Finite Group Databasebeta

← #33012751

Subgroups — #33012751

Order 18688.

1830 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556556C211yesmaximal, characteristic, complemented
2C2×#17988499C211yesmaximal, characteristic
3C2×#17988481C212yesmaximal
4C2×#17988480C211yesmaximal, characteristic
5#22608201C211yesmaximal, characteristic, complemented
6#22608195C212yesmaximal, complemented
7#22608194C211yesmaximal, characteristic, complemented
8#22608174C212yesmaximal
9#22608157C212yesmaximal, complemented
10#22608156C211yesmaximal, characteristic, complemented
11C73×#1439C211yesmaximal, characteristic
12C23×Dic146C2211yescharacteristic
13C2×#14556556—24no—
14C2×#14556556—28no—
15C22×C4⋊Dic73C2211yescharacteristic
16C22×#11924007C2212yes—
17C22×#11924007C2212yes—
18C22×C22⋊Dic73C2211yescharacteristic
19C22×C22⋊Dic73C2212yes—
20C22×C22⋊Dic73C2211yescharacteristic
21C22×C4×Dic73C2211yescharacteristic
22C22×C4×Dic73C2212yes—
23#17988499C418yescomplemented
24#17988481—28no—
25#17988480—24no—
26C2×#14556498C2212yes—
27C2×#14556498C2211yescharacteristic
28C2×#14556498C2212yes—
29C2×#14556498C2211yescharacteristic
30C2×#14556498C2212yes—
31C2×#14556498C2212yes—
32C2×#14556498C2212yes—
33C2×#14556498C2211yescharacteristic
34C2×#14556498C2212yes—
35C22×C4⋊C4×C73C2211yescharacteristic
36C22×C22⋊C4×C73C2211yescharacteristic
37C22×C22⋊C4×C73C2212yes—
38C2×#95×C73C2211yescharacteristic
39C2×#95×C73C2212yes—
40C22×Dic146—24no—
41C22×Dic146—24no—
42C22×Dic146—22no—
43C22×Dic146—22no—
44C22×Dic146C2×C414yes—
45C23×Dic73C2311yescharacteristic
46C23×Dic73C2311yescharacteristic
47C23×Dic73C2312yes—
48C23×Dic73C2311yescharacteristic
49C23×Dic73C2312yes—
50C23×Dic73C2311yescharacteristic
51#14556556—232no—
52C2×C4⋊Dic73—22no—
53C2×C4⋊Dic73—42no—
54C2×#11924007—24no—
55C2×#11924007—24no—
56C2×#11924007—24no—
57C2×#11924007—24no—
58C2×#11924007—42no—
59C2×#11924007—42no—
60C2×C22⋊Dic73—24no—
61C2×C22⋊Dic73—24no—
62C2×C22⋊Dic73—22no—
63C2×C22⋊Dic73—22no—
64C2×C22⋊Dic73—24no—
65C2×C22⋊Dic73—42no—
66C2×C22⋊Dic73C2×C418yes—
67C2×C4×Dic73—24no—
68C2×C4×Dic73—24no—
69C2×C4×Dic73—24no—
70C2×C4×Dic73D414yes—
71C2×C4×Dic73D418yes—
72#14556498—22no—
73#14556498—24no—
74#14556498—24no—
75#14556498—24no—
76#14556498—22no—
77#14556498—42no—
78#14556498—42no—
79#14556498C2×C418yes—
80#14556498C2×C414yes—
81C25×C73C2311yescharacteristic
82C23×C4×C73C2312yes—
83C23×C4×C73C2311yescharacteristic
84C23×C4×C73C2312yes—
85C23×C4×C73C2311yescharacteristic
86C2×C4⋊C4×C73—24no—
87C2×C4⋊C4×C73—22no—
88C2×C22⋊C4×C73—22no—
89C2×C22⋊C4×C73—22no—
90C2×C22⋊C4×C73—24no—
91C2×C22⋊C4×C73—42no—
92C2×C22⋊C4×C73—42no—
93C2×C22⋊C4×C73C2×C414yes—
94#95×C73—22no—
95#95×C73—42no—
96C2×Dic146—28no—
97C2×Dic146—216no—
98C2×Dic146—28no—
99C2×Dic146—28no—
100C2×Dic146—216no—
101C22×Dic73—22no—
102C22×Dic73—22no—
103C22×Dic73—22no—
104C22×Dic73—22no—
105C22×Dic73—21no—
106C22×Dic73—22no—
107C22×Dic73—21no—
108C22×Dic73—22no—
109C22×Dic73—22no—
110C22×Dic73—22no—
111C22×Dic73—21no—
112C22×Dic73—22no—
113C22×Dic73—22no—
114C22×Dic73—22no—
115C22×Dic73—22no—
116C22×Dic73—21no—
117C22×Dic73—22no—
118C22×Dic73—21no—
119C22×Dic73—22no—
120C22×Dic73—21no—
121C22×Dic73—22no—
122C22×Dic73—22no—
123C22×Dic73—21no—
124C22×Dic73—22no—
125C22×Dic73—21no—
126C22×Dic73—21no—
127C22×Dic73—21no—
128C22×Dic73C22×C414yes—
129C22×Dic73C22×C414yes—
130C22×Dic73C2×D412yes—
131C22×Dic73C2×D414yes—
132C22×Dic73C2×D412yes—
133C22×Dic73C2×D414yes—
134C22×Dic73#4114yes—
135C4⋊Dic73—44no—
136#11924007—216no—
137#11924007—48no—
138C22⋊Dic73—216no—
139C22⋊Dic73—44no—
140C22⋊Dic73—44no—
141C4×Dic73—216no—
142C4×Dic73—28no—
143C24×C73—21no—
144C24×C73—22no—
145C24×C73—21no—
146C24×C73—22no—
147C24×C73—21no—
148C24×C73—22no—
149C24×C73—21no—
150C24×C73—21no—
151C24×C73—22no—
152C24×C73—21no—
153C24×C73C22×C412yes—
154C24×C73C2411yescharacteristic
155C22×C4×C73—22no—
156C22×C4×C73—22no—
157C22×C4×C73—21no—
158C22×C4×C73—22no—
159C22×C4×C73—22no—
160C22×C4×C73—21no—
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×C73—22no—
170C22×C4×C73—22no—
171C22×C4×C73—22no—
172C22×C4×C73—22no—
173C22×C4×C73—22no—
174C22×C4×C73C22×C412yes—
175C22×C4×C73C22×C412yes—
176C22×C4×C73C2×D414yes—
177C22×C4×C73C2×D412yes—
178C22×C4×C73#4112yes—
179C4⋊C4×C73—28no—
180C22⋊C4×C73—28no—
181C22⋊C4×C73—48no—
182C23×C73—22no—
183C23×C73—24no—
184C23×C73—22no—
185C23×C73—24no—
186C23×C73—22no—
187C23×C73—41no—
188C23×C73—41no—
189C23×C73—42no—
190C23×C73—42no—
191C23×C73—41no—
192C23×C73—42no—
193C23×C73—41no—
194C23×C73—41no—
195C23×C73—42no—
196C23×C73—42no—
197C23×C73—41no—
198C23×C73—42no—
199C23×C73—41no—
200C23×C73—41no—
201C23×C73—42no—
202C23×C73—42no—
203C23×C73—42no—
204C23×C73—42no—
205C23×C73C23×C411yescharacteristic
206C23×C73C22×D411yescharacteristic
207C23×C73C22×D412yes—
208C23×C73C2×#4112yes—
209C23×C73C2×#4111yescharacteristic
210C23×C73H(5,𝔽2)12yes—
211C23×C73H(5,𝔽2)11yescharacteristic
212C23×C732−1+411yescharacteristic
213C23×C732−1+411yescharacteristic
214C23×C732−1+412yes—
215C23×C732−1+411yescharacteristic
216C2×C4×C73—24no—
217C2×C4×C73—22no—
218C2×C4×C73—22no—
219C2×C4×C73—24no—
220C2×C4×C73—22no—
221C2×C4×C73—24no—
222C2×C4×C73—24no—
223C2×C4×C73—24no—
224C2×C4×C73—24no—
225C2×C4×C73—42no—
226C2×C4×C73—42no—
227C2×C4×C73—41no—
228C2×C4×C73—41no—
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×C73—42no—
238C2×C4×C73—42no—
239C2×C4×C73C4×D414yes—
240C2×Dic73—24no—
241C2×Dic73—24no—
242C2×Dic73—24no—
243C2×Dic73—24no—
244C2×Dic73—24no—
245C2×Dic73—24no—
246C2×Dic73—24no—
247C2×Dic73—22no—
248C2×Dic73—22no—
249C2×Dic73—24no—
250C2×Dic73—24no—
251C2×Dic73—24no—
252C2×Dic73—24no—
253C2×Dic73—22no—
254C2×Dic73—22no—
255C2×Dic73—22no—
256C2×Dic73—42no—
257C2×Dic73—42no—
258C2×Dic73—42no—
259C2×Dic73—42no—
260C2×Dic73—42no—
261C2×Dic73—42no—
262C2×Dic73—41no—
263C2×Dic73—41no—
264C2×Dic73—41no—
265C2×Dic73—42no—
266C2×Dic73—41no—
267C2×Dic73—41no—
268C2×Dic73—42no—
269C2×Dic73—41no—
270C2×Dic73—42no—
271C2×Dic73—42no—
272C2×Dic73C4×D418yes—
273C2×Dic73#12014yes—
274Dic146—232no—
275C22×C73—44no—
276C22×C73—42no—
277C22×C73—44no—
278C22×C73—42no—
279C22×C73—42no—
280C22×C73—81no—
281C22×C73—81no—
282C22×C73—81no—
283C22×C73—82no—
284C22×C73—81no—
285C22×C73—82no—
286C22×C73C2×C4×D412yes—
287C22×C73C2×C4×D411yescharacteristic
288C22×C73#51911yescharacteristic
289C22×C73#52012yes—
290C22×C73#52011yescharacteristic
291C22×C73C2×#12011yescharacteristic
292C22×C73#52511yescharacteristic
293C22×C73#52512yes—
294C22×C73#53511yescharacteristic
295C22×C73#53512yes—
296C22×C73#53612yes—
297C22×C73#53611yescharacteristic
298C22×C73#53611yescharacteristic
299C22×C73#53612yes—
300C22×C73#53712yes—
301C22×C73#53711yescharacteristic
302C22×C73#54112yes—
303C22×C73#54111yescharacteristic
304C22×C73#54211yescharacteristic
305C22×C73#54212yes—
306C22×C73#54212yes—
307C22×C73#54212yes—
308C22×C73#54212yes—
309C4×C73—24no—
310C4×C73—42no—
311C4×C73—44no—
312C4×C73—82no—
313Dic73—24no—
314Dic73—28no—
315Dic73—44no—
316Dic73—42no—
317Dic73—81no—
318Dic73—81no—
319#14301—731nomaximal, Sylow
320C2×C73—82no—
321C2×C73C4×#12011yescharacteristic
322C2×C73#232312yes—
323C2×C73#232311yescharacteristic
324C2×C73#232612yes—
325C2×C73#232611yescharacteristic
326C2×C73#261411yescharacteristic
327C2×C73#261511yescharacteristic
328C2×C73#263211yescharacteristic
329C2×C73#263212yes—
330C2×C73#263711yescharacteristic
331C2×C73#263712yes—
332C22×C4×Q8—731no—
333C2×#393—731no—
334C2×#388—731no—
335C2×#388—732no—
336#1446—731no—
337#1444—732no—
338#1444—731no—
339#1439—732no—
340#1439D7311yescharacteristic
341#1434—731no—
342#1434—732no—
343C73#1430111yescharacteristic, complemented, Sylow
344C23×Q8—731no—
345C2×C4×Q8—1464no—
346C2×C4×Q8—1468no—
347C22×C4⋊C4—732no—
348C22×C4⋊C4—731no—
349C22×C4⋊C4—732no—
350C22×C4⋊C4C2×D7311yescharacteristic
351C22×C22⋊C4—731no—
352C22×C22⋊C4—732no—
353C22×C22⋊C4—731no—
354C22×C22⋊C4C2×D7311yescharacteristic
355C22×C22⋊C4C2×D7312yes—
356C22×C42—732no—
357C22×C42—731no—
358#393—738no—
359#388—1464no—
360#388—1468no—
361C2×#95—732no—
362C2×#95—732no—
363C2×#95—732no—
364C2×#95—731no—
365C2×#95—732no—
366C2×#95—731no—
367C2×#95—732no—
368C2×#95—732no—
369C2×#95—731no—
370C2×#95C2×D7311yescharacteristic
371C2×#95C2×D7312yes—
372C25C22×D7311yescharacteristic
373C22×Q8—734no—
374C22×Q8—1464no—
375C22×Q8—1464no—
376C22×Q8—1462no—
377C22×Q8—1462no—
378C23×C4—732no—
379C23×C4—731no—
380C23×C4—731no—
381C23×C4—732no—
382C23×C4—731no—
383C23×C4—731no—
384C23×C4C22×D7312yes—
385C23×C4C22×D7311yescharacteristic
386C23×C4C22×D7312yes—
387C23×C4C22×D7311yescharacteristic
388C4×Q8—14632no—
389C2×C4⋊C4—24no—
390C2×C4⋊C4—22no—
391C2×C4⋊C4—2922no—
392C2×C4⋊C4—2922no—
393C2×C4⋊C4—2922no—
394C2×C4⋊C4—1462no—
395C2×C4⋊C4—1464no—
396C2×C4⋊C4—1464no—
397C2×C4⋊C4—1464no—
398C2×C4⋊C4—1464no—
399C2×C22⋊C4—22no—
400C2×C22⋊C4—22no—
401C2×C22⋊C4—24no—
402C2×C22⋊C4—42no—
403C2×C22⋊C4—42no—
404C2×C22⋊C4—2922no—
405C2×C22⋊C4—738no—
406C2×C22⋊C4—1462no—
407C2×C22⋊C4—1464no—
408C2×C22⋊C4—1462no—
409C2×C22⋊C4—1464no—
410C2×C22⋊C4—1464no—
411C2×C22⋊C4C4×D7314yes—
412C2×C42—738no—
413C2×C42—734no—
414C2×C42—1464no—
415C2×C42—1464no—
416C2×C42—1464no—
417#95—22no—
418#95—42no—
419#95—2922no—
420#95—2922no—
421#95—738no—
422#95—734no—
423#95—1464no—
424#95—1462no—
425#95—1464no—
426#95—1464no—
427#95—1462no—
428C24—22no—
429C24—21no—
430C24—22no—
431C24—21no—
432C24—21no—
433C24—22no—
434C24—21no—
435C24—21no—
436C24—22no—
437C24—21no—
438C24C2×C4×D7312yes—
439C24C23×D7311yescharacteristic
440C2×Q8—14616no—
441C2×Q8—1468no—
442C2×Q8—1468no—
443C2×Q8—14616no—
444C2×Q8—1468no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—734no—
465C22×C4—732no—
466C22×C4—734no—
467C22×C4—734no—
468C22×C4—734no—
469C22×C4—734no—
470C22×C4—732no—
471C22×C4—1461no—
472C22×C4—1461no—
473C22×C4—1462no—
474C22×C4—1462no—
475C22×C4—1462no—
476C22×C4—1462no—
477C22×C4—1461no—
478C22×C4—1461no—
479C22×C4—1461no—
480C22×C4—1461no—
481C22×C4—1462no—
482C22×C4—1462no—
483C22×C4—1462no—
484C22×C4—1461no—
485C22×C4—1462no—
486C22×C4—1462no—
487C22×C4—1462no—
488C22×C4—1462no—
489C22×C4—1461no—
490C22×C4—1462no—
491C22×C4—1462no—
492C22×C4—1462no—
493C22×C4—1462no—
494C22×C4—1462no—
495C22×C4—1462no—
496C22×C4—1461no—
497C22×C4—1461no—
498C22×C4C2×C4×D7312yes—
499C22×C4C2×C4×D7312yes—
500C22×C4D4×D7314yes—
501C22×C4D4×D7312yes—
502C22×C4#1192402712yes—
503C4⋊C4—28no—
504C4⋊C4—2928no—
505C4⋊C4—2924no—
506C4⋊C4—14616no—
507C22⋊C4—28no—
508C22⋊C4—48no—
509C22⋊C4—2924no—
510C22⋊C4—2924no—
511C22⋊C4—14616no—
512C42—1468no—
513C42—14616no—
514C23—24no—
515C23—22no—
516C23—22no—
517C23—24no—
518C23—22no—
519C23—41no—
520C23—42no—
521C23—42no—
522C23—41no—
523C23—41no—
524C23—42no—
525C23—42no—
526C23—41no—
527C23—42no—
528C23—41no—
529C23—41no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—41no—
535C23—41no—
536C23—42no—
537C23C22×C4×D7311yescharacteristic
538C23C2×D4×D7311yescharacteristic
539C23C2×D4×D7312yes—
540C23#41×D7312yes—
541C23C2×#1192402711yescharacteristic
542C23#1455663612yes—
543C23#1455663611yescharacteristic
544C23#1455663812yes—
545C23#1455663811yescharacteristic
546C23#1455663811yescharacteristic
547C23#1455663911yescharacteristic
548Q8—14632no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—2922no—
573C2×C4—2922no—
574C2×C4—2922no—
575C2×C4—2922no—
576C2×C4—2922no—
577C2×C4—2922no—
578C2×C4—2921no—
579C2×C4—2921no—
580C2×C4—2922no—
581C2×C4—2921no—
582C2×C4—2922no—
583C2×C4—2921no—
584C2×C4—2922no—
585C2×C4—2922no—
586C2×C4—2921no—
587C2×C4—2921no—
588C2×C4—738no—
589C2×C4—734no—
590C2×C4—1462no—
591C2×C4—1462no—
592C2×C4—1464no—
593C2×C4—1464no—
594C2×C4—1464no—
595C2×C4—1462no—
596C2×C4—1462no—
597C2×C4—1464no—
598C2×C4—1464no—
599C2×C4—1462no—
600C2×C4—1464no—
601C2×C4—1464no—
602C2×C4—1464no—
603C2×C4—1464no—
604C2×C4—1464no—
605C2×C4—1464no—
606C2×C4#1455655114yes—
607C22—44no—
608C22—42no—
609C22—44no—
610C22—42no—
611C22—42no—
612C22—81no—
613C22—82no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—82no—
618C22C4×D4×D7312yes—
619C22C2×#1455655111yescharacteristic
620C22#1798906411yescharacteristic
621C22#1798906612yes—
622C22#1798906811yescharacteristic
623C22#120×D7311yescharacteristic
624C22#1798909511yescharacteristic
625C22#1798909712yes—
626C22#1798915011yescharacteristic
627C22#1798915212yes—
628C22#1798915711yescharacteristic
629C22#1798915712yes—
630C22#1798916012yes—
631C22#1798916011yescharacteristic
632C22#1798916412yes—
633C22#1798916611yescharacteristic
634C22#1798919412yes—
635C22#1798919711yescharacteristic
636C22#1798920412yes—
637C22#1798920411yescharacteristic
638C22#1798920512yes—
639C22#1798920512yes—
640C22#1798920712yes—
641C4—24no—
642C4—44no—
643C4—42no—
644C4—82no—
645C4—2922no—
646C4—2924no—
647C4—5841no—
648C4—5841no—
649C4—1464no—
650C4—1468no—
651C2—82no—
652C2#2261243611yescharacteristic
653C2#2261260612yes—
654C2#2261260911yescharacteristic
655C2#2261263712yes—
656C2#2261264111yescharacteristic
657C2#2261564911yescharacteristic
658C2#2261565411yescharacteristic
659C2#2261579311yescharacteristic
660C2#2261579612yes—
661C2#2261584011yescharacteristic
662C2#2261584412yes—
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC73×#1439C2
socle, cosocleC24×C73C24
prime radicalC25×C73C23
hypercenter#1439D73
hypocenterC73#14301
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
1. C22×#14556556C2semidirect
5. #22608201C2semidirect
6. #22608195C2semidirect
7. #22608194C2semidirect
9. #22608157C2semidirect
10. #22608156C2semidirect
23. #17988499C4semidirect
343. C73#14301semidirect

Maximal subgroups

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

subgroupindex
C22×#145565562
C2×#179884992
C2×#179884812
C2×#179884802
#226082012
#226081952
#226081942
#226081742
#226081572
#226081562
C73×#14392
#1430173