Finite Group Databasebeta

← #33148916

Subgroups — #33148916

Order 18688.

804 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988891C211yesmaximal, characteristic, complemented
2C2×#17988627C211yesmaximal, characteristic
3#22612956C211yesmaximal, characteristic
4#22611482C211yesmaximal, characteristic, complemented
5#22611427C211yesmaximal, characteristic, complemented
6#22611323C211yesmaximal, characteristic, complemented
7#22611266C211yesmaximal, characteristic, complemented
8#22610697C211yesmaximal, characteristic, complemented
9#22610506C211yesmaximal, characteristic, complemented
10#22610362C211yesmaximal, characteristic
11#22610238C211yesmaximal, characteristic
12#22610198C211yesmaximal, characteristic, complemented
13#22610128C211yesmaximal, characteristic
14#22610116C211yesmaximal, characteristic
15C73×#2442C211yesmaximal, characteristic
16C22×#11924012C2211yescharacteristic
17C2×#14556570C2211yescharacteristic
18C2×C4×Dic146C2211yescharacteristic
19C2×#14556550C2211yescharacteristic
20#17988891—24no—
21#17988627—24no—
22C2×#14556524C2211yescharacteristic
23C2×#14556524C2211yescharacteristic
24C2×#14556524C2211yescharacteristic
25C2×#14556522C2211yescharacteristic
26C2×#14556522C2211yescharacteristic
27C2×#14556522C2211yescharacteristic
28C2×C4⋊#10596831C2211yescharacteristic
29C2×C4⋊#10596831C2211yescharacteristic
30C2×#14556514C2211yescharacteristic
31C2×#14556514C2211yescharacteristic
32C2×#14556514C2211yescharacteristic
33C2×#14556514C2211yescharacteristic
34C2×C22⋊#10596831C2211yescharacteristic
35#17988493C2211yescharacteristic
36#17988485C2211yescharacteristic
37#17988481—21no—
38#17988475—21no—
39#17988469—21no—
40C4×C4⋊Dic73C2211yescharacteristic
41#17988438—21no—
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988342C2211yescharacteristic
49#17988342C2211yescharacteristic
50C2×#125×C73C2211yescharacteristic
51C22×C4⋊C4×C73C2211yescharacteristic
52#400×C73—21no—
53#398×C73—21no—
54#389×C73C2211yescharacteristic
55#389×C73C2211yescharacteristic
56#385×C73—21no—
57#382×C73C2211yescharacteristic
58#380×C73—21no—
59#380×C73C2211yescharacteristic
60C4×C22⋊C4×C73C2211yescharacteristic
61C22×Dic146C2311yescharacteristic
62C2×#11924012—22no—
63C2×#11924012—22no—
64C2×#11924012—22no—
65C22×#10596831C2311yescharacteristic
66C22×#10596831C2311yescharacteristic
67C22×#10596831C2311yescharacteristic
68C22×#10596831C2311yescharacteristic
69#14556570—22no—
70#14556570Q814yes—
71C4×Dic146—22no—
72C4×Dic146—41no—
73#14556550—22no—
74#14556550—41no—
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2311yescharacteristic
77C2×C4⋊Dic73C2311yescharacteristic
78C2×#11924007—21no—
79C2×#11924007—21no—
80C2×#11924007—21no—
81C2×C22⋊Dic73—21no—
82C2×C22⋊Dic73—21no—
83C2×C4×Dic73—21no—
84C2×C4×Dic73—21no—
85C2×C4×Dic73—21no—
86#14556524—22no—
87#14556524—22no—
88#14556524—22no—
89#14556522—22no—
90#14556522—41no—
91#14556522D414yes—
92C4⋊#10596831—22no—
93C4⋊#10596831—22no—
94#14556514—22no—
95#14556514—22no—
96#14556514—22no—
97#14556514—41no—
98C22⋊#10596831—22no—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102#14556498—21no—
103C23×C4×C73—21no—
104C23×C4×C73C2311yescharacteristic
105#125×C73—22no—
106#125×C73—41no—
107C2×C4⋊C4×C73—22no—
108C2×C4⋊C4×C73—21no—
109C2×C4⋊C4×C73—21no—
110C2×C4⋊C4×C73—21no—
111C2×C4⋊C4×C73—21no—
112C2×C4⋊C4×C73—21no—
113C2×C4⋊C4×C73—41no—
114C2×C4⋊C4×C73C2311yescharacteristic
115C2×C4⋊C4×C73C2311yescharacteristic
116C2×C4⋊C4×C73C2311yescharacteristic
117C2×C4⋊C4×C73C2311yescharacteristic
118C2×C22⋊C4×C73—21no—
119C2×C22⋊C4×C73D411yescharacteristic
120C2×C22⋊C4×C73D411yescharacteristic
121C2×C42×C73C2311yescharacteristic
122C2×C42×C73C2311yescharacteristic
123#95×C73—21no—
124#95×C73—21no—
125#95×C73—21no—
126#95×C73—21no—
127#95×C73—21no—
128#95×C73D411yescharacteristic
129#95×C73D411yescharacteristic
130C2×Dic146—21no—
131C2×Dic146—21no—
132C2×Dic146—21no—
133C2×Dic146—21no—
134C2×Dic146—21no—
135C2×Dic146C2×Q812yes—
136C22×Dic73—21no—
137C22×Dic73—21no—
138C22×Dic73—21no—
139C22×Dic73—21no—
140#11924012—28no—
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×#10596831C2×D412yes—
152C2×#10596831#4112yes—
153C4⋊Dic73—21no—
154C4⋊Dic73—21no—
155C4⋊Dic73—21no—
156C4⋊Dic73—21no—
157C4⋊Dic73C2×D412yes—
158C4⋊Dic73C2×Q812yes—
159#11924007—24no—
160#11924007—42no—
161#11924007—42no—
162C22⋊Dic73—24no—
163C22⋊Dic73—42no—
164C4×Dic73—42no—
165C4×Dic73—42no—
166C4×Dic73—42no—
167C24×C73C2×D411yescharacteristic
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—42no—
177C22×C4×C73—42no—
178C22×C4×C73—81no—
179C22×C4×C73C2×D411yescharacteristic
180C22×C4×C73C2×D411yescharacteristic
181C22×C4×C73C2×D411yescharacteristic
182C22×C4×C73C2×D411yescharacteristic
183C22×C4×C73C2×D411yescharacteristic
184C22×C4×C73C2×Q812yes—
185C22×C4×C73#4112yes—
186C22×C4×C73C2411yescharacteristic
187C4⋊C4×C73—21no—
188C4⋊C4×C73—21no—
189C4⋊C4×C73—21no—
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—42no—
197C4⋊C4×C73—81no—
198C4⋊C4×C73—81no—
199C4⋊C4×C73C2×D412yes—
200C22⋊C4×C73—41no—
201C22⋊C4×C73—41no—
202C22⋊C4×C73—42no—
203C42×C73—21no—
204C42×C73—21no—
205C42×C73—21no—
206C42×C73—21no—
207C23×C73—21no—
208C23×C73—21no—
209C23×C73—21no—
210C23×C73—21no—
211C23×C73—21no—
212C23×C73—21no—
213C23×C73C22⋊Q812yes—
214C23×C73C22×D411yescharacteristic
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—22no—
218C2×C4×C73—22no—
219C2×C4×C73—22no—
220C2×C4×C73—22no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—22no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—21no—
228C2×C4×C73—21no—
229C2×C4×C73—21no—
230C2×C4×C73—21no—
231C2×C4×C73—21no—
232C2×C4×C73—21no—
233C2×C4×C73—21no—
234C2×C4×C73—41no—
235C2×C4×C73—41no—
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—84no—
246C2×C4×C73C22⋊Q812yes—
247C2×C4×C73C22×D411yescharacteristic
248C2×C4×C73C22×Q811yescharacteristic
249C2×C4×C73C2×#4111yescharacteristic
250C2×C4×C73C2×#4111yescharacteristic
251C2×C4×C73H(5,𝔽2)11yescharacteristic
252C2×C4×C732−1+411yescharacteristic
253C2×Dic73—22no—
254C2×Dic73—22no—
255C2×Dic73—22no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260C2×Dic73—41no—
261C2×Dic73—41no—
262C2×Dic73—41no—
263C2×Dic73—41no—
264C2×Dic73—41no—
265Dic146—22no—
266Dic146—22no—
267Dic146—41no—
268Dic146—41no—
269#10596831—22no—
270#10596831—22no—
271#10596831—41no—
272#10596831—41no—
273C22×C73—22no—
274C22×C73—22no—
275C22×C73—22no—
276C22×C73—41no—
277C22×C73—41no—
278C22×C73—41no—
279C22×C73—41no—
280C22×C73C2×C22⋊Q811yescharacteristic
281C22×C73D4211yescharacteristic
282C22×C73#54711yescharacteristic
283C22×C73#57511yescharacteristic
284C22×C73#57511yescharacteristic
285C22×C73#57711yescharacteristic
286C22×C73#57811yescharacteristic
287C4×C73—22no—
288C4×C73—41no—
289C4×C73—41no—
290C4×C73—41no—
291C4×C73—41no—
292C4×C73—42no—
293C4×C73—81no—
294C4×C73—81no—
295C4×C73—81no—
296C4×C73#52511yescharacteristic
297C4×C73#53911yescharacteristic
298C4×C73D4×Q811yescharacteristic
299C4×C73#55411yescharacteristic
300Dic73—42no—
301Dic73—42no—
302Dic73—42no—
303Dic73—81no—
304#26367—731nomaximal, Sylow
305C2×C73—42no—
306C2×C73#244911yescharacteristic
307C2×C73#308011yescharacteristic
308C2×C73#308211yescharacteristic
309C2×C73#326911yescharacteristic
310C2×C73#327511yescharacteristic
311C2×C73#334911yescharacteristic
312C2×C73#335111yescharacteristic
313C2×#479—731no—
314C2×#428—731no—
315#2442D7311yescharacteristic
316#2362—731no—
317#2097—731no—
318#2087—731no—
319#2068—731no—
320#2059—731no—
321#1949—731no—
322#1919—731no—
323#1894—731no—
324#1872—731no—
325#1865—731no—
326#1854—731no—
327#1851—731no—
328C73#2636711yescharacteristic, complemented, Sylow
329C22×M16—731no—
330C2×#125C2×D7311yescharacteristic
331C2×C22⋊Q8—731no—
332C2×C4×Q8—731no—
333C2×#117—731no—
334C22×C4⋊C4C2×D7311yescharacteristic
335#479—1464no—
336#428—1464no—
337C2×#107—731no—
338C2×#107—731no—
339C2×#107—731no—
340C2×#106—731no—
341C2×#106—731no—
342C2×#106—731no—
343C2×C4⋊C8—731no—
344C2×C4⋊C8—731no—
345C2×#103—731no—
346C2×#103—731no—
347C2×#103—731no—
348C2×#103—731no—
349C2×C22⋊C8—731no—
350#400—21no—
351#398—21no—
352#391—731no—
353#389—731no—
354#389C2×D7311yescharacteristic
355#389C2×D7311yescharacteristic
356#388—1461no—
357#388—1461no—
358#387—1461no—
359#385—21no—
360#382C2×D7311yescharacteristic
361#380—21no—
362#380C2×D7311yescharacteristic
363C4×C4⋊C4—731no—
364C4×C4⋊C4—1461no—
365C4×C22⋊C4C2×D7311yescharacteristic
366#340—731no—
367#340—731no—
368#340—731no—
369#340—731no—
370#340—731no—
371#340—731no—
372#336—731no—
373#336—731no—
374C22×Q8—731no—
375C23×C4—21no—
376C23×C4C22×D7311yescharacteristic
377C2×M16—1462no—
378C2×M16—1462no—
379C2×M16—1462no—
380C22×C8—731no—
381C22×C8—731no—
382C22×C8—731no—
383C22×C8—731no—
384#125—22no—
385#125—41no—
386C22⋊Q8—734no—
387C22⋊Q8—1462no—
388C4×Q8—2921no—
389C4×Q8—1462no—
390#117—2921no—
391#117—1462no—
392C2×C4⋊C4—22no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—41no—
399C2×C4⋊C4—731no—
400C2×C4⋊C4—731no—
401C2×C4⋊C4—731no—
402C2×C4⋊C4—1461no—
403C2×C4⋊C4—1461no—
404C2×C4⋊C4—1461no—
405C2×C4⋊C4C22×D7311yescharacteristic
406C2×C4⋊C4C22×D7311yescharacteristic
407C2×C4⋊C4C22×D7311yescharacteristic
408C2×C4⋊C4C22×D7311yescharacteristic
409C2×C22⋊C4—21no—
410C2×C22⋊C4—1461no—
411C2×C22⋊C4—1461no—
412C2×C22⋊C4C22⋊D7311yescharacteristic
413C2×C22⋊C4C22⋊D7311yescharacteristic
414C2×C42—1461no—
415C2×C42—1461no—
416C2×C42—1461no—
417C2×C42C22×D7311yescharacteristic
418C2×C42C22×D7311yescharacteristic
419#107—1462no—
420#107—1462no—
421#107—1462no—
422#106—2921no—
423#106—734no—
424#106—1462no—
425C4⋊C8—1462no—
426C4⋊C8—1462no—
427#103—2921no—
428#103—1462no—
429#103—1462no—
430#103—1462no—
431C22⋊C8—1462no—
432#95—21no—
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#95C22⋊D7311yescharacteristic
442#95C22⋊D7311yescharacteristic
443C24C2×C22⋊D7311yescharacteristic
444C2×Q8—732no—
445C2×Q8—1461no—
446C2×Q8—1461no—
447C2×Q8—1461no—
448C2×Q8—1461no—
449C2×Q8—1461no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—81no—
461C22×C4—1461no—
462C22×C4—1461no—
463C22×C4—1461no—
464C22×C4—1461no—
465C22×C4C2×C22⋊D7311yescharacteristic
466C22×C4C2×C22⋊D7311yescharacteristic
467C22×C4C2×C22⋊D7311yescharacteristic
468C22×C4C2×C22⋊D7311yescharacteristic
469C22×C4C2×C22⋊D7311yescharacteristic
470C22×C4Q8×D7312yes—
471C22×C4#1192402912yes—
472C22×C4C23×D7311yescharacteristic
473M16—1468no—
474C2×C8—732no—
475C2×C8—732no—
476C2×C8—1461no—
477C2×C8—1461no—
478C2×C8—1461no—
479C2×C8—1461no—
480C2×C8—1461no—
481C2×C8—1461no—
482C2×C8—1461no—
483C2×C8—1461no—
484C2×C8—1461no—
485C2×C8—1461no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—81no—
497C4⋊C4—81no—
498C4⋊C4—2922no—
499C4⋊C4—2922no—
500C4⋊C4—732no—
501C4⋊C4—732no—
502C4⋊C4—1461no—
503C4⋊C4—1461no—
504C4⋊C4—1464no—
505C4⋊C4—1461no—
506C4⋊C4—1461no—
507C4⋊C4D4×D7312yes—
508C22⋊C4—41no—
509C22⋊C4—41no—
510C22⋊C4—42no—
511C22⋊C4—2922no—
512C22⋊C4—1464no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—21no—
517C42—2922no—
518C42—2922no—
519C42—2922no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23#1455656712yes—
527C23C22×C22⋊D7311yescharacteristic
528Q8—2921no—
529Q8—2921no—
530Q8—1462no—
531Q8—1462no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—84no—
563C2×C4—2921no—
564C2×C4—2921no—
565C2×C4—2921no—
566C2×C4—2921no—
567C2×C4—2921no—
568C2×C4—2921no—
569C2×C4—2921no—
570C2×C4—2921no—
571C2×C4—2921no—
572C2×C4—1462no—
573C2×C4—1462no—
574C2×C4—1462no—
575C2×C4#1455656712yes—
576C2×C4C2×D4×D7311yescharacteristic
577C2×C4C2×Q8×D7311yescharacteristic
578C2×C4#41×D7311yescharacteristic
579C2×C4C2×#1192402911yescharacteristic
580C2×C4#1455663611yescharacteristic
581C2×C4#1455663811yescharacteristic
582C8—2921no—
583C8—2921no—
584C8—1462no—
585C8—1462no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22C2×#1455656711yescharacteristic
594C22D4×C22⋊D7311yescharacteristic
595C22#1798923611yescharacteristic
596C22#1798938511yescharacteristic
597C22#1798938511yescharacteristic
598C22#1798939811yescharacteristic
599C22#1798940211yescharacteristic
600C4—22no—
601C4—41no—
602C4—41no—
603C4—41no—
604C4—42no—
605C4—41no—
606C4—81no—
607C4—81no—
608C4—81no—
609C4—2922no—
610C4—2922no—
611C4—2922no—
612C4—5841no—
613C4#1798910011yescharacteristic
614C4#1798918611yescharacteristic
615C4#1798925011yescharacteristic
616C4#1798929111yescharacteristic
617C2—42no—
618C2#2261373611yescharacteristic
619C2#2261974911yescharacteristic
620C2#2261976711yescharacteristic
621C2#2262148711yescharacteristic
622C2#2262155811yescharacteristic
623C2#2262246111yescharacteristic
624C2#2262248511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2442C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
hypercenter#2442D73
hypocenterC73#26367
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2442

Complements

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

NH ≅ G/Ntype
1. C2×#17988891C2semidirect
4. #22611482C2semidirect
5. #22611427C2semidirect
6. #22611323C2semidirect
7. #22611266C2semidirect
8. #22610697C2semidirect
9. #22610506C2semidirect
12. #22610198C2semidirect
328. C73#26367semidirect

Maximal subgroups

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

subgroupindex
C2×#179888912
C2×#179886272
#226129562
#226114822
#226114272
#226113232
#226112662
#226106972
#226105062
#226103622
#226102382
#226101982
#226101282
#226101162
C73×#24422
#2636773