Finite Group Databasebeta

← #33136074

Subgroups — #33136074

Order 18688.

824 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988946C211yesmaximal, characteristic
2C2×#17988806C211yesmaximal, characteristic
3#22615146C211yesmaximal, characteristic, complemented
4#22611349C211yesmaximal, characteristic, complemented
5#22611344C211yesmaximal, characteristic
6#22611253C211yesmaximal, characteristic, complemented
7#22611248C211yesmaximal, characteristic
8#22610879C211yesmaximal, characteristic, complemented
9#22610655C211yesmaximal, characteristic, complemented
10#22610272C211yesmaximal, characteristic, complemented
11#22610265C211yesmaximal, characteristic
12#22609946C211yesmaximal, characteristic, complemented
13#22609941C211yesmaximal, characteristic
14#22609764C211yesmaximal, characteristic, complemented
15C73×#2367C211yesmaximal, characteristic
16C22×#11924018C2211yescharacteristic
17C22×#11924016C2211yescharacteristic
18C2×#14556591C2211yescharacteristic
19C2×C4×Dic146C2211yescharacteristic
20#17988946—24no—
21#17988806—24no—
22C2×C4⋊#10596831C2211yescharacteristic
23C2×C4⋊#10596831C2211yescharacteristic
24C2×C4⋊#10596831C2211yescharacteristic
25C2×#14556515C2211yescharacteristic
26C2×#14556515C2211yescharacteristic
27C2×#14556515C2211yescharacteristic
28C2×#14556514C2211yescharacteristic
29C2×#14556514C2211yescharacteristic
30C2×#14556512C2211yescharacteristic
31C2×#14556512C2211yescharacteristic
32C2×#14556512C2211yescharacteristic
33C2×#14556502C2211yescharacteristic
34#17988527—21no—
35#17988527—21no—
36#17988493C2211yescharacteristic
37#17988493C2211yescharacteristic
38#17988492—21no—
39#17988492—21no—
40#17988453C2211yescharacteristic
41#17988453C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988342C2211yescharacteristic
49#17988342C2211yescharacteristic
50C2×#124×C73C2211yescharacteristic
51C2×C4×Q8×C73C2211yescharacteristic
52C2×C4×D4×C73C2211yescharacteristic
53#391×C73C2211yescharacteristic
54#390×C73C2211yescharacteristic
55#388×C73—21no—
56#388×C73—21no—
57#385×C73—21no—
58C4×C4⋊C4×C73C2211yescharacteristic
59C4×C4⋊C4×C73C2211yescharacteristic
60C4×C22⋊C4×C73—21no—
61C22×Dic146—21no—
62C22×Dic146C2311yescharacteristic
63C2×#11924018—22no—
64C2×#11924018—22no—
65C2×#11924018—41no—
66C2×#11924016—22no—
67C2×#11924016—22no—
68C2×#11924016—41no—
69C22×#10596831C2311yescharacteristic
70C22×#10596831C2311yescharacteristic
71C22×#10596831C2311yescharacteristic
72C22×#10596831C2311yescharacteristic
73#14556591—22no—
74#14556591—22no—
75C4×Dic146—22no—
76C4×Dic146—22no—
77C2×C4⋊Dic73—21no—
78C2×C4⋊Dic73C2311yescharacteristic
79C2×C4⋊Dic73C2311yescharacteristic
80C2×C4⋊Dic73C2311yescharacteristic
81C2×#11924007—21no—
82C2×C4×Dic73—21no—
83C4⋊#10596831—22no—
84C4⋊#10596831—41no—
85C4⋊#10596831—41no—
86#14556515—22no—
87#14556515—22no—
88#14556515—22no—
89#14556514—22no—
90#14556514—22no—
91#14556512—22no—
92#14556512—22no—
93#14556512—22no—
94#14556502—22no—
95#14556498—21no—
96#14556498—21no—
97#14556498—21no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102#14556498—21no—
103C22×Q8×C73C2311yescharacteristic
104C22×D4×C73C2311yescharacteristic
105C23×C4×C73—21no—
106#124×C73—22no—
107#124×C73D414yes—
108C4×Q8×C73—22no—
109C4×Q8×C73—41no—
110C4×D4×C73—22no—
111C4×D4×C73—41no—
112C2×C4⋊C4×C73—21no—
113C2×C4⋊C4×C73D411yescharacteristic
114C2×C4⋊C4×C73D411yescharacteristic
115C2×C4⋊C4×C73C2311yescharacteristic
116C2×C4⋊C4×C73C2311yescharacteristic
117C2×C22⋊C4×C73—21no—
118C2×C22⋊C4×C73—21no—
119C2×C22⋊C4×C73—21no—
120C2×C22⋊C4×C73—21no—
121C2×C22⋊C4×C73—21no—
122C2×C42×C73—21no—
123C2×C42×C73—21no—
124C2×C42×C73C2311yescharacteristic
125C2×C42×C73C2311yescharacteristic
126C2×C42×C73C2311yescharacteristic
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×Dic146—42no—
138C2×Dic146—42no—
139C2×Dic146—42no—
140C2×Dic146#4112yes—
141C22×Dic73—21no—
142C22×Dic73—21no—
143C22×Dic73—21no—
144C22×Dic73—21no—
145#11924018—44no—
146#11924016—44no—
147C2×#10596831—21no—
148C2×#10596831—21no—
149C2×#10596831—21no—
150C2×#10596831—21no—
151C2×#10596831—21no—
152C2×#10596831—21no—
153C2×#10596831—21no—
154C2×#10596831—21no—
155C2×#10596831—21no—
156C2×#10596831—21no—
157C2×#10596831—21no—
158C2×#10596831—21no—
159C4⋊Dic73—21no—
160C4⋊Dic73—21no—
161C4⋊Dic73—21no—
162C4⋊Dic73—21no—
163C4⋊Dic73—21no—
164C4⋊Dic73—21no—
165C4⋊Dic73—42no—
166#11924007—42no—
167C4×Dic73—42no—
168C24×C73—21no—
169C2×Q8×C73—21no—
170C2×Q8×C73—21no—
171C2×Q8×C73—21no—
172C2×Q8×C73—21no—
173C2×Q8×C73—21no—
174C2×Q8×C73C2×D412yes—
175C2×D4×C73—21no—
176C2×D4×C73—21no—
177C2×D4×C73—21no—
178C2×D4×C73—21no—
179C2×D4×C73—21no—
180C2×D4×C73C2×D412yes—
181C22×C4×C73—21no—
182C22×C4×C73—21no—
183C22×C4×C73—21no—
184C22×C4×C73—42no—
185C22×C4×C73—42no—
186C22×C4×C73—81no—
187C22×C4×C73C2×D411yescharacteristic
188C22×C4×C73C2×D411yescharacteristic
189C22×C4×C73C2×D411yescharacteristic
190C22×C4×C73C2×D411yescharacteristic
191C22×C4×C73C2×D411yescharacteristic
192C22×C4×C73C2×D411yescharacteristic
193C22×C4×C73C2411yescharacteristic
194C4⋊C4×C73—21no—
195C4⋊C4×C73—21no—
196C4⋊C4×C73—21no—
197C4⋊C4×C73—41no—
198C4⋊C4×C73—41no—
199C4⋊C4×C73—42no—
200C4⋊C4×C73#4112yes—
201C22⋊C4×C73—24no—
202C22⋊C4×C73—24no—
203C22⋊C4×C73—42no—
204C22⋊C4×C73—42no—
205C22⋊C4×C73—81no—
206C42×C73—21no—
207C42×C73—21no—
208C42×C73—21no—
209C42×C73—21no—
210C42×C73—42no—
211C42×C73—81no—
212C42×C73C2×D412yes—
213C42×C73#4112yes—
214C23×C73—22no—
215C23×C73—41no—
216C23×C73—41no—
217C23×C73—41no—
218C23×C73—41no—
219C23×C73—41no—
220C23×C73—41no—
221C23×C73C22×D411yescharacteristic
222Q8×C73—22no—
223Q8×C73—22no—
224Q8×C73—41no—
225Q8×C73—41no—
226D4×C73—22no—
227D4×C73—22no—
228D4×C73—41no—
229D4×C73—41no—
230C2×C4×C73—21no—
231C2×C4×C73—21no—
232C2×C4×C73—21no—
233C2×C4×C73—21no—
234C2×C4×C73—22no—
235C2×C4×C73—21no—
236C2×C4×C73—21no—
237C2×C4×C73—21no—
238C2×C4×C73—21no—
239C2×C4×C73—22no—
240C2×C4×C73—21no—
241C2×C4×C73—21no—
242C2×C4×C73—21no—
243C2×C4×C73—21no—
244C2×C4×C73—21no—
245C2×C4×C73—21no—
246C2×C4×C73—21no—
247C2×C4×C73—21no—
248C2×C4×C73—41no—
249C2×C4×C73—41no—
250C2×C4×C73—41no—
251C2×C4×C73—41no—
252C2×C4×C73—41no—
253C2×C4×C73—41no—
254C2×C4×C73—41no—
255C2×C4×C73—84no—
256C2×C4×C73#12112yes—
257C2×C4×C73#12112yes—
258C2×C4×C73C22×D411yescharacteristic
259C2×C4×C73C2×#4111yescharacteristic
260C2×C4×C73C2×#4111yescharacteristic
261C2×C4×C73C2×#4111yescharacteristic
262C2×C4×C73H(5,𝔽2)11yescharacteristic
263C2×C4×C732−1+411yescharacteristic
264C2×Dic73—22no—
265C2×Dic73—41no—
266C2×Dic73—41no—
267C2×Dic73—41no—
268C2×Dic73—41no—
269C2×Dic73—41no—
270C2×Dic73—41no—
271C2×Dic73—41no—
272C2×Dic73—41no—
273C2×Dic73—41no—
274C2×Dic73—41no—
275C2×Dic73—41no—
276Dic146—22no—
277Dic146—22no—
278Dic146—41no—
279Dic146—41no—
280Dic146—84no—
281#10596831—41no—
282#10596831—41no—
283#10596831—41no—
284#10596831—41no—
285C22×C73—42no—
286C22×C73—42no—
287C22×C73—42no—
288C22×C73—81no—
289C22×C73—81no—
290C22×C73—81no—
291C22×C73—81no—
292C22×C73C2×#12111yescharacteristic
293C22×C73#54611yescharacteristic
294C22×C73#54811yescharacteristic
295C22×C73#57511yescharacteristic
296C22×C73#57511yescharacteristic
297C22×C73#57711yescharacteristic
298C22×C73#57811yescharacteristic
299C4×C73—22no—
300C4×C73—22no—
301C4×C73—42no—
302C4×C73—41no—
303C4×C73—41no—
304C4×C73—42no—
305C4×C73—41no—
306C4×C73—41no—
307C4×C73—81no—
308C4×C73#52911yescharacteristic
309C4×C73#54211yescharacteristic
310C4×C73#54611yescharacteristic
311C4×C73#54711yescharacteristic
312Dic73—42no—
313Dic73—81no—
314Dic73—81no—
315Dic73—81no—
316#25337—731nomaximal, Sylow
317C2×C73—82no—
318C2×C73#242711yescharacteristic
319C2×C73#304011yescharacteristic
320C2×C73#304611yescharacteristic
321C2×C73#331111yescharacteristic
322C2×C73#331311yescharacteristic
323C2×C73#337111yescharacteristic
324C2×C73#337711yescharacteristic
325C2×#489—731no—
326C2×#464—731no—
327#2568—731no—
328#2367D7311yescharacteristic
329#2074—731no—
330#2073—731no—
331#2056—731no—
332#2055—731no—
333#1987—731no—
334#1940—731no—
335#1878—731no—
336#1877—731no—
337#1805—731no—
338#1804—731no—
339#1763—731no—
340C73#2533711yescharacteristic, complemented, Sylow
341C22×Q16—731no—
342C22×SD16—731no—
343C2×C4⋊Q8—731no—
344C2×#124C2×D7311yescharacteristic
345C2×C4×Q8—731no—
346C2×C4×Q8C2×D7311yescharacteristic
347C2×C4×D4C2×D7311yescharacteristic
348#489—1464no—
349#464—1464no—
350C2×C4⋊C8—731no—
351C2×C4⋊C8—731no—
352C2×C4⋊C8—731no—
353C2×#103—731no—
354C2×#103—731no—
355C2×#103—731no—
356C2×#103—731no—
357C2×#103—731no—
358C2×#102—731no—
359C2×#102—731no—
360C2×#102—731no—
361C2×#97—731no—
362#400—1461no—
363#400—1461no—
364#391—731no—
365#391—731no—
366#391—1461no—
367#391—1461no—
368#391C2×D7311yescharacteristic
369#390C2×D7311yescharacteristic
370#388—21no—
371#388—21no—
372#385—21no—
373#382—731no—
374#382—731no—
375C4×C4⋊C4C2×D7311yescharacteristic
376C4×C4⋊C4C2×D7311yescharacteristic
377C4×C22⋊C4—21no—
378#340—731no—
379#340—731no—
380#340—731no—
381#340—731no—
382#340—731no—
383#340—731no—
384#336—731no—
385#336—731no—
386C22×Q8—731no—
387C22×Q8—1461no—
388C22×Q8C22×D7311yescharacteristic
389C22×D4C22×D7311yescharacteristic
390C23×C4—21no—
391C2×Q16—2921no—
392C2×Q16—1462no—
393C2×Q16—1462no—
394C2×SD16—2921no—
395C2×SD16—1462no—
396C2×SD16—1462no—
397C22×C8—731no—
398C22×C8—731no—
399C22×C8—731no—
400C22×C8—731no—
401C4⋊Q8—1462no—
402C4⋊Q8—1462no—
403#124—22no—
404#124D29214yes—
405C4×Q8—22no—
406C4×Q8—41no—
407C4×Q8—1462no—
408C4×Q8—1462no—
409C4×D4—22no—
410C4×D4—41no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—731no—
413C2×C4⋊C4—731no—
414C2×C4⋊C4—731no—
415C2×C4⋊C4—1461no—
416C2×C4⋊C4—1461no—
417C2×C4⋊C4C22⋊D7311yescharacteristic
418C2×C4⋊C4C22⋊D7311yescharacteristic
419C2×C4⋊C4C22×D7311yescharacteristic
420C2×C4⋊C4C22×D7311yescharacteristic
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—21no—
425C2×C22⋊C4—21no—
426C2×C42—21no—
427C2×C42—21no—
428C2×C42—1461no—
429C2×C42C22×D7311yescharacteristic
430C2×C42C22×D7311yescharacteristic
431C2×C42C22×D7311yescharacteristic
432C4⋊C8—2921no—
433C4⋊C8—2921no—
434C4⋊C8—1462no—
435#103—1462no—
436#103—1462no—
437#103—1462no—
438#103—1462no—
439#103—1462no—
440#102—1462no—
441#102—1462no—
442#102—1462no—
443#97—1462no—
444#95—21no—
445#95—21no—
446#95—21no—
447#95—1461no—
448#95—1461no—
449#95—1461no—
450#95—1461no—
451#95—1461no—
452#95—1461no—
453#95—1461no—
454#95—1461no—
455#95C22⋊D7311yescharacteristic
456#95C22⋊D7311yescharacteristic
457C24—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—2922no—
464C2×Q8—2922no—
465C2×Q8—2922no—
466C2×Q8—732no—
467C2×Q8—1461no—
468C2×Q8—1461no—
469C2×Q8—1461no—
470C2×Q8—1461no—
471C2×Q8—1461no—
472C2×Q8C2×D29212yes—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4C2×D29212yes—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—81no—
485C22×C4—1461no—
486C22×C4—1461no—
487C22×C4—1461no—
488C22×C4—1461no—
489C22×C4C2×C22⋊D7311yescharacteristic
490C22×C4C2×C22⋊D7311yescharacteristic
491C22×C4C2×C22⋊D7311yescharacteristic
492C22×C4C2×C22⋊D7311yescharacteristic
493C22×C4C2×C22⋊D7311yescharacteristic
494C22×C4C2×C22⋊D7311yescharacteristic
495C22×C4C23×D7311yescharacteristic
496Q16—2924no—
497SD16—2924no—
498C2×C8—1461no—
499C2×C8—1461no—
500C2×C8—1461no—
501C2×C8—1461no—
502C2×C8—1461no—
503C2×C8—1461no—
504C2×C8—1461no—
505C2×C8—1461no—
506C2×C8—1461no—
507C2×C8—1461no—
508C2×C8—1461no—
509C2×C8—1461no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—41no—
514C4⋊C4—41no—
515C4⋊C4—42no—
516C4⋊C4—2922no—
517C4⋊C4—2922no—
518C4⋊C4—1461no—
519C4⋊C4—1461no—
520C4⋊C4—1461no—
521C4⋊C4—1461no—
522C4⋊C4—1461no—
523C4⋊C4—1461no—
524C4⋊C4#1192402712yes—
525C22⋊C4—24no—
526C22⋊C4—24no—
527C22⋊C4—42no—
528C22⋊C4—42no—
529C22⋊C4—81no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—42no—
535C42—81no—
536C42—2922no—
537C42C2×D29212yes—
538C42#1192402812yes—
539C23—22no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23C22×C22⋊D7311yescharacteristic
547Q8—22no—
548Q8—22no—
549Q8—41no—
550Q8—41no—
551Q8—2921no—
552Q8—2921no—
553Q8—5844no—
554Q8—1462no—
555Q8—1462no—
556D4—22no—
557D4—22no—
558D4—41no—
559D4—41no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—84no—
586C2×C4—2921no—
587C2×C4—2921no—
588C2×C4—2921no—
589C2×C4—2921no—
590C2×C4—2921no—
591C2×C4—2921no—
592C2×C4—2921no—
593C2×C4—2921no—
594C2×C4—2921no—
595C2×C4—2921no—
596C2×C4—2921no—
597C2×C4—1462no—
598C2×C4#1455656512yes—
599C2×C4#1455656512yes—
600C2×C4C22×D29211yescharacteristic
601C2×C4#41×D7311yescharacteristic
602C2×C4C2×#1192402711yescharacteristic
603C2×C4C2×#1192402811yescharacteristic
604C2×C4#1455663611yescharacteristic
605C2×C4#1455663811yescharacteristic
606C8—2921no—
607C8—2921no—
608C8—2921no—
609C8—2921no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22C2×#1455656511yescharacteristic
618C22#1798923111yescharacteristic
619C22#1798924411yescharacteristic
620C22#1798938511yescharacteristic
621C22#1798938511yescharacteristic
622C22#1798939811yescharacteristic
623C22#1798940211yescharacteristic
624C4—22no—
625C4—22no—
626C4—41no—
627C4—41no—
628C4—42no—
629C4—41no—
630C4—41no—
631C4—42no—
632C4—81no—
633C4—2922no—
634C4—5841no—
635C4—5841no—
636C4—5841no—
637C4#1798911911yescharacteristic
638C4#1798920211yescharacteristic
639C4#1798923211yescharacteristic
640C4#1798924111yescharacteristic
641C2—82no—
642C2#2261350211yescharacteristic
643C2#2261942511yescharacteristic
644C2#2261948111yescharacteristic
645C2#2262196711yescharacteristic
646C2#2262199711yescharacteristic
647C2#2262269311yescharacteristic
648C2#2262276711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2367C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
hypercenter#2367D73
hypocenterC73#25337
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2367

Complements

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

NH ≅ G/Ntype
3. #22615146C2semidirect
4. #22611349C2semidirect
6. #22611253C2semidirect
8. #22610879C2semidirect
9. #22610655C2semidirect
10. #22610272C2semidirect
12. #22609946C2semidirect
14. #22609764C2semidirect
340. C73#25337semidirect

Maximal subgroups

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

subgroupindex
C2×#179889462
C2×#179888062
#226151462
#226113492
#226113442
#226112532
#226112482
#226108792
#226106552
#226102722
#226102652
#226099462
#226099412
#226097642
C73×#23672
#2533773