Finite Group Databasebeta

← #33144283

Subgroups — #33144283

Order 18688.

692 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22614963C211yesmaximal, characteristic, complemented
2#22612903C211yesmaximal, characteristic
3#22611317C211yesmaximal, characteristic
4#22611305C211yesmaximal, characteristic
5#22611075C211yesmaximal, characteristic
6#22610889C211yesmaximal, characteristic, complemented
7#22610885C211yesmaximal, characteristic, complemented
8#22610702C211yesmaximal, characteristic, complemented
9#22610653C211yesmaximal, characteristic, complemented
10#22610514C211yesmaximal, characteristic, complemented
11#22610503C211yesmaximal, characteristic, complemented
12#22610063C211yesmaximal, characteristic, complemented
13#22609929C211yesmaximal, characteristic
14#22609758C211yesmaximal, characteristic
15C73×#2025C211yesmaximal, characteristic
16C2×#14556582C2211yescharacteristic
17C2×D4×Dic73C2211yescharacteristic
18C2×#14556525C2211yescharacteristic
19C2×#14556522C2211yescharacteristic
20C2×C4⋊#10596831C2211yescharacteristic
21C2×#14556520C2211yescharacteristic
22C2×#14556520C2211yescharacteristic
23C2×#14556512C2211yescharacteristic
24C2×#14556512C2211yescharacteristic
25C2×#14556501C2211yescharacteristic
26#17988518—21no—
27#17988517—21no—
28#17988490C2211yescharacteristic
29#17988485C2211yescharacteristic
30#17988483C2211yescharacteristic
31#17988482—21no—
32#17988481—21no—
33#17988475—21no—
34#17988456—21no—
35#17988456C2211yescharacteristic
36#17988452C2211yescharacteristic
37#17988448—21no—
38C4⋊C4×Dic73C2211yescharacteristic
39C4⋊C4×Dic73C2211yescharacteristic
40C4×#11924007C2211yescharacteristic
41C4×C4⋊Dic73C2211yescharacteristic
42C22⋊C4×Dic73—21no—
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—
66C2×C4⋊Dic73—21no—
67C2×C4⋊Dic73C2311yescharacteristic
68C2×C4⋊Dic73C2311yescharacteristic
69C2×C4⋊Dic73C2311yescharacteristic
70C2×C4⋊Dic73C2311yescharacteristic
71C2×#11924007—21no—
72C2×#11924007—21no—
73C2×#11924007—21no—
74C2×#11924007—21no—
75C2×#11924007—21no—
76C2×#11924007D411yescharacteristic
77C2×#11924007D411yescharacteristic
78C2×C22⋊Dic73—21no—
79C2×C22⋊Dic73—21no—
80C2×C22⋊Dic73—21no—
81C2×C4×Dic73—21no—
82C2×C4×Dic73—21no—
83C2×C4×Dic73—21no—
84C2×C4×Dic73C2311yescharacteristic
85C2×C4×Dic73C2311yescharacteristic
86#14556525—22no—
87#14556522—22no—
88C4⋊#10596831—41no—
89#14556520—22no—
90#14556520—41no—
91#14556512—22no—
92#14556512—41no—
93#14556501—22no—
94#14556498—21no—
95#14556498—21no—
96#14556498—21no—
97#14556498—21no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101#14556498D411yescharacteristic
102#14556498D411yescharacteristic
103C22×D4×C73C2311yescharacteristic
104C22×C8×C73C2311yescharacteristic
105C22×C8×C73C2311yescharacteristic
106#125×C73—22no—
107#125×C73—41no—
108C2×C4⋊C4×C73—21no—
109C2×C4⋊C4×C73—21no—
110C2×C4⋊C4×C73C2311yescharacteristic
111C2×C4⋊C4×C73C2311yescharacteristic
112C2×C4⋊C4×C73C2311yescharacteristic
113C2×C22⋊C4×C73—21no—
114C2×C22⋊C4×C73—21no—
115C2×C42×C73C2311yescharacteristic
116#102×C73—22no—
117#102×C73—41no—
118C22×Dic73—21no—
119C22×Dic73—21no—
120C22×Dic73—24no—
121C22×Dic73—21no—
122C22×Dic73—21no—
123C22×Dic73—81no—
124C22×Dic73—81no—
125C22×Dic73C2×D411yescharacteristic
126C22×Dic73C2×D411yescharacteristic
127C22×Dic73C2×D411yescharacteristic
128C22×Dic73C2×D411yescharacteristic
129C2×#10596831—21no—
130C2×#10596831—21no—
131C2×#10596831—21no—
132C2×#10596831—21no—
133C2×#10596831—21no—
134C2×#10596831—21no—
135C4⋊Dic73—21no—
136C4⋊Dic73—21no—
137C4⋊Dic73—21no—
138C4⋊Dic73—21no—
139C4⋊Dic73—21no—
140C4⋊Dic73—21no—
141C4⋊Dic73—42no—
142C4⋊Dic73C2×Q812yes—
143C4⋊Dic73#4112yes—
144#11924007—41no—
145#11924007—41no—
146#11924007—42no—
147#11924007—42no—
148#11924007—81no—
149#11924007—81no—
150#11924007—81no—
151C22⋊Dic73—24no—
152C22⋊Dic73—42no—
153C22⋊Dic73—81no—
154C4×Dic73—21no—
155C4×Dic73—21no—
156C4×Dic73—42no—
157C4×Dic73—81no—
158C4×Dic73—81no—
159C4×Dic73C2×Q812yes—
160C4×Dic73#4112yes—
161C24×C73—21no—
162C2×D4×C73—21no—
163C2×D4×C73—21no—
164C2×D4×C73—21no—
165C2×D4×C73—21no—
166C2×D4×C73—21no—
167C2×D4×C73C2×Q812yes—
168C22×C4×C73—21no—
169C22×C4×C73—21no—
170C22×C4×C73—21no—
171C22×C4×C73C2×D411yescharacteristic
172C22×C4×C73C2×D411yescharacteristic
173C22×C4×C73C2411yescharacteristic
174C2×C8×C73—21no—
175C2×C8×C73—21no—
176C2×C8×C73—21no—
177C2×C8×C73—21no—
178C2×C8×C73—21no—
179C2×C8×C73—21no—
180C4⋊C4×C73—21no—
181C4⋊C4×C73—21no—
182C4⋊C4×C73—21no—
183C4⋊C4×C73—21no—
184C4⋊C4×C73—21no—
185C4⋊C4×C73—42no—
186C4⋊C4×C73—42no—
187C4⋊C4×C73#4112yes—
188C22⋊C4×C73—42no—
189C22⋊C4×C73—81no—
190C42×C73—21no—
191C42×C73—21no—
192C23×C73—22no—
193C23×C73—41no—
194C23×C73—41no—
195C23×C73—41no—
196C23×C73—41no—
197C23×C73—41no—
198C23×C73—41no—
199C23×C73C22×D411yescharacteristic
200D4×C73—22no—
201D4×C73—22no—
202D4×C73—41no—
203D4×C73—41no—
204C2×C4×C73—21no—
205C2×C4×C73—21no—
206C2×C4×C73—22no—
207C2×C4×C73—21no—
208C2×C4×C73—21no—
209C2×C4×C73—21no—
210C2×C4×C73—21no—
211C2×C4×C73—41no—
212C2×C4×C73—41no—
213C2×C4×C73—41no—
214C2×C4×C73—41no—
215C2×C4×C73—41no—
216C2×C4×C73—41no—
217C2×C4×C73—41no—
218C2×C4×C73—41no—
219C2×C4×C73C22×Q811yescharacteristic
220C2×C4×C73C2×#4111yescharacteristic
221C2×C4×C73C2×#4111yescharacteristic
222C2×C4×C73C2×#4111yescharacteristic
223C2×C4×C73H(5,𝔽2)11yescharacteristic
224C2×C4×C732−1+411yescharacteristic
225C2×Dic73—21no—
226C2×Dic73—21no—
227C2×Dic73—21no—
228C2×Dic73—21no—
229C2×Dic73—22no—
230C2×Dic73—21no—
231C2×Dic73—21no—
232C2×Dic73—22no—
233C2×Dic73—21no—
234C2×Dic73—21no—
235C2×Dic73—21no—
236C2×Dic73—21no—
237C2×Dic73—41no—
238C2×Dic73—41no—
239C2×Dic73—41no—
240C2×Dic73—41no—
241C2×Dic73—41no—
242C2×Dic73—41no—
243C2×Dic73—41no—
244C2×Dic73—41no—
245C2×Dic73—41no—
246C2×Dic73—41no—
247C2×Dic73—84no—
248C2×Dic73C22⋊Q812yes—
249C2×Dic73C22⋊Q812yes—
250C8×C73—41no—
251C8×C73—41no—
252#10596831—41no—
253#10596831—41no—
254C22×C73—42no—
255C22×C73—42no—
256C22×C73—42no—
257C22×C73—81no—
258C22×C73—81no—
259C22×C73—81no—
260C22×C73—81no—
261C22×C73C2×C22⋊Q811yescharacteristic
262C22×C73#54611yescharacteristic
263C22×C73#54811yescharacteristic
264C22×C73#57511yescharacteristic
265C22×C73#57511yescharacteristic
266C22×C73#57611yescharacteristic
267C22×C73#57711yescharacteristic
268C4×C73—41no—
269C4×C73—42no—
270C4×C73—41no—
271C4×C73—81no—
272C4×C73—81no—
273C4×C73#53311yescharacteristic
274C4×C73#54411yescharacteristic
275C4×C73#55511yescharacteristic
276C4×C73#55611yescharacteristic
277Dic73—22no—
278Dic73—22no—
279Dic73—42no—
280Dic73—42no—
281Dic73—41no—
282Dic73—41no—
283Dic73—81no—
284Dic73—81no—
285#26037—731nomaximal, Sylow
286C2×C73—82no—
287C2×C73#243911yescharacteristic
288C2×C73#308011yescharacteristic
289C2×C73#308111yescharacteristic
290C2×C73#330711yescharacteristic
291C2×C73#330911yescharacteristic
292C2×C73#337011yescharacteristic
293C2×C73#337111yescharacteristic
294#2552—731no—
295#2357—731no—
296#2067—731no—
297#2065—731no—
298#2025—731no—
299#2025D7311yescharacteristic
300#1989—731no—
301#1989—731no—
302#1949—731no—
303#1940—731no—
304#1920—731no—
305#1918—731no—
306#1836—731no—
307#1802—731no—
308#1762—731no—
309C73#2603711yescharacteristic, complemented, Sylow
310C2×#125—731no—
311C2×#125C2×D7311yescharacteristic
312C2×C4×D4—731no—
313C2×#107—731no—
314C2×#106—731no—
315C2×C4⋊C8—731no—
316C2×C4⋊C8—731no—
317C2×C4⋊C8—731no—
318C2×#102—731no—
319C2×#102—731no—
320C2×#102C2×D7311yescharacteristic
321C2×#102C2×D7311yescharacteristic
322C2×#97—731no—
323#398—1461no—
324#398—1461no—
325#390—731no—
326#390C2×D7311yescharacteristic
327#389—731no—
328#389—731no—
329#389—1461no—
330#388—1461no—
331#388—1461no—
332#384—731no—
333#384—1461no—
334#382—731no—
335#381—1461no—
336C4×C4⋊C4—731no—
337C4×C4⋊C4—731no—
338C4×C4⋊C4—731no—
339C4×C4⋊C4—731no—
340C4×C22⋊C4—1461no—
341#340—731no—
342#340—731no—
343#340—731no—
344#340—731no—
345#340—731no—
346#340—731no—
347#340—731no—
348#340—731no—
349#340C2×D7311yescharacteristic
350#340C2×D7311yescharacteristic
351#336—731no—
352#336C2×D7311yescharacteristic
353C22×D4C22×D7311yescharacteristic
354C23×C4—1461no—
355C22×C8—731no—
356C22×C8—731no—
357C22×C8C22×D7311yescharacteristic
358C22×C8C22×D7311yescharacteristic
359#125—22no—
360#125—41no—
361#125—2921no—
362#125—1462no—
363C4×D4—734no—
364C4×D4—1462no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—731no—
368C2×C4⋊C4—731no—
369C2×C4⋊C4—731no—
370C2×C4⋊C4—731no—
371C2×C4⋊C4—731no—
372C2×C4⋊C4—731no—
373C2×C4⋊C4—1461no—
374C2×C4⋊C4—1461no—
375C2×C4⋊C4—1461no—
376C2×C4⋊C4—1461no—
377C2×C4⋊C4—1461no—
378C2×C4⋊C4—1461no—
379C2×C4⋊C4C22×D7311yescharacteristic
380C2×C4⋊C4C22×D7311yescharacteristic
381C2×C4⋊C4C22×D7311yescharacteristic
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—1461no—
385C2×C22⋊C4—1461no—
386C2×C22⋊C4—1461no—
387C2×C42—731no—
388C2×C42—731no—
389C2×C42—1461no—
390C2×C42—1461no—
391C2×C42—1461no—
392C2×C42C22×D7311yescharacteristic
393#107—1462no—
394#106—1462no—
395C4⋊C8—2921no—
396C4⋊C8—2921no—
397C4⋊C8—1462no—
398#102—22no—
399#102—41no—
400#102—2921no—
401#102—1462no—
402#97—1462no—
403#95—731no—
404#95—731no—
405#95—1461no—
406#95—1461no—
407#95—1461no—
408#95—1461no—
409#95—1461no—
410#95—1461no—
411#95—1461no—
412C24—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4Q8×D7312yes—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—5841no—
423C22×C4—5841no—
424C22×C4—731no—
425C22×C4—731no—
426C22×C4—731no—
427C22×C4—731no—
428C22×C4—1461no—
429C22×C4—1461no—
430C22×C4—1461no—
431C22×C4—1461no—
432C22×C4—1464no—
433C22×C4D4×D7311yescharacteristic
434C22×C4D4×D7311yescharacteristic
435C22×C4C23×D7311yescharacteristic
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—1461no—
443C2×C8—1461no—
444C2×C8—1461no—
445C2×C8—1461no—
446C2×C8—1461no—
447C2×C8—1461no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—2922no—
456C4⋊C4—2922no—
457C4⋊C4—2921no—
458C4⋊C4—2921no—
459C4⋊C4—2922no—
460C4⋊C4—5841no—
461C4⋊C4—5841no—
462C4⋊C4—5841no—
463C4⋊C4—732no—
464C4⋊C4—732no—
465C4⋊C4—1461no—
466C4⋊C4—1461no—
467C4⋊C4—1461no—
468C4⋊C4—1461no—
469C4⋊C4—1461no—
470C4⋊C4—1461no—
471C4⋊C4#1192402712yes—
472C22⋊C4—42no—
473C22⋊C4—81no—
474C22⋊C4—2922no—
475C22⋊C4—5841no—
476C22⋊C4—1464no—
477C42—21no—
478C42—21no—
479C42—2922no—
480C42—5841no—
481C42—5841no—
482C42—732no—
483C42—732no—
484C42—1461no—
485C42—1461no—
486C23—22no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23C2×D4×D7311yescharacteristic
494D4—22no—
495D4—22no—
496D4—41no—
497D4—41no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—2921no—
514C2×C4—2921no—
515C2×C4—2921no—
516C2×C4—2921no—
517C2×C4—2921no—
518C2×C4—2921no—
519C2×C4—2921no—
520C2×C4—2921no—
521C2×C4—2921no—
522C2×C4—2921no—
523C2×C4—5844no—
524C2×C4—732no—
525C2×C4—732no—
526C2×C4—1461no—
527C2×C4—1461no—
528C2×C4—1461no—
529C2×C4—1461no—
530C2×C4—1461no—
531C2×C4—1461no—
532C2×C4—1462no—
533C2×C4—1462no—
534C2×C4—1461no—
535C2×C4—1461no—
536C2×C4—1461no—
537C2×C4—1461no—
538C2×C4C2×Q8×D7311yescharacteristic
539C2×C4#41×D7311yescharacteristic
540C2×C4#41×D7311yescharacteristic
541C2×C4C2×#1192402711yescharacteristic
542C2×C4#1455663611yescharacteristic
543C2×C4#1455663811yescharacteristic
544C8—41no—
545C8—41no—
546C8—2921no—
547C8—2921no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22C22⋊Q8×D7311yescharacteristic
556C22#1798922411yescharacteristic
557C22#1798924511yescharacteristic
558C22#1798938311yescharacteristic
559C22#1798938411yescharacteristic
560C22#1798938911yescharacteristic
561C22#1798939711yescharacteristic
562C4—41no—
563C4—42no—
564C4—41no—
565C4—81no—
566C4—81no—
567C4—2922no—
568C4—2921no—
569C4—2922no—
570C4—2921no—
571C4—5841no—
572C4—5841no—
573C4—1462no—
574C4—1462no—
575C4#1798913911yescharacteristic
576C4#1798921611yescharacteristic
577C4#1798929911yescharacteristic
578C4#1798930611yescharacteristic
579C2—82no—
580C2#2261361511yescharacteristic
581C2#2261974811yescharacteristic
582C2#2261975911yescharacteristic
583C2#2262191111yescharacteristic
584C2#2262193011yescharacteristic
585C2#2262269011yescharacteristic
586C2#2262269411yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2025C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
hypercenter#2025D73
hypocenterC73#26037
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. #22610885C2semidirect
8. #22610702C2semidirect
9. #22610653C2semidirect
10. #22610514C2semidirect
11. #22610503C2semidirect
12. #22610063C2semidirect
309. C73#26037semidirect

Maximal subgroups

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

subgroupindex
#226149632
#226129032
#226113172
#226113052
#226110752
#226108892
#226108852
#226107022
#226106532
#226105142
#226105032
#226100632
#226099292
#226097582
C73×#20252
#2603773