Finite Group Databasebeta

← #33135744

Subgroups — #33135744

Order 18688.

788 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988806C211yesmaximal, characteristic
2C2×#17988718C211yesmaximal, characteristic, complemented
3#22615301C211yesmaximal, characteristic, complemented
4#22611312C211yesmaximal, characteristic, complemented
5#22611305C211yesmaximal, characteristic
6#22611253C211yesmaximal, characteristic, complemented
7#22611248C211yesmaximal, characteristic
8#22610801C211yesmaximal, characteristic, complemented
9#22610784C211yesmaximal, characteristic
10#22610748C211yesmaximal, characteristic
11#22610694C211yesmaximal, characteristic, complemented
12#22610631C211yesmaximal, characteristic
13#22610610C211yesmaximal, characteristic, complemented
14#22609764C211yesmaximal, characteristic, complemented
15C73×#2592C211yesmaximal, characteristic
16C22×#11924018C2211yescharacteristic
17C22×#11924016C2211yescharacteristic
18C2×C4×Dic146C2211yescharacteristic
19#17988806—24no—
20#17988718—24no—
21C2×#14556522C2211yescharacteristic
22C2×#14556522C2211yescharacteristic
23C2×C4⋊#10596831C2211yescharacteristic
24C2×C4⋊#10596831C2211yescharacteristic
25C2×C4⋊#10596831C2211yescharacteristic
26C2×#14556515C2211yescharacteristic
27C2×#14556515C2211yescharacteristic
28C2×#14556515C2211yescharacteristic
29C2×#14556514C2211yescharacteristic
30C2×#14556514C2211yescharacteristic
31C2×#14556512C2211yescharacteristic
32C2×#14556512C2211yescharacteristic
33C2×#14556512C2211yescharacteristic
34C2×#14556502C2211yescharacteristic
35C2×#14556502C2211yescharacteristic
36C2×C4×#10596831C2211yescharacteristic
37#17988520—21no—
38#17988520—21no—
39#17988494—21no—
40#17988493C2211yescharacteristic
41#17988493C2211yescharacteristic
42#17988485C2211yescharacteristic
43#17988485C2211yescharacteristic
44#17988483—21no—
45#17988457C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50C2×#124×C73C2211yescharacteristic
51C2×C4×Q8×C73C2211yescharacteristic
52#397×C73—21no—
53#393×C73—21no—
54#391×C73C2211yescharacteristic
55#391×C73C2211yescharacteristic
56#390×C73C2211yescharacteristic
57#390×C73C2211yescharacteristic
58#388×C73—21no—
59#388×C73—21no—
60C4×C4⋊C4×C73C2211yescharacteristic
61C22×Dic146C2311yescharacteristic
62C2×#11924018—22no—
63C2×#11924018—22no—
64C2×#11924018—41no—
65C2×#11924016—22no—
66C2×#11924016—22no—
67C2×#11924016—41no—
68C22×#10596831C2311yescharacteristic
69C22×#10596831C2311yescharacteristic
70C22×#10596831C2311yescharacteristic
71C22×#10596831C2311yescharacteristic
72C4×Dic146—22no—
73C4×Dic146—22no—
74C2×C4⋊Dic73—21no—
75C2×C4⋊Dic73—21no—
76C2×C4⋊Dic73C2311yescharacteristic
77C2×C4⋊Dic73C2311yescharacteristic
78C2×C4⋊Dic73C2311yescharacteristic
79C2×#11924007—21no—
80C2×#11924007—21no—
81C2×#11924007—21no—
82C2×C4×Dic73—21no—
83#14556522—41no—
84#14556522—41no—
85C4⋊#10596831—22no—
86C4⋊#10596831—22no—
87C4⋊#10596831D414yes—
88#14556515—22no—
89#14556515—22no—
90#14556515—41no—
91#14556514—22no—
92#14556514—22no—
93#14556512—22no—
94#14556512—22no—
95#14556512—41no—
96#14556502—22no—
97#14556502—22no—
98C4×#10596831—22no—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102#14556498—21no—
103#14556498—21no—
104#14556498—21no—
105C22×Q8×C73D411yescharacteristic
106C22×Q8×C73C2311yescharacteristic
107C22×D4×C73C2311yescharacteristic
108#124×C73—22no—
109#124×C73—22no—
110C4×Q8×C73—22no—
111C4×Q8×C73—22no—
112C2×C4⋊C4×C73D411yescharacteristic
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×C22⋊C4×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×C73—21no—
131#95×C73—21no—
132C2×Dic146—21no—
133C2×Dic146—21no—
134C2×Dic146—21no—
135C2×Dic146—21no—
136C2×Dic146—21no—
137C2×Dic146#4112yes—
138C22×Dic73—21no—
139C22×Dic73—21no—
140C22×Dic73—21no—
141C22×Dic73—21no—
142#11924018—44no—
143#11924016—44no—
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×#10596831C2×D412yes—
154C2×#10596831C2×D412yes—
155C2×#10596831#4112yes—
156C4⋊Dic73—21no—
157C4⋊Dic73—21no—
158C4⋊Dic73—21no—
159C4⋊Dic73—21no—
160C4⋊Dic73—21no—
161C4⋊Dic73—21no—
162C4⋊Dic73—42no—
163C4⋊Dic73—81no—
164#11924007—42no—
165#11924007—81no—
166#11924007—81no—
167C4×Dic73—42no—
168C24×C73—21no—
169C2×Q8×C73—22no—
170C2×Q8×C73—22no—
171C2×Q8×C73—21no—
172C2×Q8×C73—21no—
173C2×Q8×C73—21no—
174C2×Q8×C73—21no—
175C2×Q8×C73—21no—
176C2×Q8×C73—21no—
177C2×Q8×C73—41no—
178C2×D4×C73—21no—
179C2×D4×C73—21no—
180C2×D4×C73—21no—
181C2×D4×C73—21no—
182C2×D4×C73—21no—
183C2×D4×C73—21no—
184C22×C4×C73—21no—
185C22×C4×C73—21no—
186C22×C4×C73—21no—
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—22no—
195C4⋊C4×C73—22no—
196C4⋊C4×C73—22no—
197C4⋊C4×C73—21no—
198C4⋊C4×C73—21no—
199C4⋊C4×C73—21no—
200C4⋊C4×C73—21no—
201C22⋊C4×C73—42no—
202C22⋊C4×C73—42no—
203C22⋊C4×C73—42no—
204C22⋊C4×C73—42no—
205C22⋊C4×C73—81no—
206C22⋊C4×C73—81no—
207C42×C73—21no—
208C42×C73—21no—
209C42×C73—21no—
210C42×C73—21no—
211C42×C73—81no—
212C42×C73C2×D412yes—
213C42×C73#4112yes—
214C23×C73—41no—
215C23×C73—41no—
216C23×C73—41no—
217C23×C73—41no—
218C23×C73—41no—
219C23×C73—41no—
220C23×C73—41no—
221C23×C73C22×D411yescharacteristic
222Q8×C73—44no—
223Q8×C73—41no—
224Q8×C73—41no—
225Q8×C73—41no—
226Q8×C73—41no—
227D4×C73—41no—
228D4×C73—41no—
229D4×C73—41no—
230D4×C73—41no—
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—21no—
241C2×C4×C73—21no—
242C2×C4×C73—21no—
243C2×C4×C73—21no—
244C2×C4×C73—21no—
245C2×C4×C73—21no—
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—41no—
253C2×C4×C73—41no—
254C2×C4×C73—41no—
255C2×C4×C73#12112yes—
256C2×C4×C73#12112yes—
257C2×C4×C73#13712yes—
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—
280#10596831—22no—
281#10596831—22no—
282#10596831—22no—
283#10596831—41no—
284C22×C73—81no—
285C22×C73—81no—
286C22×C73—81no—
287C22×C73—81no—
288C22×C73—81no—
289C22×C73—81no—
290C22×C73—81no—
291C22×C73C2×#12111yescharacteristic
292C22×C73#54811yescharacteristic
293C22×C73#55011yescharacteristic
294C22×C73C2×#13711yescharacteristic
295C22×C73#57511yescharacteristic
296C22×C73#57711yescharacteristic
297C22×C73#57711yescharacteristic
298C4×C73—22no—
299C4×C73—22no—
300C4×C73—22no—
301C4×C73—41no—
302C4×C73—41no—
303C4×C73—41no—
304C4×C73—81no—
305C4×C73—81no—
306C4×C73—81no—
307C4×C73#52911yescharacteristic
308C4×C73#54211yescharacteristic
309C4×C73#54611yescharacteristic
310C4×C73#54811yescharacteristic
311Dic73—42no—
312Dic73—81no—
313Dic73—81no—
314Dic73—81no—
315#25311—731nomaximal, Sylow
316C2×C73—161no—
317C2×C73#242611yescharacteristic
318C2×C73#305611yescharacteristic
319C2×C73#306011yescharacteristic
320C2×C73#336911yescharacteristic
321C2×C73#337111yescharacteristic
322C2×C73#340011yescharacteristic
323C2×C73#340111yescharacteristic
324C2×#464—731no—
325C2×#446—731no—
326#2592D7311yescharacteristic
327#2582—731no—
328#2066—731no—
329#2065—731no—
330#2056—731no—
331#2055—731no—
332#1971—731no—
333#1968—731no—
334#1959—731no—
335#1948—731no—
336#1936—731no—
337#1933—731no—
338#1763—731no—
339C73#2531111yescharacteristic, complemented, Sylow
340C22×Q16—731no—
341C22×SD16—731no—
342C2×#124C2×D7311yescharacteristic
343C2×C4×Q8—731no—
344C2×C4×Q8C2×D7311yescharacteristic
345#464—1464no—
346#446—1464no—
347C2×#106—731no—
348C2×#106—731no—
349C2×C4⋊C8—731no—
350C2×C4⋊C8—731no—
351C2×C4⋊C8—731no—
352C2×#103—731no—
353C2×#103—731no—
354C2×#103—731no—
355C2×#103—731no—
356C2×#103—731no—
357C2×#102—731no—
358C2×#102—731no—
359C2×#102—731no—
360C2×#97—731no—
361C2×#97—731no—
362C2×C4×C8—731no—
363#398—1461no—
364#398—1461no—
365#397—21no—
366#393—21no—
367#391—731no—
368#391—731no—
369#391—1461no—
370#391C2×D7311yescharacteristic
371#391C2×D7311yescharacteristic
372#390C2×D7311yescharacteristic
373#390C2×D7311yescharacteristic
374#389—731no—
375#389—731no—
376#389—1461no—
377#388—21no—
378#388—21no—
379#384—731no—
380C4×C4⋊C4C2×D7311yescharacteristic
381#340—731no—
382#340—731no—
383#340—731no—
384#340—731no—
385C22×Q8—731no—
386C22×Q8C22⋊D7311yescharacteristic
387C22×Q8C22×D7311yescharacteristic
388C22×D4C22×D7311yescharacteristic
389C2×Q16—2921no—
390C2×Q16—1462no—
391C2×Q16—1462no—
392C2×SD16—2921no—
393C2×SD16—1462no—
394C2×SD16—1462no—
395C22×C8—731no—
396C22×C8—731no—
397C22×C8—731no—
398C22×C8—731no—
399#124—22no—
400#124—22no—
401C4×Q8—22no—
402C4×Q8—22no—
403C4×Q8—1462no—
404C4×Q8—1462no—
405C2×C4⋊C4—731no—
406C2×C4⋊C4—731no—
407C2×C4⋊C4—731no—
408C2×C4⋊C4—1461no—
409C2×C4⋊C4—1461no—
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×C4⋊C4C22×D7311yescharacteristic
417C2×C4⋊C4C22×D7311yescharacteristic
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C42—21no—
425C2×C42—1461no—
426C2×C42C22×D7311yescharacteristic
427C2×C42C22×D7311yescharacteristic
428C2×C42C22×D7311yescharacteristic
429#106—2921no—
430#106—2921no—
431C4⋊C8—734no—
432C4⋊C8—1462no—
433C4⋊C8—1462no—
434#103—2921no—
435#103—1462no—
436#103—1462no—
437#103—1462no—
438#103—1462no—
439#102—2921no—
440#102—1462no—
441#102—1462no—
442#97—1462no—
443#97—1462no—
444C4×C8—1462no—
445#95—21no—
446#95—21no—
447#95—21no—
448#95—21no—
449#95—21no—
450#95—1461no—
451#95—1461no—
452#95—1461no—
453#95—1461no—
454#95—1461no—
455#95—1461no—
456C24—21no—
457C2×Q8—22no—
458C2×Q8—22no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—41no—
466C2×Q8—732no—
467C2×Q8—1461no—
468C2×Q8—1461no—
469C2×Q8—1461no—
470C2×Q8—1461no—
471C2×Q8—1461no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—1461no—
482C22×C4—1461no—
483C22×C4—1461no—
484C22×C4—1461no—
485C22×C4C2×C22⋊D7311yescharacteristic
486C22×C4C2×C22⋊D7311yescharacteristic
487C22×C4C2×C22⋊D7311yescharacteristic
488C22×C4C2×C22⋊D7311yescharacteristic
489C22×C4C2×C22⋊D7311yescharacteristic
490C22×C4C2×C22⋊D7311yescharacteristic
491C22×C4C23×D7311yescharacteristic
492Q16—2924no—
493SD16—2924no—
494C2×C8—732no—
495C2×C8—732no—
496C2×C8—732no—
497C2×C8—1461no—
498C2×C8—1461no—
499C2×C8—1461no—
500C2×C8—1461no—
501C2×C8—1461no—
502C2×C8—1461no—
503C2×C8—1461no—
504C2×C8—1461no—
505C2×C8—1461no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—2922no—
514C4⋊C4—2922no—
515C4⋊C4—5841no—
516C4⋊C4—5841no—
517C4⋊C4—5841no—
518C4⋊C4—1461no—
519C4⋊C4—1461no—
520C4⋊C4—1461no—
521C4⋊C4—1461no—
522C4⋊C4—1461no—
523C4⋊C4—1461no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C22⋊C4—42no—
528C22⋊C4—81no—
529C22⋊C4—81no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—81no—
535C42—2922no—
536C42D4×D7312yes—
537C42#1192402712yes—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23C22×C22⋊D7311yescharacteristic
546Q8—44no—
547Q8—41no—
548Q8—41no—
549Q8—41no—
550Q8—41no—
551Q8—2921no—
552Q8—2921no—
553Q8—1462no—
554Q8—1462no—
555D4—41no—
556D4—41no—
557D4—41no—
558D4—41no—
559C2×C4—21no—
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—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—2921no—
584C2×C4—2921no—
585C2×C4—2921no—
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—1462no—
595C2×C4#1455656312yes—
596C2×C4#1455656312yes—
597C2×C4#1455662112yes—
598C2×C4C2×D4×D7311yescharacteristic
599C2×C4#41×D7311yescharacteristic
600C2×C4#41×D7311yescharacteristic
601C2×C4C2×#1192402711yescharacteristic
602C2×C4#1455663711yescharacteristic
603C2×C4#1455663811yescharacteristic
604C8—2921no—
605C8—1462no—
606C8—1462no—
607C8—1462no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22C2×#1455656311yescharacteristic
616C22#1798924411yescharacteristic
617C22#1798925611yescharacteristic
618C22C2×#1455662111yescharacteristic
619C22#1798938511yescharacteristic
620C22#1798939511yescharacteristic
621C22#1798939811yescharacteristic
622C4—22no—
623C4—22no—
624C4—22no—
625C4—41no—
626C4—41no—
627C4—41no—
628C4—81no—
629C4—81no—
630C4—81no—
631C4—2922no—
632C4—5841no—
633C4—5841no—
634C4—5841no—
635C4#1798911611yescharacteristic
636C4#1798920611yescharacteristic
637C4#1798923011yescharacteristic
638C4#1798924511yescharacteristic
639C2—161no—
640C2#2261350011yescharacteristic
641C2#2261956311yescharacteristic
642C2#2261959011yescharacteristic
643C2#2262266711yescharacteristic
644C2#2262269311yescharacteristic
645C2#2262298411yescharacteristic
646C2#2262299711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2592C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
hypercenter#2592D73
hypocenterC73#25311
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2592

Complements

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

NH ≅ G/Ntype
2. C2×#17988718C2semidirect
3. #22615301C2semidirect
4. #22611312C2semidirect
6. #22611253C2semidirect
8. #22610801C2semidirect
11. #22610694C2semidirect
13. #22610610C2semidirect
14. #22609764C2semidirect
339. C73#25311semidirect

Maximal subgroups

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

subgroupindex
C2×#179888062
C2×#179887182
#226153012
#226113122
#226113052
#226112532
#226112482
#226108012
#226107842
#226107482
#226106942
#226106312
#226106102
#226097642
C73×#25922
#2531173