Finite Group Databasebeta

← #33089130

Subgroups — #33089130

Order 18688.

890 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988672C211yesmaximal, characteristic
2#22612576C211yesmaximal, characteristic, complemented
3#22612291C211yesmaximal, characteristic
4#22610574C211yesmaximal, characteristic, complemented
5#22610516C211yesmaximal, characteristic, complemented
6#22610514C211yesmaximal, characteristic, complemented
7#22610268C211yesmaximal, characteristic
8#22610262C211yesmaximal, characteristic
9#22610181C211yesmaximal, characteristic
10#22610175C211yesmaximal, characteristic
11#107×Dic73C211yesmaximal, characteristic, complemented
12#22609763C211yesmaximal, characteristic, complemented
13#22609755C211yesmaximal, characteristic
14#22609681C211yesmaximal, characteristic, complemented
15C73×#1928C211yesmaximal, characteristic, complemented
16C22×#11924015C2211yescharacteristic
17C2×Q8×Dic73C2211yescharacteristic
18C2×C4×D292C2211yescharacteristic
19C2×#14556553C2211yescharacteristic
20C2×#14556553C2211yescharacteristic
21#17988672C418yescomplemented
22C2×#14556525C2211yescharacteristic
23C2×#14556522C2211yescharacteristic
24C2×#14556515C2211yescharacteristic
25C2×#14556513C2211yescharacteristic
26C2×#14556513C2211yescharacteristic
27C2×#14556511C2211yescharacteristic
28C2×#14556501C2211yescharacteristic
29C2×C8×Dic73C2211yescharacteristic
30#17988494C2211yescharacteristic
31#17988485C2211yescharacteristic
32#17988483C2211yescharacteristic
33#17988472—21no—
34#17988470—21no—
35#17988454—21no—
36#17988451—21no—
37C4⋊C4×Dic73C2211yescharacteristic
38C4⋊C4×Dic73C2211yescharacteristic
39C4⋊C4×Dic73C2211yescharacteristic
40C4×#11924005—21no—
41C4×#11924005—21no—
42#17988434—21no—
43#17988433C2211yescharacteristic
44#95×D73—21no—
45#17988350C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988342C2211yescharacteristic
51#17988341C2211yescharacteristic
52#17988341C2211yescharacteristic
53C2×#107×C73C2211yescharacteristic
54C2×#103×C73C2211yescharacteristic
55C2×#103×C73C2211yescharacteristic
56#391×C73C2211yescharacteristic
57C4×C4⋊C4×C73C2211yescharacteristic
58#340×C73C2211yescharacteristic
59#336×C73C2211yescharacteristic
60C22×D292C2311yescharacteristic
61C22×C4×D73—21no—
62C22×C4×D73—21no—
63C22×C4×D73—21no—
64C2×#11924015—22no—
65C2×#11924015—22no—
66C2×#11924015C2×C414yes—
67C22×#10596831C2311yescharacteristic
68C22×#10596831C2311yescharacteristic
69Q8×Dic73—22no—
70Q8×Dic73C2×C414yescomplemented
71C4×D292—22no—
72C4×D292—22no—
73#14556553—22no—
74#14556553—22no—
75#14556553—22no—
76#14556553C2×C414yes—
77C2×C4⋊Dic73—21no—
78C2×C4⋊Dic73C2311yescharacteristic
79C2×C4⋊Dic73C2311yescharacteristic
80C2×C4⋊Dic73C2311yescharacteristic
81C2×#11924007—21no—
82C2×#11924007—21no—
83C2×#11924005—21no—
84C2×#11924005—21no—
85C2×#11924005—21no—
86C2×C4×Dic73—21no—
87C2×C4×Dic73—21no—
88C2×C4×Dic73—21no—
89C2×C4×Dic73—21no—
90C2×C4×Dic73C2311yescharacteristic
91C2×C4×Dic73C2311yescharacteristic
92#14556525—22no—
93#14556522C2×C414yescomplemented
94#14556515—22no—
95#14556513—22no—
96#14556513C2×C414yes—
97#14556511—22no—
98#14556501—22no—
99C8×Dic73C2×C414yescomplemented
100#14556498—21no—
101#14556498—21no—
102#14556498—21no—
103#14556498—21no—
104#14556498—21no—
105#14556498—21no—
106#14556498—21no—
107#14556498D411yescharacteristic
108#14556498D411yescharacteristic
109#14556498D411yescharacteristic
110#14556498D411yescharacteristic
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×D73—21no—
126C2×D292—21no—
127C2×D292—21no—
128C2×D292—21no—
129C2×D292C22×C412yes—
130C2×D292#4112yes—
131C2×D292#4112yes—
132C22×Dic73—21no—
133C22×Dic73—21no—
134C22×Dic73—21no—
135C22×Dic73C2×D411yescharacteristic
136C22×Dic73C2×D411yescharacteristic
137C22×Dic73C2×D411yescharacteristic
138C22×Dic73C2×D411yescharacteristic
139C2×C4×D73—24no—
140C2×C4×D73—24no—
141C2×C4×D73—24no—
142C2×C4×D73—81no—
143C2×C4×D73—81no—
144C2×C4×D73—81no—
145C2×C4×D73—81no—
146C2×C4×D73—81no—
147C2×C4×D73—81no—
148#11924015—28no—
149C2×#10596831—21no—
150C2×#10596831—21no—
151C2×#10596831—21no—
152C2×#10596831—21no—
153C2×#10596831—21no—
154C2×#10596831C22×C412yes—
155C4⋊Dic73—21no—
156C4⋊Dic73—21no—
157C4⋊Dic73—24no—
158C4⋊Dic73—21no—
159C4⋊Dic73—21no—
160C4⋊Dic73C22×C412yes—
161C4⋊Dic73#4112yes—
162#11924007—81no—
163#11924007—81no—
164#11924005—24no—
165#11924005—24no—
166#11924005—24no—
167C4×Dic73—21no—
168C4×Dic73—21no—
169C4×Dic73—24no—
170C4×Dic73—81no—
171C4×Dic73—81no—
172C4×Dic73—81no—
173C4×Dic73C22×C412yes—
174C4×Dic73#4112yes—
175C2×Q8×C73—21no—
176C2×Q8×C73—21no—
177C2×Q8×C73—21no—
178C2×Q8×C73—21no—
179C2×Q8×C73—21no—
180C2×Q8×C73C22×C412yes—
181C22×C4×C73—21no—
182C22×C4×C73—21no—
183C22×C4×C73—21no—
184C22×C4×C73—21no—
185C22×C4×C73C2×D411yescharacteristic
186C22×C4×C73C2×D411yescharacteristic
187C22×C4×C73C2411yescharacteristic
188C2×C8×C73—21no—
189C2×C8×C73—21no—
190C2×C8×C73—21no—
191C2×C8×C73—21no—
192C2×C8×C73—21no—
193C2×C8×C73C22×C412yes—
194C4⋊C4×C73—21no—
195C4⋊C4×C73—21no—
196C4⋊C4×C73—21no—
197C4⋊C4×C73—21no—
198C4⋊C4×C73C22×C412yes—
199C4⋊C4×C73#4112yes—
200C42×C73—21no—
201C42×C73—21no—
202C42×C73—81no—
203C23×C73C22×D411yescharacteristic
204C22×D73—22no—
205C22×D73—22no—
206C22×D73—22no—
207C22×D73—41no—
208C22×D73—41no—
209C22×D73—41no—
210C22×D73—41no—
211Q8×C73—22no—
212Q8×C73—22no—
213Q8×C73—41no—
214Q8×C73—41no—
215C2×C4×C73—22no—
216C2×C4×C73—22no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—22no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—41no—
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×C73C23×C411yescharacteristic
234C2×C4×C73C2×#4111yescharacteristic
235C2×C4×C73C2×#4111yescharacteristic
236C2×C4×C73C2×#4111yescharacteristic
237C2×C4×C73C2×#4111yescharacteristic
238C2×C4×C73C2×#4111yescharacteristic
239C2×Dic73—21no—
240C2×Dic73—21no—
241C2×Dic73—21no—
242C2×Dic73—21no—
243C2×Dic73—22no—
244C2×Dic73—21no—
245C2×Dic73—21no—
246C2×Dic73—22no—
247C2×Dic73—21no—
248C2×Dic73—21no—
249C2×Dic73—21no—
250C2×Dic73—21no—
251C2×Dic73—41no—
252C2×Dic73—41no—
253C2×Dic73—41no—
254C2×Dic73—41no—
255C2×Dic73—41no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73C4×D412yes—
259C2×Dic73C4×D412yes—
260D292—22no—
261D292—22no—
262D292—22no—
263D292#11714yes—
264C4×D73—84no—
265C4×D73—84no—
266C4×D73—84no—
267C8×C73—22no—
268C8×C73—41no—
269#10596831—22no—
270#10596831—41no—
271C22×C73C2×C4×D411yescharacteristic
272C22×C73#52511yescharacteristic
273C22×C73#52511yescharacteristic
274C22×C73#57611yescharacteristic
275C22×C73#57711yescharacteristic
276C22×C73#57711yescharacteristic
277C22×C73#57811yescharacteristic
278C2×D73—24no—
279C2×D73—42no—
280C2×D73—42no—
281C2×D73—42no—
282C2×D73—42no—
283C2×D73—42no—
284C2×D73—42no—
285C4×C73—41no—
286C4×C73—42no—
287C4×C73—41no—
288C4×C73—42no—
289C4×C73—42no—
290C4×C73—81no—
291C4×C73C2×#11711yescharacteristic
292C4×C73C4×#4111yescharacteristic
293C4×C73#52911yescharacteristic
294C4×C73#52911yescharacteristic
295Dic73—22no—
296Dic73—22no—
297Dic73—41no—
298Dic73—42no—
299Dic73—41no—
300Dic73—42no—
301Dic73—81no—
302#21130—731nomaximal, Sylow
303C2×C73#229011yescharacteristic
304C2×C73#294911yescharacteristic
305C2×C73#295111yescharacteristic
306C2×C73#311811yescharacteristic
307C2×C73#311811yescharacteristic
308C2×C73#312011yescharacteristic
309C2×C73#312011yescharacteristic
310D73—44no—
311C2×C4×SD16—731no—
312#2320—731no—
313#2275—731no—
314#1928—731no—
315#1928D7311yescharacteristic, complemented
316#1920—731no—
317#1920—731no—
318#1877—731no—
319#1877—731no—
320#1862—731no—
321#1862—731no—
322C4×#107—731no—
323#1763—731no—
324#1762—731no—
325#1746—731no—
326C73#2113011yescharacteristic, complemented, Sylow
327C22×SD16—731no—
328C2×C4×Q8—731no—
329C2×C4×D4—731no—
330C2×C4×D4—731no—
331C2×C4×D4—731no—
332C4×SD16—738no—
333C2×#107—731no—
334C2×#107C2×D7311yescharacteristic
335C2×#106—731no—
336C2×#103—731no—
337C2×#103C2×D7311yescharacteristic
338C2×#103C2×D7311yescharacteristic
339C2×#102—731no—
340C2×#102—731no—
341C2×#102—731no—
342C2×#97—731no—
343C2×C4×C8—731no—
344#391—731no—
345#391C2×D7311yescharacteristic
346#389—731no—
347#389—731no—
348#387—1461no—
349#387—1461no—
350#383—1461no—
351#382—1461no—
352C4×C4⋊C4—731no—
353C4×C4⋊C4—731no—
354C4×C4⋊C4—731no—
355C4×C4⋊C4C2×D7311yescharacteristic
356C4×C22⋊C4—1461no—
357C4×C22⋊C4—1461no—
358C4×C22⋊C4—1461no—
359#376—731no—
360C2×#95—1461no—
361#340—731no—
362#340—731no—
363#340—731no—
364#340—731no—
365#340—731no—
366#340C2×D7311yescharacteristic
367#336—731no—
368#336—731no—
369#336—731no—
370#336C2×D7311yescharacteristic
371C22×Q8C22×D7311yescharacteristic
372C22×D4—731no—
373C23×C4—1461no—
374C23×C4—1461no—
375C23×C4—1461no—
376C2×SD16—734no—
377C2×SD16—1462no—
378C2×SD16—1462no—
379C22×C8—731no—
380C22×C8—731no—
381C22×C8C22×D7311yescharacteristic
382C22×C8C22×D7311yescharacteristic
383C4×Q8—734no—
384C4×Q8—1462no—
385C4×D4—734no—
386C4×D4—1462no—
387C4×D4—1462no—
388C4×D4—1462no—
389C4×D4—1462no—
390C4×D4—1462no—
391C2×C4⋊C4—731no—
392C2×C4⋊C4—731no—
393C2×C4⋊C4—731no—
394C2×C4⋊C4—1461no—
395C2×C4⋊C4—1461no—
396C2×C4⋊C4—1461no—
397C2×C4⋊C4C22×D7311yescharacteristic
398C2×C4⋊C4C22×D7311yescharacteristic
399C2×C4⋊C4C22×D7311yescharacteristic
400C2×C22⋊C4—1461no—
401C2×C22⋊C4—1461no—
402C2×C22⋊C4—1461no—
403C2×C42—21no—
404C2×C42—731no—
405C2×C42—731no—
406C2×C42—1461no—
407C2×C42—1461no—
408C2×C42—1461no—
409C2×C42—1461no—
410C2×C42C22×D7311yescharacteristic
411#107—22no—
412#107—1462no—
413#106—734no—
414#103—22no—
415#103—1462no—
416#103C4×D7314yescomplemented
417#102—734no—
418#102—1462no—
419#102—1462no—
420#97—1462no—
421C4×C8—734no—
422#95—21no—
423#95—21no—
424#95—21no—
425#95—731no—
426#95—731no—
427#95—731no—
428#95—731no—
429#95—1461no—
430#95—1461no—
431#95—1461no—
432#95—1461no—
433#95—1461no—
434#95—1461no—
435#95—1461no—
436C24—1461no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8C2×C4×D7312yes—
443C2×D4—732no—
444C2×D4—732no—
445C2×D4—732no—
446C2×D4—1461no—
447C2×D4—1461no—
448C2×D4—1461no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—5841no—
454C22×C4—5841no—
455C22×C4—5841no—
456C22×C4—5841no—
457C22×C4—5841no—
458C22×C4—5841no—
459C22×C4—731no—
460C22×C4—731no—
461C22×C4—731no—
462C22×C4—731no—
463C22×C4—1464no—
464C22×C4—1464no—
465C22×C4—1464no—
466C22×C4—1461no—
467C22×C4—1461no—
468C22×C4—1461no—
469C22×C4D4×D7311yescharacteristic
470C22×C4D4×D7311yescharacteristic
471C22×C4C23×D7311yescharacteristic
472SD16—1468no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—732no—
479C2×C8—1461no—
480C2×C8—1461no—
481C2×C8—1461no—
482C2×C8—1461no—
483C2×C8—1461no—
484C2×C8C2×C4×D7312yes—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—5841no—
490C4⋊C4—5841no—
491C4⋊C4—732no—
492C4⋊C4—732no—
493C4⋊C4—1464no—
494C4⋊C4—1461no—
495C4⋊C4—1461no—
496C4⋊C4—1461no—
497C4⋊C4—1461no—
498C4⋊C4C2×C4×D7312yes—
499C4⋊C4#1192402712yes—
500C22⋊C4—1464no—
501C22⋊C4—1464no—
502C22⋊C4—1464no—
503C42—21no—
504C42—21no—
505C42—81no—
506C42—5841no—
507C42—5841no—
508C42—5841no—
509C42—732no—
510C42—732no—
511C42—1464no—
512C42—1461no—
513C42—1461no—
514C23—2921no—
515C23—2921no—
516C23—2921no—
517C23—2921no—
518C23—1462no—
519C23—1462no—
520C23—1462no—
521C23C2×D4×D7311yescharacteristic
522Q8—22no—
523Q8—22no—
524Q8—41no—
525Q8—41no—
526D4—734no—
527D4—1462no—
528D4—1462no—
529D4—1462no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—2921no—
549C2×C4—2921no—
550C2×C4—2921no—
551C2×C4—2921no—
552C2×C4—2921no—
553C2×C4—2921no—
554C2×C4—2921no—
555C2×C4—5844no—
556C2×C4—5844no—
557C2×C4—5844no—
558C2×C4—732no—
559C2×C4—732no—
560C2×C4—1461no—
561C2×C4—1461no—
562C2×C4—1461no—
563C2×C4—1461no—
564C2×C4—1462no—
565C2×C4—1462no—
566C2×C4—1461no—
567C2×C4—1461no—
568C2×C4—1461no—
569C2×C4—1461no—
570C2×C4—1461no—
571C2×C4—1461no—
572C2×C4C22×C4×D7311yescharacteristic
573C2×C4#41×D7311yescharacteristic
574C2×C4#41×D7311yescharacteristic
575C2×C4#41×D7311yescharacteristic
576C2×C4#41×D7311yescharacteristic
577C2×C4C2×#1192402711yescharacteristic
578C8—22no—
579C8—41no—
580C8—2921no—
581C8—1462no—
582C22—2922no—
583C22—2922no—
584C22—2922no—
585C22—2922no—
586C22—2922no—
587C22—2922no—
588C22—1464no—
589C22C4×D4×D7311yescharacteristic
590C22#1798909511yescharacteristic
591C22#1798909711yescharacteristic
592C22#1798938911yescharacteristic
593C22#1798939611yescharacteristic
594C22#1798939711yescharacteristic
595C22#1798940011yescharacteristic
596C4—42no—
597C4—42no—
598C4—42no—
599C4—41no—
600C4—41no—
601C4—81no—
602C4—2922no—
603C4—2921no—
604C4—2922no—
605C4—2921no—
606C4—5841no—
607C4—1462no—
608C4—1462no—
609C4#117×D7311yescharacteristic
610C4C4×#1192402711yescharacteristic
611C4#1798911611yescharacteristic
612C4#1798912011yescharacteristic
613C2—2924no—
614C2#2261237311yescharacteristic
615C2#2261878711yescharacteristic
616C2#2261880611yescharacteristic
617C2#2262011211yescharacteristic
618C2#2262011811yescharacteristic
619C2#2262014011yescharacteristic
620C2#2262014411yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1928C2
socleC23×C73C22×D4
prime radicalC22×D292C23
cosocleC22×C4×C73C24
hypercenter#1928D73
hypocenterC73#21130
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
2. #22612576C2semidirect
4. #22610574C2semidirect
5. #22610516C2semidirect
6. #22610514C2semidirect
11. #107×Dic73C2semidirect
12. #22609763C2semidirect
14. #22609681C2semidirect
15. C73×#1928C2semidirect
21. #17988672C4semidirect
70. Q8×Dic73C2×C4semidirect
93. #14556522C2×C4semidirect
99. C8×Dic73C2×C4semidirect
121. #103×C73C2×C4semidirect
315. #1928D73semidirect
326. C73#21130semidirect
416. #103C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179886722
#226125762
#226122912
#226105742
#226105162
#226105142
#226102682
#226102622
#226101812
#226101752
#107×Dic732
#226097632
#226097552
#226096812
C73×#19282
#2113073