Finite Group Databasebeta

← C4×C61×#562 #29537785

Subgroups — C4×C61×#562 #29537785

Order 15616.

2248 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C61×#562C211yesmaximal, characteristic
2C61×#2382C211yesmaximal, characteristic, complemented
3C61×#2379C212yesmaximal, complemented
4C61×#2378C214yesmaximal, complemented
5C61×#2377C214yesmaximal, complemented
6C61×#2376C214yesmaximal, complemented
7C4×#125×C61C211yesmaximal, characteristic, complemented
8C4×#124×C61C212yesmaximal, complemented
9C4×C22⋊Q8×C61C214yesmaximal, complemented
10C4×#123×C61C214yesmaximal, complemented
11C4×#121×C61C214yesmaximal, complemented
12C61×#562C4116yesdirect product factor
13C2×#125×C61C2211yescharacteristic
14C2×#124×C61C2212yes—
15C2×#123×C61C2214yes—
16C2×C22⋊Q8×C61C2214yes—
17C2×#121×C61C2214yes—
18C2×C4×Q8×C61C2212yes—
19C2×C4×D4×C61C2214yes—
20C2×C4×D4×C61C2212yes—
21C22×C42×C61C2214yes—
22C61×#391C2214yes—
23C61×#391C2212yes—
24C61×#390C2214yes—
25C61×#390C2212yes—
26C61×#389C2214yes—
27C61×#389C2214yes—
28C61×#389C2212yes—
29C61×#388C2214yes—
30C61×#388C2214yes—
31C61×#388C2214yes—
32C61×#388C2214yes—
33C61×#388C2214yes—
34C61×#388C2214yes—
35C61×#387C2214yes—
36C61×#387C2214yes—
37C61×#387C2214yes—
38C61×#387C2214yes—
39C61×#387C2214yes—
40C61×#386C2214yes—
41C61×#386C2214yes—
42C61×#386C2214yes—
43C61×#385C2214yes—
44C61×#385C2214yes—
45C61×#385C2214yes—
46C61×#382C2212yes—
47C61×#382C2211yescharacteristic
48C61×#381C2214yes—
49C61×#380C2214yes—
50C61×#380C2214yes—
51C4×C4⋊C4×C61C2214yes—
52C4×C4⋊C4×C61C2212yes—
53C4×C4⋊C4×C61C2214yes—
54C4×C22⋊C4×C61C2214yes—
55C4×C22⋊C4×C61C2214yes—
56C4×C22⋊C4×C61C2214yes—
57C43×C61C2211yescharacteristic
58C22×Q8×C61C2312yes—
59C22×D4×C61C2314yes—
60C22×D4×C61C2312yes—
61C23×C4×C61C2314yes—
62C23×C4×C61C2314yes—
63C23×C4×C61C2314yes—
64#125×C61C2×C418yescomplemented
65#124×C61C2×C4116yescomplemented
66#123×C61C2×C4132yescomplemented
67C22⋊Q8×C61C2×C4132yescomplemented
68#121×C61C2×C4132yescomplemented
69C4×Q8×C61—44no—
70C4×D4×C61—48no—
71C4×D4×C61—44no—
72C2×C4⋊C4×C61C2314yes—
73C2×C4⋊C4×C61C2314yes—
74C2×C4⋊C4×C61C2312yes—
75C2×C4⋊C4×C61C2314yes—
76C2×C4⋊C4×C61C2314yes—
77C2×C4⋊C4×C61C2312yes—
78C2×C4⋊C4×C61C2312yes—
79C2×C4⋊C4×C61C2314yes—
80C2×C4⋊C4×C61C2314yes—
81C2×C4⋊C4×C61C2312yes—
82C2×C22⋊C4×C61C2314yes—
83C2×C22⋊C4×C61C2314yes—
84C2×C22⋊C4×C61C2314yes—
85C2×C22⋊C4×C61C2314yes—
86C2×C22⋊C4×C61C2314yes—
87C2×C22⋊C4×C61C2314yes—
88C2×C22⋊C4×C61C2314yes—
89C2×C22⋊C4×C61C2314yes—
90C2×C22⋊C4×C61C2314yes—
91C2×C42×C61—48no—
92C2×C42×C61C2311yescharacteristic
93C2×C42×C61C2312yes—
94C2×C42×C61C2314yes—
95C2×C42×C61C2314yes—
96C2×C42×C61C2312yes—
97C2×C42×C61C2311yescharacteristic
98C2×C42×C61C2314yes—
99C2×C42×C61C2314yes—
100C2×C42×C61C2311yescharacteristic
101#95×C61C2314yes—
102#95×C61C2314yes—
103#95×C61C2314yes—
104#95×C61C2314yes—
105#95×C61C2314yes—
106#95×C61C2314yes—
107#95×C61C2314yes—
108#95×C61C2314yes—
109#95×C61C2314yes—
110#95×C61C2314yes—
111#95×C61C2314yes—
112C24×C61C2414yes—
113C2×Q8×C61—42no—
114C2×Q8×C61—42no—
115C2×Q8×C61C22×C418yes—
116C2×D4×C61—44no—
117C2×D4×C61—42no—
118C2×D4×C61—42no—
119C2×D4×C61—44no—
120C2×D4×C61C22×C4116yes—
121C2×D4×C61C22×C418yes—
122C22×C4×C61—28no—
123C22×C4×C61—28no—
124C22×C4×C61—28no—
125C22×C4×C61—44no—
126C22×C4×C61—44no—
127C22×C4×C61—44no—
128C22×C4×C61—44no—
129C22×C4×C61—44no—
130C22×C4×C61C22×C4116yes—
131C22×C4×C61C2411yescharacteristic
132C22×C4×C61C2414yes—
133C22×C4×C61C2414yes—
134C22×C4×C61C2414yes—
135C22×C4×C61C2412yes—
136C22×C4×C61C2411yescharacteristic
137C22×C4×C61C2412yes—
138C22×C4×C61C2414yes—
139C22×C4×C61C2414yes—
140C22×C4×C61C2411yescharacteristic
141C4⋊C4×C61—42no—
142C4⋊C4×C61—44no—
143C4⋊C4×C61—44no—
144C4⋊C4×C61—44no—
145C4⋊C4×C61—42no—
146C4⋊C4×C61—44no—
147C4⋊C4×C61—42no—
148C4⋊C4×C61C22×C4116yes—
149C4⋊C4×C61C22×C4116yes—
150C4⋊C4×C61C22×C418yes—
151C22⋊C4×C61—44no—
152C22⋊C4×C61—44no—
153C22⋊C4×C61—44no—
154C22⋊C4×C61—44no—
155C22⋊C4×C61—44no—
156C22⋊C4×C61—44no—
157C22⋊C4×C61C22×C4116yes—
158C22⋊C4×C61C22×C4116yes—
159C22⋊C4×C61C22×C4116yes—
160C42×C61—28no—
161C42×C61—28no—
162C42×C61—24no—
163C42×C61—22no—
164C42×C61—44no—
165C42×C61—44no—
166C42×C61—41no—
167C42×C61—42no—
168C42×C61C22×C414yes—
169C23×C61—24no—
170C23×C61—24no—
171C23×C61—24no—
172C23×C61—24no—
173C23×C61—24no—
174C23×C61—24no—
175C23×C61C23×C418yes—
176C23×C61C2511yescharacteristic
177Q8×C61—48no—
178D4×C61—416no—
179D4×C61—48no—
180C2×C4×C61—24no—
181C2×C4×C61—24no—
182C2×C4×C61—22no—
183C2×C4×C61—21no—
184C2×C4×C61—21no—
185C2×C4×C61—22no—
186C2×C4×C61—24no—
187C2×C4×C61—24no—
188C2×C4×C61—24no—
189C2×C4×C61—24no—
190C2×C4×C61—22no—
191C2×C4×C61—24no—
192C2×C4×C61—24no—
193C2×C4×C61—21no—
194C2×C4×C61—24no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—24no—
199C2×C4×C61—24no—
200C2×C4×C61—24no—
201C2×C4×C61—24no—
202C2×C4×C61—416no—
203C2×C4×C61—416no—
204C2×C4×C61—416no—
205C2×C4×C61C23×C412yes—
206C2×C4×C61C23×C414yes—
207C2×C4×C61C23×C418yes—
208C2×C4×C61C23×C418yes—
209C2×C4×C61H(5,𝔽2)14yes—
210C2×C4×C612−1+412yes—
211C4×#562C6111yesmaximal, characteristic, direct product factor, Sylow
212C22×C61—28no—
213C22×C61—28no—
214C22×C61—28no—
215C22×C61—44no—
216C22×C61—44no—
217C22×C61—44no—
218C22×C61—44no—
219C22×C61C24×C411yescharacteristic
220C22×C61C2×H(5,𝔽2)12yes—
221C22×C61C2×2−1+411yescharacteristic
222C22×C61#58512yes—
223C22×C61#58511yescharacteristic
224C4×C61—28no—
225C4×C61—28no—
226C4×C61—24no—
227C4×C61—22no—
228C4×C61—44no—
229C4×C61—42no—
230C4×C61—44no—
231C4×C61—44no—
232C4×C61—41no—
233C4×C61#56214yesdirect product factor
234C2×#562C2×C6111yescharacteristic
235#2382C2×C6111yescharacteristic, complemented
236#2379C2×C6112yescomplemented
237#2378C2×C6114yescomplemented
238#2377C2×C6114yescomplemented
239#2376C2×C6114yescomplemented
240C4×#125C2×C6111yescharacteristic, complemented
241C4×#124C2×C6112yescomplemented
242C4×C22⋊Q8C2×C6114yescomplemented
243C4×#123C2×C6114yescomplemented
244C4×#121C2×C6114yescomplemented
245C2×C61—48no—
246C2×C61C4×H(5,𝔽2)12yes—
247C2×C61C4×2−1+411yescharacteristic
248C2×C61C2×#56211yescharacteristic
249C2×C61#352912yes—
250C2×C61#353111yescharacteristic
251#562C4×C61116yesdirect product factor
252C2×#125C22×C6111yescharacteristic
253C2×#124C22×C6112yes—
254C2×#123C22×C6114yes—
255C2×C22⋊Q8C22×C6114yes—
256C2×#121C22×C6114yes—
257C2×C4×Q8C22×C6112yes—
258C2×C4×D4C22×C6114yes—
259C2×C4×D4C22×C6112yes—
260C22×C42C22×C6114yes—
261#391C22×C6112yes—
262#391C22×C6114yes—
263#390C22×C6114yes—
264#390C22×C6112yes—
265#389C22×C6112yes—
266#389C22×C6114yes—
267#389C22×C6114yes—
268#388C22×C6114yes—
269#388C22×C6114yes—
270#388C22×C6114yes—
271#388C22×C6114yes—
272#388C22×C6114yes—
273#388C22×C6114yes—
274#387C22×C6114yes—
275#387C22×C6114yes—
276#387C22×C6114yes—
277#387C22×C6114yes—
278#387C22×C6114yes—
279#386C22×C6114yes—
280#386C22×C6114yes—
281#386C22×C6114yes—
282#385C22×C6114yes—
283#385C22×C6114yes—
284#385C22×C6114yes—
285#382C22×C6111yescharacteristic
286#382C22×C6112yes—
287#381C22×C6114yes—
288#380C22×C6114yes—
289#380C22×C6114yes—
290C4×C4⋊C4C22×C6114yes—
291C4×C4⋊C4C22×C6112yes—
292C4×C4⋊C4C22×C6114yes—
293C4×C22⋊C4C22×C6114yes—
294C4×C22⋊C4C22×C6114yes—
295C4×C22⋊C4C22×C6114yes—
296C43C22×C6111yescharacteristic
297C61C4×#56211yescharacteristic, direct product factor, Sylow
298C22×Q8C23×C6112yes—
299C22×D4C23×C6114yes—
300C22×D4C23×C6112yes—
301C23×C4C23×C6114yes—
302C23×C4C23×C6114yes—
303C23×C4C23×C6114yes—
304#125C2×C4×C6118yescomplemented
305#124C2×C4×C61116yescomplemented
306#123C2×C4×C61132yescomplemented
307C22⋊Q8C2×C4×C61132yescomplemented
308#121C2×C4×C61132yescomplemented
309C4×Q8—44no—
310C4×D4—48no—
311C4×D4—44no—
312C2×C4⋊C4C23×C6114yes—
313C2×C4⋊C4C23×C6112yes—
314C2×C4⋊C4C23×C6114yes—
315C2×C4⋊C4C23×C6112yes—
316C2×C4⋊C4C23×C6114yes—
317C2×C4⋊C4C23×C6114yes—
318C2×C4⋊C4C23×C6112yes—
319C2×C4⋊C4C23×C6114yes—
320C2×C4⋊C4C23×C6114yes—
321C2×C4⋊C4C23×C6112yes—
322C2×C22⋊C4C23×C6114yes—
323C2×C22⋊C4C23×C6114yes—
324C2×C22⋊C4C23×C6114yes—
325C2×C22⋊C4C23×C6114yes—
326C2×C22⋊C4C23×C6114yes—
327C2×C22⋊C4C23×C6114yes—
328C2×C22⋊C4C23×C6114yes—
329C2×C22⋊C4C23×C6114yes—
330C2×C22⋊C4C23×C6114yes—
331C2×C42—48no—
332C2×C42C23×C6112yes—
333C2×C42C23×C6114yes—
334C2×C42C23×C6114yes—
335C2×C42C23×C6111yescharacteristic
336C2×C42C23×C6112yes—
337C2×C42C23×C6111yescharacteristic
338C2×C42C23×C6114yes—
339C2×C42C23×C6114yes—
340C2×C42C23×C6111yescharacteristic
341#95C23×C6114yes—
342#95C23×C6114yes—
343#95C23×C6114yes—
344#95C23×C6114yes—
345#95C23×C6114yes—
346#95C23×C6114yes—
347#95C23×C6114yes—
348#95C23×C6114yes—
349#95C23×C6114yes—
350#95C23×C6114yes—
351#95C23×C6114yes—
352C24C24×C6114yes—
353C2×Q8—42no—
354C2×Q8—42no—
355C2×Q8C22×C4×C6118yes—
356C2×D4—44no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4—44no—
360C2×D4C22×C4×C61116yes—
361C2×D4C22×C4×C6118yes—
362C22×C4—28no—
363C22×C4—28no—
364C22×C4—28no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4C22×C4×C61116yes—
371C22×C4C24×C6111yescharacteristic
372C22×C4C24×C6114yes—
373C22×C4C24×C6114yes—
374C22×C4C24×C6114yes—
375C22×C4C24×C6112yes—
376C22×C4C24×C6111yescharacteristic
377C22×C4C24×C6112yes—
378C22×C4C24×C6114yes—
379C22×C4C24×C6114yes—
380C22×C4C24×C6111yescharacteristic
381C4⋊C4—44no—
382C4⋊C4—42no—
383C4⋊C4—44no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C4⋊C4C22×C4×C61116yes—
389C4⋊C4C22×C4×C61116yes—
390C4⋊C4C22×C4×C6118yes—
391C22⋊C4—44no—
392C22⋊C4—44no—
393C22⋊C4—44no—
394C22⋊C4—44no—
395C22⋊C4—44no—
396C22⋊C4—44no—
397C22⋊C4C22×C4×C61116yes—
398C22⋊C4C22×C4×C61116yes—
399C22⋊C4C22×C4×C61116yes—
400C42—28no—
401C42—28no—
402C42—24no—
403C42—22no—
404C42—44no—
405C42—44no—
406C42—41no—
407C42—42no—
408C42C22×C4×C6114yes—
409C23—24no—
410C23—24no—
411C23—24no—
412C23—24no—
413C23—24no—
414C23—24no—
415C23C23×C4×C6118yes—
416C23C25×C6111yescharacteristic
417Q8—48no—
418D4—416no—
419D4—48no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—22no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—21no—
434C2×C4—24no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—416no—
443C2×C4—416no—
444C2×C4—416no—
445C2×C4C23×C4×C6112yes—
446C2×C4C23×C4×C6114yes—
447C2×C4C23×C4×C6118yes—
448C2×C4C23×C4×C6118yes—
449C2×C4H(5,𝔽2)×C6114yes—
450C2×C42−1+4×C6112yes—
451C22—28no—
452C22—28no—
453C22—28no—
454C22—44no—
455C22—44no—
456C22—44no—
457C22—44no—
458C22C24×C4×C6111yescharacteristic
459C22C2×H(5,𝔽2)×C6112yes—
460C22C2×2−1+4×C6111yescharacteristic
461C22C61×#58512yes—
462C22C61×#58511yescharacteristic
463C4—28no—
464C4—28no—
465C4—24no—
466C4—22no—
467C4—44no—
468C4—42no—
469C4—44no—
470C4—44no—
471C4—41no—
472C4C61×#56214yesdirect product factor
473C2—48no—
474C2C4×H(5,𝔽2)×C6112yes—
475C2C4×2−1+4×C6111yescharacteristic
476C2C2×C61×#56211yescharacteristic
477C2C61×#352912yes—
478C2C61×#353111yescharacteristic
subgroupG/H
centerC22×C4×C61C24
commutatorC22C24×C4×C61
Frattini, cosocleC23C25×C61
socleC23×C61C25
prime radicalC2×C61×#562C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C4×C61 ◁ G

Complements

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

NH ≅ G/Ntype
2. C61×#2382C2semidirect
3. C61×#2379C2semidirect
4. C61×#2378C2semidirect
5. C61×#2377C2semidirect
6. C61×#2376C2semidirect
7. C4×#125×C61C2semidirect
8. C4×#124×C61C2semidirect
9. C4×C22⋊Q8×C61C2semidirect
10. C4×#123×C61C2semidirect
11. C4×#121×C61C2semidirect
12. C61×#562C4direct (N × H)
64. #125×C61C2×C4semidirect
65. #124×C61C2×C4semidirect
66. #123×C61C2×C4semidirect
67. C22⋊Q8×C61C2×C4semidirect
68. #121×C61C2×C4semidirect
211. C4×#562C61direct (N × H)
233. C4×C61#562direct (N × H)
235. #2382C2×C61semidirect
236. #2379C2×C61semidirect
237. #2378C2×C61semidirect
238. #2377C2×C61semidirect
239. #2376C2×C61semidirect
240. C4×#125C2×C61semidirect
241. C4×#124C2×C61semidirect
242. C4×C22⋊Q8C2×C61semidirect
243. C4×#123C2×C61semidirect
244. C4×#121C2×C61semidirect
251. #562C4×C61direct (N × H)
297. C61C4×#562direct (N × H)
304. #125C2×C4×C61semidirect
305. #124C2×C4×C61semidirect
306. #123C2×C4×C61semidirect
307. C22⋊Q8C2×C4×C61semidirect
308. #121C2×C4×C61semidirect
472. C4C61×#562direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C61×#5622
C61×#23822
C61×#23792
C61×#23782
C61×#23772
C61×#23762
C4×#125×C612
C4×#124×C612
C4×C22⋊Q8×C612
C4×#123×C612
C4×#121×C612
C4×#56261