Finite Group Databasebeta

← C47×#26083 #26065781

Subgroups — C47×#26083 #26065781

Order 12032.

690 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C47×#3164C211yesmaximal, characteristic, complemented
2C47×#3083C212yesmaximal
3C47×#2913C211yesmaximal, characteristic
4C47×#2089C212yesmaximal, complemented
5C47×#2060C212yesmaximal, complemented
6C47×#1952C211yesmaximal, characteristic, complemented
7C47×#1945C211yesmaximal, characteristic, complemented
8C47×#1859C212yesmaximal
9C47×#1845C211yesmaximal, characteristic, complemented
10C47×#1816C211yesmaximal, characteristic
11C47×#1764C211yesmaximal, characteristic
12C2×#131×C47C2211yescharacteristic
13C47×#544C2212yes—
14C47×#520C2212yes—
15C47×#501—22no—
16C47×#500—21no—
17C47×#499—21no—
18C47×#479—22no—
19C47×#478—22no—
20C47×#477—22no—
21C47×#474—22no—
22C2×#108×C47C2211yescharacteristic
23C47×#428—22no—
24C47×#428C2212yes—
25C47×#428C2212yes—
26C47×#428C2212yes—
27C47×#427—21no—
28C47×#427C2211yescharacteristic
29C2×#107×C47—21no—
30C2×#107×C47C2212yes—
31C2×#107×C47C2211yescharacteristic
32C2×#107×C47C2211yescharacteristic
33C47×#424C2212yes—
34C2×#104×C47C2211yescharacteristic
35C47×#417C2212yes—
36C47×#416C2211yescharacteristic
37C47×#405C2211yescharacteristic
38C2×C4×C8×C47C2211yescharacteristic
39C47×#384C2212yes—
40C47×#343C2212yes—
41C47×#343C2212yes—
42C47×#341C2211yescharacteristic
43C47×#339C2211yescharacteristic
44C47×#339C2212yes—
45C2×#41×C47C2311yescharacteristic
46#131×C47—22no—
47#131×C47Q814yescomplemented
48C2×M16×C47—21no—
49C2×M16×C47C2312yes—
50C2×M16×C47C2311yescharacteristic
51C2×M16×C47C2311yescharacteristic
52C22×C8×C47—21no—
53C22×C8×C47C2311yescharacteristic
54C22×C8×C47C2311yescharacteristic
55C4⋊Q8×C47—22no—
56C4⋊Q8×C47—22no—
57#125×C47—22no—
58#125×C47—22no—
59C22⋊Q8×C47—42no—
60C4×Q8×C47—22no—
61C4×D4×C47—22no—
62C4×D4×C47—42no—
63#117×C47—22no—
64#117×C47C2312yes—
65#117×C47C2311yescharacteristic
66C2×C4⋊C4×C47—22no—
67C2×C4⋊C4×C47—42no—
68C2×C4⋊C4×C47C2312yes—
69C2×C4⋊C4×C47C2312yes—
70C2×C42×C47C2311yescharacteristic
71#108×C47—22no—
72#107×C47—22no—
73#107×C47—22no—
74#107×C47—21no—
75#107×C47—21no—
76#107×C47—21no—
77#107×C47—21no—
78#107×C47—22no—
79#107×C47—22no—
80#107×C47—22no—
81#107×C47D411yescharacteristic
82#107×C47D411yescharacteristic
83#106×C47—22no—
84#106×C47—22no—
85#106×C47—22no—
86#106×C47D412yes—
87C4⋊C8×C47—22no—
88C4⋊C8×C47—22no—
89#104×C47Q814yescomplemented
90#103×C47—22no—
91#103×C47D411yescharacteristic
92#103×C47D411yescharacteristic
93#102×C47—22no—
94#102×C47D411yescharacteristic
95#102×C47D411yescharacteristic
96#97×C47—21no—
97C4×C8×C47—21no—
98C4×C8×C47—22no—
99#41×C47—21no—
100#41×C47—21no—
101#41×C47C2×Q812yes—
102#41×C47C2×Q812yes—
103C2×Q8×C47—42no—
104C2×Q8×C47C2×D411yescharacteristic
105C2×D4×C47—21no—
106C2×D4×C47C2×D411yescharacteristic
107C22×C4×C47—21no—
108C22×C4×C47—21no—
109C22×C4×C47—22no—
110C22×C4×C47—22no—
111C22×C4×C47—42no—
112C22×C4×C47C2411yescharacteristic
113M16×C47—21no—
114M16×C47—21no—
115M16×C47—24no—
116M16×C47—22no—
117M16×C47—22no—
118M16×C47C2×Q812yes—
119M16×C47C2×Q812yes—
120C2×C8×C47—21no—
121C2×C8×C47—21no—
122C2×C8×C47—21no—
123C2×C8×C47—21no—
124C2×C8×C47—24no—
125C2×C8×C47—22no—
126C2×C8×C47C2×D411yescharacteristic
127C2×C8×C47C2×D411yescharacteristic
128C2×C8×C47C2×D411yescharacteristic
129C2×C8×C47C2×D411yescharacteristic
130C2×C8×C47C2×Q812yes—
131C2×C8×C47#4112yes—
132C4⋊C4×C47—22no—
133C4⋊C4×C47—22no—
134C4⋊C4×C47—22no—
135C4⋊C4×C47—22no—
136C4⋊C4×C47—42no—
137C4⋊C4×C47—42no—
138C4⋊C4×C47—42no—
139C4⋊C4×C47—42no—
140C4⋊C4×C47—84no—
141C4⋊C4×C47C2×D411yescharacteristic
142C4⋊C4×C47C2×D411yescharacteristic
143C4⋊C4×C47C2×D412yes—
144C4⋊C4×C47C2×D412yes—
145C22⋊C4×C47—22no—
146C22⋊C4×C47—21no—
147C22⋊C4×C47—42no—
148C42×C47—22no—
149C42×C47—21no—
150C42×C47—21no—
151C42×C47—42no—
152C42×C47C2×Q812yes—
153C23×C47—21no—
154C23×C47H(5,𝔽2)11yescharacteristic
155Q8×C47—21no—
156Q8×C47—162no—
157Q8×C47C4⋊Q812yes—
158D4×C47—22no—
159D4×C47—22no—
160D4×C47—21no—
161D4×C47C4⋊Q812yes—
162C2×C4×C47—22no—
163C2×C4×C47—22no—
164C2×C4×C47—21no—
165C2×C4×C47—22no—
166C2×C4×C47—22no—
167C2×C4×C47—22no—
168C2×C4×C47—22no—
169C2×C4×C47—21no—
170C2×C4×C47—22no—
171C2×C4×C47—21no—
172C2×C4×C47—42no—
173C2×C4×C47—42no—
174C2×C4×C47—82no—
175C2×C4×C47—82no—
176C2×C4×C47—162no—
177C2×C4×C47C22×D411yescharacteristic
178C2×C4×C47C22×D411yescharacteristic
179C2×C4×C47C22×Q811yescharacteristic
180C2×C4×C47C22×Q811yescharacteristic
181C2×C4×C47C2×#4111yescharacteristic
182C2×C4×C472−1+411yescharacteristic
183C8×C47—22no—
184C8×C47—21no—
185C8×C47—21no—
186C8×C47—22no—
187C8×C47—42no—
188C8×C47C22⋊Q812yes—
189C8×C47C22⋊Q812yes—
190#26083C4711yesmaximal, characteristic, direct product factor, Sylow
191C22×C47—21no—
192C22×C47—21no—
193C22×C47—22no—
194C22×C47—41no—
195C22×C47—41no—
196C22×C47#54411yescharacteristic
197C22×C47D4211yescharacteristic
198C22×C47#55411yescharacteristic
199C4×C47—24no—
200C4×C47—22no—
201C4×C47—42no—
202C4×C47—41no—
203C4×C47—42no—
204C4×C47—41no—
205C4×C47—82no—
206C4×C47—82no—
207C4×C47C2×C22⋊Q811yescharacteristic
208C4×C47C2×C4⋊Q811yescharacteristic
209C4×C47#54711yescharacteristic
210C4×C47D4×Q811yescharacteristic
211#3164C2×C4711yescharacteristic, complemented
212#3083C2×C4712yes—
213#2913C2×C4711yescharacteristic
214#2089C2×C4712yescomplemented
215#2060C2×C4712yescomplemented
216#1952C2×C4711yescharacteristic, complemented
217#1945C2×C4711yescharacteristic, complemented
218#1859C2×C4712yes—
219#1845C2×C4711yescharacteristic, complemented
220#1816C2×C4711yescharacteristic
221#1764C2×C4711yescharacteristic
222C2×C47—21no—
223C2×C47—21no—
224C2×C47—42no—
225C2×C47#244011yescharacteristic
226C2×C47#329211yescharacteristic
227C2×C47#329311yescharacteristic
228C2×#131C22×C4711yescharacteristic
229#544C22×C4712yes—
230#520C22×C4712yes—
231#501—22no—
232#500—21no—
233#499—21no—
234#479—22no—
235#478—22no—
236#477—22no—
237#474—22no—
238C2×#108C22×C4711yescharacteristic
239#428—22no—
240#428C22×C4712yes—
241#428C22×C4712yes—
242#428C22×C4712yes—
243#427—21no—
244#427C22×C4711yescharacteristic
245C2×#107—21no—
246C2×#107C22×C4712yes—
247C2×#107C22×C4711yescharacteristic
248C2×#107C22×C4711yescharacteristic
249#424C22×C4712yes—
250C2×#104C22×C4711yescharacteristic
251#417C22×C4712yes—
252#416C22×C4711yescharacteristic
253#405C22×C4711yescharacteristic
254C2×C4×C8C22×C4711yescharacteristic
255#384C22×C4712yes—
256#343C22×C4712yes—
257#343C22×C4712yes—
258#341C22×C4711yescharacteristic
259#339C22×C4711yescharacteristic
260#339C22×C4712yes—
261C47#2608311yescharacteristic, direct product factor, Sylow
262C2×#41C23×C4711yescharacteristic
263#131—22no—
264#131Q8×C4714yescomplemented
265C2×M16—21no—
266C2×M16C23×C4712yes—
267C2×M16C23×C4711yescharacteristic
268C2×M16C23×C4711yescharacteristic
269C22×C8—21no—
270C22×C8C23×C4711yescharacteristic
271C22×C8C23×C4711yescharacteristic
272C4⋊Q8—22no—
273C4⋊Q8—22no—
274#125—22no—
275#125—22no—
276C22⋊Q8—42no—
277C4×Q8—22no—
278C4×D4—22no—
279C4×D4—42no—
280#117—22no—
281#117C23×C4712yes—
282#117C23×C4711yescharacteristic
283C2×C4⋊C4—22no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4C23×C4712yes—
286C2×C4⋊C4C23×C4712yes—
287C2×C42C23×C4711yescharacteristic
288#108—22no—
289#107—21no—
290#107—21no—
291#107—22no—
292#107—22no—
293#107—21no—
294#107—21no—
295#107—22no—
296#107—22no—
297#107—22no—
298#107D4×C4711yescharacteristic
299#107D4×C4711yescharacteristic
300#106—22no—
301#106—22no—
302#106—22no—
303#106D4×C4712yes—
304C4⋊C8—22no—
305C4⋊C8—22no—
306#104Q8×C4714yescomplemented
307#103—22no—
308#103D4×C4711yescharacteristic
309#103D4×C4711yescharacteristic
310#102—22no—
311#102D4×C4711yescharacteristic
312#102D4×C4711yescharacteristic
313#97—21no—
314C4×C8—21no—
315C4×C8—22no—
316#41—21no—
317#41—21no—
318#41C2×Q8×C4712yes—
319#41C2×Q8×C4712yes—
320C2×Q8—42no—
321C2×Q8C2×D4×C4711yescharacteristic
322C2×D4—21no—
323C2×D4C2×D4×C4711yescharacteristic
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—42no—
329C22×C4C24×C4711yescharacteristic
330M16—21no—
331M16—21no—
332M16—24no—
333M16—22no—
334M16—22no—
335M16C2×Q8×C4712yes—
336M16C2×Q8×C4712yes—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—24no—
342C2×C8—22no—
343C2×C8C2×D4×C4711yescharacteristic
344C2×C8C2×D4×C4711yescharacteristic
345C2×C8C2×D4×C4711yescharacteristic
346C2×C8C2×D4×C4711yescharacteristic
347C2×C8C2×Q8×C4712yes—
348C2×C8#41×C4712yes—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—84no—
358C4⋊C4C2×D4×C4711yescharacteristic
359C4⋊C4C2×D4×C4711yescharacteristic
360C4⋊C4C2×D4×C4712yes—
361C4⋊C4C2×D4×C4712yes—
362C22⋊C4—22no—
363C22⋊C4—21no—
364C22⋊C4—42no—
365C42—21no—
366C42—22no—
367C42—21no—
368C42—42no—
369C42C2×Q8×C4712yes—
370C23—21no—
371C23H(5,𝔽2)×C4711yescharacteristic
372Q8—21no—
373Q8—162no—
374Q8C4⋊Q8×C4712yes—
375D4—22no—
376D4—22no—
377D4—21no—
378D4C4⋊Q8×C4712yes—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—21no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—21no—
387C2×C4—22no—
388C2×C4—21no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—162no—
394C2×C4C22×D4×C4711yescharacteristic
395C2×C4C22×D4×C4711yescharacteristic
396C2×C4C22×Q8×C4711yescharacteristic
397C2×C4C22×Q8×C4711yescharacteristic
398C2×C4C2×#41×C4711yescharacteristic
399C2×C42−1+4×C4711yescharacteristic
400C8—22no—
401C8—21no—
402C8—21no—
403C8—22no—
404C8—42no—
405C8C22⋊Q8×C4712yes—
406C8C22⋊Q8×C4712yes—
407C22—21no—
408C22—21no—
409C22—22no—
410C22—41no—
411C22—41no—
412C22C47×#54411yescharacteristic
413C22D42×C4711yescharacteristic
414C22C47×#55411yescharacteristic
415C4—24no—
416C4—22no—
417C4—42no—
418C4—41no—
419C4—42no—
420C4—41no—
421C4—82no—
422C4—82no—
423C4C2×C22⋊Q8×C4711yescharacteristic
424C4C2×C4⋊Q8×C4711yescharacteristic
425C4C47×#54711yescharacteristic
426C4D4×Q8×C4711yescharacteristic
427C2—21no—
428C2—21no—
429C2—42no—
430C2C47×#244011yescharacteristic
431C2C47×#329211yescharacteristic
432C2C47×#329311yescharacteristic
subgroupG/H
center, socleC22×C47D42
commutator, Frattini, cosocleC22×C4C24×C47
prime radicalC2×#41×C47C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C47 ◁ C22×C4×C47 ◁ G

Complements

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

NH ≅ G/Ntype
1. C47×#3164C2semidirect
4. C47×#2089C2semidirect
5. C47×#2060C2semidirect
6. C47×#1952C2semidirect
7. C47×#1945C2semidirect
9. C47×#1845C2semidirect
47. #131×C47Q8semidirect
89. #104×C47Q8semidirect
190. #26083C47direct (N × H)
211. #3164C2×C47semidirect
214. #2089C2×C47semidirect
215. #2060C2×C47semidirect
216. #1952C2×C47semidirect
217. #1945C2×C47semidirect
219. #1845C2×C47semidirect
261. C47#26083direct (N × H)
264. #131Q8×C47semidirect
306. #104Q8×C47semidirect

Maximal subgroups

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

subgroupindex
C47×#31642
C47×#30832
C47×#29132
C47×#20892
C47×#20602
C47×#19522
C47×#19452
C47×#18592
C47×#18452
C47×#18162
C47×#17642
#2608347