Finite Group Databasebeta

← #33144071

Subgroups — #33144071

Order 18688.

734 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22614963C211yesmaximal, characteristic, complemented
2#22612879C211yesmaximal, characteristic
3#22611305C211yesmaximal, characteristic
4#22611303C211yesmaximal, characteristic
5#22611075C211yesmaximal, characteristic
6#22610889C211yesmaximal, characteristic, complemented
7#22610887C211yesmaximal, characteristic, complemented
8#22610693C211yesmaximal, characteristic, complemented
9#22610648C211yesmaximal, characteristic, complemented
10#22610510C211yesmaximal, characteristic, complemented
11#22610507C211yesmaximal, characteristic, complemented
12#22610057C211yesmaximal, characteristic, complemented
13#22609929C211yesmaximal, characteristic
14#102×Dic73C211yesmaximal, characteristic
15C73×#2025C211yesmaximal, characteristic
16C2×#14556582C2211yescharacteristic
17C2×D4×Dic73C2211yescharacteristic
18C2×#14556547—21no—
19C2×#14556523C2211yescharacteristic
20C2×#14556522C2211yescharacteristic
21C2×C4⋊#10596831C2211yescharacteristic
22C2×#14556520C2211yescharacteristic
23C2×#14556520C2211yescharacteristic
24C2×#14556512C2211yescharacteristic
25C2×#14556512C2211yescharacteristic
26C2×C8×Dic73C2211yescharacteristic
27#17988518—21no—
28#17988517—21no—
29#17988490C2211yescharacteristic
30#17988485C2211yescharacteristic
31#17988483C2211yescharacteristic
32#17988482—21no—
33#17988481—21no—
34#17988475—21no—
35#17988456—21no—
36#17988456C2211yescharacteristic
37#17988456C2211yescharacteristic
38#17988445—21no—
39C4⋊C4×Dic73C2211yescharacteristic
40C4⋊C4×Dic73C2211yescharacteristic
41C4×#11924007C2211yescharacteristic
42C4×C4⋊Dic73C2211yescharacteristic
43#17988350C2211yescharacteristic
44#17988350C2211yescharacteristic
45#17988350C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988341C2211yescharacteristic
52C2×#125×C73C2211yescharacteristic
53C2×#102×C73C2211yescharacteristic
54C2×#102×C73C2211yescharacteristic
55#390×C73C2211yescharacteristic
56#340×C73C2211yescharacteristic
57#340×C73C2211yescharacteristic
58#336×C73C2211yescharacteristic
59C23×Dic73—21no—
60C22×#10596831C2311yescharacteristic
61C22×#10596831C2311yescharacteristic
62#14556582—22no—
63#14556582—41no—
64D4×Dic73—22no—
65D4×Dic73Q814yes—
66#14556547—44no—
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×#11924007—21no—
77C2×#11924007—21no—
78C2×#11924007—21no—
79C2×#11924007D411yescharacteristic
80C2×#11924007D411yescharacteristic
81C2×C22⋊Dic73—21no—
82C2×C22⋊Dic73—21no—
83C2×C22⋊Dic73—21no—
84C2×C4×Dic73—21no—
85C2×C4×Dic73—21no—
86C2×C4×Dic73—21no—
87C2×C4×Dic73C2311yescharacteristic
88C2×C4×Dic73C2311yescharacteristic
89#14556523—22no—
90#14556522—22no—
91C4⋊#10596831—41no—
92#14556520—22no—
93#14556520—41no—
94#14556512—22no—
95#14556512—41no—
96C8×Dic73—22no—
97#14556498—21no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102#14556498D411yescharacteristic
103#14556498D411yescharacteristic
104C22×D4×C73C2311yescharacteristic
105C22×C8×C73C2311yescharacteristic
106C22×C8×C73C2311yescharacteristic
107#125×C73—22no—
108#125×C73—41no—
109C2×C4⋊C4×C73—21no—
110C2×C4⋊C4×C73—21no—
111C2×C4⋊C4×C73C2311yescharacteristic
112C2×C4⋊C4×C73C2311yescharacteristic
113C2×C4⋊C4×C73C2311yescharacteristic
114C2×C22⋊C4×C73—21no—
115C2×C22⋊C4×C73—21no—
116C2×C42×C73C2311yescharacteristic
117#102×C73—22no—
118#102×C73—41no—
119C22×Dic73—21no—
120C22×Dic73—21no—
121C22×Dic73—24no—
122C22×Dic73—21no—
123C22×Dic73—21no—
124C22×Dic73—42no—
125C22×Dic73—42no—
126C22×Dic73C2×D411yescharacteristic
127C22×Dic73C2×D411yescharacteristic
128C22×Dic73C2×D411yescharacteristic
129C22×Dic73C2×D411yescharacteristic
130C2×#10596831—21no—
131C2×#10596831—21no—
132C2×#10596831—21no—
133C2×#10596831—21no—
134C2×#10596831—21no—
135C2×#10596831—21no—
136C4⋊Dic73—21no—
137C4⋊Dic73—21no—
138C4⋊Dic73—21no—
139C4⋊Dic73—21no—
140C4⋊Dic73—21no—
141C4⋊Dic73—21no—
142C4⋊Dic73—42no—
143C4⋊Dic73C2×Q812yes—
144C4⋊Dic73#4112yes—
145#11924007—22no—
146#11924007—22no—
147#11924007—42no—
148#11924007—42no—
149#11924007—42no—
150#11924007—81no—
151#11924007—81no—
152#11924007—81no—
153#11924007—81no—
154C22⋊Dic73—24no—
155C22⋊Dic73—42no—
156C22⋊Dic73—81no—
157C4×Dic73—21no—
158C4×Dic73—21no—
159C4×Dic73—42no—
160C4×Dic73—42no—
161C4×Dic73—42no—
162C4×Dic73C2×Q812yes—
163C4×Dic73#4112yes—
164C24×C73—21no—
165C2×D4×C73—21no—
166C2×D4×C73—21no—
167C2×D4×C73—21no—
168C2×D4×C73—21no—
169C2×D4×C73—21no—
170C2×D4×C73C2×Q812yes—
171C22×C4×C73—21no—
172C22×C4×C73—21no—
173C22×C4×C73—21no—
174C22×C4×C73C2×D411yescharacteristic
175C22×C4×C73C2×D411yescharacteristic
176C22×C4×C73C2411yescharacteristic
177C2×C8×C73—21no—
178C2×C8×C73—21no—
179C2×C8×C73—21no—
180C2×C8×C73—21no—
181C2×C8×C73—21no—
182C2×C8×C73—21no—
183C4⋊C4×C73—21no—
184C4⋊C4×C73—21no—
185C4⋊C4×C73—21no—
186C4⋊C4×C73—21no—
187C4⋊C4×C73—21no—
188C4⋊C4×C73—42no—
189C4⋊C4×C73—42no—
190C4⋊C4×C73#4112yes—
191C22⋊C4×C73—42no—
192C22⋊C4×C73—81no—
193C42×C73—21no—
194C42×C73—21no—
195C23×C73—22no—
196C23×C73—41no—
197C23×C73—41no—
198C23×C73—41no—
199C23×C73—41no—
200C23×C73—41no—
201C23×C73—41no—
202C23×C73C22×D411yescharacteristic
203D4×C73—22no—
204D4×C73—22no—
205D4×C73—41no—
206D4×C73—41no—
207C2×C4×C73—21no—
208C2×C4×C73—21no—
209C2×C4×C73—22no—
210C2×C4×C73—21no—
211C2×C4×C73—21no—
212C2×C4×C73—21no—
213C2×C4×C73—21no—
214C2×C4×C73—41no—
215C2×C4×C73—41no—
216C2×C4×C73—41no—
217C2×C4×C73—41no—
218C2×C4×C73—41no—
219C2×C4×C73—41no—
220C2×C4×C73—41no—
221C2×C4×C73—41no—
222C2×C4×C73C22×Q811yescharacteristic
223C2×C4×C73C2×#4111yescharacteristic
224C2×C4×C73C2×#4111yescharacteristic
225C2×C4×C73C2×#4111yescharacteristic
226C2×C4×C73H(5,𝔽2)11yescharacteristic
227C2×C4×C732−1+411yescharacteristic
228C2×Dic73—21no—
229C2×Dic73—21no—
230C2×Dic73—22no—
231C2×Dic73—21no—
232C2×Dic73—21no—
233C2×Dic73—22no—
234C2×Dic73—21no—
235C2×Dic73—21no—
236C2×Dic73—21no—
237C2×Dic73—21no—
238C2×Dic73—41no—
239C2×Dic73—41no—
240C2×Dic73—41no—
241C2×Dic73—41no—
242C2×Dic73—48no—
243C2×Dic73—41no—
244C2×Dic73—41no—
245C2×Dic73—41no—
246C2×Dic73—41no—
247C2×Dic73—41no—
248C2×Dic73—41no—
249C2×Dic73C22⋊Q812yes—
250C2×Dic73C22⋊Q812yes—
251C2×Dic73#13512yes—
252C2×Dic73AGL(1,ℤ/8)12yes—
253C8×C73—41no—
254C8×C73—41no—
255#10596831—41no—
256#10596831—41no—
257C22×C73—42no—
258C22×C73—42no—
259C22×C73—42no—
260C22×C73—81no—
261C22×C73—81no—
262C22×C73—81no—
263C22×C73—81no—
264C22×C73C2×C22⋊Q811yescharacteristic
265C22×C73#54611yescharacteristic
266C22×C73#54811yescharacteristic
267C22×C73C2×#13511yescharacteristic
268C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
269C22×C73#57711yescharacteristic
270C22×C73#57711yescharacteristic
271C4×C73—41no—
272C4×C73—41no—
273C4×C73—42no—
274C4×C73—81no—
275C4×C73—81no—
276C4×C73#53311yescharacteristic
277C4×C73#54411yescharacteristic
278C4×C73#55511yescharacteristic
279C4×C73#55611yescharacteristic
280Dic73—22no—
281Dic73—42no—
282Dic73—41no—
283Dic73—42no—
284Dic73—41no—
285Dic73—81no—
286Dic73—81no—
287Dic73#47814yes—
288#26021—731nomaximal, Sylow
289C2×C73—82no—
290C2×C73#243911yescharacteristic
291C2×C73C2×#47811yescharacteristic
292C2×C73#308411yescharacteristic
293C2×C73#330311yescharacteristic
294C2×C73#333211yescharacteristic
295C2×C73#336011yescharacteristic
296C2×C73#336811yescharacteristic
297#2552—731no—
298#2354—731no—
299#2065—731no—
300#2065—731no—
301#2025—731no—
302#2025D7311yescharacteristic
303#1989—731no—
304#1989—731no—
305#1948—731no—
306#1939—731no—
307#1919—731no—
308#1919—731no—
309#1834—731no—
310#1802—731no—
311C4×#102—731no—
312C73#2602111yescharacteristic, complemented, Sylow
313C2×#125—731no—
314C2×#125C2×D7311yescharacteristic
315C2×C4×D4—731no—
316C2×#117—1461no—
317C2×#106—731no—
318C2×#106—731no—
319C2×C4⋊C8—731no—
320C2×C4⋊C8—731no—
321C2×C4⋊C8—731no—
322C2×#102—731no—
323C2×#102—731no—
324C2×#102C2×D7311yescharacteristic
325C2×#102C2×D7311yescharacteristic
326C2×C4×C8—731no—
327#398—1461no—
328#398—1461no—
329#390—731no—
330#390C2×D7311yescharacteristic
331#389—731no—
332#389—731no—
333#389—1461no—
334#388—1461no—
335#388—1461no—
336#384—731no—
337#384—731no—
338#384—1461no—
339#380—1461no—
340C4×C4⋊C4—731no—
341C4×C4⋊C4—731no—
342C4×C4⋊C4—731no—
343C4×C4⋊C4—731no—
344#340—731no—
345#340—731no—
346#340—731no—
347#340—731no—
348#340—731no—
349#340—731no—
350#340—731no—
351#340—731no—
352#340C2×D7311yescharacteristic
353#340C2×D7311yescharacteristic
354#336—731no—
355#336C2×D7311yescharacteristic
356C22×D4C22×D7311yescharacteristic
357C23×C4—1461no—
358C22×C8—731no—
359C22×C8—731no—
360C22×C8C22×D7311yescharacteristic
361C22×C8C22×D7311yescharacteristic
362#125—22no—
363#125—41no—
364#125—2921no—
365#125—1462no—
366C4×D4—734no—
367C4×D4—1462no—
368#117—2924no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—731no—
372C2×C4⋊C4—731no—
373C2×C4⋊C4—731no—
374C2×C4⋊C4—731no—
375C2×C4⋊C4—731no—
376C2×C4⋊C4—731no—
377C2×C4⋊C4—1461no—
378C2×C4⋊C4—1461no—
379C2×C4⋊C4—1461no—
380C2×C4⋊C4—1461no—
381C2×C4⋊C4—1461no—
382C2×C4⋊C4—1461no—
383C2×C4⋊C4—1461no—
384C2×C4⋊C4—1461no—
385C2×C4⋊C4C22×D7311yescharacteristic
386C2×C4⋊C4C22×D7311yescharacteristic
387C2×C4⋊C4C22×D7311yescharacteristic
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—1461no—
391C2×C22⋊C4—1461no—
392C2×C22⋊C4—1461no—
393C2×C42—731no—
394C2×C42—731no—
395C2×C42—1461no—
396C2×C42—1461no—
397C2×C42—1461no—
398C2×C42C22×D7311yescharacteristic
399#106—1462no—
400#106—1462no—
401C4⋊C8—2921no—
402C4⋊C8—2921no—
403C4⋊C8—1462no—
404#102—22no—
405#102—41no—
406#102—2921no—
407#102—1462no—
408C4×C8—1462no—
409#95—731no—
410#95—731no—
411#95—1461no—
412#95—1461no—
413#95—1461no—
414#95—1461no—
415#95—1461no—
416C24—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4Q8×D7312yes—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—2922no—
427C22×C4—2922no—
428C22×C4—731no—
429C22×C4—731no—
430C22×C4—731no—
431C22×C4—731no—
432C22×C4—1461no—
433C22×C4—1461no—
434C22×C4—1464no—
435C22×C4—1461no—
436C22×C4—1461no—
437C22×C4D4×D7311yescharacteristic
438C22×C4D4×D7311yescharacteristic
439C22×C4C23×D7311yescharacteristic
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—1461no—
447C2×C8—1461no—
448C2×C8—1461no—
449C2×C8—1461no—
450C2×C8—1461no—
451C2×C8—1461no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—2922no—
460C4⋊C4—2922no—
461C4⋊C4—2922no—
462C4⋊C4—2922no—
463C4⋊C4—5841no—
464C4⋊C4—5841no—
465C4⋊C4—5841no—
466C4⋊C4—5841no—
467C4⋊C4—732no—
468C4⋊C4—732no—
469C4⋊C4—1462no—
470C4⋊C4—1462no—
471C4⋊C4—1461no—
472C4⋊C4—1461no—
473C4⋊C4—1461no—
474C4⋊C4—1461no—
475C4⋊C4—1461no—
476C4⋊C4—1461no—
477C4⋊C4#1192402712yes—
478C22⋊C4—42no—
479C22⋊C4—81no—
480C22⋊C4—2922no—
481C22⋊C4—5841no—
482C22⋊C4—1464no—
483C42—21no—
484C42—21no—
485C42—2922no—
486C42—2922no—
487C42—2922no—
488C42—732no—
489C42—732no—
490C42—1461no—
491C42—1461no—
492C23—22no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23C2×D4×D7311yescharacteristic
500D4—22no—
501D4—22no—
502D4—41no—
503D4—41no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—2921no—
520C2×C4—2921no—
521C2×C4—2921no—
522C2×C4—2921no—
523C2×C4—2928no—
524C2×C4—2921no—
525C2×C4—2921no—
526C2×C4—2921no—
527C2×C4—2921no—
528C2×C4—2921no—
529C2×C4—2921no—
530C2×C4—732no—
531C2×C4—732no—
532C2×C4—732no—
533C2×C4—732no—
534C2×C4—1461no—
535C2×C4—1461no—
536C2×C4—1461no—
537C2×C4—1461no—
538C2×C4—1462no—
539C2×C4—1462no—
540C2×C4—1461no—
541C2×C4—1461no—
542C2×C4—1461no—
543C2×C4—1461no—
544C2×C4C2×Q8×D7311yescharacteristic
545C2×C4#41×D7311yescharacteristic
546C2×C4#41×D7311yescharacteristic
547C2×C4C2×#1192402711yescharacteristic
548C2×C4#1455663611yescharacteristic
549C2×C4#1455663811yescharacteristic
550C8—41no—
551C8—41no—
552C8—2921no—
553C8—2921no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—81no—
558C22—81no—
559C22—81no—
560C22—81no—
561C22C22⋊Q8×D7311yescharacteristic
562C22#1798922411yescharacteristic
563C22#1798924511yescharacteristic
564C22#135×D7311yescharacteristic
565C22AGL(1,ℤ/8)×D7311yescharacteristic
566C22#1798939411yescharacteristic
567C22#1798939711yescharacteristic
568C4—41no—
569C4—41no—
570C4—42no—
571C4—81no—
572C4—81no—
573C4—2922no—
574C4—2921no—
575C4—2922no—
576C4—2921no—
577C4—5841no—
578C4—5841no—
579C4—734no—
580C4—1462no—
581C4#1798913911yescharacteristic
582C4#1798921611yescharacteristic
583C4#1798929911yescharacteristic
584C4#1798930611yescharacteristic
585C2—82no—
586C2#2261361511yescharacteristic
587C2#478×D7311yescharacteristic
588C2#2261979411yescharacteristic
589C2#2262185211yescharacteristic
590C2#2262225111yescharacteristic
591C2#2262257711yescharacteristic
592C2#2262265711yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2025C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
hypercenter#2025D73
hypocenterC73#26021
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2025

Complements

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

NH ≅ G/Ntype
1. #22614963C2semidirect
6. #22610889C2semidirect
7. #22610887C2semidirect
8. #22610693C2semidirect
9. #22610648C2semidirect
10. #22610510C2semidirect
11. #22610507C2semidirect
12. #22610057C2semidirect
312. C73#26021semidirect

Maximal subgroups

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

subgroupindex
#226149632
#226128792
#226113052
#226113032
#226110752
#226108892
#226108872
#226106932
#226106482
#226105102
#226105072
#226100572
#226099292
#102×Dic732
C73×#20252
#2602173