Finite Group Databasebeta

← #33098230

Subgroups — #33098230

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×#17988890C211yesmaximal, characteristic
2#22612668C211yesmaximal, characteristic, complemented
3#22612570C211yesmaximal, characteristic
4#22610887C211yesmaximal, characteristic, complemented
5#22610688C211yesmaximal, characteristic, complemented
6#22610673C211yesmaximal, characteristic, complemented
7#22610665C211yesmaximal, characteristic, complemented
8#22610629C211yesmaximal, characteristic
9#22610612C211yesmaximal, characteristic
10#22610552C211yesmaximal, characteristic
11#22610551C211yesmaximal, characteristic
12#22610506C211yesmaximal, characteristic, complemented
13#22609930C211yesmaximal, characteristic
14#22609787C211yesmaximal, characteristic, complemented
15C73×#1921C211yesmaximal, characteristic, complemented
16C2×#14556582C2211yescharacteristic
17C2×#14556553C2211yescharacteristic
18#17988890C418yescomplemented
19C2×#14556525C2211yescharacteristic
20C2×#14556524C2211yescharacteristic
21C2×#14556522C2211yescharacteristic
22C2×#14556520C2211yescharacteristic
23C2×#14556520C2211yescharacteristic
24C2×#14556513C2211yescharacteristic
25C2×#14556513C2211yescharacteristic
26C2×#14556511C2211yescharacteristic
27C2×#14556511C2211yescharacteristic
28C2×#14556501C2211yescharacteristic
29C2×C8×Dic73C2211yescharacteristic
30#17988489C2211yescharacteristic
31#17988488C2211yescharacteristic
32#17988486C2211yescharacteristic
33#17988486C2211yescharacteristic
34#17988485—21no—
35#17988484—21no—
36#17988483C2211yescharacteristic
37#17988479—21no—
38#17988477—21no—
39#17988471—21no—
40#17988470—21no—
41#17988456C2211yescharacteristic
42#17988449—21no—
43#17988444—21no—
44C4⋊C4×Dic73C2211yescharacteristic
45C4×C4⋊Dic73C2211yescharacteristic
46#17988433C2211yescharacteristic
47#17988350C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988349C2211yescharacteristic
52#17988342C2211yescharacteristic
53C2×#107×C73C2211yescharacteristic
54#389×C73C2211yescharacteristic
55C4×C4⋊C4×C73C2211yescharacteristic
56#340×C73C2211yescharacteristic
57#340×C73C2211yescharacteristic
58#340×C73C2211yescharacteristic
59#336×C73C2211yescharacteristic
60C22×D292C2311yescharacteristic
61C22×C4×D73—21no—
62C22×#10596831C2311yescharacteristic
63C22×#10596831C2311yescharacteristic
64#14556582—22no—
65#14556582C2×C414yes—
66#14556553—22no—
67#14556553C2×C414yes—
68C2×C4⋊Dic73—21no—
69C2×C4⋊Dic73—21no—
70C2×C4⋊Dic73—21no—
71C2×C4⋊Dic73C2311yescharacteristic
72C2×C4⋊Dic73C2311yescharacteristic
73C2×C4⋊Dic73C2311yescharacteristic
74C2×#11924007—21no—
75C2×#11924007—21no—
76C2×#11924007—21no—
77C2×#11924007—21no—
78C2×#11924005—21no—
79C2×#11924005—21no—
80C2×#11924005—21no—
81C2×#11924005—21no—
82C2×#11924005—21no—
83C2×C4×Dic73—21no—
84C2×C4×Dic73—21no—
85C2×C4×Dic73C2311yescharacteristic
86C2×C4×Dic73C2311yescharacteristic
87#14556525—41no—
88#14556524—41no—
89#14556522C2×C414yes—
90#14556520—22no—
91#14556520C2×C414yes—
92#14556513—41no—
93#14556513C2×C414yes—
94#14556511—41no—
95#14556511C2×C414yes—
96#14556501—41no—
97C8×Dic73—41no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102#14556498—21no—
103#14556498D411yescharacteristic
104#14556498D411yescharacteristic
105#14556498D411yescharacteristic
106#14556498D411yescharacteristic
107C22×C8×C73C2311yescharacteristic
108C22×C8×C73C2311yescharacteristic
109C2×C4⋊C4×C73—21no—
110C2×C4⋊C4×C73C2311yescharacteristic
111C2×C4⋊C4×C73C2311yescharacteristic
112C2×C4⋊C4×C73C2311yescharacteristic
113C2×C4⋊C4×C73C2311yescharacteristic
114C2×C42×C73—21no—
115C2×C42×C73C2311yescharacteristic
116#107×C73C2×C414yes—
117#95×C73—21no—
118#95×C73—21no—
119C23×D73—21no—
120C2×D292—21no—
121C2×D292—21no—
122C2×D292—21no—
123C2×D292—21no—
124C2×D292—21no—
125C2×D292C22×C412yes—
126C22×Dic73—21no—
127C22×Dic73—21no—
128C22×Dic73—21no—
129C22×Dic73C2×D411yescharacteristic
130C22×Dic73C2×D411yescharacteristic
131C22×Dic73C2×D411yescharacteristic
132C22×Dic73C2×D411yescharacteristic
133C2×C4×D73—24no—
134C2×C4×D73—81no—
135C2×C4×D73—81no—
136C2×#10596831—21no—
137C2×#10596831—21no—
138C2×#10596831—21no—
139C2×#10596831—21no—
140C2×#10596831—21no—
141C2×#10596831C22×C412yes—
142C4⋊Dic73—21no—
143C4⋊Dic73—24no—
144C4⋊Dic73—21no—
145C4⋊Dic73—21no—
146C4⋊Dic73—21no—
147C4⋊Dic73—21no—
148C4⋊Dic73—81no—
149C4⋊Dic73—81no—
150C4⋊Dic73C22×C412yes—
151#11924007—24no—
152#11924007—81no—
153#11924007—81no—
154#11924007—81no—
155#11924005—24no—
156#11924005—81no—
157#11924005—81no—
158#11924005—81no—
159#11924005—81no—
160C4×Dic73—21no—
161C4×Dic73—21no—
162C4×Dic73—81no—
163C4×Dic73—81no—
164C4×Dic73C22×C412yes—
165C4×Dic73#4112yes—
166C22×C4×C73—21no—
167C22×C4×C73—21no—
168C22×C4×C73—21no—
169C22×C4×C73—21no—
170C22×C4×C73C2×D411yescharacteristic
171C22×C4×C73C2×D411yescharacteristic
172C22×C4×C73C2411yescharacteristic
173C2×C8×C73—21no—
174C2×C8×C73—21no—
175C2×C8×C73—21no—
176C2×C8×C73—21no—
177C2×C8×C73—21no—
178C2×C8×C73C22×C412yes—
179C4⋊C4×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×C73C22×C412yes—
187C4⋊C4×C73C22×C412yes—
188C42×C73—21no—
189C42×C73—21no—
190C42×C73—81no—
191C23×C73C22×D411yescharacteristic
192C22×D73—22no—
193C22×D73—41no—
194C22×D73—41no—
195C22×D73—41no—
196C22×D73—41no—
197C22×D73—41no—
198C22×D73—41no—
199C2×C4×C73—22no—
200C2×C4×C73—22no—
201C2×C4×C73—21no—
202C2×C4×C73—21no—
203C2×C4×C73—21no—
204C2×C4×C73—21no—
205C2×C4×C73—21no—
206C2×C4×C73—21no—
207C2×C4×C73—41no—
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—41no—
217C2×C4×C73C23×C411yescharacteristic
218C2×C4×C73C2×#4111yescharacteristic
219C2×C4×C73H(5,𝔽2)11yescharacteristic
220C2×C4×C732−1+411yescharacteristic
221C2×C4×C732−1+411yescharacteristic
222C2×C4×C732−1+411yescharacteristic
223C2×Dic73—21no—
224C2×Dic73—21no—
225C2×Dic73—21no—
226C2×Dic73—21no—
227C2×Dic73—22no—
228C2×Dic73—21no—
229C2×Dic73—21no—
230C2×Dic73—21no—
231C2×Dic73—21no—
232C2×Dic73—21no—
233C2×Dic73—21no—
234C2×Dic73—41no—
235C2×Dic73—41no—
236C2×Dic73—41no—
237C2×Dic73—41no—
238C2×Dic73—41no—
239C2×Dic73—41no—
240C2×Dic73—41no—
241C2×Dic73—41no—
242C2×Dic73C4×D412yes—
243C2×Dic73C4×D412yes—
244D292—22no—
245D292—22no—
246D292—41no—
247D292—41no—
248C4×D73—84no—
249C8×C73—22no—
250C8×C73—41no—
251#10596831—22no—
252#10596831—41no—
253C22×C73C2×C4×D411yescharacteristic
254C22×C73#53611yescharacteristic
255C22×C73#53711yescharacteristic
256C22×C73#57611yescharacteristic
257C22×C73#57611yescharacteristic
258C22×C73#57711yescharacteristic
259C22×C73#57811yescharacteristic
260C2×D73—42no—
261C2×D73—42no—
262C2×D73—42no—
263C2×D73—81no—
264C2×D73—81no—
265C2×D73—81no—
266C2×D73—81no—
267C4×C73—42no—
268C4×C73—42no—
269C4×C73—41no—
270C4×C73—41no—
271C4×C73—81no—
272C4×C73—81no—
273C4×C73#51911yescharacteristic
274C4×C73#52011yescharacteristic
275C4×C73#54111yescharacteristic
276C4×C73#54211yescharacteristic
277Dic73—22no—
278Dic73—22no—
279Dic73—42no—
280Dic73—41no—
281Dic73—41no—
282Dic73—81no—
283Dic73—81no—
284#22126—731nomaximal, Sylow
285C2×C73#232311yescharacteristic
286C2×C73#294611yescharacteristic
287C2×C73#295111yescharacteristic
288C2×C73#324111yescharacteristic
289C2×C73#324311yescharacteristic
290C2×C73#325811yescharacteristic
291C2×C73#325911yescharacteristic
292D73—82no—
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#1921D7311yescharacteristic, complemented
305#1919—731no—
306#1802—731no—
307#1768—731no—
308C73#2212611yescharacteristic, complemented, Sylow
309C2×#125—731no—
310C2×C4×D4—731no—
311#478—738no—
312C2×#107—731no—
313C2×#107—731no—
314C2×#107C2×D7311yescharacteristic
315C2×#106—731no—
316C2×C4⋊C8—731no—
317C2×C4⋊C8—731no—
318C2×#102—731no—
319C2×#102—731no—
320C2×#102—731no—
321C2×#102—731no—
322C2×#97—731no—
323C2×C4×C8—731no—
324#390—731no—
325#390—731no—
326#389—731no—
327#389—731no—
328#389—731no—
329#389—1461no—
330#389—1461no—
331#389C2×D7311yescharacteristic
332#388—1461no—
333#388—1461no—
334#387—1461no—
335#387—1461no—
336#384—731no—
337#381—1461no—
338#380—1461no—
339C4×C4⋊C4—731no—
340C4×C4⋊C4—731no—
341C4×C4⋊C4C2×D7311yescharacteristic
342#376—731no—
343#340—731no—
344#340—731no—
345#340—731no—
346#340—731no—
347#340—731no—
348#340C2×D7311yescharacteristic
349#340C2×D7311yescharacteristic
350#340C2×D7311yescharacteristic
351#336—731no—
352#336C2×D7311yescharacteristic
353C22×D4—731no—
354C23×C4—1461no—
355C22×C8—731no—
356C22×C8—731no—
357C22×C8C22×D7311yescharacteristic
358C22×C8C22×D7311yescharacteristic
359#125—734no—
360#125—1462no—
361C4×D4—734no—
362C4×D4—1462no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—731no—
365C2×C4⋊C4—731no—
366C2×C4⋊C4—731no—
367C2×C4⋊C4—1461no—
368C2×C4⋊C4—1461no—
369C2×C4⋊C4—1461no—
370C2×C4⋊C4—1461no—
371C2×C4⋊C4—1461no—
372C2×C4⋊C4—1461no—
373C2×C4⋊C4—1461no—
374C2×C4⋊C4C22×D7311yescharacteristic
375C2×C4⋊C4C22×D7311yescharacteristic
376C2×C4⋊C4C22×D7311yescharacteristic
377C2×C4⋊C4C22×D7311yescharacteristic
378C2×C22⋊C4—1461no—
379C2×C22⋊C4—1461no—
380C2×C22⋊C4—1461no—
381C2×C22⋊C4—1461no—
382C2×C22⋊C4—1461no—
383C2×C42—21no—
384C2×C42—731no—
385C2×C42—731no—
386C2×C42—1461no—
387C2×C42—1461no—
388C2×C42C22×D7311yescharacteristic
389#107—2921no—
390#107—2921no—
391#107C4×D7314yes—
392#106—734no—
393C4⋊C8—734no—
394C4⋊C8—1462no—
395#102—2921no—
396#102—2921no—
397#102—734no—
398#102—734no—
399#97—2921no—
400C4×C8—2921no—
401#95—21no—
402#95—21no—
403#95—731no—
404#95—731no—
405#95—731no—
406#95—731no—
407#95—1461no—
408#95—1461no—
409#95—1461no—
410#95—1461no—
411#95—1461no—
412C24—1461no—
413C2×D4—732no—
414C2×D4—1461no—
415C2×D4—1461no—
416C2×D4—1461no—
417C2×D4—1461no—
418C2×D4—1461no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—5841no—
424C22×C4—5841no—
425C22×C4—731no—
426C22×C4—731no—
427C22×C4—731no—
428C22×C4—731no—
429C22×C4—1464no—
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—732no—
442C2×C8—1461no—
443C2×C8—1461no—
444C2×C8—1461no—
445C2×C8—1461no—
446C2×C8—1461no—
447C2×C8C2×C4×D7312yes—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—81no—
455C4⋊C4—5841no—
456C4⋊C4—5841no—
457C4⋊C4—5841no—
458C4⋊C4—5841no—
459C4⋊C4—5841no—
460C4⋊C4—732no—
461C4⋊C4—1464no—
462C4⋊C4—1464no—
463C4⋊C4—1461no—
464C4⋊C4—1461no—
465C4⋊C4—1461no—
466C4⋊C4—1461no—
467C4⋊C4—1461no—
468C4⋊C4C2×C4×D7312yes—
469C4⋊C4C2×C4×D7312yes—
470C22⋊C4—5841no—
471C22⋊C4—5841no—
472C22⋊C4—5841no—
473C22⋊C4—5841no—
474C22⋊C4—1464no—
475C42—21no—
476C42—21no—
477C42—81no—
478C42—5841no—
479C42—5841no—
480C42—732no—
481C42—732no—
482C42—1461no—
483C42—1461no—
484C23—2921no—
485C23—2921no—
486C23—2921no—
487C23—2921no—
488C23—2921no—
489C23—2921no—
490C23—1462no—
491C23C2×D4×D7311yescharacteristic
492D4—2921no—
493D4—2921no—
494D4—1462no—
495D4—1462no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—41no—
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—41no—
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—5844no—
523C2×C4—732no—
524C2×C4—732no—
525C2×C4—1461no—
526C2×C4—1461no—
527C2×C4—1461no—
528C2×C4—1461no—
529C2×C4—1461no—
530C2×C4—1461no—
531C2×C4—1462no—
532C2×C4—1461no—
533C2×C4—1461no—
534C2×C4—1461no—
535C2×C4—1461no—
536C2×C4C22×C4×D7311yescharacteristic
537C2×C4#41×D7311yescharacteristic
538C2×C4#1455663711yescharacteristic
539C2×C4#1455663811yescharacteristic
540C2×C4#1455663811yescharacteristic
541C2×C4#1455663911yescharacteristic
542C8—22no—
543C8—41no—
544C8—2921no—
545C8—1462no—
546C22—2922no—
547C22—2922no—
548C22—2922no—
549C22—5841no—
550C22—5841no—
551C22—5841no—
552C22—5841no—
553C22C4×D4×D7311yescharacteristic
554C22#1798915911yescharacteristic
555C22#1798916711yescharacteristic
556C22#1798938911yescharacteristic
557C22#1798939111yescharacteristic
558C22#1798939711yescharacteristic
559C22#1798940011yescharacteristic
560C4—41no—
561C4—41no—
562C4—42no—
563C4—42no—
564C4—81no—
565C4—81no—
566C4—2921no—
567C4—2922no—
568C4—2921no—
569C4—5841no—
570C4—5841no—
571C4—1462no—
572C4—1462no—
573C4#1798906311yescharacteristic
574C4#1798906711yescharacteristic
575C4#1798919611yescharacteristic
576C4#1798920311yescharacteristic
577C2—5842no—
578C2#2261261211yescharacteristic
579C2#2261876011yescharacteristic
580C2#2261880611yescharacteristic
581C2#2262119011yescharacteristic
582C2#2262121811yescharacteristic
583C2#2262135611yescharacteristic
584C2#2262136911yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1921C2
socleC23×C73C22×D4
prime radicalC22×D292C23
cosocleC22×C4×C73C24
hypercenter#1921D73
hypocenterC73#22126
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1921

Complements

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

NH ≅ G/Ntype
2. #22612668C2semidirect
4. #22610887C2semidirect
5. #22610688C2semidirect
6. #22610673C2semidirect
7. #22610665C2semidirect
12. #22610506C2semidirect
14. #22609787C2semidirect
15. C73×#1921C2semidirect
18. #17988890C4semidirect
304. #1921D73semidirect
308. C73#22126semidirect

Maximal subgroups

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

subgroupindex
C2×#179888902
#226126682
#226125702
#226108872
#226106882
#226106732
#226106652
#226106292
#226106122
#226105522
#226105512
#226105062
#226099302
#226097872
C73×#19212
#2212673