Finite Group Databasebeta

← #33088994

Subgroups — #33088994

Order 18688.

966 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988685C211yesmaximal, characteristic
2C42×D292C211yesmaximal, characteristic
3#22610569C211yesmaximal, characteristic, complemented
4#22610562C211yesmaximal, characteristic, complemented
5#22610520C211yesmaximal, characteristic, complemented
6#22610499C211yesmaximal, characteristic, complemented
7#22610268C211yesmaximal, characteristic
8#22610268C211yesmaximal, characteristic
9#22610191C211yesmaximal, characteristic
10#22610177C211yesmaximal, characteristic
11#22609829C211yesmaximal, characteristic, complemented
12#22609761C211yesmaximal, characteristic, complemented
13#22609759C211yesmaximal, characteristic
14#22609680C211yesmaximal, characteristic, complemented
15C73×#2295C211yesmaximal, characteristic, complemented
16C22×#11924015C2211yescharacteristic
17C2×C4×D292C2211yescharacteristic
18C2×C4×D292C2211yescharacteristic
19C2×C4×D292C2211yescharacteristic
20C2×C42×D73—21no—
21#17988685C418yescomplemented
22C2×#14556524C2211yescharacteristic
23C2×#14556524C2211yescharacteristic
24C2×#14556522C2211yescharacteristic
25C2×#14556515C2211yescharacteristic
26C2×#14556515C2211yescharacteristic
27C2×#14556515C2211yescharacteristic
28C2×#14556511C2211yescharacteristic
29C2×#14556511C2211yescharacteristic
30C2×#14556511C2211yescharacteristic
31C2×#14556502C2211yescharacteristic
32C2×#14556502C2211yescharacteristic
33C4×C4⋊Dic73C2211yescharacteristic
34C4×C4⋊Dic73C2211yescharacteristic
35C4×C4⋊Dic73C2211yescharacteristic
36C4×#11924005—21no—
37C4×#11924005—21no—
38C4×#11924005—21no—
39#17988349C2211yescharacteristic
40#17988349C2211yescharacteristic
41#17988349C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988342C2211yescharacteristic
46#17988342C2211yescharacteristic
47#17988342C2211yescharacteristic
48#17988342C2211yescharacteristic
49C2×C4×Q8×C73C2211yescharacteristic
50#391×C73C2211yescharacteristic
51#391×C73C2211yescharacteristic
52#389×C73C2211yescharacteristic
53#389×C73C2211yescharacteristic
54#387×C73—21no—
55#387×C73—21no—
56C4×C4⋊C4×C73—21no—
57C4×C4⋊C4×C73C2211yescharacteristic
58#376×C73—21no—
59C43×C73C2211yescharacteristic
60C22×D292C2311yescharacteristic
61C22×C4×D73—21no—
62C22×C4×D73—21no—
63C22×C4×D73—21no—
64C2×#11924015—22no—
65C2×#11924015—22no—
66C2×#11924015C2×C414yes—
67C22×#10596831C2311yescharacteristic
68C22×#10596831C2311yescharacteristic
69C22×#10596831C2311yescharacteristic
70C22×#10596831C2311yescharacteristic
71C4×D292—22no—
72C4×D292—22no—
73C4×D292—22no—
74C4×D292—22no—
75C4×D292—22no—
76C4×D292C2×C414yes—
77C2×C4⋊Dic73C2311yescharacteristic
78C2×C4⋊Dic73C2311yescharacteristic
79C2×C4⋊Dic73C2311yescharacteristic
80C2×#11924005—21no—
81C2×#11924005—21no—
82C2×#11924005—21no—
83C2×C4×Dic73—21no—
84C2×C4×Dic73—21no—
85C2×C4×Dic73—21no—
86C42×D73—44no—
87#14556524—22no—
88#14556524C2×C414yescomplemented
89#14556522—22no—
90#14556515—22no—
91#14556515—22no—
92#14556515C2×C414yescomplemented
93#14556511—22no—
94#14556511—22no—
95#14556511C2×C414yes—
96#14556502—22no—
97#14556502C2×C414yescomplemented
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101C22×Q8×C73C2311yescharacteristic
102C4×Q8×C73—22no—
103C4×Q8×C73C2×C414yescomplemented
104C2×C4⋊C4×C73—21no—
105C2×C4⋊C4×C73—21no—
106C2×C4⋊C4×C73—21no—
107C2×C4⋊C4×C73C2311yescharacteristic
108C2×C4⋊C4×C73C2311yescharacteristic
109C2×C4⋊C4×C73C2311yescharacteristic
110C2×C42×C73—21no—
111C2×C42×C73—21no—
112C2×C42×C73D411yescharacteristic
113C2×C42×C73D411yescharacteristic
114C2×C42×C73D411yescharacteristic
115C2×C42×C73D411yescharacteristic
116C2×C42×C73C2311yescharacteristic
117C2×C42×C73C2311yescharacteristic
118C2×C42×C73C2311yescharacteristic
119#95×C73—21no—
120#95×C73—21no—
121#95×C73—21no—
122#95×C73—21no—
123#95×C73—21no—
124#95×C73—21no—
125#95×C73—21no—
126C23×D73—21no—
127C2×D292—21no—
128C2×D292—21no—
129C2×D292—21no—
130C2×D292C22×C412yes—
131C2×D292#4112yes—
132C2×D292#4112yes—
133C22×Dic73—21no—
134C22×Dic73—21no—
135C22×Dic73—21no—
136C2×C4×D73—24no—
137C2×C4×D73—24no—
138C2×C4×D73—24no—
139C2×C4×D73—42no—
140C2×C4×D73—42no—
141C2×C4×D73—42no—
142C2×C4×D73—42no—
143C2×C4×D73—42no—
144C2×C4×D73—42no—
145#11924015—28no—
146C2×#10596831—21no—
147C2×#10596831—21no—
148C2×#10596831—21no—
149C2×#10596831—21no—
150C2×#10596831—21no—
151C2×#10596831—21no—
152C2×#10596831—21no—
153C2×#10596831—21no—
154C2×#10596831—21no—
155C2×#10596831—21no—
156C2×#10596831C22×C412yes—
157C2×#10596831C22×C412yes—
158C4⋊Dic73—21no—
159C4⋊Dic73—21no—
160C4⋊Dic73—21no—
161C4⋊Dic73C22×C412yes—
162C4⋊Dic73#4112yes—
163C4⋊Dic73#4112yes—
164#11924005—24no—
165#11924005—24no—
166#11924005—24no—
167C4×Dic73—42no—
168C4×Dic73—42no—
169C4×Dic73—42no—
170C2×Q8×C73—21no—
171C2×Q8×C73—21no—
172C2×Q8×C73—21no—
173C2×Q8×C73—21no—
174C2×Q8×C73—21no—
175C2×Q8×C73C22×C412yes—
176C22×C4×C73—21no—
177C22×C4×C73—21no—
178C22×C4×C73—21no—
179C22×C4×C73—21no—
180C22×C4×C73C2×D411yescharacteristic
181C22×C4×C73C2×D411yescharacteristic
182C22×C4×C73C2×D411yescharacteristic
183C22×C4×C73C2×D411yescharacteristic
184C22×C4×C73C2×D411yescharacteristic
185C22×C4×C73C2×D411yescharacteristic
186C22×C4×C73C2411yescharacteristic
187C4⋊C4×C73—24no—
188C4⋊C4×C73—21no—
189C4⋊C4×C73—21no—
190C4⋊C4×C73—21no—
191C4⋊C4×C73—21no—
192C4⋊C4×C73—21no—
193C4⋊C4×C73—42no—
194C4⋊C4×C73—42no—
195C4⋊C4×C73C22×C412yes—
196C42×C73—24no—
197C42×C73—22no—
198C42×C73—22no—
199C42×C73—21no—
200C42×C73—21no—
201C42×C73—21no—
202C42×C73—21no—
203C42×C73—41no—
204C42×C73—41no—
205C42×C73—42no—
206C42×C73C22×C412yes—
207C42×C73#4112yes—
208C23×C73C22×D411yescharacteristic
209C22×D73—22no—
210C22×D73—22no—
211C22×D73—22no—
212C22×D73—41no—
213C22×D73—41no—
214C22×D73—41no—
215C22×D73—41no—
216Q8×C73—22no—
217Q8×C73—22no—
218Q8×C73—41no—
219Q8×C73—41no—
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×C73—21no—
228C2×C4×C73—22no—
229C2×C4×C73—22no—
230C2×C4×C73—21no—
231C2×C4×C73—21no—
232C2×C4×C73—21no—
233C2×C4×C73—21no—
234C2×C4×C73—21no—
235C2×C4×C73—21no—
236C2×C4×C73—41no—
237C2×C4×C73—41no—
238C2×C4×C73—41no—
239C2×C4×C73—41no—
240C2×C4×C73—41no—
241C2×C4×C73—41no—
242C2×C4×C73—41no—
243C2×C4×C73—41no—
244C2×C4×C73—41no—
245C2×C4×C73—41no—
246C2×C4×C73C4×D412yes—
247C2×C4×C73C4×D412yes—
248C2×C4×C73AGL(1,ℤ/8)12yes—
249C2×C4×C73#13712yes—
250C2×C4×C73C23×C411yescharacteristic
251C2×C4×C73C2×#4111yescharacteristic
252C2×C4×C73C2×#4111yescharacteristic
253C2×C4×C73C2×#4111yescharacteristic
254C2×C4×C73C2×#4111yescharacteristic
255C2×C4×C73C2×#4111yescharacteristic
256C2×Dic73—22no—
257C2×Dic73—22no—
258C2×Dic73—22no—
259C2×Dic73—41no—
260C2×Dic73—41no—
261C2×Dic73—41no—
262C2×Dic73—41no—
263C2×Dic73—41no—
264C2×Dic73—41no—
265D292—22no—
266D292—22no—
267D292—22no—
268D292#11714yes—
269C4×D73—48no—
270C4×D73—48no—
271C4×D73—48no—
272#10596831—22no—
273#10596831—22no—
274#10596831—41no—
275#10596831—41no—
276C22×C73C2×C4×D411yescharacteristic
277C22×C73#52511yescharacteristic
278C22×C73#52511yescharacteristic
279C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
280C22×C73C2×#13711yescharacteristic
281C22×C73#57511yescharacteristic
282C22×C73#57511yescharacteristic
283C2×D73—24no—
284C2×D73—42no—
285C2×D73—42no—
286C2×D73—42no—
287C2×D73—42no—
288C2×D73—42no—
289C2×D73—42no—
290C4×C73—22no—
291C4×C73—22no—
292C4×C73—22no—
293C4×C73—22no—
294C4×C73—41no—
295C4×C73—41no—
296C4×C73—42no—
297C4×C73—42no—
298C4×C73—81no—
299C4×C73—81no—
300C4×C73C2×#11711yescharacteristic
301C4×C73C4×#4111yescharacteristic
302C4×C73#52911yescharacteristic
303C4×C73#52911yescharacteristic
304Dic73—42no—
305Dic73—42no—
306Dic73—42no—
307#21121—731nomaximal, Sylow
308C2×C73#229011yescharacteristic
309C2×C73C4×AGL(1,ℤ/8)11yescharacteristic
310C2×C73C4×#13711yescharacteristic
311C2×C73#311011yescharacteristic
312C2×C73#311111yescharacteristic
313C2×C73#311411yescharacteristic
314C2×C73#311611yescharacteristic
315D73—44no—
316C2×#440—731no—
317#2295D7311yescharacteristic, complemented
318C42×D4—731no—
319#1928—731no—
320#1927—731no—
321#1921—731no—
322#1918—731no—
323#1877—731no—
324#1877—731no—
325#1864—731no—
326#1862—731no—
327#1776—731no—
328#1763—731no—
329#1762—731no—
330#1745—731no—
331C73#2112111yescharacteristic, complemented, Sylow
332C22×SD16—731no—
333C2×C4×Q8C2×D7311yescharacteristic
334C2×C4×D4—731no—
335C2×C4×D4—731no—
336C2×C4×D4—731no—
337C22×C42—1461no—
338#440—738no—
339C2×#107—731no—
340C2×#107—731no—
341C2×#106—731no—
342C2×#103—731no—
343C2×#103—731no—
344C2×#103—731no—
345C2×#102—731no—
346C2×#102—731no—
347C2×#102—731no—
348C2×#97—731no—
349C2×#97—731no—
350#391C2×D7311yescharacteristic
351#391C2×D7311yescharacteristic
352#389C2×D7311yescharacteristic
353#389C2×D7311yescharacteristic
354#387—21no—
355#387—21no—
356C4×C4⋊C4—21no—
357C4×C4⋊C4—731no—
358C4×C4⋊C4—731no—
359C4×C4⋊C4—731no—
360C4×C4⋊C4C2×D7311yescharacteristic
361C4×C22⋊C4—1461no—
362C4×C22⋊C4—1461no—
363C4×C22⋊C4—1461no—
364#376—21no—
365C43C2×D7311yescharacteristic
366#340—731no—
367#340—731no—
368#340—731no—
369#340—731no—
370#340—731no—
371#340—731no—
372#336—731no—
373#336—731no—
374#336—731no—
375#336—731no—
376C22×Q8C22×D7311yescharacteristic
377C22×D4—731no—
378C23×C4—1461no—
379C23×C4—1461no—
380C23×C4—1461no—
381C2×SD16—734no—
382C2×SD16—1462no—
383C2×SD16—1462no—
384C22×C8—731no—
385C22×C8—731no—
386C22×C8—731no—
387C22×C8—731no—
388C4×Q8—22no—
389C4×Q8C4×D7314yescomplemented
390C4×D4—734no—
391C4×D4—1462no—
392C4×D4—1462no—
393C4×D4—1462no—
394C4×D4—1462no—
395C4×D4—1462no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—731no—
400C2×C4⋊C4—731no—
401C2×C4⋊C4—731no—
402C2×C4⋊C4C22×D7311yescharacteristic
403C2×C4⋊C4C22×D7311yescharacteristic
404C2×C4⋊C4C22×D7311yescharacteristic
405C2×C22⋊C4—1461no—
406C2×C22⋊C4—1461no—
407C2×C22⋊C4—1461no—
408C2×C42—21no—
409C2×C42—21no—
410C2×C42—2924no—
411C2×C42—1461no—
412C2×C42—1461no—
413C2×C42—1461no—
414C2×C42C22⋊D7311yescharacteristic
415C2×C42C22⋊D7311yescharacteristic
416C2×C42C22⋊D7311yescharacteristic
417C2×C42C22⋊D7311yescharacteristic
418C2×C42C22×D7311yescharacteristic
419C2×C42C22×D7311yescharacteristic
420C2×C42C22×D7311yescharacteristic
421#107—734no—
422#107—1462no—
423#106—1462no—
424#103—734no—
425#103—1462no—
426#103—1462no—
427#102—734no—
428#102—1462no—
429#102—1462no—
430#97—734no—
431#97—1462no—
432#95—21no—
433#95—21no—
434#95—21no—
435#95—21no—
436#95—21no—
437#95—21no—
438#95—21no—
439#95—1461no—
440#95—1461no—
441#95—1461no—
442C24—1461no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8C2×C4×D7312yes—
449C2×D4—732no—
450C2×D4—732no—
451C2×D4—732no—
452C2×D4—1461no—
453C2×D4—1461no—
454C2×D4—1461no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—2922no—
460C22×C4—2922no—
461C22×C4—2922no—
462C22×C4—2922no—
463C22×C4—2922no—
464C22×C4—2922no—
465C22×C4—1464no—
466C22×C4—1464no—
467C22×C4—1461no—
468C22×C4—1461no—
469C22×C4—1461no—
470C22×C4—1464no—
471C22×C4C2×C22⋊D7311yescharacteristic
472C22×C4C2×C22⋊D7311yescharacteristic
473C22×C4C2×C22⋊D7311yescharacteristic
474C22×C4C2×C22⋊D7311yescharacteristic
475C22×C4C2×C22⋊D7311yescharacteristic
476C22×C4C2×C22⋊D7311yescharacteristic
477C22×C4C23×D7311yescharacteristic
478SD16—1468no—
479C2×C8—732no—
480C2×C8—732no—
481C2×C8—1461no—
482C2×C8—1461no—
483C2×C8—1461no—
484C2×C8—1461no—
485C2×C8—1461no—
486C2×C8—1461no—
487C2×C8—1461no—
488C2×C8—1461no—
489C2×C8—1461no—
490C2×C8—1461no—
491C4⋊C4—24no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—732no—
500C4⋊C4—732no—
501C4⋊C4—732no—
502C4⋊C4—1461no—
503C4⋊C4—1461no—
504C4⋊C4—1461no—
505C4⋊C4C2×C4×D7312yes—
506C22⋊C4—1464no—
507C22⋊C4—1464no—
508C22⋊C4—1464no—
509C42—24no—
510C42—22no—
511C42—22no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—41no—
517C42—41no—
518C42—42no—
519C42—2922no—
520C42—2922no—
521C42—2922no—
522C42C2×C4×D7312yes—
523C42#1192402812yes—
524C23—2921no—
525C23—2921no—
526C23—2921no—
527C23—2921no—
528C23—1462no—
529C23—1462no—
530C23—1462no—
531C23C22×C22⋊D7311yescharacteristic
532Q8—22no—
533Q8—22no—
534Q8—41no—
535Q8—41no—
536D4—734no—
537D4—1462no—
538D4—1462no—
539D4—1462no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—2928no—
567C2×C4—2928no—
568C2×C4—2928no—
569C2×C4—2921no—
570C2×C4—2921no—
571C2×C4—2921no—
572C2×C4—2921no—
573C2×C4—2921no—
574C2×C4—2921no—
575C2×C4—1462no—
576C2×C4—1462no—
577C2×C4—1462no—
578C2×C4C4×C22⋊D7312yes—
579C2×C4C4×C22⋊D7312yes—
580C2×C4#1455661612yes—
581C2×C4#1455662212yes—
582C2×C4C22×C4×D7311yescharacteristic
583C2×C4#41×D7311yescharacteristic
584C2×C4#41×D7311yescharacteristic
585C2×C4#41×D7311yescharacteristic
586C2×C4#41×D7311yescharacteristic
587C2×C4C2×#1192402811yescharacteristic
588C8—2921no—
589C8—2921no—
590C8—1462no—
591C8—1462no—
592C22—2922no—
593C22—2922no—
594C22—2922no—
595C22—2922no—
596C22—2922no—
597C22—2922no—
598C22—1464no—
599C22C2×C4×C22⋊D7311yescharacteristic
600C22#1798909411yescharacteristic
601C22#1798909411yescharacteristic
602C22C2×#1455661611yescharacteristic
603C22C2×#1455662211yescharacteristic
604C22#1798938511yescharacteristic
605C22#1798938511yescharacteristic
606C4—22no—
607C4—22no—
608C4—22no—
609C4—22no—
610C4—42no—
611C4—41no—
612C4—42no—
613C4—41no—
614C4—81no—
615C4—81no—
616C4—2922no—
617C4—2922no—
618C4—2922no—
619C4#117×D7311yescharacteristic
620C4#1798905511yescharacteristic
621C4#1798911511yescharacteristic
622C4#1798911511yescharacteristic
623C2—2924no—
624C2#2261236811yescharacteristic
625C2C4×#1455661611yescharacteristic
626C2C4×#1455662211yescharacteristic
627C2#2262002611yescharacteristic
628C2#2262003411yescharacteristic
629C2#2262006511yescharacteristic
630C2#2262008711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#2295C2
socleC23×C73C22×D4
prime radicalC22×D292C23
cosocleC22×C4×C73C24
hypercenter#2295D73
hypocenterC73#21121
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C43 ◁ #2295

Complements

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

NH ≅ G/Ntype
3. #22610569C2semidirect
4. #22610562C2semidirect
5. #22610520C2semidirect
6. #22610499C2semidirect
11. #22609829C2semidirect
12. #22609761C2semidirect
14. #22609680C2semidirect
15. C73×#2295C2semidirect
21. #17988685C4semidirect
88. #14556524C2×C4semidirect
92. #14556515C2×C4semidirect
97. #14556502C2×C4semidirect
103. C4×Q8×C73C2×C4semidirect
317. #2295D73semidirect
331. C73#21121semidirect
389. C4×Q8C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179886852
C42×D2922
#226105692
#226105622
#226105202
#226104992
#226102682
#226101912
#226101772
#226098292
#226097612
#226097592
#226096802
C73×#22952
#2112173