Finite Group Databasebeta

← #33112452

Subgroups — #33112452

Order 18688.

622 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22613207C211yesmaximal, characteristic
2#22610870C211yesmaximal, characteristic, complemented
3#22610865C211yesmaximal, characteristic, complemented
4#22610653C211yesmaximal, characteristic, complemented
5#22610544C211yesmaximal, characteristic
6#22610543C211yesmaximal, characteristic
7#22610542C211yesmaximal, characteristic
8#22610536C211yesmaximal, characteristic
9#22610497C211yesmaximal, characteristic, complemented
10#22610495C211yesmaximal, characteristic, complemented
11#22610492C211yesmaximal, characteristic, complemented
12#22610485C211yesmaximal, characteristic, complemented
13#22610001C211yesmaximal, characteristic
14#22609912C211yesmaximal, characteristic
15C73×#1940C211yesmaximal, characteristic, complemented
16C2×C4×Dic146C2211yescharacteristic
17C2×#14556548C2211yescharacteristic
18C2×C4⋊#10596831C2211yescharacteristic
19C2×#14556520C2211yescharacteristic
20C2×#14556520C2211yescharacteristic
21C2×#14556503C2211yescharacteristic
22C2×#14556503C2211yescharacteristic
23C2×#14556503C2211yescharacteristic
24C2×#14556503C2211yescharacteristic
25C2×#14556502C2211yescharacteristic
26C2×#14556501C2211yescharacteristic
27C2×C8×Dic73C2211yescharacteristic
28#17988528C411yescharacteristic
29#17988522C411yescharacteristic
30#17988521C411yescharacteristic
31#17988518C411yescharacteristic
32#17988511C411yescharacteristic
33#17988509C411yescharacteristic
34#17988507C411yescharacteristic
35#17988500C411yescharacteristic
36C4×#11924007C2211yescharacteristic
37C4×#11924007C2211yescharacteristic
38C4×#11924005C2211yescharacteristic
39C4×#11924005C2211yescharacteristic
40#17988342C2211yescharacteristic
41#17988342C2211yescharacteristic
42#17988342C2211yescharacteristic
43#17988342C2211yescharacteristic
44#17988341C2211yescharacteristic
45#17988341C2211yescharacteristic
46#17988341C2211yescharacteristic
47#17988341C2211yescharacteristic
48#17988341C2211yescharacteristic
49#17988341C2211yescharacteristic
50#17988341C2211yescharacteristic
51#17988341C2211yescharacteristic
52C2×C4⋊C8×C73C2211yescharacteristic
53C2×C4⋊C8×C73C2211yescharacteristic
54C2×C4⋊C8×C73C2211yescharacteristic
55C2×#97×C73C2211yescharacteristic
56C4×C4⋊C4×C73C2211yescharacteristic
57#336×C73C2211yescharacteristic
58#336×C73C2211yescharacteristic
59C22×Dic146C2×C411yescharacteristic
60C22×C4×D73C2311yescharacteristic
61C22×#10596831C2311yescharacteristic
62C22×#10596831C2311yescharacteristic
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65C4×Dic146—41no—
66C4×Dic146—41no—
67#14556548—41no—
68#14556548—41no—
69C2×C4⋊Dic73C2×C411yescharacteristic
70C2×#11924007C2×C411yescharacteristic
71C2×#11924007C2×C411yescharacteristic
72C2×#11924007C2×C411yescharacteristic
73C2×#11924007C2×C411yescharacteristic
74C2×#11924007C2×C411yescharacteristic
75C2×#11924007C2×C411yescharacteristic
76C2×#11924007C2×C411yescharacteristic
77C2×#11924007C2×C411yescharacteristic
78C2×#11924005C2×C411yescharacteristic
79C2×#11924005C2×C411yescharacteristic
80C2×#11924005C2×C411yescharacteristic
81C2×#11924005C2×C411yescharacteristic
82C2×#11924005C2×C411yescharacteristic
83C2×#11924005C2×C411yescharacteristic
84C2×C4×Dic73C2311yescharacteristic
85C2×C4×Dic73C2311yescharacteristic
86C2×C4×Dic73C2311yescharacteristic
87C4⋊#10596831—41no—
88#14556520—41no—
89#14556520—41no—
90#14556503—41no—
91#14556503—41no—
92#14556503—41no—
93#14556503—41no—
94#14556502—41no—
95#14556501—41no—
96C8×Dic73—41no—
97#14556498C2×C411yescharacteristic
98#14556498C2×C411yescharacteristic
99#14556498C2×C411yescharacteristic
100#14556498C2×C411yescharacteristic
101#14556498C2×C411yescharacteristic
102#14556498C2×C411yescharacteristic
103#14556498C2×C411yescharacteristic
104#14556498C2×C411yescharacteristic
105C22×C8×C73C2311yescharacteristic
106C22×C8×C73C2311yescharacteristic
107C22×C8×C73C2311yescharacteristic
108C22×C8×C73C2311yescharacteristic
109C2×C4⋊C4×C73C2×C411yescharacteristic
110C2×C4⋊C4×C73C2×C411yescharacteristic
111C2×C42×C73C2311yescharacteristic
112C2×C42×C73C2311yescharacteristic
113C2×C42×C73C2311yescharacteristic
114C4⋊C8×C73—41no—
115C4⋊C8×C73—41no—
116C4⋊C8×C73—41no—
117#97×C73—41no—
118#95×C73C2×C411yescharacteristic
119#95×C73C2×C411yescharacteristic
120C23×D73C22×C411yescharacteristic
121C2×Dic146—22no—
122C2×Dic146—41no—
123C2×Dic146—41no—
124C22×Dic73C22×C411yescharacteristic
125C22×Dic73C22×C411yescharacteristic
126C22×Dic73C22×C411yescharacteristic
127C22×Dic73C22×C411yescharacteristic
128C22×Dic73C22×C411yescharacteristic
129C22×Dic73C22×C411yescharacteristic
130C22×Dic73C22×C411yescharacteristic
131C2×C4×D73—21no—
132C2×C4×D73—21no—
133C2×C4×D73—21no—
134C2×C4×D73—21no—
135C2×C4×D73—21no—
136C2×C4×D73—21no—
137C2×#10596831—21no—
138C2×#10596831—21no—
139C2×#10596831—21no—
140C2×#10596831—21no—
141C2×#10596831—21no—
142C2×#10596831—21no—
143C2×#10596831—21no—
144C2×#10596831—21no—
145C2×#10596831—21no—
146C2×#10596831—21no—
147C2×#10596831—21no—
148C2×#10596831—21no—
149C4⋊Dic73—41no—
150#11924007—22no—
151#11924007—41no—
152#11924007—41no—
153#11924007—41no—
154#11924007—41no—
155#11924007—41no—
156#11924007—41no—
157#11924007—41no—
158#11924005—22no—
159#11924005—41no—
160#11924005—41no—
161#11924005—41no—
162#11924005—41no—
163#11924005—41no—
164C4×Dic73—21no—
165C4×Dic73—21no—
166C4×Dic73—21no—
167C4×Dic73—21no—
168C4×Dic73—21no—
169C4×Dic73—21no—
170C22×C4×C73C22×C411yescharacteristic
171C22×C4×C73C22×C411yescharacteristic
172C22×C4×C73C22×C411yescharacteristic
173C22×C4×C73C22×C411yescharacteristic
174C22×C4×C73C22×C411yescharacteristic
175C22×C4×C73C22×C411yescharacteristic
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—21no—
189C4⋊C4×C73—41no—
190C4⋊C4×C73—41no—
191C42×C73—21no—
192C42×C73—21no—
193C42×C73—21no—
194C42×C73—21no—
195C42×C73—21no—
196C42×C73—21no—
197C23×C73C23×C411yescharacteristic
198C22×D73—21no—
199C22×D73—21no—
200C22×D73—21no—
201C22×D73—21no—
202C22×D73—21no—
203C22×D73—21no—
204C22×D73—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—21no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73H(5,𝔽2)11yescharacteristic
224C2×C4×C73H(5,𝔽2)11yescharacteristic
225C2×C4×C73H(5,𝔽2)11yescharacteristic
226C2×C4×C732−1+411yescharacteristic
227C2×C4×C732−1+411yescharacteristic
228C2×C4×C732−1+411yescharacteristic
229C2×Dic73—21no—
230C2×Dic73—21no—
231C2×Dic73—21no—
232C2×Dic73—21no—
233C2×Dic73—21no—
234C2×Dic73—21no—
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—21no—
244C2×Dic73—21no—
245C2×Dic73—21no—
246C2×Dic73—21no—
247C2×Dic73—21no—
248C2×Dic73—21no—
249C2×Dic73#13112yes—
250C4×D73—41no—
251C4×D73—41no—
252C4×D73—41no—
253C4×D73—41no—
254Dic146—44no—
255C8×C73—41no—
256C8×C73—41no—
257C8×C73—41no—
258C8×C73—41no—
259#10596831—41no—
260#10596831—41no—
261#10596831—41no—
262#10596831—41no—
263C22×C73#51811yescharacteristic
264C22×C73#51911yescharacteristic
265C22×C73#52011yescharacteristic
266C22×C73C2×#13111yescharacteristic
267C22×C73#56811yescharacteristic
268C22×C73#56811yescharacteristic
269C22×C73#56811yescharacteristic
270C2×D73—41no—
271C2×D73—41no—
272C2×D73—41no—
273C2×D73—41no—
274C2×D73—41no—
275C2×D73—41no—
276C2×D73—41no—
277C4×C73—41no—
278C4×C73—41no—
279C4×C73—41no—
280C4×C73—41no—
281C4×C73—41no—
282C4×C73—41no—
283C4×C73#56211yescharacteristic
284C4×C73#56211yescharacteristic
285C4×C73#56311yescharacteristic
286C4×C73#56311yescharacteristic
287Dic73—22no—
288Dic73—41no—
289Dic73—41no—
290Dic73—41no—
291Dic73—41no—
292Dic73—41no—
293Dic73—41no—
294#23345—731nomaximal, Sylow
295C2×C73#237811yescharacteristic
296C2×C73#297611yescharacteristic
297C2×C73#298011yescharacteristic
298C2×C73#298511yescharacteristic
299C2×C73#298711yescharacteristic
300C2×C73#299311yescharacteristic
301C2×C73#299511yescharacteristic
302D73—81no—
303#2396—731no—
304#1985—731no—
305#1984—731no—
306#1940—731no—
307#1940D7311yescharacteristic, complemented
308#1924—731no—
309#1924—731no—
310#1924—731no—
311#1923—731no—
312#1917—731no—
313#1917—731no—
314#1917—731no—
315#1916—731no—
316#1818—731no—
317#1798—731no—
318C73#2334511yescharacteristic, complemented, Sylow
319C2×C4×Q8—731no—
320C2×#117—731no—
321C2×C4⋊C8—731no—
322C2×C4⋊C8—731no—
323C2×C4⋊C8—731no—
324C2×C4⋊C8C2×D7311yescharacteristic
325C2×C4⋊C8C2×D7311yescharacteristic
326C2×C4⋊C8C2×D7311yescharacteristic
327C2×C22⋊C8—731no—
328C2×C22⋊C8—731no—
329C2×C22⋊C8—731no—
330C2×C22⋊C8—731no—
331C2×#97—731no—
332C2×#97—731no—
333C2×#97C2×D7311yescharacteristic
334C2×C4×C8—731no—
335#400—731no—
336#399—731no—
337#398—731no—
338#398—731no—
339#396—731no—
340#396—731no—
341#395—731no—
342#393—731no—
343C4×C4⋊C4—731no—
344C4×C4⋊C4—731no—
345C4×C4⋊C4C2×D7311yescharacteristic
346C4×C22⋊C4—731no—
347C4×C22⋊C4—731no—
348#336—731no—
349#336—731no—
350#336—731no—
351#336—731no—
352#336—731no—
353#336—731no—
354#336—731no—
355#336—731no—
356#336—731no—
357#336—731no—
358#336—731no—
359#336—731no—
360#336C2×D7311yescharacteristic
361#336C2×D7311yescharacteristic
362C22×Q8—731no—
363C23×C4—731no—
364C22×C8—731no—
365C22×C8—731no—
366C22×C8—731no—
367C22×C8—731no—
368C22×C8C22×D7311yescharacteristic
369C22×C8C22×D7311yescharacteristic
370C22×C8C22×D7311yescharacteristic
371C22×C8C22×D7311yescharacteristic
372C4×Q8—2921no—
373C4×Q8—2921no—
374#117—2921no—
375#117—2921no—
376C2×C4⋊C4—731no—
377C2×C4⋊C4—731no—
378C2×C4⋊C4—731no—
379C2×C4⋊C4—731no—
380C2×C4⋊C4—731no—
381C2×C4⋊C4—731no—
382C2×C4⋊C4—731no—
383C2×C4⋊C4—731no—
384C2×C4⋊C4—731no—
385C2×C4⋊C4C4×D7311yescharacteristic
386C2×C4⋊C4C4×D7311yescharacteristic
387C2×C22⋊C4—731no—
388C2×C22⋊C4—731no—
389C2×C22⋊C4—731no—
390C2×C22⋊C4—731no—
391C2×C22⋊C4—731no—
392C2×C22⋊C4—731no—
393C2×C42—731no—
394C2×C42—731no—
395C2×C42—731no—
396C2×C42C22×D7311yescharacteristic
397C2×C42C22×D7311yescharacteristic
398C2×C42C22×D7311yescharacteristic
399C4⋊C8—41no—
400C4⋊C8—41no—
401C4⋊C8—41no—
402C4⋊C8—2921no—
403C4⋊C8—2921no—
404C4⋊C8—2921no—
405C22⋊C8—2921no—
406C22⋊C8—2921no—
407C22⋊C8—2921no—
408C22⋊C8—2921no—
409#97—41no—
410#97—2921no—
411#97—2921no—
412C4×C8—2921no—
413#95—731no—
414#95—731no—
415#95—731no—
416#95—731no—
417#95—731no—
418#95—731no—
419#95—731no—
420#95—731no—
421#95C4×D7311yescharacteristic
422#95C4×D7311yescharacteristic
423C24—731no—
424C2×Q8—2921no—
425C2×Q8—2921no—
426C2×Q8—1462no—
427C22×C4—731no—
428C22×C4—731no—
429C22×C4—731no—
430C22×C4—731no—
431C22×C4—731no—
432C22×C4—731no—
433C22×C4—731no—
434C22×C4—1461no—
435C22×C4—1461no—
436C22×C4—1461no—
437C22×C4—1461no—
438C22×C4—1461no—
439C22×C4—1461no—
440C22×C4C2×C4×D7311yescharacteristic
441C22×C4C2×C4×D7311yescharacteristic
442C22×C4C2×C4×D7311yescharacteristic
443C22×C4C2×C4×D7311yescharacteristic
444C22×C4C2×C4×D7311yescharacteristic
445C22×C4C2×C4×D7311yescharacteristic
446C22×C4C23×D7311yescharacteristic
447C2×C8—21no—
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—1461no—
460C2×C8—1461no—
461C2×C8—1461no—
462C2×C8—1461no—
463C2×C8—1461no—
464C2×C8—1461no—
465C2×C8—1461no—
466C2×C8—1461no—
467C2×C8—1461no—
468C2×C8—1461no—
469C2×C8—1461no—
470C2×C8—1461no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—2921no—
474C4⋊C4—2921no—
475C4⋊C4—2921no—
476C4⋊C4—2921no—
477C4⋊C4—2921no—
478C4⋊C4—2921no—
479C4⋊C4—2921no—
480C4⋊C4—2921no—
481C4⋊C4—1462no—
482C22⋊C4—2921no—
483C22⋊C4—2921no—
484C22⋊C4—2921no—
485C22⋊C4—2921no—
486C22⋊C4—2921no—
487C22⋊C4—1462no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—1461no—
495C42—1461no—
496C42—1461no—
497C42—1461no—
498C42—1461no—
499C42—1461no—
500C23—1461no—
501C23—1461no—
502C23—1461no—
503C23—1461no—
504C23—1461no—
505C23—1461no—
506C23—1461no—
507C23C22×C4×D7311yescharacteristic
508Q8—2924no—
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—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—2921no—
528C2×C4—2921no—
529C2×C4—2921no—
530C2×C4—2921no—
531C2×C4—732no—
532C2×C4—1461no—
533C2×C4—1461no—
534C2×C4—1461no—
535C2×C4—1461no—
536C2×C4—1461no—
537C2×C4—1461no—
538C2×C4—1461no—
539C2×C4—1461no—
540C2×C4—1461no—
541C2×C4—1461no—
542C2×C4—1461no—
543C2×C4—1461no—
544C2×C4—1461no—
545C2×C4—1461no—
546C2×C4—1461no—
547C2×C4—1461no—
548C2×C4—1461no—
549C2×C4—1461no—
550C2×C4—1461no—
551C2×C4—1461no—
552C2×C4#1455663611yescharacteristic
553C2×C4#1455663611yescharacteristic
554C2×C4#1455663711yescharacteristic
555C2×C4#1455663811yescharacteristic
556C2×C4#1455663911yescharacteristic
557C2×C4#1455663911yescharacteristic
558C8—41no—
559C8—41no—
560C8—41no—
561C8—41no—
562C8—2921no—
563C8—2921no—
564C8—2921no—
565C8—2921no—
566C22—2921no—
567C22—2921no—
568C22—2921no—
569C22—2921no—
570C22—2921no—
571C22—2921no—
572C22—2921no—
573C22#1798905911yescharacteristic
574C22#1798906311yescharacteristic
575C22#1798906811yescharacteristic
576C22#131×D7311yescharacteristic
577C22#1798935211yescharacteristic
578C22#1798935311yescharacteristic
579C22#1798935311yescharacteristic
580C4—41no—
581C4—41no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—41no—
586C4—2921no—
587C4—2921no—
588C4—2921no—
589C4—2921no—
590C4—2921no—
591C4—2921no—
592C4—1462no—
593C4#1798933211yescharacteristic
594C4#1798933311yescharacteristic
595C4#1798933611yescharacteristic
596C4#1798934111yescharacteristic
597C2—5841no—
598C2#2261309311yescharacteristic
599C2#2261897811yescharacteristic
600C2#2261899911yescharacteristic
601C2#2261902411yescharacteristic
602C2#2261903411yescharacteristic
603C2#2261907811yescharacteristic
604C2#2261909811yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D73
commutator, socleC23×C73C23×C4
FittingC73×#1940C2
prime radicalC23×D73C22×C4
cosocleC22×C4×C73C24
hypercenter#1940D73
hypocenterC73#23345
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C22×C4 ◁ #1940

Complements

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

NH ≅ G/Ntype
2. #22610870C2semidirect
3. #22610865C2semidirect
4. #22610653C2semidirect
9. #22610497C2semidirect
10. #22610495C2semidirect
11. #22610492C2semidirect
12. #22610485C2semidirect
15. C73×#1940C2semidirect
307. #1940D73semidirect
318. C73#23345semidirect

Maximal subgroups

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

subgroupindex
#226132072
#226108702
#226108652
#226106532
#226105442
#226105432
#226105422
#226105362
#226104972
#226104952
#226104922
#226104852
#226100012
#226099122
C73×#19402
#2334573