Finite Group Databasebeta

← #33108075

Subgroups — #33108075

Order 18688.

692 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619719C211yesmaximal, characteristic
2#22618874C211yesmaximal, characteristic, complemented
3#22610911C211yesmaximal, characteristic, complemented
4#22610704C211yesmaximal, characteristic, complemented
5#22610681C211yesmaximal, characteristic, complemented
6#22610676C211yesmaximal, characteristic, complemented
7#22610527C211yesmaximal, characteristic, complemented
8#22610527C211yesmaximal, characteristic, complemented
9#22610206C211yesmaximal, characteristic
10#22610206C211yesmaximal, characteristic
11#22609790C211yesmaximal, characteristic, complemented
12#22609773C211yesmaximal, characteristic
13#22609766C211yesmaximal, characteristic
14#22609741C211yesmaximal, characteristic
15C73×#2906C211yesmaximal, characteristic
16#17989144C2211yescharacteristic
17#17988894C411yescharacteristic, complemented
18#17988894C411yescharacteristic, complemented
19#17988887C411yescharacteristic, complemented
20#17988887C411yescharacteristic, complemented
21#17988882C411yescharacteristic, complemented
22#17988877C411yescharacteristic, complemented
23#17988872C411yescharacteristic, complemented
24#17988867C411yescharacteristic, complemented
25#17988716—21no—
26#17988716—21no—
27#17988714—21no—
28#17988711—21no—
29C2×#14556526C2211yescharacteristic
30C2×#14556526C2211yescharacteristic
31#17988623C2211yescharacteristic
32#17988623C2211yescharacteristic
33#17988613C2211yescharacteristic
34#17988613C2211yescharacteristic
35C2×C4⋊#10596831C2211yescharacteristic
36C2×#14556518C2211yescharacteristic
37C2×#14556518C2211yescharacteristic
38#17988583C2211yescharacteristic
39#17988583C2211yescharacteristic
40#17988542C2211yescharacteristic
41#17988542C2211yescharacteristic
42C4×#11924012C2211yescharacteristic
43C2×#14556501C2211yescharacteristic
44C2×C8×Dic73C2211yescharacteristic
45#17988433C2211yescharacteristic
46#17988351C2211yescharacteristic
47#17988351C2211yescharacteristic
48#17988348C2211yescharacteristic
49#17988348C2211yescharacteristic
50#17988348C2211yescharacteristic
51#17988348C2211yescharacteristic
52#17988346C2211yescharacteristic
53#17988346C2211yescharacteristic
54#17988341C2211yescharacteristic
55#17988341C2211yescharacteristic
56C2×#131×C73C2211yescharacteristic
57C2×#131×C73C2211yescharacteristic
58C4×#41×C73C2211yescharacteristic
59#424×C73—21no—
60#424×C73—21no—
61#424×C73C2211yescharacteristic
62#424×C73C2211yescharacteristic
63#423×C73—21no—
64#423×C73C2211yescharacteristic
65C2×C4⋊C8×C73—21no—
66C2×C4⋊C8×C73C2211yescharacteristic
67C2×#11924012C2311yescharacteristic
68C2×#11924012C2311yescharacteristic
69C22×#10596831C2311yescharacteristic
70C22×#10596831C2311yescharacteristic
71#14556591C2×C411yescharacteristic
72#14556591C2×C411yescharacteristic
73#14556583C2×C411yescharacteristic
74#14556583C2×C411yescharacteristic
75#14556570—21no—
76#14556570—21no—
77C4×Dic146—21no—
78C4×Dic146—21no—
79#14556550C2311yescharacteristic
80#14556550C2311yescharacteristic
81C2×C4×Dic73C2311yescharacteristic
82C2×C4×Dic73C2311yescharacteristic
83#14556526—41no—
84#14556526—41no—
85#14556524C2×C411yescharacteristic
86#14556524C2×C411yescharacteristic
87#14556524C2×C411yescharacteristic
88#14556524C2×C411yescharacteristic
89#14556522C2×C411yescharacteristic
90#14556522C2×C411yescharacteristic
91#14556522C2×C411yescharacteristic
92#14556522C2×C411yescharacteristic
93C4⋊#10596831C2×C411yescharacteristic
94C4⋊#10596831C2×C411yescharacteristic
95C4⋊#10596831C2×C411yescharacteristic
96C4⋊#10596831C2×C411yescharacteristic
97#14556520—21no—
98#14556520—21no—
99#14556520—21no—
100#14556520—21no—
101#14556518—22no—
102#14556518Q814yescomplemented
103#14556515C2×C411yescharacteristic
104#14556515C2×C411yescharacteristic
105#14556515C2×C411yescharacteristic
106#14556515C2×C411yescharacteristic
107#14556512C2×C411yescharacteristic
108#14556512C2×C411yescharacteristic
109#14556512C2×C411yescharacteristic
110#14556512C2×C411yescharacteristic
111#14556502—21no—
112#14556501—21no—
113#14556501—21no—
114#14556501—41no—
115C4×#10596831—21no—
116C8×Dic73—21no—
117C8×Dic73—21no—
118C8×Dic73—41no—
119#14556498—21no—
120#14556498—21no—
121C2×#41×C73C2311yescharacteristic
122#131×C73—22no—
123#131×C73—22no—
124#131×C73—41no—
125#131×C73—41no—
126C2×M16×C73—21no—
127C2×M16×C73—21no—
128C2×M16×C73C2311yescharacteristic
129C2×M16×C73C2311yescharacteristic
130C22×C8×C73—21no—
131C22×C8×C73—21no—
132C22×C8×C73C2311yescharacteristic
133C22×C8×C73C2311yescharacteristic
134C4×Q8×C73C2×C411yescharacteristic
135C4×Q8×C73C2×C411yescharacteristic
136C4×D4×C73—21no—
137C4×D4×C73—21no—
138C4×D4×C73C2×C411yescharacteristic
139C4×D4×C73C2×C411yescharacteristic
140#117×C73—21no—
141#117×C73C2311yescharacteristic
142C2×C42×C73—21no—
143C2×C42×C73C2311yescharacteristic
144C4⋊C8×C73—21no—
145C4⋊C8×C73—21no—
146C4⋊C8×C73—21no—
147C4⋊C8×C73—21no—
148C4⋊C8×C73—21no—
149C4⋊C8×C73—21no—
150C4⋊C8×C73D411yescharacteristic
151C4⋊C8×C73D411yescharacteristic
152C4⋊C8×C73D411yescharacteristic
153C4⋊C8×C73D411yescharacteristic
154C2×Dic146—21no—
155C2×Dic146—21no—
156C22×Dic73—21no—
157C22×Dic73—21no—
158#11924012—21no—
159#11924012—21no—
160#11924012—21no—
161#11924012C2×Q812yes—
162C2×#10596831—21no—
163C2×#10596831—21no—
164C2×#10596831—21no—
165C2×#10596831—21no—
166C2×#10596831—21no—
167C2×#10596831—21no—
168C2×#10596831C22×C411yescharacteristic
169C2×#10596831C22×C411yescharacteristic
170C2×#10596831C22×C411yescharacteristic
171C2×#10596831C22×C411yescharacteristic
172C4⋊Dic73—21no—
173C4⋊Dic73—21no—
174C4⋊Dic73C22×C411yescharacteristic
175C4⋊Dic73C22×C411yescharacteristic
176C4⋊Dic73C22×C411yescharacteristic
177C4⋊Dic73C22×C411yescharacteristic
178#11924007—21no—
179#11924007—21no—
180#11924007—21no—
181#11924007—21no—
182C22⋊Dic73—21no—
183C22⋊Dic73—21no—
184C4×Dic73—21no—
185C4×Dic73—21no—
186C4×Dic73—21no—
187C4×Dic73—21no—
188C4×Dic73—21no—
189C4×Dic73C2×Q812yes—
190#41×C73—21no—
191#41×C73—21no—
192#41×C73C2×Q812yes—
193#41×C73#4112yes—
194C2×Q8×C73C22×C411yescharacteristic
195C2×D4×C73—21no—
196C2×D4×C73C22×C411yescharacteristic
197C22×C4×C73—21no—
198C22×C4×C73—21no—
199C22×C4×C73—21no—
200C22×C4×C73#4111yescharacteristic
201C22×C4×C73#4111yescharacteristic
202C22×C4×C73C2411yescharacteristic
203M16×C73—21no—
204M16×C73—21no—
205M16×C73—21no—
206M16×C73—21no—
207M16×C73—42no—
208M16×C73—42no—
209C2×C8×C73—21no—
210C2×C8×C73—21no—
211C2×C8×C73—21no—
212C2×C8×C73—21no—
213C2×C8×C73—21no—
214C2×C8×C73—21no—
215C2×C8×C73—42no—
216C2×C8×C73—42no—
217C2×C8×C73C2×D411yescharacteristic
218C2×C8×C73C2×D411yescharacteristic
219C2×C8×C73C2×D411yescharacteristic
220C2×C8×C73C2×D411yescharacteristic
221C4⋊C4×C73—21no—
222C4⋊C4×C73—21no—
223C4⋊C4×C73C22×C411yescharacteristic
224C4⋊C4×C73C22×C411yescharacteristic
225C22⋊C4×C73—21no—
226C22⋊C4×C73—21no—
227C22⋊C4×C73—21no—
228C42×C73—21no—
229C42×C73—21no—
230C42×C73—21no—
231C42×C73C22×C411yescharacteristic
232C42×C73C22×C411yescharacteristic
233C42×C73C2×D411yescharacteristic
234C42×C73C2×D411yescharacteristic
235C23×C73—21no—
236C23×C73C2×#4111yescharacteristic
237Q8×C73—21no—
238Q8×C73C4×Q812yes—
239D4×C73—22no—
240D4×C73—22no—
241D4×C73—21no—
242D4×C73C4×Q812yes—
243C2×C4×C73—21no—
244C2×C4×C73—22no—
245C2×C4×C73—22no—
246C2×C4×C73—21no—
247C2×C4×C73—21no—
248C2×C4×C73—41no—
249C2×C4×C73—41no—
250C2×C4×C73—42no—
251C2×C4×C73—81no—
252C2×C4×C73C4×D411yescharacteristic
253C2×C4×C73C4×D411yescharacteristic
254C2×C4×C73C4×D411yescharacteristic
255C2×C4×C73C4×D411yescharacteristic
256C2×C4×C73C23×C411yescharacteristic
257C2×C4×C73C22×D411yescharacteristic
258C2×C4×C73C22×Q811yescharacteristic
259C2×C4×C73C2×#4111yescharacteristic
260C2×C4×C73H(5,𝔽2)11yescharacteristic
261C2×C4×C732−1+411yescharacteristic
262C2×Dic73—21no—
263C2×Dic73—21no—
264C2×Dic73—21no—
265C2×Dic73—21no—
266C2×Dic73—22no—
267C2×Dic73—41no—
268C2×Dic73—41no—
269C2×Dic73—41no—
270C2×Dic73—41no—
271C2×Dic73—41no—
272Dic146—42no—
273Dic146—81no—
274C8×C73—21no—
275C8×C73—21no—
276C8×C73—21no—
277C8×C73—21no—
278C8×C73—41no—
279C8×C73—41no—
280#10596831—21no—
281#10596831—21no—
282#10596831—21no—
283#10596831—21no—
284#10596831—22no—
285#10596831—41no—
286C22×C73—21no—
287C22×C73—21no—
288C22×C73—22no—
289C22×C73—41no—
290C22×C73—41no—
291C22×C73C2×C4×D411yescharacteristic
292C22×C73#55211yescharacteristic
293C22×C73#55411yescharacteristic
294C4×C73—22no—
295C4×C73—21no—
296C4×C73—21no—
297C4×C73—21no—
298C4×C73—21no—
299C4×C73—41no—
300C4×C73—41no—
301C4×C73C2×C4×Q811yescharacteristic
302C4×C73#52011yescharacteristic
303C4×C73#54611yescharacteristic
304C4×C73D4×Q811yescharacteristic
305Dic73—42no—
306Dic73—41no—
307Dic73—41no—
308Dic73—41no—
309Dic73—41no—
310Dic73—81no—
311#22984—731nomaximal, Sylow
312C2×C73—21no—
313C2×C73—21no—
314C2×C73—42no—
315C2×C73#236011yescharacteristic
316C2×C73#295511yescharacteristic
317C2×C73#295711yescharacteristic
318#3076—731no—
319#2962—731no—
320#2906D7311yescharacteristic
321#1995—731no—
322#1950—731no—
323#1946—731no—
324#1945—731no—
325#1922—731no—
326#1922—731no—
327#1866—731no—
328#1866—731no—
329#1768—731no—
330#1765—731no—
331#1764—731no—
332#1759—731no—
333C73#2298411yescharacteristic, complemented, Sylow
334C2×#131C2×D7311yescharacteristic
335C2×#131C2×D7311yescharacteristic
336#533—731no—
337C4×#41C2×D7311yescharacteristic
338#479—731no—
339#479—731no—
340#478—731no—
341#478—731no—
342#477—731no—
343#476—731no—
344#475—731no—
345#474—731no—
346#446—1461no—
347#446—1461no—
348#446—1461no—
349C8×Q8—1461no—
350C2×#108—731no—
351C2×#108—731no—
352#427—731no—
353#427—731no—
354#424—21no—
355#424—21no—
356#424—731no—
357#424—731no—
358#424C2×D7311yescharacteristic
359#424C2×D7311yescharacteristic
360#423—21no—
361#423C2×D7311yescharacteristic
362C2×C4⋊C8—21no—
363C2×C4⋊C8—731no—
364C2×C4⋊C8C2×D7311yescharacteristic
365C2×#104—731no—
366C2×#104—731no—
367#416—731no—
368#416—731no—
369#405—731no—
370#405—731no—
371C4×M16—731no—
372C2×#97—731no—
373C2×C4×C8—731no—
374#376—731no—
375#341—731no—
376#341—731no—
377#339—731no—
378#339—731no—
379#339—731no—
380#339—731no—
381#339—731no—
382#339—731no—
383#336—731no—
384#336—731no—
385C2×#41C22×D7311yescharacteristic
386#131—22no—
387#131—22no—
388#131—41no—
389#131—41no—
390C2×M16—21no—
391C2×M16—21no—
392C2×M16—731no—
393C2×M16—731no—
394C2×M16C22×D7311yescharacteristic
395C2×M16C22×D7311yescharacteristic
396C22×C8—21no—
397C22×C8—21no—
398C22×C8—731no—
399C22×C8—731no—
400C22×C8C22×D7311yescharacteristic
401C22×C8C22×D7311yescharacteristic
402C4⋊Q8—731no—
403C4⋊Q8—731no—
404#125—731no—
405#125—731no—
406C22⋊Q8—1461no—
407C22⋊Q8—1461no—
408C4×Q8—1461no—
409C4×Q8—1461no—
410C4×Q8C4×D7311yescharacteristic
411C4×Q8C4×D7311yescharacteristic
412C4×D4—21no—
413C4×D4—21no—
414C4×D4C4×D7311yescharacteristic
415C4×D4C4×D7311yescharacteristic
416#117—21no—
417#117—731no—
418#117—731no—
419#117C22×D7311yescharacteristic
420C2×C42—21no—
421C2×C42—731no—
422C2×C42—731no—
423C2×C42C22×D7311yescharacteristic
424#108—2921no—
425#108—2921no—
426#107—731no—
427#107—731no—
428#107—731no—
429#107—731no—
430#106—731no—
431#106—731no—
432#106—731no—
433#106—731no—
434C4⋊C8—21no—
435C4⋊C8—21no—
436C4⋊C8—21no—
437C4⋊C8—21no—
438C4⋊C8—21no—
439C4⋊C8—21no—
440C4⋊C8—731no—
441C4⋊C8—731no—
442C4⋊C8—731no—
443C4⋊C8—731no—
444C4⋊C8—1461no—
445C4⋊C8—1461no—
446C4⋊C8—1461no—
447C4⋊C8—1461no—
448C4⋊C8C22⋊D7311yescharacteristic
449C4⋊C8C22⋊D7311yescharacteristic
450C4⋊C8C22⋊D7311yescharacteristic
451C4⋊C8C22⋊D7311yescharacteristic
452#104—734no—
453#104—1462no—
454#103—731no—
455#103—731no—
456#103—731no—
457#103—731no—
458#102—731no—
459#102—731no—
460#102—731no—
461#102—731no—
462#97—2921no—
463#97—1461no—
464#97—1461no—
465#97—1461no—
466C4×C8—2921no—
467C4×C8—1461no—
468C4×C8—1461no—
469C4×C8—1461no—
470#95—1461no—
471#95—1461no—
472#41—21no—
473#41—21no—
474#41Q8×D7312yes—
475#41#1192402712yes—
476C2×Q8—1461no—
477C2×Q8—1461no—
478C2×Q8C2×C4×D7311yescharacteristic
479C2×D4—21no—
480C2×D4C2×C4×D7311yescharacteristic
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—1461no—
485C22×C4—1461no—
486C22×C4#1192402811yescharacteristic
487C22×C4#1192402811yescharacteristic
488C22×C4C23×D7311yescharacteristic
489M16—21no—
490M16—21no—
491M16—21no—
492M16—21no—
493M16—42no—
494M16—42no—
495M16—732no—
496M16—1461no—
497M16—1461no—
498M16—1461no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—42no—
506C2×C8—42no—
507C2×C8—731no—
508C2×C8—731no—
509C2×C8—731no—
510C2×C8—731no—
511C2×C8—1461no—
512C2×C8—1461no—
513C2×C8—1461no—
514C2×C8—1461no—
515C2×C8—1461no—
516C2×C8—1461no—
517C2×C8C2×C22⋊D7311yescharacteristic
518C2×C8C2×C22⋊D7311yescharacteristic
519C2×C8C2×C22⋊D7311yescharacteristic
520C2×C8C2×C22⋊D7311yescharacteristic
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—731no—
524C4⋊C4—731no—
525C4⋊C4—731no—
526C4⋊C4—731no—
527C4⋊C4—1461no—
528C4⋊C4—1461no—
529C4⋊C4—1461no—
530C4⋊C4—1461no—
531C4⋊C4—1461no—
532C4⋊C4—1461no—
533C4⋊C4C2×C4×D7311yescharacteristic
534C4⋊C4C2×C4×D7311yescharacteristic
535C22⋊C4—21no—
536C22⋊C4—21no—
537C22⋊C4—21no—
538C22⋊C4—1461no—
539C22⋊C4—1461no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—732no—
544C42—1461no—
545C42—1461no—
546C42—1461no—
547C42—1461no—
548C42—1461no—
549C42C2×C4×D7311yescharacteristic
550C42C2×C4×D7311yescharacteristic
551C42C2×C22⋊D7311yescharacteristic
552C42C2×C22⋊D7311yescharacteristic
553C23—21no—
554C23C2×#1192402811yescharacteristic
555Q8—21no—
556Q8—2922no—
557Q8—5841no—
558Q8#1455655612yes—
559D4—22no—
560D4—22no—
561D4—21no—
562D4#1455655612yes—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—81no—
572C2×C4—2921no—
573C2×C4—2921no—
574C2×C4—2921no—
575C2×C4—2921no—
576C2×C4—2921no—
577C2×C4—1462no—
578C2×C4—1461no—
579C2×C4—1461no—
580C2×C4—1461no—
581C2×C4—1461no—
582C2×C4C4×C22⋊D7311yescharacteristic
583C2×C4C4×C22⋊D7311yescharacteristic
584C2×C4C4×C22⋊D7311yescharacteristic
585C2×C4C4×C22⋊D7311yescharacteristic
586C2×C4C22×C4×D7311yescharacteristic
587C2×C4C22×C22⋊D7311yescharacteristic
588C2×C4C2×Q8×D7311yescharacteristic
589C2×C4C2×#1192402711yescharacteristic
590C2×C4#1455663611yescharacteristic
591C2×C4#1455663911yescharacteristic
592C8—21no—
593C8—21no—
594C8—21no—
595C8—21no—
596C8—41no—
597C8—41no—
598C8—2921no—
599C8—1462no—
600C8—1461no—
601C8—1461no—
602C8—1461no—
603C8—1461no—
604C22—21no—
605C22—21no—
606C22—22no—
607C22—41no—
608C22—41no—
609C22C2×C4×C22⋊D7311yescharacteristic
610C22#1798927211yescharacteristic
611C22#1798929211yescharacteristic
612C4—22no—
613C4—21no—
614C4—21no—
615C4—21no—
616C4—21no—
617C4—41no—
618C4—41no—
619C4—2922no—
620C4—2921no—
621C4—2921no—
622C4—2921no—
623C4—2921no—
624C4—5841no—
625C4C2×#1455655611yescharacteristic
626C4#1798906811yescharacteristic
627C4#1798923111yescharacteristic
628C4Q8×C22⋊D7311yescharacteristic
629C2—21no—
630C2—21no—
631C2—42no—
632C2#2261293611yescharacteristic
633C2#2261883211yescharacteristic
634C2#2261884611yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#2906C2
socleC22×C73C2×C4×D4
prime radicalC2×#41×C73C23
cosocleC22×C4×C73C24
hypercenter#2906D73
hypocenterC73#22984
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C4⋊C8 ◁ #2906

Complements

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

NH ≅ G/Ntype
2. #22618874C2semidirect
3. #22610911C2semidirect
4. #22610704C2semidirect
5. #22610681C2semidirect
6. #22610676C2semidirect
7. #22610527C2semidirect
8. #22610527C2semidirect
11. #22609790C2semidirect
17. #17988894C4semidirect
18. #17988894C4semidirect
19. #17988887C4semidirect
20. #17988887C4semidirect
21. #17988882C4semidirect
22. #17988877C4semidirect
23. #17988872C4semidirect
24. #17988867C4semidirect
102. #14556518Q8semidirect
333. C73#22984semidirect

Maximal subgroups

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

subgroupindex
#226197192
#226188742
#226109112
#226107042
#226106812
#226106762
#226105272
#226102062
#226097902
#226097732
#226097662
#226097412
C73×#29062
#2298473