Finite Group Databasebeta

← #33158728

Subgroups — #33158728

Order 18688.

708 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22614366C211yesmaximal, characteristic
2#22612936C211yesmaximal, characteristic
3#22611343C211yesmaximal, characteristic, complemented
4#22611321C211yesmaximal, characteristic, complemented
5#22611135C211yesmaximal, characteristic
6#22611134C211yesmaximal, characteristic
7#22610688C211yesmaximal, characteristic, complemented
8#22610662C211yesmaximal, characteristic, complemented
9#22610540C211yesmaximal, characteristic
10#22610525C211yesmaximal, characteristic, complemented
11#22610521C211yesmaximal, characteristic, complemented
12#22610182C211yesmaximal, characteristic, complemented
13#22610021C211yesmaximal, characteristic
14#22609750C211yesmaximal, characteristic, complemented
15C73×#1852C211yesmaximal, characteristic
16C2×#14556570C2211yescharacteristic
17C2×#14556556C2211yescharacteristic
18C2×#14556525C2211yescharacteristic
19C2×#14556524C2211yescharacteristic
20C2×#14556520C2211yescharacteristic
21C2×#14556516C2211yescharacteristic
22C2×#14556516C2211yescharacteristic
23C2×#14556514C2211yescharacteristic
24C2×#14556514C2211yescharacteristic
25C2×C22⋊#10596831C2211yescharacteristic
26C2×C8×Dic73C2211yescharacteristic
27#17988528—21no—
28#17988510—21no—
29#17988492C2211yescharacteristic
30#17988486C2211yescharacteristic
31#17988483C2211yescharacteristic
32#17988483C2211yescharacteristic
33#17988482—21no—
34#17988481—21no—
35#17988472—21no—
36#17988468—21no—
37#17988460—21no—
38#17988456C2211yescharacteristic
39#17988447C2211yescharacteristic
40C4⋊C4×Dic73C2211yescharacteristic
41C4⋊C4×Dic73C2211yescharacteristic
42C4×C22⋊Dic73—21no—
43C4×C22⋊Dic73C2211yescharacteristic
44#17988350C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988341C2211yescharacteristic
51#17988341C2211yescharacteristic
52C22×C4⋊C4×C73C2211yescharacteristic
53C2×#107×C73C2211yescharacteristic
54C2×#107×C73C2211yescharacteristic
55C2×C22⋊C8×C73C2211yescharacteristic
56#380×C73C2211yescharacteristic
57#340×C73C2211yescharacteristic
58#340×C73C2211yescharacteristic
59C22×Dic146C2311yescharacteristic
60C22×#10596831C2311yescharacteristic
61C22×#10596831C2311yescharacteristic
62#14556570—22no—
63#14556570—22no—
64#14556556—22no—
65#14556556—41no—
66C2×C4⋊Dic73—21no—
67C2×C4⋊Dic73C2311yescharacteristic
68C2×C4⋊Dic73C2311yescharacteristic
69C2×C4⋊Dic73C2311yescharacteristic
70C2×#11924007—21no—
71C2×#11924007—21no—
72C2×#11924007—21no—
73C2×#11924007—21no—
74C2×#11924007—21no—
75C2×#11924007—21no—
76C2×C22⋊Dic73—21no—
77C2×C22⋊Dic73—21no—
78C2×C22⋊Dic73D411yescharacteristic
79C2×C22⋊Dic73D411yescharacteristic
80C2×C4×Dic73—21no—
81C2×C4×Dic73—21no—
82C2×C4×Dic73—21no—
83C2×C4×Dic73C2311yescharacteristic
84C2×C4×Dic73C2311yescharacteristic
85#14556525—22no—
86#14556524—22no—
87#14556520—22no—
88#14556516—22no—
89#14556516—22no—
90#14556514—22no—
91#14556514—41no—
92C22⋊#10596831—41no—
93C8×Dic73—41no—
94#14556498—21no—
95#14556498—21no—
96#14556498—21no—
97#14556498—21no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102#14556498D411yescharacteristic
103#14556498D411yescharacteristic
104C23×C4×C73—21no—
105C23×C4×C73C2311yescharacteristic
106C22×C8×C73C2311yescharacteristic
107C22×C8×C73C2311yescharacteristic
108C2×C4⋊C4×C73—22no—
109C2×C4⋊C4×C73—21no—
110C2×C4⋊C4×C73—41no—
111C2×C4⋊C4×C73C2311yescharacteristic
112C2×C4⋊C4×C73C2311yescharacteristic
113C2×C4⋊C4×C73C2311yescharacteristic
114C2×C4⋊C4×C73C2311yescharacteristic
115C2×C22⋊C4×C73—21no—
116#107×C73—22no—
117#107×C73—41no—
118C22⋊C8×C73—22no—
119#95×C73—21no—
120C2×Dic146—21no—
121C2×Dic146—21no—
122C2×Dic146—21no—
123C2×Dic146—21no—
124C2×Dic146—21no—
125C2×Dic146#4112yes—
126C22×Dic73—21no—
127C22×Dic73—21no—
128C22×Dic73—21no—
129C22×Dic73—21no—
130C22×Dic73C2×D411yescharacteristic
131C22×Dic73C2×D411yescharacteristic
132C22×Dic73C2×D411yescharacteristic
133C22×Dic73C2×D411yescharacteristic
134C2×#10596831—21no—
135C2×#10596831—21no—
136C2×#10596831—21no—
137C2×#10596831—21no—
138C2×#10596831—21no—
139C2×#10596831—21no—
140C4⋊Dic73—21no—
141C4⋊Dic73—21no—
142C4⋊Dic73—21no—
143C4⋊Dic73—21no—
144C4⋊Dic73—21no—
145C4⋊Dic73—42no—
146C4⋊Dic73#4112yes—
147#11924007—24no—
148#11924007—42no—
149#11924007—81no—
150#11924007—81no—
151#11924007—81no—
152#11924007—81no—
153C22⋊Dic73—24no—
154C22⋊Dic73—41no—
155C22⋊Dic73—81no—
156C22⋊Dic73Q1614yes—
157C4×Dic73—21no—
158C4×Dic73—21no—
159C4×Dic73—21no—
160C4×Dic73—21no—
161C4×Dic73—42no—
162C4×Dic73—81no—
163C4×Dic73—81no—
164C24×C73C2×D411yescharacteristic
165C22×C4×C73—21no—
166C22×C4×C73—21no—
167C22×C4×C73—21no—
168C22×C4×C73—21no—
169C22×C4×C73—21no—
170C22×C4×C73—21no—
171C22×C4×C73—21no—
172C22×C4×C73—21no—
173C22×C4×C73—21no—
174C22×C4×C73—21no—
175C22×C4×C73—42no—
176C22×C4×C73—42no—
177C22×C4×C73—81no—
178C22×C4×C73C2×D411yescharacteristic
179C22×C4×C73C2411yescharacteristic
180C2×C8×C73—21no—
181C2×C8×C73—21no—
182C2×C8×C73—21no—
183C2×C8×C73—21no—
184C2×C8×C73—21no—
185C2×C8×C73#4112yes—
186C4⋊C4×C73—21no—
187C4⋊C4×C73—21no—
188C4⋊C4×C73—21no—
189C4⋊C4×C73—21no—
190C4⋊C4×C73—21no—
191C4⋊C4×C73—21no—
192C4⋊C4×C73—21no—
193C4⋊C4×C73—42no—
194C4⋊C4×C73#4112yes—
195C22⋊C4×C73—81no—
196C23×C73—21no—
197C23×C73—21no—
198C23×C73—21no—
199C23×C73—21no—
200C23×C73—21no—
201C23×C73—21no—
202C23×C73C2×Q1612yes—
203C23×C73C22×D411yescharacteristic
204C2×C4×C73—21no—
205C2×C4×C73—22no—
206C2×C4×C73—21no—
207C2×C4×C73—21no—
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×C73—41no—
218C2×C4×C73—41no—
219C2×C4×C73—41no—
220C2×C4×C73—41no—
221C2×C4×C73—41no—
222C2×C4×C73—41no—
223C2×C4×C73—84no—
224C2×C4×C73C2×#4111yescharacteristic
225C2×C4×C73C2×#4111yescharacteristic
226C2×C4×C73C2×#4111yescharacteristic
227C2×C4×C73C2×#4111yescharacteristic
228C2×C4×C73H(5,𝔽2)11yescharacteristic
229C2×C4×C732−1+411yescharacteristic
230C2×Dic73—21no—
231C2×Dic73—21no—
232C2×Dic73—21no—
233C2×Dic73—21no—
234C2×Dic73—22no—
235C2×Dic73—22no—
236C2×Dic73—21no—
237C2×Dic73—21no—
238C2×Dic73—21no—
239C2×Dic73—21no—
240C2×Dic73—21no—
241C2×Dic73—21no—
242C2×Dic73—41no—
243C2×Dic73—41no—
244C2×Dic73—41no—
245C2×Dic73—41no—
246C2×Dic73—41no—
247C2×Dic73—41no—
248C2×Dic73—41no—
249C2×Dic73—41no—
250C2×Dic73—41no—
251C2×Dic73—41no—
252C2×Dic73C2×Q1612yes—
253C2×Dic73C2×Q1612yes—
254Dic146—22no—
255Dic146—22no—
256Dic146—41no—
257Dic146—41no—
258C8×C73—22no—
259C8×C73—41no—
260#10596831—41no—
261#10596831—41no—
262C22×C73—22no—
263C22×C73—22no—
264C22×C73—22no—
265C22×C73—41no—
266C22×C73—41no—
267C22×C73—41no—
268C22×C73—41no—
269C22×C73#52511yescharacteristic
270C22×C73#54611yescharacteristic
271C22×C73#54711yescharacteristic
272C22×C73C22×Q1611yescharacteristic
273C22×C73#57511yescharacteristic
274C22×C73#57611yescharacteristic
275C22×C73#57711yescharacteristic
276C4×C73—41no—
277C4×C73—42no—
278C4×C73—81no—
279C4×C73—81no—
280C4×C73—81no—
281C4×C73#52911yescharacteristic
282C4×C73#53911yescharacteristic
283C4×C73#55311yescharacteristic
284C4×C73#55611yescharacteristic
285Dic73—22no—
286Dic73—22no—
287Dic73—41no—
288Dic73—42no—
289Dic73—42no—
290Dic73—41no—
291Dic73—81no—
292Dic73—81no—
293#27036—731nomaximal, Sylow
294C2×C73—42no—
295C2×C73#246611yescharacteristic
296C2×C73#310411yescharacteristic
297C2×C73#311911yescharacteristic
298C2×C73#329911yescharacteristic
299C2×C73#331911yescharacteristic
300C2×C73#333811yescharacteristic
301C2×C73#335111yescharacteristic
302#2497—731no—
303#2360—731no—
304#2072—731no—
305#2068—731no—
306#2036—731no—
307#2036—731no—
308#1947—731no—
309#1942—731no—
310#1923—731no—
311#1921—731no—
312#1921—731no—
313#1863—731no—
314#1852D7311yescharacteristic
315#1824—731no—
316C4×#103—731no—
317C73#2703611yescharacteristic, complemented, Sylow
318C2×C22⋊Q8—731no—
319C2×C4×Q8—731no—
320C22×C4⋊C4C2×D7311yescharacteristic
321C2×#107—731no—
322C2×#107—731no—
323C2×#107C2×D7311yescharacteristic
324C2×#107C2×D7311yescharacteristic
325C2×C4⋊C8—731no—
326C2×#103—731no—
327C2×#103—731no—
328C2×#103—731no—
329C2×#103—731no—
330C2×C22⋊C8—731no—
331C2×C22⋊C8C2×D7311yescharacteristic
332C2×C4×C8—731no—
333#400—1461no—
334#396—1461no—
335#391—731no—
336#389—731no—
337#389—731no—
338#389—731no—
339#389—1461no—
340#388—1461no—
341#387—1461no—
342#387—1461no—
343#385—1461no—
344#384—731no—
345#380—731no—
346#380C2×D7311yescharacteristic
347C4×C4⋊C4—731no—
348C4×C4⋊C4—731no—
349C4×C22⋊C4—731no—
350C4×C22⋊C4—1461no—
351#340—731no—
352#340—731no—
353#340—731no—
354#340—731no—
355#340—731no—
356#340—731no—
357#340C2×D7311yescharacteristic
358#340C2×D7311yescharacteristic
359#336—731no—
360#336—731no—
361C22×Q8—731no—
362C23×C4—21no—
363C23×C4C22×D7311yescharacteristic
364C22×C8—731no—
365C22×C8—731no—
366C22×C8C22×D7311yescharacteristic
367C22×C8C22×D7311yescharacteristic
368C22⋊Q8—1462no—
369C22⋊Q8—1462no—
370C4×Q8—2921no—
371C4×Q8—1462no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—41no—
375C2×C4⋊C4—731no—
376C2×C4⋊C4—731no—
377C2×C4⋊C4—731no—
378C2×C4⋊C4—1461no—
379C2×C4⋊C4—1461no—
380C2×C4⋊C4—1461no—
381C2×C4⋊C4—1461no—
382C2×C4⋊C4—1461no—
383C2×C4⋊C4—1461no—
384C2×C4⋊C4—1461no—
385C2×C4⋊C4C22×D7311yescharacteristic
386C2×C4⋊C4C22×D7311yescharacteristic
387C2×C4⋊C4C22×D7311yescharacteristic
388C2×C4⋊C4C22×D7311yescharacteristic
389C2×C22⋊C4—21no—
390C2×C22⋊C4—731no—
391C2×C22⋊C4—731no—
392C2×C22⋊C4—1461no—
393C2×C22⋊C4—1461no—
394C2×C42—731no—
395C2×C42—731no—
396C2×C42—1461no—
397C2×C42—1461no—
398C2×C42—1461no—
399#107—22no—
400#107—41no—
401#107—1462no—
402#107—1462no—
403C4⋊C8—1462no—
404#103—2921no—
405#103—1462no—
406#103—1462no—
407#103—1462no—
408C22⋊C8—22no—
409C22⋊C8—2921no—
410C4×C8—2921no—
411#95—21no—
412#95—731no—
413#95—731no—
414#95—1461no—
415#95—1461no—
416#95—1461no—
417#95—1461no—
418#95—1461no—
419#95—1461no—
420#95—1461no—
421#95—1461no—
422C24D4×D7311yescharacteristic
423C2×Q8—732no—
424C2×Q8—1461no—
425C2×Q8—1461no—
426C2×Q8—1461no—
427C2×Q8—1461no—
428C2×Q8—1461no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—81no—
442C22×C4—731no—
443C22×C4—731no—
444C22×C4—731no—
445C22×C4—731no—
446C22×C4—1461no—
447C22×C4—1461no—
448C22×C4—1461no—
449C22×C4—1461no—
450C22×C4D4×D7311yescharacteristic
451C22×C4C23×D7311yescharacteristic
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—1461no—
458C2×C8—1461no—
459C2×C8—1461no—
460C2×C8—1461no—
461C2×C8—1461no—
462C2×C8—1461no—
463C2×C8#1192402812yes—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—42no—
472C4⋊C4—2922no—
473C4⋊C4—2922no—
474C4⋊C4—5841no—
475C4⋊C4—5841no—
476C4⋊C4—5841no—
477C4⋊C4—5841no—
478C4⋊C4—732no—
479C4⋊C4—1464no—
480C4⋊C4—1461no—
481C4⋊C4—1461no—
482C4⋊C4—1461no—
483C4⋊C4—1461no—
484C4⋊C4—1461no—
485C4⋊C4#1192402712yes—
486C22⋊C4—81no—
487C22⋊C4—2921no—
488C22⋊C4—5841no—
489C22⋊C4—734no—
490C22⋊C4—1464no—
491C42—2922no—
492C42—5841no—
493C42—5841no—
494C42—1461no—
495C42—1461no—
496C42—1461no—
497C42—1461no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23Q16×D7312yes—
505C23C2×D4×D7311yescharacteristic
506Q8—2921no—
507Q8—2921no—
508Q8—1462no—
509Q8—1462no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—84no—
530C2×C4—2921no—
531C2×C4—2921no—
532C2×C4—2921no—
533C2×C4—2921no—
534C2×C4—2921no—
535C2×C4—2921no—
536C2×C4—2921no—
537C2×C4—2921no—
538C2×C4—2921no—
539C2×C4—2921no—
540C2×C4—732no—
541C2×C4—732no—
542C2×C4—1461no—
543C2×C4—1461no—
544C2×C4—1461no—
545C2×C4—1461no—
546C2×C4—1461no—
547C2×C4—1461no—
548C2×C4—1462no—
549C2×C4—1462no—
550C2×C4—1461no—
551C2×C4—1461no—
552C2×C4—1461no—
553C2×C4—1461no—
554C2×C4#41×D7311yescharacteristic
555C2×C4#41×D7311yescharacteristic
556C2×C4C2×#1192402711yescharacteristic
557C2×C4C2×#1192402811yescharacteristic
558C2×C4#1455663611yescharacteristic
559C2×C4#1455663911yescharacteristic
560C8—22no—
561C8—41no—
562C8—2921no—
563C8—2921no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22#1798909511yescharacteristic
572C22#1798922411yescharacteristic
573C22#1798923811yescharacteristic
574C22C2×Q16×D7311yescharacteristic
575C22#1798938311yescharacteristic
576C22#1798938811yescharacteristic
577C22#1798939611yescharacteristic
578C4—42no—
579C4—41no—
580C4—81no—
581C4—81no—
582C4—81no—
583C4—2921no—
584C4—2921no—
585C4—2922no—
586C4—2922no—
587C4—5841no—
588C4—5841no—
589C4—1462no—
590C4—1462no—
591C4#1798911911yescharacteristic
592C4#1798918311yescharacteristic
593C4#1798928611yescharacteristic
594C4#1798930811yescharacteristic
595C2—42no—
596C2#2261396411yescharacteristic
597C2#2261996711yescharacteristic
598C2#2262013311yescharacteristic
599C2#2262180011yescharacteristic
600C2#2262208011yescharacteristic
601C2#2262232111yescharacteristic
602C2#2262248711yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#1852C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
hypercenter#1852D73
hypocenterC73#27036
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1852

Complements

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

NH ≅ G/Ntype
3. #22611343C2semidirect
4. #22611321C2semidirect
7. #22610688C2semidirect
8. #22610662C2semidirect
10. #22610525C2semidirect
11. #22610521C2semidirect
12. #22610182C2semidirect
14. #22609750C2semidirect
317. C73#27036semidirect

Maximal subgroups

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

subgroupindex
#226143662
#226129362
#226113432
#226113212
#226111352
#226111342
#226106882
#226106622
#226105402
#226105252
#226105212
#226101822
#226100212
#226097502
C73×#18522
#2703673