Finite Group Databasebeta

← #33008564

Subgroups — #33008564

Order 18688.

1846 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
1C22×#14556548C211yesmaximal, characteristic
2C2×#17988469C212yesmaximal, complemented
3#22608197C212yesmaximal, complemented
4#22608175C212yesmaximal
5#22608167C212yesmaximal
6#22608159C211yesmaximal, characteristic, complemented
7#22608157C211yesmaximal, characteristic, complemented
8#22608152C211yesmaximal, characteristic
9#22608152C211yesmaximal, characteristic
10C4×#14556498C211yesmaximal, characteristic, complemented
11C73×#1434C211yesmaximal, characteristic, complemented
12C23×C4×D73C2211yescharacteristic
13C2×#14556548—28no—
14C2×#14556548C418yescomplemented
15C22×#11924007C2212yes—
16C22×#11924007C2211yescharacteristic
17C22×#11924007C2211yescharacteristic
18C22×#11924005C2212yes—
19C22×#11924005C2212yes—
20C22×#11924005C2211yescharacteristic
21C22×#11924005C2211yescharacteristic
22C22×C4×Dic73C2212yes—
23C22×C4×Dic73C2211yescharacteristic
24#17988469—28no—
25C2×#14556498C2211yescharacteristic
26C2×#14556498C2211yescharacteristic
27C2×#14556498C2212yes—
28C2×#14556498C2212yes—
29C2×#14556498C2212yes—
30C2×#14556498C2212yes—
31C2×#14556498C2212yes—
32C2×#14556498C2212yes—
33C22×C42×C73C2211yescharacteristic
34C2×#95×C73C2211yescharacteristic
35C2×#95×C73C2211yescharacteristic
36C2×#95×C73C2212yes—
37C2×#95×C73C2212yes—
38C24×D73C2311yescharacteristic
39C23×Dic73C2311yescharacteristic
40C23×Dic73C2312yes—
41C23×Dic73C2312yes—
42C23×Dic73C2311yescharacteristic
43C23×Dic73C2311yescharacteristic
44C22×C4×D73—24no—
45C22×C4×D73—24no—
46C22×C4×D73—22no—
47C22×C4×D73—22no—
48C22×C4×D73C2×C414yes—
49#14556548—232no—
50C2×#11924007—24no—
51C2×#11924007—24no—
52C2×#11924007—24no—
53C2×#11924007—42no—
54C2×#11924007C2×C414yes—
55C2×#11924007C2×C414yes—
56C2×#11924007D418yes—
57C2×#11924005—24no—
58C2×#11924005—24no—
59C2×#11924005—24no—
60C2×#11924005—24no—
61C2×#11924005—24no—
62C2×#11924005—24no—
63C2×#11924005—42no—
64C2×#11924005—42no—
65C2×#11924005C2×C414yes—
66C2×#11924005C2×C414yes—
67C2×C4×Dic73—24no—
68C2×C4×Dic73—24no—
69C2×C4×Dic73—24no—
70C2×C4×Dic73—42no—
71C2×C4×Dic73C2×C414yes—
72#14556498—24no—
73#14556498—24no—
74#14556498—24no—
75#14556498—24no—
76#14556498—24no—
77#14556498—22no—
78#14556498—22no—
79#14556498—42no—
80C23×C4×C73C2312yes—
81C23×C4×C73C2312yes—
82C23×C4×C73C2311yescharacteristic
83C23×C4×C73C2311yescharacteristic
84C23×C4×C73C2311yescharacteristic
85C2×C42×C73—24no—
86C2×C42×C73C2×C414yes—
87#95×C73—24no—
88#95×C73—22no—
89#95×C73—22no—
90#95×C73—42no—
91C23×D73—22no—
92C23×D73—22no—
93C23×D73—22no—
94C23×D73—21no—
95C23×D73—21no—
96C23×D73—21no—
97C23×D73—21no—
98C23×D73—21no—
99C23×D73—21no—
100C23×D73C22×C412yes—
101C23×D73#4114yes—
102C22×Dic73—22no—
103C22×Dic73—22no—
104C22×Dic73—21no—
105C22×Dic73—21no—
106C22×Dic73—21no—
107C22×Dic73—21no—
108C22×Dic73—22no—
109C22×Dic73—22no—
110C22×Dic73—22no—
111C22×Dic73—22no—
112C22×Dic73—22no—
113C22×Dic73—22no—
114C22×Dic73—22no—
115C22×Dic73—22no—
116C22×Dic73—22no—
117C22×Dic73—22no—
118C22×Dic73—21no—
119C22×Dic73—22no—
120C22×Dic73—22no—
121C22×Dic73—22no—
122C22×Dic73—21no—
123C22×Dic73—22no—
124C22×Dic73C22×C412yes—
125C22×Dic73C22×C412yes—
126C22×Dic73C22×C412yes—
127C22×Dic73C2×D414yes—
128C22×Dic73C2×D414yes—
129C22×Dic73#4114yes—
130C22×Dic73#4114yes—
131C2×C4×D73—216no—
132C2×C4×D73—216no—
133C2×C4×D73—28no—
134C2×C4×D73—28no—
135C2×C4×D73—28no—
136#11924007—28no—
137#11924007—48no—
138#11924007C22⋊C4116yescomplemented
139#11924005—28no—
140#11924005—28no—
141#11924005—48no—
142#11924005—48no—
143C4×Dic73—28no—
144C4×Dic73—48no—
145C24×C73C2411yescharacteristic
146C22×C4×C73—22no—
147C22×C4×C73—22no—
148C22×C4×C73—22no—
149C22×C4×C73—21no—
150C22×C4×C73—21no—
151C22×C4×C73—22no—
152C22×C4×C73—22no—
153C22×C4×C73—21no—
154C22×C4×C73—21no—
155C22×C4×C73—22no—
156C22×C4×C73—22no—
157C22×C4×C73—22no—
158C22×C4×C73—22no—
159C22×C4×C73—22no—
160C22×C4×C73—22no—
161C22×C4×C73—22no—
162C22×C4×C73—22no—
163C22×C4×C73—22no—
164C22×C4×C73—22no—
165C22×C4×C73—22no—
166C22×C4×C73—22no—
167C22×C4×C73—22no—
168C22×C4×C73—22no—
169C22×C4×C73C22×C412yes—
170C22×C4×C73C22×C412yes—
171C22×C4×C73C22×C412yes—
172C22×C4×C73C2×D414yes—
173C22×C4×C73#4112yes—
174C22×C4×C73#4112yes—
175C42×C73—28no—
176C23×C73C23×C411yescharacteristic
177C23×C73C22×D412yes—
178C23×C73C2×#4112yes—
179C23×C73C2×#4112yes—
180C23×C73C2×#4112yes—
181C23×C73C2×#4111yescharacteristic
182C23×C73C2×#4111yescharacteristic
183C23×C73H(5,𝔽2)11yescharacteristic
184C23×C73H(5,𝔽2)11yescharacteristic
185C23×C732−1+411yescharacteristic
186C23×C732−1+411yescharacteristic
187C22×D73—22no—
188C22×D73—22no—
189C22×D73—22no—
190C22×D73—22no—
191C22×D73—22no—
192C22×D73—22no—
193C22×D73—24no—
194C22×D73—24no—
195C22×D73—24no—
196C22×D73—24no—
197C22×D73—24no—
198C22×D73—24no—
199C22×D73—42no—
200C22×D73—42no—
201C22×D73—41no—
202C22×D73—41no—
203C22×D73—41no—
204C22×D73—41no—
205C22×D73—42no—
206C22×D73—42no—
207C22×D73—42no—
208C22×D73—42no—
209C22×D73#11714yes—
210C2×C4×C73—22no—
211C2×C4×C73—22no—
212C2×C4×C73—22no—
213C2×C4×C73—22no—
214C2×C4×C73—24no—
215C2×C4×C73—24no—
216C2×C4×C73—24no—
217C2×C4×C73—24no—
218C2×C4×C73—22no—
219C2×C4×C73—22no—
220C2×C4×C73—22no—
221C2×C4×C73—42no—
222C2×C4×C73—42no—
223C2×C4×C73—42no—
224C2×C4×C73—42no—
225C2×C4×C73—42no—
226C2×C4×C73—42no—
227C2×C4×C73—42no—
228C2×C4×C73—42no—
229C2×C4×C73—42no—
230C2×C4×C73—42no—
231C2×C4×C73—42no—
232C2×C4×C73—42no—
233C2×C4×C73—42no—
234C2×C4×C73—42no—
235C2×C4×C73—42no—
236C2×C4×C73—42no—
237C2×C4×C73C2×C22⋊C414yes—
238C2×C4×C73#11714yes—
239C2×Dic73—22no—
240C2×Dic73—22no—
241C2×Dic73—22no—
242C2×Dic73—22no—
243C2×Dic73—24no—
244C2×Dic73—24no—
245C2×Dic73—24no—
246C2×Dic73—24no—
247C2×Dic73—22no—
248C2×Dic73—22no—
249C2×Dic73—22no—
250C2×Dic73—24no—
251C2×Dic73—24no—
252C2×Dic73—24no—
253C2×Dic73—24no—
254C2×Dic73—24no—
255C2×Dic73—24no—
256C2×Dic73—42no—
257C2×Dic73—42no—
258C2×Dic73—42no—
259C2×Dic73—42no—
260C2×Dic73—42no—
261C2×Dic73—42no—
262C2×Dic73—42no—
263C2×Dic73—42no—
264C2×Dic73—42no—
265C2×Dic73—42no—
266C2×Dic73C2×C22⋊C414yes—
267C2×Dic73C2×C22⋊C414yes—
268C4×D73—232no—
269C22×C73C22×C22⋊C411yescharacteristic
270C22×C73C2×#11711yescharacteristic
271C22×C73C2×#11711yescharacteristic
272C22×C73C4×#4111yescharacteristic
273C22×C73C4×#4111yescharacteristic
274C22×C73#52011yescharacteristic
275C22×C73#52011yescharacteristic
276C22×C73#52512yes—
277C22×C73#52512yes—
278C22×C73#52912yes—
279C22×C73#52912yes—
280C22×C73#54011yescharacteristic
281C22×C73#54111yescharacteristic
282C22×C73#54211yescharacteristic
283C22×C73#54211yescharacteristic
284C22×C73#54612yes—
285C22×C73#54712yes—
286C22×C73#54812yes—
287C22×C73#55012yes—
288C22×C73#55112yes—
289C22×C73#55112yes—
290C22×C73#55212yes—
291C22×C73#55212yes—
292C2×D73—24no—
293C2×D73—24no—
294C2×D73—24no—
295C2×D73—42no—
296C2×D73—42no—
297C2×D73—42no—
298C2×D73—44no—
299C2×D73—44no—
300C2×D73—44no—
301C2×D73—44no—
302C2×D73—42no—
303C4×C73—24no—
304C4×C73—24no—
305C4×C73—42no—
306C4×C73—82no—
307C4×C73—82no—
308Dic73—24no—
309Dic73—24no—
310Dic73—42no—
311Dic73—44no—
312Dic73—44no—
313#13944—731nomaximal, Sylow
314C2×C73#228011yescharacteristic
315C2×C73#229811yescharacteristic
316C2×C73#231411yescharacteristic
317C2×C73#233911yescharacteristic
318C2×C73#234111yescharacteristic
319C2×C73#234311yescharacteristic
320C2×C73#234311yescharacteristic
321C2×C73#247112yes—
322C2×C73#247512yes—
323C2×C73#247712yes—
324C2×C73#247912yes—
325D73—44no—
326C22×#117—731no—
327C2×#387—732no—
328#1445—732no—
329#1439—732no—
330#1437—732no—
331C4⋊#95—731no—
332#1434—731no—
333#1434D7311yescharacteristic, complemented
334C22⋊#95—731no—
335C22⋊#95—731no—
336C4×#95—731no—
337C73#1394411yescharacteristic, complemented, Sylow
338C24×C4—731no—
339C2×#117—738no—
340C2×#117—1468no—
341C22×C4⋊C4—732no—
342C22×C4⋊C4—731no—
343C22×C4⋊C4—731no—
344C22×C22⋊C4—732no—
345C22×C22⋊C4—732no—
346C22×C22⋊C4—731no—
347C22×C22⋊C4—731no—
348C22×C42—732no—
349C22×C42—731no—
350C22×C42C2×D7311yescharacteristic
351#387—1468no—
352C2×#95—732no—
353C2×#95—731no—
354C2×#95—731no—
355C2×#95—732no—
356C2×#95—732no—
357C2×#95—732no—
358C2×#95—732no—
359C2×#95—732no—
360C2×#95C2×D7311yescharacteristic
361C2×#95C2×D7311yescharacteristic
362C2×#95C2×D7312yes—
363C2×#95C2×D7312yes—
364C25—731no—
365C23×C4—731no—
366C23×C4—731no—
367C23×C4—731no—
368C23×C4—732no—
369C23×C4—732no—
370C23×C4—734no—
371C23×C4—1464no—
372C23×C4—1464no—
373C23×C4—1462no—
374C23×C4—1462no—
375C23×C4C22×D7312yes—
376C23×C4C22×D7312yes—
377C23×C4C22×D7311yescharacteristic
378C23×C4C22×D7311yescharacteristic
379C23×C4C22×D7311yescharacteristic
380#117—14632no—
381C2×C4⋊C4—2922no—
382C2×C4⋊C4—734no—
383C2×C4⋊C4—738no—
384C2×C4⋊C4—734no—
385C2×C4⋊C4—1464no—
386C2×C4⋊C4—1464no—
387C2×C4⋊C4—1464no—
388C2×C22⋊C4—2922no—
389C2×C22⋊C4—2922no—
390C2×C22⋊C4—734no—
391C2×C22⋊C4—734no—
392C2×C22⋊C4—1464no—
393C2×C22⋊C4—1464no—
394C2×C22⋊C4—1464no—
395C2×C22⋊C4—1464no—
396C2×C22⋊C4—1464no—
397C2×C22⋊C4—1464no—
398C2×C42—24no—
399C2×C42—2922no—
400C2×C42—734no—
401C2×C42—1464no—
402C2×C42—1464no—
403C2×C42—1464no—
404C2×C42C4×D7314yes—
405#95—24no—
406#95—22no—
407#95—22no—
408#95—42no—
409#95—2922no—
410#95—1464no—
411#95—1464no—
412#95—1464no—
413#95—1464no—
414#95—1464no—
415#95—1462no—
416#95—1462no—
417C24—732no—
418C24—734no—
419C24—1462no—
420C24—1462no—
421C24—1462no—
422C24—1461no—
423C24—1461no—
424C24—1461no—
425C24—1461no—
426C24—1461no—
427C24—1461no—
428C24C23×D7311yescharacteristic
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—732no—
453C22×C4—732no—
454C22×C4—734no—
455C22×C4—734no—
456C22×C4—732no—
457C22×C4—734no—
458C22×C4—734no—
459C22×C4—1468no—
460C22×C4—1468no—
461C22×C4—14616no—
462C22×C4—14616no—
463C22×C4—1462no—
464C22×C4—1462no—
465C22×C4—1462no—
466C22×C4—1462no—
467C22×C4—1462no—
468C22×C4—1462no—
469C22×C4—1462no—
470C22×C4—1462no—
471C22×C4—1462no—
472C22×C4—1462no—
473C22×C4—1461no—
474C22×C4—1462no—
475C22×C4—1462no—
476C22×C4—1462no—
477C22×C4—1462no—
478C22×C4—1461no—
479C22×C4—1462no—
480C22×C4—1462no—
481C22×C4—1461no—
482C22×C4—1461no—
483C22×C4—1461no—
484C22×C4—1461no—
485C22×C4—1468no—
486C22×C4C2×C4×D7312yes—
487C22×C4C2×C4×D7312yes—
488C22×C4C2×C4×D7312yes—
489C22×C4D4×D7314yes—
490C22×C4#1192402812yes—
491C22×C4#1192402812yes—
492C4⋊C4—2928no—
493C4⋊C4—7316no—
494C4⋊C4—1468no—
495C22⋊C4—2928no—
496C22⋊C4—2928no—
497C22⋊C4—1468no—
498C22⋊C4—1468no—
499C42—28no—
500C42—2928no—
501C42—1468no—
502C23—2921no—
503C23—2921no—
504C23—2921no—
505C23—2921no—
506C23—2922no—
507C23—2922no—
508C23—2922no—
509C23—2922no—
510C23—2922no—
511C23—2922no—
512C23—734no—
513C23—1464no—
514C23—1464no—
515C23—1464no—
516C23—1464no—
517C23—1462no—
518C23—1462no—
519C23—1462no—
520C23—1462no—
521C23—1462no—
522C23—1462no—
523C23—1464no—
524C23—1464no—
525C23C22×C4×D7311yescharacteristic
526C23C2×D4×D7312yes—
527C23#41×D7312yes—
528C23#41×D7312yes—
529C23#41×D7312yes—
530C23C2×#1192402811yescharacteristic
531C23C2×#1192402811yescharacteristic
532C23#1455663611yescharacteristic
533C23#1455663711yescharacteristic
534C23#1455663811yescharacteristic
535C23#1455663911yescharacteristic
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—2922no—
564C2×C4—2922no—
565C2×C4—2922no—
566C2×C4—2922no—
567C2×C4—2922no—
568C2×C4—2922no—
569C2×C4—2922no—
570C2×C4—2922no—
571C2×C4—2922no—
572C2×C4—2922no—
573C2×C4—734no—
574C2×C4—734no—
575C2×C4—14632no—
576C2×C4—1464no—
577C2×C4—1464no—
578C2×C4—1464no—
579C2×C4—1464no—
580C2×C4—1462no—
581C2×C4—1462no—
582C2×C4—1462no—
583C2×C4—1464no—
584C2×C4—1464no—
585C2×C4—1464no—
586C2×C4—1464no—
587C2×C4—1462no—
588C2×C4—1462no—
589C2×C4—1462no—
590C2×C4—1462no—
591C2×C4—1464no—
592C2×C4—1464no—
593C2×C4C22⋊C4×D7314yes—
594C2×C4#1455654814yes—
595C22—2922no—
596C22—2924no—
597C22—2922no—
598C22—2922no—
599C22—2924no—
600C22—2924no—
601C22—2924no—
602C22—2922no—
603C22—1464no—
604C22—1464no—
605C22—1464no—
606C22C2×C22⋊C4×D7311yescharacteristic
607C22#117×D7311yescharacteristic
608C22C2×#1455654811yescharacteristic
609C22#1798905511yescharacteristic
610C22#1798905511yescharacteristic
611C22#1798906711yescharacteristic
612C22#1798906811yescharacteristic
613C22#1798909612yes—
614C22#1798909612yes—
615C22#1798911512yes—
616C22#1798911512yes—
617C22#1798918811yescharacteristic
618C22#1798919511yescharacteristic
619C22#1798920211yescharacteristic
620C22#1798920911yescharacteristic
621C22#1798922912yes—
622C22#1798923512yes—
623C22#1798924712yes—
624C22#1798925512yes—
625C22#1798926212yes—
626C22#1798926712yes—
627C22#1798926912yes—
628C22#1798927312yes—
629C4—24no—
630C4—24no—
631C4—42no—
632C4—82no—
633C4—82no—
634C4—2924no—
635C4—2922no—
636C4—2924no—
637C4—1464no—
638C4—1464no—
639C2—2924no—
640C2#2261231211yescharacteristic
641C2#2261242611yescharacteristic
642C2#2261253611yescharacteristic
643C2#2261275511yescharacteristic
644C2#2261276811yescharacteristic
645C2#2261278411yescharacteristic
646C2#2261278811yescharacteristic
647C2#2261404412yes—
648C2#2261410412yes—
649C2#2261413412yes—
650C2#2261416412yes—
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC73×#1434C2
socle, cosocleC24×C73C24
prime radicalC24×D73C23
hypercenter#1434D73
hypocenterC73#13944
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
2. C2×#17988469C2semidirect
3. #22608197C2semidirect
6. #22608159C2semidirect
7. #22608157C2semidirect
10. C4×#14556498C2semidirect
11. C73×#1434C2semidirect
14. C2×#14556548C4semidirect
138. #11924007C22⋊C4semidirect
333. #1434D73semidirect
337. C73#13944semidirect

Maximal subgroups

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

subgroupindex
C22×#145565482
C2×#179884692
#226081972
#226081752
#226081672
#226081592
#226081572
#226081522
C4×#145564982
C73×#14342
#1394473