Finite Group Databasebeta

← #30544356

Subgroups — #30544356

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#21460530C211yesmaximal, characteristic, complemented
3#21454734C216yesmaximal, complemented
4#21454726C216yesmaximal, complemented
5#21454720C216yesmaximal, complemented
6C2×#16839882C216yesmaximal, complemented
7#21454428C211yesmaximal, characteristic, complemented
8#21454366C211yesmaximal, characteristic, complemented
9#21454364C211yesmaximal, characteristic, complemented
10#21454245C211yesmaximal, characteristic, complemented
11C61×#3573C211yesmaximal, characteristic, complemented
12#16840598C2211yescharacteristic
13#16840577C414yescomplemented
14#16840573C414yescomplemented
15#16840570C414yescomplemented
16#16840565C414yescomplemented
17#16840560C2216yes—
18C2×#14519797C2216yes—
19#16840529C2211yescharacteristic
20#16840528C2216yescomplemented
21#16840527—24no—
22C2×#14519784C2211yescharacteristic
23C2×#14519783C2216yescomplemented
24#131×D61—24no—
25#41×Dic61C2211yescharacteristic
26#16839882C2214yescomplemented
27#16839882C22112yescomplemented
28#16839882C22112yescomplemented
29#16839882C22112yescomplemented
30#16839877C22124yescomplemented
31#16839810C2216yescomplemented
32C2×#14519712C2216yescomplemented
33#16839779C2216yescomplemented
34#16839778C2216yescomplemented
35C2×#14519704C2216yescomplemented
36C2×#14519703C2216yescomplemented
37#16839720C2213yescomplemented
38#16839717C2213yescomplemented
39M16×Dic61C2213yescomplemented
40C2×#14519687C2213yescomplemented
41C61×#585C2211yescharacteristic
42C61×#568—24no—
43C61×#568C2211yescharacteristic
44C61×#568C2216yescomplemented
45C2×#131×C61—24no—
46C2×#131×C61C2211yescharacteristic
47C2×#131×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—212no—
67#14519784—24no—
68#14519784C2×C414yes—
69#14519784C2×C414yes—
70#14519783—26no—
71#14519783—26no—
72#14519783C2314yes—
73#14519783C23112yes—
74C2×#11758194C2316yes—
75C2×#11758194C2313yes—
76C2×#11758189—26no—
77M16×D61—212no—
78C22×#92418C2313yes—
79C2×C8×D61—26no—
80Q8×Dic61—21no—
81D4×Dic61—23no—
82#14519736C2313yes—
83C2×C4×Dic61C2313yes—
84#14519712C2314yes—
85#14519712C23112yes—
86#14519704C2314yes—
87#14519704C23112yes—
88#14519703C2314yes—
89#14519703C23112yes—
90#14519687C2311yescharacteristic
91#14519687C2313yes—
92#14519687C2313yes—
93#14519687C2313yes—
94C8×Dic61C2316yes—
952−1+4×C61—21no—
962−1+4×C61C2×C414yes—
97H(5,𝔽2)×C61—23no—
98H(5,𝔽2)×C61C2×C414yes—
99C2×#41×C61—24no—
100C2×#41×C61C2316yes—
101C2×#41×C61C2311yescharacteristic
102#131×C61—24no—
103#131×C61—212no—
104#131×C61—26no—
105#131×C61—26no—
106#131×C61D414yes—
107#131×C61D414yes—
108#131×C61C2314yes—
109#131×C61C23112yes—
110C2×M16×C61—212no—
111C2×M16×C61—26no—
112C2×M16×C61C2316yes—
113C2×M16×C61C2313yes—
114C22×C8×C61—26no—
115C22×C8×C61C2313yes—
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×C61C2413yes—
133C2×C22⋊D61—26no—
134C22×Dic61—23no—
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×C61—24no—
148M16×C61—212no—
149M16×C61C2×D4112yes—
150M16×C61C2414yes—
151C2×C8×C61—26no—
152C2×C8×C61—24no—
153C2×C8×C61—212no—
154C2×C8×C61C2×D416yes—
155C2×C8×C61C2×D416yes—
156C2×C8×C61C2411yescharacteristic
157C2×C8×C61C2413yes—
158C22⋊Dic61—23no—
159#11758202C22×C4112yes—
160#11758202C22×C414yes—
161#11758201C22×C4112yes—
162#11758201C22×C414yes—
163#11758200C22×C414yes—
164#11758200C22×C4112yes—
165#11758199C22×C414yes—
166#11758199C22×C4112yes—
167C4⋊Dic61—23no—
168C4×Dic61—23no—
169C4×Dic61C2413yes—
170C4×Dic61C2411yescharacteristic
171#11758194—26no—
172#11758194C22×C4112yes—
173#11758194C2414yes—
174C2×#92418—26no—
175C2×#92418C22×C416yes—
176C2×#92418C22×C416yes—
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×C61C23×C416yes—
198C2×C4×C61C22×D416yes—
199C2×C4×C61C2×#4113yes—
200C2×C4×C61C2511yescharacteristic
201C22⋊D61—212no—
202C22⋊D61—24no—
203C2×Dic61—26no—
204C2×Dic61—21no—
205C2×Dic61—23no—
206C2×Dic61—83no—
207D244—26no—
208D244C23×C414yes—
209C4×D61—212no—
210C4×D61—24no—
211Dic122—26no—
212Dic122C23×C414yes—
213C8×C61—24no—
214C8×C61C22×D411yescharacteristic
215C8×C61C22×D416yes—
216C8×C61C22×D411yescharacteristic
217#92418—24no—
218#92418C23×C411yescharacteristic
219#92418C23×C411yescharacteristic
220#92418C23×C416yes—
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×C61C2×C4×D416yes—
231C4×C61C24×C411yescharacteristic
232C4×C61C23×D411yescharacteristic
233Dic61—24no—
234Dic61—43no—
235Dic61—41no—
236#3584—611no—
237#3573—611no—
238#3573D6111yescharacteristic, complemented
239#2957—616no—
240#2956—616no—
241#2955—616no—
242C2×#444—616no—
243#2916—611no—
244#2901—611no—
245#2901—611no—
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—24no—
261#568—611no—
262#568—616no—
263#568—1224no—
264#568C2×D6111yescharacteristic
265#568C2×D6116yescomplemented
266C2×#131—24no—
267C2×#131—611no—
268C2×#131—616no—
269C2×#131—1224no—
270C2×#131C2×D6111yescharacteristic
271C2×#131C2×D6116yescomplemented
272C4×#41—611no—
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#421—616no—
282C2×#104—616no—
283C2×#104—616no—
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—212no—
316#131—26no—
317#131—26no—
318#131—614no—
319#131—614no—
320#131—614no—
321#131—6112no—
322#131—1224no—
323#131—12212no—
324#131—1226no—
325#131—1226no—
326#131C22⋊D6114yes—
327#131C22⋊D6114yes—
328#131C22×D6114yes—
329#131C22×D61112yes—
330C2×M16—212no—
331C2×M16—26no—
332C2×M16—613no—
333C2×M16—616no—
334C2×M16—12212no—
335C2×M16—1226no—
336C2×M16C22×D6116yes—
337C2×M16C22×D6113yes—
338C22×C8—26no—
339C22×C8—613no—
340C22×C8—1226no—
341C22×C8C22×D6113yes—
342C4×Q8—1221no—
343C4×D4—1223no—
344#117—613no—
345C2×C42—613no—
346#108—614no—
347#108—6112no—
348#104—614no—
349#104—6112no—
350#104—614no—
351#104—6112no—
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#41C2×C22⋊D6114yes—
370#41C23×D6114yes—
371C2×Q8—24no—
372C2×Q8—616no—
373C2×Q8—1224no—
374C2×Q8C2×C4×D6116yes—
375C2×Q8#1175822211yescharacteristic
376C2×D4—26no—
377C2×D4—212no—
378C2×D4—616no—
379C2×D4—1226no—
380C2×D4—12212no—
381C2×D4C2×C4×D6116yes—
382C2×D4#1175822213yes—
383C22×C4—26no—
384C22×C4—1226no—
385C22×C4—1223no—
386C22×C4C23×D6113yes—
387Q16—614no—
388Q16—6112no—
389SD16—614no—
390SD16—6112no—
391SD16—614no—
392SD16—6112no—
393D8—614no—
394D8—6112no—
395M16—26no—
396M16—212no—
397M16—24no—
398M16—6112no—
399M16—614no—
400M16—1226no—
401M16—1224no—
402M16—12212no—
403M16C2×C22⋊D61112yes—
404M16C23×D6114yes—
405C2×C8—26no—
406C2×C8—24no—
407C2×C8—212no—
408C2×C8—616no—
409C2×C8—616no—
410C2×C8—613no—
411C2×C8—611no—
412C2×C8—1226no—
413C2×C8—1224no—
414C2×C8—12212no—
415C2×C8C2×C22⋊D6116yes—
416C2×C8C2×C22⋊D6116yes—
417C2×C8C23×D6113yes—
418C2×C8C23×D6111yescharacteristic
419C4⋊C4—1223no—
420C22⋊C4—1223no—
421C42—613no—
422C42—611no—
423C42—1223no—
424C23—26no—
425C23—1226no—
426C23C2×#1175822213yes—
427Q8—26no—
428Q8—614no—
429Q8—1226no—
430Q8C4×C22⋊D6114yes—
431Q8C22×C4×D6114yes—
432D4—212no—
433D4—24no—
434D4—26no—
435D4—614no—
436D4—12212no—
437D4—1224no—
438D4—1226no—
439D4C4×C22⋊D61112yes—
440D4C22×C4×D6114yes—
441C2×C4—26no—
442C2×C4—212no—
443C2×C4—24no—
444C2×C4—1226no—
445C2×C4—12212no—
446C2×C4—1224no—
447C2×C4—1221no—
448C2×C4—1223no—
449C2×C4—4883no—
450C2×C4C22×C4×D6116yes—
451C2×C4C22×C22⋊D6116yes—
452C2×C4C2×#1175822213yes—
453C2×C4C24×D6111yescharacteristic
454C8—24no—
455C8—611no—
456C8—611no—
457C8—616no—
458C8—1224no—
459C8C22×C22⋊D6111yescharacteristic
460C8C22×C22⋊D6116yes—
461C8C22×C22⋊D6111yescharacteristic
462C22—24no—
463C22—43no—
464C22—86no—
465C22—1224no—
466C22—4886no—
467C22C2×C4×C22⋊D6116yes—
468C22C22×#1175822211yescharacteristic
469C4—24no—
470C4—1224no—
471C4—2443no—
472C4—2441no—
473C4C2×C4×C22⋊D6116yes—
474C4C23×C4×D6111yescharacteristic
475C4C23×C22⋊D6111yescharacteristic
476C2—21no—
477C2—26no—
478C2—44no—
479C2—2444no—
480C2C22×C4×C22⋊D6111yescharacteristic
subgroupG/H
centerC4C23×C22⋊D61
commutatorC4×C61C24×C4
FrattiniC2×C4C24×D61
FittingC61×#3573C2
socleC2×C61C22×C4×D4
prime radical#21460590C2
cosocleC2×C4×C61C25
hypercenter#3573D61
hypocenterC61#60184
LCSG ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C2×#131 ◁ #3573

Complements

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

NH ≅ G/Ntype
2. #21460530C2semidirect
3. #21454734C2semidirect
4. #21454726C2semidirect
5. #21454720C2semidirect
6. C2×#16839882C2semidirect
7. #21454428C2semidirect
8. #21454366C2semidirect
9. #21454364C2semidirect
10. #21454245C2semidirect
11. C61×#3573C2semidirect
13. #16840577C4semidirect
14. #16840573C4semidirect
15. #16840570C4semidirect
16. #16840565C4semidirect
20. #16840528C22semidirect
23. C2×#14519783C22semidirect
26. #16839882C22semidirect
27. #16839882C22semidirect
28. #16839882C22semidirect
29. #16839882C22semidirect
30. #16839877C22semidirect
31. #16839810C22semidirect
32. C2×#14519712C22semidirect
33. #16839779C22semidirect
34. #16839778C22semidirect
35. C2×#14519704C22semidirect
36. C2×#14519703C22semidirect
37. #16839720C22semidirect
38. #16839717C22semidirect
39. M16×Dic61C22semidirect
40. C2×#14519687C22semidirect
44. C61×#568C22semidirect
47. C2×#131×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
238. #3573D61semidirect
265. #568C2×D61semidirect
271. C2×#131C2×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
#214605302
#214547342
#214547262
#214547202
C2×#168398822
#214544282
#214543662
#214543642
#214542452
C61×#35732
#6018461