Finite Group Databasebeta

← #33154099

Subgroups — #33154099

Order 18688.

764 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22614305C211yesmaximal, characteristic
2#22611253C211yesmaximal, characteristic, complemented
3#22611131C211yesmaximal, characteristic
4#22611072C211yesmaximal, characteristic, complemented
5#22610939C211yesmaximal, characteristic
6#22610879C211yesmaximal, characteristic, complemented
7#22610606C211yesmaximal, characteristic, complemented
8#22610572C211yesmaximal, characteristic, complemented
9#22610539C211yesmaximal, characteristic
10#22610520C211yesmaximal, characteristic, complemented
11#22610272C211yesmaximal, characteristic, complemented
12#22610238C211yesmaximal, characteristic
13#22610021C211yesmaximal, characteristic
14C4×#14556515C211yesmaximal, characteristic, complemented
15C73×#2360C211yesmaximal, characteristic
16C22×#11924018C2211yescharacteristic
17C2×#14556591C2211yescharacteristic
18C2×#14556570C2211yescharacteristic
19C2×#14556524C2211yescharacteristic
20C2×C4⋊#10596831C2211yescharacteristic
21C2×#14556515C2211yescharacteristic
22C2×#14556515C2211yescharacteristic
23C2×#14556515C2211yescharacteristic
24C2×#14556514C2211yescharacteristic
25C2×#14556514C2211yescharacteristic
26C2×#14556514C2211yescharacteristic
27C2×C22⋊#10596831C2211yescharacteristic
28C2×C22⋊#10596831C2211yescharacteristic
29C2×C4×#10596831C2211yescharacteristic
30#17988529—21no—
31#17988499—21no—
32#17988493C2211yescharacteristic
33#17988493C2211yescharacteristic
34#17988491—21no—
35#17988480—21no—
36#17988447C2211yescharacteristic
37C4×C4⋊Dic73C2211yescharacteristic
38#17988349C2211yescharacteristic
39#17988349C2211yescharacteristic
40#17988349C2211yescharacteristic
41#17988349C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988342C2211yescharacteristic
47#17988342C2211yescharacteristic
48C2×C22⋊Q8×C73C2211yescharacteristic
49C2×C4×Q8×C73C2211yescharacteristic
50C22×C4⋊C4×C73C2211yescharacteristic
51#391×C73C2211yescharacteristic
52#389×C73—21no—
53#389×C73C2211yescharacteristic
54#387×C73—21no—
55#385×C73—21no—
56C4×C4⋊C4×C73C2211yescharacteristic
57C4×C22⋊C4×C73—21no—
58C4×C22⋊C4×C73C2211yescharacteristic
59C22×Dic146—21no—
60C22×Dic146C2311yescharacteristic
61C2×#11924018—22no—
62C2×#11924018—41no—
63C2×#11924018—41no—
64C22×#10596831C2311yescharacteristic
65C22×#10596831C2311yescharacteristic
66C22×#10596831C2311yescharacteristic
67C22×#10596831C2311yescharacteristic
68#14556591—22no—
69#14556591—41no—
70#14556570—22no—
71#14556570—41no—
72C2×C4⋊Dic73—21no—
73C2×C4⋊Dic73C2311yescharacteristic
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×#11924007—21no—
77C2×C22⋊Dic73—21no—
78C2×C22⋊Dic73—21no—
79C2×C4×Dic73—21no—
80#14556524—22no—
81C4⋊#10596831—41no—
82#14556515—22no—
83#14556515—22no—
84#14556515—41no—
85#14556514—22no—
86#14556514—22no—
87#14556514—41no—
88C22⋊#10596831—41no—
89C22⋊#10596831—41no—
90C4×#10596831—41no—
91#14556498—21no—
92#14556498—21no—
93#14556498—21no—
94#14556498—21no—
95#14556498—21no—
96#14556498—21no—
97C22×Q8×C73C2311yescharacteristic
98C23×C4×C73—21no—
99C23×C4×C73C2311yescharacteristic
100C22⋊Q8×C73—22no—
101C22⋊Q8×C73—22no—
102C4×Q8×C73—22no—
103C4×Q8×C73—41no—
104C2×C4⋊C4×C73—21no—
105C2×C4⋊C4×C73—22no—
106C2×C4⋊C4×C73—21no—
107C2×C4⋊C4×C73—21no—
108C2×C4⋊C4×C73—21no—
109C2×C4⋊C4×C73—41no—
110C2×C4⋊C4×C73C2311yescharacteristic
111C2×C4⋊C4×C73C2311yescharacteristic
112C2×C4⋊C4×C73C2311yescharacteristic
113C2×C22⋊C4×C73—21no—
114C2×C22⋊C4×C73D411yescharacteristic
115C2×C22⋊C4×C73D411yescharacteristic
116C2×C42×C73—21no—
117C2×C42×C73—21no—
118C2×C42×C73C2311yescharacteristic
119C2×C42×C73C2311yescharacteristic
120#95×C73—21no—
121#95×C73—21no—
122#95×C73—21no—
123#95×C73—21no—
124#95×C73D411yescharacteristic
125#95×C73D411yescharacteristic
126C2×Dic146—21no—
127C2×Dic146—21no—
128C2×Dic146—21no—
129C2×Dic146—21no—
130C2×Dic146—21no—
131C2×Dic146—21no—
132C2×Dic146—42no—
133C2×Dic146—42no—
134C2×Dic146—81no—
135C22×Dic73—21no—
136C22×Dic73—21no—
137C22×Dic73—21no—
138C22×Dic73—21no—
139#11924018—44no—
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—
149C2×#10596831—21no—
150C2×#10596831—21no—
151C2×#10596831—21no—
152C4⋊Dic73—21no—
153C4⋊Dic73—21no—
154C4⋊Dic73—21no—
155C4⋊Dic73—21no—
156C4⋊Dic73—21no—
157C4⋊Dic73—42no—
158C4⋊Dic73#4112yes—
159#11924007—42no—
160C22⋊Dic73—42no—
161C22⋊Dic73—81no—
162C4×Dic73—81no—
163C24×C73C2×D411yescharacteristic
164C2×Q8×C73—21no—
165C2×Q8×C73—21no—
166C2×Q8×C73—21no—
167C2×Q8×C73—21no—
168C2×Q8×C73—21no—
169C2×Q8×C73#4112yes—
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×C73—21no—
178C22×C4×C73—21no—
179C22×C4×C73—21no—
180C22×C4×C73—42no—
181C22×C4×C73—42no—
182C22×C4×C73—81no—
183C22×C4×C73C2×D411yescharacteristic
184C22×C4×C73C2×D411yescharacteristic
185C22×C4×C73C2×D411yescharacteristic
186C22×C4×C73C2×D411yescharacteristic
187C22×C4×C73C2×D411yescharacteristic
188C22×C4×C73C2411yescharacteristic
189C4⋊C4×C73—24no—
190C4⋊C4×C73—21no—
191C4⋊C4×C73—21no—
192C4⋊C4×C73—21no—
193C4⋊C4×C73—21no—
194C4⋊C4×C73—42no—
195C4⋊C4×C73—42no—
196C4⋊C4×C73—81no—
197C4⋊C4×C73#4112yes—
198C4⋊C4×C73#4112yes—
199C22⋊C4×C73—24no—
200C22⋊C4×C73—41no—
201C22⋊C4×C73Q1614yescomplemented
202C42×C73—21no—
203C42×C73—21no—
204C42×C73—21no—
205C42×C73—21no—
206C42×C73—42no—
207C42×C73—81no—
208C23×C73—21no—
209C23×C73—21no—
210C23×C73—21no—
211C23×C73—21no—
212C23×C73—21no—
213C23×C73—21no—
214C23×C73C2×Q1612yes—
215C23×C73C22×D411yescharacteristic
216Q8×C73—22no—
217Q8×C73—22no—
218Q8×C73—41no—
219Q8×C73—41no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—22no—
225C2×C4×C73—22no—
226C2×C4×C73—21no—
227C2×C4×C73—21no—
228C2×C4×C73—21no—
229C2×C4×C73—22no—
230C2×C4×C73—21no—
231C2×C4×C73—21no—
232C2×C4×C73—21no—
233C2×C4×C73—21no—
234C2×C4×C73—21no—
235C2×C4×C73—21no—
236C2×C4×C73—41no—
237C2×C4×C73—41no—
238C2×C4×C73—41no—
239C2×C4×C73—41no—
240C2×C4×C73—41no—
241C2×C4×C73—41no—
242C2×C4×C73—41no—
243C2×C4×C73—41no—
244C2×C4×C73—41no—
245C2×C4×C73—41no—
246C2×C4×C73—41no—
247C2×C4×C73—41no—
248C2×C4×C73—41no—
249C2×C4×C73—84no—
250C2×C4×C73C2×Q1612yes—
251C2×C4×C73C2×Q1612yes—
252C2×C4×C73C2×#4111yescharacteristic
253C2×C4×C73C2×#4111yescharacteristic
254C2×C4×C73C2×#4111yescharacteristic
255C2×C4×C73C2×#4111yescharacteristic
256C2×C4×C73H(5,𝔽2)11yescharacteristic
257C2×C4×C73H(5,𝔽2)11yescharacteristic
258C2×Dic73—22no—
259C2×Dic73—41no—
260C2×Dic73—41no—
261C2×Dic73—41no—
262C2×Dic73—41no—
263C2×Dic73—41no—
264C2×Dic73—41no—
265C2×Dic73—41no—
266C2×Dic73—41no—
267C2×Dic73—41no—
268C2×Dic73—41no—
269C2×Dic73—41no—
270Dic146—41no—
271Dic146—41no—
272Dic146—41no—
273Dic146—41no—
274Dic146—84no—
275#10596831—41no—
276#10596831—41no—
277#10596831—41no—
278#10596831—41no—
279C22×C73—22no—
280C22×C73—22no—
281C22×C73—22no—
282C22×C73—41no—
283C22×C73—41no—
284C22×C73—41no—
285C22×C73—41no—
286C22×C73#52511yescharacteristic
287C22×C73#54611yescharacteristic
288C22×C73#54611yescharacteristic
289C22×C73C22×Q1611yescharacteristic
290C22×C73#57511yescharacteristic
291C22×C73#57711yescharacteristic
292C22×C73#57711yescharacteristic
293C4×C73—22no—
294C4×C73—22no—
295C4×C73—42no—
296C4×C73—41no—
297C4×C73—42no—
298C4×C73—41no—
299C4×C73—42no—
300C4×C73—41no—
301C4×C73—81no—
302C4×C73#52911yescharacteristic
303C4×C73#53811yescharacteristic
304C4×C73#55511yescharacteristic
305C4×C73#55911yescharacteristic
306Dic73—42no—
307Dic73—81no—
308Dic73—81no—
309Dic73—81no—
310#26726—731nomaximal, Sylow
311C2×C73—42no—
312C2×C73#245911yescharacteristic
313C2×C73#310411yescharacteristic
314C2×C73#312511yescharacteristic
315C2×C73#329911yescharacteristic
316C2×C73#329911yescharacteristic
317C2×C73#331111yescharacteristic
318C2×C73#332311yescharacteristic
319#2492—731no—
320#2360D7311yescharacteristic
321#2056—731no—
322#2036—731no—
323#2024—731no—
324#2001—731no—
325#1987—731no—
326#1933—731no—
327#1928—731no—
328#1923—731no—
329#1921—731no—
330#1878—731no—
331#1872—731no—
332#1824—731no—
333C4×#103—731no—
334C73#2672611yescharacteristic, complemented, Sylow
335C22×Q16—731no—
336C2×C4⋊Q8—731no—
337C2×C22⋊Q8—731no—
338C2×C22⋊Q8C2×D7311yescharacteristic
339C2×C4×Q8C2×D7311yescharacteristic
340C22×C4⋊C4C2×D7311yescharacteristic
341C2×#107—731no—
342C2×C4⋊C8—731no—
343C2×#103—731no—
344C2×#103—731no—
345C2×#103—731no—
346C2×#103—731no—
347C2×#103—731no—
348C2×#103—731no—
349C2×C22⋊C8—731no—
350C2×C22⋊C8—731no—
351C2×C4×C8—731no—
352#400—1461no—
353#393—1461no—
354#391—731no—
355#391—731no—
356#391—1461no—
357#391C2×D7311yescharacteristic
358#389—21no—
359#389C2×D7311yescharacteristic
360#388—1461no—
361#387—21no—
362#385—21no—
363#380—731no—
364C4×C4⋊C4—731no—
365C4×C4⋊C4C2×D7311yescharacteristic
366C4×C22⋊C4—21no—
367C4×C22⋊C4C2×D7311yescharacteristic
368#340—731no—
369#340—731no—
370#340—731no—
371#340—731no—
372#340—731no—
373#340—731no—
374#340—731no—
375#340—731no—
376#336—731no—
377#336—731no—
378C22×Q8—731no—
379C22×Q8—1461no—
380C22×Q8C22×D7311yescharacteristic
381C23×C4—21no—
382C23×C4C22×D7311yescharacteristic
383C2×Q16—2921no—
384C2×Q16—2921no—
385C2×Q16—1462no—
386C22×C8—731no—
387C22×C8—731no—
388C22×C8—731no—
389C22×C8—731no—
390C4⋊Q8—2921no—
391C4⋊Q8—1462no—
392C22⋊Q8—22no—
393C22⋊Q8—22no—
394C22⋊Q8—2921no—
395C22⋊Q8—1462no—
396C4×Q8—22no—
397C4×Q8—41no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—22no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—41no—
404C2×C4⋊C4—731no—
405C2×C4⋊C4—731no—
406C2×C4⋊C4—731no—
407C2×C4⋊C4—1461no—
408C2×C4⋊C4—1461no—
409C2×C4⋊C4C22×D7311yescharacteristic
410C2×C4⋊C4C22×D7311yescharacteristic
411C2×C4⋊C4C22×D7311yescharacteristic
412C2×C22⋊C4—21no—
413C2×C22⋊C4—1461no—
414C2×C22⋊C4—1461no—
415C2×C22⋊C4C22⋊D7311yescharacteristic
416C2×C22⋊C4C22⋊D7311yescharacteristic
417C2×C42—21no—
418C2×C42—21no—
419C2×C42—1461no—
420C2×C42C22×D7311yescharacteristic
421C2×C42C22×D7311yescharacteristic
422#107—1462no—
423C4⋊C8—2921no—
424#103—2921no—
425#103—2921no—
426#103—1462no—
427#103—1462no—
428#103—1462no—
429#103—1462no—
430C22⋊C8—2921no—
431C22⋊C8—2921no—
432C4×C8—2921no—
433#95—21no—
434#95—21no—
435#95—21no—
436#95—21no—
437#95—1461no—
438#95—1461no—
439#95—1461no—
440#95—1461no—
441#95—1461no—
442#95—1461no—
443#95C22⋊D7311yescharacteristic
444#95C22⋊D7311yescharacteristic
445C24C2×C22⋊D7311yescharacteristic
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—2922no—
452C2×Q8—2922no—
453C2×Q8—5841no—
454C2×Q8—1461no—
455C2×Q8—1461no—
456C2×Q8—1461no—
457C2×Q8—1461no—
458C2×Q8—1461no—
459C2×Q8—1461no—
460C2×Q8#1192402812yes—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—81no—
474C22×C4—1461no—
475C22×C4—1461no—
476C22×C4—1461no—
477C22×C4—1461no—
478C22×C4C2×C22⋊D7311yescharacteristic
479C22×C4C2×C22⋊D7311yescharacteristic
480C22×C4C2×C22⋊D7311yescharacteristic
481C22×C4C2×C22⋊D7311yescharacteristic
482C22×C4C2×C22⋊D7311yescharacteristic
483C22×C4C23×D7311yescharacteristic
484Q16—2924no—
485C2×C8—1461no—
486C2×C8—1461no—
487C2×C8—1461no—
488C2×C8—1461no—
489C2×C8—1461no—
490C2×C8—1461no—
491C2×C8—1461no—
492C2×C8—1461no—
493C2×C8—1461no—
494C2×C8—1461no—
495C2×C8—1461no—
496C2×C8—1461no—
497C4⋊C4—24no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—81no—
505C4⋊C4—2922no—
506C4⋊C4—2922no—
507C4⋊C4—732no—
508C4⋊C4—1461no—
509C4⋊C4—1461no—
510C4⋊C4—1461no—
511C4⋊C4—1461no—
512C4⋊C4—1461no—
513C4⋊C4#1192402712yes—
514C4⋊C4#1192402812yes—
515C22⋊C4—24no—
516C22⋊C4—41no—
517C22⋊C4—2922no—
518C22⋊C4—5841no—
519C22⋊C4#1192401814yescomplemented
520C42—21no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—42no—
525C42—81no—
526C42—5841no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23C2×#1192401812yes—
534C23C22×C22⋊D7311yescharacteristic
535Q8—22no—
536Q8—22no—
537Q8—41no—
538Q8—41no—
539Q8—2921no—
540Q8—2921no—
541Q8—2921no—
542Q8—2921no—
543Q8—5844no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—84no—
574C2×C4—2921no—
575C2×C4—2921no—
576C2×C4—2921no—
577C2×C4—2921no—
578C2×C4—2921no—
579C2×C4—2921no—
580C2×C4—2921no—
581C2×C4—2921no—
582C2×C4—2921no—
583C2×C4—2921no—
584C2×C4—2921no—
585C2×C4—1462no—
586C2×C4C2×#1192401812yes—
587C2×C4C2×#1192401812yes—
588C2×C4#41×D7311yescharacteristic
589C2×C4C2×#1192402711yescharacteristic
590C2×C4C2×#1192402811yescharacteristic
591C2×C4C2×#1192402811yescharacteristic
592C2×C4#1455663611yescharacteristic
593C2×C4#1455663711yescharacteristic
594C8—2921no—
595C8—2921no—
596C8—2921no—
597C8—2921no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22#1798909811yescharacteristic
606C22#1798922511yescharacteristic
607C22#1798923111yescharacteristic
608C22C22×#1192401811yescharacteristic
609C22#1798938511yescharacteristic
610C22#1798939511yescharacteristic
611C22#1798939811yescharacteristic
612C4—22no—
613C4—22no—
614C4—41no—
615C4—42no—
616C4—42no—
617C4—41no—
618C4—42no—
619C4—41no—
620C4—81no—
621C4—2922no—
622C4—5841no—
623C4—5841no—
624C4—5841no—
625C4#1798911711yescharacteristic
626C4#1798917011yescharacteristic
627C4#1798930211yescharacteristic
628C4#1798932011yescharacteristic
629C2—42no—
630C2#2261386211yescharacteristic
631C2#2261997011yescharacteristic
632C2#2262020811yescharacteristic
633C2#2262180111yescharacteristic
634C2#2262180711yescharacteristic
635C2#2262196711yescharacteristic
636C2#2262214111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2360C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
hypercenter#2360D73
hypocenterC73#26726
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2360

Complements

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

NH ≅ G/Ntype
2. #22611253C2semidirect
4. #22611072C2semidirect
6. #22610879C2semidirect
7. #22610606C2semidirect
8. #22610572C2semidirect
10. #22610520C2semidirect
11. #22610272C2semidirect
14. C4×#14556515C2semidirect
201. C22⋊C4×C73Q16semidirect
334. C73#26726semidirect
519. C22⋊C4#11924018semidirect

Maximal subgroups

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

subgroupindex
#226143052
#226112532
#226111312
#226110722
#226109392
#226108792
#226106062
#226105722
#226105392
#226105202
#226102722
#226102382
#226100212
C4×#145565152
C73×#23602
#2672673