Finite Group Databasebeta

← #33154489

Subgroups — #33154489

Order 18688.

832 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22614447C211yesmaximal, characteristic, complemented
2#22611334C211yesmaximal, characteristic, complemented
3#22611319C211yesmaximal, characteristic, complemented
4#22611162C211yesmaximal, characteristic
5#22611162C211yesmaximal, characteristic
6#22610617C211yesmaximal, characteristic, complemented
7#22610548C211yesmaximal, characteristic, complemented
8#22610502C211yesmaximal, characteristic, complemented
9#22610502C211yesmaximal, characteristic, complemented
10#22610490C211yesmaximal, characteristic, complemented
11#22610178C211yesmaximal, characteristic, complemented
12#22610178C211yesmaximal, characteristic, complemented
13#22609931C211yesmaximal, characteristic, complemented
14C4×#14556512C211yesmaximal, characteristic, complemented
15C73×#2366C211yesmaximal, characteristic
16C2×#14556581—21no—
17C2×#14556524C2211yescharacteristic
18C2×#14556524C2211yescharacteristic
19C2×C4⋊#10596831C2211yescharacteristic
20C2×#14556512C2211yescharacteristic
21C2×#14556512C2211yescharacteristic
22C2×#14556512C2211yescharacteristic
23C2×#14556512C2211yescharacteristic
24C2×#14556512C2211yescharacteristic
25C2×#14556512C2211yescharacteristic
26C2×#14556512C2211yescharacteristic
27C2×#14556512C2211yescharacteristic
28C2×C4×#10596831C2211yescharacteristic
29#17988527—21no—
30#17988517—21no—
31#17988485C2211yescharacteristic
32#17988485C2211yescharacteristic
33#17988485C2211yescharacteristic
34#17988485C2211yescharacteristic
35#17988472—21no—
36#17988457C2211yescharacteristic
37C4×C4⋊Dic73C2211yescharacteristic
38#17988349C2211yescharacteristic
39#17988349C2211yescharacteristic
40#17988349C2211yescharacteristic
41#17988349C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988342C2211yescharacteristic
45#17988342C2211yescharacteristic
46#17988342C2211yescharacteristic
47#17988342C2211yescharacteristic
48C2×#127×C73C2211yescharacteristic
49C2×C4×D4×C73—21no—
50C2×C4×D4×C73C2211yescharacteristic
51C2×C4×D4×C73C2211yescharacteristic
52#390×C73—21no—
53#390×C73C2211yescharacteristic
54#390×C73C2211yescharacteristic
55#386×C73—21no—
56#386×C73—21no—
57C4×C4⋊C4×C73C2211yescharacteristic
58C4×C4⋊C4×C73C2211yescharacteristic
59C22×#10596831C2311yescharacteristic
60C22×#10596831C2311yescharacteristic
61C22×#10596831C2311yescharacteristic
62C22×#10596831C2311yescharacteristic
63#14556581—44no—
64C2×C4⋊Dic73—21no—
65C2×C4⋊Dic73—21no—
66C2×C4⋊Dic73C2311yescharacteristic
67C2×C4⋊Dic73C2311yescharacteristic
68C2×C4⋊Dic73C2311yescharacteristic
69C2×C4⋊Dic73C2311yescharacteristic
70C2×#11924007—21no—
71C2×#11924007—21no—
72C2×#11924007—21no—
73C2×#11924007—21no—
74C2×C4×Dic73—21no—
75#14556524—22no—
76#14556524—22no—
77C4⋊#10596831—41no—
78#14556512—22no—
79#14556512—22no—
80#14556512—22no—
81#14556512—22no—
82#14556512—22no—
83#14556512—22no—
84#14556512—41no—
85#14556512—41no—
86C4×#10596831—41no—
87#14556498—21no—
88#14556498—21no—
89#14556498—21no—
90#14556498—21no—
91#14556498—21no—
92C22×D4×C73—21no—
93C22×D4×C73—21no—
94C22×D4×C73C2311yescharacteristic
95C22×D4×C73C2311yescharacteristic
96C23×C4×C73—21no—
97C23×C4×C73—21no—
98#127×C73—22no—
99#127×C73—22no—
100C4×D4×C73—22no—
101C4×D4×C73—22no—
102C4×D4×C73—41no—
103C4×D4×C73—41no—
104C4×D4×C73—44no—
105C2×C4⋊C4×C73D411yescharacteristic
106C2×C4⋊C4×C73D411yescharacteristic
107C2×C4⋊C4×C73C2311yescharacteristic
108C2×C4⋊C4×C73C2311yescharacteristic
109C2×C22⋊C4×C73—21no—
110C2×C22⋊C4×C73—21no—
111C2×C22⋊C4×C73—21no—
112C2×C22⋊C4×C73—21no—
113C2×C22⋊C4×C73—21no—
114C2×C22⋊C4×C73—21no—
115C2×C42×C73—21no—
116C2×C42×C73C2311yescharacteristic
117C2×C42×C73C2311yescharacteristic
118C2×C42×C73C2311yescharacteristic
119#95×C73—21no—
120#95×C73D411yescharacteristic
121#95×C73D411yescharacteristic
122C22×Dic73—21no—
123C22×Dic73—21no—
124C22×Dic73—21no—
125C22×Dic73—21no—
126C2×#10596831—21no—
127C2×#10596831—21no—
128C2×#10596831—21no—
129C2×#10596831—21no—
130C2×#10596831—21no—
131C2×#10596831—21no—
132C2×#10596831—21no—
133C2×#10596831—21no—
134C2×#10596831—21no—
135C2×#10596831—21no—
136C2×#10596831—21no—
137C2×#10596831—21no—
138C4⋊Dic73—21no—
139C4⋊Dic73—21no—
140C4⋊Dic73—21no—
141C4⋊Dic73—21no—
142C4⋊Dic73—21no—
143C4⋊Dic73—21no—
144C4⋊Dic73—21no—
145C4⋊Dic73—42no—
146C4⋊Dic73—42no—
147C4⋊Dic73#4112yes—
148#11924007—42no—
149#11924007—42no—
150#11924007—42no—
151#11924007—42no—
152C4×Dic73—42no—
153C24×C73—21no—
154C24×C73—21no—
155C2×D4×C73—24no—
156C2×D4×C73—24no—
157C2×D4×C73—21no—
158C2×D4×C73—21no—
159C2×D4×C73—21no—
160C2×D4×C73—21no—
161C2×D4×C73—21no—
162C2×D4×C73—21no—
163C2×D4×C73—21no—
164C2×D4×C73—21no—
165C2×D4×C73—21no—
166C2×D4×C73—21no—
167C2×D4×C73—42no—
168C2×D4×C73—42no—
169C2×D4×C73—81no—
170C2×D4×C73—81no—
171C2×D4×C73#4112yes—
172C2×D4×C73#4112yes—
173C22×C4×C73—21no—
174C22×C4×C73—21no—
175C22×C4×C73—42no—
176C22×C4×C73—42no—
177C22×C4×C73—42no—
178C22×C4×C73—42no—
179C22×C4×C73—81no—
180C22×C4×C73—81no—
181C22×C4×C73C2×D411yescharacteristic
182C22×C4×C73C2×D411yescharacteristic
183C22×C4×C73C2×D411yescharacteristic
184C22×C4×C73C2×D411yescharacteristic
185C22×C4×C73C2×D411yescharacteristic
186C22×C4×C73C2×D411yescharacteristic
187C22×C4×C73C2411yescharacteristic
188C4⋊C4×C73—21no—
189C4⋊C4×C73—21no—
190C4⋊C4×C73—21no—
191C4⋊C4×C73—41no—
192C4⋊C4×C73D814yes—
193C4⋊C4×C73#4112yes—
194C22⋊C4×C73—42no—
195C22⋊C4×C73—42no—
196C22⋊C4×C73—42no—
197C22⋊C4×C73—42no—
198C22⋊C4×C73—81no—
199C22⋊C4×C73—81no—
200C42×C73—21no—
201C42×C73—21no—
202C42×C73—21no—
203C42×C73—21no—
204C42×C73—21no—
205C42×C73—21no—
206C42×C73—42no—
207C23×C73—22no—
208C23×C73—22no—
209C23×C73—41no—
210C23×C73—41no—
211C23×C73—41no—
212C23×C73—41no—
213C23×C73—41no—
214C23×C73—41no—
215C23×C73—41no—
216C23×C73—41no—
217C23×C73—41no—
218C23×C73—41no—
219C23×C73—41no—
220C23×C73—41no—
221C23×C73C22×D411yescharacteristic
222D4×C73—22no—
223D4×C73—22no—
224D4×C73—22no—
225D4×C73—22no—
226D4×C73—48no—
227D4×C73—41no—
228D4×C73—41no—
229D4×C73—41no—
230D4×C73—41no—
231D4×C73—84no—
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—22no—
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—84no—
254C2×C4×C73—84no—
255C2×C4×C73C2×D812yes—
256C2×C4×C73C2×D812yes—
257C2×C4×C73C2×D812yes—
258C2×C4×C73C2×#4111yescharacteristic
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—
276#10596831—41no—
277#10596831—41no—
278#10596831—41no—
279#10596831—41no—
280C22×C73—42no—
281C22×C73—42no—
282C22×C73—42no—
283C22×C73—42no—
284C22×C73—42no—
285C22×C73—42no—
286C22×C73—81no—
287C22×C73—81no—
288C22×C73—81no—
289C22×C73—81no—
290C22×C73—81no—
291C22×C73—81no—
292C22×C73—81no—
293C22×C73—81no—
294C22×C73#52511yescharacteristic
295C22×C73#54611yescharacteristic
296C22×C73#54811yescharacteristic
297C22×C73C22×D811yescharacteristic
298C22×C73#57511yescharacteristic
299C22×C73#57711yescharacteristic
300C22×C73#57711yescharacteristic
301C4×C73—22no—
302C4×C73—22no—
303C4×C73—22no—
304C4×C73—41no—
305C4×C73—42no—
306C4×C73—41no—
307C4×C73—41no—
308C4×C73—81no—
309C4×C73#52911yescharacteristic
310C4×C73#53911yescharacteristic
311C4×C73#55211yescharacteristic
312C4×C73#55311yescharacteristic
313Dic73—42no—
314Dic73—81no—
315Dic73—81no—
316Dic73—81no—
317#26752—731nomaximal, Sylow
318C2×C73—82no—
319C2×C73—82no—
320C2×C73#246011yescharacteristic
321C2×C73#310011yescharacteristic
322C2×C73#312311yescharacteristic
323C2×C73#329411yescharacteristic
324C2×C73#331711yescharacteristic
325C2×C73#335811yescharacteristic
326C2×C73#337111yescharacteristic
327#2506—731no—
328#2366D7311yescharacteristic
329#2071—731no—
330#2067—731no—
331#2041—731no—
332#2041—731no—
333#1934—731no—
334#1925—731no—
335#1918—731no—
336#1918—731no—
337#1916—731no—
338#1862—731no—
339#1862—731no—
340#1802—731no—
341C4×#102—731no—
342C73#2675211yescharacteristic, complemented, Sylow
343C2×#127C2×D7311yescharacteristic
344C2×#125—1461no—
345C2×C4×D4—21no—
346C2×C4×D4C2×D7311yescharacteristic
347C2×C4×D4C2×D7311yescharacteristic
348C2×#107—731no—
349C2×#107—731no—
350C2×C4⋊C8—731no—
351C2×#102—731no—
352C2×#102—731no—
353C2×#102—731no—
354C2×#102—731no—
355C2×#102—731no—
356C2×#102—731no—
357C2×#102—731no—
358C2×#102—731no—
359C2×C4×C8—731no—
360#400—1461no—
361#398—1461no—
362#390—21no—
363#390C2×D7311yescharacteristic
364#390C2×D7311yescharacteristic
365#389—731no—
366#389—731no—
367#389—731no—
368#389—731no—
369#387—1461no—
370#386—21no—
371#386—21no—
372#384—731no—
373C4×C4⋊C4—731no—
374C4×C4⋊C4C2×D7311yescharacteristic
375C4×C4⋊C4C2×D7311yescharacteristic
376#340—731no—
377#340—731no—
378#340—731no—
379#340—731no—
380#340—731no—
381#340—731no—
382#336—731no—
383#336—731no—
384#336—731no—
385#336—731no—
386C22×D4—21no—
387C22×D4—21no—
388C22×D4C22×D7311yescharacteristic
389C22×D4C22×D7311yescharacteristic
390C23×C4—21no—
391C23×C4—21no—
392C22×C8—731no—
393C22×C8—731no—
394C22×C8—731no—
395C22×C8—731no—
396#127—22no—
397#127—22no—
398#125—2924no—
399C4×D4—22no—
400C4×D4—22no—
401C4×D4—41no—
402C4×D4—41no—
403C4×D4—44no—
404C2×C4⋊C4—731no—
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⋊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—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#107—1462no—
430#107—1462no—
431C4⋊C8—2921no—
432#102—2921no—
433#102—2921no—
434#102—1462no—
435#102—1462no—
436#102—1462no—
437#102—1462no—
438#102—1462no—
439#102—1462no—
440C4×C8—2921no—
441#95—21no—
442#95—1461no—
443#95—1461no—
444#95—1461no—
445#95—1461no—
446#95—1461no—
447#95C22⋊D7311yescharacteristic
448#95C22⋊D7311yescharacteristic
449C24—21no—
450C24—21no—
451C2×D4—24no—
452C2×D4—24no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—81no—
466C2×D4—81no—
467C2×D4#1192402812yes—
468C2×D4#1192402812yes—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—81no—
476C22×C4—81no—
477C22×C4—1461no—
478C22×C4—1461no—
479C22×C4—1461no—
480C22×C4—1461no—
481C22×C4C2×C22⋊D7311yescharacteristic
482C22×C4C2×C22⋊D7311yescharacteristic
483C22×C4C2×C22⋊D7311yescharacteristic
484C22×C4C2×C22⋊D7311yescharacteristic
485C22×C4C2×C22⋊D7311yescharacteristic
486C22×C4C2×C22⋊D7311yescharacteristic
487C22×C4C23×D7311yescharacteristic
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—
499C2×C8—1461no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—41no—
504C4⋊C4—2922no—
505C4⋊C4—2922no—
506C4⋊C4—2922no—
507C4⋊C4—2922no—
508C4⋊C4—2922no—
509C4⋊C4—2922no—
510C4⋊C4—732no—
511C4⋊C4—1461no—
512C4⋊C4—1461no—
513C4⋊C4—1461no—
514C4⋊C4—1461no—
515C4⋊C4—1461no—
516C4⋊C4—1461no—
517C4⋊C4—1461no—
518C4⋊C4#1192401314yes—
519C4⋊C4#1192402712yes—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—81no—
525C22⋊C4—81no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—42no—
533C42—2922no—
534C23—22no—
535C23—22no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23C22×C22⋊D7311yescharacteristic
549D4—22no—
550D4—22no—
551D4—22no—
552D4—22no—
553D4—48no—
554D4—41no—
555D4—41no—
556D4—41no—
557D4—41no—
558D4—84no—
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—22no—
574C2×C4—21no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—84no—
581C2×C4—84no—
582C2×C4—2921no—
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—1462no—
594C2×C4C2×#1192401312yes—
595C2×C4C2×#1192401312yes—
596C2×C4C2×#1192401312yes—
597C2×C4#41×D7311yescharacteristic
598C2×C4C2×#1192402711yescharacteristic
599C2×C4C2×#1192402811yescharacteristic
600C2×C4C2×#1192402811yescharacteristic
601C2×C4#1455663611yescharacteristic
602C2×C4#1455663811yescharacteristic
603C8—2921no—
604C8—2921no—
605C8—2921no—
606C8—2921no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22#1798909811yescharacteristic
622C22#1798923111yescharacteristic
623C22#1798924411yescharacteristic
624C22C22×#1192401311yescharacteristic
625C22#1798938511yescharacteristic
626C22#1798939811yescharacteristic
627C22#1798939811yescharacteristic
628C4—22no—
629C4—22no—
630C4—22no—
631C4—41no—
632C4—42no—
633C4—41no—
634C4—41no—
635C4—81no—
636C4—2922no—
637C4—5841no—
638C4—5841no—
639C4—5841no—
640C4#1798911711yescharacteristic
641C4#1798917911yescharacteristic
642C4#1798927711yescharacteristic
643C4#1798928611yescharacteristic
644C2—82no—
645C2—82no—
646C2#2261387711yescharacteristic
647C2#2261993111yescharacteristic
648C2#2262018311yescharacteristic
649C2#2262174411yescharacteristic
650C2#2262205711yescharacteristic
651C2#2262256211yescharacteristic
652C2#2262269311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2366C2
socleC23×C73C22×D4
prime radicalC2×#127×C73C22
hypercenter#2366D73
hypocenterC73#26752
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2366

Complements

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

NH ≅ G/Ntype
1. #22614447C2semidirect
2. #22611334C2semidirect
3. #22611319C2semidirect
6. #22610617C2semidirect
7. #22610548C2semidirect
8. #22610502C2semidirect
9. #22610502C2semidirect
10. #22610490C2semidirect
11. #22610178C2semidirect
12. #22610178C2semidirect
13. #22609931C2semidirect
14. C4×#14556512C2semidirect
342. C73#26752semidirect

Maximal subgroups

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

subgroupindex
#226144472
#226113342
#226113192
#226111622
#226106172
#226105482
#226105022
#226104902
#226101782
#226099312
C4×#145565122
C73×#23662
#2675273