Finite Group Databasebeta

← #33093831

Subgroups — #33093831

Order 18688.

730 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619721C211yesmaximal, characteristic
2#22618868C211yesmaximal, characteristic, complemented
3#22618360C211yesmaximal, characteristic
4#22610875C211yesmaximal, characteristic, complemented
5#22610529C211yesmaximal, characteristic, complemented
6#22610529C211yesmaximal, characteristic, complemented
7#22610203C211yesmaximal, characteristic
8#22610203C211yesmaximal, characteristic
9#22609948C211yesmaximal, characteristic
10#22609818C211yesmaximal, characteristic, complemented
11#22609813C211yesmaximal, characteristic, complemented
12#22609808C211yesmaximal, characteristic, complemented
13#22609774C211yesmaximal, characteristic
14#22609767C211yesmaximal, characteristic
15C73×#1950C211yesmaximal, characteristic, complemented
16C2×#14556599C2211yescharacteristic
17C2×#14556599C2211yescharacteristic
18#17989144C2211yescharacteristic
19C4×#11924028C2211yescharacteristic
20#17988895C411yescharacteristic, complemented
21#17988895C411yescharacteristic, complemented
22#17988888C411yescharacteristic, complemented
23#17988888C411yescharacteristic, complemented
24#17988883C411yescharacteristic, complemented
25#17988878C411yescharacteristic, complemented
26#17988873C411yescharacteristic, complemented
27#17988868C411yescharacteristic, complemented
28#17988716—21no—
29#17988716—21no—
30#17988716—21no—
31#17988716—21no—
32#17988625C2211yescharacteristic
33#17988625C2211yescharacteristic
34#17988615C2211yescharacteristic
35#17988615C2211yescharacteristic
36C2×#14556517C2211yescharacteristic
37C2×#14556517C2211yescharacteristic
38#17988584C2211yescharacteristic
39#17988584C2211yescharacteristic
40#17988545C2211yescharacteristic
41#17988545C2211yescharacteristic
42#17988545C2211yescharacteristic
43#17988545C2211yescharacteristic
44#17988541—21no—
45#17988541—21no—
46#17988539—21no—
47C2×#14556502C2211yescharacteristic
48C2×#14556501C2211yescharacteristic
49#97×D73—21no—
50C2×C8×Dic73C2211yescharacteristic
51C4×C4⋊Dic73C2211yescharacteristic
52#17988348C2211yescharacteristic
53#17988348C2211yescharacteristic
54#17988348C2211yescharacteristic
55#17988348C2211yescharacteristic
56#17988348C2211yescharacteristic
57#17988348C2211yescharacteristic
58#17988341C2211yescharacteristic
59#17988341C2211yescharacteristic
60C2×#108×C73C2211yescharacteristic
61C2×#108×C73C2211yescharacteristic
62#423×C73C2211yescharacteristic
63C2×C4⋊C8×C73C2211yescharacteristic
64C4×M16×C73C2211yescharacteristic
65#341×C73C2211yescharacteristic
66#341×C73C2211yescharacteristic
67C2×#11924028C2311yescharacteristic
68#14556599—22no—
69#14556599—22no—
70#14556599—22no—
71#14556599Q814yescomplemented
72C2×#11924011—21no—
73C2×#11924011—21no—
74C22×#10596831C2311yescharacteristic
75C22×#10596831C2311yescharacteristic
76C2×C8×D73—21no—
77C2×C8×D73—21no—
78#14556591C2×C411yescharacteristic
79#14556591C2×C411yescharacteristic
80#14556583C2×C411yescharacteristic
81#14556583C2×C411yescharacteristic
82#14556570—21no—
83#14556570—21no—
84C4×Dic146—21no—
85C4×Dic146—21no—
86C4×Dic146C2×C411yescharacteristic
87C4×Dic146C2×C411yescharacteristic
88C4×D292C2×C411yescharacteristic
89C4×D292C2×C411yescharacteristic
90C4×C22⋊D73—21no—
91C4×C22⋊D73—21no—
92#14556550C2311yescharacteristic
93#14556550C2311yescharacteristic
94#14556550C2311yescharacteristic
95#14556548—21no—
96C2×C4⋊Dic73—21no—
97C2×C4×Dic73C2311yescharacteristic
98C2×C4×Dic73C2311yescharacteristic
99C42×D73—21no—
100#14556525C2×C411yescharacteristic
101#14556525C2×C411yescharacteristic
102#14556525C2×C411yescharacteristic
103#14556525C2×C411yescharacteristic
104#14556523C2×C411yescharacteristic
105#14556523C2×C411yescharacteristic
106#14556523C2×C411yescharacteristic
107#14556523C2×C411yescharacteristic
108#14556520—21no—
109#14556520—21no—
110#14556520—21no—
111#14556520—21no—
112#14556517—22no—
113#14556517—22no—
114#14556516C2×C411yescharacteristic
115#14556516C2×C411yescharacteristic
116#14556516C2×C411yescharacteristic
117#14556516C2×C411yescharacteristic
118#14556513C2×C411yescharacteristic
119#14556513C2×C411yescharacteristic
120#14556513C2×C411yescharacteristic
121#14556513C2×C411yescharacteristic
122#14556502D411yescharacteristic
123#14556502D411yescharacteristic
124#14556502D411yescharacteristic
125#14556502D411yescharacteristic
126#14556501—21no—
127#14556501—21no—
128#14556501—21no—
129#14556501—21no—
130#14556501—22no—
131C8×Dic73—21no—
132C8×Dic73—21no—
133C8×Dic73—21no—
134C8×Dic73—21no—
135C8×Dic73—22no—
136#14556498—21no—
137C2×M16×C73C2311yescharacteristic
138C2×M16×C73C2311yescharacteristic
139C2×M16×C73C2311yescharacteristic
140C2×M16×C73C2311yescharacteristic
141C22×C8×C73C2311yescharacteristic
142C22×C8×C73C2311yescharacteristic
143C2×C42×C73C2311yescharacteristic
144#108×C73—22no—
145#108×C73—22no—
146C4⋊C8×C73—21no—
147C4⋊C8×C73—21no—
148C4⋊C8×C73C2×C411yescharacteristic
149C4⋊C8×C73C2×C411yescharacteristic
150C4⋊C8×C73C2×C411yescharacteristic
151C4⋊C8×C73C2×C411yescharacteristic
152#97×C73—21no—
153C4×C8×C73—21no—
154#11924028—21no—
155#11924028—21no—
156#11924028C2×Q812yes—
157#11924028#4112yes—
158C2×Dic146—21no—
159C2×Dic146—21no—
160C2×Dic146C22×C411yescharacteristic
161C2×D292C22×C411yescharacteristic
162C2×C22⋊D73—21no—
163C22×Dic73—21no—
164C22×Dic73—21no—
165C2×C4×D73—21no—
166C2×C4×D73—21no—
167C2×C4×D73—21no—
168#11924011—24no—
169#11924011—24no—
170C2×#10596831—21no—
171C2×#10596831—21no—
172C2×#10596831C2×D411yescharacteristic
173C2×#10596831C2×D411yescharacteristic
174C2×#10596831C2×D411yescharacteristic
175C2×#10596831C2×D411yescharacteristic
176C2×#10596831C2×Q812yes—
177C2×#10596831#4112yes—
178C8×D73—24no—
179C8×D73—24no—
180C4⋊Dic73—21no—
181C4⋊Dic73—21no—
182C4⋊Dic73—42no—
183C4⋊Dic73C22×C411yescharacteristic
184C4⋊Dic73C22×C411yescharacteristic
185C4⋊Dic73C22×C411yescharacteristic
186C4⋊Dic73C22×C411yescharacteristic
187C4⋊Dic73C22×C411yescharacteristic
188C4⋊Dic73C22×C411yescharacteristic
189#11924007—21no—
190#11924007—21no—
191#11924007—21no—
192#11924007—21no—
193#11924007—21no—
194#11924007—21no—
195C22⋊Dic73—21no—
196C22⋊Dic73—21no—
197C22⋊Dic73—21no—
198#11924005—21no—
199#11924005—21no—
200C4×Dic73—21no—
201C4×Dic73—21no—
202C4×Dic73—21no—
203C4×Dic73—21no—
204C4×Dic73—21no—
205C4×Dic73—21no—
206C4×Dic73—21no—
207C4×Dic73—21no—
208C4×Dic73—21no—
209C22×C4×C73#4111yescharacteristic
210C22×C4×C73#4111yescharacteristic
211C22×C4×C73C2411yescharacteristic
212M16×C73—21no—
213M16×C73—21no—
214M16×C73—21no—
215M16×C73—21no—
216M16×C73—21no—
217M16×C73—21no—
218M16×C73C2×Q812yes—
219M16×C73#4112yes—
220C2×C8×C73—21no—
221C2×C8×C73—21no—
222C2×C8×C73—21no—
223C2×C8×C73—21no—
224C2×C8×C73—21no—
225C2×C8×C73—21no—
226C2×C8×C73—21no—
227C2×C8×C73—21no—
228C2×C8×C73C22×C411yescharacteristic
229C2×C8×C73C22×C411yescharacteristic
230C2×C8×C73C22×C411yescharacteristic
231C2×C8×C73C22×C411yescharacteristic
232C42×C73C22×C411yescharacteristic
233C42×C73C22×C411yescharacteristic
234C42×C73C2×D411yescharacteristic
235C42×C73C2×D411yescharacteristic
236C23×C73C2×#4111yescharacteristic
237C22×D73—21no—
238C2×C4×C73—41no—
239C2×C4×C73—41no—
240C2×C4×C73C4×D411yescharacteristic
241C2×C4×C73C4×D411yescharacteristic
242C2×C4×C73C4×D411yescharacteristic
243C2×C4×C73C4×D411yescharacteristic
244C2×C4×C73C23×C411yescharacteristic
245C2×C4×C73C22×D411yescharacteristic
246C2×C4×C73C22×Q811yescharacteristic
247C2×C4×C73C2×#4111yescharacteristic
248C2×C4×C73C2×#4111yescharacteristic
249C2×C4×C73C2×#4111yescharacteristic
250C22⋊D73—22no—
251C22⋊D73—22no—
252C2×Dic73—21no—
253C2×Dic73—22no—
254C2×Dic73—21no—
255C2×Dic73—21no—
256C2×Dic73—21no—
257C2×Dic73—21no—
258C2×Dic73—21no—
259C2×Dic73—21no—
260C2×Dic73—41no—
261C2×Dic73—41no—
262C2×Dic73—41no—
263C2×Dic73—41no—
264C2×Dic73—41no—
265D292—21no—
266D292C4×Q812yes—
267C4×D73—22no—
268C4×D73—22no—
269C4×D73—81no—
270C4×D73—81no—
271Dic146—21no—
272Dic146—81no—
273Dic146—81no—
274Dic146C4×Q812yes—
275C8×C73—21no—
276C8×C73—21no—
277C8×C73—21no—
278C8×C73—21no—
279C8×C73—22no—
280C8×C73—22no—
281C8×C73—41no—
282C8×C73—41no—
283#10596831—21no—
284#10596831—21no—
285#10596831C22⋊Q812yes—
286#10596831C22⋊Q812yes—
287C22×C73—21no—
288C22×C73—21no—
289C22×C73C2×C4×D411yescharacteristic
290C22×C73#52911yescharacteristic
291C22×C73#53311yescharacteristic
292C2×D73—22no—
293C2×D73—41no—
294C2×D73—41no—
295C4×C73—21no—
296C4×C73—21no—
297C4×C73—21no—
298C4×C73—21no—
299C4×C73C2×C4×Q811yescharacteristic
300C4×C73C4×#4111yescharacteristic
301C4×C73C2×C22⋊Q811yescharacteristic
302C4×C73#52511yescharacteristic
303Dic73—22no—
304Dic73—42no—
305Dic73—41no—
306Dic73—41no—
307Dic73—41no—
308Dic73—41no—
309Dic73—41no—
310Dic73—41no—
311Dic73—81no—
312#21629—731nomaximal, Sylow
313C2×C73—21no—
314C2×C73—21no—
315C2×C73C4×C22⋊Q811yescharacteristic
316C2×C73#295611yescharacteristic
317C2×C73#295711yescharacteristic
318D73—42no—
319#3076—731no—
320#2961—731no—
321#2875—731no—
322#1986—731no—
323#1950D7311yescharacteristic, complemented
324#1922—731no—
325#1922—731no—
326#1866—731no—
327#1866—731no—
328#1806—731no—
329#1773—731no—
330#1772—731no—
331#1771—731no—
332#1765—731no—
333#1764—731no—
334C73#2162911yescharacteristic, complemented, Sylow
335C2×#131—731no—
336C2×#131—731no—
337#533—731no—
338C4×#41—731no—
339#479—731no—
340#479—731no—
341#478—731no—
342#478—731no—
343#477—731no—
344#476—731no—
345#475—731no—
346#474—731no—
347#446—1461no—
348#446—1461no—
349#446—1461no—
350#446—1461no—
351C2×#108C2×D7311yescharacteristic
352C2×#108C2×D7311yescharacteristic
353#427—731no—
354#427—731no—
355#424—731no—
356#424—731no—
357#423C2×D7311yescharacteristic
358C2×C4⋊C8C2×D7311yescharacteristic
359C2×#104—731no—
360C2×#104—731no—
361#416—731no—
362#416—731no—
363#405—731no—
364#405—731no—
365#405—731no—
366#405—731no—
367#405—1461no—
368#405—1461no—
369C4×M16—1461no—
370C4×M16C2×D7311yescharacteristic
371C2×#97—731no—
372C2×#97—731no—
373C2×#97—1461no—
374C2×C4×C8—731no—
375C4×C4⋊C4—731no—
376#341C2×D7311yescharacteristic
377#341C2×D7311yescharacteristic
378#339—731no—
379#339—731no—
380#339—731no—
381#339—731no—
382#339—731no—
383#339—731no—
384#336—731no—
385#336—731no—
386C2×#41—731no—
387#131—734no—
388#131—1462no—
389#131—1462no—
390#131—1462no—
391C2×M16—1461no—
392C2×M16—1461no—
393C2×M16C22×D7311yescharacteristic
394C2×M16C22×D7311yescharacteristic
395C2×M16C22×D7311yescharacteristic
396C2×M16C22×D7311yescharacteristic
397C22×C8—731no—
398C22×C8—731no—
399C22×C8—1461no—
400C22×C8—1461no—
401C22×C8C22×D7311yescharacteristic
402C22×C8C22×D7311yescharacteristic
403C4⋊Q8—731no—
404C4⋊Q8—731no—
405#125—731no—
406#125—731no—
407C22⋊Q8—1461no—
408C22⋊Q8—1461no—
409C4×Q8—731no—
410C4×Q8—731no—
411C4×Q8—1461no—
412C4×Q8—1461no—
413C4×D4—731no—
414C4×D4—731no—
415C4×D4—1461no—
416C4×D4—1461no—
417#117—731no—
418#117—731no—
419#117—731no—
420#117—1461no—
421C2×C4⋊C4—1461no—
422C2×C42—731no—
423C2×C42—731no—
424C2×C42—1461no—
425C2×C42C22×D7311yescharacteristic
426#108—22no—
427#108—22no—
428#107—731no—
429#107—731no—
430#107—731no—
431#107—731no—
432#106—731no—
433#106—731no—
434#106—731no—
435#106—731no—
436C4⋊C8—21no—
437C4⋊C8—21no—
438C4⋊C8—1461no—
439C4⋊C8—1461no—
440C4⋊C8—1461no—
441C4⋊C8—1461no—
442C4⋊C8C4×D7311yescharacteristic
443C4⋊C8C4×D7311yescharacteristic
444C4⋊C8C4×D7311yescharacteristic
445C4⋊C8C4×D7311yescharacteristic
446#104—1462no—
447#104—1462no—
448#103—731no—
449#103—731no—
450#103—731no—
451#103—731no—
452#102—731no—
453#102—731no—
454#102—731no—
455#102—731no—
456#97—21no—
457#97—731no—
458#97—731no—
459#97—731no—
460#97—731no—
461#97—1462no—
462#97—1461no—
463#97—1461no—
464#97—1461no—
465#97—1461no—
466C4×C8—21no—
467C4×C8—1462no—
468C4×C8—1461no—
469C4×C8—1461no—
470C4×C8—1461no—
471C4×C8—1461no—
472#95—1461no—
473#41—732no—
474#41—732no—
475#41—1461no—
476#41—1461no—
477C2×Q8—731no—
478C2×Q8—1461no—
479C2×Q8—1461no—
480C2×D4—731no—
481C2×D4—1461no—
482C22×C4—1461no—
483C22×C4—1461no—
484C22×C4—1461no—
485C22×C4—1461no—
486C22×C4—1461no—
487C22×C4#1192402911yescharacteristic
488C22×C4#1192402911yescharacteristic
489C22×C4C23×D7311yescharacteristic
490M16—21no—
491M16—21no—
492M16—21no—
493M16—21no—
494M16—21no—
495M16—21no—
496M16—1464no—
497M16—1464no—
498M16Q8×D7312yes—
499M16#1192402712yes—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—731no—
509C2×C8—731no—
510C2×C8—732no—
511C2×C8—732no—
512C2×C8—731no—
513C2×C8—731no—
514C2×C8—1464no—
515C2×C8—1464no—
516C2×C8—1461no—
517C2×C8—1461no—
518C2×C8C2×C4×D7311yescharacteristic
519C2×C8C2×C4×D7311yescharacteristic
520C2×C8C2×C4×D7311yescharacteristic
521C2×C8C2×C4×D7311yescharacteristic
522C4⋊C4—2922no—
523C4⋊C4—731no—
524C4⋊C4—731no—
525C4⋊C4—731no—
526C4⋊C4—731no—
527C4⋊C4—731no—
528C4⋊C4—731no—
529C4⋊C4—1461no—
530C4⋊C4—1461no—
531C4⋊C4—1461no—
532C4⋊C4—1461no—
533C4⋊C4—1461no—
534C4⋊C4—1461no—
535C4⋊C4—1461no—
536C4⋊C4—1461no—
537C22⋊C4—1461no—
538C22⋊C4—1461no—
539C22⋊C4—1461no—
540C22⋊C4—1461no—
541C22⋊C4—1461no—
542C42—1461no—
543C42—1461no—
544C42—1461no—
545C42—1461no—
546C42—1461no—
547C42—1461no—
548C42—1461no—
549C42—1461no—
550C42—1461no—
551C42C2×C4×D7311yescharacteristic
552C42C2×C4×D7311yescharacteristic
553C42D4×D7311yescharacteristic
554C42D4×D7311yescharacteristic
555C23—1461no—
556C23C2×#1192402911yescharacteristic
557Q8—5841no—
558Q8—5841no—
559Q8—732no—
560Q8—1461no—
561D4—732no—
562D4—1462no—
563D4—1462no—
564D4—1461no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—2921no—
568C2×C4—2921no—
569C2×C4—2921no—
570C2×C4—2921no—
571C2×C4—2921no—
572C2×C4—5841no—
573C2×C4—5841no—
574C2×C4—1462no—
575C2×C4—1461no—
576C2×C4—1461no—
577C2×C4—1461no—
578C2×C4—1461no—
579C2×C4—1461no—
580C2×C4—1461no—
581C2×C4—1461no—
582C2×C4—1462no—
583C2×C4—1462no—
584C2×C4#1455655311yescharacteristic
585C2×C4#1455655311yescharacteristic
586C2×C4#1455655311yescharacteristic
587C2×C4#1455655311yescharacteristic
588C2×C4C22×C4×D7311yescharacteristic
589C2×C4C2×D4×D7311yescharacteristic
590C2×C4C2×Q8×D7311yescharacteristic
591C2×C4#41×D7311yescharacteristic
592C2×C4#41×D7311yescharacteristic
593C2×C4C2×#1192402711yescharacteristic
594C8—21no—
595C8—21no—
596C8—21no—
597C8—21no—
598C8—22no—
599C8—22no—
600C8—41no—
601C8—41no—
602C8—732no—
603C8—732no—
604C8—1461no—
605C8—1461no—
606C22—21no—
607C22—21no—
608C22—2921no—
609C22—2921no—
610C22—1462no—
611C22C2×#1455655311yescharacteristic
612C22#1798912211yescharacteristic
613C22#1798914211yescharacteristic
614C4—21no—
615C4—21no—
616C4—21no—
617C4—21no—
618C4—2922no—
619C4—2921no—
620C4—2921no—
621C4—2921no—
622C4—2921no—
623C4—2921no—
624C4—2921no—
625C4—5841no—
626C4—1462no—
627C4C4×Q8×D7311yescharacteristic
628C4C4×#1192402711yescharacteristic
629C4C22⋊Q8×D7311yescharacteristic
630C4#1798909711yescharacteristic
631C2—21no—
632C2—21no—
633C2—2922no—
634C2#2261246111yescharacteristic
635C2#2261884011yescharacteristic
636C2#2261884711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1950C2
socleC22×C73C2×C4×D4
prime radicalC2×#11924028C23
cosocleC22×C4×C73C24
hypercenter#1950D73
hypocenterC73#21629
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1950

Complements

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

NH ≅ G/Ntype
2. #22618868C2semidirect
4. #22610875C2semidirect
5. #22610529C2semidirect
6. #22610529C2semidirect
10. #22609818C2semidirect
11. #22609813C2semidirect
12. #22609808C2semidirect
15. C73×#1950C2semidirect
20. #17988895C4semidirect
21. #17988895C4semidirect
22. #17988888C4semidirect
23. #17988888C4semidirect
24. #17988883C4semidirect
25. #17988878C4semidirect
26. #17988873C4semidirect
27. #17988868C4semidirect
71. #14556599Q8semidirect
323. #1950D73semidirect
334. C73#21629semidirect

Maximal subgroups

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

subgroupindex
#226197212
#226188682
#226183602
#226108752
#226105292
#226102032
#226099482
#226098182
#226098132
#226098082
#226097742
#226097672
C73×#19502
#2162973