Finite Group Databasebeta

← #33950682

Subgroups — #33950682

Order 18688.

1950 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22623529C211yesmaximal, characteristic
2#22618950C211yesmaximal, characteristic, complemented
3#22618938C213yesmaximal, complemented
4#22618935C213yesmaximal, complemented
5#22618909C213yesmaximal, complemented
6#22618900C213yesmaximal, complemented
7#41×#10596831C211yesmaximal, characteristic, complemented
8C8×#11924029C211yesmaximal, characteristic, complemented
9Q8×#11924011C211yesmaximal, characteristic, complemented
10C8×Q8×D73C211yesmaximal, characteristic, complemented
11#22618676C213yesmaximal, complemented
12#22618640C213yesmaximal, complemented
13#22618598C213yesmaximal, complemented
14C4×#14556599C213yesmaximal, complemented
15C73×#2971C211yesmaximal, characteristic, complemented
16C2×#14556599C2213yes—
17C2×#14556599C2213yes—
18#17989320C413yes—
19#17989313C411yescharacteristic
20#17989305C413yes—
21#17989287C413yes—
22#17989281C413yes—
23#17989256C411yescharacteristic
24#17989249C411yescharacteristic
25#17989244C411yescharacteristic
26#17989134C2213yes—
27#17989121C2213yes—
28#17989119C2213yes—
29C4×#11924029C2211yescharacteristic
30C4×#11924028C2213yes—
31C4×Q8×D73C2211yescharacteristic
32#17988717C2213yes—
33#17988714C2213yes—
34Q8×#10596831C2211yescharacteristic
35Q8×#10596831C2211yescharacteristic
36C8×Dic146C2213yes—
37#17988711C2213yes—
38#17988666C2213yes—
39#17988664C2213yes—
40#17988663C2216yes—
41#17988656C2216yes—
42#17988655C2216yes—
43C8×D292C2213yes—
44D4×#10596831C2213yes—
45D4×#10596831C2213yes—
46#17988652C2213yes—
47#17988650C2216yes—
48#17988649C2216yes—
49#17988645C2216yes—
50#17988643C2213yes—
51#17988640C2213yes—
52#17988639C2213yes—
53#17988609C2213yes—
54C4⋊C8×D73C2213yes—
55#17988552C2216yes—
56#17988545C2213yes—
57#17988545C2213yes—
58#17988544C2213yes—
59#17988544C2213yes—
60#17988544C2213yes—
61#17988544C2211yescharacteristic
62C4×#11924011C2213yes—
63C4×#11924011C2213yes—
64C2×C4×#10596831C2213yes—
65C4×C8×D73C2213yes—
66C4×C8×D73C2213yes—
67C4×#41×C73C2211yescharacteristic
68C8×Q8×C73C2211yescharacteristic
69C8×Q8×C73C2211yescharacteristic
70#436×C73C2213yes—
71#436×C73C2213yes—
72#432×C73C2213yes—
73C4×M16×C73C2213yes—
74C2×#11924029C2×C411yescharacteristic
75C2×#11924028C2313yes—
76C2×Q8×D73C2×C411yescharacteristic
77#14556599—212no—
78#14556599—212no—
79C2×#11924011C2313yes—
80C2×#11924011C2313yes—
81C2×#11924011C2313yes—
82C2×#11924011C2311yescharacteristic
83C22×#10596831C2313yes—
84C22×#10596831C2313yes—
85C2×C8×D73C2313yes—
86C2×C8×D73C2313yes—
87C2×C8×D73C2313yes—
88C2×C8×D73C2311yescharacteristic
89#14556590C2×C413yes—
90#14556587C2×C413yes—
91#14556585C2×C416yes—
92#14556584C2×C416yes—
93#14556583C2×C413yes—
94#14556582C2×C413yes—
95#14556579C2×C413yes—
96#14556578C2×C413yes—
97#14556577C2×C416yes—
98#14556574C2×C416yes—
99#14556573C2×C416yes—
100#14556571C2×C413yes—
101#14556570C2×C413yes—
102#14556566C2×C413yes—
103#14556565C2×C413yes—
104#14556564C2×C416yes—
105#14556562C2×C413yes—
106#14556562C2×C413yes—
107Q8×Dic73C2×C411yescharacteristic
108Q8×Dic73C2×C411yescharacteristic
109C4×Dic146C2×C413yes—
110C4×Dic146C2×C413yes—
111C4×Dic146C2313yes—
112#14556556C2×C413yes—
113C4×D292C2×C413yes—
114C4×D292C2×C413yes—
115C4×D292C2313yes—
116#14556553C2×C413yes—
117C4×C22⋊D73C2×C413yes—
118C4×C22⋊D73C2×C413yes—
119C4×C22⋊D73C2×C413yes—
120C4×C22⋊D73C2×C413yes—
121C4×C22⋊D73C2316yes—
122#14556550C2313yes—
123#14556549C2×C413yes—
124#14556548C2313yes—
125#14556548C2311yescharacteristic
126C4⋊C4×D73C2×C413yes—
127C42×D73C2313yes—
128C42×D73C2313yes—
129C4⋊#10596831C2313yes—
130C4⋊#10596831C2313yes—
131#14556520C2316yes—
132#14556520C2316yes—
133C22⋊#10596831C2316yes—
134#14556503C2316yes—
135#14556503C2316yes—
136#14556501C2313yes—
137#14556501C2313yes—
138#14556501C2313yes—
139#14556501C2311yescharacteristic
140C4×#10596831C2313yes—
141C4×#10596831C2313yes—
142C4×#10596831C2313yes—
143C4×#10596831C2311yescharacteristic
144C8×Dic73C2313yes—
145C8×Dic73C2313yes—
146C8×Dic73C2313yes—
147C8×Dic73C2311yescharacteristic
148C2×#41×C73C2×C411yescharacteristic
149C2×M16×C73C2313yes—
150C2×M16×C73C2313yes—
151C4×Q8×C73C2×C411yescharacteristic
152C4×Q8×C73C2311yescharacteristic
153C4×D4×C73C2×C413yes—
154C4×D4×C73C2313yes—
155#117×C73C2×C413yes—
156C2×C42×C73C2313yes—
157C4⋊C8×C73C2313yes—
158C4⋊C8×C73C2313yes—
159C22⋊C8×C73C2316yes—
160#97×C73C2311yescharacteristic
161#97×C73C2313yes—
162C4×C8×C73C2313yes—
163C4×C8×C73C2313yes—
164#11924029—24no—
165#11924028—26no—
166#11924028—26no—
167C2×Dic146C22×C413yes—
168C2×Dic146C22×C413yes—
169Q8×D73—24no—
170C2×D292C22×C413yes—
171C2×D292C22×C413yes—
172C2×C22⋊D73C22×C413yes—
173C2×C22⋊D73C22×C413yes—
174C2×C22⋊D73C22×C416yes—
175C2×C4×D73C22×C413yes—
176C2×C4×D73C22×C413yes—
177C2×C4×D73C22×C413yes—
178C2×C4×D73C22×C413yes—
179C2×C4×D73C2413yes—
180C2×C4×D73C2411yescharacteristic
181#11924011—26no—
182#11924011—26no—
183#11924011—26no—
184#11924011#4114yescomplemented
185C2×#10596831—26no—
186C2×#10596831—26no—
187C2×#10596831C2413yes—
188C2×#10596831C2413yes—
189C2×#10596831C2411yescharacteristic
190C2×#10596831C2411yescharacteristic
191C8×D73—26no—
192C8×D73—26no—
193C8×D73—26no—
194C8×D73#4114yescomplemented
195C4⋊Dic73C22×C413yes—
196C4⋊Dic73C22×C413yes—
197C4⋊Dic73C22×C413yes—
198C4⋊Dic73C22×C413yes—
199#11924007C22×C416yes—
200#11924007C22×C411yescharacteristic
201#11924007C22×C411yescharacteristic
202#11924007C22×C413yes—
203#11924007C22×C413yes—
204#11924007C22×C416yes—
205C22⋊Dic73C22×C413yes—
206C22⋊Dic73C22×C413yes—
207C22⋊Dic73C22×C416yes—
208#11924005C22×C411yescharacteristic
209#11924005C22×C411yescharacteristic
210#11924005C22×C413yes—
211#11924005C22×C413yes—
212#11924005C22×C416yes—
213#11924005C22×C416yes—
214C4×Dic73C22×C413yes—
215C4×Dic73C22×C413yes—
216C4×Dic73C22×C413yes—
217C4×Dic73C22×C413yes—
218C4×Dic73C2411yescharacteristic
219C4×Dic73C2413yes—
220#41×C73—24no—
221C2×Q8×C73C22×C411yescharacteristic
222C2×D4×C73C22×C413yes—
223C22×C4×C73C22×C413yes—
224C22×C4×C73C22×C413yes—
225C22×C4×C73C2413yes—
226M16×C73—26no—
227M16×C73—26no—
228C2×C8×C73C2413yes—
229C2×C8×C73C2413yes—
230C2×C8×C73C2411yescharacteristic
231C2×C8×C73C2411yescharacteristic
232C4⋊C4×C73C22×C413yes—
233C4⋊C4×C73C22×C413yes—
234C22⋊C4×C73C22×C416yes—
235C42×C73C22×C413yes—
236C42×C73C22×C413yes—
237C42×C73C2413yes—
238C42×C73C2411yescharacteristic
239C23×C73C23×C413yes—
240C22×D73C23×C413yes—
241C22×D73C23×C411yescharacteristic
242Q8×C73#13114yescomplemented
243D4×C73—26no—
244C2×C4×C73—26no—
245C2×C4×C73—23no—
246C2×C4×C73—23no—
247C2×C4×C73—43no—
248C2×C4×C73C23×C411yescharacteristic
249C2×C4×C73C23×C411yescharacteristic
250C2×C4×C73C23×C413yes—
251C2×C4×C73C23×C413yes—
252C2×C4×C73C23×C413yes—
253C2×C4×C73C2511yescharacteristic
254C22⋊D73—26no—
255C22⋊D73—26no—
256C22⋊D73—46no—
257C2×Dic73C23×C411yescharacteristic
258C2×Dic73C23×C411yescharacteristic
259C2×Dic73C23×C413yes—
260C2×Dic73C23×C413yes—
261C2×Dic73C23×C413yes—
262C2×Dic73C23×C411yescharacteristic
263D292—26no—
264D292—26no—
265C4×D73—26no—
266C4×D73—26no—
267C4×D73—23no—
268C4×D73—23no—
269C4×D73—43no—
270C4×D73—43no—
271C4×D73C2×#4112yes—
272C4×D73C2×#4112yes—
273Dic146—26no—
274Dic146—26no—
275C8×C73—23no—
276C8×C73—23no—
277C8×C73C2×#4112yes—
278C8×C73C2×#4112yes—
279#10596831—23no—
280#10596831—23no—
281#10596831C2×#4112yes—
282#10596831C2×#4112yes—
283C22×C73—23no—
284C22×C73—23no—
285C22×C73—23no—
286C22×C73C24×C411yescharacteristic
287C2×D73—21no—
288C2×D73—21no—
289C2×D73—23no—
290C2×D73—23no—
291C2×D73—23no—
292C2×D73C4×#4112yes—
293C4×C73—23no—
294C4×C73—21no—
295C4×C73—23no—
296C4×C73—21no—
297C4×C73C2×#13116yes—
298C4×C73C22×#4111yescharacteristic
299C4×C73C22×#4111yescharacteristic
300Dic73—23no—
301Dic73—23no—
302Dic73—21no—
303Dic73—21no—
304Dic73—23no—
305Dic73C4×#4112yes—
306#60218—731nomaximal, Sylow
307C2×C73—43no—
308C2×C73C2×C4×#4111yescharacteristic
309C2×C73C22×#13111yescharacteristic
310C2×C73#357311yescharacteristic
311D73—22no—
312D73—43no—
313#3480—731no—
314#2971—731no—
315#2971D7311yescharacteristic, complemented
316#2969—733no—
317#2969—733no—
318#2966—733no—
319#2966—733no—
320C8×#41—731no—
321C8×#41—731no—
322Q8×M16—731no—
323C2×C8×Q8—731no—
324#2935—733no—
325#2931—733no—
326#2925—733no—
327C4×#131—733no—
328C73#6021811yescharacteristic, complemented, Sylow
329C2×#131—733no—
330C2×#131—733no—
331#559—733no—
332#557—731no—
333#556—733no—
334#553—733no—
335#553—733no—
336#550—731no—
337#548—731no—
338#548—731no—
339#532—733no—
340#529—733no—
341#529—733no—
342C4×#41—731no—
343C4×#41—733no—
344C4×#41C2×D7311yescharacteristic
345C2×C4×Q8—731no—
346#446—733no—
347#446—733no—
348C8×Q8—731no—
349C8×Q8—731no—
350C8×Q8—733no—
351C8×Q8—733no—
352C8×Q8C2×D7311yescharacteristic
353C8×Q8C2×D7311yescharacteristic
354#436—736no—
355#436—733no—
356#436—733no—
357#436C2×D7313yes—
358#436C2×D7313yes—
359#435—736no—
360#435—736no—
361C8×D4—733no—
362C8×D4—733no—
363C8×D4—733no—
364C8×D4—736no—
365C8×D4—733no—
366#433—736no—
367#433—736no—
368#432—733no—
369#432C2×D7313yes—
370#431—733no—
371#431—733no—
372#423—733no—
373C2×C4⋊C8—733no—
374#408—736no—
375#405—733no—
376#405—733no—
377#405—731no—
378#405—733no—
379#405—733no—
380#405—733no—
381C4×M16—733no—
382C4×M16—733no—
383C4×M16C2×D7313yes—
384C2×C4×C8—733no—
385C2×C4×C8—733no—
386C2×C4×C8—733no—
387C2×#41—731no—
388C2×#41—733no—
389C2×#41C4×D7311yescharacteristic
390C22×Q8—731no—
391#131—14612no—
392#131—14612no—
393C2×M16—731no—
394C2×M16—733no—
395C2×M16—733no—
396C2×M16—733no—
397C2×M16C22×D7313yes—
398C2×M16C22×D7313yes—
399C22×C8—731no—
400C22×C8—733no—
401C22×C8—733no—
402C22×C8—733no—
403C22×C8—733no—
404C22×C8—733no—
405C4⋊Q8—733no—
406#127—733no—
407#126—736no—
408#126—736no—
409#125—733no—
410#125—733no—
411#124—733no—
412#124—736no—
413#124—733no—
414#123—736no—
415#123—736no—
416C22⋊Q8—733no—
417C22⋊Q8—733no—
418#121—733no—
419#121—733no—
420#121—733no—
421#121—733no—
422#121—736no—
423C4×Q8—731no—
424C4×Q8—731no—
425C4×Q8—733no—
426C4×Q8—733no—
427C4×Q8—733no—
428C4×Q8—733no—
429C4×Q8C4×D7311yescharacteristic
430C4×Q8C22×D7311yescharacteristic
431C4×D4—733no—
432C4×D4—733no—
433C4×D4—733no—
434C4×D4—733no—
435C4×D4—736no—
436C4×D4—733no—
437C4×D4—733no—
438C4×D4—733no—
439C4×D4—733no—
440C4×D4C4×D7313yes—
441C4×D4C22×D7313yes—
442#117—733no—
443#117—733no—
444#117—731no—
445#117—733no—
446#117C4×D7313yes—
447C2×C4⋊C4—733no—
448C2×C42—733no—
449C2×C42—733no—
450C2×C42C22×D7313yes—
451C4⋊C8—736no—
452C4⋊C8—736no—
453C4⋊C8—733no—
454C4⋊C8—733no—
455C4⋊C8C22×D7313yes—
456C4⋊C8C22×D7313yes—
457C22⋊C8—736no—
458C22⋊C8—736no—
459C22⋊C8—736no—
460C22⋊C8C22×D7316yes—
461#97—731no—
462#97—733no—
463#97—733no—
464#97—733no—
465#97C22×D7311yescharacteristic
466#97C22×D7313yes—
467C4×C8—731no—
468C4×C8—733no—
469C4×C8—733no—
470C4×C8—733no—
471C4×C8—731no—
472C4×C8—733no—
473C4×C8—733no—
474C4×C8—733no—
475C4×C8C22×D7313yes—
476C4×C8C22×D7313yes—
477#41—24no—
478#41—1466no—
479#41—1466no—
480#41—1464no—
481C2×Q8—733no—
482C2×Q8—733no—
483C2×Q8—1464no—
484C2×Q8C2×C4×D7311yescharacteristic
485C2×D4—736no—
486C2×D4—733no—
487C2×D4—733no—
488C2×D4—733no—
489C2×D4—733no—
490C2×D4C2×C4×D7313yes—
491C22×C4—733no—
492C22×C4—733no—
493C22×C4—731no—
494C22×C4—733no—
495C22×C4—733no—
496C22×C4—733no—
497C22×C4C2×C4×D7313yes—
498C22×C4C2×C4×D7313yes—
499C22×C4C23×D7313yes—
500M16—26no—
501M16—26no—
502M16—734no—
503M16—1466no—
504M16—1466no—
505M16—1466no—
506C2×C8—731no—
507C2×C8—731no—
508C2×C8—734no—
509C2×C8—733no—
510C2×C8—733no—
511C2×C8—1466no—
512C2×C8—1466no—
513C2×C8—1466no—
514C2×C8—1466no—
515C2×C8—1466no—
516C2×C8C23×D7313yes—
517C2×C8C23×D7313yes—
518C2×C8C23×D7311yescharacteristic
519C2×C8C23×D7311yescharacteristic
520C4⋊C4—736no—
521C4⋊C4—731no—
522C4⋊C4—731no—
523C4⋊C4—733no—
524C4⋊C4—733no—
525C4⋊C4—736no—
526C4⋊C4—733no—
527C4⋊C4—733no—
528C4⋊C4—733no—
529C4⋊C4—733no—
530C4⋊C4C2×C4×D7313yes—
531C4⋊C4C2×C4×D7313yes—
532C22⋊C4—733no—
533C22⋊C4—733no—
534C22⋊C4—736no—
535C22⋊C4—731no—
536C22⋊C4—731no—
537C22⋊C4—733no—
538C22⋊C4—736no—
539C22⋊C4—733no—
540C22⋊C4—736no—
541C22⋊C4C2×C4×D7316yes—
542C42—733no—
543C42—733no—
544C42—733no—
545C42—733no—
546C42—731no—
547C42—733no—
548C42C2×C4×D7313yes—
549C42C2×C4×D7313yes—
550C42C23×D7311yescharacteristic
551C42C23×D7313yes—
552C23—731no—
553C23—733no—
554C23C22×C4×D7313yes—
555Q8—1466no—
556Q8—1466no—
557Q8#1455659914yescomplemented
558D4—26no—
559D4—2926no—
560D4—1466no—
561D4—1466no—
562D4—1466no—
563D4—1466no—
564C2×C4—23no—
565C2×C4—23no—
566C2×C4—26no—
567C2×C4—43no—
568C2×C4—2923no—
569C2×C4—2923no—
570C2×C4—731no—
571C2×C4—732no—
572C2×C4—732no—
573C2×C4—731no—
574C2×C4—731no—
575C2×C4—733no—
576C2×C4—733no—
577C2×C4—733no—
578C2×C4—1463no—
579C2×C4—1463no—
580C2×C4—1466no—
581C2×C4—1466no—
582C2×C4C22×C4×D7313yes—
583C2×C4C22×C4×D7311yescharacteristic
584C2×C4C22×C4×D7311yescharacteristic
585C2×C4C22×C4×D7313yes—
586C2×C4C22×C4×D7313yes—
587C2×C4C24×D7311yescharacteristic
588C8—23no—
589C8—23no—
590C8—732no—
591C8—732no—
592C8—1463no—
593C8—1463no—
594C8#41×D7312yes—
595C8#41×D7312yes—
596C22—23no—
597C22—23no—
598C22—23no—
599C22—732no—
600C22—1463no—
601C22—1463no—
602C22—1463no—
603C22—1461no—
604C22—1461no—
605C22C23×C4×D7311yescharacteristic
606C4—21no—
607C4—23no—
608C4—21no—
609C4—23no—
610C4—732no—
611C4—1461no—
612C4—1463no—
613C4—1463no—
614C4—1461no—
615C4—1463no—
616C4C2×#1455659916yes—
617C4C2×#41×D7311yescharacteristic
618C4C2×#41×D7311yescharacteristic
619C2—43no—
620C2—2923no—
621C2—1462no—
622C2C4×#41×D7311yescharacteristic
623C2C22×#1455659911yescharacteristic
624C2#2262464411yescharacteristic
subgroupG/H
center, FrattiniC2×C4C24×D73
commutator, socleC22×C73C24×C4
FittingC73×#2971C2
prime radical#22623529C2
cosocleC2×C4×C73C25
hypercenter#2971D73
hypocenterC73#60218
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ #2971

Complements

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

NH ≅ G/Ntype
2. #22618950C2semidirect
3. #22618938C2semidirect
4. #22618935C2semidirect
5. #22618909C2semidirect
6. #22618900C2semidirect
7. #41×#10596831C2semidirect
8. C8×#11924029C2semidirect
9. Q8×#11924011C2semidirect
10. C8×Q8×D73C2semidirect
11. #22618676C2semidirect
12. #22618640C2semidirect
13. #22618598C2semidirect
14. C4×#14556599C2semidirect
15. C73×#2971C2semidirect
184. #11924011#41semidirect
194. C8×D73#41semidirect
242. Q8×C73#131semidirect
315. #2971D73semidirect
328. C73#60218semidirect
557. Q8#14556599semidirect

Maximal subgroups

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

subgroupindex
#226235292
#226189502
#226189382
#226189352
#226189092
#226189002
#41×#105968312
C8×#119240292
Q8×#119240112
C8×Q8×D732
#226186762
#226186402
#226185982
C4×#145565992
C73×#29712
#6021873