Finite Group Databasebeta

← #33089110

Subgroups — #33089110

Order 18688.

862 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988690C211yesmaximal, characteristic
2#22612570C211yesmaximal, characteristic
3#22612296C211yesmaximal, characteristic, complemented
4#22610559C211yesmaximal, characteristic
5#22610550C211yesmaximal, characteristic
6#22610526C211yesmaximal, characteristic, complemented
7#22610499C211yesmaximal, characteristic, complemented
8#22610286C211yesmaximal, characteristic
9#22610284C211yesmaximal, characteristic
10#22610201C211yesmaximal, characteristic, complemented
11#22609830C211yesmaximal, characteristic, complemented
12#103×Dic73C211yesmaximal, characteristic, complemented
13#22609745C211yesmaximal, characteristic
14#22609681C211yesmaximal, characteristic, complemented
15C73×#1863C211yesmaximal, characteristic, complemented
16C22×#11924015C2211yescharacteristic
17C2×Q8×Dic73C2211yescharacteristic
18C2×Q8×Dic73C2211yescharacteristic
19C2×#14556553C2211yescharacteristic
20#17988690C418yescomplemented
21C2×#14556524C2211yescharacteristic
22C2×#14556523C2211yescharacteristic
23C2×#14556515C2211yescharacteristic
24C2×#14556513C2211yescharacteristic
25C2×#14556513C2211yescharacteristic
26C2×#14556511C2211yescharacteristic
27C2×#14556501C2211yescharacteristic
28C2×C8×Dic73C2211yescharacteristic
29#17988489C2211yescharacteristic
30#17988488C2211yescharacteristic
31#17988486C2211yescharacteristic
32#17988479—21no—
33#17988477—21no—
34#17988449—21no—
35#17988444—21no—
36C4⋊C4×Dic73—21no—
37C4⋊C4×Dic73—21no—
38C4⋊C4×Dic73C2211yescharacteristic
39C4⋊C4×Dic73C2211yescharacteristic
40C4⋊C4×Dic73C2211yescharacteristic
41C4×C4⋊Dic73—21no—
42C4×C4⋊Dic73C2211yescharacteristic
43#17988433—21no—
44#17988433C2211yescharacteristic
45#17988350C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988342C2211yescharacteristic
51#17988341C2211yescharacteristic
52#17988341C2211yescharacteristic
53C2×C4×Q8×C73C2211yescharacteristic
54C2×#107×C73C2211yescharacteristic
55C2×#103×C73C2211yescharacteristic
56C2×#103×C73C2211yescharacteristic
57#389×C73C2211yescharacteristic
58#340×C73C2211yescharacteristic
59#336×C73C2211yescharacteristic
60C22×D292C2311yescharacteristic
61C22×C4×D73—21no—
62C2×#11924015—22no—
63C2×#11924015—22no—
64C2×#11924015C2×C414yes—
65C22×#10596831C2311yescharacteristic
66C22×#10596831C2311yescharacteristic
67Q8×Dic73—22no—
68Q8×Dic73—22no—
69Q8×Dic73—22no—
70Q8×Dic73C2×C414yes—
71#14556553—22no—
72#14556553C2×C414yes—
73C2×C4⋊Dic73—21no—
74C2×C4⋊Dic73—21no—
75C2×C4⋊Dic73—21no—
76C2×C4⋊Dic73C2311yescharacteristic
77C2×C4⋊Dic73C2311yescharacteristic
78C2×C4⋊Dic73C2311yescharacteristic
79C2×#11924005—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×Dic73C2311yescharacteristic
91C2×C4×Dic73C2311yescharacteristic
92#14556524C2×C414yes—
93#14556523—22no—
94#14556515—22no—
95#14556513—22no—
96#14556513C2×C414yes—
97#14556511—22no—
98#14556501C2×C414yes—
99C8×Dic73—22no—
100#14556498—21no—
101#14556498—21no—
102#14556498—21no—
103#14556498—21no—
104#14556498—21no—
105#14556498D411yescharacteristic
106#14556498D411yescharacteristic
107#14556498D411yescharacteristic
108#14556498D411yescharacteristic
109C22×Q8×C73C2311yescharacteristic
110C22×C8×C73C2311yescharacteristic
111C22×C8×C73C2311yescharacteristic
112C4×Q8×C73—22no—
113C4×Q8×C73—22no—
114C2×C4⋊C4×C73—21no—
115C2×C4⋊C4×C73—21no—
116C2×C4⋊C4×C73C2311yescharacteristic
117C2×C4⋊C4×C73C2311yescharacteristic
118C2×C4⋊C4×C73C2311yescharacteristic
119C2×C42×C73—21no—
120C2×C42×C73C2311yescharacteristic
121#107×C73—22no—
122#103×C73—22no—
123#103×C73C2×C414yes—
124#95×C73—21no—
125C23×D73—21no—
126C2×D292—21no—
127C2×D292—21no—
128C2×D292—21no—
129C2×D292—21no—
130C2×D292—21no—
131C2×D292C22×C412yes—
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—81no—
141C2×C4×D73—81no—
142#11924015—28no—
143C2×#10596831—21no—
144C2×#10596831—21no—
145C2×#10596831—21no—
146C2×#10596831—21no—
147C2×#10596831—21no—
148C2×#10596831C22×C412yes—
149C4⋊Dic73—24no—
150C4⋊Dic73—21no—
151C4⋊Dic73—21no—
152C4⋊Dic73—24no—
153C4⋊Dic73—21no—
154C4⋊Dic73—21no—
155C4⋊Dic73—81no—
156C4⋊Dic73C22×C412yes—
157C4⋊Dic73#4112yes—
158#11924005—24no—
159#11924005—81no—
160#11924005—81no—
161#11924005—81no—
162#11924005—81no—
163C4×Dic73—24no—
164C4×Dic73—21no—
165C4×Dic73—21no—
166C4×Dic73—24no—
167C4×Dic73—81no—
168C4×Dic73—81no—
169C4×Dic73—81no—
170C4×Dic73—81no—
171C4×Dic73C22×C412yes—
172C4×Dic73#4112yes—
173C2×Q8×C73—21no—
174C2×Q8×C73—21no—
175C2×Q8×C73—21no—
176C2×Q8×C73C22×C412yes—
177C2×Q8×C73#4112yes—
178C2×Q8×C73#4112yes—
179C22×C4×C73—21no—
180C22×C4×C73—21no—
181C22×C4×C73—21no—
182C22×C4×C73—21no—
183C22×C4×C73C2×D411yescharacteristic
184C22×C4×C73C2×D411yescharacteristic
185C22×C4×C73C2411yescharacteristic
186C2×C8×C73—21no—
187C2×C8×C73—21no—
188C2×C8×C73—21no—
189C2×C8×C73—21no—
190C2×C8×C73—21no—
191C2×C8×C73C22×C412yes—
192C4⋊C4×C73—24no—
193C4⋊C4×C73—21no—
194C4⋊C4×C73—21no—
195C4⋊C4×C73—21no—
196C4⋊C4×C73—21no—
197C4⋊C4×C73—81no—
198C4⋊C4×C73C22×C412yes—
199C4⋊C4×C73#4112yes—
200C42×C73—24no—
201C42×C73—21no—
202C42×C73—21no—
203C23×C73C22×D411yescharacteristic
204C22×D73—22no—
205C22×D73—41no—
206C22×D73—41no—
207C22×D73—41no—
208C22×D73—41no—
209C22×D73—41no—
210C22×D73—41no—
211Q8×C73—22no—
212Q8×C73—22no—
213Q8×C73—22no—
214Q8×C73#11714yes—
215C2×C4×C73—22no—
216C2×C4×C73—22no—
217C2×C4×C73—22no—
218C2×C4×C73—22no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—22no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
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—41no—
263D292—41no—
264C4×D73—84no—
265C8×C73—22no—
266C8×C73—41no—
267#10596831—22no—
268#10596831—41no—
269C22×C73C2×C4×D411yescharacteristic
270C22×C73#52511yescharacteristic
271C22×C73#52511yescharacteristic
272C22×C73#57611yescharacteristic
273C22×C73#57711yescharacteristic
274C22×C73#57711yescharacteristic
275C22×C73#57811yescharacteristic
276C2×D73—42no—
277C2×D73—42no—
278C2×D73—42no—
279C2×D73—81no—
280C2×D73—81no—
281C2×D73—81no—
282C2×D73—81no—
283C4×C73—24no—
284C4×C73—41no—
285C4×C73—42no—
286C4×C73—41no—
287C4×C73—42no—
288C4×C73—81no—
289C4×C73C2×#11711yescharacteristic
290C4×C73C4×#4111yescharacteristic
291C4×C73#52911yescharacteristic
292C4×C73#52911yescharacteristic
293Dic73—22no—
294Dic73—22no—
295Dic73—41no—
296Dic73—42no—
297Dic73—41no—
298Dic73—42no—
299Dic73—81no—
300#21128—731nomaximal, Sylow
301C2×C73#229011yescharacteristic
302C2×C73#294611yescharacteristic
303C2×C73#294711yescharacteristic
304C2×C73#311911yescharacteristic
305C2×C73#312111yescharacteristic
306C2×C73#312411yescharacteristic
307C2×C73#312611yescharacteristic
308D73—82no—
309C2×#440—731no—
310#2319—731no—
311#2276—731no—
312#1926—731no—
313#1925—731no—
314#1921—731no—
315#1918—731no—
316#1880—731no—
317#1880—731no—
318#1865—731no—
319#1863D7311yescharacteristic, complemented
320#1776—731no—
321C4×#103—731no—
322C4×#102—731no—
323#1746—731no—
324C73#2112811yescharacteristic, complemented, Sylow
325C22×SD16—731no—
326C2×C4×Q8—731no—
327C2×C4×Q8—731no—
328C2×C4×Q8C2×D7311yescharacteristic
329C2×C4×D4—731no—
330#440—738no—
331C2×#107—731no—
332C2×#107C2×D7311yescharacteristic
333C2×#106—731no—
334C2×#103—731no—
335C2×#103C2×D7311yescharacteristic
336C2×#103C2×D7311yescharacteristic
337C2×#102—731no—
338C2×#102—731no—
339C2×#102—731no—
340C2×#97—731no—
341C2×C4×C8—731no—
342#390—731no—
343#390—731no—
344#389—731no—
345#389C2×D7311yescharacteristic
346#388—1461no—
347#388—1461no—
348#381—1461no—
349#380—1461no—
350C4×C4⋊C4—731no—
351C4×C4⋊C4—731no—
352C4×C4⋊C4—731no—
353C4×C4⋊C4—731no—
354C4×C4⋊C4—1461no—
355C4×C4⋊C4—1461no—
356C4×C4⋊C4—1461no—
357#376—731no—
358#376—1461no—
359#340—731no—
360#340—731no—
361#340—731no—
362#340—731no—
363#340—731no—
364#340C2×D7311yescharacteristic
365#336—731no—
366#336—731no—
367#336—731no—
368#336C2×D7311yescharacteristic
369C22×Q8C22×D7311yescharacteristic
370C22×D4—731no—
371C23×C4—1461no—
372C2×SD16—734no—
373C2×SD16—1462no—
374C2×SD16—1462no—
375C22×C8—731no—
376C22×C8—731no—
377C22×C8C22×D7311yescharacteristic
378C22×C8C22×D7311yescharacteristic
379C4×Q8—22no—
380C4×Q8—22no—
381C4×Q8—734no—
382C4×Q8—1462no—
383C4×Q8—1462no—
384C4×Q8—1462no—
385C4×D4—734no—
386C4×D4—1462no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
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×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×C42—1461no—
411C2×C42—1461no—
412C2×C42C22×D7311yescharacteristic
413#107—22no—
414#107—734no—
415#106—1462no—
416#103—22no—
417#103—1462no—
418#103C4×D7314yes—
419#102—734no—
420#102—1462no—
421#102—1462no—
422#97—734no—
423C4×C8—1462no—
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—
434C24—1461no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8C2×C4×D7312yes—
439C2×Q8#1192402712yes—
440C2×Q8#1192402912yes—
441C2×D4—732no—
442C2×D4—1461no—
443C2×D4—1461no—
444C2×D4—1461no—
445C2×D4—1461no—
446C2×D4—1461no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—5841no—
452C22×C4—5841no—
453C22×C4—731no—
454C22×C4—731no—
455C22×C4—731no—
456C22×C4—731no—
457C22×C4—1464no—
458C22×C4—1461no—
459C22×C4—1461no—
460C22×C4—1461no—
461C22×C4D4×D7311yescharacteristic
462C22×C4D4×D7311yescharacteristic
463C22×C4C23×D7311yescharacteristic
464SD16—1468no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—732no—
471C2×C8—1461no—
472C2×C8—1461no—
473C2×C8—1461no—
474C2×C8—1461no—
475C2×C8—1461no—
476C2×C8C2×C4×D7312yes—
477C4⋊C4—24no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—81no—
483C4⋊C4—5841no—
484C4⋊C4—732no—
485C4⋊C4—732no—
486C4⋊C4—1464no—
487C4⋊C4—1464no—
488C4⋊C4—1461no—
489C4⋊C4—1461no—
490C4⋊C4—1461no—
491C4⋊C4—1461no—
492C4⋊C4C2×C4×D7312yes—
493C4⋊C4#1192402712yes—
494C22⋊C4—5841no—
495C22⋊C4—5841no—
496C22⋊C4—5841no—
497C22⋊C4—5841no—
498C22⋊C4—1464no—
499C42—24no—
500C42—21no—
501C42—21no—
502C42—5841no—
503C42—5841no—
504C42—5841no—
505C42—5841no—
506C42—732no—
507C42—732no—
508C42—1464no—
509C42—1464no—
510C42—1461no—
511C42—1461no—
512C23—2921no—
513C23—2921no—
514C23—2921no—
515C23—2921no—
516C23—2921no—
517C23—2921no—
518C23—1462no—
519C23C2×D4×D7311yescharacteristic
520Q8—22no—
521Q8—22no—
522Q8—22no—
523Q8#1455654914yes—
524D4—2921no—
525D4—2921no—
526D4—1462no—
527D4—1462no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
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—2921no—
547C2×C4—2921no—
548C2×C4—2921no—
549C2×C4—2921no—
550C2×C4—2921no—
551C2×C4—2921no—
552C2×C4—2921no—
553C2×C4—5844no—
554C2×C4—732no—
555C2×C4—732no—
556C2×C4—1461no—
557C2×C4—1461no—
558C2×C4—1461no—
559C2×C4—1461no—
560C2×C4—1462no—
561C2×C4—1461no—
562C2×C4—1462no—
563C2×C4—1461no—
564C2×C4—1461no—
565C2×C4—1461no—
566C2×C4—1461no—
567C2×C4—1461no—
568C2×C4C22×C4×D7311yescharacteristic
569C2×C4#41×D7311yescharacteristic
570C2×C4#41×D7311yescharacteristic
571C2×C4C2×#1192402711yescharacteristic
572C2×C4C2×#1192402711yescharacteristic
573C2×C4C2×#1192402911yescharacteristic
574C8—22no—
575C8—41no—
576C8—2921no—
577C8—1462no—
578C22—2922no—
579C22—2922no—
580C22—2922no—
581C22—5841no—
582C22—5841no—
583C22—5841no—
584C22—5841no—
585C22C4×D4×D7311yescharacteristic
586C22#1798909311yescharacteristic
587C22#1798910011yescharacteristic
588C22#1798939111yescharacteristic
589C22#1798939411yescharacteristic
590C22#1798939611yescharacteristic
591C22#1798940011yescharacteristic
592C4—24no—
593C4—41no—
594C4—42no—
595C4—41no—
596C4—42no—
597C4—81no—
598C4—2922no—
599C4—2921no—
600C4—2921no—
601C4—2922no—
602C4—5841no—
603C4—1462no—
604C4—1462no—
605C4C2×#1455654911yescharacteristic
606C4C4×#1192402711yescharacteristic
607C4#1798911411yescharacteristic
608C4#1798912311yescharacteristic
609C2—5842no—
610C2#2261236711yescharacteristic
611C2#2261876211yescharacteristic
612C2#2261876911yescharacteristic
613C2#2262013111yescharacteristic
614C2#2262016111yescharacteristic
615C2#2262020011yescharacteristic
616C2#2262022411yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1863C2
socleC23×C73C22×D4
prime radicalC22×D292C23
cosocleC22×C4×C73C24
hypercenter#1863D73
hypocenterC73#21128
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1863

Complements

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

NH ≅ G/Ntype
3. #22612296C2semidirect
6. #22610526C2semidirect
7. #22610499C2semidirect
10. #22610201C2semidirect
11. #22609830C2semidirect
12. #103×Dic73C2semidirect
14. #22609681C2semidirect
15. C73×#1863C2semidirect
20. #17988690C4semidirect
319. #1863D73semidirect
324. C73#21128semidirect

Maximal subgroups

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

subgroupindex
C2×#179886902
#226125702
#226122962
#226105592
#226105502
#226105262
#226104992
#226102862
#226102842
#226102012
#226098302
#103×Dic732
#226097452
#226096812
C73×#18632
#2112873