Finite Group Databasebeta

← #33090192

Subgroups — #33090192

Order 18688.

898 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×#17988678C211yesmaximal, characteristic
2#22612402C211yesmaximal, characteristic, complemented
3#22612393C211yesmaximal, characteristic
4#22610786C211yesmaximal, characteristic
5#22610784C211yesmaximal, characteristic
6#22610658C211yesmaximal, characteristic, complemented
7#22610656C211yesmaximal, characteristic, complemented
8#22610616C211yesmaximal, characteristic
9#22610607C211yesmaximal, characteristic, complemented
10#22610605C211yesmaximal, characteristic, complemented
11#22609819C211yesmaximal, characteristic, complemented
12#22609750C211yesmaximal, characteristic, complemented
13#102×Dic73C211yesmaximal, characteristic
14C4×C8×Dic73C211yesmaximal, characteristic, complemented
15C73×#1934C211yesmaximal, characteristic
16C22×#11924016C2211yescharacteristic
17C2×#14556556C2211yescharacteristic
18C2×D4×Dic73C2211yescharacteristic
19C2×#14556547—21no—
20#17988678C418yescomplemented
21C2×#14556522C2211yescharacteristic
22C2×#14556522C2211yescharacteristic
23C2×#14556522C2211yescharacteristic
24C2×#14556516C2211yescharacteristic
25C2×#14556516C2211yescharacteristic
26C2×#14556514C2211yescharacteristic
27C2×#14556512C2211yescharacteristic
28C2×C4×#10596831C2211yescharacteristic
29C2×C8×Dic73C2211yescharacteristic
30C2×C8×Dic73C2211yescharacteristic
31C2×C8×Dic73C2211yescharacteristic
32C2×C8×Dic73C2211yescharacteristic
33#17988493C2211yescharacteristic
34#17988492C2211yescharacteristic
35#17988490C2211yescharacteristic
36#17988486C2211yescharacteristic
37#17988485C2211yescharacteristic
38#17988483C2211yescharacteristic
39#17988481—21no—
40#17988475—21no—
41#17988474—21no—
42#17988468—21no—
43C4⋊C4×Dic73—21no—
44C4⋊C4×Dic73C2211yescharacteristic
45C4⋊C4×Dic73C2211yescharacteristic
46C22⋊C4×Dic73—21no—
47#17988432—21no—
48C42×Dic73C2211yescharacteristic
49#17988350C2211yescharacteristic
50#17988350C2211yescharacteristic
51#17988349C2211yescharacteristic
52#17988349C2211yescharacteristic
53C2×#102×C73C2211yescharacteristic
54C2×#102×C73C2211yescharacteristic
55C2×C4×C8×C73C2211yescharacteristic
56#390×C73C2211yescharacteristic
57#389×C73C2211yescharacteristic
58#340×C73C2211yescharacteristic
59#340×C73C2211yescharacteristic
60C22×Dic146C2311yescharacteristic
61C23×Dic73—21no—
62C2×#11924016—22no—
63C2×#11924016—22no—
64C2×#11924016C2×C414yes—
65C22×#10596831C2311yescharacteristic
66C22×#10596831C2311yescharacteristic
67#14556556—22no—
68#14556556C2×C414yes—
69D4×Dic73—22no—
70D4×Dic73C2×C414yes—
71#14556547—44no—
72C2×C4⋊Dic73—21no—
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×C22⋊Dic73—21no—
82C2×C4×Dic73—21no—
83C2×C4×Dic73—21no—
84C2×C4×Dic73—21no—
85C2×C4×Dic73D411yescharacteristic
86C2×C4×Dic73D411yescharacteristic
87C2×C4×Dic73D411yescharacteristic
88C2×C4×Dic73D411yescharacteristic
89C2×C4×Dic73C2311yescharacteristic
90C2×C4×Dic73C2311yescharacteristic
91#14556522—22no—
92#14556522—22no—
93#14556522C2×C414yes—
94#14556516—22no—
95#14556516C2×C414yes—
96#14556514—22no—
97#14556512—22no—
98C4×#10596831—22no—
99C8×Dic73—22no—
100C8×Dic73—22no—
101C8×Dic73—22no—
102C8×Dic73C2×C414yes—
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×D4×C73C2311yescharacteristic
113C22×C8×C73C2311yescharacteristic
114C22×C8×C73C2311yescharacteristic
115C2×C4⋊C4×C73—21no—
116C2×C4⋊C4×C73C2311yescharacteristic
117C2×C4⋊C4×C73C2311yescharacteristic
118C2×C4⋊C4×C73C2311yescharacteristic
119C2×C22⋊C4×C73—21no—
120C2×C22⋊C4×C73—21no—
121C2×C42×C73C2311yescharacteristic
122#102×C73—22no—
123#102×C73C2×C414yes—
124C4×C8×C73—22no—
125#95×C73—21no—
126C2×Dic146—21no—
127C2×Dic146—21no—
128C2×Dic146—21no—
129C2×Dic146—21no—
130C2×Dic146—21no—
131C2×Dic146C22×C412yes—
132C22×Dic73—21no—
133C22×Dic73—21no—
134C22×Dic73—24no—
135C22×Dic73—21no—
136C22×Dic73—21no—
137C22×Dic73—42no—
138C22×Dic73—42no—
139C22×Dic73C2×D411yescharacteristic
140C22×Dic73C2×D411yescharacteristic
141C22×Dic73C2×D411yescharacteristic
142C22×Dic73C2×D411yescharacteristic
143#11924016—28no—
144C2×#10596831—21no—
145C2×#10596831—21no—
146C2×#10596831C22×C412yes—
147C2×#10596831#4112yes—
148C2×#10596831#4112yes—
149C2×#10596831#4112yes—
150C4⋊Dic73—21no—
151C4⋊Dic73—21no—
152C4⋊Dic73—21no—
153C4⋊Dic73—21no—
154C4⋊Dic73—21no—
155C4⋊Dic73—42no—
156C4⋊Dic73C22×C412yes—
157#11924007—24no—
158#11924007—42no—
159#11924007—42no—
160C22⋊Dic73—24no—
161C22⋊Dic73—42no—
162C22⋊Dic73—42no—
163C4×Dic73—22no—
164C4×Dic73—21no—
165C4×Dic73—21no—
166C4×Dic73—24no—
167C4×Dic73—22no—
168C4×Dic73—22no—
169C4×Dic73—42no—
170C4×Dic73—42no—
171C4×Dic73SD1614yescomplemented
172C4×Dic73C22×C412yes—
173C4×Dic73#4112yes—
174C24×C73—21no—
175C2×D4×C73—21no—
176C2×D4×C73—21no—
177C2×D4×C73—21no—
178C2×D4×C73—21no—
179C2×D4×C73—21no—
180C2×D4×C73C22×C412yes—
181C22×C4×C73—21no—
182C22×C4×C73—21no—
183C22×C4×C73—21no—
184C22×C4×C73C2×D411yescharacteristic
185C22×C4×C73C2×D411yescharacteristic
186C22×C4×C73C2411yescharacteristic
187C2×C8×C73—21no—
188C2×C8×C73—21no—
189C2×C8×C73—21no—
190C2×C8×C73—21no—
191C2×C8×C73C22×C412yes—
192C2×C8×C73#4112yes—
193C4⋊C4×C73—21no—
194C4⋊C4×C73—21no—
195C4⋊C4×C73—21no—
196C4⋊C4×C73—21no—
197C4⋊C4×C73—21no—
198C4⋊C4×C73—42no—
199C4⋊C4×C73C22×C412yes—
200C22⋊C4×C73—42no—
201C22⋊C4×C73—42no—
202C42×C73—21no—
203C42×C73—21no—
204C23×C73—22no—
205C23×C73—41no—
206C23×C73—41no—
207C23×C73—41no—
208C23×C73—41no—
209C23×C73—41no—
210C23×C73—41no—
211C23×C73C22×D411yescharacteristic
212D4×C73—22no—
213D4×C73—22no—
214D4×C73—41no—
215D4×C73—41no—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—22no—
221C2×C4×C73—41no—
222C2×C4×C73—41no—
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×C73C2×SD1612yes—
230C2×C4×C73#13512yes—
231C2×C4×C73C23×C411yescharacteristic
232C2×C4×C73C2×#4111yescharacteristic
233C2×C4×C73C2×#4111yescharacteristic
234C2×C4×C73C2×#4111yescharacteristic
235C2×C4×C73C2×#4111yescharacteristic
236C2×C4×C73C2×#4111yescharacteristic
237C2×Dic73—21no—
238C2×Dic73—21no—
239C2×Dic73—21no—
240C2×Dic73—21no—
241C2×Dic73—21no—
242C2×Dic73—21no—
243C2×Dic73—22no—
244C2×Dic73—22no—
245C2×Dic73—41no—
246C2×Dic73—41no—
247C2×Dic73—41no—
248C2×Dic73—41no—
249C2×Dic73—48no—
250C2×Dic73—41no—
251C2×Dic73—41no—
252C2×Dic73—41no—
253C2×Dic73—41no—
254C2×Dic73—41no—
255C2×Dic73—41no—
256C2×Dic73C4×D412yes—
257C2×Dic73C4×D412yes—
258C2×Dic73C2×SD1612yes—
259C2×Dic73C2×SD1612yes—
260C2×Dic73#13512yes—
261C2×Dic73#13512yes—
262Dic146—22no—
263Dic146—22no—
264Dic146—41no—
265Dic146—41no—
266C8×C73—22no—
267C8×C73—22no—
268#10596831—22no—
269#10596831#11714yescomplemented
270C22×C73—42no—
271C22×C73—42no—
272C22×C73—42no—
273C22×C73—81no—
274C22×C73—81no—
275C22×C73—81no—
276C22×C73—81no—
277C22×C73C2×C4×D411yescharacteristic
278C22×C73#53211yescharacteristic
279C22×C73#53211yescharacteristic
280C22×C73C22×SD1611yescharacteristic
281C22×C73C2×#13511yescharacteristic
282C22×C73C2×#13511yescharacteristic
283C22×C73C2×#13511yescharacteristic
284C4×C73—22no—
285C4×C73—22no—
286C4×C73—42no—
287C4×C73—81no—
288C4×C73—81no—
289C4×C73C2×#11711yescharacteristic
290C4×C73C4×#4111yescharacteristic
291C4×C73#52911yescharacteristic
292C4×C73#52911yescharacteristic
293Dic73—22no—
294Dic73—22no—
295Dic73—22no—
296Dic73—42no—
297Dic73—42no—
298Dic73—81no—
299Dic73—81no—
300Dic73C4×SD1614yes—
301#21274—731nomaximal, Sylow
302C2×C73—82no—
303C2×C73#229311yescharacteristic
304C2×C73C2×C4×SD1611yescharacteristic
305C2×C73C4×#13511yescharacteristic
306C2×C73#316711yescharacteristic
307C2×C73#316711yescharacteristic
308C2×C73#316811yescharacteristic
309C2×C73#316811yescharacteristic
310C2×C4×SD16—731no—
311#2295—731no—
312#2293—731no—
313#1968—731no—
314#1968—731no—
315#1941—731no—
316#1941—731no—
317#1934—731no—
318#1934D7311yescharacteristic
319#1933—731no—
320#1933—731no—
321C4×#106—731no—
322C4×#103—731no—
323C4×#102—731no—
324C42×C8—731no—
325C73#2127411yescharacteristic, complemented, Sylow
326C22×SD16—731no—
327C2×C4×Q8—731no—
328C2×C4×D4—731no—
329C2×#117—1461no—
330C4×SD16—738no—
331C2×#106—731no—
332C2×#106—731no—
333C2×#106—731no—
334C2×#103—731no—
335C2×#103—731no—
336C2×#103—731no—
337C2×#102—731no—
338C2×#102C2×D7311yescharacteristic
339C2×#102C2×D7311yescharacteristic
340C2×C4×C8—731no—
341C2×C4×C8—731no—
342C2×C4×C8—731no—
343C2×C4×C8—731no—
344C2×C4×C8—731no—
345C2×C4×C8C2×D7311yescharacteristic
346#391—731no—
347#391—731no—
348#390—731no—
349#390C2×D7311yescharacteristic
350#389—731no—
351#389—731no—
352#389—731no—
353#389C2×D7311yescharacteristic
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—
363C43—731no—
364#340—731no—
365#340—731no—
366#340—731no—
367#340—731no—
368#340C2×D7311yescharacteristic
369#340C2×D7311yescharacteristic
370C22×Q8—731no—
371C22×D4C22×D7311yescharacteristic
372C23×C4—1461no—
373C2×SD16—734no—
374C2×SD16—1462no—
375C2×SD16—1462no—
376C22×C8—731no—
377C22×C8—731no—
378C22×C8C22×D7311yescharacteristic
379C22×C8C22×D7311yescharacteristic
380C4×Q8—734no—
381C4×Q8—1462no—
382C4×D4—734no—
383C4×D4—1462no—
384#117—2924no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—731no—
387C2×C4⋊C4—731no—
388C2×C4⋊C4—731no—
389C2×C4⋊C4—1461no—
390C2×C4⋊C4—1461no—
391C2×C4⋊C4—1461no—
392C2×C4⋊C4—1461no—
393C2×C4⋊C4C22×D7311yescharacteristic
394C2×C4⋊C4C22×D7311yescharacteristic
395C2×C4⋊C4C22×D7311yescharacteristic
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—1461no—
399C2×C22⋊C4—1461no—
400C2×C22⋊C4—1461no—
401C2×C42—731no—
402C2×C42—731no—
403C2×C42—731no—
404C2×C42—731no—
405C2×C42—731no—
406C2×C42—731no—
407C2×C42—1461no—
408C2×C42—1461no—
409C2×C42—1461no—
410C2×C42C22×D7311yescharacteristic
411#106—734no—
412#106—1462no—
413#106—1462no—
414#103—734no—
415#103—1462no—
416#103—1462no—
417#102—22no—
418#102—1462no—
419#102C4×D7314yes—
420C4×C8—22no—
421C4×C8—734no—
422C4×C8—1462no—
423C4×C8—1462no—
424C4×C8—1462no—
425C4×C8—1462no—
426#95—21no—
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—21no—
437C2×Q8—732no—
438C2×Q8—1461no—
439C2×Q8—1461no—
440C2×Q8—1461no—
441C2×Q8—1461no—
442C2×Q8—1461no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4C2×C4×D7312yes—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—2922no—
453C22×C4—2922no—
454C22×C4—731no—
455C22×C4—731no—
456C22×C4—731no—
457C22×C4—731no—
458C22×C4—1461no—
459C22×C4—1461no—
460C22×C4—1464no—
461C22×C4—1461no—
462C22×C4—1461no—
463C22×C4D4×D7311yescharacteristic
464C22×C4D4×D7311yescharacteristic
465C22×C4C23×D7311yescharacteristic
466SD16—1468no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—732no—
472C2×C8—732no—
473C2×C8—732no—
474C2×C8—732no—
475C2×C8—1461no—
476C2×C8—1461no—
477C2×C8C2×C4×D7312yes—
478C2×C8#1192402712yes—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—42no—
485C4⋊C4—2922no—
486C4⋊C4—2922no—
487C4⋊C4—2922no—
488C4⋊C4—732no—
489C4⋊C4—1464no—
490C4⋊C4—1461no—
491C4⋊C4—1461no—
492C4⋊C4—1461no—
493C4⋊C4—1461no—
494C4⋊C4—1461no—
495C4⋊C4C2×C4×D7312yes—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—2922no—
499C22⋊C4—2922no—
500C22⋊C4—1464no—
501C42—21no—
502C42—21no—
503C42—2922no—
504C42—2922no—
505C42—734no—
506C42—732no—
507C42—732no—
508C42—1462no—
509C42—1462no—
510C42—1464no—
511C42—1462no—
512C42—1461no—
513C42—1461no—
514C23—22no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23C2×D4×D7311yescharacteristic
522Q8—2921no—
523Q8—2921no—
524Q8—1462no—
525Q8—1462no—
526D4—22no—
527D4—22no—
528D4—41no—
529D4—41no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—2921no—
544C2×C4—2921no—
545C2×C4—2921no—
546C2×C4—2921no—
547C2×C4—2928no—
548C2×C4—2921no—
549C2×C4—2921no—
550C2×C4—2921no—
551C2×C4—2921no—
552C2×C4—2921no—
553C2×C4—2921no—
554C2×C4—732no—
555C2×C4—732no—
556C2×C4—732no—
557C2×C4—732no—
558C2×C4—732no—
559C2×C4—732no—
560C2×C4—1462no—
561C2×C4—1462no—
562C2×C4—1461no—
563C2×C4—1461no—
564C2×C4—1461no—
565C2×C4—1461no—
566C2×C4—1461no—
567C2×C4—1461no—
568C2×C4SD16×D7312yes—
569C2×C4#1455661012yes—
570C2×C4C22×C4×D7311yescharacteristic
571C2×C4#41×D7311yescharacteristic
572C2×C4#41×D7311yescharacteristic
573C2×C4#41×D7311yescharacteristic
574C2×C4#41×D7311yescharacteristic
575C2×C4C2×#1192402711yescharacteristic
576C8—22no—
577C8—22no—
578C8—734no—
579C8—1462no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22C4×D4×D7311yescharacteristic
588C22#1798913211yescharacteristic
589C22#1798913611yescharacteristic
590C22C2×SD16×D7311yescharacteristic
591C22#135×D7311yescharacteristic
592C22#135×D7311yescharacteristic
593C22C2×#1455661011yescharacteristic
594C4—22no—
595C4—22no—
596C4—42no—
597C4—81no—
598C4—81no—
599C4—2922no—
600C4—2922no—
601C4—5841no—
602C4—5841no—
603C4—734no—
604C4—1462no—
605C4—1462no—
606C4—1462no—
607C4#117×D7311yescharacteristic
608C4C4×#1192402711yescharacteristic
609C4#1798911411yescharacteristic
610C4#1798912011yescharacteristic
611C2—82no—
612C2#2261239111yescharacteristic
613C2C4×SD16×D7311yescharacteristic
614C2C4×#1455661011yescharacteristic
615C2#2262048511yescharacteristic
616C2#2262048911yescharacteristic
617C2#2262049611yescharacteristic
618C2#2262050011yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1934C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
cosocleC22×C4×C73C24
hypercenter#1934D73
hypocenterC73#21274
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1934

Complements

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

NH ≅ G/Ntype
2. #22612402C2semidirect
6. #22610658C2semidirect
7. #22610656C2semidirect
9. #22610607C2semidirect
10. #22610605C2semidirect
11. #22609819C2semidirect
12. #22609750C2semidirect
14. C4×C8×Dic73C2semidirect
20. #17988678C4semidirect
171. C4×Dic73SD16semidirect
269. #10596831#117semidirect
325. C73#21274semidirect

Maximal subgroups

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

subgroupindex
C2×#179886782
#226124022
#226123932
#226107862
#226107842
#226106582
#226106562
#226106162
#226106072
#226106052
#226098192
#226097502
#102×Dic732
C4×C8×Dic732
C73×#19342
#2127473