Finite Group Databasebeta

← #33089062

Subgroups — #33089062

Order 18688.

890 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988689C211yesmaximal, characteristic
2#22612583C211yesmaximal, characteristic, complemented
3#22612293C211yesmaximal, characteristic
4#22610573C211yesmaximal, characteristic, complemented
5#22610566C211yesmaximal, characteristic, complemented
6#22610524C211yesmaximal, characteristic, complemented
7#22610503C211yesmaximal, characteristic, complemented
8#22610264C211yesmaximal, characteristic
9#22610264C211yesmaximal, characteristic
10#22610195C211yesmaximal, characteristic
11#22610181C211yesmaximal, characteristic
12#22609828C211yesmaximal, characteristic, complemented
13C4×#14556516C211yesmaximal, characteristic, complemented
14C4×#14556513C211yesmaximal, characteristic
15C73×#1746C211yesmaximal, characteristic, complemented
16C22×#11924017C2211yescharacteristic
17C2×C4×Dic146C2211yescharacteristic
18C2×C4×D292C2211yescharacteristic
19C2×C4×D292C2211yescharacteristic
20C2×C4×D292C2211yescharacteristic
21#17988689C418yescomplemented
22C2×#14556525C2211yescharacteristic
23C2×#14556525C2211yescharacteristic
24C2×#14556523C2211yescharacteristic
25C2×#14556516C2211yescharacteristic
26C2×#14556516C2211yescharacteristic
27C2×#14556516C2211yescharacteristic
28C2×#14556513C2211yescharacteristic
29C2×#14556513C2211yescharacteristic
30C2×#14556513C2211yescharacteristic
31#17988493C2211yescharacteristic
32#17988493C2211yescharacteristic
33#17988485C2211yescharacteristic
34#17988485C2211yescharacteristic
35#17988469—21no—
36#17988469—21no—
37#17988454—21no—
38#17988451—21no—
39C4×C4⋊Dic73C2211yescharacteristic
40C4×C4⋊Dic73C2211yescharacteristic
41C4×C4⋊Dic73C2211yescharacteristic
42C4×C4⋊Dic73C2211yescharacteristic
43#17988434—21no—
44#17988434—21no—
45#17988434—21no—
46#95×D73—21no—
47#17988350C2211yescharacteristic
48#17988350C2211yescharacteristic
49#17988350C2211yescharacteristic
50#17988350C2211yescharacteristic
51#17988350C2211yescharacteristic
52#17988350C2211yescharacteristic
53C2×#97×C73C2211yescharacteristic
54C2×C4×C8×C73C2211yescharacteristic
55#376×C73C2211yescharacteristic
56#336×C73C2211yescharacteristic
57#336×C73C2211yescharacteristic
58#336×C73C2211yescharacteristic
59#336×C73C2211yescharacteristic
60C22×Dic146C2311yescharacteristic
61C22×D292C2311yescharacteristic
62C22×C4×D73—21no—
63C22×C4×D73—21no—
64C22×C4×D73—21no—
65C2×#11924017—22no—
66C2×#11924017—22no—
67C2×#11924017C2×C414yes—
68C4×Dic146—22no—
69C4×Dic146C2×C414yescomplemented
70C4×D292—22no—
71C4×D292—22no—
72C4×D292—22no—
73C4×D292—22no—
74C4×D292—22no—
75C4×D292C2×C414yes—
76C2×C4⋊Dic73C2311yescharacteristic
77C2×C4⋊Dic73C2311yescharacteristic
78C2×C4⋊Dic73C2311yescharacteristic
79C2×C4⋊Dic73C2311yescharacteristic
80C2×C4⋊Dic73C2311yescharacteristic
81C2×C4⋊Dic73C2311yescharacteristic
82C2×#11924007—21no—
83C2×#11924007—21no—
84C2×#11924007—21no—
85C2×#11924005—21no—
86C2×#11924005—21no—
87C2×#11924005—21no—
88C2×C4×Dic73—21no—
89C2×C4×Dic73—21no—
90C2×C4×Dic73—21no—
91C2×C4×Dic73—21no—
92C2×C4×Dic73—21no—
93#14556525—22no—
94#14556525C2×C414yescomplemented
95#14556523—22no—
96#14556516—22no—
97#14556516—22no—
98#14556516C2×C414yescomplemented
99#14556513—22no—
100#14556513—22no—
101#14556513C2×C414yes—
102#14556498—21no—
103#14556498—21no—
104#14556498—21no—
105#14556498—21no—
106#14556498—21no—
107#14556498—21no—
108#14556498—21no—
109#14556498—21no—
110#14556498—21no—
111#14556498—21no—
112C22×C8×C73C2311yescharacteristic
113C22×C8×C73C2311yescharacteristic
114C22×C8×C73C2311yescharacteristic
115C22×C8×C73C2311yescharacteristic
116C2×C42×C73C2311yescharacteristic
117C2×C42×C73C2311yescharacteristic
118C2×C42×C73C2311yescharacteristic
119#97×C73C2×C414yescomplemented
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×D292C22×C412yes—
136C2×D292#4112yes—
137C2×D292#4112yes—
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—24no—
147C2×C4×D73—24no—
148C2×C4×D73—81no—
149C2×C4×D73—81no—
150C2×C4×D73—81no—
151C2×C4×D73—81no—
152C2×C4×D73—81no—
153C2×C4×D73—81no—
154#11924017—28no—
155C4⋊Dic73—21no—
156C4⋊Dic73—21no—
157C4⋊Dic73—21no—
158C4⋊Dic73—21no—
159C4⋊Dic73—21no—
160C4⋊Dic73—21no—
161C4⋊Dic73—21no—
162C4⋊Dic73—21no—
163C4⋊Dic73C22×C412yes—
164C4⋊Dic73C22×C412yes—
165C4⋊Dic73#4112yes—
166C4⋊Dic73#4112yes—
167#11924007—24no—
168#11924007—81no—
169#11924007—81no—
170#11924005—24no—
171#11924005—24no—
172#11924005—24no—
173C4×Dic73—24no—
174C4×Dic73—81no—
175C4×Dic73—81no—
176C4×Dic73—81no—
177C4×Dic73—81no—
178C22×C4×C73C2×D411yescharacteristic
179C22×C4×C73C2×D411yescharacteristic
180C22×C4×C73C2×D411yescharacteristic
181C22×C4×C73C2×D411yescharacteristic
182C22×C4×C73C2×D411yescharacteristic
183C22×C4×C73C2×D411yescharacteristic
184C22×C4×C73C2411yescharacteristic
185C2×C8×C73—21no—
186C2×C8×C73—21no—
187C2×C8×C73—21no—
188C2×C8×C73—21no—
189C2×C8×C73—21no—
190C2×C8×C73—21no—
191C2×C8×C73—21no—
192C2×C8×C73—21no—
193C2×C8×C73—21no—
194C2×C8×C73—21no—
195C2×C8×C73C22×C412yes—
196C2×C8×C73C22×C412yes—
197C42×C73—21no—
198C42×C73—21no—
199C42×C73—21no—
200C42×C73—21no—
201C42×C73C22×C412yes—
202C42×C73#4112yes—
203C23×C73C22×D411yescharacteristic
204C22×D73—22no—
205C22×D73—22no—
206C22×D73—22no—
207C22×D73—41no—
208C22×D73—41no—
209C22×D73—41no—
210C22×D73—41no—
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—21no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73C4×D412yes—
228C2×C4×C73C4×D412yes—
229C2×C4×C73C23×C411yescharacteristic
230C2×C4×C73C2×#4111yescharacteristic
231C2×C4×C73C2×#4111yescharacteristic
232C2×C4×C73C2×#4111yescharacteristic
233C2×C4×C73C2×#4111yescharacteristic
234C2×C4×C73C2×#4111yescharacteristic
235C2×Dic73—22no—
236C2×Dic73—22no—
237C2×Dic73—22no—
238C2×Dic73—22no—
239C2×Dic73—22no—
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—
256D292—22no—
257D292—22no—
258D292—22no—
259D292#11714yes—
260C4×D73—84no—
261C4×D73—84no—
262C4×D73—84no—
263Dic146—22no—
264Dic146—22no—
265Dic146—41no—
266Dic146—41no—
267C8×C73—22no—
268C8×C73—22no—
269C8×C73—41no—
270C8×C73—41no—
271C22×C73C2×C4×D411yescharacteristic
272C22×C73#52511yescharacteristic
273C22×C73#52511yescharacteristic
274C22×C73#57611yescharacteristic
275C22×C73#57711yescharacteristic
276C22×C73#57711yescharacteristic
277C22×C73#57811yescharacteristic
278C2×D73—24no—
279C2×D73—42no—
280C2×D73—42no—
281C2×D73—42no—
282C2×D73—42no—
283C2×D73—42no—
284C2×D73—42no—
285C4×C73—22no—
286C4×C73—22no—
287C4×C73—41no—
288C4×C73—41no—
289C4×C73—41no—
290C4×C73—41no—
291C4×C73C2×#11711yescharacteristic
292C4×C73C4×#4111yescharacteristic
293C4×C73#52911yescharacteristic
294C4×C73#52911yescharacteristic
295Dic73—42no—
296Dic73—42no—
297Dic73—42no—
298Dic73—42no—
299Dic73—42no—
300Dic73—81no—
301Dic73—81no—
302#21125—731nomaximal, Sylow
303C2×C73#229011yescharacteristic
304C2×C73#294611yescharacteristic
305C2×C73#294711yescharacteristic
306C2×C73#311811yescharacteristic
307C2×C73#312011yescharacteristic
308C2×C73#312311yescharacteristic
309C2×C73#312411yescharacteristic
310D73—44no—
311C2×#440—731no—
312#2320—731no—
313#2275—731no—
314#1928—731no—
315#1927—731no—
316#1921—731no—
317#1918—731no—
318#1877—731no—
319#1877—731no—
320#1864—731no—
321#1862—731no—
322#1776—731no—
323C4×#103—731no—
324C4×#102—731no—
325#1746D7311yescharacteristic, complemented
326C73#2112511yescharacteristic, complemented, Sylow
327C22×SD16—731no—
328C2×C4×Q8—731no—
329C2×C4×D4—731no—
330C2×C4×D4—731no—
331C2×C4×D4—731no—
332#440—738no—
333C2×#107—731no—
334C2×#107—731no—
335C2×#106—731no—
336C2×#103—731no—
337C2×#103—731no—
338C2×#103—731no—
339C2×#102—731no—
340C2×#102—731no—
341C2×#102—731no—
342C2×#97C2×D7311yescharacteristic
343C2×C4×C8C2×D7311yescharacteristic
344#391—731no—
345#391—731no—
346#389—731no—
347#389—731no—
348#387—1461no—
349#387—1461no—
350#383—1461no—
351#382—1461no—
352C4×C4⋊C4—731no—
353C4×C4⋊C4—731no—
354C4×C4⋊C4—731no—
355C4×C4⋊C4—731no—
356C4×C22⋊C4—1461no—
357C4×C22⋊C4—1461no—
358C4×C22⋊C4—1461no—
359#376C2×D7311yescharacteristic
360C2×#95—1461no—
361#340—731no—
362#340—731no—
363#340—731no—
364#340—731no—
365#340—731no—
366#340—731no—
367#336C2×D7311yescharacteristic
368#336C2×D7311yescharacteristic
369#336C2×D7311yescharacteristic
370#336C2×D7311yescharacteristic
371C22×Q8—731no—
372C22×D4—731no—
373C23×C4—1461no—
374C23×C4—1461no—
375C23×C4—1461no—
376C2×SD16—734no—
377C2×SD16—1462no—
378C2×SD16—1462no—
379C22×C8C22×D7311yescharacteristic
380C22×C8C22×D7311yescharacteristic
381C22×C8C22×D7311yescharacteristic
382C22×C8C22×D7311yescharacteristic
383C4×Q8—734no—
384C4×Q8—1462no—
385C4×D4—734no—
386C4×D4—1462no—
387C4×D4—1462no—
388C4×D4—1462no—
389C4×D4—1462no—
390C4×D4—1462no—
391C2×C4⋊C4—731no—
392C2×C4⋊C4—731no—
393C2×C4⋊C4—731no—
394C2×C4⋊C4—731no—
395C2×C4⋊C4—731no—
396C2×C4⋊C4—731no—
397C2×C4⋊C4—1461no—
398C2×C4⋊C4—1461no—
399C2×C4⋊C4—1461no—
400C2×C22⋊C4—1461no—
401C2×C22⋊C4—1461no—
402C2×C22⋊C4—1461no—
403C2×C42—1461no—
404C2×C42—1461no—
405C2×C42—1461no—
406C2×C42—1461no—
407C2×C42—1461no—
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#97C4×D7314yescomplemented
421C4×C8—22no—
422#95—1461no—
423#95—1461no—
424#95—1461no—
425#95—1461no—
426#95—1461no—
427#95—1461no—
428#95—1461no—
429#95—1461no—
430#95—1461no—
431#95—1461no—
432#95D29211yescharacteristic
433#95D29211yescharacteristic
434#95D29211yescharacteristic
435#95D29211yescharacteristic
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—732no—
445C2×D4—732no—
446C2×D4—1461no—
447C2×D4—1461no—
448C2×D4—1461no—
449C22×C4—5841no—
450C22×C4—5841no—
451C22×C4—5841no—
452C22×C4—5841no—
453C22×C4—5841no—
454C22×C4—5841no—
455C22×C4—1464no—
456C22×C4—1464no—
457C22×C4—1461no—
458C22×C4—1461no—
459C22×C4—1461no—
460C22×C4—1461no—
461C22×C4—1461no—
462C22×C4—1461no—
463C22×C4—1461no—
464C22×C4—1464no—
465C22×C4C2×D29211yescharacteristic
466C22×C4C2×D29211yescharacteristic
467C22×C4C2×D29211yescharacteristic
468C22×C4C2×D29211yescharacteristic
469C22×C4C2×D29211yescharacteristic
470C22×C4C2×D29211yescharacteristic
471C22×C4C23×D7311yescharacteristic
472SD16—1468no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8C2×C4×D7312yes—
484C2×C8C2×C4×D7312yes—
485C4⋊C4—5841no—
486C4⋊C4—5841no—
487C4⋊C4—732no—
488C4⋊C4—732no—
489C4⋊C4—732no—
490C4⋊C4—732no—
491C4⋊C4—1461no—
492C4⋊C4—1461no—
493C4⋊C4—1461no—
494C4⋊C4—1461no—
495C4⋊C4—1464no—
496C4⋊C4—1461no—
497C4⋊C4—1461no—
498C4⋊C4—1461no—
499C4⋊C4—1461no—
500C22⋊C4—1464no—
501C22⋊C4—1464no—
502C22⋊C4—1464no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—5841no—
508C42—5841no—
509C42—5841no—
510C42—5841no—
511C42—1464no—
512C42C2×C4×D7312yes—
513C42#1192402812yes—
514C23—2921no—
515C23—2921no—
516C23—2921no—
517C23—2921no—
518C23—1462no—
519C23—1462no—
520C23—1462no—
521C23C22×D29211yescharacteristic
522Q8—2921no—
523Q8—2921no—
524Q8—1462no—
525Q8—1462no—
526D4—734no—
527D4—1462no—
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—21no—
545C2×C4—21no—
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—5844no—
563C2×C4—5844no—
564C2×C4—5844no—
565C2×C4—1462no—
566C2×C4—1462no—
567C2×C4—1462no—
568C2×C4—1462no—
569C2×C4—1462no—
570C2×C4C4×D29212yes—
571C2×C4C4×D29212yes—
572C2×C4C22×C4×D7311yescharacteristic
573C2×C4#41×D7311yescharacteristic
574C2×C4#41×D7311yescharacteristic
575C2×C4#41×D7311yescharacteristic
576C2×C4#41×D7311yescharacteristic
577C2×C4C2×#1192402811yescharacteristic
578C8—22no—
579C8—22no—
580C8—41no—
581C8—41no—
582C22—2922no—
583C22—2922no—
584C22—2922no—
585C22—2922no—
586C22—2922no—
587C22—2922no—
588C22—1464no—
589C22C2×C4×D29211yescharacteristic
590C22#1798909911yescharacteristic
591C22#1798909911yescharacteristic
592C22#1798939211yescharacteristic
593C22#1798939911yescharacteristic
594C22#1798939911yescharacteristic
595C22#1798940411yescharacteristic
596C4—22no—
597C4—22no—
598C4—41no—
599C4—41no—
600C4—41no—
601C4—41no—
602C4—2922no—
603C4—2922no—
604C4—2922no—
605C4—2922no—
606C4—2922no—
607C4—5841no—
608C4—5841no—
609C4#117×D7311yescharacteristic
610C4#1798905511yescharacteristic
611C4#1798911511yescharacteristic
612C4#1798911511yescharacteristic
613C2—2924no—
614C2#2261237511yescharacteristic
615C2#2261876511yescharacteristic
616C2#2261877611yescharacteristic
617C2#2262011711yescharacteristic
618C2#2262014711yescharacteristic
619C2#2262018411yescharacteristic
620C2#2262019511yescharacteristic
subgroupG/H
centerC23C22×D292
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1746C2
socleC23×C73C22×D4
prime radicalC22×D292C23
cosocleC22×C4×C73C24
hypercenter#1746D73
hypocenterC73#21125
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ #376 ◁ #1746

Complements

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

NH ≅ G/Ntype
2. #22612583C2semidirect
4. #22610573C2semidirect
5. #22610566C2semidirect
6. #22610524C2semidirect
7. #22610503C2semidirect
12. #22609828C2semidirect
13. C4×#14556516C2semidirect
15. C73×#1746C2semidirect
21. #17988689C4semidirect
69. C4×Dic146C2×C4semidirect
94. #14556525C2×C4semidirect
98. #14556516C2×C4semidirect
119. #97×C73C2×C4semidirect
325. #1746D73semidirect
326. C73#21125semidirect
420. #97C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179886892
#226125832
#226122932
#226105732
#226105662
#226105242
#226105032
#226102642
#226101952
#226101812
#226098282
C4×#145565162
C4×#145565132
C73×#17462
#2112573