Finite Group Databasebeta

← #30544354

Subgroups — #30544354

Order 15616.

2604 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21460590C211yesmaximal, characteristic
2#21460532C211yesmaximal, characteristic, complemented
3#21460531C211yesmaximal, characteristic, complemented
4#21454732C216yesmaximal, complemented
5#21454725C216yesmaximal, complemented
6#21454718C216yesmaximal, complemented
7C2×#16839881C216yesmaximal, complemented
8#21454428C211yesmaximal, characteristic, complemented
9#21454367C211yesmaximal, characteristic, complemented
10#21454247C211yesmaximal, characteristic, complemented
11C61×#2901C211yesmaximal, characteristic, complemented
12#16840598C2211yescharacteristic
13#16840577C414yescomplemented
14#16840573C414yescomplemented
15#16840570C414yescomplemented
16#16840565C414yescomplemented
17#16840560C2216yes—
18C2×#14519797C2216yes—
19#16840529—24no—
20#16840529C2211yescharacteristic
21#16840529C2216yescomplemented
22#16840528C2216yescomplemented
23#16840527—24no—
24C2×#14519785C2216yescomplemented
25C2×#14519784—24no—
26C2×#14519784C2211yescharacteristic
27C2×#14519784C2211yescharacteristic
28C2×#14519784C2216yescomplemented
29#131×D61—24no—
30#16839881C2214yescomplemented
31#16839881C22112yescomplemented
32#16839881C22112yescomplemented
33#16839881C22112yescomplemented
34#16839876C22124yescomplemented
35#16839810C2216yescomplemented
36C2×#14519712C2216yescomplemented
37#16839780C2216yescomplemented
38C2×#14519705C2216yescomplemented
39#16839718C2213yescomplemented
40#16839718C2213yescomplemented
41C4×#11758194C2213yescomplemented
42C2×#14519688C2213yescomplemented
43C61×#585C2211yescharacteristic
44C61×#568C2211yescharacteristic
45C4×#41×C61C2211yescharacteristic
46C61×#421C2216yescomplemented
47C2×#104×C61C2216yescomplemented
48#14519825—21no—
49#14519824C2×C414yes—
50#14519823C2×C414yes—
51#14519822—23no—
52C2×#11758222C2316yes—
53#41×D61—24no—
54#14519808C2×C4112yescomplemented
55#14519807C2×C4112yescomplemented
56#14519803C2×C4112yescomplemented
57#14519802C2×C4112yescomplemented
58#14519797C2×C4112yescomplemented
59#14519797C2×C4112yescomplemented
60#14519797C2314yes—
61#14519797C23112yes—
62C2×#11758202C2×C416yescomplemented
63C2×#11758200C2×C416yescomplemented
64C2×#11758201C2×C416yescomplemented
65C2×#11758199C2×C416yescomplemented
66#14519785—26no—
67#14519785—26no—
68#14519785C2314yes—
69#14519785C23112yes—
70#14519784—24no—
71#14519784—212no—
72#14519784—26no—
73#14519784—26no—
74#14519784C2×C414yes—
75#14519784C2×C414yes—
76#14519784D414yes—
77#14519784D414yes—
78#14519784C2314yes—
79#14519784C23112yes—
80#14519783—212no—
81C2×#11758194—212no—
82C2×#11758194—26no—
83C2×#11758194C2316yes—
84C2×#11758194C2313yes—
85C2×#11758194C2313yes—
86C2×#11758189—26no—
87M16×D61—212no—
88C22×#92418—26no—
89C22×#92418C2313yes—
90C22×#92418C2313yes—
91C2×C8×D61—26no—
92#14519712C2314yes—
93#14519712C23112yes—
94#14519705C2314yes—
95#14519705C23112yes—
96#14519688C2311yescharacteristic
97#14519688C2313yes—
98#14519688C2313yes—
99#14519688C2313yes—
100C4×#92418C2316yes—
1012−1+4×C61—21no—
1022−1+4×C61C2×C414yes—
103H(5,𝔽2)×C61—23no—
104H(5,𝔽2)×C61C2×C414yes—
105C2×#41×C61—24no—
106C2×#41×C61C2316yes—
107C2×#41×C61C2311yescharacteristic
108#131×C61—24no—
109C2×M16×C61C2316yes—
110C4×Q8×C61—21no—
111C4×D4×C61—23no—
112#117×C61C2313yes—
113C2×C42×C61C2313yes—
114#104×C61C2314yes—
115#104×C61C23112yes—
116#41×C61—24no—
117#41×C61—212no—
118#41×C61—26no—
119#41×C61—26no—
120#41×C61C22×C4112yes—
121#41×C61C22×C414yes—
122#41×C61C2×D414yes—
123#41×C61C2414yes—
124C2×Q8×C61—24no—
125C2×Q8×C61C22×C416yes—
126C2×Q8×C61#4111yescharacteristic
127C2×D4×C61—26no—
128C2×D4×C61—212no—
129C2×D4×C61C22×C416yes—
130C2×D4×C61#4113yes—
131C22×C4×C61—26no—
132C22×C4×C61—23no—
133C22×C4×C61C2413yes—
134C2×C22⋊D61—26no—
135#11758226—24no—
136Q8×D61—24no—
137#11758224—212no—
138D4×D61—212no—
139#11758222—26no—
140#11758222—26no—
141#11758222C22×C4112yes—
142#11758222C2414yes—
143C2×D244C22×C416yes—
144C2×C4×D61—26no—
145C2×Dic122C22×C416yes—
146M16×C61—26no—
147M16×C61C2414yes—
148C2×C8×C61—26no—
149C4⋊C4×C61—23no—
150C22⋊C4×C61—23no—
151C42×C61—23no—
152C42×C61C2411yescharacteristic
153C42×C61C2413yes—
154#11758202C22×C4112yes—
155#11758202C22×C414yes—
156#11758201C22×C4112yes—
157#11758201C22×C414yes—
158#11758200C22×C414yes—
159#11758200C22×C4112yes—
160#11758199C22×C414yes—
161#11758199C22×C4112yes—
162#11758194—26no—
163#11758194—24no—
164#11758194—212no—
165#11758194C22×C4112yes—
166#11758194C2×D4112yes—
167#11758194C2414yes—
168C2×#92418—26no—
169C2×#92418—24no—
170C2×#92418—212no—
171C2×#92418C22×C416yes—
172C2×#92418C22×C416yes—
173C2×#92418C2×D416yes—
174C2×#92418C2×D416yes—
175C2×#92418C2413yes—
176C2×#92418C2411yescharacteristic
177C2×#92418C2413yes—
178C2×#92418C2411yescharacteristic
179#11758189—212no—
180#11758189—24no—
181C8×D61—24no—
182C8×D61—212no—
183C23×C61—26no—
184C23×C61C2×#4113yes—
185C22×D61—26no—
186Q8×C61—26no—
187Q8×C61C4×D414yes—
188Q8×C61C23×C414yes—
189D4×C61—212no—
190D4×C61—24no—
191D4×C61—26no—
192D4×C61C4×D4112yes—
193D4×C61C23×C414yes—
194C2×C4×C61—26no—
195C2×C4×C61—212no—
196C2×C4×C61—24no—
197C2×C4×C61—21no—
198C2×C4×C61—23no—
199C2×C4×C61—83no—
200C2×C4×C61C23×C416yes—
201C2×C4×C61C22×D416yes—
202C2×C4×C61C2×#4113yes—
203C2×C4×C61C2511yescharacteristic
204C22⋊D61—212no—
205C22⋊D61—24no—
206C2×Dic61—26no—
207D244—26no—
208D244C23×C414yes—
209C4×D61—212no—
210C4×D61—24no—
211Dic122—26no—
212Dic122C23×C414yes—
213C8×C61—24no—
214#92418—24no—
215#92418C23×C411yescharacteristic
216#92418C23×C411yescharacteristic
217#92418C23×C416yes—
218#92418C22×D411yescharacteristic
219#92418C22×D416yes—
220#92418C22×D411yescharacteristic
221#60184—611nomaximal, Sylow
222C22×C61—24no—
223C22×C61—43no—
224C22×C61—86no—
225C22×C61C2×C4×D416yes—
226C22×C61C22×#4111yescharacteristic
227C2×D61—24no—
228C2×D61—86no—
229C4×C61—24no—
230C4×C61—43no—
231C4×C61—41no—
232C4×C61C2×C4×D416yes—
233C4×C61C24×C411yescharacteristic
234C4×C61C23×D411yescharacteristic
235Dic61—24no—
236#3584—611no—
237#3573—611no—
238#3573—611no—
239#2957—616no—
240#2956—616no—
241#2955—616no—
242C2×#444—616no—
243#2916—611no—
244#2901—611no—
245#2901D6111yescharacteristic, complemented
246#2875—611no—
247C2×C61—21no—
248C2×C61—26no—
249C2×C61—44no—
250C2×C61C22×C4×D411yescharacteristic
251D61—44no—
252#585—611no—
253#585C2×D6111yescharacteristic
254#578—614no—
255#577—614no—
256#577—614no—
257#576—614no—
258#575—616no—
259C2×#135—616no—
260#568—611no—
261#568—616no—
262#568—616no—
263#568—1224no—
264#568—1224no—
265#568C2×D6111yescharacteristic
266C2×#131—611no—
267C2×#131—611no—
268C2×#131—616no—
269C2×#131—616no—
270C2×#131—1224no—
271C2×#131—1224no—
272C4×#41C2×D6111yescharacteristic
273#444—614no—
274#444—6112no—
275#444—6112no—
276#444—6112no—
277#443—6124no—
278#430—616no—
279C2×#108—616no—
280#421—616no—
281#421C2×D6116yescomplemented
282C2×#104—616no—
283C2×#104C2×D6116yescomplemented
284#405—613no—
285#405—613no—
286C4×M16—613no—
287C2×#97—613no—
288C61#6018411yescharacteristic, complemented, Sylow
2892−1+4—21no—
2902−1+4—614no—
2912−1+4—1221no—
2922−1+4C4×D6114yes—
293H(5,𝔽2)—23no—
294H(5,𝔽2)—614no—
295H(5,𝔽2)—1223no—
296H(5,𝔽2)C4×D6114yes—
297C2×#41—24no—
298C2×#41—616no—
299C2×#41—1224no—
300C2×#41C22×D6116yes—
301C2×#41C22×D6111yescharacteristic
302#137—6112no—
303#137—6112no—
304AGL(1,ℤ/8)—6112no—
305AGL(1,ℤ/8)—6112no—
306#135—6112no—
307#135—6112no—
308#135—614no—
309#135—6112no—
310C2×Q16—616no—
311C2×SD16—616no—
312C2×SD16—616no—
313C2×D8—616no—
314#131—24no—
315#131—614no—
316#131—614no—
317#131—614no—
318#131—614no—
319#131—614no—
320#131—6112no—
321#131—614no—
322#131—6112no—
323#131—12212no—
324#131—1226no—
325#131—1226no—
326#131—1226no—
327#131—1226no—
328#131—12212no—
329#131—1224no—
330C2×M16—613no—
331C2×M16—613no—
332C2×M16—616no—
333C2×M16—1226no—
334C2×M16—1226no—
335C2×M16—12212no—
336C2×M16—12212no—
337C2×M16C22×D6116yes—
338C22×C8—613no—
339C22×C8—613no—
340C22×C8—1226no—
341C22×C8—1226no—
342C4×Q8—21no—
343C4×D4—23no—
344#117C22×D6113yes—
345C2×C42C22×D6113yes—
346#108—614no—
347#108—6112no—
348#104—614no—
349#104—6112no—
350#104C22×D6114yes—
351#104C22×D61112yes—
352#97—611no—
353#97—613no—
354#97—613no—
355#97—613no—
356C4×C8—616no—
357#41—24no—
358#41—212no—
359#41—26no—
360#41—26no—
361#41—6112no—
362#41—614no—
363#41—1224no—
364#41—12212no—
365#41—1226no—
366#41—1226no—
367#41C2×C4×D61112yes—
368#41C2×C4×D6114yes—
369#41D4×D6114yes—
370#41C23×D6114yes—
371C2×Q8—24no—
372C2×Q8—616no—
373C2×Q8—1224no—
374C2×Q8C2×C4×D6116yes—
375C2×Q8#1175822411yescharacteristic
376C2×D4—26no—
377C2×D4—212no—
378C2×D4—616no—
379C2×D4—1226no—
380C2×D4—12212no—
381C2×D4C2×C4×D6116yes—
382C2×D4#1175822413yes—
383C22×C4—26no—
384C22×C4—23no—
385C22×C4—1226no—
386C22×C4C23×D6113yes—
387Q16—614no—
388Q16—6112no—
389SD16—614no—
390SD16—6112no—
391SD16—614no—
392SD16—6112no—
393D8—614no—
394D8—6112no—
395M16—26no—
396M16—6112no—
397M16—6112no—
398M16—614no—
399M16—1224no—
400M16—12212no—
401M16—1224no—
402M16—12212no—
403M16—1226no—
404M16C23×D6114yes—
405C2×C8—26no—
406C2×C8—616no—
407C2×C8—616no—
408C2×C8—616no—
409C2×C8—616no—
410C2×C8—613no—
411C2×C8—613no—
412C2×C8—611no—
413C2×C8—611no—
414C2×C8—1224no—
415C2×C8—12212no—
416C2×C8—1224no—
417C2×C8—12212no—
418C2×C8—1226no—
419C4⋊C4—23no—
420C22⋊C4—23no—
421C42—23no—
422C42C23×D6113yes—
423C42C23×D6111yescharacteristic
424C23—26no—
425C23—1226no—
426C23C2×#1175822413yes—
427Q8—26no—
428Q8—614no—
429Q8—1226no—
430Q8#1451973714yes—
431Q8C22×C4×D6114yes—
432D4—212no—
433D4—24no—
434D4—26no—
435D4—614no—
436D4—12212no—
437D4—1224no—
438D4—1226no—
439D4#14519737112yes—
440D4C22×C4×D6114yes—
441C2×C4—26no—
442C2×C4—212no—
443C2×C4—24no—
444C2×C4—21no—
445C2×C4—23no—
446C2×C4—83no—
447C2×C4—1226no—
448C2×C4—12212no—
449C2×C4—1224no—
450C2×C4C22×C4×D6116yes—
451C2×C4C2×D4×D6116yes—
452C2×C4C2×#1175822413yes—
453C2×C4C24×D6111yescharacteristic
454C8—24no—
455C8—611no—
456C8—611no—
457C8—616no—
458C8—611no—
459C8—611no—
460C8—616no—
461C8—1224no—
462C22—24no—
463C22—43no—
464C22—86no—
465C22—1224no—
466C22—4886no—
467C22C2×#1451973716yes—
468C22C22×#1175822411yescharacteristic
469C4—24no—
470C4—43no—
471C4—41no—
472C4—1224no—
473C4C2×#1451973716yes—
474C4C23×C4×D6111yescharacteristic
475C4C22×D4×D6111yescharacteristic
476C2—21no—
477C2—26no—
478C2—44no—
479C2—2444no—
480C2C22×#1451973711yescharacteristic
subgroupG/H
centerC4C22×D4×D61
commutatorC4×C61C24×C4
FrattiniC2×C4C24×D61
FittingC61×#2901C2
socleC2×C61C22×C4×D4
prime radical#21460590C2
cosocleC2×C4×C61C25
hypercenter#2901D61
hypocenterC61#60184
LCSG ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C2×#41 ◁ #2901

Complements

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

NH ≅ G/Ntype
2. #21460532C2semidirect
3. #21460531C2semidirect
4. #21454732C2semidirect
5. #21454725C2semidirect
6. #21454718C2semidirect
7. C2×#16839881C2semidirect
8. #21454428C2semidirect
9. #21454367C2semidirect
10. #21454247C2semidirect
11. C61×#2901C2semidirect
13. #16840577C4semidirect
14. #16840573C4semidirect
15. #16840570C4semidirect
16. #16840565C4semidirect
21. #16840529C22semidirect
22. #16840528C22semidirect
24. C2×#14519785C22semidirect
28. C2×#14519784C22semidirect
30. #16839881C22semidirect
31. #16839881C22semidirect
32. #16839881C22semidirect
33. #16839881C22semidirect
34. #16839876C22semidirect
35. #16839810C22semidirect
36. C2×#14519712C22semidirect
37. #16839780C22semidirect
38. C2×#14519705C22semidirect
39. #16839718C22semidirect
40. #16839718C22semidirect
41. C4×#11758194C22semidirect
42. C2×#14519688C22semidirect
46. C61×#421C22semidirect
47. C2×#104×C61C22semidirect
54. #14519808C2×C4semidirect
55. #14519807C2×C4semidirect
56. #14519803C2×C4semidirect
57. #14519802C2×C4semidirect
58. #14519797C2×C4semidirect
59. #14519797C2×C4semidirect
62. C2×#11758202C2×C4semidirect
63. C2×#11758200C2×C4semidirect
64. C2×#11758201C2×C4semidirect
65. C2×#11758199C2×C4semidirect
245. #2901D61semidirect
281. #421C2×D61semidirect
283. C2×#104C2×D61semidirect
288. C61#60184semidirect

Maximal subgroups

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

subgroupindex
#214605902
#214605322
#214605312
#214547322
#214547252
#214547182
C2×#168398812
#214544282
#214543672
#214542472
C61×#29012
#6018461