928no—
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—2921no—
555C2×C4—2921no—
556C2×C4—2921no—
557C2×C4—2921no—
558C2×C4—2921no—
559C2×C4—2921no—
560C2×C4—2921no—
561C2×C4—2921no—
562C2×C4—2921no—
563C2×C4—1462no—
564C2×C4—1462no—
565C2×C4—1462no—
566C2×C4C4×D29212yes—
567C2×C4C4×D29212yes—
568C2×C4#1455661912yes—
569C2×C4#1455662412yes—
570C2×C4C22×C4×D7311yescharacteristic
571C2×C4C2×#1192402811yescharacteristic
572C2×C4C2×#1192402811yescharacteristic
573C2×C4C2×#1192402811yescharacteristic
574C2×C4C2×#1192402811yescharacteristic
575C2×C4C2×#1192402811yescharacteristic
576C8—22no—
577C8—22no—
578C8—22no—
579C8#1455654814yescomplemented
580C22—2922no—
581C22—2922no—
582C22—2922no—
583C22—5841no—
584C22—5841no—
585C22—5841no—
586C22—5841no—
587C22C2×C4×D29211yescharacteristic
588C22#1798913711yescharacteristic
589C22#1798913711yescharacteristic
590C22C2×#1455661911yescharacteristic
591C22C2×#1455662411yescharacteristic
592C22#1798938711yescharacteristic
593C22#1798938711yescharacteristic
594C4—22no—
595C4—22no—
596C4—22no—
597C4—22no—
598C4—41no—
599C4—41no—
600C4—2922no—
601C4—2922no—
602C4—2922no—
603C4—5841no—
604C4—5841no—
605C4—5841no—
606C4—5841no—
607C4C2×#1455654811yescharacteristic
608C4C4×#1192402811yescharacteristic
609C4#1798912111yescharacteristic
610C4#1798912111yescharacteristic
611C2—5842no—
612C2#2261239511yescharacteristic
613C2C4×#1455661911yescharacteristic
614C2C4×#1455662411yescharacteristic
615C2#2262062911yescharacteristic
616C2#2262064011yescharacteristic
617C2#2262064911yescharacteristic
618C2#2262066011yescharacteristic
subgroupG/H
centerC23C22×D292
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC4×#97×C73C2
socleC23×C73C22×D4
prime radicalC22×D292C23
cosocleC22×C4×C73C24
hypercenterC4×#97D73
hypocenterC73#21288
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C43 ◁ C4×#97

Complements

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

NH ≅ G/Ntype
2. #22612408C2semidirect
6. #22610669C2semidirect
7. #22610664C2semidirect
9. #22610621C2semidirect
11. #22610607C2semidirect
12. #22609828C2semidirect
13. #22609762C2semidirect
15. C4×#97×C73C2semidirect
20. #17988689C4semidirect
269. C8×C73#117semidirect
324. C4×#97D73semidirect
325. C73#21288semidirect
579. C8#14556548semidirect

Maximal subgroups

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

subgroupindex
C2×#179886892
#226124082
#226123952
#226108122
#226107852
#226106692
#226106642
#226106282
#226106212
#226106142
#226106072
#226098282
#226097622
#226097572
C4×#97×C732
#2128873