Finite Group Databasebeta

← #33008743

Subgroups — #33008743

Order 18688.

1862 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988444C211yesmaximal, characteristic
2C2×#17988434C211yesmaximal, characteristic
3C2×#95×D73C211yesmaximal, characteristic
4#22608190C211yesmaximal, characteristic, complemented
5#22608188C211yesmaximal, characteristic, complemented
6#22608187C211yesmaximal, characteristic, complemented
7#22608156C212yesmaximal, complemented
8#22608155C212yesmaximal, complemented
9#22608152C212yesmaximal
10#22608152C212yesmaximal
11C73×#1442C211yesmaximal, characteristic, complemented
12C23×C4×D73C2211yescharacteristic
13C23×C4×D73C2211yescharacteristic
14C23×C4×D73C2211yescharacteristic
15C22×C4⋊Dic73C2211yescharacteristic
16C22×#11924005C2212yes—
17C22×#11924005C2212yes—
18C22×C4×Dic73C2212yes—
19#17988444—24no—
20#17988434C418yescomplemented
21C2×#14556498C2212yes—
22C2×#14556498C2212yes—
23C2×#14556498C2212yes—
24C2×#14556498C2211yescharacteristic
25C2×#14556498C2212yes—
26C2×#14556498C2211yescharacteristic
27C2×#14556498C2211yescharacteristic
28C2×#14556498C2212yes—
29C2×#14556498C2212yes—
30C2×#14556498C2212yes—
31C2×#14556498C2211yescharacteristic
32#95×D73—24no—
33C22×C4⋊C4×C73C2211yescharacteristic
34C2×#95×C73C2212yes—
35C2×#95×C73C2211yescharacteristic
36C2×#95×C73C2212yes—
37C2×#95×C73C2211yescharacteristic
38C24×D73C2311yescharacteristic
39C23×Dic73C2312yes—
40C23×Dic73C2312yes—
41C23×Dic73C2311yescharacteristic
42C23×Dic73C2311yescharacteristic
43C23×Dic73C2311yescharacteristic
44C22×C4×D73—22no—
45C22×C4×D73—22no—
46C22×C4×D73—22no—
47C22×C4×D73—22no—
48C22×C4×D73—24no—
49C22×C4×D73—24no—
50C22×C4×D73—22no—
51C22×C4×D73—22no—
52C22×C4×D73—42no—
53C22×C4×D73—42no—
54C22×C4×D73—42no—
55C22×C4×D73—42no—
56C22×C4×D73C2×C414yes—
57C22×C4×D73D414yes—
58C22×C4×D73Q814yes—
59C2×C4⋊Dic73—22no—
60C2×C4⋊Dic73—42no—
61C2×#11924005—24no—
62C2×#11924005—24no—
63C2×#11924005—24no—
64C2×#11924005—24no—
65C2×#11924005—24no—
66C2×#11924005C2×C418yes—
67C2×C4×Dic73—42no—
68C2×C4×Dic73—42no—
69C2×C4×Dic73C2×C418yescomplemented
70#14556498—22no—
71#14556498—22no—
72#14556498—24no—
73#14556498—24no—
74#14556498—24no—
75#14556498—22no—
76#14556498—24no—
77#14556498—42no—
78#14556498—42no—
79#14556498—42no—
80#14556498C2×C414yescomplemented
81C23×C4×C73C2312yes—
82C23×C4×C73C2312yes—
83C23×C4×C73C2311yescharacteristic
84C23×C4×C73C2311yescharacteristic
85C23×C4×C73C2311yescharacteristic
86C2×C4⋊C4×C73—22no—
87C2×C4⋊C4×C73—42no—
88#95×C73—22no—
89#95×C73—24no—
90#95×C73—42no—
91#95×C73C2×C414yescomplemented
92C23×D73—21no—
93C23×D73—21no—
94C23×D73—21no—
95C23×D73—21no—
96C23×D73—22no—
97C23×D73—22no—
98C23×D73C22×C412yes—
99C23×D73C2×D412yes—
100C23×D73C2×Q812yes—
101C23×D73#4114yes—
102C23×D73#4114yes—
103C22×Dic73—22no—
104C22×Dic73—22no—
105C22×Dic73—21no—
106C22×Dic73—21no—
107C22×Dic73—22no—
108C22×Dic73—22no—
109C22×Dic73—22no—
110C22×Dic73—22no—
111C22×Dic73—22no—
112C22×Dic73—22no—
113C22×Dic73—21no—
114C22×Dic73—21no—
115C22×Dic73—22no—
116C22×Dic73—22no—
117C22×Dic73—22no—
118C22×Dic73—22no—
119C22×Dic73—22no—
120C22×Dic73—22no—
121C22×Dic73—22no—
122C22×Dic73—22no—
123C22×Dic73—22no—
124C22×Dic73—22no—
125C22×Dic73—22no—
126C22×Dic73C22×C414yes—
127C22×Dic73C22×C412yes—
128C22×Dic73C2×D412yes—
129C22×Dic73C2×Q812yes—
130C22×Dic73#4112yes—
131C22×Dic73#4112yes—
132C2×C4×D73—28no—
133C2×C4×D73—28no—
134C2×C4×D73—48no—
135C2×C4×D73—44no—
136C2×C4×D73—44no—
137C2×C4×D73—48no—
138C2×C4×D73—48no—
139C2×C4×D73—48no—
140C2×C4×D73—48no—
141C2×C4×D73—48no—
142C2×C4×D73—44no—
143C2×C4×D73—44no—
144C2×C4×D73—44no—
145C2×C4×D73—44no—
146C2×C4×D73C4⋊C4116yes—
147C4⋊Dic73—44no—
148#11924005—216no—
149#11924005—48no—
150C4×Dic73—48no—
151C24×C73C2411yescharacteristic
152C22×C4×C73—22no—
153C22×C4×C73—22no—
154C22×C4×C73—21no—
155C22×C4×C73—21no—
156C22×C4×C73—22no—
157C22×C4×C73—22no—
158C22×C4×C73—22no—
159C22×C4×C73—22no—
160C22×C4×C73—21no—
161C22×C4×C73—21no—
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×C73—22no—
175C22×C4×C73C22×C414yes—
176C22×C4×C73C22×C412yes—
177C22×C4×C73C2×D412yes—
178C22×C4×C73C2×Q812yes—
179C22×C4×C73#4112yes—
180C22×C4×C73#4112yes—
181C4⋊C4×C73—44no—
182C23×C73C23×C411yescharacteristic
183C23×C73C22×D411yescharacteristic
184C23×C73C22×Q811yescharacteristic
185C23×C73C2×#4112yes—
186C23×C73C2×#4112yes—
187C23×C73C2×#4111yescharacteristic
188C23×C73C2×#4111yescharacteristic
189C23×C73C2×#4111yescharacteristic
190C23×C73C2×#4111yescharacteristic
191C23×C73H(5,𝔽2)12yes—
192C23×C73H(5,𝔽2)12yes—
193C22×D73—22no—
194C22×D73—22no—
195C22×D73—24no—
196C22×D73—22no—
197C22×D73—22no—
198C22×D73—22no—
199C22×D73—22no—
200C22×D73—22no—
201C22×D73—22no—
202C22×D73—24no—
203C22×D73—24no—
204C22×D73—24no—
205C22×D73—24no—
206C22×D73—24no—
207C22×D73—24no—
208C22×D73—24no—
209C22×D73—24no—
210C22×D73—24no—
211C22×D73C2×C4⋊C414yes—
212C22×D73#11718yes—
213C22×D73#12414yes—
214C22×D73#12414yes—
215C22×D73#12518yes—
216C2×C4×C73—24no—
217C2×C4×C73—22no—
218C2×C4×C73—22no—
219C2×C4×C73—24no—
220C2×C4×C73—24no—
221C2×C4×C73—24no—
222C2×C4×C73—24no—
223C2×C4×C73—22no—
224C2×C4×C73—22no—
225C2×C4×C73—24no—
226C2×C4×C73—24no—
227C2×C4×C73—41no—
228C2×C4×C73—41no—
229C2×C4×C73—41no—
230C2×C4×C73—41no—
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×C73—42no—
240C2×C4×C73—42no—
241C2×C4×C73—42no—
242C2×C4×C73—42no—
243C2×C4×C73—42no—
244C2×C4×C73C2×C4⋊C414yes—
245C2×Dic73—24no—
246C2×Dic73—22no—
247C2×Dic73—22no—
248C2×Dic73—24no—
249C2×Dic73—24no—
250C2×Dic73—24no—
251C2×Dic73—24no—
252C2×Dic73—22no—
253C2×Dic73—22no—
254C2×Dic73—24no—
255C2×Dic73—24no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260C2×Dic73—42no—
261C2×Dic73—42no—
262C2×Dic73—42no—
263C2×Dic73—42no—
264C2×Dic73—42no—
265C2×Dic73—42no—
266C2×Dic73—42no—
267C2×Dic73—42no—
268C2×Dic73—42no—
269C2×Dic73—42no—
270C2×Dic73—42no—
271C2×Dic73—42no—
272C2×Dic73—42no—
273C2×Dic73C2×C4⋊C414yes—
274C4×D73—416no—
275C4×D73—416no—
276C4×D73—416no—
277C22×C73C22×C4⋊C411yescharacteristic
278C22×C73C2×#11712yes—
279C22×C73C4×#4111yescharacteristic
280C22×C73C4×#4111yescharacteristic
281C22×C73#51812yes—
282C22×C73#52511yescharacteristic
283C22×C73#52511yescharacteristic
284C22×C73C2×#12411yescharacteristic
285C22×C73C2×#12411yescharacteristic
286C22×C73C2×#12512yes—
287C22×C73#53311yescharacteristic
288C22×C73#53311yescharacteristic
289C22×C73#53411yescharacteristic
290C22×C73#53411yescharacteristic
291C22×C73#54312yes—
292C22×C73#55112yes—
293C22×C73#55112yes—
294C22×C73#55112yes—
295C22×C73#55112yes—
296C22×C73#55312yes—
297C22×C73#55312yes—
298C22×C73#55312yes—
299C22×C73#55312yes—
300C2×D73—24no—
301C2×D73—24no—
302C2×D73—28no—
303C2×D73—28no—
304C2×D73—24no—
305C2×D73—24no—
306C2×D73—24no—
307C2×D73—28no—
308C2×D73—28no—
309C2×D73—24no—
310C2×D73#38218yes—
311C4×C73—24no—
312C4×C73—42no—
313C4×C73—42no—
314C4×C73—44no—
315C4×C73—82no—
316Dic73—24no—
317Dic73—42no—
318Dic73—42no—
319Dic73—44no—
320Dic73—82no—
321#13966—731nomaximal, Sylow
322C2×C73C2×#38211yescharacteristic
323C2×C73#229611yescharacteristic
324C2×C73#231511yescharacteristic
325C2×C73#235312yes—
326C2×C73#235312yes—
327C2×C73#256111yescharacteristic
328C2×C73#256111yescharacteristic
329C2×C73#256912yes—
330C2×C73#256912yes—
331C2×C73#259111yescharacteristic
332C2×C73#259111yescharacteristic
333D73—28no—
334C2×#380—731no—
335C2×C4×C22⋊C4—731no—
336C22×#95—731no—
337#1442—731no—
338#1442—731no—
339#1442—731no—
340#1442D7311yescharacteristic, complemented
341#1434—732no—
342#1434—732no—
343C22⋊#95—732no—
344C22⋊#95—732no—
345C73#1396611yescharacteristic, complemented, Sylow
346C24×C4—731no—
347C24×C4—731no—
348C24×C4—731no—
349C22×C4⋊C4—731no—
350C22×C4⋊C4C2×D7311yescharacteristic
351C22×C22⋊C4—732no—
352C22×C22⋊C4—732no—
353C22×C42—732no—
354#380—1464no—
355C4×C22⋊C4—738no—
356C2×#95—732no—
357C2×#95—732no—
358C2×#95—732no—
359C2×#95—731no—
360C2×#95—732no—
361C2×#95—732no—
362C2×#95—732no—
363C2×#95—731no—
364C2×#95—731no—
365C2×#95—732no—
366C2×#95—731no—
367C2×#95—1464no—
368C2×#95C2×D7312yes—
369C2×#95C2×D7311yescharacteristic
370C2×#95C2×D7312yes—
371C2×#95C2×D7311yescharacteristic
372C25—731no—
373C23×C4—2922no—
374C23×C4—2922no—
375C23×C4—2922no—
376C23×C4—2922no—
377C23×C4—734no—
378C23×C4—734no—
379C23×C4—734no—
380C23×C4—731no—
381C23×C4—732no—
382C23×C4—731no—
383C23×C4—731no—
384C23×C4—732no—
385C23×C4—1464no—
386C23×C4—1464no—
387C23×C4—1462no—
388C23×C4—1462no—
389C23×C4—1462no—
390C23×C4—1462no—
391C23×C4—1462no—
392C23×C4—1462no—
393C23×C4C22×D7312yes—
394C23×C4C22×D7312yes—
395C23×C4C22×D7311yescharacteristic
396C23×C4C22×D7311yescharacteristic
397C23×C4C22×D7311yescharacteristic
398C2×C4⋊C4—22no—
399C2×C4⋊C4—42no—
400C2×C4⋊C4—2922no—
401C2×C4⋊C4—1462no—
402C2×C22⋊C4—738no—
403C2×C22⋊C4—1464no—
404C2×C22⋊C4—1464no—
405C2×C22⋊C4—1464no—
406C2×C22⋊C4—1464no—
407C2×C22⋊C4—1464no—
408C2×C42—2922no—
409C2×C42—2922no—
410C2×C42—738no—
411#95—22no—
412#95—24no—
413#95—42no—
414#95—2922no—
415#95—2922no—
416#95—2922no—
417#95—734no—
418#95—1464no—
419#95—1464no—
420#95—1462no—
421#95—1462no—
422#95—1464no—
423#95—1462no—
424#95—1464no—
425#95C4×D7314yescomplemented
426C24—732no—
427C24—732no—
428C24—732no—
429C24—734no—
430C24—734no—
431C24—1461no—
432C24—1461no—
433C24—1461no—
434C24—1461no—
435C24—1462no—
436C24—1462no—
437C24C23×D7311yescharacteristic
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
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—2928no—
462C22×C4—2928no—
463C22×C4—2924no—
464C22×C4—2924no—
465C22×C4—2928no—
466C22×C4—2928no—
467C22×C4—2924no—
468C22×C4—2924no—
469C22×C4—2924no—
470C22×C4—2924no—
471C22×C4—2928no—
472C22×C4—2928no—
473C22×C4—7316no—
474C22×C4—732no—
475C22×C4—732no—
476C22×C4—732no—
477C22×C4—734no—
478C22×C4—732no—
479C22×C4—732no—
480C22×C4—1468no—
481C22×C4—1468no—
482C22×C4—1462no—
483C22×C4—1462no—
484C22×C4—1462no—
485C22×C4—1462no—
486C22×C4—1462no—
487C22×C4—1462no—
488C22×C4—1462no—
489C22×C4—1461no—
490C22×C4—1461no—
491C22×C4—1462no—
492C22×C4—1462no—
493C22×C4—1462no—
494C22×C4—1462no—
495C22×C4—1462no—
496C22×C4—1461no—
497C22×C4—1462no—
498C22×C4—1461no—
499C22×C4—1462no—
500C22×C4—1462no—
501C22×C4—1462no—
502C22×C4—1462no—
503C22×C4—1462no—
504C22×C4—1462no—
505C22×C4C2×C4×D7314yes—
506C22×C4C2×C4×D7312yes—
507C22×C4D4×D7312yes—
508C22×C4Q8×D7312yes—
509C22×C4#1192402712yes—
510C22×C4#1192402912yes—
511C4⋊C4—44no—
512C4⋊C4—2924no—
513C22⋊C4—2928no—
514C22⋊C4—14616no—
515C42—2928no—
516C23—734no—
517C23—738no—
518C23—734no—
519C23—734no—
520C23—738no—
521C23—1462no—
522C23—1462no—
523C23—1462no—
524C23—1462no—
525C23—1464no—
526C23—1464no—
527C23—1464no—
528C23—1464no—
529C23—1464no—
530C23—1464no—
531C23—1464no—
532C23—1464no—
533C23—1464no—
534C23—1462no—
535C23—1462no—
536C23—1464no—
537C23—1462no—
538C23—1462no—
539C23C22×C4×D7311yescharacteristic
540C23C2×D4×D7311yescharacteristic
541C23C2×Q8×D7311yescharacteristic
542C23#41×D7311yescharacteristic
543C23#41×D7311yescharacteristic
544C23#41×D7312yes—
545C23#41×D7312yes—
546C23C2×#1192402711yescharacteristic
547C23C2×#1192402911yescharacteristic
548C23#1455663612yes—
549C23#1455663712yes—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—29216no—
579C2×C4—29216no—
580C2×C4—29216no—
581C2×C4—2921no—
582C2×C4—2921no—
583C2×C4—2922no—
584C2×C4—2922no—
585C2×C4—2922no—
586C2×C4—2922no—
587C2×C4—2922no—
588C2×C4—2922no—
589C2×C4—2921no—
590C2×C4—2921no—
591C2×C4—2922no—
592C2×C4—2922no—
593C2×C4—2922no—
594C2×C4—2922no—
595C2×C4—2922no—
596C2×C4—2922no—
597C2×C4—2922no—
598C2×C4—734no—
599C2×C4—1464no—
600C2×C4—1464no—
601C2×C4—1464no—
602C2×C4—1464no—
603C2×C4—1464no—
604C2×C4—1462no—
605C2×C4—1462no—
606C2×C4—1462no—
607C2×C4—1462no—
608C2×C4—1464no—
609C2×C4—1464no—
610C2×C4C4⋊C4×D7314yes—
611C22—738no—
612C22—1464no—
613C22—1464no—
614C22—1464no—
615C22—1468no—
616C22—1464no—
617C22—1468no—
618C22—1464no—
619C22—1468no—
620C22—1468no—
621C22—1464no—
622C22C2×C4⋊C4×D7311yescharacteristic
623C22#117×D7312yes—
624C22C4×#1192402711yescharacteristic
625C22C4×#1192402911yescharacteristic
626C22#1798905912yes—
627C22#1798909711yescharacteristic
628C22#1798910011yescharacteristic
629C22#124×D7311yescharacteristic
630C22#124×D7311yescharacteristic
631C22#125×D7312yes—
632C22#1798914211yescharacteristic
633C22#1798914311yescharacteristic
634C22#1798914711yescharacteristic
635C22#1798914811yescharacteristic
636C22#1798921212yes—
637C22#1798926012yes—
638C22#1798926212yes—
639C22#1798926412yes—
640C22#1798926712yes—
641C22#1798928212yes—
642C22#1798928412yes—
643C22#1798928812yes—
644C22#1798929012yes—
645C4—24no—
646C4—42no—
647C4—42no—
648C4—44no—
649C4—82no—
650C4—2922no—
651C4—2922no—
652C4—2924no—
653C4—5842no—
654C4—1464no—
655C2—1468no—
656C2#382×D7311yescharacteristic
657C2#2261241211yescharacteristic
658C2#2261254211yescharacteristic
659C2#2261286612yes—
660C2#2261287012yes—
661C2#2261508311yescharacteristic
662C2#2261508511yescharacteristic
663C2#2261514912yes—
664C2#2261515112yes—
665C2#2261540511yescharacteristic
666C2#2261540711yescharacteristic
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC73×#1442C2
socle, cosocleC24×C73C24
prime radicalC24×D73C23
hypercenter#1442D73
hypocenterC73#13966
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ #1442

Complements

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

NH ≅ G/Ntype
4. #22608190C2semidirect
5. #22608188C2semidirect
6. #22608187C2semidirect
7. #22608156C2semidirect
8. #22608155C2semidirect
11. C73×#1442C2semidirect
20. #17988434C4semidirect
69. C2×C4×Dic73C2×C4semidirect
80. #14556498C2×C4semidirect
91. #95×C73C2×C4semidirect
340. #1442D73semidirect
345. C73#13966semidirect
425. #95C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179884442
C2×#179884342
C2×#95×D732
#226081902
#226081882
#226081872
#226081562
#226081552
#226081522
C73×#14422
#1396673