Finite Group Databasebeta

← #33144982

Subgroups — #33144982

Order 18688.

750 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22614964C211yesmaximal, characteristic, complemented
2#22612879C211yesmaximal, characteristic
3#22611462C211yesmaximal, characteristic
4#22611305C211yesmaximal, characteristic
5#22611301C211yesmaximal, characteristic
6#22610692C211yesmaximal, characteristic, complemented
7#22610691C211yesmaximal, characteristic, complemented
8#22610691C211yesmaximal, characteristic, complemented
9#22610648C211yesmaximal, characteristic, complemented
10#22610518C211yesmaximal, characteristic, complemented
11#22610517C211yesmaximal, characteristic, complemented
12#22610086C211yesmaximal, characteristic, complemented
13#22609929C211yesmaximal, characteristic
14#102×Dic73C211yesmaximal, characteristic
15C73×#2094C211yesmaximal, characteristic
16C2×#14556581—21no—
17C2×D4×Dic73C2211yescharacteristic
18C2×#14556547—21no—
19C2×#14556523C2211yescharacteristic
20C2×#14556522C2211yescharacteristic
21C2×#14556522C2211yescharacteristic
22C2×#14556522C2211yescharacteristic
23C2×C4⋊#10596831C2211yescharacteristic
24C2×#14556520C2211yescharacteristic
25C2×#14556520C2211yescharacteristic
26C2×#14556512C2211yescharacteristic
27C2×#14556512C2211yescharacteristic
28C2×C8×Dic73C2211yescharacteristic
29#17988520—21no—
30#17988518—21no—
31#17988490C2211yescharacteristic
32#17988486C2211yescharacteristic
33#17988485C2211yescharacteristic
34#17988482—21no—
35#17988481—21no—
36#17988475—21no—
37#17988456C2211yescharacteristic
38#17988456C2211yescharacteristic
39#17988456C2211yescharacteristic
40#17988445—21no—
41C4⋊C4×Dic73C2211yescharacteristic
42C4⋊C4×Dic73C2211yescharacteristic
43C4×#11924007C2211yescharacteristic
44C4×C4⋊Dic73C2211yescharacteristic
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988350C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988341C2211yescharacteristic
52C2×#106×C73C2211yescharacteristic
53C2×#106×C73C2211yescharacteristic
54C2×#102×C73C2211yescharacteristic
55C2×#102×C73C2211yescharacteristic
56#390×C73C2211yescharacteristic
57#384×C73C2211yescharacteristic
58#336×C73C2211yescharacteristic
59C23×Dic73—21no—
60C22×#10596831C2311yescharacteristic
61C22×#10596831C2311yescharacteristic
62#14556581—44no—
63D4×Dic73—22no—
64D4×Dic73Q814yes—
65#14556547—44no—
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×#11924007—21no—
77C2×#11924007—21no—
78C2×#11924007D411yescharacteristic
79C2×#11924007D411yescharacteristic
80C2×C22⋊Dic73—21no—
81C2×C22⋊Dic73—21no—
82C2×C22⋊Dic73—21no—
83C2×C4×Dic73—21no—
84C2×C4×Dic73—21no—
85C2×C4×Dic73—21no—
86C2×C4×Dic73C2311yescharacteristic
87C2×C4×Dic73C2311yescharacteristic
88#14556523—22no—
89#14556522—41no—
90#14556522—41no—
91#14556522—41no—
92C4⋊#10596831—22no—
93#14556520—22no—
94#14556520—22no—
95#14556512—22no—
96#14556512—41no—
97C8×Dic73—22no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102#14556498—21no—
103#14556498D411yescharacteristic
104#14556498D411yescharacteristic
105C22×D4×C73C2311yescharacteristic
106C22×C8×C73C2311yescharacteristic
107C22×C8×C73C2311yescharacteristic
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#106×C73—22no—
117#106×C73—41no—
118#102×C73—22no—
119#102×C73—41no—
120C22×Dic73—21no—
121C22×Dic73—24no—
122C22×Dic73—21no—
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×#10596831#4112yes—
137C4⋊Dic73—21no—
138C4⋊Dic73—21no—
139C4⋊Dic73—21no—
140C4⋊Dic73—21no—
141C4⋊Dic73—21no—
142C4⋊Dic73—21no—
143C4⋊Dic73—21no—
144C4⋊Dic73—42no—
145C4⋊Dic73C2×Q812yes—
146#11924007—22no—
147#11924007—42no—
148#11924007—42no—
149#11924007—42no—
150#11924007—42no—
151#11924007—81no—
152#11924007—81no—
153#11924007—81no—
154#11924007SD1614yes—
155C22⋊Dic73—24no—
156C22⋊Dic73—42no—
157C22⋊Dic73—81no—
158C4×Dic73—21no—
159C4×Dic73—21no—
160C4×Dic73—42no—
161C4×Dic73—42no—
162C4×Dic73—42no—
163C4×Dic73C2×Q812yes—
164C4×Dic73#4112yes—
165C24×C73—21no—
166C2×D4×C73—21no—
167C2×D4×C73—21no—
168C2×D4×C73—21no—
169C2×D4×C73—21no—
170C2×D4×C73—21no—
171C2×D4×C73C2×Q812yes—
172C22×C4×C73—21no—
173C22×C4×C73—21no—
174C22×C4×C73—21no—
175C22×C4×C73C2×D411yescharacteristic
176C22×C4×C73C2×D411yescharacteristic
177C22×C4×C73C2411yescharacteristic
178C2×C8×C73—21no—
179C2×C8×C73—21no—
180C2×C8×C73—21no—
181C2×C8×C73—21no—
182C2×C8×C73—21no—
183C2×C8×C73#4112yes—
184C4⋊C4×C73—21no—
185C4⋊C4×C73—21no—
186C4⋊C4×C73—21no—
187C4⋊C4×C73—21no—
188C4⋊C4×C73—21no—
189C4⋊C4×C73—21no—
190C4⋊C4×C73—42no—
191C4⋊C4×C73—81no—
192C22⋊C4×C73—42no—
193C22⋊C4×C73—81no—
194C42×C73—21no—
195C42×C73—21no—
196C23×C73—22no—
197C23×C73—41no—
198C23×C73—41no—
199C23×C73—41no—
200C23×C73—41no—
201C23×C73—41no—
202C23×C73—41no—
203C23×C73C22×D411yescharacteristic
204D4×C73—22no—
205D4×C73—22no—
206D4×C73—41no—
207D4×C73—41no—
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×C73—41no—
222C2×C4×C73C2×SD1612yes—
223C2×C4×C73C22×Q811yescharacteristic
224C2×C4×C73C2×#4111yescharacteristic
225C2×C4×C73C2×#4111yescharacteristic
226C2×C4×C73C2×#4111yescharacteristic
227C2×C4×C73H(5,𝔽2)11yescharacteristic
228C2×C4×C732−1+411yescharacteristic
229C2×Dic73—21no—
230C2×Dic73—21no—
231C2×Dic73—21no—
232C2×Dic73—21no—
233C2×Dic73—22no—
234C2×Dic73—21no—
235C2×Dic73—21no—
236C2×Dic73—21no—
237C2×Dic73—41no—
238C2×Dic73—41no—
239C2×Dic73—48no—
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×Dic73C22⋊Q812yes—
250C2×Dic73C22⋊Q812yes—
251C2×Dic73C2×SD1612yes—
252C2×Dic73C2×SD1612yes—
253C2×Dic73AGL(1,ℤ/8)12yes—
254C8×C73—22no—
255C8×C73—41no—
256#10596831—22no—
257#10596831—41no—
258C22×C73—42no—
259C22×C73—42no—
260C22×C73—42no—
261C22×C73—81no—
262C22×C73—81no—
263C22×C73—81no—
264C22×C73—81no—
265C22×C73C2×C22⋊Q811yescharacteristic
266C22×C73#54711yescharacteristic
267C22×C73#55011yescharacteristic
268C22×C73C22×SD1611yescharacteristic
269C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
270C22×C73#57711yescharacteristic
271C22×C73#57711yescharacteristic
272C4×C73—22no—
273C4×C73—41no—
274C4×C73—81no—
275C4×C73—81no—
276C4×C73—81no—
277C4×C73#53311yescharacteristic
278C4×C73#54411yescharacteristic
279C4×C73#55211yescharacteristic
280C4×C73#55311yescharacteristic
281Dic73—22no—
282Dic73—22no—
283Dic73—41no—
284Dic73—42no—
285Dic73—81no—
286Dic73—81no—
287Dic73—81no—
288Dic73#47614yes—
289#26090—731nomaximal, Sylow
290C2×C73—82no—
291C2×C73#244111yescharacteristic
292C2×C73C2×#47611yescharacteristic
293C2×C73#308411yescharacteristic
294C2×C73#333511yescharacteristic
295C2×C73#335411yescharacteristic
296C2×C73#339211yescharacteristic
297C2×C73#339711yescharacteristic
298#2552—731no—
299#2354—731no—
300#2094—731no—
301#2094D7311yescharacteristic
302#2065—731no—
303#2065—731no—
304#1948—731no—
305#1948—731no—
306#1948—731no—
307#1939—731no—
308#1920—731no—
309#1920—731no—
310#1844—731no—
311#1802—731no—
312C4×#102—731no—
313C73#2609011yescharacteristic, complemented, Sylow
314C2×#125—1461no—
315C2×C4×D4—731no—
316C2×#117—1461no—
317C2×#106—731no—
318C2×#106—731no—
319C2×#106—731no—
320C2×#106—731no—
321C2×#106C2×D7311yescharacteristic
322C2×#106C2×D7311yescharacteristic
323C2×C4⋊C8—731no—
324C2×C4⋊C8—731no—
325C2×C4⋊C8—731no—
326C2×#102—731no—
327C2×#102—731no—
328C2×#102C2×D7311yescharacteristic
329C2×#102C2×D7311yescharacteristic
330C2×C4×C8—731no—
331#398—1461no—
332#398—1461no—
333#390—731no—
334#390C2×D7311yescharacteristic
335#389—731no—
336#389—731no—
337#389—1461no—
338#388—1461no—
339#388—1461no—
340#384—731no—
341#384—731no—
342#384—731no—
343#384C2×D7311yescharacteristic
344#380—1461no—
345C4×C4⋊C4—731no—
346C4×C4⋊C4—731no—
347C4×C4⋊C4—731no—
348C4×C4⋊C4—731no—
349#340—731no—
350#340—731no—
351#340—731no—
352#340—731no—
353#340—731no—
354#340—731no—
355#336—731no—
356#336C2×D7311yescharacteristic
357C22×D4C22×D7311yescharacteristic
358C23×C4—1461no—
359C22×C8—731no—
360C22×C8—731no—
361C22×C8C22×D7311yescharacteristic
362C22×C8C22×D7311yescharacteristic
363#125—2924no—
364C4×D4—734no—
365C4×D4—1462no—
366#117—2924no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—731no—
370C2×C4⋊C4—731no—
371C2×C4⋊C4—731no—
372C2×C4⋊C4—731no—
373C2×C4⋊C4—731no—
374C2×C4⋊C4—731no—
375C2×C4⋊C4—1461no—
376C2×C4⋊C4—1461no—
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⋊C4C22×D7311yescharacteristic
384C2×C4⋊C4C22×D7311yescharacteristic
385C2×C4⋊C4C22×D7311yescharacteristic
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—1461no—
389C2×C22⋊C4—1461no—
390C2×C22⋊C4—1461no—
391C2×C42—731no—
392C2×C42—731no—
393C2×C42—1461no—
394C2×C42—1461no—
395C2×C42—1461no—
396C2×C42C22×D7311yescharacteristic
397#106—22no—
398#106—41no—
399#106—2921no—
400#106—2921no—
401#106—2921no—
402#106—1462no—
403C4⋊C8—1462no—
404C4⋊C8—1462no—
405C4⋊C8—1462no—
406#102—22no—
407#102—41no—
408#102—2921no—
409#102—1462no—
410C4×C8—1462no—
411#95—731no—
412#95—731no—
413#95—1461no—
414#95—1461no—
415#95—1461no—
416#95—1461no—
417#95—1461no—
418C24—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4Q8×D7312yes—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—2922no—
429C22×C4—2922no—
430C22×C4—731no—
431C22×C4—731no—
432C22×C4—731no—
433C22×C4—731no—
434C22×C4—1461no—
435C22×C4—1461no—
436C22×C4—1461no—
437C22×C4—1464no—
438C22×C4—1461no—
439C22×C4D4×D7311yescharacteristic
440C22×C4D4×D7311yescharacteristic
441C22×C4C23×D7311yescharacteristic
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—732no—
448C2×C8—1461no—
449C2×C8—1461no—
450C2×C8—1461no—
451C2×C8—1461no—
452C2×C8—1461no—
453C2×C8#1192402712yes—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—42no—
461C4⋊C4—81no—
462C4⋊C4—2922no—
463C4⋊C4—2922no—
464C4⋊C4—2922no—
465C4⋊C4—2922no—
466C4⋊C4—2922no—
467C4⋊C4—5841no—
468C4⋊C4—5841no—
469C4⋊C4—5841no—
470C4⋊C4—734no—
471C4⋊C4—732no—
472C4⋊C4—1462no—
473C4⋊C4—1461no—
474C4⋊C4—1461no—
475C4⋊C4—1461no—
476C4⋊C4—1461no—
477C4⋊C4—1461no—
478C4⋊C4—1461no—
479C4⋊C4—1461no—
480C22⋊C4—42no—
481C22⋊C4—81no—
482C22⋊C4—2922no—
483C22⋊C4—5841no—
484C22⋊C4—1464no—
485C42—21no—
486C42—21no—
487C42—2922no—
488C42—2922no—
489C42—2922no—
490C42—732no—
491C42—732no—
492C42—1461no—
493C42—1461no—
494C23—22no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23C2×D4×D7311yescharacteristic
502D4—22no—
503D4—22no—
504D4—41no—
505D4—41no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
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—41no—
520C2×C4—2921no—
521C2×C4—2921no—
522C2×C4—2928no—
523C2×C4—2921no—
524C2×C4—2921no—
525C2×C4—2921no—
526C2×C4—2921no—
527C2×C4—2921no—
528C2×C4—2921no—
529C2×C4—2921no—
530C2×C4—2921no—
531C2×C4—2921no—
532C2×C4—732no—
533C2×C4—732no—
534C2×C4—732no—
535C2×C4—732no—
536C2×C4—732no—
537C2×C4—1461no—
538C2×C4—1461no—
539C2×C4—1462no—
540C2×C4—1461no—
541C2×C4—1461no—
542C2×C4—1461no—
543C2×C4—1461no—
544C2×C4—1461no—
545C2×C4SD16×D7312yes—
546C2×C4C2×Q8×D7311yescharacteristic
547C2×C4#41×D7311yescharacteristic
548C2×C4#41×D7311yescharacteristic
549C2×C4C2×#1192402711yescharacteristic
550C2×C4#1455663611yescharacteristic
551C2×C4#1455663811yescharacteristic
552C8—22no—
553C8—41no—
554C8—2921no—
555C8—1462no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—81no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22C22⋊Q8×D7311yescharacteristic
564C22#1798923511yescharacteristic
565C22#1798925711yescharacteristic
566C22C2×SD16×D7311yescharacteristic
567C22AGL(1,ℤ/8)×D7311yescharacteristic
568C22#1798939311yescharacteristic
569C22#1798939711yescharacteristic
570C4—22no—
571C4—41no—
572C4—81no—
573C4—81no—
574C4—81no—
575C4—2922no—
576C4—2921no—
577C4—5841no—
578C4—5841no—
579C4—5841no—
580C4—734no—
581C4—1462no—
582C4—1462no—
583C4#1798913911yescharacteristic
584C4#1798921611yescharacteristic
585C4#1798927411yescharacteristic
586C4#1798928211yescharacteristic
587C2—82no—
588C2#2261363711yescharacteristic
589C2#476×D7311yescharacteristic
590C2#2261978811yescharacteristic
591C2#2262228611yescharacteristic
592C2#2262251911yescharacteristic
593C2#2262290911yescharacteristic
594C2#2262294811yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2094C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
hypercenter#2094D73
hypocenterC73#26090
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2094

Complements

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

NH ≅ G/Ntype
1. #22614964C2semidirect
6. #22610692C2semidirect
7. #22610691C2semidirect
8. #22610691C2semidirect
9. #22610648C2semidirect
10. #22610518C2semidirect
11. #22610517C2semidirect
12. #22610086C2semidirect
313. C73#26090semidirect

Maximal subgroups

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

subgroupindex
#226149642
#226128792
#226114622
#226113052
#226113012
#226106922
#226106912
#226106482
#226105182
#226105172
#226100862
#226099292
#102×Dic732
C73×#20942
#2609073