Finite Group Databasebeta

← #33159059

Subgroups — #33159059

Order 18688.

700 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22614353C211yesmaximal, characteristic
2#22612956C211yesmaximal, characteristic
3#22611341C211yesmaximal, characteristic, complemented
4#22611311C211yesmaximal, characteristic, complemented
5#22611135C211yesmaximal, characteristic
6#22611135C211yesmaximal, characteristic
7#22610690C211yesmaximal, characteristic, complemented
8#22610669C211yesmaximal, characteristic, complemented
9#22610524C211yesmaximal, characteristic, complemented
10#22610503C211yesmaximal, characteristic, complemented
11#22610188C211yesmaximal, characteristic, complemented
12#22610132C211yesmaximal, characteristic
13#22610027C211yesmaximal, characteristic
14#22609762C211yesmaximal, characteristic, complemented
15C73×#1924C211yesmaximal, characteristic
16C2×#14556584—21no—
17C2×#14556570C2211yescharacteristic
18C2×C4×Dic146C2211yescharacteristic
19C2×#14556550C2211yescharacteristic
20C2×#14556525C2211yescharacteristic
21C2×#14556525C2211yescharacteristic
22C2×#14556525C2211yescharacteristic
23C2×#14556523C2211yescharacteristic
24C2×#14556516C2211yescharacteristic
25C2×#14556516C2211yescharacteristic
26C2×#14556516C2211yescharacteristic
27C2×#14556516C2211yescharacteristic
28#17988527—21no—
29#17988512—21no—
30#17988493C2211yescharacteristic
31#17988485C2211yescharacteristic
32#17988485C2211yescharacteristic
33#17988485C2211yescharacteristic
34#17988481—21no—
35#17988475—21no—
36#17988472—21no—
37#17988469—21no—
38#17988457C2211yescharacteristic
39#17988447C2211yescharacteristic
40#17988447C2211yescharacteristic
41C4×C4⋊Dic73C2211yescharacteristic
42C4×C4⋊Dic73C2211yescharacteristic
43#17988438—21no—
44#17988350C2211yescharacteristic
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988350C2211yescharacteristic
48#17988350C2211yescharacteristic
49#17988350C2211yescharacteristic
50#17988350C2211yescharacteristic
51#17988350C2211yescharacteristic
52C2×C4⋊C8×C73C2211yescharacteristic
53C2×C22⋊C8×C73C2211yescharacteristic
54C2×C22⋊C8×C73C2211yescharacteristic
55C2×#97×C73C2211yescharacteristic
56C4×C22⋊C4×C73C2211yescharacteristic
57#336×C73C2211yescharacteristic
58#336×C73C2211yescharacteristic
59C22×Dic146C2311yescharacteristic
60#14556584—44no—
61#14556570—22no—
62#14556570—22no—
63C4×Dic146—22no—
64C4×Dic146—41no—
65#14556550—22no—
66#14556550—41no—
67C2×C4⋊Dic73—21no—
68C2×C4⋊Dic73—21no—
69C2×C4⋊Dic73C2311yescharacteristic
70C2×C4⋊Dic73C2311yescharacteristic
71C2×C4⋊Dic73C2311yescharacteristic
72C2×C4⋊Dic73C2311yescharacteristic
73C2×C4⋊Dic73C2311yescharacteristic
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×#11924007—21no—
77C2×#11924007—21no—
78C2×#11924007—21no—
79C2×#11924007—21no—
80C2×#11924007—21no—
81C2×C22⋊Dic73—21no—
82C2×C22⋊Dic73—21no—
83C2×C22⋊Dic73—21no—
84C2×C22⋊Dic73—21no—
85C2×C4×Dic73—21no—
86C2×C4×Dic73—21no—
87C2×C4×Dic73—21no—
88C2×C4×Dic73—21no—
89#14556525—22no—
90#14556525—22no—
91#14556525—22no—
92#14556523—41no—
93#14556516—22no—
94#14556516—22no—
95#14556516—22no—
96#14556516—41no—
97#14556498—21no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102#14556498—21no—
103#14556498—21no—
104#14556498—21no—
105#14556498—21no—
106C23×C4×C73C2311yescharacteristic
107C22×C8×C73C2311yescharacteristic
108C22×C8×C73C2311yescharacteristic
109C22×C8×C73C2311yescharacteristic
110C22×C8×C73C2311yescharacteristic
111C2×C22⋊C4×C73D411yescharacteristic
112C2×C22⋊C4×C73D411yescharacteristic
113C2×C42×C73C2311yescharacteristic
114C2×C42×C73C2311yescharacteristic
115C4⋊C8×C73—22no—
116C22⋊C8×C73—22no—
117C22⋊C8×C73—41no—
118#97×C73—41no—
119#95×C73D411yescharacteristic
120#95×C73D411yescharacteristic
121C2×Dic146—21no—
122C2×Dic146—21no—
123C2×Dic146—21no—
124C2×Dic146—21no—
125C2×Dic146—21no—
126C2×Dic146#4112yes—
127C22×Dic73—21no—
128C22×Dic73—21no—
129C22×Dic73—21no—
130C22×Dic73—21no—
131C22×Dic73—21no—
132C22×Dic73—21no—
133C22×Dic73—21no—
134C22×Dic73—21no—
135C4⋊Dic73—21no—
136C4⋊Dic73—21no—
137C4⋊Dic73—21no—
138C4⋊Dic73—21no—
139C4⋊Dic73—21no—
140C4⋊Dic73—21no—
141C4⋊Dic73—21no—
142C4⋊Dic73—21no—
143C4⋊Dic73—21no—
144C4⋊Dic73—21no—
145C4⋊Dic73—21no—
146C4⋊Dic73—21no—
147C4⋊Dic73—42no—
148C4⋊Dic73—42no—
149C4⋊Dic73#4112yes—
150C4⋊Dic73#4112yes—
151#11924007—24no—
152#11924007—42no—
153#11924007—42no—
154#11924007—42no—
155#11924007—42no—
156C22⋊Dic73—24no—
157C22⋊Dic73—42no—
158C22⋊Dic73—42no—
159C22⋊Dic73—42no—
160C4×Dic73—42no—
161C4×Dic73—42no—
162C4×Dic73—42no—
163C4×Dic73—42no—
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×C73C2×D411yescharacteristic
172C22×C4×C73C2×D411yescharacteristic
173C22×C4×C73C2×D411yescharacteristic
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×C73—21no—
183C2×C8×C73—21no—
184C2×C8×C73—21no—
185C2×C8×C73—21no—
186C2×C8×C73—21no—
187C2×C8×C73—21no—
188C2×C8×C73#4112yes—
189C22⋊C4×C73—22no—
190C22⋊C4×C73—41no—
191C42×C73—21no—
192C42×C73—21no—
193C42×C73—21no—
194C42×C73—21no—
195C23×C73—21no—
196C23×C73—21no—
197C23×C73—21no—
198C23×C73—21no—
199C23×C73—21no—
200C23×C73—21no—
201C23×C73—21no—
202C23×C73C22×D411yescharacteristic
203C2×C4×C73—21no—
204C2×C4×C73—21no—
205C2×C4×C73—21no—
206C2×C4×C73—21no—
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—21no—
214C2×C4×C73—21no—
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—41no—
218C2×C4×C73—41no—
219C2×C4×C73—41no—
220C2×C4×C73—41no—
221C2×C4×C73AGL(1,ℤ/8)12yes—
222C2×C4×C73C2×#4111yescharacteristic
223C2×C4×C73C2×#4111yescharacteristic
224C2×C4×C73C2×#4111yescharacteristic
225C2×C4×C73C2×#4111yescharacteristic
226C2×C4×C73H(5,𝔽2)11yescharacteristic
227C2×C4×C732−1+411yescharacteristic
228C2×Dic73—22no—
229C2×Dic73—22no—
230C2×Dic73—22no—
231C2×Dic73—41no—
232C2×Dic73—41no—
233C2×Dic73—41no—
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×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—
252Dic146—22no—
253Dic146—22no—
254Dic146—41no—
255Dic146—41no—
256C8×C73—22no—
257C8×C73—41no—
258C8×C73—41no—
259C8×C73—41no—
260C22×C73—41no—
261C22×C73—41no—
262C22×C73—41no—
263C22×C73—41no—
264C22×C73—41no—
265C22×C73—41no—
266C22×C73—41no—
267C22×C73#52511yescharacteristic
268C22×C73#54611yescharacteristic
269C22×C73#54711yescharacteristic
270C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
271C22×C73#57511yescharacteristic
272C22×C73#57811yescharacteristic
273C22×C73#57811yescharacteristic
274C4×C73—22no—
275C4×C73—41no—
276C4×C73—41no—
277C4×C73—41no—
278C4×C73—41no—
279C4×C73#52911yescharacteristic
280C4×C73#53911yescharacteristic
281C4×C73#55311yescharacteristic
282C4×C73#55611yescharacteristic
283Dic73—42no—
284Dic73—42no—
285Dic73—42no—
286Dic73—81no—
287Dic73—81no—
288Dic73—81no—
289Dic73—81no—
290Dic73—81no—
291#27058—731nomaximal, Sylow
292C2×C73—81no—
293C2×C73#246611yescharacteristic
294C2×C73#311511yescharacteristic
295C2×C73#312711yescharacteristic
296C2×C73#331311yescharacteristic
297C2×C73#333011yescharacteristic
298C2×C73#334311yescharacteristic
299C2×C73#334911yescharacteristic
300#2496—731no—
301#2362—731no—
302#2072—731no—
303#2066—731no—
304#2036—731no—
305#2036—731no—
306#1947—731no—
307#1943—731no—
308#1924D7311yescharacteristic
309#1921—731no—
310#1918—731no—
311#1863—731no—
312#1854—731no—
313#1825—731no—
314#1763—731no—
315C73#2705811yescharacteristic, complemented, Sylow
316C2×#126—1461no—
317C2×C22⋊Q8—731no—
318C2×C4×Q8—731no—
319C2×#117—731no—
320C2×#107—731no—
321C2×#107—731no—
322C2×#107—731no—
323C2×#106—731no—
324C2×C4⋊C8C2×D7311yescharacteristic
325C2×#103—731no—
326C2×#103—731no—
327C2×#103—731no—
328C2×#103—731no—
329C2×C22⋊C8C2×D7311yescharacteristic
330C2×C22⋊C8C2×D7311yescharacteristic
331C2×#97C2×D7311yescharacteristic
332#400—1461no—
333#397—1461no—
334#391—731no—
335#389—731no—
336#389—731no—
337#389—731no—
338#388—1461no—
339#388—1461no—
340#387—1461no—
341#387—1461no—
342#384—731no—
343#380—731no—
344#380—731no—
345C4×C4⋊C4—731no—
346C4×C4⋊C4—731no—
347C4×C4⋊C4—1461no—
348C4×C22⋊C4C2×D7311yescharacteristic
349#340—731no—
350#340—731no—
351#340—731no—
352#340—731no—
353#340—731no—
354#340—731no—
355#340—731no—
356#340—731no—
357#336C2×D7311yescharacteristic
358#336C2×D7311yescharacteristic
359C22×Q8—731no—
360C23×C4C22×D7311yescharacteristic
361C22×C8C22×D7311yescharacteristic
362C22×C8C22×D7311yescharacteristic
363C22×C8C22×D7311yescharacteristic
364C22×C8C22×D7311yescharacteristic
365#126—2924no—
366C22⋊Q8—1462no—
367C22⋊Q8—1462no—
368C4×Q8—2921no—
369C4×Q8—1462no—
370#117—2921no—
371#117—1462no—
372C2×C4⋊C4—731no—
373C2×C4⋊C4—731no—
374C2×C4⋊C4—731no—
375C2×C4⋊C4—731no—
376C2×C4⋊C4—731no—
377C2×C4⋊C4—731no—
378C2×C4⋊C4—731no—
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⋊C4—1461no—
386C2×C22⋊C4—1461no—
387C2×C22⋊C4—1461no—
388C2×C22⋊C4—1461no—
389C2×C22⋊C4—1461no—
390C2×C22⋊C4D29211yescharacteristic
391C2×C22⋊C4D29211yescharacteristic
392C2×C42—1461no—
393C2×C42—1461no—
394C2×C42—1461no—
395C2×C42—1461no—
396C2×C42C22×D7311yescharacteristic
397C2×C42C22×D7311yescharacteristic
398#107—1462no—
399#107—1462no—
400#107—1462no—
401#106—2921no—
402C4⋊C8—22no—
403#103—2921no—
404#103—1462no—
405#103—1462no—
406#103—1462no—
407C22⋊C8—22no—
408C22⋊C8—41no—
409#97—41no—
410#95—1461no—
411#95—1461no—
412#95—1461no—
413#95—1461no—
414#95—1461no—
415#95—1461no—
416#95—1461no—
417#95—1461no—
418#95—1461no—
419#95D29211yescharacteristic
420#95D29211yescharacteristic
421C24C2×D29211yescharacteristic
422C2×Q8—732no—
423C2×Q8—1461no—
424C2×Q8—1461no—
425C2×Q8—1461no—
426C2×Q8—1461no—
427C2×Q8—1461no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—1461no—
435C22×C4—1461no—
436C22×C4—1461no—
437C22×C4—1461no—
438C22×C4—1461no—
439C22×C4—1461no—
440C22×C4—1461no—
441C22×C4—1461no—
442C22×C4C2×D29211yescharacteristic
443C22×C4C2×D29211yescharacteristic
444C22×C4C2×D29211yescharacteristic
445C22×C4C2×D29211yescharacteristic
446C22×C4C2×D29211yescharacteristic
447C22×C4C23×D7311yescharacteristic
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8#1192402712yes—
460C4⋊C4—2922no—
461C4⋊C4—2922no—
462C4⋊C4—2922no—
463C4⋊C4—2922no—
464C4⋊C4—2922no—
465C4⋊C4—2922no—
466C4⋊C4—732no—
467C4⋊C4—732no—
468C4⋊C4—1464no—
469C4⋊C4—1461no—
470C4⋊C4—1461no—
471C4⋊C4—1461no—
472C4⋊C4—1461no—
473C4⋊C4—1461no—
474C4⋊C4—1461no—
475C4⋊C4—1461no—
476C4⋊C4—1461no—
477C4⋊C4—1461no—
478C4⋊C4—1461no—
479C4⋊C4—1461no—
480C4⋊C4—1461no—
481C22⋊C4—22no—
482C22⋊C4—41no—
483C22⋊C4—2922no—
484C22⋊C4—2922no—
485C22⋊C4—2922no—
486C22⋊C4—1464no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—2922no—
492C42—2922no—
493C42—2922no—
494C42—2922no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23C22×D29211yescharacteristic
503Q8—2921no—
504Q8—2921no—
505Q8—1462no—
506Q8—1462no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—2921no—
526C2×C4—2921no—
527C2×C4—2921no—
528C2×C4—2921no—
529C2×C4—2921no—
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—2921no—
541C2×C4—2921no—
542C2×C4—2921no—
543C2×C4—2921no—
544C2×C4—2921no—
545C2×C4—2921no—
546C2×C4—1462no—
547C2×C4—1462no—
548C2×C4—1462no—
549C2×C4#1455661912yes—
550C2×C4#41×D7311yescharacteristic
551C2×C4#41×D7311yescharacteristic
552C2×C4#41×D7311yescharacteristic
553C2×C4C2×#1192402711yescharacteristic
554C2×C4#1455663611yescharacteristic
555C2×C4#1455663811yescharacteristic
556C8—22no—
557C8—41no—
558C8—41no—
559C8—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22#1798909911yescharacteristic
568C22#1798923211yescharacteristic
569C22#1798924111yescharacteristic
570C22C2×#1455661911yescharacteristic
571C22#1798938711yescharacteristic
572C22#1798940411yescharacteristic
573C22#1798940411yescharacteristic
574C4—22no—
575C4—41no—
576C4—41no—
577C4—41no—
578C4—41no—
579C4—2922no—
580C4—2922no—
581C4—2922no—
582C4—5841no—
583C4—5841no—
584C4—5841no—
585C4—5841no—
586C4—5841no—
587C4#1798911611yescharacteristic
588C4#1798918011yescharacteristic
589C4#1798928011yescharacteristic
590C4#1798930411yescharacteristic
591C2—81no—
592C2#2261396511yescharacteristic
593C2#2262008311yescharacteristic
594C2#2262023211yescharacteristic
595C2#2262199911yescharacteristic
596C2#2262223711yescharacteristic
597C2#2262237311yescharacteristic
598C2#2262246311yescharacteristic
subgroupG/H
centerC23C22×D292
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#1924C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
hypercenter#1924D73
hypocenterC73#27058
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1924

Complements

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

NH ≅ G/Ntype
3. #22611341C2semidirect
4. #22611311C2semidirect
7. #22610690C2semidirect
8. #22610669C2semidirect
9. #22610524C2semidirect
10. #22610503C2semidirect
11. #22610188C2semidirect
14. #22609762C2semidirect
315. C73#27058semidirect

Maximal subgroups

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

subgroupindex
#226143532
#226129562
#226113412
#226113112
#226111352
#226106902
#226106692
#226105242
#226105032
#226101882
#226101322
#226100272
#226097622
C73×#19242
#2705873