Finite Group Databasebeta

← #33097115

Subgroups — #33097115

Order 18688.

1308 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988721C211yesmaximal, characteristic
2C4×#14556565C211yesmaximal, characteristic, complemented
3#22610796C212yesmaximal, complemented
4#22610617C212yesmaximal, complemented
5#22610548C212yesmaximal, complemented
6#22610282C212yesmaximal, complemented
7#22610248C211yesmaximal, characteristic, complemented
8#22610119C211yesmaximal, characteristic, complemented
9#22609862C211yesmaximal, characteristic, complemented
10C4×C22⋊#10596831C211yesmaximal, characteristic, complemented
11C2×#390×C73C211yesmaximal, characteristic, complemented
12C22×#11924013C2212yes—
13C2×#14556565C2211yescharacteristic
14C2×C4×D292C2211yescharacteristic, complemented
15C2×C4×C22⋊D73—21no—
16#17988721C418yescomplemented
17C2×#14556524C2212yescomplemented
18C2×#14556512C2212yes—
19C2×#14556512C2212yes—
20C2×#14556512C2212yes—
21C2×#14556511C2212yescomplemented
22C2×C22⋊#10596831C2211yescharacteristic
23C2×C22⋊#10596831C2211yescharacteristic
24C2×C4×#10596831C2212yescomplemented
25#17988489C2211yescharacteristic, complemented
26#17988485C2211yescharacteristic, complemented
27#17988478—21no—
28#17988466—21no—
29#17988447C2211yescharacteristic
30C4×C4⋊Dic73C2211yescharacteristic, complemented
31C4×#11924005—21no—
32#17988349C2212yescomplemented
33#17988349C2212yescomplemented
34#17988342C2212yescomplemented
35C23×D4×C73C2211yescharacteristic
36C22×C4⋊C4×C73C2211yescharacteristic
37C22×C22⋊C4×C73—21no—
38C22×C22⋊C4×C73—21no—
39C22×C42×C73C2211yescharacteristic
40#390×C73—21no—
41#390×C73—21no—
42#390×C73C2212yes—
43#390×C73C2212yes—
44C22×D292C2311yescharacteristic
45C22×C22⋊D73—21no—
46C22×C4×D73—21no—
47C2×#11924013—42no—
48C2×#11924013—42no—
49C2×#11924013C2×C418yescomplemented
50C22×#10596831C2312yes—
51C22×#10596831C2312yes—
52#14556565—22no—
53#14556565C2×C414yes—
54C4×D292—42no—
55C4×C22⋊D73—44no—
56C2×C4⋊Dic73C2311yescharacteristic
57C2×C4⋊Dic73C2311yescharacteristic
58C2×C4⋊Dic73C2311yescharacteristic
59C2×#11924007—21no—
60C2×C22⋊Dic73—21no—
61C2×#11924005—21no—
62C2×#11924005—21no—
63C2×#11924005—21no—
64C2×#11924005—21no—
65C2×C4×Dic73—21no—
66#14556524—42no—
67#14556512—42no—
68#14556512—42no—
69#14556512C2×C418yescomplemented
70#14556511—42no—
71C22⋊#10596831—22no—
72C22⋊#10596831C2×C414yescomplemented
73C4×#10596831—42no—
74#14556498—21no—
75#14556498—21no—
76#14556498—21no—
77C25×C73—21no—
78C22×D4×C73—21no—
79C22×D4×C73—22no—
80C22×D4×C73—42no—
81C22×D4×C73—42no—
82C22×D4×C73C2×C414yescomplemented
83C22×D4×C73C2312yes—
84C23×C4×C73—21no—
85C23×C4×C73D411yescharacteristic
86C23×C4×C73D411yescharacteristic
87C23×C4×C73C2311yescharacteristic
88C2×C4⋊C4×C73—21no—
89C2×C4⋊C4×C73—42no—
90C2×C4⋊C4×C73C2312yes—
91C2×C22⋊C4×C73—22no—
92C2×C22⋊C4×C73—22no—
93C2×C22⋊C4×C73—22no—
94C2×C22⋊C4×C73—22no—
95C2×C22⋊C4×C73—44no—
96C2×C22⋊C4×C73—44no—
97C2×C42×C73—42no—
98C2×C42×C73D411yescharacteristic
99C2×C42×C73D411yescharacteristic
100C2×C42×C73C2311yescharacteristic
101C2×C42×C73C2311yescharacteristic
102C23×D73—21no—
103C2×D292—22no—
104C2×D292—22no—
105C2×D292—21no—
106C2×D292C22×C412yes—
107C2×C22⋊D73—24no—
108C2×C22⋊D73—42no—
109C2×C22⋊D73—42no—
110C22×Dic73—21no—
111C22×Dic73—21no—
112C22×Dic73—21no—
113C2×C4×D73—42no—
114C2×C4×D73—42no—
115C2×C4×D73—42no—
116#11924013—48no—
117C2×#10596831—22no—
118C2×#10596831—22no—
119C2×#10596831—22no—
120C2×#10596831—22no—
121C2×#10596831—22no—
122C2×#10596831C22×C414yes—
123C4⋊Dic73—21no—
124C4⋊Dic73—22no—
125C4⋊Dic73—22no—
126C4⋊Dic73C22×C412yes—
127#11924007—42no—
128C22⋊Dic73—42no—
129#11924005—24no—
130#11924005—42no—
131#11924005—42no—
132#11924005—42no—
133C4×Dic73—42no—
134C24×C73—24no—
135C24×C73—22no—
136C24×C73—42no—
137C24×C73—42no—
138C24×C73—42no—
139C24×C73—42no—
140C24×C73—42no—
141C24×C73—42no—
142C24×C73C2×D411yescharacteristic
143C2×D4×C73—24no—
144C2×D4×C73—22no—
145C2×D4×C73—22no—
146C2×D4×C73—22no—
147C2×D4×C73—22no—
148C2×D4×C73—22no—
149C2×D4×C73—48no—
150C2×D4×C73—42no—
151C2×D4×C73—42no—
152C2×D4×C73—48no—
153C2×D4×C73C22×C414yes—
154C22×C4×C73—22no—
155C22×C4×C73—22no—
156C22×C4×C73—22no—
157C22×C4×C73—22no—
158C22×C4×C73—22no—
159C22×C4×C73—41no—
160C22×C4×C73—41no—
161C22×C4×C73—41no—
162C22×C4×C73—42no—
163C22×C4×C73—42no—
164C22×C4×C73—42no—
165C22×C4×C73D814yescomplemented
166C22×C4×C73C22×C412yes—
167C22×C4×C73C2×D411yescharacteristic
168C22×C4×C73C2×D411yescharacteristic
169C22×C4×C73C2×D411yescharacteristic
170C22×C4×C73C2×D411yescharacteristic
171C22×C4×C73C2×D411yescharacteristic
172C22×C4×C73#4112yes—
173C22×C4×C73C2411yescharacteristic
174C4⋊C4×C73—22no—
175C4⋊C4×C73—22no—
176C4⋊C4×C73—42no—
177C22⋊C4×C73—44no—
178C22⋊C4×C73—44no—
179C22⋊C4×C73—44no—
180C22⋊C4×C73—44no—
181C42×C73—22no—
182C42×C73—22no—
183C42×C73—41no—
184C42×C73—41no—
185C23×C73—21no—
186C23×C73—21no—
187C23×C73—21no—
188C23×C73—21no—
189C23×C73—24no—
190C23×C73—48no—
191C23×C73—48no—
192C23×C73—42no—
193C23×C73—42no—
194C23×C73—42no—
195C23×C73—42no—
196C23×C73—42no—
197C23×C73—42no—
198C23×C73—48no—
199C23×C73—84no—
200C23×C73—84no—
201C23×C73—84no—
202C23×C73—84no—
203C23×C73C4×D412yes—
204C23×C73C2×D812yes—
205C23×C73#13512yes—
206C23×C73C22×D411yescharacteristic
207C22×D73—22no—
208C22×D73—41no—
209C22×D73—41no—
210C22×D73—41no—
211C22×D73—41no—
212C22×D73—41no—
213C22×D73—41no—
214D4×C73—24no—
215D4×C73—24no—
216D4×C73—48no—
217D4×C73—42no—
218D4×C73—42no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—24no—
228C2×C4×C73—24no—
229C2×C4×C73—21no—
230C2×C4×C73—21no—
231C2×C4×C73—21no—
232C2×C4×C73—44no—
233C2×C4×C73—44no—
234C2×C4×C73—42no—
235C2×C4×C73—42no—
236C2×C4×C73—42no—
237C2×C4×C73—84no—
238C2×C4×C73C4×D412yes—
239C2×C4×C73C2×D812yes—
240C2×C4×C73C2×D812yes—
241C2×C4×C73AGL(1,ℤ/8)12yes—
242C2×C4×C73C23×C411yescharacteristic
243C2×C4×C73C2×#4111yescharacteristic
244C2×C4×C73H(5,𝔽2)12yes—
245C2×C4×C73H(5,𝔽2)12yes—
246C22⋊D73—48no—
247C2×Dic73—22no—
248C2×Dic73—41no—
249C2×Dic73—41no—
250C2×Dic73—41no—
251C2×Dic73—41no—
252C2×Dic73—41no—
253C2×Dic73—41no—
254C2×Dic73—41no—
255C2×Dic73—41no—
256D292—24no—
257D292—42no—
258C4×D73—84no—
259#10596831—24no—
260#10596831—42no—
261C22×C73—22no—
262C22×C73—22no—
263C22×C73—22no—
264C22×C73—22no—
265C22×C73—22no—
266C22×C73—22no—
267C22×C73—44no—
268C22×C73—44no—
269C22×C73—44no—
270C22×C73—816no—
271C22×C73—82no—
272C22×C73—82no—
273C22×C73—82no—
274C22×C73—82no—
275C22×C73C4×D814yes—
276C22×C73C2×C4×D411yescharacteristic
277C22×C73#53411yescharacteristic
278C22×C73#53511yescharacteristic
279C22×C73C22×D811yescharacteristic
280C22×C73C2×#13511yescharacteristic
281C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
282C22×C73#57511yescharacteristic
283C2×D73—42no—
284C2×D73—42no—
285C2×D73—42no—
286C2×D73—81no—
287C2×D73—81no—
288C2×D73—81no—
289C2×D73—81no—
290C4×C73—22no—
291C4×C73—22no—
292C4×C73—22no—
293C4×C73—22no—
294C4×C73—41no—
295C4×C73—82no—
296C4×C73#51812yes—
297C4×C73#53812yes—
298Dic73—42no—
299Dic73—81no—
300Dic73—81no—
301#22040—731nomaximal, Sylow
302C2×C73—24no—
303C2×C73—84no—
304C2×C73#232111yescharacteristic
305C2×C73C2×C4×D811yescharacteristic
306C2×C73C4×AGL(1,ℤ/8)11yescharacteristic
307C2×C73#318611yescharacteristic
308C2×C73#318811yescharacteristic
309C2×C73#320511yescharacteristic
310C2×C73#321111yescharacteristic
311D73—82no—
312C2×#447—731no—
313C4×#121—731no—
314C2×#390D7311yescharacteristic, complemented
315#1970—732no—
316#1934—732no—
317#1925—732no—
318#1879—732no—
319#1874—731no—
320#1852—731no—
321#1786—731no—
322C4×C22⋊C8—731no—
323C73#2204011yescharacteristic, complemented, Sylow
324C23×D4C2×D7311yescharacteristic
325C22×D8—732no—
326C2×#121—731no—
327C2×C4×D4—731no—
328C2×C4×D4—1461no—
329C22×C4⋊C4C2×D7311yescharacteristic
330C22×C22⋊C4—21no—
331C22×C22⋊C4—21no—
332C22×C42C2×D7311yescharacteristic
333#447—738no—
334C2×#107—732no—
335C2×#102—732no—
336C2×#102—732no—
337C2×#102—732no—
338C2×#102—732no—
339C2×C22⋊C8—731no—
340C2×C22⋊C8—731no—
341C2×C4×C8—732no—
342#390—21no—
343#390—21no—
344#390—731no—
345#390C2×D7312yes—
346#390C2×D7312yes—
347#389—731no—
348#388—1461no—
349#386—1461no—
350#380—731no—
351C4×C4⋊C4—731no—
352C4×C22⋊C4—1461no—
353#340—732no—
354#340—732no—
355#336—732no—
356C25—21no—
357C22×D4—21no—
358C22×D4—22no—
359C22×D4—42no—
360C22×D4—42no—
361C22×D4—731no—
362C22×D4—1461no—
363C22×D4C4×D7314yescomplemented
364C22×D4C22×D7312yes—
365C23×C4—21no—
366C23×C4—1461no—
367C23×C4C22⋊D7311yescharacteristic
368C23×C4C22⋊D7311yescharacteristic
369C23×C4C22×D7311yescharacteristic
370C2×D8—2922no—
371C2×D8—2922no—
372C2×D8—738no—
373C22×C8—732no—
374C22×C8—732no—
375#121—734no—
376#121—1462no—
377C4×D4—2922no—
378C4×D4—2924no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—42no—
381C2×C4⋊C4—731no—
382C2×C4⋊C4—731no—
383C2×C4⋊C4—731no—
384C2×C4⋊C4—1461no—
385C2×C4⋊C4C22×D7312yes—
386C2×C22⋊C4—22no—
387C2×C22⋊C4—22no—
388C2×C22⋊C4—22no—
389C2×C22⋊C4—22no—
390C2×C22⋊C4—44no—
391C2×C22⋊C4—44no—
392C2×C22⋊C4—1461no—
393C2×C22⋊C4—1461no—
394C2×C22⋊C4—1461no—
395C2×C22⋊C4—1461no—
396C2×C22⋊C4—1461no—
397C2×C42—42no—
398C2×C42—1461no—
399C2×C42C22⋊D7311yescharacteristic
400C2×C42C22⋊D7311yescharacteristic
401C2×C42C22×D7311yescharacteristic
402C2×C42C22×D7311yescharacteristic
403#107—2922no—
404#102—2922no—
405#102—2922no—
406#102—2922no—
407#102—738no—
408C22⋊C8—734no—
409C22⋊C8—1462no—
410C4×C8—2922no—
411#95—1461no—
412#95—1461no—
413#95—1461no—
414C24—24no—
415C24—22no—
416C24—42no—
417C24—42no—
418C24—42no—
419C24—42no—
420C24—42no—
421C24—42no—
422C24—1461no—
423C24C2×C22⋊D7311yescharacteristic
424C2×D4—24no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—48no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—48no—
434C2×D4—2922no—
435C2×D4—2922no—
436C2×D4—732no—
437C2×D4—1464no—
438C2×D4—1461no—
439C2×D4—1462no—
440C2×D4—1462no—
441C2×D4C2×C4×D7314yes—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—2922no—
454C22×C4—2922no—
455C22×C4—2922no—
456C22×C4—1461no—
457C22×C4—1461no—
458C22×C4—1461no—
459C22×C4#1192401314yescomplemented
460C22×C4C2×C4×D7312yes—
461C22×C4C2×C22⋊D7311yescharacteristic
462C22×C4C2×C22⋊D7311yescharacteristic
463C22×C4C2×C22⋊D7311yescharacteristic
464C22×C4C2×C22⋊D7311yescharacteristic
465C22×C4C2×C22⋊D7311yescharacteristic
466C22×C4#1192402812yes—
467C22×C4C23×D7311yescharacteristic
468D8—2928no—
469C2×C8—734no—
470C2×C8—1462no—
471C2×C8—1462no—
472C2×C8—1462no—
473C2×C8—1462no—
474C2×C8—1462no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—42no—
478C4⋊C4—2922no—
479C4⋊C4—732no—
480C4⋊C4—1462no—
481C4⋊C4—1462no—
482C4⋊C4—1461no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—2922no—
488C22⋊C4—2922no—
489C22⋊C4—2922no—
490C22⋊C4—2922no—
491C22⋊C4—1464no—
492C42—22no—
493C42—22no—
494C42—41no—
495C42—41no—
496C42—2922no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—24no—
502C23—48no—
503C23—48no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—48no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—84no—
512C23—84no—
513C23—84no—
514C23—84no—
515C23—2921no—
516C23—2921no—
517C23—2921no—
518C23—2921no—
519C23—2921no—
520C23—2921no—
521C23—1462no—
522C23C4×C22⋊D7312yes—
523C23C2×#1192401312yes—
524C23#1455661112yes—
525C23C22×C22⋊D7311yescharacteristic
526D4—24no—
527D4—24no—
528D4—48no—
529D4—42no—
530D4—42no—
531D4—2928no—
532D4—2922no—
533D4—1464no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—84no—
553C2×C4—2921no—
554C2×C4—2921no—
555C2×C4—2921no—
556C2×C4—2921no—
557C2×C4—2921no—
558C2×C4—2921no—
559C2×C4—2921no—
560C2×C4—2921no—
561C2×C4—5844no—
562C2×C4—1462no—
563C2×C4C4×C22⋊D7312yes—
564C2×C4C2×#1192401312yes—
565C2×C4C2×#1192401312yes—
566C2×C4#1455661712yes—
567C2×C4C22×C4×D7311yescharacteristic
568C2×C4C2×#1192402811yescharacteristic
569C2×C4#1455663612yes—
570C2×C4#1455663612yes—
571C8—2922no—
572C8—1464no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22—816no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22—82no—
587C22—2922no—
588C22—2922no—
589C22—2922no—
590C22—5841no—
591C22—5841no—
592C22—5841no—
593C22—5841no—
594C22C4×#1192401314yes—
595C22C2×C4×C22⋊D7311yescharacteristic
596C22#1798914611yescharacteristic
597C22#1798915411yescharacteristic
598C22C22×#1192401311yescharacteristic
599C22C2×#1455661111yescharacteristic
600C22C2×#1455661711yescharacteristic
601C22#1798938511yescharacteristic
602C4—22no—
603C4—22no—
604C4—22no—
605C4—22no—
606C4—41no—
607C4—82no—
608C4—2922no—
609C4—5841no—
610C4—5841no—
611C4#1798906012yes—
612C4#1798917512yes—
613C2—24no—
614C2—84no—
615C2—5842no—
616C2#2261258911yescharacteristic
617C2C2×C4×#1192401311yescharacteristic
618C2C4×#1455661711yescharacteristic
619C2#2262067911yescharacteristic
620C2#2262069611yescharacteristic
621C2#2262084911yescharacteristic
622C2#2262089611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC2×#390×C73C2
socleC23×C73C22×D4
prime radicalC2×#17988721C2
cosocleC22×C4×C73C24
hypercenterC2×#390D73
hypocenterC73#22040
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
2. C4×#14556565C2semidirect
3. #22610796C2semidirect
4. #22610617C2semidirect
5. #22610548C2semidirect
6. #22610282C2semidirect
7. #22610248C2semidirect
8. #22610119C2semidirect
9. #22609862C2semidirect
10. C4×C22⋊#10596831C2semidirect
11. C2×#390×C73C2semidirect
14. C2×C4×D292C22semidirect
16. #17988721C4semidirect
17. C2×#14556524C22semidirect
21. C2×#14556511C22semidirect
24. C2×C4×#10596831C22semidirect
25. #17988489C22semidirect
26. #17988485C22semidirect
30. C4×C4⋊Dic73C22semidirect
32. #17988349C22semidirect
33. #17988349C22semidirect
34. #17988342C22semidirect
49. C2×#11924013C2×C4semidirect
69. #14556512C2×C4semidirect
72. C22⋊#10596831C2×C4semidirect
82. C22×D4×C73C2×C4semidirect
165. C22×C4×C73D8semidirect
314. C2×#390D73semidirect
323. C73#22040semidirect
363. C22×D4C4×D73semidirect
459. C22×C4#11924013semidirect

Maximal subgroups

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

subgroupindex
C2×#179887212
C4×#145565652
#226107962
#226106172
#226105482
#226102822
#226102482
#226101192
#226098622
C4×C22⋊#105968312
C2×#390×C732
#2204073