Finite Group Databasebeta

← #33158739

Subgroups — #33158739

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#22614333C211yesmaximal, characteristic
2#22612940C211yesmaximal, characteristic
3#22611352C211yesmaximal, characteristic, complemented
4#22611308C211yesmaximal, characteristic, complemented
5#22611124C211yesmaximal, characteristic
6#22610696C211yesmaximal, characteristic, complemented
7#22610658C211yesmaximal, characteristic, complemented
8#22610534C211yesmaximal, characteristic
9#22610525C211yesmaximal, characteristic, complemented
10#22610519C211yesmaximal, characteristic, complemented
11#22610197C211yesmaximal, characteristic, complemented
12#22610123C211yesmaximal, characteristic
13#22610018C211yesmaximal, characteristic
14#103×Dic73C211yesmaximal, characteristic, complemented
15C73×#2036C211yesmaximal, characteristic
16C2×Q8×Dic73C2211yescharacteristic
17C22×C4⋊Dic73C2211yescharacteristic
18C2×#14556522C2211yescharacteristic
19C2×#14556522C2211yescharacteristic
20C2×#14556522C2211yescharacteristic
21C2×#14556520C2211yescharacteristic
22C2×#14556515C2211yescharacteristic
23C2×#14556515C2211yescharacteristic
24C2×C22⋊#10596831C2211yescharacteristic
25C2×C8×Dic73C2211yescharacteristic
26C73⋊#401—21no—
27#17988512—21no—
28#17988494C2211yescharacteristic
29#17988486—21no—
30#17988486C2211yescharacteristic
31#17988486C2211yescharacteristic
32#17988483C2211yescharacteristic
33#17988475—21no—
34#17988474—21no—
35#17988473—21no—
36#17988462—21no—
37#17988452C2211yescharacteristic
38#17988447C2211yescharacteristic
39C4⋊C4×Dic73C2211yescharacteristic
40C4⋊C4×Dic73C2211yescharacteristic
41C22⋊C4×Dic73—21no—
42C4×C22⋊Dic73C2211yescharacteristic
43#17988350C2211yescharacteristic
44#17988350C2211yescharacteristic
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988341C2211yescharacteristic
51#17988341C2211yescharacteristic
52C2×C22⋊Q8×C73C2211yescharacteristic
53C2×#107×C73C2211yescharacteristic
54C2×#103×C73C2211yescharacteristic
55C2×#103×C73C2211yescharacteristic
56C2×C22⋊C8×C73C2211yescharacteristic
57#380×C73C2211yescharacteristic
58#340×C73C2211yescharacteristic
59C23×Dic73—21no—
60C22×#10596831C2311yescharacteristic
61C22×#10596831C2311yescharacteristic
62Q8×Dic73—22no—
63Q8×Dic73—41no—
64C2×C4⋊Dic73—22no—
65C2×C4⋊Dic73—21no—
66C2×C4⋊Dic73—21no—
67C2×C4⋊Dic73—21no—
68C2×C4⋊Dic73—21no—
69C2×C4⋊Dic73—41no—
70C2×C4⋊Dic73C2311yescharacteristic
71C2×C4⋊Dic73C2311yescharacteristic
72C2×C4⋊Dic73C2311yescharacteristic
73C2×C4⋊Dic73C2311yescharacteristic
74C2×#11924007—21no—
75C2×C22⋊Dic73—21no—
76C2×C22⋊Dic73D411yescharacteristic
77C2×C22⋊Dic73D411yescharacteristic
78C2×C4×Dic73—21no—
79C2×C4×Dic73—21no—
80C2×C4×Dic73—21no—
81C2×C4×Dic73C2311yescharacteristic
82C2×C4×Dic73C2311yescharacteristic
83#14556522—22no—
84#14556522—22no—
85#14556522—41no—
86#14556520—22no—
87#14556515—22no—
88#14556515—22no—
89C22⋊#10596831—22no—
90C8×Dic73—41no—
91#14556498—21no—
92#14556498—21no—
93#14556498—21no—
94#14556498—21no—
95#14556498—21no—
96#14556498—21no—
97#14556498—21no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101#14556498D411yescharacteristic
102#14556498D411yescharacteristic
103C22×Q8×C73C2311yescharacteristic
104C23×C4×C73C2311yescharacteristic
105C22×C8×C73C2311yescharacteristic
106C22×C8×C73C2311yescharacteristic
107C22⋊Q8×C73—22no—
108C22⋊Q8×C73—22no—
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—
115#107×C73—22no—
116#103×C73—22no—
117#103×C73—41no—
118C22⋊C8×C73—41no—
119#95×C73—21no—
120C22×Dic73—21no—
121C22×Dic73—21no—
122C22×Dic73—21no—
123C22×Dic73—21no—
124C22×Dic73—42no—
125C22×Dic73—42no—
126C22×Dic73—81no—
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⋊Dic73—42no—
146C4⋊Dic73—81no—
147C4⋊Dic73—81no—
148C4⋊Dic73#4112yes—
149#11924007—42no—
150C22⋊Dic73—22no—
151C22⋊Dic73—22no—
152C22⋊Dic73—42no—
153C4×Dic73—21no—
154C4×Dic73—21no—
155C4×Dic73—21no—
156C4×Dic73—21no—
157C4×Dic73—42no—
158C4×Dic73—81no—
159C4×Dic73—81no—
160C24×C73C2×D411yescharacteristic
161C2×Q8×C73—21no—
162C2×Q8×C73—21no—
163C2×Q8×C73—21no—
164C2×Q8×C73—21no—
165C2×Q8×C73—21no—
166C2×Q8×C73#4112yes—
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—21no—
176C22×C4×C73—21no—
177C22×C4×C73C2×D411yescharacteristic
178C22×C4×C73C2411yescharacteristic
179C2×C8×C73—21no—
180C2×C8×C73—21no—
181C2×C8×C73—21no—
182C2×C8×C73—21no—
183C2×C8×C73—21no—
184C2×C8×C73—21no—
185C4⋊C4×C73—24no—
186C4⋊C4×C73—21no—
187C4⋊C4×C73—21no—
188C4⋊C4×C73—21no—
189C4⋊C4×C73—21no—
190C4⋊C4×C73—21no—
191C4⋊C4×C73#4112yes—
192C22⋊C4×C73—24no—
193C22⋊C4×C73—81no—
194C23×C73—21no—
195C23×C73—21no—
196C23×C73—21no—
197C23×C73—21no—
198C23×C73—21no—
199C23×C73—21no—
200C23×C73#13512yes—
201C23×C73C22×D411yescharacteristic
202Q8×C73—22no—
203Q8×C73—22no—
204Q8×C73—41no—
205Q8×C73—41no—
206C2×C4×C73—22no—
207C2×C4×C73—22no—
208C2×C4×C73—21no—
209C2×C4×C73—21no—
210C2×C4×C73—21no—
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—41no—
224C2×C4×C73—41no—
225C2×C4×C73C2×#4111yescharacteristic
226C2×C4×C73C2×#4111yescharacteristic
227C2×C4×C73C2×#4111yescharacteristic
228C2×C4×C73C2×#4111yescharacteristic
229C2×C4×C73H(5,𝔽2)11yescharacteristic
230C2×C4×C732−1+411yescharacteristic
231C2×Dic73—21no—
232C2×Dic73—21no—
233C2×Dic73—21no—
234C2×Dic73—22no—
235C2×Dic73—21no—
236C2×Dic73—21no—
237C2×Dic73—21no—
238C2×Dic73—21no—
239C2×Dic73—21no—
240C2×Dic73—21no—
241C2×Dic73—21no—
242C2×Dic73—21no—
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×Dic73—41no—
253C2×Dic73—41no—
254C2×Dic73—84no—
255C2×Dic73#13712yes—
256C8×C73—41no—
257C8×C73—41no—
258#10596831—22no—
259#10596831—41no—
260C22×C73—22no—
261C22×C73—22no—
262C22×C73—22no—
263C22×C73—41no—
264C22×C73—41no—
265C22×C73—41no—
266C22×C73—41no—
267C22×C73#52511yescharacteristic
268C22×C73#54611yescharacteristic
269C22×C73#54711yescharacteristic
270C22×C73C2×#13511yescharacteristic
271C22×C73C2×#13711yescharacteristic
272C22×C73#57711yescharacteristic
273C22×C73#57711yescharacteristic
274C4×C73—42no—
275C4×C73—41no—
276C4×C73—42no—
277C4×C73—81no—
278C4×C73—81no—
279C4×C73#52911yescharacteristic
280C4×C73#53911yescharacteristic
281C4×C73#55311yescharacteristic
282C4×C73#55611yescharacteristic
283Dic73—22no—
284Dic73—41no—
285Dic73—41no—
286Dic73—41no—
287Dic73—42no—
288Dic73—81no—
289Dic73—81no—
290Dic73—81no—
291#27037—731nomaximal, Sylow
292C2×C73—42no—
293C2×C73#246611yescharacteristic
294C2×C73#310811yescharacteristic
295C2×C73#312511yescharacteristic
296C2×C73#329611yescharacteristic
297C2×C73#332011yescharacteristic
298C2×C73#333711yescharacteristic
299C2×C73#335611yescharacteristic
300#2494—731no—
301#2360—731no—
302#2074—731no—
303#2066—731no—
304#2036D7311yescharacteristic
305#2035—731no—
306#1949—731no—
307#1941—731no—
308#1923—731no—
309#1921—731no—
310#1920—731no—
311#1865—731no—
312#1853—731no—
313#1823—731no—
314C4×#103—731no—
315C73#2703711yescharacteristic, complemented, Sylow
316C2×C22⋊Q8C2×D7311yescharacteristic
317C2×C4×Q8—731no—
318C22×C4⋊C4—731no—
319C2×#107C2×D7311yescharacteristic
320C2×#106—731no—
321C2×#106—731no—
322C2×#106—731no—
323C2×C4⋊C8—731no—
324C2×#103—731no—
325C2×#103—731no—
326C2×#103C2×D7311yescharacteristic
327C2×#103C2×D7311yescharacteristic
328C2×C22⋊C8—731no—
329C2×C22⋊C8C2×D7311yescharacteristic
330C2×C4×C8—731no—
331#401—1461no—
332#397—1461no—
333#391—731no—
334#389—731no—
335#389—731no—
336#389—731no—
337#389—1461no—
338#388—1461no—
339#387—1461no—
340#387—1461no—
341#385—1461no—
342#382—731no—
343#380—731no—
344#380C2×D7311yescharacteristic
345C4×C4⋊C4—731no—
346C4×C4⋊C4—731no—
347C4×C22⋊C4—731no—
348C4×C22⋊C4—1461no—
349#340—731no—
350#340—731no—
351#340—731no—
352#340—731no—
353#340—731no—
354#340—731no—
355#340—731no—
356#340C2×D7311yescharacteristic
357#336—731no—
358#336—731no—
359C22×Q8C22×D7311yescharacteristic
360C23×C4—1461no—
361C23×C4C22×D7311yescharacteristic
362C22×C8—731no—
363C22×C8—731no—
364C22×C8C22×D7311yescharacteristic
365C22×C8C22×D7311yescharacteristic
366C22⋊Q8—22no—
367C22⋊Q8—22no—
368C4×Q8—2921no—
369C4×Q8—1462no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—2921no—
372C2×C4⋊C4—731no—
373C2×C4⋊C4—731no—
374C2×C4⋊C4—731no—
375C2×C4⋊C4—731no—
376C2×C4⋊C4—1461no—
377C2×C4⋊C4—1461no—
378C2×C4⋊C4—1461no—
379C2×C4⋊C4—1462no—
380C2×C4⋊C4—1461no—
381C2×C4⋊C4—1461no—
382C2×C4⋊C4C22×D7311yescharacteristic
383C2×C4⋊C4C22×D7311yescharacteristic
384C2×C4⋊C4C22×D7311yescharacteristic
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—731no—
388C2×C22⋊C4—731no—
389C2×C22⋊C4—1461no—
390C2×C42—731no—
391C2×C42—731no—
392C2×C42—1461no—
393C2×C42—1461no—
394C2×C42—1461no—
395#107—22no—
396#106—2921no—
397#106—1462no—
398#106—1462no—
399C4⋊C8—1462no—
400#103—22no—
401#103—41no—
402#103—1462no—
403#103—1462no—
404C22⋊C8—41no—
405C22⋊C8—1462no—
406C4×C8—2921no—
407#95—21no—
408#95—731no—
409#95—731no—
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#95—1461no—
420C24D4×D7311yescharacteristic
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8#1192402712yes—
427C22×C4—21no—
428C22×C4—21no—
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—2922no—
438C22×C4—2922no—
439C22×C4—5841no—
440C22×C4—731no—
441C22×C4—731no—
442C22×C4—731no—
443C22×C4—731no—
444C22×C4—1461no—
445C22×C4—1461no—
446C22×C4—1461no—
447C22×C4—1461no—
448C22×C4D4×D7311yescharacteristic
449C22×C4C23×D7311yescharacteristic
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—732no—
457C2×C8—1461no—
458C2×C8—1461no—
459C2×C8—1461no—
460C2×C8—1461no—
461C2×C8—1461no—
462C4⋊C4—24no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—2922no—
469C4⋊C4—2922no—
470C4⋊C4—2922no—
471C4⋊C4—5841no—
472C4⋊C4—5841no—
473C4⋊C4—732no—
474C4⋊C4—1461no—
475C4⋊C4—1461no—
476C4⋊C4—1461no—
477C4⋊C4—1461no—
478C4⋊C4—1461no—
479C4⋊C4—1461no—
480C4⋊C4—1461no—
481C4⋊C4#1192402712yes—
482C22⋊C4—24no—
483C22⋊C4—81no—
484C22⋊C4—2922no—
485C22⋊C4—1462no—
486C22⋊C4—1462no—
487C42—2922no—
488C42—5841no—
489C42—5841no—
490C42—1461no—
491C42—1461no—
492C42—1461no—
493C42—1461no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23#1455661412yes—
501C23C2×D4×D7311yescharacteristic
502Q8—22no—
503Q8—22no—
504Q8—41no—
505Q8—41no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—22no—
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—41no—
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—5844no—
537C2×C4—732no—
538C2×C4—1461no—
539C2×C4—1461no—
540C2×C4—1461no—
541C2×C4—1461no—
542C2×C4—1462no—
543C2×C4—1461no—
544C2×C4—1461no—
545C2×C4—1461no—
546C2×C4—1461no—
547C2×C4—1461no—
548C2×C4—1461no—
549C2×C4—1461no—
550C2×C4#41×D7311yescharacteristic
551C2×C4#41×D7311yescharacteristic
552C2×C4C2×#1192402711yescharacteristic
553C2×C4C2×#1192402711yescharacteristic
554C2×C4#1455663611yescharacteristic
555C2×C4#1455663811yescharacteristic
556C8—41no—
557C8—41no—
558C8—2921no—
559C8—1462no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22#1798909311yescharacteristic
568C22#1798922611yescharacteristic
569C22#1798923511yescharacteristic
570C22C2×#1455661411yescharacteristic
571C22#137×D7311yescharacteristic
572C22#1798939411yescharacteristic
573C22#1798939711yescharacteristic
574C4—42no—
575C4—41no—
576C4—42no—
577C4—81no—
578C4—81no—
579C4—2922no—
580C4—2921no—
581C4—2921no—
582C4—2921no—
583C4—5841no—
584C4—5841no—
585C4—5841no—
586C4—1462no—
587C4#1798911411yescharacteristic
588C4#1798918211yescharacteristic
589C4#1798928411yescharacteristic
590C4#1798930611yescharacteristic
591C2—42no—
592C2#2261396011yescharacteristic
593C2#2262001211yescharacteristic
594C2#2262020911yescharacteristic
595C2#2262177211yescharacteristic
596C2#2262209711yescharacteristic
597C2#2262230811yescharacteristic
598C2#2262254411yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2036C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
hypercenter#2036D73
hypocenterC73#27037
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2036

Complements

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

NH ≅ G/Ntype
3. #22611352C2semidirect
4. #22611308C2semidirect
6. #22610696C2semidirect
7. #22610658C2semidirect
9. #22610525C2semidirect
10. #22610519C2semidirect
11. #22610197C2semidirect
14. #103×Dic73C2semidirect
315. C73#27037semidirect

Maximal subgroups

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

subgroupindex
#226143332
#226129402
#226113522
#226113082
#226111242
#226106962
#226106582
#226105342
#226105252
#226105192
#226101972
#226101232
#226100182
#103×Dic732
C73×#20362
#2703773