Finite Group Databasebeta

← C4×C61×#578 #29563606

Subgroups — C4×C61×#578 #29563606

Order 15616.

2658 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C61×#578C211yesmaximal, characteristic
2C4×2−1+4×C61C212yesmaximal, complemented
3C61×#2951C213yesmaximal, complemented
4C61×#2950C213yesmaximal, complemented
5C61×#2947C216yesmaximal, complemented
6C4×#137×C61C216yesmaximal, complemented
7C4×#135×C61C213yesmaximal, complemented
8C2×C4×Q16×C61C213yesmaximal, complemented
9C61×#2913C211yesmaximal, characteristic, complemented
10C61×#2898C212yesmaximal, complemented
11C4×#131×C61C211yesmaximal, characteristic, complemented
12C2×2−1+4×C61C2212yes—
13C61×#578C4116yesdirect product factor
14C2×#137×C61C2216yes—
15C2×#135×C61C2213yes—
16C22×Q16×C61C2213yes—
17C2×#131×C61C2211yescharacteristic
18C2×#131×C61C2211yescharacteristic
19C61×#520—26no—
20C61×#520C2216yes—
21C61×#520C2212yes—
22C61×#519—22no—
23C61×#519C2216yes—
24C4×#41×C61—26no—
25C4×#41×C61C2216yes—
26C4×#41×C61C2211yescharacteristic
27C2×C4×Q8×C61—22no—
28C2×C4×Q8×C61C2216yes—
29C61×#441C22112yes—
30C61×#440C22112yes—
31C4×Q16×C61C2213yes—
32C4×Q16×C61C2216yes—
33C4×Q16×C61C2216yes—
34C4×Q16×C61C2216yes—
35C4×Q16×C61C2213yes—
36C4×SD16×C61C2216yes—
37C4×SD16×C61C2216yes—
38C4×D8×C61C2211yescharacteristic
39C4×D8×C61C2213yes—
40C61×#428C2216yes—
41C61×#427C2213yes—
42C2×#107×C61C2213yes—
43C61×#419C2216yes—
44C61×#417C2216yes—
45C61×#416C2216yes—
46C2×#103×C61C2216yes—
47C61×#405C2213yes—
48C61×#405C2213yes—
49C4×M16×C61C2213yes—
50C2×C4×C8×C61C2213yes—
512−1+4×C61—28no—
522−1+4×C61C2×C4116yescomplemented
53C2×#41×C61—26no—
54C2×#41×C61C2316yes—
55C2×#41×C61C2311yescharacteristic
56C22×Q8×C61—22no—
57C22×Q8×C61C2316yes—
58#137×C61C2×C4148yescomplemented
59#135×C61C2×C4124yescomplemented
60C2×Q16×C61C2×C4124yescomplemented
61C2×Q16×C61C2313yes—
62C2×Q16×C61C2316yes—
63C2×SD16×C61C2316yes—
64C2×D8×C61C2311yescharacteristic
65#131×C61—24no—
66#131×C61C2×C418yescomplemented
67C2×M16×C61C2313yes—
68C2×M16×C61C2313yes—
69C22×C8×C61C2313yes—
70C22×C8×C61C2313yes—
71C4×Q8×C61—22no—
72C4×Q8×C61—26no—
73C4×Q8×C61—26no—
74C4×Q8×C61—26no—
75C4×Q8×C61—26no—
76C4×Q8×C61D411yescharacteristic
77C4×Q8×C61D411yescharacteristic
78C4×Q8×C61C2316yes—
79C4×Q8×C61C2316yes—
80C4×Q8×C61C23112yes—
81C4×D4×C61—26no—
82C4×D4×C61—26no—
83C4×D4×C61—26no—
84C4×D4×C61—212no—
85C4×D4×C61D413yes—
86C4×D4×C61D413yes—
87C4×D4×C61C2312yes—
88C4×D4×C61C2316yes—
89#117×C61—26no—
90#117×C61—212no—
91#117×C61—26no—
92#117×C61C2316yes—
93#117×C61C2313yes—
94C2×C4⋊C4×C61—26no—
95C2×C4⋊C4×C61—26no—
96C2×C4⋊C4×C61C2316yes—
97C2×C42×C61—26no—
98C2×C42×C61—26no—
99C2×C42×C61C2313yes—
100#107×C61C2313yes—
101#107×C61C2316yes—
102#107×C61C2311yescharacteristic
103#106×C61C2316yes—
104#103×C61C2316yes—
105#103×C61C23112yes—
106#103×C61C2316yes—
107#102×C61C2312yes—
108#102×C61C2316yes—
109#97×C61C2316yes—
110C4×C8×C61C2311yescharacteristic
111C4×C8×C61C2313yes—
112C4×C8×C61C2313yes—
113C4×C8×C61C2313yes—
114#41×C61—248no—
115#41×C61—212no—
116#41×C61C22×C4124yes—
117#41×C61C22×C414yes—
118#41×C61#4114yes—
119C2×Q8×C61—216no—
120C2×Q8×C61—26no—
121C2×Q8×C61—212no—
122C2×Q8×C61C22×C4124yes—
123C2×Q8×C61C2×D411yescharacteristic
124C2×Q8×C61C2416yes—
125C2×D4×C61—26no—
126C2×D4×C61C2×D413yes—
127C2×D4×C61C2412yes—
128C22×C4×C61—26no—
129C22×C4×C61—26no—
130C22×C4×C61—26no—
131C22×C4×C61—22no—
132C22×C4×C61—26no—
133C22×C4×C61C2×D413yes—
134C22×C4×C61C2×D413yes—
135C22×C4×C61C2413yes—
136Q16×C61C22×C4112yes—
137Q16×C61C22×C4124yes—
138SD16×C61C22×C4124yes—
139D8×C61C22×C414yes—
140M16×C61—26no—
141M16×C61C22×C4112yes—
142C2×C8×C61—26no—
143C2×C8×C61C22×C4112yes—
144C2×C8×C61C2411yescharacteristic
145C2×C8×C61C2413yes—
146C2×C8×C61C2411yescharacteristic
147C2×C8×C61C2413yes—
148C4⋊C4×C61—212no—
149C4⋊C4×C61—26no—
150C4⋊C4×C61—26no—
151C4⋊C4×C61—26no—
152C4⋊C4×C61C2×D413yes—
153C4⋊C4×C61C2×D413yes—
154C4⋊C4×C61C2412yes—
155C4⋊C4×C61C2416yes—
156C22⋊C4×C61—26no—
157C22⋊C4×C61—22no—
158C22⋊C4×C61—26no—
159C22⋊C4×C61C2×D416yes—
160C42×C61—26no—
161C42×C61—212no—
162C42×C61—26no—
163C42×C61—26no—
164C42×C61C2×D413yes—
165C42×C61C2×D413yes—
166C42×C61C2413yes—
167C42×C61C2411yescharacteristic
168C23×C61—22no—
169C23×C61C22×D413yes—
170Q8×C61—224no—
171Q8×C61—26no—
172Q8×C61C4×D414yes—
173Q8×C61C23×C4112yes—
174D4×C61—224no—
175D4×C61—22no—
176D4×C61C4×D4112yes—
177D4×C61C23×C414yes—
178C2×C4×C61—224no—
179C2×C4×C61—26no—
180C2×C4×C61—224no—
181C2×C4×C61—26no—
182C2×C4×C61—22no—
183C2×C4×C61—26no—
184C2×C4×C61—43no—
185C2×C4×C61—44no—
186C2×C4×C61—86no—
187C2×C4×C61—86no—
188C2×C4×C61C23×C416yes—
189C2×C4×C61C22×D413yes—
190C2×C4×C61C22×D413yes—
191C2×C4×C61C22×D411yescharacteristic
192C2×C4×C61C22×D413yes—
193C2×C4×C61C22×D411yescharacteristic
194C2×C4×C61C2×#4116yes—
195C2×C4×C61C2511yescharacteristic
196C8×C61—21no—
197C8×C61—23no—
198C8×C61C23×C412yes—
199C8×C61C23×C416yes—
200C4×#578C6111yesmaximal, characteristic, direct product factor, Sylow
201C22×C61—23no—
202C22×C61—23no—
203C22×C61—24no—
204C22×C61—42no—
205C22×C61—42no—
206C22×C61C2×C4×D416yes—
207C22×C61C23×D411yescharacteristic
208C4×C61—21no—
209C4×C61—23no—
210C4×C61—212no—
211C4×C61—23no—
212C4×C61—46no—
213C4×C61—42no—
214C4×C61C2×C4×D416yes—
215C4×C61#57812yesdirect product factor
216C4×C61C24×C411yescharacteristic
217C4×C61C22×#4111yescharacteristic
218C2×#578C2×C6111yescharacteristic
219C4×2−1+4C2×C6112yescomplemented
220#2951C2×C6113yescomplemented
221#2950C2×C6113yescomplemented
222#2947C2×C6116yescomplemented
223C4×#137C2×C6116yescomplemented
224C4×#135C2×C6113yescomplemented
225C2×C4×Q16C2×C6113yescomplemented
226#2913C2×C6111yescharacteristic, complemented
227#2898C2×C6112yescomplemented
228C4×#131C2×C6111yescharacteristic, complemented
229C2×C61—26no—
230C2×C61—44no—
231C2×C61C22×C4×D411yescharacteristic
232C2×C61C2×#57811yescharacteristic
233C2×C61#358411yescharacteristic
234C2×2−1+4C22×C6112yes—
235#578C4×C61116yesdirect product factor
236C2×#137C22×C6116yes—
237C2×#135C22×C6113yes—
238C22×Q16C22×C6113yes—
239C2×#131C22×C6111yescharacteristic
240C2×#131C22×C6111yescharacteristic
241#520—26no—
242#520C22×C6116yes—
243#520C22×C6112yes—
244#519—22no—
245#519C22×C6116yes—
246C4×#41—26no—
247C4×#41C22×C6116yes—
248C4×#41C22×C6111yescharacteristic
249C2×C4×Q8—22no—
250C2×C4×Q8C22×C6116yes—
251#441C22×C61112yes—
252#440C22×C61112yes—
253C4×Q16C22×C6113yes—
254C4×Q16C22×C6116yes—
255C4×Q16C22×C6116yes—
256C4×Q16C22×C6116yes—
257C4×Q16C22×C6113yes—
258C4×SD16C22×C6116yes—
259C4×SD16C22×C6116yes—
260C4×D8C22×C6111yescharacteristic
261C4×D8C22×C6113yes—
262#428C22×C6116yes—
263#427C22×C6113yes—
264C2×#107C22×C6113yes—
265#419C22×C6116yes—
266#417C22×C6116yes—
267#416C22×C6116yes—
268C2×#103C22×C6116yes—
269#405C22×C6113yes—
270#405C22×C6113yes—
271C4×M16C22×C6113yes—
272C2×C4×C8C22×C6113yes—
273C61C4×#57811yescharacteristic, direct product factor, Sylow
2742−1+4—28no—
2752−1+4C2×C4×C61116yescomplemented
276C2×#41—26no—
277C2×#41C23×C6116yes—
278C2×#41C23×C6111yescharacteristic
279C22×Q8—22no—
280C22×Q8C23×C6116yes—
281#137C2×C4×C61148yescomplemented
282#135C2×C4×C61124yescomplemented
283C2×Q16C2×C4×C61124yescomplemented
284C2×Q16C23×C6113yes—
285C2×Q16C23×C6116yes—
286C2×SD16C23×C6116yes—
287C2×D8C23×C6111yescharacteristic
288#131—24no—
289#131C2×C4×C6118yescomplemented
290C2×M16C23×C6113yes—
291C2×M16C23×C6113yes—
292C22×C8C23×C6113yes—
293C22×C8C23×C6113yes—
294C4×Q8—22no—
295C4×Q8—26no—
296C4×Q8—26no—
297C4×Q8—26no—
298C4×Q8—26no—
299C4×Q8D4×C6111yescharacteristic
300C4×Q8D4×C6111yescharacteristic
301C4×Q8C23×C6116yes—
302C4×Q8C23×C61112yes—
303C4×Q8C23×C6116yes—
304C4×D4—26no—
305C4×D4—26no—
306C4×D4—26no—
307C4×D4—212no—
308C4×D4D4×C6113yes—
309C4×D4D4×C6113yes—
310C4×D4C23×C6112yes—
311C4×D4C23×C6116yes—
312#117—26no—
313#117—212no—
314#117—26no—
315#117C23×C6116yes—
316#117C23×C6113yes—
317C2×C4⋊C4—26no—
318C2×C4⋊C4—26no—
319C2×C4⋊C4C23×C6116yes—
320C2×C42—26no—
321C2×C42—26no—
322C2×C42C23×C6113yes—
323#107C23×C6113yes—
324#107C23×C6116yes—
325#107C23×C6111yescharacteristic
326#106C23×C6116yes—
327#103C23×C6116yes—
328#103C23×C61112yes—
329#103C23×C6116yes—
330#102C23×C6112yes—
331#102C23×C6116yes—
332#97C23×C6116yes—
333C4×C8C23×C6111yescharacteristic
334C4×C8C23×C6113yes—
335C4×C8C23×C6113yes—
336C4×C8C23×C6113yes—
337#41—248no—
338#41—212no—
339#41C22×C4×C61124yes—
340#41C22×C4×C6114yes—
341#41#41×C6114yes—
342C2×Q8—216no—
343C2×Q8—26no—
344C2×Q8—212no—
345C2×Q8C22×C4×C61124yes—
346C2×Q8C2×D4×C6111yescharacteristic
347C2×Q8C24×C6116yes—
348C2×D4—26no—
349C2×D4C2×D4×C6113yes—
350C2×D4C24×C6112yes—
351C22×C4—26no—
352C22×C4—26no—
353C22×C4—26no—
354C22×C4—22no—
355C22×C4—26no—
356C22×C4C2×D4×C6113yes—
357C22×C4C2×D4×C6113yes—
358C22×C4C24×C6113yes—
359Q16C22×C4×C61112yes—
360Q16C22×C4×C61124yes—
361SD16C22×C4×C61124yes—
362D8C22×C4×C6114yes—
363M16—26no—
364M16C22×C4×C61112yes—
365C2×C8—26no—
366C2×C8C22×C4×C61112yes—
367C2×C8C24×C6111yescharacteristic
368C2×C8C24×C6113yes—
369C2×C8C24×C6111yescharacteristic
370C2×C8C24×C6113yes—
371C4⋊C4—212no—
372C4⋊C4—26no—
373C4⋊C4—26no—
374C4⋊C4—26no—
375C4⋊C4C2×D4×C6113yes—
376C4⋊C4C2×D4×C6113yes—
377C4⋊C4C24×C6112yes—
378C4⋊C4C24×C6116yes—
379C22⋊C4—26no—
380C22⋊C4—22no—
381C22⋊C4—26no—
382C22⋊C4C2×D4×C6116yes—
383C42—26no—
384C42—26no—
385C42—212no—
386C42—26no—
387C42C2×D4×C6113yes—
388C42C2×D4×C6113yes—
389C42C24×C6111yescharacteristic
390C42C24×C6113yes—
391C23—22no—
392C23C22×D4×C6113yes—
393Q8—26no—
394Q8—224no—
395Q8C4×D4×C6114yes—
396Q8C23×C4×C61112yes—
397D4—224no—
398D4—22no—
399D4C4×D4×C61112yes—
400D4C23×C4×C6114yes—
401C2×C4—26no—
402C2×C4—224no—
403C2×C4—26no—
404C2×C4—224no—
405C2×C4—22no—
406C2×C4—26no—
407C2×C4—43no—
408C2×C4—44no—
409C2×C4—86no—
410C2×C4—86no—
411C2×C4C23×C4×C6116yes—
412C2×C4C22×D4×C6113yes—
413C2×C4C22×D4×C6113yes—
414C2×C4C22×D4×C6111yescharacteristic
415C2×C4C22×D4×C6111yescharacteristic
416C2×C4C22×D4×C6113yes—
417C2×C4C2×#41×C6116yes—
418C2×C4C25×C6111yescharacteristic
419C8—21no—
420C8—23no—
421C8C23×C4×C6112yes—
422C8C23×C4×C6116yes—
423C22—23no—
424C22—23no—
425C22—24no—
426C22—42no—
427C22—42no—
428C22C2×C4×D4×C6116yes—
429C22C23×D4×C6111yescharacteristic
430C4—23no—
431C4—21no—
432C4—23no—
433C4—212no—
434C4—42no—
435C4—46no—
436C4C2×C4×D4×C6116yes—
437C4C61×#57812yesdirect product factor
438C4C24×C4×C6111yescharacteristic
439C4C22×#41×C6111yescharacteristic
440C2—26no—
441C2—44no—
442C2C22×C4×D4×C6111yescharacteristic
443C2C2×C61×#57811yescharacteristic
444C2C61×#358411yescharacteristic
subgroupG/H
centerC2×C4×C61C22×D4
commutatorC4C24×C4×C61
Frattini, cosocleC2×C4C25×C61
socleC22×C61C23×D4
prime radicalC2×C61×#578C2
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C61 ◁ C4×#41×C61 ◁ G

Complements

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

NH ≅ G/Ntype
2. C4×2−1+4×C61C2semidirect
3. C61×#2951C2semidirect
4. C61×#2950C2semidirect
5. C61×#2947C2semidirect
6. C4×#137×C61C2semidirect
7. C4×#135×C61C2semidirect
8. C2×C4×Q16×C61C2semidirect
9. C61×#2913C2semidirect
10. C61×#2898C2semidirect
11. C4×#131×C61C2semidirect
13. C61×#578C4direct (N × H)
52. 2−1+4×C61C2×C4semidirect
58. #137×C61C2×C4semidirect
59. #135×C61C2×C4semidirect
60. C2×Q16×C61C2×C4semidirect
66. #131×C61C2×C4semidirect
200. C4×#578C61direct (N × H)
215. C4×C61#578direct (N × H)
219. C4×2−1+4C2×C61semidirect
220. #2951C2×C61semidirect
221. #2950C2×C61semidirect
222. #2947C2×C61semidirect
223. C4×#137C2×C61semidirect
224. C4×#135C2×C61semidirect
225. C2×C4×Q16C2×C61semidirect
226. #2913C2×C61semidirect
227. #2898C2×C61semidirect
228. C4×#131C2×C61semidirect
235. #578C4×C61direct (N × H)
273. C61C4×#578direct (N × H)
275. 2−1+4C2×C4×C61semidirect
281. #137C2×C4×C61semidirect
282. #135C2×C4×C61semidirect
283. C2×Q16C2×C4×C61semidirect
289. #131C2×C4×C61semidirect
437. C4C61×#578direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C61×#5782
C4×2−1+4×C612
C61×#29512
C61×#29502
C61×#29472
C4×#137×C612
C4×#135×C612
C2×C4×Q16×C612
C61×#29132
C61×#28982
C4×#131×C612
C4×#57861