Finite Group Databasebeta

← #33154041

Subgroups — #33154041

Order 18688.

848 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22612945C211yesmaximal, characteristic
2#22611246C211yesmaximal, characteristic, complemented
3#22611120C211yesmaximal, characteristic
4#22611064C211yesmaximal, characteristic, complemented
5#22610932C211yesmaximal, characteristic
6#22610880C211yesmaximal, characteristic, complemented
7#22610610C211yesmaximal, characteristic, complemented
8#22610562C211yesmaximal, characteristic, complemented
9#22610539C211yesmaximal, characteristic, complemented
10#22610509C211yesmaximal, characteristic, complemented
11#22610268C211yesmaximal, characteristic, complemented
12#22610239C211yesmaximal, characteristic, complemented
13#22610015C211yesmaximal, characteristic, complemented
14C4×#14556511C211yesmaximal, characteristic, complemented
15C73×#2492C211yesmaximal, characteristic, complemented
16C22×#11924015C2211yescharacteristic
17C2×#14556565C2211yescharacteristic
18C2×C4×D292C2211yescharacteristic
19C2×#14556551—21no—
20C22×C4⋊Dic73C2211yescharacteristic
21C2×#14556522C2211yescharacteristic
22C2×C4⋊#10596831C2211yescharacteristic
23C2×#14556515C2211yescharacteristic
24C2×#14556515C2211yescharacteristic
25C2×#14556515C2211yescharacteristic
26C2×#14556511C2211yescharacteristic
27C2×#14556511C2211yescharacteristic
28C2×#14556511C2211yescharacteristic
29C2×C22⋊#10596831C2211yescharacteristic
30C2×C22⋊#10596831C2211yescharacteristic
31C2×C4×#10596831C2211yescharacteristic
32#17988489C2211yescharacteristic
33#17988485C2211yescharacteristic
34#17988479—21no—
35#17988464—21no—
36#17988461—21no—
37C4×C4⋊Dic73C2211yescharacteristic
38#17988349C2211yescharacteristic
39#17988349C2211yescharacteristic
40#17988349C2211yescharacteristic
41#17988349C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988342C2211yescharacteristic
47#17988342C2211yescharacteristic
48C2×C4⋊Q8×C73C2211yescharacteristic
49C2×C22⋊Q8×C73C2211yescharacteristic
50#400×C73—21no—
51#393×C73—21no—
52#391×C73—21no—
53#391×C73C2211yescharacteristic
54#391×C73C2211yescharacteristic
55#388×C73—21no—
56#380×C73C2211yescharacteristic
57C4×C4⋊C4×C73C2211yescharacteristic
58C4×C22⋊C4×C73C2211yescharacteristic
59C22×D292C2311yescharacteristic
60C22×C22⋊D73—21no—
61C23×Dic73—21no—
62C22×C4×D73—21no—
63C2×#11924015—22no—
64C2×#11924015—41no—
65C2×#11924015—41no—
66C22×#10596831C2311yescharacteristic
67C22×#10596831C2311yescharacteristic
68C22×#10596831C2311yescharacteristic
69C22×#10596831C2311yescharacteristic
70#14556565—22no—
71#14556565—22no—
72C4×D292—22no—
73C4×D292—41no—
74#14556551—44no—
75C2×C4⋊Dic73—22no—
76C2×C4⋊Dic73—21no—
77C2×C4⋊Dic73—41no—
78C2×C4⋊Dic73C2311yescharacteristic
79C2×C4⋊Dic73C2311yescharacteristic
80C2×C4⋊Dic73C2311yescharacteristic
81C2×#11924007—21no—
82C2×#11924005—21no—
83C2×#11924005—21no—
84C2×#11924005—21no—
85C2×#11924005—21no—
86C2×C4×Dic73—21no—
87#14556522—22no—
88C4⋊#10596831—41no—
89#14556515—22no—
90#14556515—22no—
91#14556515—41no—
92#14556511—22no—
93#14556511—22no—
94#14556511—41no—
95C22⋊#10596831—41no—
96C22⋊#10596831—41no—
97C4×#10596831—41no—
98#14556498—21no—
99#14556498—21no—
100C22×Q8×C73—21no—
101C22×Q8×C73C2311yescharacteristic
102C23×C4×C73C2311yescharacteristic
103C4⋊Q8×C73—22no—
104C4⋊Q8×C73—41no—
105C22⋊Q8×C73—22no—
106C22⋊Q8×C73—41no—
107C2×C4⋊C4×C73—21no—
108C2×C4⋊C4×C73—21no—
109C2×C4⋊C4×C73C2311yescharacteristic
110C2×C4⋊C4×C73C2311yescharacteristic
111C2×C4⋊C4×C73C2311yescharacteristic
112C2×C22⋊C4×C73—21no—
113C2×C22⋊C4×C73—21no—
114C2×C22⋊C4×C73D411yescharacteristic
115C2×C22⋊C4×C73D411yescharacteristic
116C2×C42×C73—21no—
117C2×C42×C73C2311yescharacteristic
118C2×C42×C73C2311yescharacteristic
119#95×C73—21no—
120#95×C73—21no—
121#95×C73—21no—
122#95×C73—21no—
123#95×C73—21no—
124#95×C73—21no—
125#95×C73D411yescharacteristic
126#95×C73D411yescharacteristic
127C23×D73—21no—
128C2×D292—21no—
129C2×D292—21no—
130C2×D292—21no—
131C2×D292—21no—
132C2×D292—21no—
133C2×D292#4112yes—
134C2×C22⋊D73—24no—
135C2×C22⋊D73—42no—
136C2×C22⋊D73—42no—
137C22×Dic73—21no—
138C22×Dic73—21no—
139C22×Dic73—21no—
140C22×Dic73—42no—
141C22×Dic73—42no—
142C22×Dic73—81no—
143C2×C4×D73—42no—
144C2×C4×D73—42no—
145C2×C4×D73—81no—
146#11924015—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—42no—
164C4⋊Dic73#4112yes—
165C4⋊Dic73#4112yes—
166#11924007—42no—
167#11924005—24no—
168#11924005—42no—
169#11924005—42no—
170#11924005—81no—
171C4×Dic73—42no—
172C24×C73C2×D411yescharacteristic
173C2×Q8×C73—21no—
174C2×Q8×C73—21no—
175C2×Q8×C73—21no—
176C2×Q8×C73—21no—
177C2×Q8×C73—21no—
178C2×Q8×C73—21no—
179C2×Q8×C73—42no—
180C2×Q8×C73—42no—
181C2×Q8×C73—81no—
182C22×C4×C73—21no—
183C22×C4×C73—21no—
184C22×C4×C73—21no—
185C22×C4×C73—21no—
186C22×C4×C73—21no—
187C22×C4×C73—21no—
188C22×C4×C73—21no—
189C22×C4×C73—21no—
190C22×C4×C73—21no—
191C22×C4×C73—21no—
192C22×C4×C73C2×D411yescharacteristic
193C22×C4×C73C2×D411yescharacteristic
194C22×C4×C73C2×D411yescharacteristic
195C22×C4×C73C2×D411yescharacteristic
196C22×C4×C73C2×D411yescharacteristic
197C22×C4×C73C2411yescharacteristic
198C4⋊C4×C73—21no—
199C4⋊C4×C73—21no—
200C4⋊C4×C73—21no—
201C4⋊C4×C73—21no—
202C4⋊C4×C73—21no—
203C4⋊C4×C73—42no—
204C4⋊C4×C73—42no—
205C4⋊C4×C73#4112yes—
206C22⋊C4×C73—41no—
207C22⋊C4×C73—42no—
208C22⋊C4×C73—81no—
209C22⋊C4×C73SD1614yescomplemented
210C42×C73—21no—
211C42×C73—21no—
212C42×C73—21no—
213C42×C73—21no—
214C42×C73—81no—
215C23×C73—21no—
216C23×C73—21no—
217C23×C73—21no—
218C23×C73—21no—
219C23×C73—21no—
220C23×C73—21no—
221C23×C73C2×SD1612yes—
222C23×C73C22×D411yescharacteristic
223C22×D73—22no—
224C22×D73—41no—
225C22×D73—41no—
226C22×D73—41no—
227C22×D73—41no—
228C22×D73—41no—
229C22×D73—41no—
230Q8×C73—41no—
231Q8×C73—41no—
232Q8×C73—41no—
233Q8×C73—41no—
234Q8×C73—84no—
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—22no—
243C2×C4×C73—21no—
244C2×C4×C73—21no—
245C2×C4×C73—21no—
246C2×C4×C73—21no—
247C2×C4×C73—21no—
248C2×C4×C73—21no—
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—41no—
256C2×C4×C73—41no—
257C2×C4×C73—41no—
258C2×C4×C73—41no—
259C2×C4×C73—41no—
260C2×C4×C73—41no—
261C2×C4×C73—41no—
262C2×C4×C73—41no—
263C2×C4×C73—41no—
264C2×C4×C73C2×SD1612yes—
265C2×C4×C73C2×SD1612yes—
266C2×C4×C73C2×#4111yescharacteristic
267C2×C4×C73C2×#4111yescharacteristic
268C2×C4×C73C2×#4111yescharacteristic
269C2×C4×C73C2×#4111yescharacteristic
270C2×C4×C73H(5,𝔽2)11yescharacteristic
271C2×C4×C73H(5,𝔽2)11yescharacteristic
272C22⋊D73—48no—
273C2×Dic73—22no—
274C2×Dic73—22no—
275C2×Dic73—41no—
276C2×Dic73—41no—
277C2×Dic73—41no—
278C2×Dic73—41no—
279C2×Dic73—41no—
280C2×Dic73—41no—
281C2×Dic73—41no—
282C2×Dic73—84no—
283D292—22no—
284D292—22no—
285D292—41no—
286D292—41no—
287C4×D73—84no—
288#10596831—41no—
289#10596831—41no—
290#10596831—41no—
291#10596831—41no—
292C22×C73—22no—
293C22×C73—22no—
294C22×C73—22no—
295C22×C73—41no—
296C22×C73—41no—
297C22×C73—41no—
298C22×C73—41no—
299C22×C73#52511yescharacteristic
300C22×C73#54611yescharacteristic
301C22×C73#54611yescharacteristic
302C22×C73C22×SD1611yescharacteristic
303C22×C73#57511yescharacteristic
304C22×C73#57611yescharacteristic
305C22×C73#57811yescharacteristic
306C2×D73—42no—
307C2×D73—42no—
308C2×D73—42no—
309C2×D73—81no—
310C2×D73—81no—
311C2×D73—81no—
312C2×D73—81no—
313C4×C73—22no—
314C4×C73—22no—
315C4×C73—41no—
316C4×C73—41no—
317C4×C73—41no—
318C4×C73—42no—
319C4×C73—81no—
320C4×C73—81no—
321C4×C73—81no—
322C4×C73#52911yescharacteristic
323C4×C73#53811yescharacteristic
324C4×C73#55511yescharacteristic
325C4×C73#55911yescharacteristic
326Dic73—42no—
327Dic73—42no—
328Dic73—81no—
329#26722—731nomaximal, Sylow
330C2×C73—42no—
331C2×C73#245911yescharacteristic
332C2×C73#310111yescharacteristic
333C2×C73#312411yescharacteristic
334C2×C73#329511yescharacteristic
335C2×C73#329811yescharacteristic
336C2×C73#331311yescharacteristic
337C2×C73#332211yescharacteristic
338D73—82no—
339#2492D7311yescharacteristic, complemented
340#2361—731no—
341#2055—731no—
342#2034—731no—
343#2023—731no—
344#2000—731no—
345#1987—731no—
346#1933—731no—
347#1927—731no—
348#1923—731no—
349#1919—731no—
350#1877—731no—
351#1872—731no—
352#1823—731no—
353C4×#102—731no—
354C73#2672211yescharacteristic, complemented, Sylow
355C22×SD16—731no—
356C2×C4⋊Q8C2×D7311yescharacteristic
357C2×C22⋊Q8C2×D7311yescharacteristic
358C2×#121—731no—
359C2×C4×D4—731no—
360C2×C4×D4—1461no—
361C22×C4⋊C4—731no—
362C2×#106—731no—
363C2×C4⋊C8—731no—
364C2×#103—731no—
365C2×#103—731no—
366C2×#103—731no—
367C2×#102—731no—
368C2×#102—731no—
369C2×#102—731no—
370C2×C22⋊C8—731no—
371C2×C22⋊C8—731no—
372C2×C4×C8—731no—
373#400—21no—
374#393—21no—
375#391—21no—
376#391C2×D7311yescharacteristic
377#391C2×D7311yescharacteristic
378#390—731no—
379#389—731no—
380#388—21no—
381#388—1461no—
382#386—1461no—
383#385—1461no—
384#380C2×D7311yescharacteristic
385C4×C4⋊C4—731no—
386C4×C4⋊C4C2×D7311yescharacteristic
387C4×C22⋊C4C2×D7311yescharacteristic
388#340—731no—
389#340—731no—
390#340—731no—
391#340—731no—
392#340—731no—
393#340—731no—
394#340—731no—
395#340—731no—
396#336—731no—
397#336—731no—
398C22×Q8—21no—
399C22×Q8C22×D7311yescharacteristic
400C22×D4—731no—
401C22×D4—1461no—
402C23×C4—1461no—
403C23×C4—1461no—
404C23×C4C22×D7311yescharacteristic
405C2×SD16—2921no—
406C2×SD16—2921no—
407C2×SD16—1462no—
408C22×C8—731no—
409C22×C8—731no—
410C22×C8—731no—
411C22×C8—731no—
412C4⋊Q8—22no—
413C4⋊Q8—41no—
414C22⋊Q8—22no—
415C22⋊Q8—41no—
416#121—1462no—
417#121—1462no—
418C4×D4—2921no—
419C4×D4—2924no—
420C4×D4—1462no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—2921no—
424C2×C4⋊C4—731no—
425C2×C4⋊C4—731no—
426C2×C4⋊C4—731no—
427C2×C4⋊C4—1461no—
428C2×C4⋊C4—1461no—
429C2×C4⋊C4—1462no—
430C2×C4⋊C4C22×D7311yescharacteristic
431C2×C4⋊C4C22×D7311yescharacteristic
432C2×C4⋊C4C22×D7311yescharacteristic
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—1461no—
436C2×C22⋊C4—1461no—
437C2×C22⋊C4—1461no—
438C2×C22⋊C4—1461no—
439C2×C22⋊C4C22⋊D7311yescharacteristic
440C2×C22⋊C4C22⋊D7311yescharacteristic
441C2×C42—21no—
442C2×C42—1461no—
443C2×C42C22×D7311yescharacteristic
444C2×C42C22×D7311yescharacteristic
445#106—1462no—
446C4⋊C8—2921no—
447#103—2921no—
448#103—1462no—
449#103—1462no—
450#102—2921no—
451#102—1462no—
452#102—1462no—
453C22⋊C8—2921no—
454C22⋊C8—2921no—
455C4×C8—2921no—
456#95—21no—
457#95—21no—
458#95—21no—
459#95—21no—
460#95—21no—
461#95—21no—
462#95—1461no—
463#95—1461no—
464#95C22⋊D7311yescharacteristic
465#95C22⋊D7311yescharacteristic
466C24—1461no—
467C24C2×C22⋊D7311yescharacteristic
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—42no—
475C2×Q8—42no—
476C2×Q8—81no—
477C2×D4—2922no—
478C2×D4—2922no—
479C2×D4—732no—
480C2×D4—1464no—
481C2×D4—1461no—
482C2×D4—1461no—
483C2×D4—1461no—
484C2×D4—1461no—
485C2×D4—1461no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—2922no—
497C22×C4—2922no—
498C22×C4—2922no—
499C22×C4—2922no—
500C22×C4—5841no—
501C22×C4—5841no—
502C22×C4—1461no—
503C22×C4—1461no—
504C22×C4—1461no—
505C22×C4C2×C22⋊D7311yescharacteristic
506C22×C4C2×C22⋊D7311yescharacteristic
507C22×C4C2×C22⋊D7311yescharacteristic
508C22×C4C2×C22⋊D7311yescharacteristic
509C22×C4C2×C22⋊D7311yescharacteristic
510C22×C4C23×D7311yescharacteristic
511SD16—2924no—
512C2×C8—1461no—
513C2×C8—1461no—
514C2×C8—1461no—
515C2×C8—1461no—
516C2×C8—1461no—
517C2×C8—1461no—
518C2×C8—1461no—
519C2×C8—1461no—
520C2×C8—1461no—
521C2×C8—1461no—
522C2×C8—1461no—
523C2×C8—1461no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—2922no—
532C4⋊C4—2922no—
533C4⋊C4—732no—
534C4⋊C4—732no—
535C4⋊C4—1461no—
536C4⋊C4—1461no—
537C4⋊C4—1461no—
538C4⋊C4—1461no—
539C4⋊C4#1192402712yes—
540C22⋊C4—41no—
541C22⋊C4—42no—
542C22⋊C4—81no—
543C22⋊C4—2922no—
544C22⋊C4—2922no—
545C22⋊C4—5841no—
546C22⋊C4—1464no—
547C22⋊C4#1192401514yescomplemented
548C42—21no—
549C42—21no—
550C42—21no—
551C42—21no—
552C42—81no—
553C42—2922no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—2921no—
561C23—2921no—
562C23—2921no—
563C23—2921no—
564C23—2921no—
565C23—2921no—
566C23—1462no—
567C23C2×#1192401512yes—
568C23C22×C22⋊D7311yescharacteristic
569Q8—41no—
570Q8—41no—
571Q8—41no—
572Q8—41no—
573Q8—84no—
574D4—2928no—
575D4—2921no—
576D4—2921no—
577D4—1462no—
578D4—1462no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—2921no—
609C2×C4—2921no—
610C2×C4—2921no—
611C2×C4—2921no—
612C2×C4—2921no—
613C2×C4—2921no—
614C2×C4—2921no—
615C2×C4—5844no—
616C2×C4—5844no—
617C2×C4—1462no—
618C2×C4—1462no—
619C2×C4C2×#1192401512yes—
620C2×C4C2×#1192401512yes—
621C2×C4#41×D7311yescharacteristic
622C2×C4#41×D7311yescharacteristic
623C2×C4#41×D7311yescharacteristic
624C2×C4C2×#1192402711yescharacteristic
625C2×C4#1455663611yescharacteristic
626C2×C4#1455663711yescharacteristic
627C8—2921no—
628C8—2921no—
629C8—2921no—
630C8—2921no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—2922no—
639C22—2922no—
640C22—2922no—
641C22—5841no—
642C22—5841no—
643C22—5841no—
644C22—5841no—
645C22#1798909411yescharacteristic
646C22#1798922511yescharacteristic
647C22#1798923111yescharacteristic
648C22C22×#1192401511yescharacteristic
649C22#1798938511yescharacteristic
650C22#1798939011yescharacteristic
651C22#1798940211yescharacteristic
652C4—22no—
653C4—22no—
654C4—41no—
655C4—41no—
656C4—41no—
657C4—42no—
658C4—81no—
659C4—81no—
660C4—81no—
661C4—2922no—
662C4—2922no—
663C4—5841no—
664C4#1798911611yescharacteristic
665C4#1798917111yescharacteristic
666C4#1798929911yescharacteristic
667C4#1798931911yescharacteristic
668C2—42no—
669C2—5842no—
670C2#2261386411yescharacteristic
671C2#2261993511yescharacteristic
672C2#2262018711yescharacteristic
673C2#2262174911yescharacteristic
674C2#2262179611yescharacteristic
675C2#2262199111yescharacteristic
676C2#2262213211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2492C2
socleC23×C73C22×D4
prime radicalC2×#14556565C22
hypercenter#2492D73
hypocenterC73#26722
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2492

Complements

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

NH ≅ G/Ntype
2. #22611246C2semidirect
4. #22611064C2semidirect
6. #22610880C2semidirect
7. #22610610C2semidirect
8. #22610562C2semidirect
9. #22610539C2semidirect
10. #22610509C2semidirect
11. #22610268C2semidirect
12. #22610239C2semidirect
13. #22610015C2semidirect
14. C4×#14556511C2semidirect
15. C73×#2492C2semidirect
209. C22⋊C4×C73SD16semidirect
339. #2492D73semidirect
354. C73#26722semidirect
547. C22⋊C4#11924015semidirect

Maximal subgroups

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

subgroupindex
#226129452
#226112462
#226111202
#226110642
#226109322
#226108802
#226106102
#226105622
#226105392
#226105092
#226102682
#226102392
#226100152
C4×#145565112
C73×#24922
#2672273