Finite Group Databasebeta

← #33134148

Subgroups — #33134148

Order 18688.

806 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988928C211yesmaximal, characteristic
2C2×#17988826C211yesmaximal, characteristic
3#22612956C211yesmaximal, characteristic
4#22611403C211yesmaximal, characteristic, complemented
5#22611350C211yesmaximal, characteristic, complemented
6#22611312C211yesmaximal, characteristic, complemented
7#22611253C211yesmaximal, characteristic, complemented
8#22610697C211yesmaximal, characteristic, complemented
9#22610569C211yesmaximal, characteristic, complemented
10#22610275C211yesmaximal, characteristic, complemented
11#22610244C211yesmaximal, characteristic
12#22610111C211yesmaximal, characteristic
13#22609933C211yesmaximal, characteristic, complemented
14#22609647C211yesmaximal, characteristic
15C73×#2499C211yesmaximal, characteristic
16C22×#11924018C2211yescharacteristic
17C23×#10596831C2211yescharacteristic
18C2×#14556570C2211yescharacteristic
19C2×C4×Dic146C2211yescharacteristic
20C2×#14556550C2211yescharacteristic
21#17988928—24no—
22#17988826—24no—
23C2×#14556524C2211yescharacteristic
24C2×#14556524C2211yescharacteristic
25C2×#14556522C2211yescharacteristic
26C2×C4⋊#10596831C2211yescharacteristic
27C2×C4⋊#10596831C2211yescharacteristic
28C2×#14556515C2211yescharacteristic
29C2×#14556515C2211yescharacteristic
30C2×#14556515C2211yescharacteristic
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×C22⋊Q8×C73C2211yescharacteristic
51C2×#117×C73C2211yescharacteristic
52#397×C73—21no—
53#397×C73—21no—
54#391×C73C2211yescharacteristic
55#391×C73C2211yescharacteristic
56#387×C73—21no—
57#387×C73—21no—
58#384×C73C2211yescharacteristic
59#382×C73C2211yescharacteristic
60C4×C22⋊C4×C73C2211yescharacteristic
61C22×Dic146C2311yescharacteristic
62C2×#11924018—22no—
63C2×#11924018—22no—
64C2×#11924018—41no—
65C22×#10596831—22no—
66C22×#10596831—22no—
67C22×#10596831—41no—
68C22×#10596831C2311yescharacteristic
69C22×#10596831C2311yescharacteristic
70C22×#10596831C2311yescharacteristic
71C22×#10596831C2311yescharacteristic
72#14556570—22no—
73#14556570D414yes—
74C4×Dic146—22no—
75C4×Dic146—41no—
76#14556550—22no—
77#14556550—41no—
78C2×C4⋊Dic73C2311yescharacteristic
79C2×C4⋊Dic73C2311yescharacteristic
80C2×C4⋊Dic73C2311yescharacteristic
81C2×#11924007—21no—
82C2×#11924007—21no—
83C2×#11924007—21no—
84C2×C22⋊Dic73—21no—
85C2×C22⋊Dic73—21no—
86C2×C4×Dic73—21no—
87C2×C4×Dic73—21no—
88C2×C4×Dic73—21no—
89#14556524—22no—
90#14556524—22no—
91#14556522—22no—
92C4⋊#10596831—22no—
93C4⋊#10596831—41no—
94#14556515—22no—
95#14556515—22no—
96#14556515—41no—
97#14556514—22no—
98#14556514—22no—
99#14556514—22no—
100C22⋊#10596831—22no—
101#14556498—21no—
102#14556498—21no—
103#14556498—21no—
104#14556498—21no—
105C22×Q8×C73C2311yescharacteristic
106C23×C4×C73C2311yescharacteristic
107C22⋊Q8×C73—22no—
108C22⋊Q8×C73—22no—
109#117×C73—22no—
110#117×C73—22no—
111C2×C4⋊C4×C73—21no—
112C2×C4⋊C4×C73—21no—
113C2×C4⋊C4×C73—21no—
114C2×C4⋊C4×C73C2311yescharacteristic
115C2×C4⋊C4×C73C2311yescharacteristic
116C2×C4⋊C4×C73C2311yescharacteristic
117C2×C22⋊C4×C73—21no—
118C2×C22⋊C4×C73—21no—
119C2×C22⋊C4×C73D411yescharacteristic
120C2×C22⋊C4×C73D411yescharacteristic
121C2×C42×C73—21no—
122C2×C42×C73C2311yescharacteristic
123C2×C42×C73C2311yescharacteristic
124#95×C73—21no—
125#95×C73—21no—
126#95×C73—21no—
127#95×C73—21no—
128#95×C73—21no—
129#95×C73—21no—
130#95×C73D411yescharacteristic
131#95×C73D411yescharacteristic
132C2×Dic146—21no—
133C2×Dic146—21no—
134C2×Dic146—21no—
135C2×Dic146—21no—
136C2×Dic146—21no—
137C2×Dic146C2×D412yes—
138C22×Dic73—21no—
139C22×Dic73—21no—
140C22×Dic73—21no—
141C22×Dic73—21no—
142#11924018—44no—
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—
152C2×#10596831—21no—
153C2×#10596831—21no—
154C2×#10596831—44no—
155C2×#10596831#4112yes—
156C4⋊Dic73—21no—
157C4⋊Dic73—21no—
158C4⋊Dic73—21no—
159C4⋊Dic73—21no—
160C4⋊Dic73C2×D412yes—
161C4⋊Dic73#4112yes—
162#11924007—24no—
163#11924007—42no—
164#11924007—42no—
165C22⋊Dic73—24no—
166C22⋊Dic73—42no—
167C4×Dic73—42no—
168C4×Dic73—42no—
169C4×Dic73—42no—
170C24×C73C2×D411yescharacteristic
171C2×Q8×C73—21no—
172C2×Q8×C73—21no—
173C2×Q8×C73—21no—
174C2×Q8×C73—21no—
175C2×Q8×C73—21no—
176C2×Q8×C73—21no—
177C22×C4×C73—21no—
178C22×C4×C73—21no—
179C22×C4×C73—21no—
180C22×C4×C73—21no—
181C22×C4×C73—21no—
182C22×C4×C73—21no—
183C22×C4×C73—21no—
184C22×C4×C73—21no—
185C22×C4×C73C2×D412yes—
186C22×C4×C73C2×D411yescharacteristic
187C22×C4×C73C2×D411yescharacteristic
188C22×C4×C73C2×D411yescharacteristic
189C22×C4×C73C2×D411yescharacteristic
190C22×C4×C73C2×D411yescharacteristic
191C22×C4×C73#4112yes—
192C22×C4×C73C2411yescharacteristic
193C4⋊C4×C73—21no—
194C4⋊C4×C73—21no—
195C4⋊C4×C73—21no—
196C4⋊C4×C73—21no—
197C4⋊C4×C73—21no—
198C4⋊C4×C73—21no—
199C4⋊C4×C73—42no—
200C4⋊C4×C73—42no—
201C4⋊C4×C73—42no—
202C22⋊C4×C73—22no—
203C22⋊C4×C73—41no—
204C22⋊C4×C73—42no—
205C22⋊C4×C73—42no—
206C42×C73—21no—
207C42×C73—21no—
208C42×C73—21no—
209C42×C73—21no—
210C42×C73—42no—
211C23×C73—21no—
212C23×C73—21no—
213C23×C73—21no—
214C23×C73—21no—
215C23×C73—21no—
216C23×C73—21no—
217C23×C73#12112yes—
218C23×C73C22×D411yescharacteristic
219Q8×C73—41no—
220Q8×C73—41no—
221Q8×C73—41no—
222Q8×C73—41no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—22no—
228C2×C4×C73—22no—
229C2×C4×C73—22no—
230C2×C4×C73—22no—
231C2×C4×C73—21no—
232C2×C4×C73—21no—
233C2×C4×C73—21no—
234C2×C4×C73—21no—
235C2×C4×C73—21no—
236C2×C4×C73—21no—
237C2×C4×C73—21no—
238C2×C4×C73—21no—
239C2×C4×C73—21no—
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—41no—
250C2×C4×C73—41no—
251C2×C4×C73—41no—
252C2×C4×C73#12112yes—
253C2×C4×C73#13512yes—
254C2×C4×C73C22×D411yescharacteristic
255C2×C4×C73C2×#4111yescharacteristic
256C2×C4×C73C2×#4111yescharacteristic
257C2×C4×C73C2×#4111yescharacteristic
258C2×C4×C73H(5,𝔽2)11yescharacteristic
259C2×C4×C732−1+411yescharacteristic
260C2×Dic73—22no—
261C2×Dic73—22no—
262C2×Dic73—22no—
263C2×Dic73—41no—
264C2×Dic73—41no—
265C2×Dic73—41no—
266C2×Dic73—41no—
267C2×Dic73—41no—
268C2×Dic73—41no—
269C2×Dic73—41no—
270C2×Dic73—41no—
271C2×Dic73—41no—
272Dic146—22no—
273Dic146—22no—
274Dic146—41no—
275Dic146—41no—
276#10596831—22no—
277#10596831—41no—
278#10596831—41no—
279#10596831—41no—
280C22×C73—22no—
281C22×C73—22no—
282C22×C73—22no—
283C22×C73—41no—
284C22×C73—41no—
285C22×C73—41no—
286C22×C73—41no—
287C22×C73C2×#12111yescharacteristic
288C22×C73#54611yescharacteristic
289C22×C73#54711yescharacteristic
290C22×C73C2×#13511yescharacteristic
291C22×C73#57511yescharacteristic
292C22×C73#57811yescharacteristic
293C22×C73#57811yescharacteristic
294C4×C73—22no—
295C4×C73—22no—
296C4×C73—41no—
297C4×C73—41no—
298C4×C73—41no—
299C4×C73—81no—
300C4×C73—81no—
301C4×C73—81no—
302C4×C73—81no—
303C4×C73#52911yescharacteristic
304C4×C73#53911yescharacteristic
305C4×C73#54611yescharacteristic
306C4×C73#54711yescharacteristic
307Dic73—42no—
308Dic73—42no—
309Dic73—42no—
310Dic73—81no—
311#25201—731nomaximal, Sylow
312C2×C73—42no—
313C2×C73#242511yescharacteristic
314C2×C73#303511yescharacteristic
315C2×C73#304411yescharacteristic
316C2×C73#330011yescharacteristic
317C2×C73#331511yescharacteristic
318C2×C73#334211yescharacteristic
319C2×C73#334911yescharacteristic
320C2×#485—731no—
321C2×#467—731no—
322#2499D7311yescharacteristic
323#2362—731no—
324#2083—731no—
325#2074—731no—
326#2066—731no—
327#2056—731no—
328#1949—731no—
329#1928—731no—
330#1878—731no—
331#1873—731no—
332#1850—731no—
333#1803—731no—
334#1735—731no—
335C73#2520111yescharacteristic, complemented, Sylow
336C22×Q16—731no—
337C23×C8—731no—
338C2×C22⋊Q8—731no—
339C2×C22⋊Q8C2×D7311yescharacteristic
340C2×C4×Q8—731no—
341C2×#117—731no—
342C2×#117C2×D7311yescharacteristic
343#485—1464no—
344#467—1464no—
345C2×#107—731no—
346C2×#107—731no—
347C2×#106—731no—
348C2×C4⋊C8—731no—
349C2×C4⋊C8—731no—
350C2×#103—731no—
351C2×#103—731no—
352C2×#103—731no—
353C2×#103—731no—
354C2×#103—731no—
355C2×#103—731no—
356C2×C22⋊C8—731no—
357#397—21no—
358#397—21no—
359#391—731no—
360#391C2×D7311yescharacteristic
361#391C2×D7311yescharacteristic
362#389—731no—
363#388—1461no—
364#388—1461no—
365#387—21no—
366#387—21no—
367#387—1461no—
368#384C2×D7311yescharacteristic
369#382C2×D7311yescharacteristic
370C4×C4⋊C4—731no—
371C4×C4⋊C4—1461no—
372C4×C22⋊C4C2×D7311yescharacteristic
373#340—731no—
374#340—731no—
375#340—731no—
376#340—731no—
377#340—731no—
378#340—731no—
379#336—731no—
380#336—731no—
381C22×Q8—731no—
382C22×Q8C22×D7311yescharacteristic
383C23×C4C22×D7311yescharacteristic
384C2×Q16—2921no—
385C2×Q16—1462no—
386C2×Q16—1462no—
387C22×C8—2921no—
388C22×C8—731no—
389C22×C8—731no—
390C22×C8—731no—
391C22×C8—731no—
392C22×C8—1462no—
393C22×C8—1462no—
394C22⋊Q8—22no—
395C22⋊Q8—22no—
396C22⋊Q8—734no—
397C22⋊Q8—1462no—
398C4×Q8—2921no—
399C4×Q8—1462no—
400#117—22no—
401#117—22no—
402#117—2921no—
403#117—1462no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—731no—
408C2×C4⋊C4—731no—
409C2×C4⋊C4—731no—
410C2×C4⋊C4—1461no—
411C2×C4⋊C4—1461no—
412C2×C4⋊C4—1461no—
413C2×C4⋊C4C22×D7311yescharacteristic
414C2×C4⋊C4C22×D7311yescharacteristic
415C2×C4⋊C4C22×D7311yescharacteristic
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—1461no—
419C2×C22⋊C4—1461no—
420C2×C22⋊C4C22⋊D7311yescharacteristic
421C2×C22⋊C4C22⋊D7311yescharacteristic
422C2×C42—21no—
423C2×C42—1461no—
424C2×C42—1461no—
425C2×C42—1461no—
426C2×C42C22×D7311yescharacteristic
427C2×C42C22×D7311yescharacteristic
428#107—1462no—
429#107—1462no—
430#106—1462no—
431C4⋊C8—2921no—
432C4⋊C8—1462no—
433#103—2921no—
434#103—1462no—
435#103—1462no—
436#103—1462no—
437#103—1462no—
438#103—1462no—
439C22⋊C8—1462no—
440#95—21no—
441#95—21no—
442#95—21no—
443#95—21no—
444#95—21no—
445#95—21no—
446#95—1461no—
447#95—1461no—
448#95—1461no—
449#95—1461no—
450#95C22⋊D7311yescharacteristic
451#95C22⋊D7311yescharacteristic
452C24C2×C22⋊D7311yescharacteristic
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—732no—
460C2×Q8—1461no—
461C2×Q8—1461no—
462C2×Q8—1461no—
463C2×Q8—1461no—
464C2×Q8—1461no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—1461no—
474C22×C4—1461no—
475C22×C4—1461no—
476C22×C4—1461no—
477C22×C4D4×D7312yes—
478C22×C4C2×C22⋊D7311yescharacteristic
479C22×C4C2×C22⋊D7311yescharacteristic
480C22×C4C2×C22⋊D7311yescharacteristic
481C22×C4C2×C22⋊D7311yescharacteristic
482C22×C4C2×C22⋊D7311yescharacteristic
483C22×C4#1192402712yes—
484C22×C4C23×D7311yescharacteristic
485Q16—2924no—
486C2×C8—2924no—
487C2×C8—732no—
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—
497C2×C8—1461no—
498C2×C8—1461no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—2922no—
509C4⋊C4—2922no—
510C4⋊C4—732no—
511C4⋊C4—732no—
512C4⋊C4—1461no—
513C4⋊C4—1461no—
514C4⋊C4—1464no—
515C4⋊C4—1461no—
516C4⋊C4—1461no—
517C22⋊C4—22no—
518C22⋊C4—41no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—2922no—
522C22⋊C4—1464no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—42no—
528C42—2922no—
529C42—2922no—
530C42—2922no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23#1455656212yes—
538C23C22×C22⋊D7311yescharacteristic
539Q8—41no—
540Q8—41no—
541Q8—41no—
542Q8—41no—
543Q8—2921no—
544Q8—2921no—
545Q8—1462no—
546Q8—1462no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
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—41no—
574C2×C4—41no—
575C2×C4—41no—
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×C4—1462no—
587C2×C4—1462no—
588C2×C4#1455656212yes—
589C2×C4#1455661112yes—
590C2×C4C2×D4×D7311yescharacteristic
591C2×C4#41×D7311yescharacteristic
592C2×C4#41×D7311yescharacteristic
593C2×C4C2×#1192402711yescharacteristic
594C2×C4#1455663711yescharacteristic
595C2×C4#1455663811yescharacteristic
596C8—2921no—
597C8—2921no—
598C8—2921no—
599C8—1462no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22C2×#1455656211yescharacteristic
608C22#1798922511yescharacteristic
609C22#1798923611yescharacteristic
610C22C2×#1455661111yescharacteristic
611C22#1798938511yescharacteristic
612C22#1798940211yescharacteristic
613C22#1798940211yescharacteristic
614C4—22no—
615C4—22no—
616C4—41no—
617C4—41no—
618C4—41no—
619C4—81no—
620C4—81no—
621C4—81no—
622C4—81no—
623C4—2922no—
624C4—2922no—
625C4—2922no—
626C4—5841no—
627C4#1798911411yescharacteristic
628C4#1798918111yescharacteristic
629C4#1798922911yescharacteristic
630C4#1798923511yescharacteristic
631C2—42no—
632C2#2261347811yescharacteristic
633C2#2261938011yescharacteristic
634C2#2261945711yescharacteristic
635C2#2262181211yescharacteristic
636C2#2262202111yescharacteristic
637C2#2262235711yescharacteristic
638C2#2262245511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2499C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
hypercenter#2499D73
hypocenterC73#25201
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2499

Complements

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

NH ≅ G/Ntype
4. #22611403C2semidirect
5. #22611350C2semidirect
6. #22611312C2semidirect
7. #22611253C2semidirect
8. #22610697C2semidirect
9. #22610569C2semidirect
10. #22610275C2semidirect
13. #22609933C2semidirect
335. C73#25201semidirect

Maximal subgroups

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

subgroupindex
C2×#179889282
C2×#179888262
#226129562
#226114032
#226113502
#226113122
#226112532
#226106972
#226105692
#226102752
#226102442
#226101112
#226099332
#226096472
C73×#24992
#2520173