Finite Group Databasebeta

← #33127351

Subgroups — #33127351

Order 18688.

746 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×#17988882C211yesmaximal, characteristic, complemented
2C2×#17988648C211yesmaximal, characteristic
3#22615752C211yesmaximal, characteristic
4#22611362C211yesmaximal, characteristic
5#22611357C211yesmaximal, characteristic
6#22611131C211yesmaximal, characteristic
7#22611124C211yesmaximal, characteristic
8#22610883C211yesmaximal, characteristic, complemented
9#22610699C211yesmaximal, characteristic, complemented
10#22610671C211yesmaximal, characteristic, complemented
11#22610661C211yesmaximal, characteristic, complemented
12#22610624C211yesmaximal, characteristic, complemented
13#22610610C211yesmaximal, characteristic, complemented
14#22609946C211yesmaximal, characteristic, complemented
15C73×#2323C211yesmaximal, characteristic
16C2×#14556591C2211yescharacteristic
17#17988882—24no—
18#17988648—24no—
19C2×#14556524C2211yescharacteristic
20C2×#14556524C2211yescharacteristic
21C2×#14556524C2211yescharacteristic
22C2×#14556522C2211yescharacteristic
23C2×C4⋊#10596831C2211yescharacteristic
24C2×C4⋊#10596831C2211yescharacteristic
25C2×#14556515C2211yescharacteristic
26C2×#14556515C2211yescharacteristic
27C2×#14556515C2211yescharacteristic
28C2×#14556515C2211yescharacteristic
29C2×C22⋊#10596831C2211yescharacteristic
30C2×C22⋊#10596831C2211yescharacteristic
31C2×#14556502C2211yescharacteristic
32C2×C4×#10596831C2211yescharacteristic
33#17988526—21no—
34#17988523—21no—
35#17988507—21no—
36#17988499—21no—
37#17988485C2211yescharacteristic
38#17988485C2211yescharacteristic
39#17988453C2211yescharacteristic
40#17988447C2211yescharacteristic
41#17988447C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50C2×C22⋊Q8×C73C2211yescharacteristic
51C2×C4×Q8×C73C2211yescharacteristic
52C2×#117×C73C2211yescharacteristic
53#391×C73C2211yescharacteristic
54#389×C73—21no—
55#389×C73C2211yescharacteristic
56#388×C73—21no—
57#388×C73—21no—
58#387×C73—21no—
59#380×C73C2211yescharacteristic
60C4×C4⋊C4×C73C2211yescharacteristic
61C22×Dic146—21no—
62C22×#10596831C2311yescharacteristic
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65C22×#10596831C2311yescharacteristic
66#14556591—22no—
67#14556591—22no—
68C2×C4⋊Dic73—21no—
69C2×C4⋊Dic73C2311yescharacteristic
70C2×C4⋊Dic73C2311yescharacteristic
71C2×C4⋊Dic73C2311yescharacteristic
72C2×C4⋊Dic73C2311yescharacteristic
73C2×#11924007—21no—
74C2×#11924007—21no—
75C2×C22⋊Dic73—21no—
76C2×C22⋊Dic73—21no—
77#14556524—22no—
78#14556524—22no—
79#14556524—22no—
80#14556522—22no—
81C4⋊#10596831—22no—
82C4⋊#10596831D414yes—
83#14556515—22no—
84#14556515—22no—
85#14556515—22no—
86#14556515—22no—
87C22⋊#10596831—22no—
88C22⋊#10596831—22no—
89#14556502—22no—
90C4×#10596831—22no—
91#14556498—21no—
92#14556498—21no—
93#14556498—21no—
94#14556498—21no—
95#14556498—21no—
96#14556498—21no—
97C22×Q8×C73C2311yescharacteristic
98C23×C4×C73C2311yescharacteristic
99C22⋊Q8×C73—22no—
100C22⋊Q8×C73—22no—
101C4×Q8×C73—22no—
102C4×Q8×C73—22no—
103#117×C73—22no—
104#117×C73—22no—
105C2×C4⋊C4×C73—21no—
106C2×C4⋊C4×C73—21no—
107C2×C4⋊C4×C73—21no—
108C2×C4⋊C4×C73D411yescharacteristic
109C2×C4⋊C4×C73D411yescharacteristic
110C2×C4⋊C4×C73C2311yescharacteristic
111C2×C4⋊C4×C73C2311yescharacteristic
112C2×C4⋊C4×C73C2311yescharacteristic
113C2×C22⋊C4×C73—21no—
114C2×C22⋊C4×C73—21no—
115C2×C22⋊C4×C73D411yescharacteristic
116C2×C22⋊C4×C73D411yescharacteristic
117C2×C42×C73—21no—
118C2×C42×C73—21no—
119C2×C42×C73C2311yescharacteristic
120C2×C42×C73C2311yescharacteristic
121#95×C73—21no—
122#95×C73—21no—
123#95×C73—21no—
124#95×C73—21no—
125#95×C73—21no—
126C2×Dic146—42no—
127C2×Dic146—81no—
128C2×Dic146—81no—
129C22×Dic73—21no—
130C22×Dic73—21no—
131C22×Dic73—21no—
132C22×Dic73—21no—
133C2×#10596831—21no—
134C2×#10596831—21no—
135C2×#10596831—21no—
136C2×#10596831—21no—
137C2×#10596831—21no—
138C2×#10596831—21no—
139C2×#10596831—21no—
140C2×#10596831—21no—
141C2×#10596831—21no—
142C2×#10596831—21no—
143C2×#10596831C2×D412yes—
144C2×#10596831C2×D412yes—
145C4⋊Dic73—21no—
146C4⋊Dic73—21no—
147C4⋊Dic73—21no—
148C4⋊Dic73—21no—
149C4⋊Dic73—21no—
150C4⋊Dic73—21no—
151C4⋊Dic73—21no—
152C4⋊Dic73—21no—
153C4⋊Dic73—42no—
154#11924007—81no—
155#11924007—81no—
156C22⋊Dic73—81no—
157C22⋊Dic73—81no—
158C24×C73C2×D411yescharacteristic
159C2×Q8×C73—21no—
160C2×Q8×C73—21no—
161C2×Q8×C73—21no—
162C2×Q8×C73—21no—
163C2×Q8×C73—21no—
164C2×Q8×C73#4112yes—
165C22×C4×C73—21no—
166C22×C4×C73—21no—
167C22×C4×C73—21no—
168C22×C4×C73—21no—
169C22×C4×C73—21no—
170C22×C4×C73—21no—
171C22×C4×C73—21no—
172C22×C4×C73—21no—
173C22×C4×C73—21no—
174C22×C4×C73—21no—
175C22×C4×C73C2×D411yescharacteristic
176C22×C4×C73C2×D411yescharacteristic
177C22×C4×C73C2×D411yescharacteristic
178C22×C4×C73C2×D411yescharacteristic
179C22×C4×C73C2×D411yescharacteristic
180C22×C4×C73C2411yescharacteristic
181C4⋊C4×C73—24no—
182C4⋊C4×C73—22no—
183C4⋊C4×C73—22no—
184C4⋊C4×C73—21no—
185C4⋊C4×C73—21no—
186C4⋊C4×C73—21no—
187C4⋊C4×C73—21no—
188C4⋊C4×C73—21no—
189C4⋊C4×C73—42no—
190C4⋊C4×C73—81no—
191C4⋊C4×C73#4112yes—
192C22⋊C4×C73—24no—
193C22⋊C4×C73—22no—
194C22⋊C4×C73—22no—
195C22⋊C4×C73—81no—
196C42×C73—21no—
197C42×C73—21no—
198C42×C73—42no—
199C42×C73—81no—
200C42×C73C2×D412yes—
201C42×C73#4112yes—
202C23×C73—21no—
203C23×C73—21no—
204C23×C73—21no—
205C23×C73—21no—
206C23×C73—21no—
207C23×C73—21no—
208C23×C73—21no—
209C23×C73C22×D411yescharacteristic
210Q8×C73—22no—
211Q8×C73—22no—
212Q8×C73—41no—
213Q8×C73—41no—
214C2×C4×C73—21no—
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—22no—
222C2×C4×C73—22no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—21no—
228C2×C4×C73—21no—
229C2×C4×C73—41no—
230C2×C4×C73—41no—
231C2×C4×C73—41no—
232C2×C4×C73—41no—
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—41no—
242C2×C4×C73—41no—
243C2×C4×C73#12112yes—
244C2×C4×C73#12112yes—
245C2×C4×C73C22×D411yescharacteristic
246C2×C4×C73C2×#4111yescharacteristic
247C2×C4×C73C2×#4111yescharacteristic
248C2×C4×C73C2×#4111yescharacteristic
249C2×C4×C732−1+411yescharacteristic
250C2×C4×C732−1+411yescharacteristic
251C2×Dic73—41no—
252C2×Dic73—41no—
253C2×Dic73—41no—
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—
263Dic146—84no—
264#10596831—22no—
265#10596831—22no—
266#10596831—41no—
267#10596831—41no—
268C22×C73—41no—
269C22×C73—41no—
270C22×C73—41no—
271C22×C73—41no—
272C22×C73—41no—
273C22×C73—41no—
274C22×C73—41no—
275C22×C73C2×#12111yescharacteristic
276C22×C73#52511yescharacteristic
277C22×C73#53611yescharacteristic
278C22×C73#57611yescharacteristic
279C22×C73#57711yescharacteristic
280C22×C73#57811yescharacteristic
281C22×C73#57811yescharacteristic
282C4×C73—22no—
283C4×C73—22no—
284C4×C73—42no—
285C4×C73—41no—
286C4×C73—42no—
287C4×C73—41no—
288C4×C73—41no—
289C4×C73—81no—
290C4×C73—81no—
291C4×C73#54711yescharacteristic
292C4×C73#54811yescharacteristic
293C4×C73#55211yescharacteristic
294C4×C73#55611yescharacteristic
295Dic73—81no—
296Dic73—81no—
297Dic73—81no—
298Dic73—81no—
299#24714—731nomaximal, Sylow
300C2×C73—81no—
301C2×C73#241511yescharacteristic
302C2×C73#305911yescharacteristic
303C2×C73#306211yescharacteristic
304C2×C73#312111yescharacteristic
305C2×C73#312411yescharacteristic
306C2×C73#324211yescharacteristic
307C2×C73#324311yescharacteristic
308C2×#477—731no—
309C2×#433—731no—
310#2628—731no—
311#2323D7311yescharacteristic
312#2076—731no—
313#2075—731no—
314#2036—731no—
315#2035—731no—
316#1988—731no—
317#1949—731no—
318#1944—731no—
319#1942—731no—
320#1935—731no—
321#1933—731no—
322#1805—731no—
323C73#2471411yescharacteristic, complemented, Sylow
324C2×C4⋊Q8—731no—
325C2×C22⋊Q8C2×D7311yescharacteristic
326C2×C4×Q8C2×D7311yescharacteristic
327C2×#117C2×D7311yescharacteristic
328#477—1464no—
329#433—1464no—
330C2×#107—731no—
331C2×#107—731no—
332C2×#107—731no—
333C2×#106—731no—
334C2×C4⋊C8—731no—
335C2×C4⋊C8—731no—
336C2×#103—731no—
337C2×#103—731no—
338C2×#103—731no—
339C2×#103—731no—
340C2×C22⋊C8—731no—
341C2×C22⋊C8—731no—
342C2×#97—731no—
343C2×C4×C8—731no—
344#400—1461no—
345#399—1461no—
346#395—1461no—
347#393—1461no—
348#391C2×D7311yescharacteristic
349#389—21no—
350#389—731no—
351#389—731no—
352#389C2×D7311yescharacteristic
353#388—21no—
354#388—21no—
355#387—21no—
356#382—731no—
357#380—731no—
358#380—731no—
359#380C2×D7311yescharacteristic
360C4×C4⋊C4C2×D7311yescharacteristic
361#340—731no—
362#340—731no—
363#340—731no—
364#340—731no—
365#340—731no—
366#340—731no—
367#340—731no—
368#340—731no—
369C22×Q8—1461no—
370C22×Q8C22×D7311yescharacteristic
371C23×C4C22×D7311yescharacteristic
372C22×C8—731no—
373C22×C8—731no—
374C22×C8—731no—
375C22×C8—731no—
376C4⋊Q8—1462no—
377C4⋊Q8—1462no—
378C22⋊Q8—22no—
379C22⋊Q8—22no—
380C4×Q8—22no—
381C4×Q8—22no—
382#117—22no—
383#117—22no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—731no—
388C2×C4⋊C4—731no—
389C2×C4⋊C4—731no—
390C2×C4⋊C4—731no—
391C2×C4⋊C4—1461no—
392C2×C4⋊C4—1461no—
393C2×C4⋊C4—1461no—
394C2×C4⋊C4C22⋊D7311yescharacteristic
395C2×C4⋊C4C22⋊D7311yescharacteristic
396C2×C4⋊C4C22×D7311yescharacteristic
397C2×C4⋊C4C22×D7311yescharacteristic
398C2×C4⋊C4C22×D7311yescharacteristic
399C2×C22⋊C4—21no—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—1461no—
402C2×C22⋊C4—1461no—
403C2×C22⋊C4C22⋊D7311yescharacteristic
404C2×C22⋊C4C22⋊D7311yescharacteristic
405C2×C42—21no—
406C2×C42—21no—
407C2×C42C22×D7311yescharacteristic
408C2×C42C22×D7311yescharacteristic
409#107—1462no—
410#107—1462no—
411#107—1462no—
412#106—1462no—
413C4⋊C8—734no—
414C4⋊C8—1462no—
415#103—1462no—
416#103—1462no—
417#103—1462no—
418#103—1462no—
419C22⋊C8—1462no—
420C22⋊C8—1462no—
421#97—1462no—
422C4×C8—1462no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95—21no—
428#95—1461no—
429#95—1461no—
430#95—1461no—
431#95—1461no—
432#95—1461no—
433#95—1461no—
434C24C2×C22⋊D7311yescharacteristic
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—2922no—
441C2×Q8—5841no—
442C2×Q8—5841no—
443C2×Q8#1192402812yes—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—1461no—
455C22×C4—1461no—
456C22×C4—1461no—
457C22×C4—1461no—
458C22×C4C2×C22⋊D7311yescharacteristic
459C22×C4C2×C22⋊D7311yescharacteristic
460C22×C4C2×C22⋊D7311yescharacteristic
461C22×C4C2×C22⋊D7311yescharacteristic
462C22×C4C2×C22⋊D7311yescharacteristic
463C22×C4C23×D7311yescharacteristic
464C2×C8—732no—
465C2×C8—732no—
466C2×C8—1461no—
467C2×C8—1461no—
468C2×C8—1461no—
469C2×C8—1461no—
470C2×C8—1461no—
471C2×C8—1461no—
472C2×C8—1461no—
473C2×C8—1461no—
474C2×C8—1461no—
475C2×C8—1461no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—42no—
485C4⋊C4—81no—
486C4⋊C4—2922no—
487C4⋊C4—5841no—
488C4⋊C4—5841no—
489C4⋊C4—1461no—
490C4⋊C4—1461no—
491C4⋊C4—1461no—
492C4⋊C4—1461no—
493C4⋊C4—1461no—
494C4⋊C4—1461no—
495C4⋊C4—1461no—
496C4⋊C4—1461no—
497C4⋊C4#1192402812yes—
498C22⋊C4—24no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—81no—
502C22⋊C4—5841no—
503C22⋊C4—5841no—
504C42—21no—
505C42—21no—
506C42—42no—
507C42—81no—
508C42D4×D7312yes—
509C42#1192402712yes—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23C22×C22⋊D7311yescharacteristic
518Q8—22no—
519Q8—22no—
520Q8—41no—
521Q8—41no—
522Q8—5844no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
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—22no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
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—41no—
550C2×C4—41no—
551C2×C4—41no—
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—2921no—
561C2×C4—2921no—
562C2×C4—2921no—
563C2×C4—2921no—
564C2×C4#1455656312yes—
565C2×C4#1455656312yes—
566C2×C4C2×D4×D7311yescharacteristic
567C2×C4C2×#1192402711yescharacteristic
568C2×C4C2×#1192402811yescharacteristic
569C2×C4C2×#1192402811yescharacteristic
570C2×C4#1455663811yescharacteristic
571C2×C4#1455663811yescharacteristic
572C8—2921no—
573C8—2921no—
574C8—1462no—
575C8—1462no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22C2×#1455656311yescharacteristic
584C22#1798909811yescharacteristic
585C22#1798915811yescharacteristic
586C22#1798939011yescharacteristic
587C22#1798939511yescharacteristic
588C22#1798940211yescharacteristic
589C22#1798940211yescharacteristic
590C4—22no—
591C4—22no—
592C4—42no—
593C4—41no—
594C4—42no—
595C4—41no—
596C4—41no—
597C4—81no—
598C4—81no—
599C4—5841no—
600C4—5841no—
601C4—5841no—
602C4—5841no—
603C4#1798924211yescharacteristic
604C4#1798924611yescharacteristic
605C4#1798927711yescharacteristic
606C4#1798930711yescharacteristic
607C2—81no—
608C2#2261336011yescharacteristic
609C2#2261958311yescharacteristic
610C2#2261961311yescharacteristic
611C2#2262016011yescharacteristic
612C2#2262019311yescharacteristic
613C2#2262119311yescharacteristic
614C2#2262120811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2323C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
hypercenter#2323D73
hypocenterC73#24714
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2323

Complements

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

NH ≅ G/Ntype
1. C2×#17988882C2semidirect
8. #22610883C2semidirect
9. #22610699C2semidirect
10. #22610671C2semidirect
11. #22610661C2semidirect
12. #22610624C2semidirect
13. #22610610C2semidirect
14. #22609946C2semidirect
323. C73#24714semidirect

Maximal subgroups

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

subgroupindex
C2×#179888822
C2×#179886482
#226157522
#226113622
#226113572
#226111312
#226111242
#226108832
#226106992
#226106712
#226106612
#226106242
#226106102
#226099462
C73×#23232
#2471473