Finite Group Databasebeta

← #33098227

Subgroups — #33098227

Order 18688.

732 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988887C211yesmaximal, characteristic
2#22612670C211yesmaximal, characteristic, complemented
3#22612574C211yesmaximal, characteristic
4#22610886C211yesmaximal, characteristic, complemented
5#22610687C211yesmaximal, characteristic, complemented
6#22610673C211yesmaximal, characteristic, complemented
7#22610663C211yesmaximal, characteristic, complemented
8#22610625C211yesmaximal, characteristic
9#22610611C211yesmaximal, characteristic
10#22610554C211yesmaximal, characteristic
11#22610554C211yesmaximal, characteristic
12#22610526C211yesmaximal, characteristic, complemented
13#22610512C211yesmaximal, characteristic, complemented
14#22609790C211yesmaximal, characteristic, complemented
15C73×#1802C211yesmaximal, characteristic
16C2×#14556583C2211yescharacteristic
17#17988887C418yescomplemented
18C2×#14556524C2211yescharacteristic
19C2×#14556524C2211yescharacteristic
20C2×#14556524C2211yescharacteristic
21C2×#14556522C2211yescharacteristic
22C2×C4⋊#10596831C2211yescharacteristic
23C2×#14556512C2211yescharacteristic
24C2×#14556512C2211yescharacteristic
25C2×#14556501C2211yescharacteristic
26C2×C8×Dic73C2211yescharacteristic
27#17988490C2211yescharacteristic
28#17988490C2211yescharacteristic
29#17988486C2211yescharacteristic
30#17988486C2211yescharacteristic
31#17988483C2211yescharacteristic
32#17988483C2211yescharacteristic
33#17988482—21no—
34#17988482—21no—
35#17988480—21no—
36#17988480—21no—
37#17988468—21no—
38#17988468—21no—
39#17988450—21no—
40#17988447—21no—
41C4⋊C4×Dic73C2211yescharacteristic
42C4⋊C4×Dic73C2211yescharacteristic
43C4×C4⋊Dic73C2211yescharacteristic
44#17988433C2211yescharacteristic
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988350C2211yescharacteristic
48#17988350C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988341C2211yescharacteristic
52#17988341C2211yescharacteristic
53C2×C4×D4×C73C2211yescharacteristic
54C2×C4⋊C8×C73C2211yescharacteristic
55C2×#102×C73C2211yescharacteristic
56C2×#102×C73C2211yescharacteristic
57#384×C73C2211yescharacteristic
58#340×C73C2211yescharacteristic
59#340×C73C2211yescharacteristic
60C22×#10596831C2311yescharacteristic
61C22×#10596831C2311yescharacteristic
62#14556583—22no—
63#14556583C2×C414yes—
64C2×C4⋊Dic73—21no—
65C2×C4⋊Dic73—21no—
66C2×C4⋊Dic73C2311yescharacteristic
67C2×C4⋊Dic73C2311yescharacteristic
68C2×C4⋊Dic73C2311yescharacteristic
69C2×C4⋊Dic73C2311yescharacteristic
70C2×#11924007—21no—
71C2×#11924007—21no—
72C2×#11924007—21no—
73C2×#11924007—21no—
74C2×C22⋊Dic73—21no—
75C2×C22⋊Dic73—21no—
76C2×C22⋊Dic73—21no—
77C2×C22⋊Dic73—21no—
78C2×C4×Dic73—21no—
79C2×C4×Dic73—21no—
80C2×C4×Dic73—21no—
81C2×C4×Dic73C2311yescharacteristic
82C2×C4×Dic73C2311yescharacteristic
83#14556524—41no—
84#14556524—41no—
85#14556524C2×C414yes—
86#14556522C2×C414yes—
87C4⋊#10596831C2×C414yes—
88#14556512C2×C414yes—
89#14556512C2×C414yes—
90#14556501—41no—
91C8×Dic73—41no—
92#14556498—21no—
93#14556498—21no—
94#14556498—21no—
95#14556498—21no—
96#14556498—21no—
97#14556498—21no—
98#14556498—21no—
99#14556498D411yescharacteristic
100#14556498D411yescharacteristic
101#14556498D411yescharacteristic
102#14556498D411yescharacteristic
103C22×D4×C73C2311yescharacteristic
104C23×C4×C73—21no—
105C22×C8×C73C2311yescharacteristic
106C22×C8×C73C2311yescharacteristic
107C4×D4×C73—22no—
108C4×D4×C73C2×C414yes—
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×C42×C73C2311yescharacteristic
116C4⋊C8×C73—22no—
117#102×C73—41no—
118#102×C73—41no—
119C22×Dic73—21no—
120C22×Dic73—21no—
121C22×Dic73—21no—
122C22×Dic73—21no—
123C22×Dic73C2×D411yescharacteristic
124C22×Dic73C2×D411yescharacteristic
125C22×Dic73C2×D411yescharacteristic
126C22×Dic73C2×D411yescharacteristic
127C2×#10596831—21no—
128C2×#10596831—21no—
129C2×#10596831—21no—
130C2×#10596831—21no—
131C2×#10596831C22×C412yes—
132C2×#10596831C22×C412yes—
133C4⋊Dic73—21no—
134C4⋊Dic73—21no—
135C4⋊Dic73—21no—
136C4⋊Dic73—21no—
137C4⋊Dic73—21no—
138C4⋊Dic73—21no—
139C4⋊Dic73—81no—
140C4⋊Dic73—81no—
141C4⋊Dic73C22×C412yes—
142C4⋊Dic73C22×C412yes—
143#11924007—24no—
144#11924007—24no—
145#11924007—81no—
146#11924007—81no—
147C22⋊Dic73—81no—
148C22⋊Dic73—81no—
149C22⋊Dic73—81no—
150C22⋊Dic73—81no—
151C4×Dic73—21no—
152C4×Dic73—21no—
153C4×Dic73—21no—
154C4×Dic73—21no—
155C4×Dic73—81no—
156C4×Dic73—81no—
157C4×Dic73—81no—
158C24×C73—21no—
159C2×D4×C73—21no—
160C2×D4×C73—21no—
161C2×D4×C73—21no—
162C2×D4×C73—21no—
163C2×D4×C73—21no—
164C2×D4×C73C22×C412yes—
165C22×C4×C73—21no—
166C22×C4×C73—24no—
167C22×C4×C73—21no—
168C22×C4×C73—21no—
169C22×C4×C73—81no—
170C22×C4×C73—81no—
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—81no—
186C4⋊C4×C73—81no—
187C4⋊C4×C73C22×C412yes—
188C22⋊C4×C73—24no—
189C42×C73C22×C412yes—
190C42×C73#4112yes—
191C23×C73—22no—
192C23×C73—41no—
193C23×C73—41no—
194C23×C73—41no—
195C23×C73—41no—
196C23×C73—41no—
197C23×C73—41no—
198C23×C73C22×D411yescharacteristic
199D4×C73—22no—
200D4×C73—22no—
201D4×C73—41no—
202D4×C73—41no—
203C2×C4×C73—21no—
204C2×C4×C73—21no—
205C2×C4×C73—21no—
206C2×C4×C73—21no—
207C2×C4×C73—22no—
208C2×C4×C73—41no—
209C2×C4×C73—41no—
210C2×C4×C73—41no—
211C2×C4×C73—41no—
212C2×C4×C73—41no—
213C2×C4×C73—41no—
214C2×C4×C73—41no—
215C2×C4×C73—41no—
216C2×C4×C73—84no—
217C2×C4×C73C4×D412yes—
218C2×C4×C73C4×D412yes—
219C2×C4×C73C23×C411yescharacteristic
220C2×C4×C73C2×#4111yescharacteristic
221C2×C4×C73H(5,𝔽2)11yescharacteristic
222C2×C4×C732−1+411yescharacteristic
223C2×C4×C732−1+411yescharacteristic
224C2×C4×C732−1+411yescharacteristic
225C2×Dic73—22no—
226C2×Dic73—22no—
227C2×Dic73—21no—
228C2×Dic73—21no—
229C2×Dic73—21no—
230C2×Dic73—21no—
231C2×Dic73—21no—
232C2×Dic73—21no—
233C2×Dic73—21no—
234C2×Dic73—21no—
235C2×Dic73—21no—
236C2×Dic73—21no—
237C2×Dic73—21no—
238C2×Dic73—21no—
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—41no—
248C2×Dic73—41no—
249C8×C73—41no—
250C8×C73—41no—
251#10596831—22no—
252#10596831—22no—
253C22×C73—42no—
254C22×C73—42no—
255C22×C73—42no—
256C22×C73—81no—
257C22×C73—81no—
258C22×C73—81no—
259C22×C73—81no—
260C22×C73C2×C4×D411yescharacteristic
261C22×C73#53611yescharacteristic
262C22×C73#53711yescharacteristic
263C22×C73#57611yescharacteristic
264C22×C73#57611yescharacteristic
265C22×C73#57711yescharacteristic
266C22×C73#57811yescharacteristic
267C4×C73—22no—
268C4×C73—22no—
269C4×C73—42no—
270C4×C73—81no—
271C4×C73—81no—
272C4×C73#51911yescharacteristic
273C4×C73#52011yescharacteristic
274C4×C73#54111yescharacteristic
275C4×C73#54211yescharacteristic
276Dic73—42no—
277Dic73—42no—
278Dic73—41no—
279Dic73—41no—
280Dic73—41no—
281Dic73—41no—
282Dic73—81no—
283Dic73—81no—
284#22126—731nomaximal, Sylow
285C2×C73—82no—
286C2×C73#232311yescharacteristic
287C2×C73#294611yescharacteristic
288C2×C73#295111yescharacteristic
289C2×C73#324111yescharacteristic
290C2×C73#324311yescharacteristic
291C2×C73#325811yescharacteristic
292C2×C73#325911yescharacteristic
293C2×#478—731no—
294#2330—731no—
295#2319—731no—
296#1989—731no—
297#1947—731no—
298#1944—731no—
299#1942—731no—
300#1936—731no—
301#1934—731no—
302#1925—731no—
303#1925—731no—
304#1921—731no—
305#1919—731no—
306#1802D7311yescharacteristic
307#1768—731no—
308C73#2212611yescharacteristic, complemented, Sylow
309C2×#125—731no—
310C2×C4×D4C2×D7311yescharacteristic
311#478—738no—
312C2×#107—731no—
313C2×#107—731no—
314C2×#107—731no—
315C2×#106—731no—
316C2×C4⋊C8—731no—
317C2×C4⋊C8C2×D7311yescharacteristic
318C2×#102—731no—
319C2×#102—731no—
320C2×#102C2×D7311yescharacteristic
321C2×#102C2×D7311yescharacteristic
322C2×#97—731no—
323C2×C4×C8—731no—
324#390—731no—
325#390—731no—
326#389—731no—
327#389—731no—
328#389—731no—
329#389—731no—
330#389—1461no—
331#389—1461no—
332#388—1461no—
333#388—1461no—
334#387—1461no—
335#387—1461no—
336#384C2×D7311yescharacteristic
337#381—1461no—
338#380—1461no—
339C4×C4⋊C4—731no—
340C4×C4⋊C4—731no—
341C4×C4⋊C4—731no—
342#376—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#336—731no—
353C22×D4C22×D7311yescharacteristic
354C23×C4—21no—
355C22×C8—731no—
356C22×C8—731no—
357C22×C8C22×D7311yescharacteristic
358C22×C8C22×D7311yescharacteristic
359#125—734no—
360#125—1462no—
361C4×D4—22no—
362C4×D4C4×D7314yes—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—731no—
366C2×C4⋊C4—731no—
367C2×C4⋊C4—731no—
368C2×C4⋊C4—731no—
369C2×C4⋊C4—1461no—
370C2×C4⋊C4—1461no—
371C2×C4⋊C4—1461no—
372C2×C4⋊C4—1461no—
373C2×C4⋊C4—1461no—
374C2×C4⋊C4—1461no—
375C2×C4⋊C4C22×D7311yescharacteristic
376C2×C4⋊C4C22×D7311yescharacteristic
377C2×C4⋊C4C22×D7311yescharacteristic
378C2×C22⋊C4—21no—
379C2×C22⋊C4—1461no—
380C2×C22⋊C4—1461no—
381C2×C22⋊C4—1461no—
382C2×C22⋊C4—1461no—
383C2×C42—731no—
384C2×C42—731no—
385C2×C42—1461no—
386C2×C42—1461no—
387C2×C42—1461no—
388C2×C42C22×D7311yescharacteristic
389#107—2921no—
390#107—2921no—
391#107—734no—
392#106—734no—
393C4⋊C8—22no—
394C4⋊C8—734no—
395#102—41no—
396#102—41no—
397#102—734no—
398#102—734no—
399#97—2921no—
400C4×C8—2921no—
401#95—731no—
402#95—731no—
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×D4C2×C4×D7312yes—
419C22×C4—21no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—81no—
424C22×C4—81no—
425C22×C4—731no—
426C22×C4—731no—
427C22×C4—731no—
428C22×C4—731no—
429C22×C4—1461no—
430C22×C4—1461no—
431C22×C4—1461no—
432C22×C4—1461no—
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—732no—
443C2×C8—732no—
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—81no—
454C4⋊C4—81no—
455C4⋊C4—5841no—
456C4⋊C4—5841no—
457C4⋊C4—5841no—
458C4⋊C4—5841no—
459C4⋊C4—732no—
460C4⋊C4—732no—
461C4⋊C4—1461no—
462C4⋊C4—1461no—
463C4⋊C4—1461no—
464C4⋊C4—1461no—
465C4⋊C4—1464no—
466C4⋊C4—1464no—
467C4⋊C4—1461no—
468C4⋊C4—1461no—
469C4⋊C4C2×C4×D7312yes—
470C22⋊C4—24no—
471C22⋊C4—5841no—
472C22⋊C4—5841no—
473C22⋊C4—5841no—
474C22⋊C4—5841no—
475C42—5841no—
476C42—5841no—
477C42—5841no—
478C42—1461no—
479C42—1461no—
480C42—1461no—
481C42—1461no—
482C42C2×C4×D7312yes—
483C42#1192402712yes—
484C23—22no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23C2×D4×D7311yescharacteristic
492D4—22no—
493D4—22no—
494D4—41no—
495D4—41no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—84no—
510C2×C4—2921no—
511C2×C4—2921no—
512C2×C4—2921no—
513C2×C4—2921no—
514C2×C4—2921no—
515C2×C4—2921no—
516C2×C4—2921no—
517C2×C4—2921no—
518C2×C4—2921no—
519C2×C4—2921no—
520C2×C4—1461no—
521C2×C4—1461no—
522C2×C4—1461no—
523C2×C4—1461no—
524C2×C4—1461no—
525C2×C4—1461no—
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#1455655112yes—
535C2×C4#1455655112yes—
536C2×C4C22×C4×D7311yescharacteristic
537C2×C4C2×#1192402711yescharacteristic
538C2×C4#1455663611yescharacteristic
539C2×C4#1455663811yescharacteristic
540C2×C4#1455663811yescharacteristic
541C2×C4#1455663911yescharacteristic
542C8—41no—
543C8—41no—
544C8—1462no—
545C8—1462no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—81no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22C2×#1455655111yescharacteristic
554C22#1798916011yescharacteristic
555C22#1798916611yescharacteristic
556C22#1798938811yescharacteristic
557C22#1798939111yescharacteristic
558C22#1798939311yescharacteristic
559C22#1798940311yescharacteristic
560C4—22no—
561C4—22no—
562C4—42no—
563C4—81no—
564C4—81no—
565C4—2921no—
566C4—2922no—
567C4—2921no—
568C4—2922no—
569C4—2921no—
570C4—2921no—
571C4—5841no—
572C4—5841no—
573C4#1798906411yescharacteristic
574C4#1798906811yescharacteristic
575C4#1798919711yescharacteristic
576C4#1798920411yescharacteristic
577C2—82no—
578C2#2261260911yescharacteristic
579C2#2261875711yescharacteristic
580C2#2261880211yescharacteristic
581C2#2262118311yescharacteristic
582C2#2262121111yescharacteristic
583C2#2262135311yescharacteristic
584C2#2262136811yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1802C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
cosocleC22×C4×C73C24
hypercenter#1802D73
hypocenterC73#22126
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
2. #22612670C2semidirect
4. #22610886C2semidirect
5. #22610687C2semidirect
6. #22610673C2semidirect
7. #22610663C2semidirect
12. #22610526C2semidirect
13. #22610512C2semidirect
14. #22609790C2semidirect
17. #17988887C4semidirect
308. C73#22126semidirect

Maximal subgroups

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

subgroupindex
C2×#179888872
#226126702
#226125742
#226108862
#226106872
#226106732
#226106632
#226106252
#226106112
#226105542
#226105262
#226105122
#226097902
C73×#18022
#2212673