Finite Group Databasebeta

← C47×#12713 #26052411

Subgroups — C47×#12713 #26052411

Order 12032.

986 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C47×#3485C211yesmaximal, characteristic
2C47×#2135C211yesmaximal, characteristic, complemented
3C47×#2129C211yesmaximal, characteristic, complemented
4C47×#2120C212yesmaximal, complemented
5C47×#1795C212yesmaximal, complemented
6C2×H(5,𝔽2)×C47C2211yescharacteristic
7C22×#41×C47C411yescharacteristic, complemented
8C47×#547—22no—
9C47×#547—22no—
10C47×#546—22no—
11C47×#546—22no—
12C47×#536C411yescharacteristic, complemented
13C47×#536C2211yescharacteristic
14C47×#534—22no—
15C47×#525C411yescharacteristic, complemented
16C47×#525C2211yescharacteristic
17C2×#123×C47—22no—
18C47×#518C411yescharacteristic, complemented
19C47×#421—22no—
20C47×#418—22no—
21C47×#413C2212yes—
22C47×#410C2212yes—
23C47×#356—22no—
24C47×#354—22no—
25C47×#352—22no—
26C47×AΣL(1,𝔽16)—22no—
27H(5,𝔽2)×C47—44no—
28C2×#41×C47—22no—
29C2×#41×C47—21no—
30C2×#41×C47—22no—
31C2×#41×C47—44no—
32C2×#41×C47C2×C411yescharacteristic
33C2×#41×C47D412yes—
34C2×#41×C47C2311yescharacteristic
35C22×Q8×C47C2×C411yescharacteristic
36C22×D4×C47—21no—
37C22×D4×C47—22no—
38C22×D4×C47—21no—
39C22×D4×C47C2×C411yescharacteristic
40C22×D4×C47D412yes—
41C23×C4×C47—21no—
42C23×C4×C47C2×C411yescharacteristic
43C2×M16×C47—22no—
44C4⋊Q8×C47—21no—
45C4⋊Q8×C47D412yes—
46#124×C47—21no—
47#124×C47D412yes—
48#123×C47—24no—
49#123×C47—24no—
50#123×C47—24no—
51#123×C47—22no—
52#123×C47—21no—
53#123×C47—22no—
54#123×C47D412yes—
55C22⋊Q8×C47—21no—
56C22⋊Q8×C47—22no—
57C22⋊Q8×C47—21no—
58C22⋊Q8×C47—22no—
59#121×C47—24no—
60#121×C47—24no—
61#121×C47—21no—
62#121×C47—21no—
63#120×C47—24no—
64#120×C47—21no—
65#120×C47D412yes—
66C4×D4×C47—22no—
67C4×D4×C47—22no—
68C4×D4×C47—22no—
69C4×D4×C47—22no—
70#117×C47—22no—
71#117×C47C2×C411yescharacteristic
72#117×C47C2×C411yescharacteristic
73C2×C4⋊C4×C47—24no—
74C2×C22⋊C4×C47—24no—
75C2×C22⋊C4×C47—22no—
76#104×C47—44no—
77#102×C47—44no—
78#101×C47—22no—
79#100×C47—22no—
80#99×C47—22no—
81#99×C47—22no—
82C24×C47—21no—
83C24×C47—22no—
84C24×C47C22⋊C411yescharacteristic
85#41×C47—24no—
86#41×C47—42no—
87#41×C47—42no—
88#41×C47—42no—
89#41×C47—48no—
90#41×C47—44no—
91#41×C47—44no—
92#41×C47—44no—
93C2×Q8×C47—21no—
94C2×Q8×C47—42no—
95C2×Q8×C47C22⋊C412yes—
96C2×Q8×C47C22⋊C411yescharacteristic
97C2×Q8×C47C2×D411yescharacteristic
98C2×D4×C47—24no—
99C2×D4×C47—24no—
100C2×D4×C47—24no—
101C2×D4×C47—21no—
102C2×D4×C47—22no—
103C2×D4×C47—22no—
104C2×D4×C47—22no—
105C2×D4×C47—21no—
106C2×D4×C47—22no—
107C2×D4×C47—42no—
108C2×D4×C47—44no—
109C2×D4×C47—44no—
110C2×D4×C47—48no—
111C2×D4×C47C22⋊C411yescharacteristic
112C2×D4×C47C22⋊C412yes—
113C2×D4×C47C2×D411yescharacteristic
114C2×D4×C47C2×D412yes—
115C22×C4×C47—24no—
116C22×C4×C47—21no—
117C22×C4×C47—24no—
118C22×C4×C47—21no—
119C22×C4×C47—21no—
120C22×C4×C47—21no—
121C22×C4×C47—22no—
122C22×C4×C47—42no—
123C22×C4×C47—44no—
124C22×C4×C47C22⋊C411yescharacteristic
125C22×C4×C47C22×C411yescharacteristic
126C22×C4×C47C2×D412yes—
127M16×C47—42no—
128M16×C47—42no—
129C2×C8×C47—42no—
130C4⋊C4×C47—22no—
131C4⋊C4×C47—24no—
132C4⋊C4×C47—21no—
133C4⋊C4×C47—21no—
134C4⋊C4×C47—22no—
135C4⋊C4×C47—24no—
136C4⋊C4×C47—42no—
137C22⋊C4×C47—22no—
138C22⋊C4×C47—24no—
139C22⋊C4×C47—24no—
140C22⋊C4×C47—22no—
141C22⋊C4×C47—21no—
142C22⋊C4×C47—22no—
143C22⋊C4×C47—24no—
144C22⋊C4×C47—22no—
145C22⋊C4×C47—42no—
146C22⋊C4×C47C22⋊C412yes—
147C42×C47—21no—
148C42×C47—42no—
149C42×C47C22⋊C412yes—
150C23×C47—24no—
151C23×C47—22no—
152C23×C47—21no—
153C23×C47—21no—
154C23×C47—44no—
155C23×C47—41no—
156C23×C47—81no—
157C23×C47—82no—
158C23×C47—84no—
159C23×C47C2×C22⋊C411yescharacteristic
160C23×C47#12012yes—
161Q8×C47—22no—
162Q8×C47—41no—
163Q8×C47—42no—
164Q8×C47—42no—
165D4×C47—24no—
166D4×C47—22no—
167D4×C47—22no—
168D4×C47—42no—
169D4×C47—41no—
170D4×C47—44no—
171D4×C47—44no—
172D4×C47—42no—
173D4×C47—48no—
174D4×C47—82no—
175D4×C47—82no—
176D4×C47#9914yes—
177C2×C4×C47—22no—
178C2×C4×C47—22no—
179C2×C4×C47—22no—
180C2×C4×C47—22no—
181C2×C4×C47—22no—
182C2×C4×C47—21no—
183C2×C4×C47—22no—
184C2×C4×C47—21no—
185C2×C4×C47—44no—
186C2×C4×C47—42no—
187C2×C4×C47—42no—
188C2×C4×C47—42no—
189C2×C4×C47—48no—
190C2×C4×C47—42no—
191C2×C4×C47—41no—
192C2×C4×C47—84no—
193C2×C4×C47—82no—
194C2×C4×C47C2×C22⋊C411yescharacteristic
195C2×C4×C47C2×C22⋊C411yescharacteristic
196C2×C4×C47#12012yes—
197C8×C47—82no—
198#12713C4711yesmaximal, characteristic, direct product factor, Sylow
199C22×C47—21no—
200C22×C47—22no—
201C22×C47—24no—
202C22×C47—42no—
203C22×C47—41no—
204C22×C47—44no—
205C22×C47—41no—
206C22×C47—41no—
207C22×C47—42no—
208C22×C47—41no—
209C22×C47—41no—
210C22×C47—84no—
211C22×C47—82no—
212C22×C47—164no—
213C22×C47#37911yescharacteristic
214C4×C47—22no—
215C4×C47—21no—
216C4×C47—42no—
217C4×C47—42no—
218C4×C47—41no—
219C4×C47—41no—
220C4×C47—42no—
221C4×C47—162no—
222C4×C47C2×#9912yes—
223#3485C2×C4711yescharacteristic
224#2135C2×C4711yescharacteristic, complemented
225#2129C2×C4711yescharacteristic, complemented
226#2120C2×C4712yescomplemented
227#1795C2×C4712yescomplemented
228C2×C47—21no—
229C2×C47—42no—
230C2×C47—44no—
231C2×C47—41no—
232C2×C47—81no—
233C2×C47—81no—
234C2×C47#178111yescharacteristic
235C2×H(5,𝔽2)C22×C4711yescharacteristic
236C22×#41C4×C4711yescharacteristic, complemented
237#547—22no—
238#547—22no—
239#546—22no—
240#546—22no—
241#536C4×C4711yescharacteristic, complemented
242#536C22×C4711yescharacteristic
243#534—22no—
244#525C4×C4711yescharacteristic, complemented
245#525C22×C4711yescharacteristic
246C2×#123—22no—
247#518C4×C4711yescharacteristic, complemented
248#421—22no—
249#418—22no—
250#413C22×C4712yes—
251#410C22×C4712yes—
252#356—22no—
253#354—22no—
254#352—22no—
255AΣL(1,𝔽16)—22no—
256C47#1271311yescharacteristic, direct product factor, Sylow
257H(5,𝔽2)—44no—
258C2×#41—22no—
259C2×#41—21no—
260C2×#41—22no—
261C2×#41—44no—
262C2×#41C2×C4×C4711yescharacteristic
263C2×#41D4×C4712yes—
264C2×#41C23×C4711yescharacteristic
265C22×Q8C2×C4×C4711yescharacteristic
266C22×D4—21no—
267C22×D4—22no—
268C22×D4—21no—
269C22×D4C2×C4×C4711yescharacteristic
270C22×D4D4×C4712yes—
271C23×C4—21no—
272C23×C4C2×C4×C4711yescharacteristic
273C2×M16—22no—
274C4⋊Q8—21no—
275C4⋊Q8D4×C4712yes—
276#124—21no—
277#124D4×C4712yes—
278#123—24no—
279#123—24no—
280#123—24no—
281#123—22no—
282#123—21no—
283#123—22no—
284#123D4×C4712yes—
285C22⋊Q8—21no—
286C22⋊Q8—22no—
287C22⋊Q8—21no—
288C22⋊Q8—22no—
289#121—24no—
290#121—24no—
291#121—21no—
292#121—21no—
293#120—24no—
294#120—21no—
295#120D4×C4712yes—
296C4×D4—22no—
297C4×D4—22no—
298C4×D4—22no—
299C4×D4—22no—
300#117—22no—
301#117C2×C4×C4711yescharacteristic
302#117C2×C4×C4711yescharacteristic
303C2×C4⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—22no—
306#104—44no—
307#102—44no—
308#101—22no—
309#100—22no—
310#99—22no—
311#99—22no—
312C24—21no—
313C24—22no—
314C24C22⋊C4×C4711yescharacteristic
315#41—24no—
316#41—42no—
317#41—42no—
318#41—42no—
319#41—48no—
320#41—44no—
321#41—44no—
322#41—44no—
323C2×Q8—21no—
324C2×Q8—42no—
325C2×Q8C22⋊C4×C4711yescharacteristic
326C2×Q8C22⋊C4×C4712yes—
327C2×Q8C2×D4×C4711yescharacteristic
328C2×D4—24no—
329C2×D4—24no—
330C2×D4—24no—
331C2×D4—21no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—21no—
336C2×D4—22no—
337C2×D4—42no—
338C2×D4—44no—
339C2×D4—44no—
340C2×D4—48no—
341C2×D4C22⋊C4×C4711yescharacteristic
342C2×D4C22⋊C4×C4712yes—
343C2×D4C2×D4×C4711yescharacteristic
344C2×D4C2×D4×C4712yes—
345C22×C4—24no—
346C22×C4—21no—
347C22×C4—24no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—42no—
353C22×C4—44no—
354C22×C4C22⋊C4×C4711yescharacteristic
355C22×C4C22×C4×C4711yescharacteristic
356C22×C4C2×D4×C4712yes—
357M16—42no—
358M16—42no—
359C2×C8—42no—
360C4⋊C4—22no—
361C4⋊C4—24no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—22no—
365C4⋊C4—24no—
366C4⋊C4—42no—
367C22⋊C4—22no—
368C22⋊C4—24no—
369C22⋊C4—24no—
370C22⋊C4—22no—
371C22⋊C4—21no—
372C22⋊C4—22no—
373C22⋊C4—24no—
374C22⋊C4—22no—
375C22⋊C4—42no—
376C22⋊C4C22⋊C4×C4712yes—
377C42—21no—
378C42—42no—
379C42C22⋊C4×C4712yes—
380C23—24no—
381C23—22no—
382C23—21no—
383C23—21no—
384C23—44no—
385C23—41no—
386C23—81no—
387C23—82no—
388C23—84no—
389C23C2×C22⋊C4×C4711yescharacteristic
390C23#120×C4712yes—
391Q8—22no—
392Q8—41no—
393Q8—42no—
394Q8—42no—
395D4—24no—
396D4—22no—
397D4—22no—
398D4—42no—
399D4—41no—
400D4—44no—
401D4—44no—
402D4—42no—
403D4—48no—
404D4—82no—
405D4—82no—
406D4#99×C4714yes—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—21no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—48no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—84no—
423C2×C4—82no—
424C2×C4C2×C22⋊C4×C4711yescharacteristic
425C2×C4C2×C22⋊C4×C4711yescharacteristic
426C2×C4#120×C4712yes—
427C8—82no—
428C22—21no—
429C22—22no—
430C22—24no—
431C22—42no—
432C22—41no—
433C22—44no—
434C22—41no—
435C22—41no—
436C22—42no—
437C22—41no—
438C22—41no—
439C22—84no—
440C22—82no—
441C22—164no—
442C22C47×#37911yescharacteristic
443C4—22no—
444C4—21no—
445C4—42no—
446C4—42no—
447C4—41no—
448C4—41no—
449C4—42no—
450C4—162no—
451C4C2×#99×C4712yes—
452C2—21no—
453C2—42no—
454C2—44no—
455C2—41no—
456C2—81no—
457C2—81no—
458C2C47×#178111yescharacteristic
subgroupG/H
center, socleC2×C47#1781
commutatorC22×C4C22×C4×C47
Frattini, cosocleC2×#41C23×C47
prime radicalC47×#3485C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C47 ◁ C2×C4×C47 ◁ C2×H(5,𝔽2)×C47 ◁ G

Complements

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

NH ≅ G/Ntype
2. C47×#2135C2semidirect
3. C47×#2129C2semidirect
4. C47×#2120C2semidirect
5. C47×#1795C2semidirect
7. C22×#41×C47C4semidirect
12. C47×#536C4semidirect
15. C47×#525C4semidirect
18. C47×#518C4semidirect
198. #12713C47direct (N × H)
224. #2135C2×C47semidirect
225. #2129C2×C47semidirect
226. #2120C2×C47semidirect
227. #1795C2×C47semidirect
236. C22×#41C4×C47semidirect
241. #536C4×C47semidirect
244. #525C4×C47semidirect
247. #518C4×C47semidirect
256. C47#12713direct (N × H)

Maximal subgroups

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

subgroupindex
C47×#34852
C47×#21352
C47×#21292
C47×#21202
C47×#17952
#1271347