Finite Group Databasebeta

← #33093980

Subgroups — #33093980

Order 18688.

818 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988941C211yesmaximal, characteristic
2#22612672C211yesmaximal, characteristic, complemented
3#22610899C211yesmaximal, characteristic, complemented
4#22610748C211yesmaximal, characteristic
5#22610631C211yesmaximal, characteristic
6#22610624C211yesmaximal, characteristic, complemented
7#22610617C211yesmaximal, characteristic
8#22610610C211yesmaximal, characteristic, complemented
9#22610058C211yesmaximal, characteristic, complemented
10#22609761C211yesmaximal, characteristic, complemented
11#22609756C211yesmaximal, characteristic
12C4×#14556515C211yesmaximal, characteristic, complemented
13C4×#14556512C211yesmaximal, characteristic
14#22609618C211yesmaximal, characteristic, complemented
15C73×#2326C211yesmaximal, characteristic
16C2×#14556583C2211yescharacteristic
17#17988941C418yescomplemented
18C2×#14556515C2211yescharacteristic
19C2×#14556515C2211yescharacteristic
20C2×#14556515C2211yescharacteristic
21C2×#14556515C2211yescharacteristic
22C2×#14556512C2211yescharacteristic
23C2×#14556512C2211yescharacteristic
24C2×#14556512C2211yescharacteristic
25C2×#14556512C2211yescharacteristic
26C2×#14556502C2211yescharacteristic
27C2×#14556502C2211yescharacteristic
28C2×#14556502C2211yescharacteristic
29C2×C4×#10596831C2211yescharacteristic
30C2×C4×#10596831C2211yescharacteristic
31C2×C4×#10596831C2211yescharacteristic
32#17988485C2211yescharacteristic
33#17988485C2211yescharacteristic
34#17988484—21no—
35#17988484—21no—
36#17988474—21no—
37#17988474—21no—
38#17988457C2211yescharacteristic
39C4×C4⋊Dic73C2211yescharacteristic
40C4×C4⋊Dic73C2211yescharacteristic
41#17988349C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49C2×#124×C73C2211yescharacteristic
50C2×C4×Q8×C73C2211yescharacteristic
51C2×C4×D4×C73C2211yescharacteristic
52#391×C73C2211yescharacteristic
53#390×C73C2211yescharacteristic
54#388×C73—21no—
55#388×C73—21no—
56#385×C73—21no—
57#385×C73—21no—
58#384×C73C2211yescharacteristic
59#376×C73C2211yescharacteristic
60C22×#10596831C2311yescharacteristic
61C22×#10596831C2311yescharacteristic
62C22×#10596831C2311yescharacteristic
63C22×#10596831C2311yescharacteristic
64#14556583—22no—
65#14556583C2×C414yescomplemented
66C2×C4⋊Dic73—21no—
67C2×C4⋊Dic73—21no—
68C2×C4⋊Dic73C2311yescharacteristic
69C2×C4⋊Dic73C2311yescharacteristic
70C2×C4⋊Dic73C2311yescharacteristic
71C2×C4⋊Dic73C2311yescharacteristic
72C2×#11924007—21no—
73C2×#11924007—21no—
74C2×#11924007—21no—
75C2×#11924007—21no—
76C2×C4×Dic73—21no—
77C2×C4×Dic73—21no—
78#14556515—22no—
79#14556515—22no—
80#14556515C2×C414yescomplemented
81#14556515C2×C414yescomplemented
82#14556512—22no—
83#14556512—22no—
84#14556512C2×C414yes—
85#14556512C2×C414yes—
86#14556502—22no—
87#14556502—22no—
88#14556502—22no—
89C4×#10596831—22no—
90C4×#10596831—22no—
91C4×#10596831C2×C414yescomplemented
92#14556498—21no—
93#14556498—21no—
94#14556498—21no—
95#14556498—21no—
96C22×Q8×C73C2311yescharacteristic
97C22×D4×C73C2311yescharacteristic
98C23×C4×C73—21no—
99#124×C73—22no—
100#124×C73C2×C414yes—
101C4×Q8×C73—22no—
102C4×Q8×C73—22no—
103C4×D4×C73—22no—
104C4×D4×C73—22no—
105C2×C4⋊C4×C73—21no—
106C2×C4⋊C4×C73—21no—
107C2×C4⋊C4×C73—21no—
108C2×C4⋊C4×C73C2311yescharacteristic
109C2×C4⋊C4×C73C2311yescharacteristic
110C2×C22⋊C4×C73—21no—
111C2×C22⋊C4×C73—21no—
112C2×C22⋊C4×C73—21no—
113C2×C22⋊C4×C73—21no—
114C2×C22⋊C4×C73—21no—
115C2×C42×C73—21no—
116C2×C42×C73C2311yescharacteristic
117C2×C42×C73C2311yescharacteristic
118C2×C42×C73C2311yescharacteristic
119#95×C73—21no—
120#95×C73—21no—
121#95×C73D411yescharacteristic
122#95×C73D411yescharacteristic
123#95×C73D411yescharacteristic
124#95×C73D411yescharacteristic
125C22×Dic73—21no—
126C22×Dic73—21no—
127C22×Dic73—21no—
128C22×Dic73—21no—
129C2×#10596831—21no—
130C2×#10596831—21no—
131C2×#10596831—21no—
132C2×#10596831—21no—
133C2×#10596831—21no—
134C2×#10596831—21no—
135C2×#10596831—21no—
136C2×#10596831—21no—
137C2×#10596831C22×C412yes—
138C2×#10596831C22×C412yes—
139C2×#10596831#4112yes—
140C2×#10596831#4112yes—
141C4⋊Dic73—21no—
142C4⋊Dic73—21no—
143C4⋊Dic73—21no—
144C4⋊Dic73—21no—
145C4⋊Dic73—21no—
146C4⋊Dic73—21no—
147C4⋊Dic73—81no—
148C4⋊Dic73—81no—
149C4⋊Dic73C22×C412yes—
150C4⋊Dic73C22×C412yes—
151#11924007—24no—
152#11924007—24no—
153#11924007—81no—
154#11924007—81no—
155C4×Dic73—81no—
156C4×Dic73—81no—
157C24×C73—21no—
158C2×Q8×C73—21no—
159C2×Q8×C73—21no—
160C2×Q8×C73—21no—
161C2×Q8×C73—21no—
162C2×Q8×C73—21no—
163C2×Q8×C73C22×C412yes—
164C2×D4×C73—21no—
165C2×D4×C73—21no—
166C2×D4×C73—21no—
167C2×D4×C73—21no—
168C2×D4×C73—21no—
169C2×D4×C73C22×C412yes—
170C22×C4×C73—21no—
171C22×C4×C73—21no—
172C22×C4×C73—21no—
173C22×C4×C73—42no—
174C22×C4×C73—81no—
175C22×C4×C73—81no—
176C22×C4×C73C2×D411yescharacteristic
177C22×C4×C73C2×D411yescharacteristic
178C22×C4×C73C2×D411yescharacteristic
179C22×C4×C73C2×D411yescharacteristic
180C22×C4×C73C2×D411yescharacteristic
181C22×C4×C73C2×D411yescharacteristic
182C22×C4×C73C2411yescharacteristic
183C4⋊C4×C73—21no—
184C4⋊C4×C73—21no—
185C4⋊C4×C73—21no—
186C4⋊C4×C73—21no—
187C4⋊C4×C73—42no—
188C4⋊C4×C73—81no—
189C4⋊C4×C73—81no—
190C22⋊C4×C73—24no—
191C22⋊C4×C73—24no—
192C22⋊C4×C73—42no—
193C22⋊C4×C73—81no—
194C22⋊C4×C73—81no—
195C42×C73—21no—
196C42×C73—21no—
197C42×C73—42no—
198C42×C73C22×C412yes—
199C42×C73#4112yes—
200C42×C73#4112yes—
201C42×C73#4112yes—
202C23×C73—22no—
203C23×C73—41no—
204C23×C73—41no—
205C23×C73—41no—
206C23×C73—41no—
207C23×C73—41no—
208C23×C73—41no—
209C23×C73C22×D411yescharacteristic
210Q8×C73—22no—
211Q8×C73—22no—
212Q8×C73—41no—
213Q8×C73—41no—
214D4×C73—22no—
215D4×C73—22no—
216D4×C73—41no—
217D4×C73—41no—
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—22no—
227C2×C4×C73—21no—
228C2×C4×C73—21no—
229C2×C4×C73—21no—
230C2×C4×C73—21no—
231C2×C4×C73—21no—
232C2×C4×C73—21no—
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—84no—
242C2×C4×C73C4×D412yes—
243C2×C4×C73C4×D412yes—
244C2×C4×C73#12412yes—
245C2×C4×C73#12412yes—
246C2×C4×C73C23×C411yescharacteristic
247C2×C4×C73C2×#4111yescharacteristic
248C2×C4×C73C2×#4111yescharacteristic
249C2×C4×C73C2×#4111yescharacteristic
250C2×C4×C73C2×#4111yescharacteristic
251C2×C4×C73C2×#4111yescharacteristic
252C2×Dic73—22no—
253C2×Dic73—22no—
254C2×Dic73—41no—
255C2×Dic73—41no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260C2×Dic73—41no—
261C2×Dic73—41no—
262C2×Dic73—41no—
263C2×Dic73—41no—
264#10596831—22no—
265#10596831—22no—
266#10596831—22no—
267#10596831—22no—
268C22×C73—42no—
269C22×C73—42no—
270C22×C73—42no—
271C22×C73—81no—
272C22×C73—81no—
273C22×C73—81no—
274C22×C73—81no—
275C22×C73C2×C4×D411yescharacteristic
276C22×C73C2×#12411yescharacteristic
277C22×C73#53211yescharacteristic
278C22×C73#57611yescharacteristic
279C22×C73#57711yescharacteristic
280C22×C73#57711yescharacteristic
281C22×C73#57811yescharacteristic
282C4×C73—22no—
283C4×C73—22no—
284C4×C73—22no—
285C4×C73—22no—
286C4×C73—41no—
287C4×C73—41no—
288C4×C73—42no—
289C4×C73—81no—
290C4×C73—81no—
291C4×C73C4×#4111yescharacteristic
292C4×C73C4×#4111yescharacteristic
293C4×C73#52911yescharacteristic
294C4×C73#52911yescharacteristic
295Dic73—42no—
296Dic73—42no—
297Dic73—81no—
298Dic73—81no—
299#21643—731nomaximal, Sylow
300C2×C73—82no—
301C2×C73C4×#12411yescharacteristic
302C2×C73#295111yescharacteristic
303C2×C73#295211yescharacteristic
304C2×C73#313611yescharacteristic
305C2×C73#313711yescharacteristic
306C2×C73#317811yescharacteristic
307C2×C73#317911yescharacteristic
308C2×#488—731no—
309#2330—731no—
310#2326D7311yescharacteristic
311#1992—731no—
312#1959—731no—
313#1936—731no—
314#1935—731no—
315#1934—731no—
316#1933—731no—
317#1834—731no—
318#1763—731no—
319#1762—731no—
320C4×#103—731no—
321C4×#102—731no—
322#1726—731no—
323C73#2164311yescharacteristic, complemented, Sylow
324C2×#125—731no—
325C2×#124C2×D7311yescharacteristic
326C2×C4×Q8C2×D7311yescharacteristic
327C2×C4×D4C2×D7311yescharacteristic
328#488—738no—
329C2×#103—731no—
330C2×#103—731no—
331C2×#103—731no—
332C2×#103—731no—
333C2×#102—731no—
334C2×#102—731no—
335C2×#102—731no—
336C2×#102—731no—
337C2×#97—731no—
338C2×#97—731no—
339C2×#97—731no—
340C2×C4×C8—731no—
341C2×C4×C8—731no—
342C2×C4×C8—731no—
343#391C2×D7311yescharacteristic
344#390C2×D7311yescharacteristic
345#389—731no—
346#389—731no—
347#389—1461no—
348#389—1461no—
349#388—21no—
350#388—21no—
351#387—1461no—
352#387—1461no—
353#385—21no—
354#385—21no—
355#384—731no—
356#384C2×D7311yescharacteristic
357C4×C4⋊C4—731no—
358C4×C4⋊C4—731no—
359#376C2×D7311yescharacteristic
360#340—731no—
361#340—731no—
362#340—731no—
363#340—731no—
364#340—731no—
365#340—731no—
366#340—731no—
367#340—731no—
368C22×Q8C22×D7311yescharacteristic
369C22×D4C22×D7311yescharacteristic
370C23×C4—21no—
371C22×C8—731no—
372C22×C8—731no—
373C22×C8—731no—
374C22×C8—731no—
375#125—734no—
376#125—1462no—
377#124—22no—
378#124C4×D7314yes—
379C4×Q8—22no—
380C4×Q8—22no—
381C4×D4—22no—
382C4×D4—22no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—731no—
387C2×C4⋊C4—731no—
388C2×C4⋊C4—731no—
389C2×C4⋊C4—731no—
390C2×C4⋊C4—1461no—
391C2×C4⋊C4—1461no—
392C2×C4⋊C4—1461no—
393C2×C4⋊C4—1461no—
394C2×C4⋊C4—1461no—
395C2×C4⋊C4—1461no—
396C2×C4⋊C4C22×D7311yescharacteristic
397C2×C4⋊C4C22×D7311yescharacteristic
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—21no—
403C2×C42—21no—
404C2×C42—1461no—
405C2×C42—1461no—
406C2×C42C22×D7311yescharacteristic
407C2×C42C22×D7311yescharacteristic
408C2×C42C22×D7311yescharacteristic
409#103—734no—
410#103—734no—
411#103—1462no—
412#103—1462no—
413#102—734no—
414#102—734no—
415#102—1462no—
416#102—1462no—
417#97—1462no—
418#97—1462no—
419#97—1462no—
420C4×C8—734no—
421C4×C8—1462no—
422C4×C8—1462no—
423#95—21no—
424#95—21no—
425#95—1461no—
426#95—1461no—
427#95—1461no—
428#95—1461no—
429#95C22⋊D7311yescharacteristic
430#95C22⋊D7311yescharacteristic
431#95C22⋊D7311yescharacteristic
432#95C22⋊D7311yescharacteristic
433C24—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8C2×C4×D7312yes—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4C2×C4×D7312yes—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—42no—
450C22×C4—81no—
451C22×C4—81no—
452C22×C4—1461no—
453C22×C4—1461no—
454C22×C4—1461no—
455C22×C4—1461no—
456C22×C4C2×C22⋊D7311yescharacteristic
457C22×C4C2×C22⋊D7311yescharacteristic
458C22×C4C2×C22⋊D7311yescharacteristic
459C22×C4C2×C22⋊D7311yescharacteristic
460C22×C4C2×C22⋊D7311yescharacteristic
461C22×C4C2×C22⋊D7311yescharacteristic
462C22×C4C23×D7311yescharacteristic
463C2×C8—732no—
464C2×C8—732no—
465C2×C8—732no—
466C2×C8—732no—
467C2×C8—1461no—
468C2×C8—1461no—
469C2×C8—1461no—
470C2×C8—1461no—
471C2×C8—1461no—
472C2×C8—1461no—
473C2×C8—1461no—
474C2×C8—1461no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—42no—
480C4⋊C4—81no—
481C4⋊C4—81no—
482C4⋊C4—5841no—
483C4⋊C4—5841no—
484C4⋊C4—5841no—
485C4⋊C4—5841no—
486C4⋊C4—732no—
487C4⋊C4—732no—
488C4⋊C4—1461no—
489C4⋊C4—1461no—
490C4⋊C4—1461no—
491C4⋊C4—1461no—
492C4⋊C4—1464no—
493C4⋊C4—1464no—
494C4⋊C4—1461no—
495C4⋊C4—1461no—
496C22⋊C4—24no—
497C22⋊C4—24no—
498C22⋊C4—42no—
499C22⋊C4—81no—
500C22⋊C4—81no—
501C42—21no—
502C42—21no—
503C42—42no—
504C42—5841no—
505C42—5841no—
506C42C2×C4×D7312yes—
507C42#1192402712yes—
508C42#1192402712yes—
509C42#1192402812yes—
510C23—22no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23C22×C22⋊D7311yescharacteristic
518Q8—22no—
519Q8—22no—
520Q8—41no—
521Q8—41no—
522D4—22no—
523D4—22no—
524D4—41no—
525D4—41no—
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—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—84no—
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—1462no—
561C2×C4—1462no—
562C2×C4C4×C22⋊D7312yes—
563C2×C4C4×C22⋊D7312yes—
564C2×C4#1455657812yes—
565C2×C4#1455657812yes—
566C2×C4C22×C4×D7311yescharacteristic
567C2×C4#41×D7311yescharacteristic
568C2×C4#41×D7311yescharacteristic
569C2×C4C2×#1192402711yescharacteristic
570C2×C4C2×#1192402711yescharacteristic
571C2×C4C2×#1192402811yescharacteristic
572C8—1462no—
573C8—1462no—
574C8—1462no—
575C8—1462no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—81no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22C2×C4×C22⋊D7311yescharacteristic
584C22C2×#1455657811yescharacteristic
585C22#1798913311yescharacteristic
586C22#1798939011yescharacteristic
587C22#1798939511yescharacteristic
588C22#1798939811yescharacteristic
589C22#1798940211yescharacteristic
590C4—22no—
591C4—22no—
592C4—22no—
593C4—22no—
594C4—41no—
595C4—41no—
596C4—42no—
597C4—81no—
598C4—81no—
599C4—2922no—
600C4—2922no—
601C4—5841no—
602C4—5841no—
603C4C4×#1192402711yescharacteristic
604C4C4×#1192402711yescharacteristic
605C4#1798911911yescharacteristic
606C4#1798911911yescharacteristic
607C2—82no—
608C2C4×#1455657811yescharacteristic
609C2#2261880411yescharacteristic
610C2#2261881211yescharacteristic
611C2#2262028511yescharacteristic
612C2#2262029611yescharacteristic
613C2#2262059311yescharacteristic
614C2#2262060811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#2326C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
cosocleC22×C4×C73C24
hypercenter#2326D73
hypocenterC73#21643
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ #376 ◁ #2326

Complements

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

NH ≅ G/Ntype
2. #22612672C2semidirect
3. #22610899C2semidirect
6. #22610624C2semidirect
8. #22610610C2semidirect
9. #22610058C2semidirect
10. #22609761C2semidirect
12. C4×#14556515C2semidirect
14. #22609618C2semidirect
17. #17988941C4semidirect
65. #14556583C2×C4semidirect
80. #14556515C2×C4semidirect
81. #14556515C2×C4semidirect
91. C4×#10596831C2×C4semidirect
323. C73#21643semidirect

Maximal subgroups

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

subgroupindex
C2×#179889412
#226126722
#226108992
#226107482
#226106312
#226106242
#226106172
#226106102
#226100582
#226097612
#226097562
C4×#145565152
C4×#145565122
#226096182
C73×#23262
#2164373