Finite Group Databasebeta

← #33154534

Subgroups — #33154534

Order 18688.

804 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
1#22614480C211yesmaximal, characteristic, complemented
2#22611344C211yesmaximal, characteristic, complemented
3#22611305C211yesmaximal, characteristic, complemented
4#22611162C211yesmaximal, characteristic
5#22611162C211yesmaximal, characteristic
6#22610617C211yesmaximal, characteristic, complemented
7#22610555C211yesmaximal, characteristic, complemented
8#22610523C211yesmaximal, characteristic, complemented
9#22610509C211yesmaximal, characteristic, complemented
10#22610490C211yesmaximal, characteristic, complemented
11#22610178C211yesmaximal, characteristic, complemented
12#22610178C211yesmaximal, characteristic, complemented
13#22609941C211yesmaximal, characteristic, complemented
14C4×#14556512C211yesmaximal, characteristic, complemented
15C73×#2366C211yesmaximal, characteristic
16C2×#14556524C2211yescharacteristic
17C2×#14556522C2211yescharacteristic
18C2×C4⋊#10596831C2211yescharacteristic
19C2×#14556512C2211yescharacteristic
20C2×#14556512C2211yescharacteristic
21C2×#14556512C2211yescharacteristic
22C2×#14556512C2211yescharacteristic
23C2×#14556512C2211yescharacteristic
24C2×#14556512C2211yescharacteristic
25C2×#14556512C2211yescharacteristic
26C2×#14556512C2211yescharacteristic
27C2×C4×#10596831C2211yescharacteristic
28#17988528—21no—
29#17988525—21no—
30#17988485C2211yescharacteristic
31#17988485C2211yescharacteristic
32#17988485C2211yescharacteristic
33#17988485C2211yescharacteristic
34#17988482—21no—
35#17988469—21no—
36#17988453C2211yescharacteristic
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
63C2×C4⋊Dic73C2311yescharacteristic
64C2×C4⋊Dic73C2311yescharacteristic
65C2×C4⋊Dic73C2311yescharacteristic
66C2×C4⋊Dic73C2311yescharacteristic
67C2×#11924007—21no—
68C2×#11924007—21no—
69C2×#11924007—21no—
70C2×#11924007—21no—
71C2×C4×Dic73—21no—
72#14556524—22no—
73#14556522—22no—
74C4⋊#10596831—41no—
75#14556512—22no—
76#14556512—22no—
77#14556512—22no—
78#14556512—22no—
79#14556512—22no—
80#14556512—22no—
81#14556512—41no—
82#14556512—41no—
83C4×#10596831—41no—
84#14556498—21no—
85#14556498—21no—
86#14556498—21no—
87#14556498—21no—
88#14556498—21no—
89#14556498—21no—
90#14556498—21no—
91C22×D4×C73—21no—
92C22×D4×C73—21no—
93C22×D4×C73C2311yescharacteristic
94C22×D4×C73C2311yescharacteristic
95C23×C4×C73—21no—
96C23×C4×C73—21no—
97#127×C73—22no—
98#127×C73—22no—
99C4×D4×C73—22no—
100C4×D4×C73—22no—
101C4×D4×C73—41no—
102C4×D4×C73—41no—
103C4×D4×C73—44no—
104C2×C4⋊C4×C73D411yescharacteristic
105C2×C4⋊C4×C73D411yescharacteristic
106C2×C4⋊C4×C73C2311yescharacteristic
107C2×C4⋊C4×C73C2311yescharacteristic
108C2×C22⋊C4×C73—21no—
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×C42×C73—21no—
115C2×C42×C73C2311yescharacteristic
116C2×C42×C73C2311yescharacteristic
117C2×C42×C73C2311yescharacteristic
118#95×C73—21no—
119#95×C73D411yescharacteristic
120#95×C73D411yescharacteristic
121C22×Dic73—21no—
122C22×Dic73—21no—
123C22×Dic73—21no—
124C22×Dic73—21no—
125C2×#10596831—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—
137C4⋊Dic73—21no—
138C4⋊Dic73—21no—
139C4⋊Dic73—21no—
140C4⋊Dic73—21no—
141C4⋊Dic73—21no—
142C4⋊Dic73—21no—
143C4⋊Dic73—21no—
144C4⋊Dic73#4112yes—
145#11924007—42no—
146#11924007—42no—
147#11924007—81no—
148#11924007—81no—
149C4×Dic73—81no—
150C24×C73—21no—
151C24×C73—21no—
152C2×D4×C73—24no—
153C2×D4×C73—24no—
154C2×D4×C73—21no—
155C2×D4×C73—21no—
156C2×D4×C73—21no—
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—42no—
165C2×D4×C73—42no—
166C2×D4×C73—81no—
167C2×D4×C73—81no—
168C2×D4×C73#4112yes—
169C2×D4×C73#4112yes—
170C22×C4×C73—21no—
171C22×C4×C73—21no—
172C22×C4×C73—42no—
173C22×C4×C73—42no—
174C22×C4×C73—42no—
175C22×C4×C73—42no—
176C22×C4×C73—81no—
177C22×C4×C73—81no—
178C22×C4×C73C2×D411yescharacteristic
179C22×C4×C73C2×D411yescharacteristic
180C22×C4×C73C2×D411yescharacteristic
181C22×C4×C73C2×D411yescharacteristic
182C22×C4×C73C2×D411yescharacteristic
183C22×C4×C73C2×D411yescharacteristic
184C22×C4×C73C2411yescharacteristic
185C4⋊C4×C73—22no—
186C4⋊C4×C73—22no—
187C4⋊C4×C73—21no—
188C4⋊C4×C73—21no—
189C4⋊C4×C73—21no—
190C4⋊C4×C73#4112yes—
191C22⋊C4×C73—42no—
192C22⋊C4×C73—42no—
193C22⋊C4×C73—42no—
194C22⋊C4×C73—42no—
195C22⋊C4×C73—81no—
196C22⋊C4×C73—81no—
197C42×C73—21no—
198C42×C73—21no—
199C42×C73—21no—
200C42×C73—21no—
201C42×C73—21no—
202C42×C73—21no—
203C42×C73—42no—
204C23×C73—22no—
205C23×C73—22no—
206C23×C73—41no—
207C23×C73—41no—
208C23×C73—41no—
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×C73C22×D411yescharacteristic
219D4×C73—22no—
220D4×C73—22no—
221D4×C73—22no—
222D4×C73—22no—
223D4×C73—48no—
224D4×C73—41no—
225D4×C73—41no—
226D4×C73—41no—
227D4×C73—41no—
228D4×C73—84no—
229C2×C4×C73—21no—
230C2×C4×C73—21no—
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—22no—
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—84no—
252C2×C4×C73—84no—
253C2×C4×C73#13512yes—
254C2×C4×C73AGL(1,ℤ/8)12yes—
255C2×C4×C73C2×#4111yescharacteristic
256C2×C4×C73C2×#4111yescharacteristic
257C2×C4×C73C2×#4111yescharacteristic
258C2×C4×C73C2×#4111yescharacteristic
259C2×C4×C73H(5,𝔽2)11yescharacteristic
260C2×C4×C732−1+411yescharacteristic
261C2×Dic73—22no—
262C2×Dic73—41no—
263C2×Dic73—41no—
264C2×Dic73—41no—
265C2×Dic73—41no—
266C2×Dic73—41no—
267C2×Dic73—41no—
268C2×Dic73—41no—
269C2×Dic73—41no—
270C2×Dic73—41no—
271C2×Dic73—41no—
272C2×Dic73—41no—
273#10596831—41no—
274#10596831—41no—
275#10596831—41no—
276#10596831—41no—
277C22×C73—42no—
278C22×C73—42no—
279C22×C73—42no—
280C22×C73—42no—
281C22×C73—42no—
282C22×C73—42no—
283C22×C73—81no—
284C22×C73—81no—
285C22×C73—81no—
286C22×C73—81no—
287C22×C73—81no—
288C22×C73—81no—
289C22×C73—81no—
290C22×C73—81no—
291C22×C73#52511yescharacteristic
292C22×C73#54611yescharacteristic
293C22×C73#54811yescharacteristic
294C22×C73C2×#13511yescharacteristic
295C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
296C22×C73#57611yescharacteristic
297C22×C73#57711yescharacteristic
298C4×C73—22no—
299C4×C73—22no—
300C4×C73—41no—
301C4×C73—41no—
302C4×C73—41no—
303C4×C73—41no—
304C4×C73—42no—
305C4×C73—81no—
306C4×C73#52911yescharacteristic
307C4×C73#53911yescharacteristic
308C4×C73#55211yescharacteristic
309C4×C73#55311yescharacteristic
310Dic73—42no—
311Dic73—81no—
312Dic73—81no—
313Dic73—81no—
314#26755—731nomaximal, Sylow
315C2×C73—82no—
316C2×C73—82no—
317C2×C73#246011yescharacteristic
318C2×C73#310511yescharacteristic
319C2×C73#311811yescharacteristic
320C2×C73#330411yescharacteristic
321C2×C73#331811yescharacteristic
322C2×C73#336111yescharacteristic
323C2×C73#336811yescharacteristic
324#2509—731no—
325#2366D7311yescharacteristic
326#2073—731no—
327#2065—731no—
328#2041—731no—
329#2041—731no—
330#1934—731no—
331#1926—731no—
332#1921—731no—
333#1919—731no—
334#1916—731no—
335#1862—731no—
336#1862—731no—
337#1804—731no—
338C4×#102—731no—
339C73#2675511yescharacteristic, complemented, Sylow
340C2×#127C2×D7311yescharacteristic
341C2×C4×D4—21no—
342C2×C4×D4C2×D7311yescharacteristic
343C2×C4×D4C2×D7311yescharacteristic
344C2×#107—731no—
345C2×#106—731no—
346C2×C4⋊C8—731no—
347C2×#102—731no—
348C2×#102—731no—
349C2×#102—731no—
350C2×#102—731no—
351C2×#102—731no—
352C2×#102—731no—
353C2×#102—731no—
354C2×#102—731no—
355C2×C4×C8—731no—
356#400—1461no—
357#400—1461no—
358#390—21no—
359#390C2×D7311yescharacteristic
360#390C2×D7311yescharacteristic
361#389—731no—
362#389—731no—
363#389—731no—
364#389—731no—
365#389—1461no—
366#387—1461no—
367#386—21no—
368#386—21no—
369#382—731no—
370C4×C4⋊C4—731no—
371C4×C4⋊C4C2×D7311yescharacteristic
372C4×C4⋊C4C2×D7311yescharacteristic
373#340—731no—
374#340—731no—
375#340—731no—
376#340—731no—
377#340—731no—
378#340—731no—
379#336—731no—
380#336—731no—
381#336—731no—
382#336—731no—
383C22×D4—21no—
384C22×D4—21no—
385C22×D4C22×D7311yescharacteristic
386C22×D4C22×D7311yescharacteristic
387C23×C4—21no—
388C23×C4—21no—
389C22×C8—731no—
390C22×C8—731no—
391C22×C8—731no—
392C22×C8—731no—
393#127—22no—
394#127—22no—
395C4×D4—22no—
396C4×D4—22no—
397C4×D4—41no—
398C4×D4—41no—
399C4×D4—44no—
400C2×C4⋊C4—731no—
401C2×C4⋊C4—731no—
402C2×C4⋊C4—731no—
403C2×C4⋊C4—731no—
404C2×C4⋊C4—1461no—
405C2×C4⋊C4—1461no—
406C2×C4⋊C4—1461no—
407C2×C4⋊C4—1461no—
408C2×C4⋊C4C22⋊D7311yescharacteristic
409C2×C4⋊C4C22⋊D7311yescharacteristic
410C2×C4⋊C4C22×D7311yescharacteristic
411C2×C4⋊C4C22×D7311yescharacteristic
412C2×C22⋊C4—21no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—21no—
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C42—21no—
419C2×C42—1461no—
420C2×C42C22×D7311yescharacteristic
421C2×C42C22×D7311yescharacteristic
422C2×C42C22×D7311yescharacteristic
423#107—1462no—
424#106—1462no—
425C4⋊C8—2921no—
426#102—2921no—
427#102—2921no—
428#102—1462no—
429#102—1462no—
430#102—1462no—
431#102—1462no—
432#102—1462no—
433#102—1462no—
434C4×C8—2921no—
435#95—21no—
436#95—1461no—
437#95—1461no—
438#95—1461no—
439#95—1461no—
440#95—1461no—
441#95—1461no—
442#95—1461no—
443#95C22⋊D7311yescharacteristic
444#95C22⋊D7311yescharacteristic
445C24—21no—
446C24—21no—
447C2×D4—24no—
448C2×D4—24no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—81no—
462C2×D4—81no—
463C2×D4#1192402812yes—
464C2×D4#1192402812yes—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—81no—
472C22×C4—81no—
473C22×C4—1461no—
474C22×C4—1461no—
475C22×C4—1461no—
476C22×C4—1461no—
477C22×C4C2×C22⋊D7311yescharacteristic
478C22×C4C2×C22⋊D7311yescharacteristic
479C22×C4C2×C22⋊D7311yescharacteristic
480C22×C4C2×C22⋊D7311yescharacteristic
481C22×C4C2×C22⋊D7311yescharacteristic
482C22×C4C2×C22⋊D7311yescharacteristic
483C22×C4C23×D7311yescharacteristic
484C2×C8—1461no—
485C2×C8—1461no—
486C2×C8—1461no—
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—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—2922no—
502C4⋊C4—2922no—
503C4⋊C4—5841no—
504C4⋊C4—5841no—
505C4⋊C4—732no—
506C4⋊C4—1461no—
507C4⋊C4—1461no—
508C4⋊C4—1461no—
509C4⋊C4—1461no—
510C4⋊C4—1461no—
511C4⋊C4—1461no—
512C4⋊C4—1461no—
513C4⋊C4#1192402712yes—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—42no—
518C22⋊C4—81no—
519C22⋊C4—81no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42—42no—
527C42—5841no—
528C23—22no—
529C23—22no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23C22×C22⋊D7311yescharacteristic
543D4—22no—
544D4—22no—
545D4—22no—
546D4—22no—
547D4—48no—
548D4—41no—
549D4—41no—
550D4—41no—
551D4—41no—
552D4—84no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—84no—
576C2×C4—84no—
577C2×C4—2921no—
578C2×C4—2921no—
579C2×C4—2921no—
580C2×C4—2921no—
581C2×C4—2921no—
582C2×C4—2921no—
583C2×C4—2921no—
584C2×C4—2921no—
585C2×C4—2921no—
586C2×C4—2921no—
587C2×C4—2921no—
588C2×C4—1462no—
589C2×C4#1455661112yes—
590C2×C4#1455661712yes—
591C2×C4#41×D7311yescharacteristic
592C2×C4C2×#1192402711yescharacteristic
593C2×C4C2×#1192402811yescharacteristic
594C2×C4C2×#1192402811yescharacteristic
595C2×C4#1455663611yescharacteristic
596C2×C4#1455663811yescharacteristic
597C8—2921no—
598C8—2921no—
599C8—2921no—
600C8—2921no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22#1798909811yescharacteristic
616C22#1798923111yescharacteristic
617C22#1798924411yescharacteristic
618C22C2×#1455661111yescharacteristic
619C22C2×#1455661711yescharacteristic
620C22#1798939011yescharacteristic
621C22#1798939811yescharacteristic
622C4—22no—
623C4—22no—
624C4—41no—
625C4—42no—
626C4—41no—
627C4—41no—
628C4—41no—
629C4—81no—
630C4—2922no—
631C4—5841no—
632C4—5841no—
633C4—5841no—
634C4#1798911711yescharacteristic
635C4#1798917911yescharacteristic
636C4#1798927711yescharacteristic
637C4#1798928611yescharacteristic
638C2—82no—
639C2—82no—
640C2#2261387711yescharacteristic
641C2#2261997811yescharacteristic
642C2#2262011511yescharacteristic
643C2#2262187011yescharacteristic
644C2#2262207211yescharacteristic
645C2#2262258711yescharacteristic
646C2#2262265611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2366C2
socleC23×C73C22×D4
prime radicalC2×#127×C73C22
hypercenter#2366D73
hypocenterC73#26755
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. #22614480C2semidirect
2. #22611344C2semidirect
3. #22611305C2semidirect
6. #22610617C2semidirect
7. #22610555C2semidirect
8. #22610523C2semidirect
9. #22610509C2semidirect
10. #22610490C2semidirect
11. #22610178C2semidirect
12. #22610178C2semidirect
13. #22609941C2semidirect
14. C4×#14556512C2semidirect
339. C73#26755semidirect

Maximal subgroups

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

subgroupindex
#226144802
#226113442
#226113052
#226111622
#226106172
#226105552
#226105232
#226105092
#226104902
#226101782
#226099412
C4×#145565122
C73×#23662
#2675573