Finite Group Databasebeta

← #33093159

Subgroups — #33093159

Order 18688.

888 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×#17988927C211yesmaximal, characteristic
2#22612469C211yesmaximal, characteristic
3C4×#14556549C211yesmaximal, characteristic
4#22610562C211yesmaximal, characteristic, complemented
5#22610509C211yesmaximal, characteristic, complemented
6#22610503C211yesmaximal, characteristic, complemented
7#22610240C211yesmaximal, characteristic
8#22610127C211yesmaximal, characteristic
9#22610025C211yesmaximal, characteristic
10#107×Dic73C211yesmaximal, characteristic, complemented
11#22609819C211yesmaximal, characteristic, complemented
12#103×Dic73C211yesmaximal, characteristic, complemented
13#103×Dic73C211yesmaximal, characteristic, complemented
14#22609688C211yesmaximal, characteristic
15C73×#1928C211yesmaximal, characteristic, complemented
16C2×#14556567C2211yescharacteristic
17C2×Q8×Dic73C2211yescharacteristic
18C2×#14556549C2211yescharacteristic
19C2×#14556549C2211yescharacteristic
20C2×C42×D73C2211yescharacteristic
21#17988927C418yescomplemented
22C2×#14556525C2211yescharacteristic
23C2×#14556522C2211yescharacteristic
24C2×#14556522C2211yescharacteristic
25C2×#14556515C2211yescharacteristic
26C2×#14556515C2211yescharacteristic
27C2×#14556503C2211yescharacteristic
28C2×#14556503C2211yescharacteristic
29C2×C8×Dic73C2211yescharacteristic
30C2×C8×Dic73C2211yescharacteristic
31#17988485C2211yescharacteristic
32#17988474—21no—
33#17988461—21no—
34#17988444C2211yescharacteristic
35C4⋊C4×Dic73C2211yescharacteristic
36C4⋊C4×Dic73C2211yescharacteristic
37C4⋊C4×Dic73C2211yescharacteristic
38C4×C4⋊Dic73C2211yescharacteristic
39#17988438—21no—
40C4×#11924005—21no—
41#17988434—21no—
42#17988434—21no—
43#17988432—21no—
44C42×Dic73—21no—
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988350C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988349C2211yescharacteristic
52#17988342C2211yescharacteristic
53C2×#107×C73C2211yescharacteristic
54C2×#103×C73C2211yescharacteristic
55C2×#103×C73C2211yescharacteristic
56#391×C73C2211yescharacteristic
57C4×C4⋊C4×C73C2211yescharacteristic
58#340×C73C2211yescharacteristic
59#336×C73C2211yescharacteristic
60C22×C4×D73D411yescharacteristic
61C22×C4×D73D411yescharacteristic
62C22×C4×D73C2311yescharacteristic
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65#14556567—22no—
66#14556567C2×C414yes—
67Q8×Dic73—22no—
68Q8×Dic73—22no—
69#14556549—22no—
70#14556549—22no—
71#14556549—22no—
72#14556549C2×C414yes—
73C2×C4⋊Dic73—21no—
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2311yescharacteristic
77C2×C4⋊Dic73C2311yescharacteristic
78C2×#11924007—21no—
79C2×#11924007—21no—
80C2×#11924005—21no—
81C2×#11924005—21no—
82C2×#11924005—21no—
83C2×#11924005—21no—
84C2×C4×Dic73—21no—
85C2×C4×Dic73—21no—
86C2×C4×Dic73—21no—
87C2×C4×Dic73—21no—
88C2×C4×Dic73—21no—
89C2×C4×Dic73—21no—
90C2×C4×Dic73—21no—
91C2×C4×Dic73D411yescharacteristic
92C2×C4×Dic73D411yescharacteristic
93C2×C4×Dic73C2311yescharacteristic
94C42×D73—22no—
95C42×D73—22no—
96#14556525C2×C414yescomplemented
97#14556522—22no—
98#14556522C2×C414yescomplemented
99#14556515—22no—
100#14556515C2×C414yescomplemented
101#14556503—22no—
102#14556503C2×C414yes—
103C8×Dic73—22no—
104C8×Dic73—22no—
105#14556498—21no—
106#14556498—21no—
107#14556498—21no—
108#14556498—21no—
109#14556498—21no—
110#14556498—21no—
111C22×Q8×C73C2311yescharacteristic
112C22×C8×C73C2311yescharacteristic
113C22×C8×C73C2311yescharacteristic
114C2×C4⋊C4×C73C2311yescharacteristic
115C2×C4⋊C4×C73C2311yescharacteristic
116C2×C4⋊C4×C73C2311yescharacteristic
117C2×C42×C73—21no—
118C2×C42×C73C2311yescharacteristic
119#107×C73—22no—
120#103×C73—22no—
121#103×C73C2×C414yescomplemented
122#95×C73—21no—
123#95×C73—21no—
124#95×C73—21no—
125C23×D73C2×D411yescharacteristic
126C22×Dic73—21no—
127C22×Dic73—21no—
128C22×Dic73—21no—
129C22×Dic73—21no—
130C22×Dic73C2×D411yescharacteristic
131C22×Dic73C2×D411yescharacteristic
132C22×Dic73C2×D411yescharacteristic
133C2×C4×D73—21no—
134C2×C4×D73—21no—
135C2×C4×D73—21no—
136C2×C4×D73—21no—
137C2×C4×D73—22no—
138C2×C4×D73—22no—
139C2×C4×D73—22no—
140C2×C4×D73—22no—
141C2×C4×D73—22no—
142C2×C4×D73—41no—
143C2×C4×D73C22×C412yes—
144C2×C4×D73#4112yes—
145C2×#10596831—21no—
146C2×#10596831—21no—
147C2×#10596831—21no—
148C2×#10596831—21no—
149C2×#10596831—21no—
150C2×#10596831C22×C412yes—
151C4⋊Dic73—21no—
152C4⋊Dic73—21no—
153C4⋊Dic73—21no—
154C4⋊Dic73—21no—
155C4⋊Dic73—21no—
156C4⋊Dic73—42no—
157C4⋊Dic73C22×C412yes—
158C4⋊Dic73C22×C412yes—
159C4⋊Dic73#4112yes—
160#11924007—24no—
161#11924007—42no—
162#11924005—24no—
163#11924005—24no—
164#11924005—24no—
165#11924005—42no—
166C4×Dic73—24no—
167C4×Dic73—24no—
168C4×Dic73—22no—
169C4×Dic73—22no—
170C4×Dic73—42no—
171C4×Dic73—42no—
172C4×Dic73—42no—
173C4×Dic73—42no—
174C4×Dic73—42no—
175C4×Dic73#4112yes—
176C4×Dic73#4112yes—
177C2×Q8×C73—21no—
178C2×Q8×C73—21no—
179C2×Q8×C73—21no—
180C2×Q8×C73—21no—
181C2×Q8×C73—21no—
182C2×Q8×C73C22×C412yes—
183C22×C4×C73—21no—
184C22×C4×C73—21no—
185C22×C4×C73—21no—
186C22×C4×C73—21no—
187C22×C4×C73C2×D411yescharacteristic
188C22×C4×C73C2×D411yescharacteristic
189C22×C4×C73C2411yescharacteristic
190C2×C8×C73—21no—
191C2×C8×C73—21no—
192C2×C8×C73—21no—
193C2×C8×C73—21no—
194C2×C8×C73—21no—
195C2×C8×C73C22×C412yes—
196C4⋊C4×C73—21no—
197C4⋊C4×C73—21no—
198C4⋊C4×C73—21no—
199C4⋊C4×C73—21no—
200C4⋊C4×C73C22×C412yes—
201C4⋊C4×C73#4112yes—
202C42×C73—21no—
203C42×C73—21no—
204C42×C73—42no—
205C23×C73C22×D411yescharacteristic
206C22×D73—21no—
207C22×D73—21no—
208C22×D73—21no—
209C22×D73—21no—
210C22×D73—21no—
211C22×D73—21no—
212C22×D73C4×D412yes—
213Q8×C73—22no—
214Q8×C73—22no—
215Q8×C73—41no—
216Q8×C73—41no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—22no—
220C2×C4×C73—22no—
221C2×C4×C73—21no—
222C2×C4×C73—22no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—41no—
226C2×C4×C73—41no—
227C2×C4×C73—41no—
228C2×C4×C73—41no—
229C2×C4×C73—41no—
230C2×C4×C73—41no—
231C2×C4×C73—41no—
232C2×C4×C73—41no—
233C2×C4×C73—41no—
234C2×C4×C73#13512yes—
235C2×C4×C73C23×C411yescharacteristic
236C2×C4×C73C2×#4111yescharacteristic
237C2×C4×C73C2×#4111yescharacteristic
238C2×C4×C73C2×#4111yescharacteristic
239C2×C4×C73C2×#4111yescharacteristic
240C2×C4×C73C2×#4111yescharacteristic
241C2×Dic73—21no—
242C2×Dic73—21no—
243C2×Dic73—21no—
244C2×Dic73—21no—
245C2×Dic73—22no—
246C2×Dic73—22no—
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—
256C2×Dic73—41no—
257C2×Dic73C4×D412yes—
258C2×Dic73#12312yes—
259C2×Dic73#12312yes—
260C2×Dic73#13512yes—
261C2×Dic73#13712yes—
262C4×D73—22no—
263C4×D73—22no—
264C4×D73—22no—
265C4×D73—22no—
266C4×D73—44no—
267C4×D73—44no—
268C8×C73—22no—
269C8×C73—41no—
270#10596831—22no—
271#10596831—41no—
272C22×C73C2×C4×D411yescharacteristic
273C22×C73C2×#12311yescharacteristic
274C22×C73#52511yescharacteristic
275C22×C73C2×#13511yescharacteristic
276C22×C73C2×#13511yescharacteristic
277C22×C73C2×#13711yescharacteristic
278C22×C73#57511yescharacteristic
279C2×D73—22no—
280C2×D73—22no—
281C2×D73—22no—
282C2×D73—41no—
283C2×D73—41no—
284C2×D73—41no—
285C2×D73—41no—
286C4×C73—22no—
287C4×C73—41no—
288C4×C73—42no—
289C4×C73—42no—
290C4×C73—42no—
291C4×C73—81no—
292C4×C73C4×#4111yescharacteristic
293C4×C73C4×#4111yescharacteristic
294C4×C73#52911yescharacteristic
295C4×C73#52911yescharacteristic
296Dic73—22no—
297Dic73—22no—
298Dic73—42no—
299Dic73—42no—
300Dic73—42no—
301Dic73—81no—
302Dic73#48514yescomplemented
303C4×#485—731nomaximal, Sylow
304C2×C73C4×#12311yescharacteristic
305C2×C73C4×#13511yescharacteristic
306C2×C73C4×#13711yescharacteristic
307C2×C73C2×#48511yescharacteristic
308C2×C73#309111yescharacteristic
309C2×C73#310211yescharacteristic
310C2×C73#310711yescharacteristic
311D73—42no—
312C2×#485—731no—
313C4×C22⋊Q8—731no—
314C4×#117—731no—
315#1928D7311yescharacteristic, complemented
316#1927—731no—
317#1919—731no—
318#1918—731no—
319#1873—731no—
320#1854—731no—
321#1825—731no—
322C4×#107—731no—
323C4×#106—731no—
324C4×#103—731no—
325C4×#103—731no—
326C4×C22⋊C8—731no—
327C73C4×#48511yescharacteristic, complemented, Sylow
328C2×C22⋊Q8—731no—
329C2×C4×Q8—731no—
330C2×#117—731no—
331C2×#117—731no—
332C22×C42—731no—
333#485—738no—
334C2×#107—731no—
335C2×#107C2×D7311yescharacteristic
336C2×#106—731no—
337C2×#106—731no—
338C2×#103—731no—
339C2×#103—731no—
340C2×#103C2×D7311yescharacteristic
341C2×#103C2×D7311yescharacteristic
342C2×C22⋊C8—731no—
343C2×C22⋊C8—731no—
344C2×C4×C8—731no—
345C2×C4×C8—731no—
346#391C2×D7311yescharacteristic
347#389—731no—
348#387—1461no—
349#385—1461no—
350#380—731no—
351C4×C4⋊C4—731no—
352C4×C4⋊C4—731no—
353C4×C4⋊C4—731no—
354C4×C4⋊C4—731no—
355C4×C4⋊C4—1461no—
356C4×C4⋊C4C2×D7311yescharacteristic
357C4×C22⋊C4—1461no—
358C4×C22⋊C4—1461no—
359C4×C22⋊C4—1461no—
360#376—1461no—
361C43—1461no—
362#340—731no—
363#340—731no—
364#340—731no—
365#340—731no—
366#340—731no—
367#340—731no—
368#340—731no—
369#340C2×D7311yescharacteristic
370#336—731no—
371#336C2×D7311yescharacteristic
372C22×Q8C22×D7311yescharacteristic
373C23×C4—731no—
374C23×C4—731no—
375C23×C4—731no—
376C22×C8—731no—
377C22×C8—731no—
378C22×C8C22×D7311yescharacteristic
379C22×C8C22×D7311yescharacteristic
380C22⋊Q8—734no—
381C22⋊Q8—1462no—
382C4×Q8—1462no—
383C4×Q8—1462no—
384#117—734no—
385#117—1462no—
386#117—1462no—
387#117—1462no—
388C2×C4⋊C4—731no—
389C2×C4⋊C4—731no—
390C2×C4⋊C4—731no—
391C2×C4⋊C4—731no—
392C2×C4⋊C4—1461no—
393C2×C4⋊C4—1461no—
394C2×C4⋊C4—1461no—
395C2×C4⋊C4C22×D7311yescharacteristic
396C2×C4⋊C4C22×D7311yescharacteristic
397C2×C4⋊C4C22×D7311yescharacteristic
398C2×C22⋊C4—1461no—
399C2×C22⋊C4—1461no—
400C2×C22⋊C4—1461no—
401C2×C22⋊C4—1461no—
402C2×C42—21no—
403C2×C42—731no—
404C2×C42—731no—
405C2×C42—731no—
406C2×C42—1462no—
407C2×C42—1462no—
408C2×C42—1461no—
409C2×C42—1461no—
410C2×C42—1461no—
411C2×C42—1461no—
412C2×C42—1461no—
413C2×C42—1461no—
414C2×C42—1461no—
415C2×C42C22×D7311yescharacteristic
416#107—22no—
417#107—734no—
418#106—734no—
419#106—1462no—
420#103—22no—
421#103—734no—
422#103—1462no—
423#103C4×D7314yescomplemented
424C22⋊C8—734no—
425C22⋊C8—1462no—
426C4×C8—1462no—
427C4×C8—1462no—
428#95—21no—
429#95—21no—
430#95—21no—
431#95—1461no—
432#95—1461no—
433#95—1461no—
434#95—1461no—
435#95—1461no—
436#95—1461no—
437C24—731no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8C2×C4×D7312yes—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—2921no—
449C22×C4—731no—
450C22×C4—731no—
451C22×C4—731no—
452C22×C4—732no—
453C22×C4—732no—
454C22×C4—1462no—
455C22×C4—1462no—
456C22×C4—1462no—
457C22×C4—1462no—
458C22×C4—1461no—
459C22×C4—1461no—
460C22×C4—1461no—
461C22×C4—1461no—
462C22×C4—1462no—
463C22×C4—1461no—
464C22×C4—1461no—
465C22×C4—1461no—
466C22×C4—1461no—
467C22×C4D4×D7311yescharacteristic
468C22×C4D4×D7311yescharacteristic
469C22×C4C23×D7311yescharacteristic
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—732no—
476C2×C8—1461no—
477C2×C8—1461no—
478C2×C8—1461no—
479C2×C8—1461no—
480C2×C8—1461no—
481C2×C8C2×C4×D7312yes—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—2922no—
487C4⋊C4—2922no—
488C4⋊C4—732no—
489C4⋊C4—732no—
490C4⋊C4—732no—
491C4⋊C4—1464no—
492C4⋊C4—1461no—
493C4⋊C4—1461no—
494C4⋊C4—1461no—
495C4⋊C4—1461no—
496C4⋊C4—1461no—
497C4⋊C4C2×C4×D7312yes—
498C4⋊C4#1192402712yes—
499C22⋊C4—2922no—
500C22⋊C4—1464no—
501C22⋊C4—1464no—
502C22⋊C4—1464no—
503C42—21no—
504C42—21no—
505C42—42no—
506C42—2922no—
507C42—2922no—
508C42—2922no—
509C42—2922no—
510C42—2922no—
511C42—732no—
512C42—732no—
513C42—1462no—
514C42—1462no—
515C42—1464no—
516C42—1464no—
517C23—732no—
518C23—1461no—
519C23—1461no—
520C23—1461no—
521C23—1461no—
522C23—1461no—
523C23—1461no—
524C23C2×D4×D7311yescharacteristic
525Q8—22no—
526Q8—22no—
527Q8—41no—
528Q8—41no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—2924no—
547C2×C4—2924no—
548C2×C4—2921no—
549C2×C4—2921no—
550C2×C4—2921no—
551C2×C4—2921no—
552C2×C4—2921no—
553C2×C4—2921no—
554C2×C4—2921no—
555C2×C4—2921no—
556C2×C4—2921no—
557C2×C4—732no—
558C2×C4—732no—
559C2×C4—732no—
560C2×C4—732no—
561C2×C4—732no—
562C2×C4—1462no—
563C2×C4—1462no—
564C2×C4—1462no—
565C2×C4—1461no—
566C2×C4—1461no—
567C2×C4—1461no—
568C2×C4—1461no—
569C2×C4—1462no—
570C2×C4—1462no—
571C2×C4—1462no—
572C2×C4—1462no—
573C2×C4#1455661412yes—
574C2×C4C22×C4×D7311yescharacteristic
575C2×C4#41×D7311yescharacteristic
576C2×C4#41×D7311yescharacteristic
577C2×C4#41×D7311yescharacteristic
578C2×C4#41×D7311yescharacteristic
579C2×C4C2×#1192402711yescharacteristic
580C8—22no—
581C8—41no—
582C8—2921no—
583C8—1462no—
584C22—2921no—
585C22—2921no—
586C22—2921no—
587C22—2921no—
588C22—1462no—
589C22—1462no—
590C22—1462no—
591C22C4×D4×D7311yescharacteristic
592C22#123×D7311yescharacteristic
593C22#1798909711yescharacteristic
594C22#135×D7311yescharacteristic
595C22C2×#1455661411yescharacteristic
596C22#137×D7311yescharacteristic
597C22#1798938311yescharacteristic
598C4—22no—
599C4—42no—
600C4—42no—
601C4—41no—
602C4—42no—
603C4—81no—
604C4—2922no—
605C4—2922no—
606C4—2922no—
607C4—5841no—
608C4—734no—
609C4—1462no—
610C4—1462no—
611C4C4×#1192402711yescharacteristic
612C4#1798905511yescharacteristic
613C4#1798911611yescharacteristic
614C4#1798912011yescharacteristic
615C2—2922no—
616C2#2261245811yescharacteristic
617C2C4×#1455661411yescharacteristic
618C2#2261875311yescharacteristic
619C2#485×D7311yescharacteristic
620C2#2261984711yescharacteristic
621C2#2261994611yescharacteristic
622C2#2262000111yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1928C2
socleC23×C73C22×D4
prime radicalC23×D73C2×D4
cosocleC22×C4×C73C24
hypercenter#1928D73
hypocenterC73C4×#485
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1928

Complements

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

NH ≅ G/Ntype
4. #22610562C2semidirect
5. #22610509C2semidirect
6. #22610503C2semidirect
10. #107×Dic73C2semidirect
11. #22609819C2semidirect
12. #103×Dic73C2semidirect
13. #103×Dic73C2semidirect
15. C73×#1928C2semidirect
21. #17988927C4semidirect
96. #14556525C2×C4semidirect
98. #14556522C2×C4semidirect
100. #14556515C2×C4semidirect
121. #103×C73C2×C4semidirect
302. Dic73#485semidirect
315. #1928D73semidirect
327. C73C4×#485semidirect
423. #103C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179889272
#226124692
C4×#145565492
#226105622
#226105092
#226105032
#226102402
#226101272
#226100252
#107×Dic732
#226098192
#103×Dic732
#226096882
C73×#19282
C4×#48573