Finite Group Databasebeta

← C2×C47×#3517 #26102488

Subgroups — C2×C47×#3517 #26102488

Order 12032.

7854 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C47×#3517C2132yesmaximal, direct product factor
2C2×C47×#557C211yesmaximal, characteristic, complemented
3C2×C47×#553C216yesmaximal, complemented
4C2×C47×#550C213yesmaximal, complemented
5C2×C47×#550C211yesmaximal, characteristic, complemented
6C2×C47×#550C212yesmaximal, complemented
7C2×C47×#547C213yesmaximal, complemented
8C2×C47×#540C216yesmaximal, complemented
9C2×C47×#532C213yesmaximal, complemented
10C2×C47×#529C213yesmaximal, complemented
11C2×C47×#520C212yesmaximal, complemented
12C2×C4×#41×C47C211yesmaximal, characteristic, complemented
13C22×#41×C47C2211yescharacteristic
14C22×#41×C47C2212yes—
15C22×#41×C47C2216yes—
16C47×#557C22116yescomplemented
17C47×#553C22196yescomplemented
18C47×#550C22148yescomplemented
19C47×#550C22116yescomplemented
20C47×#550C22132yescomplemented
21C47×#547C22148yescomplemented
22C47×#540C22196yescomplemented
23C47×#532C22148yescomplemented
24C2×C4⋊Q8×C47C2213yes—
25C2×#127×C47C2213yes—
26C2×#127×C47C2216yes—
27C47×#529C22148yescomplemented
28C2×#126×C47C2216yes—
29C2×#125×C47C2211yescharacteristic
30C2×#125×C47C2216yes—
31C2×#124×C47C2213yes—
32C2×#123×C47C2216yes—
33C2×#123×C47C22112yes—
34C2×C22⋊Q8×C47C2213yes—
35C2×#121×C47C2213yes—
36C2×#121×C47C2216yes—
37C2×#121×C47C2216yes—
38C2×#121×C47C2216yes—
39C2×#121×C47C22112yes—
40C2×#121×C47C2216yes—
41C47×#520C22132yescomplemented
42C4×#41×C47C22116yescomplemented
43C2×C4×Q8×C47C2211yescharacteristic
44C2×C4×Q8×C47C2211yescharacteristic
45C2×C4×Q8×C47C2212yes—
46C2×C4×Q8×C47C2213yes—
47C2×C4×D4×C47C2213yes—
48C2×C4×D4×C47C2213yes—
49C2×C4×D4×C47C2216yes—
50C2×C4×D4×C47C2216yes—
51C2×C4×D4×C47C2213yes—
52C2×C4×D4×C47C2216yes—
53C2×C4×D4×C47C2216yes—
54C2×#117×C47C2213yes—
55C2×#117×C47C2216yes—
56C22×C4⋊C4×C47C2216yes—
57C22×C42×C47C2213yes—
58C2×#41×C47—28no—
59C2×#41×C47—216no—
60C2×#41×C47—424no—
61C2×#41×C47C2318yes—
62C2×#41×C47C23116yes—
63C2×#41×C47C23148yes—
64C22×Q8×C47C2313yes—
65C22×Q8×C47C2311yescharacteristic
66C22×D4×C47C2313yes—
67C22×D4×C47C2316yes—
68C22×D4×C47C2313yes—
69C22×D4×C47C2316yes—
70C22×D4×C47C2316yes—
71C22×D4×C47C23112yes—
72C23×C4×C47C2313yes—
73C23×C4×C47C2313yes—
74C23×C4×C47C2316yes—
75C23×C4×C47C2313yes—
76C23×C4×C47C23112yes—
77C4⋊Q8×C47C23124yes—
78#127×C47C23124yes—
79#127×C47C23148yes—
80#126×C47C23148yes—
81#125×C47C2318yes—
82#125×C47C23148yes—
83#124×C47C23124yes—
84#123×C47C23148yes—
85#123×C47C23196yes—
86C22⋊Q8×C47C23124yes—
87#121×C47C23124yes—
88#121×C47C23148yes—
89#121×C47C23148yes—
90#121×C47C23148yes—
91#121×C47C23196yes—
92#121×C47C23148yes—
93C4×Q8×C47C2318yes—
94C4×Q8×C47C2318yes—
95C4×Q8×C47C23116yes—
96C4×Q8×C47C23124yes—
97C4×D4×C47C23124yes—
98C4×D4×C47C23124yes—
99C4×D4×C47C23148yes—
100C4×D4×C47C23148yes—
101C4×D4×C47C23124yes—
102C4×D4×C47C23148yes—
103C4×D4×C47C23148yes—
104#117×C47C23124yes—
105#117×C47C23148yes—
106C2×C4⋊C4×C47C2313yes—
107C2×C4⋊C4×C47C2316yes—
108C2×C4⋊C4×C47C2316yes—
109C2×C4⋊C4×C47C23148yes—
110C2×C4⋊C4×C47C2312yes—
111C2×C4⋊C4×C47C2313yes—
112C2×C4⋊C4×C47C2311yescharacteristic
113C2×C4⋊C4×C47C2312yes—
114C2×C4⋊C4×C47C2316yes—
115C2×C4⋊C4×C47C2311yescharacteristic
116C2×C4⋊C4×C47C2316yes—
117C2×C22⋊C4×C47C2316yes—
118C2×C22⋊C4×C47C2316yes—
119C2×C22⋊C4×C47C2316yes—
120C2×C22⋊C4×C47C2316yes—
121C2×C22⋊C4×C47C2316yes—
122C2×C22⋊C4×C47C2316yes—
123C2×C42×C47C2311yescharacteristic
124C2×C42×C47C2313yes—
125C2×C42×C47C2313yes—
126C2×C42×C47C2313yes—
127C2×C42×C47C23124yes—
128C2×C42×C47C2316yes—
129C24×C47C2413yes—
130C24×C47C2416yes—
131#41×C47—232no—
132#41×C47—264no—
133#41×C47—496no—
134C2×Q8×C47—212no—
135C2×Q8×C47#4118yescomplemented
136C2×Q8×C47C24112yes—
137C2×Q8×C47C2414yes—
138C2×D4×C47—212no—
139C2×D4×C47—224no—
140C2×D4×C47—212no—
141C2×D4×C47—412no—
142C2×D4×C47—412no—
143C2×D4×C47—424no—
144C2×D4×C47C24112yes—
145C2×D4×C47C24124yes—
146C2×D4×C47C24112yes—
147C2×D4×C47C24124yes—
148C2×D4×C47C24124yes—
149C2×D4×C47C24148yes—
150C22×C4×C47—212no—
151C22×C4×C47—224no—
152C22×C4×C47—212no—
153C22×C4×C47—46no—
154C22×C4×C47—424no—
155C22×C4×C47C24112yes—
156C22×C4×C47C24112yes—
157C22×C4×C47C24124yes—
158C22×C4×C47C2411yescharacteristic
159C22×C4×C47C2411yescharacteristic
160C22×C4×C47C2412yes—
161C22×C4×C47C2413yes—
162C22×C4×C47C2413yes—
163C22×C4×C47C2416yes—
164C22×C4×C47C2412yes—
165C22×C4×C47C24112yes—
166C22×C4×C47C24148yes—
167C22×C4×C47C2413yes—
168C22×C4×C47C2411yescharacteristic
169C4⋊C4×C47C24112yes—
170C4⋊C4×C47C24124yes—
171C4⋊C4×C47C24124yes—
172C4⋊C4×C47C2418yes—
173C4⋊C4×C47C24112yes—
174C4⋊C4×C47C2414yes—
175C4⋊C4×C47C2418yes—
176C4⋊C4×C47C24124yes—
177C4⋊C4×C47C2414yes—
178C4⋊C4×C47C24124yes—
179C22⋊C4×C47C24124yes—
180C22⋊C4×C47C24124yes—
181C22⋊C4×C47C24124yes—
182C22⋊C4×C47C24124yes—
183C22⋊C4×C47C24124yes—
184C22⋊C4×C47C24124yes—
185C42×C47C2414yes—
186C42×C47C24112yes—
187C42×C47C24112yes—
188C42×C47C24112yes—
189C42×C47C24124yes—
190C23×C47—26no—
191C23×C47—212no—
192C23×C47—26no—
193C23×C47—212no—
194C23×C47—26no—
195C23×C47—212no—
196C23×C47C2516yes—
197C23×C47C25112yes—
198C23×C47C2511yescharacteristic
199Q8×C47—224no—
200Q8×C47C2×#41116yescomplemented
201D4×C47—224no—
202D4×C47—248no—
203D4×C47—224no—
204D4×C47—424no—
205D4×C47—424no—
206D4×C47—448no—
207C2×C4×C47—22no—
208C2×C4×C47—24no—
209C2×C4×C47—26no—
210C2×C4×C47—212no—
211C2×C4×C47—22no—
212C2×C4×C47—26no—
213C2×C4×C47—24no—
214C2×C4×C47—224no—
215C2×C4×C47—248no—
216C2×C4×C47—224no—
217C2×C4×C47—412no—
218C2×C4×C47—448no—
219C2×C4×C47C2×#41112yes—
220C2×C4×C47H(5,𝔽2)14yes—
221C2×C4×C47C2512yes—
222C2×C4×C47C2512yes—
223C2×C4×C47C2514yes—
224C2×C4×C47C2516yes—
225C2×C4×C47C2516yes—
226C2×C4×C47C25112yes—
227C2×C4×C47C2514yes—
228C2×C4×C47C2516yes—
229C2×C4×C47C2512yes—
230C2×#3517C4711yesmaximal, characteristic, direct product factor, Sylow
231C22×C47—26no—
232C22×C47—212no—
233C22×C47—26no—
234C22×C47—212no—
235C22×C47—26no—
236C22×C47—212no—
237C22×C47—412no—
238C22×C47—424no—
239C22×C47C22×#4112yes—
240C22×C47C2×H(5,𝔽2)12yes—
241C22×C47#58512yes—
242C22×C47C2611yescharacteristic
243C4×C47—22no—
244C4×C47—24no—
245C4×C47—26no—
246C4×C47—212no—
247C4×C47—22no—
248C4×C47—26no—
249C4×C47—24no—
250C4×C47C22×#41112yes—
251C4×C47C2×H(5,𝔽2)14yes—
252#3517C2×C47132yesdirect product factor
253C2×#557C2×C4711yescharacteristic, complemented
254C2×#553C2×C4716yescomplemented
255C2×#550C2×C4713yescomplemented
256C2×#550C2×C4711yescharacteristic, complemented
257C2×#550C2×C4712yescomplemented
258C2×#547C2×C4713yescomplemented
259C2×#540C2×C4716yescomplemented
260C2×#532C2×C4713yescomplemented
261C2×#529C2×C4713yescomplemented
262C2×#520C2×C4712yescomplemented
263C2×C4×#41C2×C4711yescharacteristic, complemented
264C2×C47—46no—
265C2×C47—412no—
266C2×C47#351714yesdirect product factor
267C2×C47C23×#4111yescharacteristic
268C2×C47C22×H(5,𝔽2)11yescharacteristic
269C2×C47C2×#58511yescharacteristic
270C22×#41C22×C4711yescharacteristic
271C22×#41C22×C4712yes—
272C22×#41C22×C4716yes—
273#557C22×C47116yescomplemented
274#553C22×C47196yescomplemented
275#550C22×C47148yescomplemented
276#550C22×C47116yescomplemented
277#550C22×C47132yescomplemented
278#547C22×C47148yescomplemented
279#540C22×C47196yescomplemented
280#532C22×C47148yescomplemented
281C2×C4⋊Q8C22×C4713yes—
282C2×#127C22×C4713yes—
283C2×#127C22×C4716yes—
284#529C22×C47148yescomplemented
285C2×#126C22×C4716yes—
286C2×#125C22×C4711yescharacteristic
287C2×#125C22×C4716yes—
288C2×#124C22×C4713yes—
289C2×#123C22×C4716yes—
290C2×#123C22×C47112yes—
291C2×C22⋊Q8C22×C4713yes—
292C2×#121C22×C4713yes—
293C2×#121C22×C4716yes—
294C2×#121C22×C4716yes—
295C2×#121C22×C4716yes—
296C2×#121C22×C47112yes—
297C2×#121C22×C4716yes—
298#520C22×C47132yescomplemented
299C4×#41C22×C47116yescomplemented
300C2×C4×Q8C22×C4711yescharacteristic
301C2×C4×Q8C22×C4711yescharacteristic
302C2×C4×Q8C22×C4712yes—
303C2×C4×Q8C22×C4713yes—
304C2×C4×D4C22×C4713yes—
305C2×C4×D4C22×C4713yes—
306C2×C4×D4C22×C4716yes—
307C2×C4×D4C22×C4716yes—
308C2×C4×D4C22×C4713yes—
309C2×C4×D4C22×C4716yes—
310C2×C4×D4C22×C4716yes—
311C2×#117C22×C4713yes—
312C2×#117C22×C4716yes—
313C22×C4⋊C4C22×C4716yes—
314C22×C42C22×C4713yes—
315C47C2×#351711yescharacteristic, direct product factor, Sylow
316C2×#41—28no—
317C2×#41—216no—
318C2×#41—424no—
319C2×#41C23×C4718yes—
320C2×#41C23×C47116yes—
321C2×#41C23×C47148yes—
322C22×Q8C23×C4713yes—
323C22×Q8C23×C4711yescharacteristic
324C22×D4C23×C4713yes—
325C22×D4C23×C4716yes—
326C22×D4C23×C4713yes—
327C22×D4C23×C4716yes—
328C22×D4C23×C4716yes—
329C22×D4C23×C47112yes—
330C23×C4C23×C4716yes—
331C23×C4C23×C4713yes—
332C23×C4C23×C4713yes—
333C23×C4C23×C47112yes—
334C23×C4C23×C4713yes—
335C4⋊Q8C23×C47124yes—
336#127C23×C47124yes—
337#127C23×C47148yes—
338#126C23×C47148yes—
339#125C23×C4718yes—
340#125C23×C47148yes—
341#124C23×C47124yes—
342#123C23×C47148yes—
343#123C23×C47196yes—
344C22⋊Q8C23×C47124yes—
345#121C23×C47124yes—
346#121C23×C47148yes—
347#121C23×C47148yes—
348#121C23×C47148yes—
349#121C23×C47196yes—
350#121C23×C47148yes—
351C4×Q8C23×C4718yes—
352C4×Q8C23×C4718yes—
353C4×Q8C23×C47116yes—
354C4×Q8C23×C47124yes—
355C4×D4C23×C47124yes—
356C4×D4C23×C47124yes—
357C4×D4C23×C47148yes—
358C4×D4C23×C47148yes—
359C4×D4C23×C47124yes—
360C4×D4C23×C47148yes—
361C4×D4C23×C47148yes—
362#117C23×C47124yes—
363#117C23×C47148yes—
364C2×C4⋊C4C23×C4713yes—
365C2×C4⋊C4C23×C4716yes—
366C2×C4⋊C4C23×C4716yes—
367C2×C4⋊C4C23×C47148yes—
368C2×C4⋊C4C23×C4712yes—
369C2×C4⋊C4C23×C4713yes—
370C2×C4⋊C4C23×C4711yescharacteristic
371C2×C4⋊C4C23×C4712yes—
372C2×C4⋊C4C23×C4716yes—
373C2×C4⋊C4C23×C4711yescharacteristic
374C2×C4⋊C4C23×C4716yes—
375C2×C22⋊C4C23×C4716yes—
376C2×C22⋊C4C23×C4716yes—
377C2×C22⋊C4C23×C4716yes—
378C2×C22⋊C4C23×C4716yes—
379C2×C22⋊C4C23×C4716yes—
380C2×C22⋊C4C23×C4716yes—
381C2×C42C23×C4713yes—
382C2×C42C23×C4713yes—
383C2×C42C23×C4716yes—
384C2×C42C23×C4713yes—
385C2×C42C23×C47124yes—
386C2×C42C23×C4711yescharacteristic
387C24C24×C4713yes—
388C24C24×C4716yes—
389#41—232no—
390#41—264no—
391#41—496no—
392C2×Q8—212no—
393C2×Q8#41×C4718yescomplemented
394C2×Q8C24×C47112yes—
395C2×Q8C24×C4714yes—
396C2×D4—212no—
397C2×D4—224no—
398C2×D4—212no—
399C2×D4—412no—
400C2×D4—412no—
401C2×D4—424no—
402C2×D4C24×C47112yes—
403C2×D4C24×C47124yes—
404C2×D4C24×C47112yes—
405C2×D4C24×C47124yes—
406C2×D4C24×C47124yes—
407C2×D4C24×C47148yes—
408C22×C4—224no—
409C22×C4—212no—
410C22×C4—212no—
411C22×C4—46no—
412C22×C4—424no—
413C22×C4C24×C4711yescharacteristic
414C22×C4C24×C4711yescharacteristic
415C22×C4C24×C4712yes—
416C22×C4C24×C4713yes—
417C22×C4C24×C4713yes—
418C22×C4C24×C47112yes—
419C22×C4C24×C47112yes—
420C22×C4C24×C4716yes—
421C22×C4C24×C4712yes—
422C22×C4C24×C47124yes—
423C22×C4C24×C47112yes—
424C22×C4C24×C47148yes—
425C22×C4C24×C4713yes—
426C22×C4C24×C4711yescharacteristic
427C4⋊C4C24×C47112yes—
428C4⋊C4C24×C47124yes—
429C4⋊C4C24×C47124yes—
430C4⋊C4C24×C4718yes—
431C4⋊C4C24×C47112yes—
432C4⋊C4C24×C4714yes—
433C4⋊C4C24×C4718yes—
434C4⋊C4C24×C47124yes—
435C4⋊C4C24×C4714yes—
436C4⋊C4C24×C47124yes—
437C22⋊C4C24×C47124yes—
438C22⋊C4C24×C47124yes—
439C22⋊C4C24×C47124yes—
440C22⋊C4C24×C47124yes—
441C22⋊C4C24×C47124yes—
442C22⋊C4C24×C47124yes—
443C42C24×C4714yes—
444C42C24×C47112yes—
445C42C24×C47112yes—
446C42C24×C47112yes—
447C42C24×C47124yes—
448C23—26no—
449C23—212no—
450C23—26no—
451C23—212no—
452C23—26no—
453C23—212no—
454C23C25×C4716yes—
455C23C25×C47112yes—
456C23C25×C4711yescharacteristic
457Q8—224no—
458Q8C2×#41×C47116yescomplemented
459D4—224no—
460D4—248no—
461D4—224no—
462D4—424no—
463D4—424no—
464D4—448no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—26no—
468C2×C4—212no—
469C2×C4—22no—
470C2×C4—26no—
471C2×C4—24no—
472C2×C4—224no—
473C2×C4—248no—
474C2×C4—224no—
475C2×C4—412no—
476C2×C4—448no—
477C2×C4C2×#41×C47112yes—
478C2×C4H(5,𝔽2)×C4714yes—
479C2×C4C25×C4712yes—
480C2×C4C25×C4712yes—
481C2×C4C25×C4714yes—
482C2×C4C25×C4716yes—
483C2×C4C25×C4716yes—
484C2×C4C25×C47112yes—
485C2×C4C25×C4714yes—
486C2×C4C25×C4716yes—
487C2×C4C25×C4712yes—
488C22—26no—
489C22—212no—
490C22—26no—
491C22—212no—
492C22—26no—
493C22—212no—
494C22—412no—
495C22—424no—
496C22C22×#41×C4712yes—
497C22C2×H(5,𝔽2)×C4712yes—
498C22C47×#58512yes—
499C22C26×C4711yescharacteristic
500C4—22no—
501C4—24no—
502C4—26no—
503C4—212no—
504C4—22no—
505C4—26no—
506C4—24no—
507C4C22×#41×C47112yes—
508C4C2×H(5,𝔽2)×C4714yes—
509C2—46no—
510C2—412no—
511C2C47×#351714yesdirect product factor
512C2C23×#41×C4711yescharacteristic
513C2C22×H(5,𝔽2)×C4711yescharacteristic
514C2C2×C47×#58511yescharacteristic
subgroupG/H
center, socleC23×C47C25
commutator, Frattini, cosocleC22C26×C47
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C47 ◁ G

Complements

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

NH ≅ G/Ntype
1. C47×#3517C2direct (N × H)
2. C2×C47×#557C2semidirect
3. C2×C47×#553C2semidirect
4. C2×C47×#550C2semidirect
5. C2×C47×#550C2semidirect
6. C2×C47×#550C2semidirect
7. C2×C47×#547C2semidirect
8. C2×C47×#540C2semidirect
9. C2×C47×#532C2semidirect
10. C2×C47×#529C2semidirect
11. C2×C47×#520C2semidirect
12. C2×C4×#41×C47C2semidirect
16. C47×#557C22semidirect
17. C47×#553C22semidirect
18. C47×#550C22semidirect
19. C47×#550C22semidirect
20. C47×#550C22semidirect
21. C47×#547C22semidirect
22. C47×#540C22semidirect
23. C47×#532C22semidirect
27. C47×#529C22semidirect
41. C47×#520C22semidirect
42. C4×#41×C47C22semidirect
135. C2×Q8×C47#41semidirect
200. Q8×C47C2×#41semidirect
230. C2×#3517C47direct (N × H)
252. #3517C2×C47direct (N × H)
253. C2×#557C2×C47semidirect
254. C2×#553C2×C47semidirect
255. C2×#550C2×C47semidirect
256. C2×#550C2×C47semidirect
257. C2×#550C2×C47semidirect
258. C2×#547C2×C47semidirect
259. C2×#540C2×C47semidirect
260. C2×#532C2×C47semidirect
261. C2×#529C2×C47semidirect
262. C2×#520C2×C47semidirect
263. C2×C4×#41C2×C47semidirect
266. C2×C47#3517direct (N × H)
273. #557C22×C47semidirect
274. #553C22×C47semidirect
275. #550C22×C47semidirect
276. #550C22×C47semidirect
277. #550C22×C47semidirect
278. #547C22×C47semidirect
279. #540C22×C47semidirect
280. #532C22×C47semidirect
284. #529C22×C47semidirect
298. #520C22×C47semidirect
299. C4×#41C22×C47semidirect
315. C47C2×#3517direct (N × H)
393. C2×Q8#41×C47semidirect
458. Q8C2×#41×C47semidirect
511. C2C47×#3517direct (N × H)

Maximal subgroups

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

subgroupindex
C47×#35172
C2×C47×#5572
C2×C47×#5532
C2×C47×#5502
C2×C47×#5472
C2×C47×#5402
C2×C47×#5322
C2×C47×#5292
C2×C47×#5202
C2×C4×#41×C472
C2×#351747