Finite Group Databasebeta

← #29597612

Subgroups — #29597612

Order 15616.

688 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455087C211yesmaximal, characteristic
2#21454280C211yesmaximal, characteristic, complemented
3#21447490C211yesmaximal, characteristic, complemented
4#21447470C211yesmaximal, characteristic, complemented
5#21443949C211yesmaximal, characteristic, complemented
6#21443939C211yesmaximal, characteristic, complemented
7C61×#1795C211yesmaximal, characteristic, complemented
8C2×#14519802C411yescharacteristic, complemented
9C22×#11758199C411yescharacteristic, complemented
10#16840326C2211yescharacteristic
11#16840237—21no—
12#16839907—21no—
13#16839907—21no—
14#16839896—21no—
15#16839896—21no—
16#16839767C411yescharacteristic, complemented
17#16839758C411yescharacteristic, complemented
18#16839736—21no—
19#16839736C2211yescharacteristic
20#16839736C2211yescharacteristic
21C2×#14519692—22no—
22C2×#14519692—22no—
23#16839729C2211yescharacteristic
24#16839554—22no—
25#16839550—22no—
26C2×H(5,𝔽2)×C61C2211yescharacteristic
27C61×#421—21no—
28C61×#418—21no—
29C61×#413C2211yescharacteristic
30C61×#410C2211yescharacteristic
31C22×D244C2×C411yescharacteristic, complemented
32#14519802—42no—
33C2×#11758201—21no—
34C2×#11758199—21no—
35C2×#11758199—21no—
36C2×#11758199—21no—
37C2×#11758199—42no—
38C2×#11758194C2×C411yescharacteristic, complemented
39C22×#92418C2×C411yescharacteristic, complemented
40#14519773—21no—
41#14519766—21no—
42#14519751—21no—
43#14519751—21no—
44#14519745—21no—
45#14519745—21no—
46D4×Dic61—42no—
47D4×Dic61—42no—
48#14519736—21no—
49#14519736—21no—
50#14519736C2×C411yescharacteristic, complemented
51C2×C22⋊Dic61—24no—
52#14519701—21no—
53#14519698—21no—
54#14519698—21no—
55#14519698—21no—
56#14519692—24no—
57#14519692—21no—
58#14519692—21no—
59#14519692D412yes—
60#14519692D412yes—
61C22⋊#92418—21no—
62C22⋊#92418—21no—
63H(5,𝔽2)×C61—42no—
64H(5,𝔽2)×C61—42no—
65C2×#41×C61—21no—
66C2×#41×C61C2×C411yescharacteristic
67C2×#41×C61D411yescharacteristic
68C2×#41×C61D411yescharacteristic
69C2×#41×C61C2311yescharacteristic
70C22×D4×C61—21no—
71C22×D4×C61—21no—
72C22×D4×C61—21no—
73C22×D4×C61C2×C411yescharacteristic
74C22×D4×C61D411yescharacteristic
75C22×D4×C61D411yescharacteristic
76C2×M16×C61—21no—
77#117×C61—21no—
78#104×C61—42no—
79#102×C61—42no—
80#101×C61—21no—
81#100×C61—21no—
82#99×C61—21no—
83#99×C61—21no—
84C24×C61—21no—
85C24×C61—21no—
86C24×C61—21no—
87C23×D61—21no—
88#41×C61—22no—
89#41×C61—22no—
90#41×C61—41no—
91#41×C61—41no—
92#41×C61—41no—
93#41×C61—41no—
94#41×C61—41no—
95#41×C61—41no—
96#41×C61—44no—
97C2×Q8×C61C22⋊C411yescharacteristic
98C2×Q8×C61#4111yescharacteristic
99C2×D4×C61—22no—
100C2×D4×C61—22no—
101C2×D4×C61—22no—
102C2×D4×C61—22no—
103C2×D4×C61—21no—
104C2×D4×C61—21no—
105C2×D4×C61—21no—
106C2×D4×C61—21no—
107C2×D4×C61—21no—
108C2×D4×C61—21no—
109C2×D4×C61—41no—
110C2×D4×C61—41no—
111C2×D4×C61—44no—
112C2×D4×C61—44no—
113C2×D4×C61—44no—
114C2×D4×C61C22⋊C411yescharacteristic
115C2×D4×C61C22⋊C411yescharacteristic
116C2×D4×C61C22⋊C411yescharacteristic
117C2×D4×C61C2×D411yescharacteristic
118C2×D4×C61C2×D411yescharacteristic
119C2×D4×C61#4111yescharacteristic
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61C22×C411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C2×C22⋊D61—41no—
127C2×C22⋊D61—41no—
128C22×Dic61—44no—
129C2×D244—21no—
130C2×D244—21no—
131C2×D244—41no—
132C2×D244—41no—
133M16×C61—41no—
134M16×C61—41no—
135C2×C8×C61—41no—
136C4⋊C4×C61—41no—
137C22⋊C4×C61—41no—
138C42×C61—41no—
139C22⋊Dic61—21no—
140C22⋊Dic61—22no—
141C22⋊Dic61—22no—
142C22⋊Dic61—44no—
143#11758201—42no—
144#11758199—42no—
145#11758199—42no—
146#11758199—42no—
147#11758198—41no—
148#11758198—41no—
149C4⋊Dic61—21no—
150C4⋊Dic61—41no—
151C4⋊Dic61—41no—
152C4×Dic61—21no—
153C4×Dic61—41no—
154C4×Dic61—41no—
155#11758194—41no—
156C2×#92418—21no—
157C2×#92418—21no—
158C2×#92418—41no—
159C23×C61—24no—
160C23×C61—21no—
161C23×C61—21no—
162C23×C61—21no—
163C23×C61—42no—
164C23×C61—42no—
165C23×C61—161no—
166C23×C61—161no—
167C23×C61—161no—
168C23×C61C4×D411yescharacteristic
169C23×C61#12011yescharacteristic
170C23×C61#12111yescharacteristic
171C22×D61—41no—
172C22×D61—81no—
173C22×D61—161no—
174Q8×C61—21no—
175Q8×C61—21no—
176Q8×C61—41no—
177D4×C61—24no—
178D4×C61—21no—
179D4×C61—21no—
180D4×C61—21no—
181D4×C61—21no—
182D4×C61—41no—
183D4×C61—41no—
184D4×C61—41no—
185D4×C61—42no—
186D4×C61—42no—
187D4×C61—42no—
188D4×C61—42no—
189D4×C61—42no—
190D4×C61#9912yes—
191D4×C61#9912yes—
192C2×C4×C61—21no—
193C2×C4×C61—21no—
194C2×C4×C61—21no—
195C2×C4×C61—21no—
196C2×C4×C61—21no—
197C2×C4×C61—21no—
198C2×C4×C61—21no—
199C2×C4×C61—42no—
200C2×C4×C61—42no—
201C2×C4×C61—42no—
202C2×C4×C61—81no—
203C2×C4×C61C2×C22⋊C411yescharacteristic
204C2×C4×C61C4×D411yescharacteristic
205C2×C4×C61#12111yescharacteristic
206C2×C4×C61#12311yescharacteristic
207C22⋊D61—161no—
208C22⋊D61—161no—
209C2×Dic61—42no—
210C2×Dic61—42no—
211C2×Dic61—41no—
212C2×Dic61—164no—
213D244—41no—
214D244—41no—
215D244—81no—
216C8×C61—81no—
217#92418—41no—
218#92418—41no—
219#13102—611nomaximal, Sylow
220C22×C61—21no—
221C22×C61—21no—
222C22×C61—21no—
223C22×C61—22no—
224C22×C61—22no—
225C22×C61—41no—
226C22×C61—41no—
227C22×C61—41no—
228C22×C61—41no—
229C22×C61—81no—
230C22×C61—81no—
231C22×C61—81no—
232C22×C61—84no—
233C22×C61—162no—
234C22×C61—162no—
235C22×C61#38611yescharacteristic
236C2×D61—81no—
237C2×D61—81no—
238C2×D61—81no—
239C2×D61—162no—
240C4×C61—21no—
241C4×C61—21no—
242C4×C61—21no—
243C4×C61—41no—
244C4×C61—161no—
245C4×C61C2×#9911yescharacteristic
246C4×C61C2×#9911yescharacteristic
247Dic61—81no—
248Dic61—82no—
249Dic61—82no—
250#3008—611no—
251#2883—611no—
252#2128—611no—
253#2122—611no—
254#1795D6111yescharacteristic, complemented
255#1395—611no—
256#1391—611no—
257C2×C61—21no—
258C2×C61—41no—
259C2×C61—41no—
260C2×C61—41no—
261C2×C61—42no—
262C2×C61—42no—
263C2×C61—81no—
264C2×C61#189811yescharacteristic
265D61—161no—
266C2×H(5,𝔽2)C2×D6111yescharacteristic
267C2×AGL(1,ℤ/8)—611no—
268C22×D8—611no—
269#535—611no—
270#518—1221no—
271#449—1221no—
272#449—1221no—
273#447—1221no—
274#447—1221no—
275#421—21no—
276#418—21no—
277#418—611no—
278#416—611no—
279#413C2×D6111yescharacteristic
280#410—611no—
281#410—611no—
282#410—1221no—
283#410C2×D6111yescharacteristic
284C2×#99—1222no—
285C2×#99—1222no—
286#408—611no—
287#353—1222no—
288AΣL(1,𝔽16)—1222no—
289C61#1310211yescharacteristic, complemented, Sylow
290H(5,𝔽2)—42no—
291H(5,𝔽2)—42no—
292C2×#41—21no—
293C2×#41C4×D6111yescharacteristic
294C2×#41D24411yescharacteristic
295C2×#41C22⋊D6111yescharacteristic
296C2×#41C22×D6111yescharacteristic
297C22×D4—21no—
298C22×D4—21no—
299C22×D4—21no—
300C22×D4—611no—
301C22×D4C4×D6111yescharacteristic
302C22×D4D24411yescharacteristic
303C22×D4C22⋊D6111yescharacteristic
304AGL(1,ℤ/8)—2442no—
305C2×SD16—1221no—
306C2×D8—1221no—
307C2×D8—1221no—
308C2×D8—1221no—
309C2×D8—2442no—
310C2×M16—21no—
311C2×M16—611no—
312C22×C8—611no—
313#127—1221no—
314#124—1221no—
315#121—1221no—
316#121—1221no—
317#120—1221no—
318#120—1221no—
319C4×D4—2442no—
320C4×D4—2442no—
321#117—21no—
322#117—611no—
323#117—1221no—
324#117—1221no—
325C2×C22⋊C4—1224no—
326#104—42no—
327#103—1221no—
328#102—42no—
329#102—1221no—
330#102—1221no—
331#102—1221no—
332#101—21no—
333#100—21no—
334#99—21no—
335#99—21no—
336#99—612no—
337#99—612no—
338#99—1224no—
339#99—1221no—
340#99—1221no—
341C22⋊C8—1221no—
342C22⋊C8—1221no—
343C24—21no—
344C24—21no—
345C24—21no—
346C24—1221no—
347#41—22no—
348#41—22no—
349#41—41no—
350#41—41no—
351#41—41no—
352#41—41no—
353#41—41no—
354#41—41no—
355#41—44no—
356C2×Q8#1175819811yescharacteristic
357C2×Q8#1175822411yescharacteristic
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—44no—
371C2×D4—44no—
372C2×D4—44no—
373C2×D4—1221no—
374C2×D4—1221no—
375C2×D4—2441no—
376C2×D4—2441no—
377C2×D4—2441no—
378C2×D4—2441no—
379C2×D4#1175819811yescharacteristic
380C2×D4#1175819811yescharacteristic
381C2×D4#1175819811yescharacteristic
382C2×D4D4×D6111yescharacteristic
383C2×D4D4×D6111yescharacteristic
384C2×D4#1175822411yescharacteristic
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—2444no—
389C22×C4C2×C4×D6111yescharacteristic
390C22×C4C2×D24411yescharacteristic
391C22×C4C2×C22⋊D6111yescharacteristic
392SD16—2442no—
393D8—2442no—
394D8—2442no—
395D8—2442no—
396M16—41no—
397M16—41no—
398M16—2441no—
399C2×C8—41no—
400C2×C8—1221no—
401C2×C8—1221no—
402C2×C8—2441no—
403C4⋊C4—41no—
404C4⋊C4—1221no—
405C4⋊C4—2441no—
406C4⋊C4—2441no—
407C22⋊C4—41no—
408C22⋊C4—1221no—
409C22⋊C4—1222no—
410C22⋊C4—1222no—
411C22⋊C4—2444no—
412C22⋊C4—2441no—
413C22⋊C4—2441no—
414C42—41no—
415C42—1221no—
416C42—2441no—
417C42—2441no—
418C23—24no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—42no—
423C23—42no—
424C23—161no—
425C23—161no—
426C23—161no—
427C23—9761no—
428C23—4881no—
429C23—2441no—
430C23#1451973711yescharacteristic
431C23#1451974711yescharacteristic
432C23#1451974911yescharacteristic
433Q8—21no—
434Q8—21no—
435Q8—41no—
436D4—24no—
437D4—21no—
438D4—21no—
439D4—21no—
440D4—21no—
441D4—41no—
442D4—41no—
443D4—41no—
444D4—42no—
445D4—42no—
446D4—42no—
447D4—42no—
448D4—42no—
449D4—9761no—
450D4—9761no—
451D4—4881no—
452D4—2441no—
453D4—2441no—
454D4#1451969112yes—
455D4#1451969112yes—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—81no—
467C2×C4—9764no—
468C2×C4—2442no—
469C2×C4—2442no—
470C2×C4—2441no—
471C2×C4C2×#1175819811yescharacteristic
472C2×C4#1451973711yescharacteristic
473C2×C4#1451974911yescharacteristic
474C2×C4#1451976111yescharacteristic
475C8—81no—
476C8—2441no—
477C8—2441no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—22no—
482C22—22no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—81no—
488C22—81no—
489C22—81no—
490C22—84no—
491C22—162no—
492C22—162no—
493C22—9762no—
494C22—4881no—
495C22—4881no—
496C22—4881no—
497C22#1683963911yescharacteristic
498C4—21no—
499C4—21no—
500C4—21no—
501C4—41no—
502C4—161no—
503C4—4881no—
504C4—4882no—
505C4—4882no—
506C4C2×#1451969111yescharacteristic
507C4C2×#1451969111yescharacteristic
508C2—21no—
509C2—41no—
510C2—41no—
511C2—41no—
512C2—42no—
513C2—42no—
514C2—81no—
515C2—9761no—
516C2#2144627011yescharacteristic
subgroupG/H
centerC2#21446270
commutatorC22×C4×C61C22×C4
FrattiniC2×#41C22×D61
FittingC61×#1795C2
socleC2×C61#1898
prime radical#21455087C2
cosocleC2×#41×C61C23
hypercenter#1795D61
hypocenterC61#13102
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×H(5,𝔽2) ◁ #1795

Complements

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

NH ≅ G/Ntype
2. #21454280C2semidirect
3. #21447490C2semidirect
4. #21447470C2semidirect
5. #21443949C2semidirect
6. #21443939C2semidirect
7. C61×#1795C2semidirect
8. C2×#14519802C4semidirect
9. C22×#11758199C4semidirect
16. #16839767C4semidirect
17. #16839758C4semidirect
31. C22×D244C2×C4semidirect
38. C2×#11758194C2×C4semidirect
39. C22×#92418C2×C4semidirect
50. #14519736C2×C4semidirect
254. #1795D61semidirect
289. C61#13102semidirect

Maximal subgroups

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

subgroupindex
#214550872
#214542802
#214474902
#214474702
#214439492
#214439392
C61×#17952
#1310261