Finite Group Databasebeta

← #34032644

Subgroups — #34032644

Order 18688.

1930 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626861C213yesmaximal, complemented
2#22626858C213yesmaximal, complemented
3#22626853C213yesmaximal, complemented
4#22626849C211yesmaximal, characteristic, complemented
5#22626834C211yesmaximal, characteristic, complemented
6#22626833C211yesmaximal, characteristic, complemented
7#22626828C213yesmaximal, complemented
8#22626773C213yesmaximal, complemented
9#22626772C213yesmaximal, complemented
10#22626742C213yesmaximal, complemented
11#22626737C211yesmaximal, characteristic, complemented
12Q8×#11924038C211yesmaximal, characteristic, complemented
13Q8×#11924037C211yesmaximal, characteristic, complemented
14D73⋊#446C213yesmaximal, complemented
15#22623140C211yesmaximal, characteristic
16#17989639C2213yes—
17#17989639C2216yes—
18#17989639C2213yes—
19#17989638C2213yes—
20#17989638C2213yes—
21#17989638C2213yes—
22#17989638C2213yes—
23#17989637C2213yes—
24#17989637C2213yes—
25#17989637C2213yes—
26#17989637C2213yes—
27Q8×#10596833C2211yescharacteristic
28Q8×#10596833C2211yescharacteristic
29Q8×#10596833C2211yescharacteristic
30Q8×#10596833C2211yescharacteristic
31#17989620C2213yes—
32#17989620C2211yescharacteristic
33#17989619C2216yes—
34#17989619C2213yes—
35#17989619C2213yes—
36#17989617C2216yes—
37#17989617C2213yes—
38#17989617C2213yes—
39#17989615C2213yes—
40#17989615C2213yes—
41#17989615C2213yes—
42#17989615C2213yes—
43#17989610C2216yes—
44#17989609C2216yes—
45#17989609C2213yes—
46#17989608C2213yes—
47#17989606C2213yes—
48#17989606C2213yes—
49#17989605C2213yes—
50#17989604C2213yes—
51#17989591C2213yes—
52#17989590C2213yes—
53#17989588C2213yes—
54#17989588C2216yes—
55C4×#11924038C2213yes—
56C4×#11924037C2213yes—
57#17989546C2216yes—
58C4×#11924036C2213yes—
59C2×#14556639C411yescharacteristic
60#17989069C413yes—
61#17989069C411yescharacteristic
62#17989067C413yes—
63#17989067C413yes—
64#17989063C413yes—
65#17989063C411yescharacteristic
66C4×#11924029C2211yescharacteristic
67C4×#11924029C2213yes—
68#17989055C2213yes—
69#17989055C2213yes—
70C2×Q8×Dic73C2211yescharacteristic
71C4×Q8×D73C2213yes—
72C4×Q8×D73C2211yescharacteristic
73#519×C73C411yescharacteristic
74C2×#11924038C2311yescharacteristic
75C2×#11924038C2313yes—
76C2×#11924037C2311yescharacteristic
77C2×#11924037C2313yes—
78C2×#11924036C2313yes—
79C2×#11924036C2313yes—
80C2×#11924036C2316yes—
81#14556658C2313yes—
82#14556658C2316yes—
83#14556658C2313yes—
84#14556658C2313yes—
85#14556658C2311yescharacteristic
86C4⋊#10596833C2313yes—
87C4⋊#10596833C2313yes—
88C4⋊#10596833C2313yes—
89C4⋊#10596833C2313yes—
90C4⋊#10596833C2313yes—
91C4⋊#10596833C2316yes—
92C4⋊#10596833C2313yes—
93#14556648C2313yes—
94#14556648C2313yes—
95#14556648C2311yescharacteristic
96#14556648C2313yes—
97#14556648C2316yes—
98C22⋊#10596833C2311yescharacteristic
99C22⋊#10596833C2313yes—
100#14556646C2313yes—
101#14556646C2316yes—
102#14556646C2313yes—
103#14556646C2313yes—
104#14556646C2316yes—
105#14556646C2313yes—
106C4×#10596833C2313yes—
107C4×#10596833C2313yes—
108C4×#10596833C2313yes—
109C4×#10596833C2313yes—
110C4×#10596833C2311yescharacteristic
111C4×#10596833C2313yes—
112#14556639—28no—
113C2×#11924029C2×C413yes—
114C2×#11924029C2×C411yescharacteristic
115C2×#11924028C2×C413yes—
116C2×#11924028C2×C413yes—
117C2×Q8×D73C2×C413yes—
118C2×Q8×D73C2×C411yescharacteristic
119Q8×Dic73C2×C413yes—
120Q8×Dic73C2×C411yescharacteristic
121Q8×Dic73C2311yescharacteristic
122Q8×Dic73C2313yes—
123C4×Dic146C2×C416yes—
124C4×Dic146C2×C413yes—
125C4×Dic146C2×C413yes—
126#14556556C2×C413yes—
127#14556556C2×C413yes—
128#14556556C2313yes—
129#14556556C2313yes—
130C4×D292C2×C416yes—
131C4×D292C2×C413yes—
132C4×D292C2×C413yes—
133#14556553C2×C413yes—
134#14556553C2×C413yes—
135#14556553C2313yes—
136#14556553C2313yes—
137#14556551C2×C413yes—
138#14556551C2×C413yes—
139#14556551C2×C416yes—
140#14556551C2313yes—
141#14556551C2311yescharacteristic
142#14556549C2×C416yes—
143#14556549C2×C413yes—
144#14556549C2×C413yes—
145#14556548C2×C413yes—
146#14556548C2×C416yes—
147#14556548C2×C413yes—
148C2×C4⋊Dic73C2×C413yes—
149C4⋊C4×D73C2×C416yes—
150C4⋊C4×D73C2×C413yes—
151C4⋊C4×D73C2×C413yes—
152C2×C4×Dic73C2313yes—
153C42×D73C2313yes—
154C42×D73C2316yes—
155C42×D73C2313yes—
156C22×Q8×C73C2×C411yescharacteristic
157C4×Q8×C73C2×C413yes—
158C4×Q8×C73C2×C411yescharacteristic
159C4×Q8×C73C2×C413yes—
160C4×Q8×C73C2×C411yescharacteristic
161#117×C73C2×C413yes—
162#117×C73C2×C413yes—
163#11924038—22no—
164#11924038—43no—
165#11924037—22no—
166#11924037—43no—
167#11924036—26no—
168#11924036—43no—
169#11924036—46no—
170C2×#10596833C2411yescharacteristic
171C2×#10596833C2413yes—
172C2×#10596833C2413yes—
173C2×#10596833C2413yes—
174C2×#10596833C2413yes—
175C2×#10596833C2411yescharacteristic
176C2×#10596833C2411yescharacteristic
177C2×#10596833C2411yescharacteristic
178#11924029—212no—
179#11924029M1618yescomplemented
180#11924028—212no—
181#11924028—212no—
182C2×Dic146C22×C413yes—
183C2×Dic146C22×C413yes—
184Q8×D73—24no—
185Q8×D73—212no—
186C2×D292C22×C413yes—
187C2×D292C22×C413yes—
188C2×C22⋊D73C22×C413yes—
189C2×C22⋊D73C22×C411yescharacteristic
190C22×Dic73C22×C413yes—
191C22×Dic73C2411yescharacteristic
192C2×C4×D73C22×C416yes—
193C2×C4×D73C22×C413yes—
194C2×C4×D73C22×C413yes—
195C2×C4×D73C22×C413yes—
196C2×C4×D73C22×C416yes—
197C2×C4×D73C22×C413yes—
198C2×C4×D73C2413yes—
199C2×C4×D73C2411yescharacteristic
200C4⋊Dic73C22×C416yes—
201C4⋊Dic73C22×C413yes—
202C4⋊Dic73C22×C413yes—
203#11924007C22×C416yes—
204#11924007C22×C413yes—
205#11924007C22×C413yes—
206#11924007C22×C413yes—
207#11924007C22×C411yescharacteristic
208#11924005C22×C411yescharacteristic
209#11924005C22×C413yes—
210#11924005C22×C413yes—
211#11924005C22×C416yes—
212#11924005C22×C413yes—
213C4×Dic73C22×C416yes—
214C4×Dic73C22×C413yes—
215C4×Dic73C22×C413yes—
216C4×Dic73C22×C413yes—
217C4×Dic73C22×C413yes—
218C4×Dic73C2411yescharacteristic
219C4×Dic73C2413yes—
220C4×Dic73C2413yes—
221C4×Dic73C2413yes—
222C2×Q8×C73—24no—
223C2×Q8×C73C22×C413yes—
224C2×Q8×C73C22×C411yescharacteristic
225C22×C4×C73C22×C413yes—
226C4⋊C4×C73C22×C413yes—
227C4⋊C4×C73C22×C413yes—
228C4⋊C4×C73C22×C413yes—
229C4⋊C4×C73C22×C413yes—
230C22⋊C4×C73C22×C413yes—
231C22⋊C4×C73C22×C411yescharacteristic
232C42×C73C22×C416yes—
233C42×C73C22×C413yes—
234C42×C73C22×C413yes—
235C23×C73C23×C411yescharacteristic
236C22×D73C23×C411yescharacteristic
237C22×D73C23×C413yes—
238Q8×C73—26no—
239Q8×C73C2×M1614yes—
240C2×C4×C73—26no—
241C2×C4×C73C23×C413yes—
242C2×C4×C73C23×C411yescharacteristic
243C2×C4×C73C23×C411yescharacteristic
244C2×C4×C73C23×C413yes—
245C2×C4×C73C23×C413yes—
246C2×C4×C73C23×C413yes—
247C22⋊D73—26no—
248C22⋊D73—22no—
249C2×Dic73—21no—
250C2×Dic73—21no—
251C2×Dic73—43no—
252C2×Dic73C23×C411yescharacteristic
253C2×Dic73C23×C413yes—
254C2×Dic73C23×C413yes—
255C2×Dic73C23×C413yes—
256C2×Dic73C23×C411yescharacteristic
257C2×Dic73C2511yescharacteristic
258D292—26no—
259D292C2×M16112yes—
260C4×D73—212no—
261C4×D73—26no—
262C4×D73—23no—
263C4×D73—21no—
264C4×D73—23no—
265C4×D73—21no—
266C4×D73—43no—
267C4×D73—46no—
268C4×D73—43no—
269C4×D73C2×M16112yes—
270Dic146—26no—
271Dic146—26no—
272#10596833—23no—
273#10596833—21no—
274#10596833—21no—
275#10596833—21no—
276#10596833—23no—
277#10596833—23no—
278#10596833—23no—
279#105968332−1+412yescomplemented
280C22×C73—21no—
281C22×C73—21no—
282C22×C73—21no—
283C22×C73C24×C411yescharacteristic
284C2×D73—23no—
285C2×D73—23no—
286C2×D73—21no—
287C2×D73—21no—
288C2×D73—21no—
289C2×D73C22×M1616yes—
290C4×C73—23no—
291C4×C73—21no—
292C4×C73—23no—
293C4×C73—21no—
294C4×C73—23no—
295C4×C73C22×M1616yes—
296Dic73—21no—
297Dic73—23no—
298Dic73—23no—
299Dic73—23no—
300Dic73C22×M1612yes—
301Dic73C2×2−1+411yescharacteristic
302Dic73#58511yescharacteristic
303#60343—731nomaximal, Sylow
304C2×C73—41no—
305C2×C73C4×2−1+411yescharacteristic
306C2×C73C23×M1611yescharacteristic
307C2×C73#357311yescharacteristic
308D73—26no—
309D73—41no—
310C4×2−1+4—731no—
311#2990—733no—
312#2990—733no—
313#2989—733no—
314#2988—731no—
315#2984—731no—
316#2984—731no—
317#2983—733no—
318#2970—733no—
319#2970—733no—
320#2965—733no—
321C8×#41—731no—
322Q8×M16—731no—
323Q8×M16—731no—
324C2×#446—733no—
325C73#6034311yescharacteristic, complemented, Sylow
326C2×2−1+4—731no—
327#520—731no—
328#520—733no—
329#520—733no—
330#520—733no—
331#519—733no—
332#519—731no—
333#519#6440811yescharacteristic
334C4×#41—731no—
335C4×#41—733no—
336C4×#41—733no—
337C4×#41—733no—
338C2×C4×Q8—731no—
339C2×C4×Q8—733no—
340C2×C4×Q8—731no—
341#446—733no—
342#446—733no—
343#446—733no—
344#446—733no—
345#446—733no—
346#446—733no—
347#446—733no—
348#446—733no—
349#446—733no—
350#446—736no—
351#446—733no—
352C8×Q8—731no—
353C8×Q8—731no—
354C8×Q8—731no—
355C8×Q8—731no—
356#436—731no—
357#436—733no—
358#435—733no—
359#435—733no—
360#435—733no—
361#435—733no—
362#435—736no—
363#435—736no—
364C8×D4—733no—
365C8×D4—733no—
366C8×D4—733no—
367C8×D4—733no—
368#432—736no—
369#432—736no—
370#432—733no—
371#432—733no—
372#431—733no—
373#431—733no—
374#431—733no—
375#431—733no—
376#423—733no—
377#423—733no—
378C2×C4⋊C8—733no—
379C2×C4⋊C8—736no—
380C4×M16—733no—
381C4×M16—733no—
382C2×#97—736no—
383C2×C4×C8—733no—
3842−1+4—1468no—
385C2×#41—731no—
386C2×#41—733no—
387C2×#41—733no—
388C2×#41—733no—
389C22×Q8—733no—
390C22×Q8—731no—
391C22×Q8C2×#6440811yescharacteristic
392C2×M16—733no—
393C2×M16—733no—
394C2×M16—731no—
395C2×M16—731no—
396C22×C8—733no—
397C22×C8—736no—
398C22×C8—733no—
399C4×Q8—731no—
400C4×Q8—733no—
401C4×Q8—733no—
402C4×Q8—731no—
403C4×Q8—733no—
404C4×Q8—733no—
405C4×Q8—733no—
406C4×Q8—733no—
407C4×Q8—736no—
408C4×Q8—733no—
409C4×Q8—733no—
410C4×Q8C2×#6440813yes—
411C4×Q8C2×#6440811yescharacteristic
412C4×Q8C2×#6440813yes—
413C4×Q8C2×#6440811yescharacteristic
414C4×D4—733no—
415C4×D4—733no—
416C4×D4—733no—
417C4×D4—733no—
418C4×D4—733no—
419C4×D4—731no—
420C4×D4—733no—
421C4×D4—733no—
422C4×D4—736no—
423C4×D4—733no—
424C4×D4—733no—
425C4×D4—736no—
426#117—733no—
427#117—733no—
428#117—736no—
429#117—733no—
430#117—733no—
431#117—736no—
432#117C2×#6440813yes—
433#117C2×#6440813yes—
434C2×C4⋊C4—733no—
435C2×C4⋊C4—733no—
436C2×C4⋊C4—736no—
437C2×C4⋊C4—733no—
438C2×C42—733no—
439C2×C42—733no—
440C2×C42—736no—
441C2×C42—733no—
442C4⋊C8—733no—
443C4⋊C8—733no—
444C4⋊C8—733no—
445C4⋊C8—733no—
446C4⋊C8—733no—
447C4⋊C8—736no—
448C4⋊C8—736no—
449C4⋊C8—733no—
450C4⋊C8—731no—
451C4⋊C8—733no—
452C4⋊C8—733no—
453C4⋊C8—733no—
454C22⋊C8—731no—
455C22⋊C8—731no—
456C22⋊C8—733no—
457C22⋊C8—733no—
458C22⋊C8—736no—
459C22⋊C8—733no—
460C22⋊C8—733no—
461#97—733no—
462#97—733no—
463#97—736no—
464#97—736no—
465#97—733no—
466#97—733no—
467C4×C8—731no—
468C4×C8—733no—
469C4×C8—733no—
470C4×C8—733no—
471C4×C8—733no—
472C4×C8—733no—
473#41—738no—
474#41—14612no—
475#41—14612no—
476#41—14612no—
477C2×Q8—24no—
478C2×Q8—733no—
479C2×Q8—733no—
480C2×Q8—14612no—
481C2×Q8—1464no—
482C2×Q8C22×#6440813yes—
483C2×Q8C22×#6440811yescharacteristic
484C2×D4—731no—
485C2×D4—733no—
486C2×D4—733no—
487C2×D4—733no—
488C22×C4—733no—
489C22×C4—733no—
490C22×C4—736no—
491C22×C4—733no—
492C22×C4—733no—
493C22×C4—731no—
494C22×C4—736no—
495C22×C4—733no—
496C22×C4—733no—
497C22×C4—731no—
498C22×C4C22×#6440813yes—
499M16—2923no—
500M16—2923no—
501M16—1462no—
502M16—1462no—
503C2×C8—2923no—
504C2×C8—2926no—
505C2×C8—731no—
506C2×C8—731no—
507C2×C8—731no—
508C2×C8—731no—
509C2×C8—733no—
510C2×C8—733no—
511C2×C8—733no—
512C2×C8—733no—
513C2×C8—1466no—
514C4⋊C4—733no—
515C4⋊C4—736no—
516C4⋊C4—733no—
517C4⋊C4—731no—
518C4⋊C4—733no—
519C4⋊C4—736no—
520C4⋊C4—733no—
521C4⋊C4—733no—
522C4⋊C4C22×#6440813yes—
523C4⋊C4C22×#6440813yes—
524C4⋊C4C22×#6440813yes—
525C4⋊C4C22×#6440813yes—
526C22⋊C4—733no—
527C22⋊C4—731no—
528C22⋊C4—736no—
529C22⋊C4—733no—
530C22⋊C4—733no—
531C22⋊C4C22×#6440813yes—
532C22⋊C4C22×#6440811yescharacteristic
533C42—733no—
534C42—733no—
535C42—733no—
536C42—736no—
537C42—733no—
538C42—731no—
539C42—733no—
540C42—733no—
541C42—733no—
542C42C22×#6440816yes—
543C42C22×#6440813yes—
544C42C22×#6440813yes—
545C23—733no—
546C23—731no—
547C23C23×#6440811yescharacteristic
548Q8—26no—
549Q8—1466no—
550Q8—1466no—
551Q8D73⋊M1614yes—
552D4—7312no—
553D4—1466no—
554D4—1462no—
555D4—1466no—
556C2×C4—26no—
557C2×C4—2923no—
558C2×C4—2923no—
559C2×C4—2926no—
560C2×C4—2923no—
561C2×C4—731no—
562C2×C4—731no—
563C2×C4—7312no—
564C2×C4—731no—
565C2×C4—733no—
566C2×C4—733no—
567C2×C4—733no—
568C2×C4—1463no—
569C2×C4—1463no—
570C2×C4—1461no—
571C2×C4—1461no—
572C2×C4—14612no—
573C2×C4—1461no—
574C2×C4—1461no—
575C2×C4—1466no—
576C2×C4C23×#6440813yes—
577C2×C4C23×#6440811yescharacteristic
578C2×C4C23×#6440813yes—
579C2×C4C23×#6440811yescharacteristic
580C2×C4C23×#6440813yes—
581C2×C4C23×#6440813yes—
582C8—732no—
583C8—1461no—
584C8—1463no—
585C8—1463no—
586C8—1463no—
587C8—1463no—
588C8—1461no—
589C8—1461no—
590C22—21no—
591C22—21no—
592C22—21no—
593C22—736no—
594C22—1461no—
595C22—1461no—
596C22—1461no—
597C22—1463no—
598C22—1463no—
599C22C24×#6440811yescharacteristic
600C4—23no—
601C4—21no—
602C4—23no—
603C4—23no—
604C4—21no—
605C4—731no—
606C4—731no—
607C4—732no—
608C4—1461no—
609C4—1463no—
610C4—1463no—
611C4—1463no—
612C4C2×D73⋊M1616yes—
613C2—41no—
614C2—2921no—
615C2—1466no—
616C22−1+4×#6440811yescharacteristic
617C2C22×D73⋊M1611yescharacteristic
618C2#2262692711yescharacteristic
subgroupG/H
center, FrattiniC22C24×#64408
commutator, socleC22×C73C24×C4
Fitting#519×C73C4
prime radicalC2×#14556639C4
cosocleC2×Dic73C25
hypercenter#519#64408
hypocenterC73#60343
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ #519

Complements

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

NH ≅ G/Ntype
1. #22626861C2semidirect
2. #22626858C2semidirect
3. #22626853C2semidirect
4. #22626849C2semidirect
5. #22626834C2semidirect
6. #22626833C2semidirect
7. #22626828C2semidirect
8. #22626773C2semidirect
9. #22626772C2semidirect
10. #22626742C2semidirect
11. #22626737C2semidirect
12. Q8×#11924038C2semidirect
13. Q8×#11924037C2semidirect
14. D73⋊#446C2semidirect
179. #11924029M16semidirect
279. #105968332−1+4semidirect
325. C73#60343semidirect

Maximal subgroups

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

subgroupindex
#226268612
#226268582
#226268532
#226268492
#226268342
#226268332
#226268282
#226267732
#226267722
#226267422
#226267372
Q8×#119240382
Q8×#119240372
D73⋊#4462
#226231402
#6034373