Finite Group Databasebeta

← #33144269

Subgroups — #33144269

Order 18688.

744 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22614957C211yesmaximal, characteristic, complemented
2#22611319C211yesmaximal, characteristic
3#22611305C211yesmaximal, characteristic
4#22611067C211yesmaximal, characteristic
5#22611067C211yesmaximal, characteristic
6#22610883C211yesmaximal, characteristic, complemented
7#22610879C211yesmaximal, characteristic, complemented
8#22610697C211yesmaximal, characteristic, complemented
9#22610655C211yesmaximal, characteristic, complemented
10#22610523C211yesmaximal, characteristic, complemented
11#22610509C211yesmaximal, characteristic, complemented
12#22610064C211yesmaximal, characteristic, complemented
13#22609931C211yesmaximal, characteristic
14#22609756C211yesmaximal, characteristic
15C73×#2357C211yesmaximal, characteristic
16C2×#14556591C2211yescharacteristic
17C2×#14556591C2211yescharacteristic
18C2×#14556590—21no—
19C2×#14556524C2211yescharacteristic
20C2×#14556522C2211yescharacteristic
21C2×C4⋊#10596831C2211yescharacteristic
22C2×C4⋊#10596831C2211yescharacteristic
23C2×C4⋊#10596831C2211yescharacteristic
24C2×#14556512C2211yescharacteristic
25C2×#14556512C2211yescharacteristic
26C2×#14556512C2211yescharacteristic
27C2×#14556512C2211yescharacteristic
28C2×#14556502C2211yescharacteristic
29#17988508—21no—
30#17988508—21no—
31#17988486—21no—
32#17988485C2211yescharacteristic
33#17988485C2211yescharacteristic
34#17988457C2211yescharacteristic
35C4×C4⋊Dic73C2211yescharacteristic
36#17988349C2211yescharacteristic
37#17988349C2211yescharacteristic
38#17988349C2211yescharacteristic
39#17988349C2211yescharacteristic
40#17988349C2211yescharacteristic
41#17988349C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988342C2211yescharacteristic
47#17988342C2211yescharacteristic
48C2×C4×D4×C73C2211yescharacteristic
49#390×C73C2211yescharacteristic
50#390×C73C2211yescharacteristic
51#388×C73—21no—
52#388×C73—21no—
53#382×C73C2211yescharacteristic
54#381×C73—21no—
55C4×C4⋊C4×C73C2211yescharacteristic
56C4×C4⋊C4×C73C2211yescharacteristic
57C4×C4⋊C4×C73C2211yescharacteristic
58C4×C22⋊C4×C73—21no—
59C22×Dic146—21no—
60C22×Dic146—21no—
61C22×#10596831C2311yescharacteristic
62C22×#10596831C2311yescharacteristic
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65#14556591—22no—
66#14556591—22no—
67#14556591—41no—
68#14556591—41no—
69#14556590—44no—
70C2×C4⋊Dic73—21no—
71C2×C4⋊Dic73—21no—
72C2×C4⋊Dic73—21no—
73C2×C4⋊Dic73—21no—
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2311yescharacteristic
77C2×C4⋊Dic73C2311yescharacteristic
78C2×#11924007—21no—
79C2×#11924007—21no—
80C2×C4×Dic73—21no—
81#14556524—22no—
82#14556522—22no—
83C4⋊#10596831—22no—
84C4⋊#10596831—41no—
85C4⋊#10596831—41no—
86#14556512—22no—
87#14556512—22no—
88#14556512—41no—
89#14556512—41no—
90#14556502—22no—
91#14556498—21no—
92#14556498—21no—
93#14556498—21no—
94C22×D4×C73C2311yescharacteristic
95C23×C4×C73—21no—
96C4×D4×C73—22no—
97C4×D4×C73Q814yescomplemented
98C2×C4⋊C4×C73D411yescharacteristic
99C2×C4⋊C4×C73D411yescharacteristic
100C2×C4⋊C4×C73C2311yescharacteristic
101C2×C4⋊C4×C73C2311yescharacteristic
102C2×C4⋊C4×C73C2311yescharacteristic
103C2×C22⋊C4×C73—21no—
104C2×C22⋊C4×C73—21no—
105C2×C22⋊C4×C73—21no—
106C2×C22⋊C4×C73—21no—
107C2×C22⋊C4×C73—21no—
108C2×C42×C73—21no—
109C2×C42×C73—21no—
110C2×C42×C73C2311yescharacteristic
111C2×C42×C73C2311yescharacteristic
112C2×C42×C73C2311yescharacteristic
113#95×C73—21no—
114#95×C73—21no—
115#95×C73—21no—
116#95×C73—21no—
117#95×C73D411yescharacteristic
118#95×C73D411yescharacteristic
119C2×Dic146—42no—
120C2×Dic146—42no—
121C2×Dic146—42no—
122C2×Dic146—42no—
123C2×Dic146—81no—
124C2×Dic146—81no—
125C22×Dic73—21no—
126C22×Dic73—21no—
127C22×Dic73—21no—
128C22×Dic73—21no—
129C2×#10596831—21no—
130C2×#10596831—21no—
131C2×#10596831—21no—
132C2×#10596831—21no—
133C2×#10596831—21no—
134C2×#10596831—21no—
135C2×#10596831—21no—
136C2×#10596831—21no—
137C2×#10596831—21no—
138C2×#10596831—21no—
139C2×#10596831—21no—
140C2×#10596831—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—42no—
150C4⋊Dic73—81no—
151C4⋊Dic73#4112yes—
152C4⋊Dic73#4112yes—
153#11924007—42no—
154#11924007—42no—
155C4×Dic73—42no—
156C24×C73—21no—
157C2×D4×C73—21no—
158C2×D4×C73—21no—
159C2×D4×C73—21no—
160C2×D4×C73—21no—
161C2×D4×C73—21no—
162C2×D4×C73C2×Q812yes—
163C22×C4×C73—21no—
164C22×C4×C73—24no—
165C22×C4×C73—21no—
166C22×C4×C73—21no—
167C22×C4×C73—81no—
168C22×C4×C73—81no—
169C22×C4×C73C2×D411yescharacteristic
170C22×C4×C73C2×D411yescharacteristic
171C22×C4×C73C2×D411yescharacteristic
172C22×C4×C73C2×D411yescharacteristic
173C22×C4×C73C2×D411yescharacteristic
174C22×C4×C73C2×D411yescharacteristic
175C22×C4×C73C2411yescharacteristic
176C4⋊C4×C73—22no—
177C4⋊C4×C73—22no—
178C4⋊C4×C73—21no—
179C4⋊C4×C73—21no—
180C4⋊C4×C73—21no—
181C4⋊C4×C73—21no—
182C4⋊C4×C73—21no—
183C4⋊C4×C73C2×Q812yes—
184C22⋊C4×C73—24no—
185C22⋊C4×C73—42no—
186C22⋊C4×C73—42no—
187C22⋊C4×C73—81no—
188C22⋊C4×C73—81no—
189C42×C73—21no—
190C42×C73—21no—
191C42×C73—21no—
192C42×C73—21no—
193C42×C73—81no—
194C42×C73—81no—
195C42×C73C2×Q812yes—
196C42×C73#4112yes—
197C23×C73—22no—
198C23×C73—41no—
199C23×C73—41no—
200C23×C73—41no—
201C23×C73—41no—
202C23×C73—41no—
203C23×C73—41no—
204C23×C73C22×D411yescharacteristic
205D4×C73—22no—
206D4×C73—22no—
207D4×C73—41no—
208D4×C73—41no—
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—22no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—41no—
225C2×C4×C73—41no—
226C2×C4×C73—41no—
227C2×C4×C73—41no—
228C2×C4×C73—41no—
229C2×C4×C73—41no—
230C2×C4×C73—41no—
231C2×C4×C73—41no—
232C2×C4×C73—84no—
233C2×C4×C73C22⋊Q812yes—
234C2×C4×C73C22⋊Q812yes—
235C2×C4×C73AGL(1,ℤ/8)12yes—
236C2×C4×C73#13712yes—
237C2×C4×C73C22×Q811yescharacteristic
238C2×C4×C73C2×#4111yescharacteristic
239C2×C4×C73C2×#4111yescharacteristic
240C2×C4×C73C2×#4111yescharacteristic
241C2×C4×C73H(5,𝔽2)11yescharacteristic
242C2×C4×C732−1+411yescharacteristic
243C2×Dic73—22no—
244C2×Dic73—22no—
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—41no—
255Dic146—84no—
256Dic146—84no—
257#10596831—41no—
258#10596831—41no—
259#10596831—41no—
260#10596831—41no—
261C22×C73—42no—
262C22×C73—42no—
263C22×C73—42no—
264C22×C73—81no—
265C22×C73—81no—
266C22×C73—81no—
267C22×C73—81no—
268C22×C73C2×C22⋊Q811yescharacteristic
269C22×C73#54611yescharacteristic
270C22×C73#54811yescharacteristic
271C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
272C22×C73C2×#13711yescharacteristic
273C22×C73#57611yescharacteristic
274C22×C73#57711yescharacteristic
275C4×C73—22no—
276C4×C73—22no—
277C4×C73—22no—
278C4×C73—22no—
279C4×C73—42no—
280C4×C73—41no—
281C4×C73—41no—
282C4×C73—81no—
283C4×C73—81no—
284C4×C73#53311yescharacteristic
285C4×C73#54411yescharacteristic
286C4×C73#55511yescharacteristic
287C4×C73#55611yescharacteristic
288Dic73—42no—
289Dic73—42no—
290Dic73—81no—
291Dic73—81no—
292#26036—731nomaximal, Sylow
293C2×C73—82no—
294C2×C73#243911yescharacteristic
295C2×C73#307911yescharacteristic
296C2×C73#308111yescharacteristic
297C2×C73#331811yescharacteristic
298C2×C73#332011yescharacteristic
299C2×C73#337311yescharacteristic
300C2×C73#337511yescharacteristic
301#2551—731no—
302#2357D7311yescharacteristic
303#2067—731no—
304#2065—731no—
305#2023—731no—
306#2023—731no—
307#1988—731no—
308#1987—731no—
309#1949—731no—
310#1940—731no—
311#1921—731no—
312#1919—731no—
313#1836—731no—
314#1802—731no—
315#1762—731no—
316C73#2603611yescharacteristic, complemented, Sylow
317C2×C4⋊Q8—731no—
318C2×C4⋊Q8—731no—
319C2×C4⋊Q8—1461no—
320C2×C4×D4C2×D7311yescharacteristic
321C2×#107—731no—
322C2×#106—731no—
323C2×C4⋊C8—731no—
324C2×C4⋊C8—731no—
325C2×C4⋊C8—731no—
326C2×#102—731no—
327C2×#102—731no—
328C2×#102—731no—
329C2×#102—731no—
330C2×#97—731no—
331#395—1461no—
332#395—1461no—
333#390C2×D7311yescharacteristic
334#390C2×D7311yescharacteristic
335#389—731no—
336#389—731no—
337#389—1461no—
338#388—21no—
339#388—21no—
340#384—731no—
341#382C2×D7311yescharacteristic
342#381—21no—
343C4×C4⋊C4—731no—
344C4×C4⋊C4C2×D7311yescharacteristic
345C4×C4⋊C4C2×D7311yescharacteristic
346C4×C4⋊C4C2×D7311yescharacteristic
347C4×C22⋊C4—21no—
348#340—731no—
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#340—731no—
358#336—731no—
359#336—731no—
360C22×Q8—1461no—
361C22×Q8—1461no—
362C22×D4C22×D7311yescharacteristic
363C23×C4—21no—
364C22×C8—731no—
365C22×C8—731no—
366C22×C8—731no—
367C22×C8—731no—
368C4⋊Q8—2921no—
369C4⋊Q8—2921no—
370C4⋊Q8—2924no—
371C4⋊Q8—1462no—
372C4⋊Q8—1462no—
373C4×D4—22no—
374C4×D4Dic14614yescomplemented
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⋊C4C22⋊D7311yescharacteristic
386C2×C4⋊C4C22⋊D7311yescharacteristic
387C2×C4⋊C4C22×D7311yescharacteristic
388C2×C4⋊C4C22×D7311yescharacteristic
389C2×C4⋊C4C22×D7311yescharacteristic
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—21no—
395C2×C42—21no—
396C2×C42—21no—
397C2×C42—1461no—
398C2×C42C22×D7311yescharacteristic
399C2×C42C22×D7311yescharacteristic
400C2×C42C22×D7311yescharacteristic
401#107—1462no—
402#106—1462no—
403C4⋊C8—2921no—
404C4⋊C8—2921no—
405C4⋊C8—1462no—
406#102—2921no—
407#102—2921no—
408#102—1462no—
409#102—1462no—
410#97—1462no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95—1461no—
416#95—1461no—
417#95—1461no—
418#95C22⋊D7311yescharacteristic
419#95C22⋊D7311yescharacteristic
420C24—21no—
421C2×Q8—2922no—
422C2×Q8—2922no—
423C2×Q8—2922no—
424C2×Q8—2922no—
425C2×Q8—5841no—
426C2×Q8—5841no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4C2×Dic14612yes—
433C22×C4—21no—
434C22×C4—24no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—81no—
438C22×C4—81no—
439C22×C4—1461no—
440C22×C4—1461no—
441C22×C4—1461no—
442C22×C4—1461no—
443C22×C4C2×C22⋊D7311yescharacteristic
444C22×C4C2×C22⋊D7311yescharacteristic
445C22×C4C2×C22⋊D7311yescharacteristic
446C22×C4C2×C22⋊D7311yescharacteristic
447C22×C4C2×C22⋊D7311yescharacteristic
448C22×C4C2×C22⋊D7311yescharacteristic
449C22×C4C23×D7311yescharacteristic
450C2×C8—1461no—
451C2×C8—1461no—
452C2×C8—1461no—
453C2×C8—1461no—
454C2×C8—1461no—
455C2×C8—1461no—
456C2×C8—1461no—
457C2×C8—1461no—
458C2×C8—1461no—
459C2×C8—1461no—
460C2×C8—1461no—
461C2×C8—1461no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—2922no—
470C4⋊C4—2922no—
471C4⋊C4—2922no—
472C4⋊C4—2922no—
473C4⋊C4—2922no—
474C4⋊C4—5841no—
475C4⋊C4—732no—
476C4⋊C4—732no—
477C4⋊C4—1461no—
478C4⋊C4—1461no—
479C4⋊C4—1461no—
480C4⋊C4—1461no—
481C4⋊C4—1461no—
482C4⋊C4—1461no—
483C4⋊C4C2×Dic14612yes—
484C22⋊C4—24no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—81no—
488C22⋊C4—81no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—81no—
494C42—81no—
495C42—2922no—
496C42C2×Dic14612yes—
497C42#1192402812yes—
498C23—22no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23C22×C22⋊D7311yescharacteristic
506Q8—5844no—
507Q8—5844no—
508D4—22no—
509D4—22no—
510D4—41no—
511D4—41no—
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—22no—
526C2×C4—21no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—84no—
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#1455657012yes—
549C2×C4#1455657012yes—
550C2×C4#1455661612yes—
551C2×C4#1455662112yes—
552C2×C4C22×Dic14611yescharacteristic
553C2×C4#41×D7311yescharacteristic
554C2×C4#41×D7311yescharacteristic
555C2×C4C2×#1192402811yescharacteristic
556C2×C4#1455663711yescharacteristic
557C2×C4#1455663811yescharacteristic
558C8—2921no—
559C8—2921no—
560C8—2921no—
561C8—2921no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22C2×#1455657011yescharacteristic
570C22#1798922511yescharacteristic
571C22#1798924411yescharacteristic
572C22C2×#1455661611yescharacteristic
573C22C2×#1455662111yescharacteristic
574C22#1798939011yescharacteristic
575C22#1798939811yescharacteristic
576C4—22no—
577C4—22no—
578C4—22no—
579C4—22no—
580C4—42no—
581C4—41no—
582C4—41no—
583C4—81no—
584C4—81no—
585C4—2922no—
586C4—2922no—
587C4—5841no—
588C4—5841no—
589C4#1798914011yescharacteristic
590C4#1798921511yescharacteristic
591C4#1798930011yescharacteristic
592C4#1798930511yescharacteristic
593C2—82no—
594C2#2261361611yescharacteristic
595C2#2261974611yescharacteristic
596C2#2261976211yescharacteristic
597C2#2262206611yescharacteristic
598C2#2262209611yescharacteristic
599C2#2262271911yescharacteristic
600C2#2262274111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2357C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
hypercenter#2357D73
hypocenterC73#26036
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2357

Complements

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

NH ≅ G/Ntype
1. #22614957C2semidirect
6. #22610883C2semidirect
7. #22610879C2semidirect
8. #22610697C2semidirect
9. #22610655C2semidirect
10. #22610523C2semidirect
11. #22610509C2semidirect
12. #22610064C2semidirect
97. C4×D4×C73Q8semidirect
316. C73#26036semidirect
374. C4×D4Dic146semidirect

Maximal subgroups

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

subgroupindex
#226149572
#226113192
#226113052
#226110672
#226108832
#226108792
#226106972
#226106552
#226105232
#226105092
#226100642
#226099312
#226097562
C73×#23572
#2603673