Finite Group Databasebeta

← #29689199

Subgroups — #29689199

Order 15616.

1382 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
1#21459593C211yesmaximal, characteristic
2#21454296C211yesmaximal, characteristic, complemented
3#21454278C211yesmaximal, characteristic, complemented
4#21446348C212yesmaximal, complemented
5#21446343C212yesmaximal, complemented
6#21446214C212yesmaximal, complemented
7#21446199C212yesmaximal, complemented
8#21445924C211yesmaximal, characteristic, complemented
9#21445920C211yesmaximal, characteristic, complemented
10#21445789C211yesmaximal, characteristic, complemented
11C61×#1795C211yesmaximal, characteristic, complemented
12C2×#14519823C2211yescharacteristic
13#16840528C2212yescomplemented
14#16840431—22no—
15D4×C22⋊D61—24no—
16D4×C22⋊D61—22no—
17#16840329C412yescomplemented
18#16840329C2211yescharacteristic
19#16840326C412yescomplemented
20#16840326C2211yescharacteristic
21#16840308C412yescomplemented
22#16840308C2211yescharacteristic
23C2×#14519773C412yescomplemented
24C2×#14519773C2211yescharacteristic
25#16840236C2212yescomplemented
26#41×Dic61C2211yescharacteristic
27#16839779C2212yescomplemented
28#16839767C2212yescomplemented
29#16839746C2211yescharacteristic
30#16839745—22no—
31C2×#14519693C2212yescomplemented
32#100×D61—22no—
33#16839736C2211yescharacteristic
34#16839735C2212yescomplemented
35#99×D61—24no—
36#16839523C2212yescomplemented
37#16839521C2212yescomplemented
38#16839519C2212yescomplemented
39#16839518C2212yescomplemented
40C2×H(5,𝔽2)×C61C2211yescharacteristic
41C61×#421C2212yescomplemented
42C61×#418C2212yescomplemented
43C61×#413C2211yescharacteristic
44C61×#410C2211yescharacteristic
45#14519823—42no—
46#14519823—42no—
47C2×#11758226D412yes—
48C2×#11758222C2×C412yes—
49C2×#11758222C2311yescharacteristic
50C22×D244C2×C412yes—
51C22×D244C2311yescharacteristic
52C22×C22⋊D61—24no—
53C22×C22⋊D61—22no—
54C2×D4×D61D414yes—
55C2×D4×D61D412yes—
56#14519785—42no—
57#14519783—42no—
58C2×#11758194C2312yes—
59M16×D61—24no—
60#14519775C2×C411yescharacteristic, complemented
61#14519773—24no—
62#14519773—22no—
63#14519773C2×C412yes—
64#14519773C2×C412yes—
65#14519773C2×C412yes—
66#14519773C2×C411yescharacteristic, complemented
67#14519773C2×C412yescomplemented
68#14519766C2×C412yes—
69#14519766C2×C411yescharacteristic, complemented
70#14519764C2×C412yescomplemented
71#14519764C2×C411yescharacteristic, complemented
72#14519751—24no—
73#14519751—22no—
74#14519749—24no—
75#14519749—22no—
76#14519748—24no—
77#14519748—22no—
78#14519746—24no—
79#14519746—22no—
80#14519745—24no—
81#14519745—22no—
82Q8×Dic61—21no—
83C4×D244—42no—
84D4×Dic61—22no—
85D4×Dic61—21no—
86#14519739—42no—
87C4×C22⋊D61—24no—
88C4×C22⋊D61—22no—
89#14519736C2×C412yescomplemented
90#14519736C2312yes—
91#14519736C2311yescharacteristic
92C22⋊C4×D61—24no—
93C2×C4×Dic61C2×C412yescomplemented
94C2×C4×Dic61C2311yescharacteristic
95#14519704—42no—
96#14519698—42no—
97#14519696—21no—
98#14519694—21no—
99#14519693—24no—
100#14519692—22no—
101#14519691—24no—
102H(5,𝔽2)×C61—42no—
103H(5,𝔽2)×C61—42no—
104C2×#41×C61—21no—
105C2×#41×C61C2×C412yescomplemented
106C2×#41×C61C2311yescharacteristic
107C2×#41×C61C2311yescharacteristic
108C22×D4×C61—22no—
109C22×D4×C61—21no—
110C22×D4×C61C2×C412yescomplemented
111C22×D4×C61C2311yescharacteristic
112C2×M16×C61C2312yes—
113#117×C61C2312yes—
114#104×C61—42no—
115#102×C61—42no—
116#101×C61—21no—
117#100×C61—21no—
118#99×C61—22no—
119C24×C61—22no—
120C24×C61—21no—
121C23×D61C22⋊C414yes—
122C23×D61C2×D412yes—
123#41×C61—22no—
124#41×C61—22no—
125#41×C61—22no—
126#41×C61—22no—
127#41×C61—42no—
128#41×C61—42no—
129#41×C61—44no—
130C2×Q8×C61C22×C411yescharacteristic
131C2×Q8×C61C2×D411yescharacteristic
132C2×D4×C61—22no—
133C2×D4×C61—21no—
134C2×D4×C61—22no—
135C2×D4×C61—21no—
136C2×D4×C61—22no—
137C2×D4×C61—22no—
138C2×D4×C61—42no—
139C2×D4×C61—42no—
140C2×D4×C61—48no—
141C2×D4×C61—44no—
142C2×D4×C61C22×C411yescharacteristic
143C2×D4×C61C22×C412yes—
144C2×D4×C61C2×D412yes—
145C2×D4×C61C2×D411yescharacteristic
146C22×C4×C61—22no—
147C22×C4×C61—21no—
148C22×C4×C61C22×C412yes—
149C22×C4×C61C2411yescharacteristic
150C2×C22⋊D61—24no—
151C2×C22⋊D61—22no—
152C2×C22⋊D61—24no—
153C2×C22⋊D61—24no—
154C2×C22⋊D61—24no—
155C2×C22⋊D61—84no—
156C2×C22⋊D61—88no—
157C2×C22⋊D61C22⋊C414yes—
158C2×C22⋊D61C2×D412yes—
159C22×Dic61—22no—
160C22×Dic61—21no—
161#11758226—44no—
162D4×D61—44no—
163D4×D61—48no—
164#11758222—22no—
165#11758222—22no—
166#11758222—42no—
167#11758222—42no—
168C2×D244—22no—
169C2×D244—22no—
170C2×D244—42no—
171C2×D244—42no—
172C2×D244C22⋊C414yes—
173C2×D244C22×C411yescharacteristic
174C2×D244C22×C412yes—
175C2×D244C2×D412yes—
176C2×C4×D61—44no—
177C2×C4×D61C22⋊C414yes—
178C2×C4×D61C2×D412yes—
179C2×Dic122C22×C411yescharacteristic
180M16×C61—22no—
181M16×C61—22no—
182C2×C8×C61—22no—
183C4⋊C4×C61—22no—
184C22⋊C4×C61—22no—
185C42×C61—22no—
186C22⋊Dic61—22no—
187C22⋊Dic61—21no—
188C22⋊Dic61—24no—
189C22⋊Dic61—22no—
190C22⋊Dic61—22no—
191#11758198—24no—
192#11758198—22no—
193#11758198—44no—
194C4⋊Dic61—22no—
195C4⋊Dic61—21no—
196C4⋊Dic61—22no—
197#11758196—22no—
198C4×Dic61—22no—
199C4×Dic61—21no—
200C4×Dic61—22no—
201C4×Dic61C22×C411yescharacteristic
202C4×Dic61C22×C412yes—
203C4×Dic61C22×C411yescharacteristic
204#11758194—22no—
205#11758194—22no—
206C2×#92418—22no—
207#11758189—44no—
208C8×D61—44no—
209C23×C61—22no—
210C23×C61—21no—
211C23×C61—44no—
212C23×C61—42no—
213C23×C61—82no—
214C23×C61—84no—
215C23×C61C2×C22⋊C412yes—
216C23×C61C22×D411yescharacteristic
217C22×D61—24no—
218C22×D61—44no—
219C22×D61—82no—
220C22×D61—84no—
221C22×D61C2×C22⋊C414yes—
222C22×D61C2×C22⋊C412yes—
223Q8×C61—22no—
224Q8×C61—21no—
225Q8×C61—21no—
226D4×C61—24no—
227D4×C61—22no—
228D4×C61—22no—
229D4×C61—21no—
230D4×C61—21no—
231D4×C61—22no—
232D4×C61—44no—
233D4×C61—44no—
234D4×C61—42no—
235D4×C61—42no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—21no—
239C2×C4×C61—44no—
240C2×C4×C61—42no—
241C2×C4×C61—42no—
242C2×C4×C61C2×C22⋊C412yes—
243C2×C4×C61C23×C411yescharacteristic
244C2×C4×C61C22×D411yescharacteristic
245C2×C4×C61H(5,𝔽2)12yes—
246C2×C4×C61H(5,𝔽2)12yes—
247C22⋊D61—24no—
248C22⋊D61—44no—
249C22⋊D61—48no—
250C22⋊D61—44no—
251C22⋊D61—84no—
252C22⋊D61—84no—
253C22⋊D61—84no—
254C2×Dic61—21no—
255C2×Dic61—22no—
256C2×Dic61—21no—
257C2×Dic61—42no—
258C2×Dic61—44no—
259C2×Dic61—42no—
260C2×Dic61C2×C22⋊C412yes—
261D244—22no—
262D244—22no—
263D244—21no—
264D244—21no—
265D244—24no—
266D244—44no—
267C4×D61—24no—
268C4×D61—44no—
269C4×D61—164no—
270Dic122—21no—
271Dic122—21no—
272C8×C61—42no—
273#92418—42no—
274#21782—611nomaximal, Sylow
275C22×C61—22no—
276C22×C61—41no—
277C22×C61—42no—
278C22×C61—44no—
279C22×C61—42no—
280C22×C61—42no—
281C22×C61—41no—
282C22×C61—162no—
283C22×C61—164no—
284C22×C61C22×C22⋊C411yescharacteristic
285C22×C61#53412yes—
286C2×D61—22no—
287C2×D61—24no—
288C2×D61—44no—
289C2×D61—42no—
290C2×D61—84no—
291C2×D61—164no—
292C4×C61—22no—
293C4×C61—41no—
294C4×C61—82no—
295C4×C61#51811yescharacteristic
296C4×C61#51811yescharacteristic
297C4×C61#53412yes—
298Dic61—22no—
299Dic61—22no—
300Dic61—42no—
301Dic61—41no—
302Dic61—82no—
303#3498—611no—
304#2888—611no—
305#2883—611no—
306#1911—612no—
307#1910—612no—
308#1887—612no—
309#1884—612no—
310#1827—611no—
311#1827—611no—
312#1795—611no—
313#1795D6111yescharacteristic, complemented
314C2×C61—21no—
315C2×C61—22no—
316C2×C61—42no—
317C2×C61—82no—
318C2×C61—81no—
319C2×C61#230811yescharacteristic
320D61—44no—
321D61—42no—
322C2×H(5,𝔽2)—611no—
323C2×H(5,𝔽2)C2×D6111yescharacteristic
324#568—612no—
325#550—1222no—
326D42—1224no—
327D42—1222no—
328#535—612no—
329#535—611no—
330#535—612no—
331#535—611no—
332#532—612no—
333#532—611no—
334C2×#127—612no—
335C2×#127—611no—
336#518—612no—
337C4×#41—611no—
338#421—612no—
339#421C2×D6112yescomplemented
340#418—612no—
341#418C2×D6112yescomplemented
342#413—611no—
343#413—1222no—
344#413C2×D6111yescharacteristic
345C2×#100—612no—
346C2×#100—1222no—
347#410—611no—
348#410—612no—
349#410C2×D6111yescharacteristic
350C2×#99—1224no—
351#339—612no—
352#339—612no—
353H(3,ℤ/4)—612no—
354H(3,ℤ/4)—612no—
355C61#2178211yescharacteristic, complemented, Sylow
356H(5,𝔽2)—42no—
357H(5,𝔽2)—42no—
358H(5,𝔽2)—2442no—
359H(5,𝔽2)—2442no—
360C2×#41—21no—
361C2×#41—612no—
362C2×#41—612no—
363C2×#41—611no—
364C2×#41C4×D6112yescomplemented
365C2×#41C22×D6111yescharacteristic
366C2×#41C22×D6111yescharacteristic
367C22×D4—21no—
368C22×D4—22no—
369C22×D4—614no—
370C22×D4—612no—
371C22×D4—612no—
372C22×D4—611no—
373C22×D4—1222no—
374C22×D4—1224no—
375C22×D4C4×D6112yescomplemented
376C22×D4C22×D6111yescharacteristic
377#131—2442no—
378#131—2442no—
379C2×M16—612no—
380C2×M16—1224no—
381C2×M16C22×D6112yes—
382C4⋊Q8—611no—
383#127—612no—
384#127—611no—
385#127—612no—
386#127—612no—
387#127—612no—
388#127—1222no—
389#127—1224no—
390#124—612no—
391#124—611no—
392#124—611no—
393#124—612no—
394#121—1222no—
395#121—1224no—
396#121—1222no—
397#121—1224no—
398#121—1222no—
399#121—1224no—
400#120—1222no—
401#120—1224no—
402#120—1222no—
403#120—1224no—
404C4×Q8—1221no—
405C4×D4—1222no—
406C4×D4—1224no—
407C4×D4—1221no—
408C4×D4—1222no—
409C4×D4—2442no—
410C4×D4—2442no—
411#117—612no—
412#117—611no—
413#117—612no—
414#117C22×D6112yes—
415C2×C22⋊C4—1224no—
416C2×C42—612no—
417C2×C42—611no—
418#104—42no—
419#104—2442no—
420#102—42no—
421#102—2442no—
422#101—21no—
423#101—1221no—
424#100—21no—
425#100—1224no—
426#100—1221no—
427#99—22no—
428#99—1224no—
429#99—1222no—
430C24—21no—
431C24—22no—
432C24—612no—
433C24—614no—
434#41—22no—
435#41—22no—
436#41—22no—
437#41—22no—
438#41—42no—
439#41—42no—
440#41—44no—
441#41—1222no—
442#41—1222no—
443#41—2442no—
444#41—2442no—
445#41—2444no—
446C2×Q8—611no—
447C2×Q8C2×C4×D6111yescharacteristic
448C2×Q8D4×D6111yescharacteristic
449C2×D4—21no—
450C2×D4—22no—
451C2×D4—21no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4—48no—
458C2×D4—44no—
459C2×D4—612no—
460C2×D4—614no—
461C2×D4—612no—
462C2×D4—611no—
463C2×D4—612no—
464C2×D4—614no—
465C2×D4—1224no—
466C2×D4—1224no—
467C2×D4—1224no—
468C2×D4—1224no—
469C2×D4—1222no—
470C2×D4—1222no—
471C2×D4—1222no—
472C2×D4—4888no—
473C2×D4—4884no—
474C2×D4—2442no—
475C2×D4—2442no—
476C2×D4—2444no—
477C2×D4—2448no—
478C2×D4C2×C4×D6111yescharacteristic
479C2×D4C2×C4×D6112yes—
480C2×D4D4×D6111yescharacteristic
481C2×D4D4×D6112yes—
482C22×C4—21no—
483C22×C4—22no—
484C22×C4—612no—
485C22×C4—614no—
486C22×C4—1221no—
487C22×C4—1222no—
488C22×C4—2444no—
489C22×C4C2×C4×D6112yes—
490C22×C4C23×D6111yescharacteristic
491M16—22no—
492M16—22no—
493M16—1222no—
494M16—1222no—
495M16—2444no—
496C2×C8—22no—
497C2×C8—1222no—
498C2×C8—2444no—
499C4⋊C4—22no—
500C4⋊C4—1222no—
501C4⋊C4—1221no—
502C4⋊C4—1222no—
503C4⋊C4—1222no—
504C22⋊C4—22no—
505C22⋊C4—1224no—
506C22⋊C4—1222no—
507C22⋊C4—1222no—
508C22⋊C4—1224no—
509C22⋊C4—1221no—
510C22⋊C4—1222no—
511C22⋊C4—1222no—
512C22⋊C4—2444no—
513C42—22no—
514C42—611no—
515C42—612no—
516C42—611no—
517C42—1221no—
518C42—1222no—
519C42—1222no—
520C23—21no—
521C23—22no—
522C23—44no—
523C23—42no—
524C23—82no—
525C23—84no—
526C23—612no—
527C23—614no—
528C23—1224no—
529C23—4884no—
530C23—4882no—
531C23—2444no—
532C23C22⋊C4×D6112yes—
533C23C2×D4×D6111yescharacteristic
534Q8—22no—
535Q8—21no—
536Q8—21no—
537Q8—1221no—
538Q8—1221no—
539D4—24no—
540D4—22no—
541D4—22no—
542D4—21no—
543D4—21no—
544D4—22no—
545D4—44no—
546D4—44no—
547D4—42no—
548D4—42no—
549D4—1224no—
550D4—1222no—
551D4—1222no—
552D4—1221no—
553D4—1221no—
554D4—1224no—
555D4—4884no—
556D4—4884no—
557D4—4884no—
558D4—2444no—
559D4—2444no—
560D4—2444no—
561D4—2448no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—612no—
569C2×C4—1221no—
570C2×C4—1224no—
571C2×C4—1221no—
572C2×C4—1222no—
573C2×C4—9764no—
574C2×C4—2444no—
575C2×C4—2442no—
576C2×C4—2444no—
577C2×C4—2442no—
578C2×C4C22⋊C4×D6112yes—
579C2×C4C22×C4×D6111yescharacteristic
580C2×C4C2×D4×D6111yescharacteristic
581C2×C4#1451982212yes—
582C2×C4#1451982212yes—
583C8—42no—
584C8—2442no—
585C22—22no—
586C22—41no—
587C22—42no—
588C22—44no—
589C22—42no—
590C22—42no—
591C22—41no—
592C22—162no—
593C22—164no—
594C22—1222no—
595C22—1224no—
596C22—9764no—
597C22—4884no—
598C22—2442no—
599C22—2444no—
600C22C2×C22⋊C4×D6111yescharacteristic
601C22#1684032212yes—
602C4—22no—
603C4—41no—
604C4—82no—
605C4—1222no—
606C4—1222no—
607C4—4882no—
608C4—2441no—
609C4—2442no—
610C4#1684023511yescharacteristic
611C4#1684023511yescharacteristic
612C4#1684032212yes—
613C2—22no—
614C2—21no—
615C2—42no—
616C2—82no—
617C2—81no—
618C2—2442no—
619C2—2444no—
620C2#2144839511yescharacteristic
subgroupG/H
centerC2#21448395
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1795C2
socleC2×C61#2308
prime radical#21459593C2
cosocleC22×C4×C61C24
hypercenter#1795D61
hypocenterC61#21782
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1795

Complements

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

NH ≅ G/Ntype
2. #21454296C2semidirect
3. #21454278C2semidirect
4. #21446348C2semidirect
5. #21446343C2semidirect
6. #21446214C2semidirect
7. #21446199C2semidirect
8. #21445924C2semidirect
9. #21445920C2semidirect
10. #21445789C2semidirect
11. C61×#1795C2semidirect
13. #16840528C22semidirect
17. #16840329C4semidirect
19. #16840326C4semidirect
21. #16840308C4semidirect
23. C2×#14519773C4semidirect
25. #16840236C22semidirect
27. #16839779C22semidirect
28. #16839767C22semidirect
31. C2×#14519693C22semidirect
34. #16839735C22semidirect
36. #16839523C22semidirect
37. #16839521C22semidirect
38. #16839519C22semidirect
39. #16839518C22semidirect
41. C61×#421C22semidirect
42. C61×#418C22semidirect
60. #14519775C2×C4semidirect
66. #14519773C2×C4semidirect
67. #14519773C2×C4semidirect
69. #14519766C2×C4semidirect
70. #14519764C2×C4semidirect
71. #14519764C2×C4semidirect
89. #14519736C2×C4semidirect
93. C2×C4×Dic61C2×C4semidirect
105. C2×#41×C61C2×C4semidirect
110. C22×D4×C61C2×C4semidirect
313. #1795D61semidirect
339. #421C2×D61semidirect
341. #418C2×D61semidirect
355. C61#21782semidirect
364. C2×#41C4×D61semidirect
375. C22×D4C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214595932
#214542962
#214542782
#214463482
#214463432
#214462142
#214461992
#214459242
#214459202
#214457892
C61×#17952
#2178261