Finite Group Databasebeta

← #33149546

Subgroups — #33149546

Order 18688.

802 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22612967C211yesmaximal, characteristic
2#22611498C211yesmaximal, characteristic, complemented
3#22611433C211yesmaximal, characteristic, complemented
4#22611302C211yesmaximal, characteristic, complemented
5#22611064C211yesmaximal, characteristic, complemented
6#22610884C211yesmaximal, characteristic, complemented
7#22610520C211yesmaximal, characteristic, complemented
8#22610357C211yesmaximal, characteristic
9#22610253C211yesmaximal, characteristic
10#22610177C211yesmaximal, characteristic, complemented
11#22610128C211yesmaximal, characteristic, complemented
12#22610111C211yesmaximal, characteristic, complemented
13#22609938C211yesmaximal, characteristic, complemented
14#22609647C211yesmaximal, characteristic, complemented
15C73×#2439C211yesmaximal, characteristic, complemented
16C23×#10596831C2211yescharacteristic
17C2×#14556565C2211yescharacteristic
18C2×C4×D292C2211yescharacteristic
19C2×#14556550C2211yescharacteristic
20C2×#14556524C2211yescharacteristic
21C2×#14556522C2211yescharacteristic
22C2×C4⋊#10596831C2211yescharacteristic
23C2×C4⋊#10596831C2211yescharacteristic
24C2×#14556511C2211yescharacteristic
25C2×#14556511C2211yescharacteristic
26C2×#14556511C2211yescharacteristic
27C2×#14556511C2211yescharacteristic
28C2×C22⋊#10596831C2211yescharacteristic
29#17988489C2211yescharacteristic
30#17988487—21no—
31#17988485C2211yescharacteristic
32#17988476—21no—
33#17988463—21no—
34C4×C4⋊Dic73C2211yescharacteristic
35#17988434—21no—
36#17988349C2211yescharacteristic
37#17988349C2211yescharacteristic
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×#117×C73C2211yescharacteristic
50#396×C73—21no—
51#393×C73—21no—
52#391×C73—21no—
53#389×C73C2211yescharacteristic
54#389×C73C2211yescharacteristic
55#387×C73—21no—
56#382×C73C2211yescharacteristic
57#380×C73C2211yescharacteristic
58C4×C22⋊C4×C73C2211yescharacteristic
59C22×D292C2311yescharacteristic
60C22×C22⋊D73—21no—
61C22×C4×D73—21no—
62C22×#10596831—22no—
63C22×#10596831—41no—
64C22×#10596831—41no—
65C22×#10596831C2311yescharacteristic
66C22×#10596831C2311yescharacteristic
67C22×#10596831C2311yescharacteristic
68C22×#10596831C2311yescharacteristic
69#14556565—22no—
70#14556565Q814yescomplemented
71C4×D292—22no—
72C4×D292—41no—
73#14556550—22no—
74#14556550—41no—
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2311yescharacteristic
77C2×C4⋊Dic73C2311yescharacteristic
78C2×#11924007—21no—
79C2×C22⋊Dic73—21no—
80C2×#11924005—21no—
81C2×#11924005—21no—
82C2×#11924005—21no—
83C2×#11924005—21no—
84C2×C4×Dic73—21no—
85C2×C4×Dic73—21no—
86#14556524—22no—
87#14556522—41no—
88C4⋊#10596831—41no—
89C4⋊#10596831—41no—
90#14556511—22no—
91#14556511—41no—
92#14556511—41no—
93#14556511—41no—
94C22⋊#10596831—22no—
95#14556498—21no—
96#14556498—21no—
97C22×Q8×C73—21no—
98C23×C4×C73C2311yescharacteristic
99C4⋊Q8×C73—22no—
100C4⋊Q8×C73—41no—
101#117×C73—22no—
102#117×C73—41no—
103C2×C4⋊C4×C73—21no—
104C2×C4⋊C4×C73—21no—
105C2×C4⋊C4×C73—21no—
106C2×C4⋊C4×C73C2311yescharacteristic
107C2×C4⋊C4×C73C2311yescharacteristic
108C2×C4⋊C4×C73C2311yescharacteristic
109C2×C4⋊C4×C73C2311yescharacteristic
110C2×C22⋊C4×C73—21no—
111C2×C22⋊C4×C73—21no—
112C2×C22⋊C4×C73D411yescharacteristic
113C2×C22⋊C4×C73D411yescharacteristic
114C2×C42×C73—21no—
115C2×C42×C73C2311yescharacteristic
116C2×C42×C73C2311yescharacteristic
117#95×C73—21no—
118#95×C73—21no—
119#95×C73—21no—
120#95×C73—21no—
121#95×C73—21no—
122#95×C73D411yescharacteristic
123#95×C73D411yescharacteristic
124C23×D73—21no—
125C2×D292—21no—
126C2×D292—21no—
127C2×D292—21no—
128C2×D292—21no—
129C2×D292—21no—
130C2×D292C2×Q812yes—
131C2×C22⋊D73—24no—
132C2×C22⋊D73—81no—
133C2×C22⋊D73—81no—
134C22×Dic73—21no—
135C22×Dic73—21no—
136C22×Dic73—21no—
137C2×C4×D73—42no—
138C2×C4×D73—42no—
139C2×C4×D73—81no—
140C2×#10596831—21no—
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×#10596831—21no—
152C2×#10596831—44no—
153C4⋊Dic73—21no—
154C4⋊Dic73—21no—
155C4⋊Dic73—21no—
156C4⋊Dic73—21no—
157C4⋊Dic73C2×Q812yes—
158C4⋊Dic73#4112yes—
159#11924007—81no—
160C22⋊Dic73—42no—
161#11924005—24no—
162#11924005—42no—
163#11924005—42no—
164#11924005—81no—
165C4×Dic73—42no—
166C4×Dic73—81no—
167C24×C73C2×D411yescharacteristic
168C2×Q8×C73—42no—
169C2×Q8×C73—42no—
170C2×Q8×C73—81no—
171C22×C4×C73—21no—
172C22×C4×C73—21no—
173C22×C4×C73—21no—
174C22×C4×C73—21no—
175C22×C4×C73—21no—
176C22×C4×C73—21no—
177C22×C4×C73—21no—
178C22×C4×C73—21no—
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—81no—
197C4⋊C4×C73#4112yes—
198C22⋊C4×C73—22no—
199C22⋊C4×C73—41no—
200C22⋊C4×C73—42no—
201C22⋊C4×C73—81no—
202C42×C73—21no—
203C42×C73—21no—
204C42×C73—21no—
205C42×C73—21no—
206C42×C73—42no—
207C23×C73—21no—
208C23×C73—21no—
209C23×C73—21no—
210C23×C73—21no—
211C23×C73—21no—
212C23×C73—21no—
213C23×C73C22⋊Q812yes—
214C23×C73C22×D411yescharacteristic
215C22×D73—22no—
216C22×D73—41no—
217C22×D73—41no—
218C22×D73—41no—
219C22×D73—41no—
220C22×D73—41no—
221C22×D73—41no—
222Q8×C73—84no—
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—22no—
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—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×C73C22⋊Q812yes—
253C2×C4×C73#13512yes—
254C2×C4×C73C22×Q811yescharacteristic
255C2×C4×C73C2×#4111yescharacteristic
256C2×C4×C73C2×#4111yescharacteristic
257C2×C4×C73C2×#4111yescharacteristic
258C2×C4×C73H(5,𝔽2)11yescharacteristic
259C2×C4×C73H(5,𝔽2)11yescharacteristic
260C22⋊D73—84no—
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—
270D292—22no—
271D292—22no—
272D292—41no—
273D292—41no—
274C4×D73—84no—
275#10596831—41no—
276#10596831—41no—
277#10596831—41no—
278#10596831—41no—
279C22×C73—22no—
280C22×C73—22no—
281C22×C73—22no—
282C22×C73—41no—
283C22×C73—41no—
284C22×C73—41no—
285C22×C73—41no—
286C22×C73C2×C22⋊Q811yescharacteristic
287C22×C73#54611yescharacteristic
288C22×C73#54611yescharacteristic
289C22×C73C2×#13511yescharacteristic
290C22×C73#57511yescharacteristic
291C22×C73#57611yescharacteristic
292C22×C73#57711yescharacteristic
293C2×D73—42no—
294C2×D73—42no—
295C2×D73—42no—
296C2×D73—81no—
297C2×D73—81no—
298C2×D73—81no—
299C2×D73—81no—
300C4×C73—22no—
301C4×C73—22no—
302C4×C73—42no—
303C4×C73—41no—
304C4×C73—41no—
305C4×C73—41no—
306C4×C73—81no—
307C4×C73—81no—
308C4×C73—81no—
309C4×C73#52911yescharacteristic
310C4×C73#53811yescharacteristic
311C4×C73#55411yescharacteristic
312C4×C73#55411yescharacteristic
313Dic73—42no—
314Dic73—42no—
315Dic73—81no—
316#26409—731nomaximal, Sylow
317C2×C73—42no—
318C2×C73#245011yescharacteristic
319C2×C73#307611yescharacteristic
320C2×C73#308111yescharacteristic
321C2×C73#329711yescharacteristic
322C2×C73#330411yescharacteristic
323C2×C73#331911yescharacteristic
324C2×C73#332111yescharacteristic
325D73—82no—
326#2439D7311yescharacteristic, complemented
327#2363—731no—
328#2101—731no—
329#2088—731no—
330#2065—731no—
331#2023—731no—
332#1988—731no—
333#1921—731no—
334#1893—731no—
335#1875—731no—
336#1862—731no—
337#1854—731no—
338#1850—731no—
339#1804—731no—
340#1735—731no—
341C73#2640911yescharacteristic, complemented, Sylow
342C23×C8—731no—
343C2×C4⋊Q8C2×D7311yescharacteristic
344C2×#121—731no—
345C2×C4×D4—731no—
346C2×#117—731no—
347C2×#117C2×D7311yescharacteristic
348C2×#107—731no—
349C2×#106—731no—
350C2×C4⋊C8—731no—
351C2×C4⋊C8—731no—
352C2×#102—731no—
353C2×#102—731no—
354C2×#102—731no—
355C2×#102—731no—
356C2×C22⋊C8—731no—
357#396—21no—
358#393—21no—
359#391—21no—
360#390—731no—
361#390—1461no—
362#389—731no—
363#389C2×D7311yescharacteristic
364#389C2×D7311yescharacteristic
365#388—1461no—
366#387—21no—
367#386—1461no—
368#382C2×D7311yescharacteristic
369#380C2×D7311yescharacteristic
370C4×C4⋊C4—731no—
371C4×C22⋊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#340—731no—
380#340—731no—
381#340—731no—
382#340—731no—
383#336—731no—
384#336—731no—
385C22×Q8—21no—
386C22×D4—731no—
387C22×D4—1461no—
388C23×C4—1461no—
389C23×C4C22×D7311yescharacteristic
390C22×C8—2921no—
391C22×C8—2921no—
392C22×C8—731no—
393C22×C8—731no—
394C22×C8—731no—
395C22×C8—731no—
396C22×C8—1462no—
397C4⋊Q8—22no—
398C4⋊Q8—41no—
399#121—734no—
400#121—1462no—
401C4×D4—2921no—
402C4×D4—1462no—
403#117—22no—
404#117—41no—
405#117—2921no—
406#117—1462no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—731no—
411C2×C4⋊C4—731no—
412C2×C4⋊C4—731no—
413C2×C4⋊C4—1461no—
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—1461no—
421C2×C22⋊C4—1461no—
422C2×C22⋊C4—1461no—
423C2×C22⋊C4—1461no—
424C2×C22⋊C4—1461no—
425C2×C22⋊C4C22⋊D7311yescharacteristic
426C2×C22⋊C4C22⋊D7311yescharacteristic
427C2×C42—21no—
428C2×C42—1461no—
429C2×C42—1461no—
430C2×C42C22×D7311yescharacteristic
431C2×C42C22×D7311yescharacteristic
432#107—1462no—
433#106—2921no—
434C4⋊C8—2921no—
435C4⋊C8—2921no—
436#102—2921no—
437#102—2921no—
438#102—2921no—
439#102—1462no—
440C22⋊C8—1462no—
441#95—21no—
442#95—21no—
443#95—21no—
444#95—21no—
445#95—21no—
446#95—1461no—
447#95—1461no—
448#95C22⋊D7311yescharacteristic
449#95C22⋊D7311yescharacteristic
450C24—1461no—
451C24C2×C22⋊D7311yescharacteristic
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8—81no—
455C2×D4—5841no—
456C2×D4—5841no—
457C2×D4—732no—
458C2×D4—1464no—
459C2×D4—1461no—
460C2×D4—1461no—
461C2×D4—1461no—
462C2×D4—1461no—
463C2×D4—1461no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—2922no—
473C22×C4—2922no—
474C22×C4—5841no—
475C22×C4—1461no—
476C22×C4—1461no—
477C22×C4—1461no—
478C22×C4C2×C22⋊D7311yescharacteristic
479C22×C4C2×C22⋊D7311yescharacteristic
480C22×C4C2×C22⋊D7311yescharacteristic
481C22×C4C2×C22⋊D7311yescharacteristic
482C22×C4C2×C22⋊D7311yescharacteristic
483C22×C4Q8×D7312yes—
484C22×C4#1192402912yes—
485C22×C4C23×D7311yescharacteristic
486C2×C8—2924no—
487C2×C8—1461no—
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—21no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—81no—
509C4⋊C4—5841no—
510C4⋊C4—732no—
511C4⋊C4—732no—
512C4⋊C4—1461no—
513C4⋊C4—1461no—
514C4⋊C4—1461no—
515C4⋊C4—1461no—
516C4⋊C4#1192402712yes—
517C22⋊C4—22no—
518C22⋊C4—41no—
519C22⋊C4—42no—
520C22⋊C4—81no—
521C22⋊C4—2922no—
522C22⋊C4—2922no—
523C22⋊C4—2922no—
524C22⋊C4—5841no—
525C22⋊C4—1464no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—42no—
531C42—2922no—
532C42—5841no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—2921no—
540C23—2921no—
541C23—2921no—
542C23—2921no—
543C23—2921no—
544C23—2921no—
545C23—1462no—
546C23#1455656712yes—
547C23C22×C22⋊D7311yescharacteristic
548Q8—84no—
549D4—2921no—
550D4—2921no—
551D4—5844no—
552D4—1462no—
553D4—1462no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
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—22no—
571C2×C4—21no—
572C2×C4—41no—
573C2×C4—41no—
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—5844no—
591C2×C4—1462no—
592C2×C4—1462no—
593C2×C4#1455656712yes—
594C2×C4#1455661112yes—
595C2×C4C2×Q8×D7311yescharacteristic
596C2×C4#41×D7311yescharacteristic
597C2×C4C2×#1192402711yescharacteristic
598C2×C4C2×#1192402911yescharacteristic
599C2×C4#1455663611yescharacteristic
600C2×C4#1455663711yescharacteristic
601C8—2921no—
602C8—2921no—
603C8—2921no—
604C8—2921no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—2922no—
613C22—2922no—
614C22—2922no—
615C22—5841no—
616C22—5841no—
617C22—5841no—
618C22—5841no—
619C22C2×#1455656711yescharacteristic
620C22#1798922511yescharacteristic
621C22#1798923111yescharacteristic
622C22C2×#1455661111yescharacteristic
623C22#1798938511yescharacteristic
624C22#1798939011yescharacteristic
625C22#1798939511yescharacteristic
626C4—22no—
627C4—22no—
628C4—42no—
629C4—41no—
630C4—41no—
631C4—41no—
632C4—81no—
633C4—81no—
634C4—81no—
635C4—2922no—
636C4—2922no—
637C4—5841no—
638C4#1798912311yescharacteristic
639C4#1798917711yescharacteristic
640C4#1798929111yescharacteristic
641C4#1798929511yescharacteristic
642C2—42no—
643C2—5842no—
644C2#2261375111yescharacteristic
645C2#2261971311yescharacteristic
646C2#2261975611yescharacteristic
647C2#2262178111yescharacteristic
648C2#2262186411yescharacteristic
649C2#2262208111yescharacteristic
650C2#2262211711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2439C2
socleC23×C73C22×D4
prime radicalC2×#14556565C22
hypercenter#2439D73
hypocenterC73#26409
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2439

Complements

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

NH ≅ G/Ntype
2. #22611498C2semidirect
3. #22611433C2semidirect
4. #22611302C2semidirect
5. #22611064C2semidirect
6. #22610884C2semidirect
7. #22610520C2semidirect
10. #22610177C2semidirect
11. #22610128C2semidirect
12. #22610111C2semidirect
13. #22609938C2semidirect
14. #22609647C2semidirect
15. C73×#2439C2semidirect
70. #14556565Q8semidirect
326. #2439D73semidirect
341. C73#26409semidirect

Maximal subgroups

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

subgroupindex
#226129672
#226114982
#226114332
#226113022
#226110642
#226108842
#226105202
#226103572
#226102532
#226101772
#226101282
#226101112
#226099382
#226096472
C73×#24392
#2640973