Finite Group Databasebeta

← #33098142

Subgroups — #33098142

Order 18688.

784 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988889C211yesmaximal, characteristic
2#22612490C211yesmaximal, characteristic, complemented
3#22612393C211yesmaximal, characteristic
4#22610885C211yesmaximal, characteristic, complemented
5#22610702C211yesmaximal, characteristic, complemented
6#22610674C211yesmaximal, characteristic, complemented
7#22610666C211yesmaximal, characteristic, complemented
8#22610631C211yesmaximal, characteristic
9#22610561C211yesmaximal, characteristic
10#22610553C211yesmaximal, characteristic
11#22610508C211yesmaximal, characteristic, complemented
12#22610500C211yesmaximal, characteristic, complemented
13#22609939C211yesmaximal, characteristic
14#22609785C211yesmaximal, characteristic, complemented
15C73×#1936C211yesmaximal, characteristic
16C2×#14556582C2211yescharacteristic
17C2×D4×Dic73C2211yescharacteristic
18C2×#14556547—21no—
19#17988889C418yescomplemented
20C2×#14556525C2211yescharacteristic
21C2×#14556524C2211yescharacteristic
22C2×#14556523C2211yescharacteristic
23C2×#14556522C2211yescharacteristic
24C2×#14556520C2211yescharacteristic
25C2×#14556520C2211yescharacteristic
26C2×#14556512C2211yescharacteristic
27C2×#14556512C2211yescharacteristic
28C2×#14556502C2211yescharacteristic
29#17988490C2211yescharacteristic
30#17988485C2211yescharacteristic
31#17988485C2211yescharacteristic
32#17988483C2211yescharacteristic
33#17988481—21no—
34#17988475—21no—
35#17988472—21no—
36#17988468—21no—
37#17988452C2211yescharacteristic
38C4⋊C4×Dic73—21no—
39C4⋊C4×Dic73C2211yescharacteristic
40C4⋊C4×Dic73C2211yescharacteristic
41C4⋊C4×Dic73C2211yescharacteristic
42#17988438—21no—
43C22⋊C4×Dic73—21no—
44C42×Dic73C2211yescharacteristic
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988341C2211yescharacteristic
52#17988341C2211yescharacteristic
53C2×#102×C73C2211yescharacteristic
54C2×#102×C73C2211yescharacteristic
55C2×#97×C73C2211yescharacteristic
56#390×C73C2211yescharacteristic
57#389×C73C2211yescharacteristic
58#340×C73C2211yescharacteristic
59#340×C73C2211yescharacteristic
60C23×Dic73—21no—
61C22×#10596831C2311yescharacteristic
62C22×#10596831C2311yescharacteristic
63#14556582—22no—
64#14556582C2×C414yes—
65D4×Dic73—22no—
66D4×Dic73C2×C414yes—
67#14556547—44no—
68C2×C4⋊Dic73—21no—
69C2×C4⋊Dic73C2311yescharacteristic
70C2×C4⋊Dic73C2311yescharacteristic
71C2×C4⋊Dic73C2311yescharacteristic
72C2×C4⋊Dic73C2311yescharacteristic
73C2×#11924007—21no—
74C2×#11924007—21no—
75C2×#11924007—21no—
76C2×#11924007—21no—
77C2×C22⋊Dic73—21no—
78C2×C22⋊Dic73—21no—
79C2×C22⋊Dic73—21no—
80C2×C4×Dic73—21no—
81C2×C4×Dic73—21no—
82C2×C4×Dic73—21no—
83C2×C4×Dic73D411yescharacteristic
84C2×C4×Dic73D411yescharacteristic
85C2×C4×Dic73D411yescharacteristic
86C2×C4×Dic73D411yescharacteristic
87C2×C4×Dic73C2311yescharacteristic
88C2×C4×Dic73C2311yescharacteristic
89#14556525—41no—
90#14556524C2×C414yes—
91#14556523C2×C414yes—
92#14556522—41no—
93#14556520—22no—
94#14556520C2×C414yes—
95#14556512—41no—
96#14556512C2×C414yes—
97#14556502—41no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102#14556498—21no—
103#14556498—21no—
104#14556498—21no—
105#14556498—21no—
106C22×D4×C73C2311yescharacteristic
107C22×C8×C73C2311yescharacteristic
108C22×C8×C73C2311yescharacteristic
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—41no—
117#102×C73C2×C414yes—
118#97×C73—41no—
119#95×C73—21no—
120C22×Dic73—21no—
121C22×Dic73—21no—
122C22×Dic73—24no—
123C22×Dic73—21no—
124C22×Dic73—21no—
125C22×Dic73—42no—
126C22×Dic73—42no—
127C22×Dic73C2×D411yescharacteristic
128C22×Dic73C2×D411yescharacteristic
129C22×Dic73C2×D411yescharacteristic
130C22×Dic73C2×D411yescharacteristic
131C2×#10596831—21no—
132C2×#10596831—21no—
133C2×#10596831—21no—
134C2×#10596831—21no—
135C2×#10596831—21no—
136C2×#10596831C22×C412yes—
137C4⋊Dic73—21no—
138C4⋊Dic73—21no—
139C4⋊Dic73—24no—
140C4⋊Dic73—21no—
141C4⋊Dic73—21no—
142C4⋊Dic73—21no—
143C4⋊Dic73—21no—
144C4⋊Dic73C22×C412yes—
145C4⋊Dic73C22×C412yes—
146#11924007—24no—
147#11924007—42no—
148#11924007—42no—
149#11924007—42no—
150C22⋊Dic73—24no—
151C22⋊Dic73—42no—
152C22⋊Dic73—42no—
153C4×Dic73—21no—
154C4×Dic73—21no—
155C4×Dic73—22no—
156C4×Dic73—41no—
157C4×Dic73—41no—
158C4×Dic73—41no—
159C4×Dic73—42no—
160C4×Dic73—42no—
161C4×Dic73—42no—
162C4×Dic73C22×C412yes—
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×C73C22×C412yes—
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×C73C22×C412yes—
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×C73C22×C412yes—
190C22⋊C4×C73—42no—
191C22⋊C4×C73—42no—
192C42×C73—21no—
193C42×C73—21no—
194C23×C73—22no—
195C23×C73—41no—
196C23×C73—41no—
197C23×C73—41no—
198C23×C73—41no—
199C23×C73—41no—
200C23×C73—41no—
201C23×C73C22×D411yescharacteristic
202D4×C73—22no—
203D4×C73—22no—
204D4×C73—41no—
205D4×C73—41no—
206C2×C4×C73—22no—
207C2×C4×C73—21no—
208C2×C4×C73—21no—
209C2×C4×C73—21no—
210C2×C4×C73—21no—
211C2×C4×C73—21no—
212C2×C4×C73—21no—
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×C73—41no—
220C2×C4×C73—41no—
221C2×C4×C73C23×C411yescharacteristic
222C2×C4×C73C2×#4111yescharacteristic
223C2×C4×C73H(5,𝔽2)11yescharacteristic
224C2×C4×C732−1+411yescharacteristic
225C2×C4×C732−1+411yescharacteristic
226C2×C4×C732−1+411yescharacteristic
227C2×Dic73—21no—
228C2×Dic73—21no—
229C2×Dic73—21no—
230C2×Dic73—21no—
231C2×Dic73—21no—
232C2×Dic73—21no—
233C2×Dic73—22no—
234C2×Dic73—22no—
235C2×Dic73—21no—
236C2×Dic73—21no—
237C2×Dic73—21no—
238C2×Dic73—48no—
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—
249C2×Dic73C4×D412yes—
250C2×Dic73C4×D412yes—
251C2×Dic73AGL(1,ℤ/8)12yes—
252C8×C73—22no—
253C8×C73—41no—
254#10596831—22no—
255#10596831—41no—
256C22×C73—42no—
257C22×C73—42no—
258C22×C73—42no—
259C22×C73—81no—
260C22×C73—81no—
261C22×C73—81no—
262C22×C73—81no—
263C22×C73C2×C4×D411yescharacteristic
264C22×C73#53611yescharacteristic
265C22×C73#53711yescharacteristic
266C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
267C22×C73#57511yescharacteristic
268C22×C73#57511yescharacteristic
269C22×C73#57511yescharacteristic
270C4×C73—42no—
271C4×C73—41no—
272C4×C73—41no—
273C4×C73—81no—
274C4×C73—81no—
275C4×C73#51911yescharacteristic
276C4×C73#52011yescharacteristic
277C4×C73#54111yescharacteristic
278C4×C73#54211yescharacteristic
279Dic73—22no—
280Dic73—22no—
281Dic73—22no—
282Dic73—42no—
283Dic73—42no—
284Dic73—41no—
285Dic73—81no—
286Dic73—81no—
287#22120—731nomaximal, Sylow
288C2×C73—82no—
289C2×C73#232311yescharacteristic
290C2×C73#294311yescharacteristic
291C2×C73C4×AGL(1,ℤ/8)11yescharacteristic
292C2×C73#323411yescharacteristic
293C2×C73#324011yescharacteristic
294C2×C73#325111yescharacteristic
295C2×C73#325611yescharacteristic
296C2×#478—731no—
297C4×#125—731no—
298#2293—731no—
299#1989—731no—
300#1949—731no—
301#1944—731no—
302#1943—731no—
303#1936—731no—
304#1936D7311yescharacteristic
305#1926—731no—
306#1925—731no—
307#1919—731no—
308#1918—731no—
309#1804—731no—
310#1767—731no—
311C73#2212011yescharacteristic, complemented, Sylow
312C2×#125—731no—
313C2×C4×D4—731no—
314C2×#117—1461no—
315#478—738no—
316C2×#107—731no—
317C2×#107—731no—
318C2×#106—731no—
319C2×#106—731no—
320C2×C4⋊C8—731no—
321C2×C4⋊C8—731no—
322C2×#102—731no—
323C2×#102—731no—
324C2×#102C2×D7311yescharacteristic
325C2×#102C2×D7311yescharacteristic
326C2×#97—731no—
327C2×#97C2×D7311yescharacteristic
328#390—731no—
329#390C2×D7311yescharacteristic
330#389—731no—
331#389—731no—
332#389—731no—
333#389C2×D7311yescharacteristic
334#388—1461no—
335#388—1461no—
336#387—1461no—
337#387—1461no—
338#382—731no—
339C4×C4⋊C4—731no—
340C4×C4⋊C4—731no—
341C4×C4⋊C4—731no—
342C4×C4⋊C4—1461no—
343C4×C4⋊C4—1461no—
344C4×C22⋊C4—1461no—
345C43—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#336—731no—
356C22×D4C22×D7311yescharacteristic
357C23×C4—1461no—
358C22×C8—731no—
359C22×C8—731no—
360C22×C8C22×D7311yescharacteristic
361C22×C8C22×D7311yescharacteristic
362#125—734no—
363#125—1462no—
364C4×D4—734no—
365C4×D4—1462no—
366#117—2924no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—731no—
369C2×C4⋊C4—731no—
370C2×C4⋊C4—731no—
371C2×C4⋊C4—731no—
372C2×C4⋊C4—1461no—
373C2×C4⋊C4—1461no—
374C2×C4⋊C4—1461no—
375C2×C4⋊C4—1461no—
376C2×C4⋊C4—1461no—
377C2×C4⋊C4C22×D7311yescharacteristic
378C2×C4⋊C4C22×D7311yescharacteristic
379C2×C4⋊C4C22×D7311yescharacteristic
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—1461no—
383C2×C22⋊C4—1461no—
384C2×C22⋊C4—1461no—
385C2×C42—731no—
386C2×C42—731no—
387C2×C42—731no—
388C2×C42—731no—
389C2×C42—731no—
390C2×C42—731no—
391C2×C42—1461no—
392C2×C42—1461no—
393C2×C42—1461no—
394C2×C42C22×D7311yescharacteristic
395#107—2921no—
396#107—734no—
397#106—2921no—
398#106—734no—
399C4⋊C8—734no—
400C4⋊C8—1462no—
401#102—41no—
402#102—2921no—
403#102—734no—
404#102C4×D7314yes—
405#97—41no—
406#97—2921no—
407#95—21no—
408#95—1461no—
409#95—1461no—
410#95—1461no—
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×D4C2×C4×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—732no—
446C2×C8—1461no—
447C2×C8—1461no—
448C2×C8—1461no—
449C2×C8—1461no—
450C2×C8—1461no—
451C2×C8C2×C4×D7312yes—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—42no—
458C4⋊C4—2922no—
459C4⋊C4—2922no—
460C4⋊C4—2922no—
461C4⋊C4—732no—
462C4⋊C4—732no—
463C4⋊C4—1464no—
464C4⋊C4—1464no—
465C4⋊C4—1461no—
466C4⋊C4—1461no—
467C4⋊C4—1461no—
468C4⋊C4—1461no—
469C4⋊C4—1461no—
470C4⋊C4—1461no—
471C4⋊C4C2×C4×D7312yes—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—2922no—
475C22⋊C4—2922no—
476C22⋊C4—1464no—
477C42—21no—
478C42—21no—
479C42—2921no—
480C42—2921no—
481C42—2921no—
482C42—2922no—
483C42—2922no—
484C42—2922no—
485C42—732no—
486C42—732no—
487C42—1462no—
488C42—1461no—
489C42—1461no—
490C23—22no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23C2×D4×D7311yescharacteristic
498D4—22no—
499D4—22no—
500D4—41no—
501D4—41no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—2921no—
518C2×C4—2921no—
519C2×C4—2921no—
520C2×C4—2921no—
521C2×C4—2928no—
522C2×C4—2921no—
523C2×C4—2921no—
524C2×C4—2921no—
525C2×C4—2921no—
526C2×C4—2921no—
527C2×C4—2921no—
528C2×C4—732no—
529C2×C4—732no—
530C2×C4—732no—
531C2×C4—1461no—
532C2×C4—1461no—
533C2×C4—1461no—
534C2×C4—1462no—
535C2×C4—1462no—
536C2×C4—1461no—
537C2×C4—1461no—
538C2×C4—1461no—
539C2×C4—1461no—
540C2×C4—1461no—
541C2×C4—1461no—
542C2×C4C22×C4×D7311yescharacteristic
543C2×C4#41×D7311yescharacteristic
544C2×C4#1455663611yescharacteristic
545C2×C4#1455663811yescharacteristic
546C2×C4#1455663811yescharacteristic
547C2×C4#1455663811yescharacteristic
548C8—22no—
549C8—41no—
550C8—2921no—
551C8—1462no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22C4×D4×D7311yescharacteristic
560C22#1798915711yescharacteristic
561C22#1798916411yescharacteristic
562C22AGL(1,ℤ/8)×D7311yescharacteristic
563C22#1798938311yescharacteristic
564C22#1798938311yescharacteristic
565C22#1798938611yescharacteristic
566C4—41no—
567C4—42no—
568C4—41no—
569C4—81no—
570C4—81no—
571C4—2922no—
572C4—2922no—
573C4—2921no—
574C4—5841no—
575C4—5841no—
576C4—1462no—
577C4—1462no—
578C4—1462no—
579C4#1798906411yescharacteristic
580C4#1798906611yescharacteristic
581C4#1798919411yescharacteristic
582C4#1798920511yescharacteristic
583C2—82no—
584C2#2261260611yescharacteristic
585C2#2261872611yescharacteristic
586C2#2261874211yescharacteristic
587C2#2262111611yescharacteristic
588C2#2262116611yescharacteristic
589C2#2262128811yescharacteristic
590C2#2262132611yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1936C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
cosocleC22×C4×C73C24
hypercenter#1936D73
hypocenterC73#22120
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1936

Complements

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

NH ≅ G/Ntype
2. #22612490C2semidirect
4. #22610885C2semidirect
5. #22610702C2semidirect
6. #22610674C2semidirect
7. #22610666C2semidirect
11. #22610508C2semidirect
12. #22610500C2semidirect
14. #22609785C2semidirect
19. #17988889C4semidirect
311. C73#22120semidirect

Maximal subgroups

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

subgroupindex
C2×#179888892
#226124902
#226123932
#226108852
#226107022
#226106742
#226106662
#226106312
#226105612
#226105532
#226105082
#226105002
#226099392
#226097852
C73×#19362
#2212073