Finite Group Databasebeta

← C47×#10271 #26049969

Subgroups — C47×#10271 #26049969

Order 12032.

594 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C47×#498C211yesmaximal, characteristic, complemented
2C2×C47×#476C211yesmaximal, characteristic
3C2×C47×#461C211yesmaximal, characteristic
4C47×#1954C211yesmaximal, characteristic
5C2×C47×#333C211yesmaximal, characteristic, complemented
6C2×C47×#332C211yesmaximal, characteristic, complemented
7C47×#1277C211yesmaximal, characteristic, complemented
8C22×SD16×C47—21no—
9C2×C4⋊Q8×C47C2211yescharacteristic
10C2×C4⋊Q8×C47C2211yescharacteristic
11C2×C4×D4×C47C2211yescharacteristic
12C47×#498—22no—
13C47×#498—22no—
14C47×#476—22no—
15C47×#476—22no—
16C47×#461—22no—
17C47×#461—22no—
18C2×#106×C47—22no—
19C2×#106×C47—21no—
20C2×C4⋊C8×C47C2211yescharacteristic
21C2×C4⋊C8×C47C2211yescharacteristic
22C2×C4⋊C8×C47C2211yescharacteristic
23C2×#103×C47—21no—
24C2×#102×C47—21no—
25C2×C22⋊C8×C47—22no—
26C2×C4×C8×C47C2211yescharacteristic
27C47×#333—22no—
28C47×#332—22no—
29C22×Q8×C47—21no—
30C22×Q8×C47—21no—
31C22×D4×C47D411yescharacteristic
32C23×C4×C47—21no—
33C2×SD16×C47—24no—
34C2×SD16×C47—42no—
35C2×SD16×C47—42no—
36C22×C8×C47—21no—
37C22×C8×C47—21no—
38C22×C8×C47—21no—
39C22×C8×C47D412yes—
40C4⋊Q8×C47—21no—
41C4⋊Q8×C47—21no—
42C4⋊Q8×C47—21no—
43C4⋊Q8×C47—21no—
44C4⋊Q8×C47—21no—
45C4⋊Q8×C47—21no—
46C4⋊Q8×C47D412yes—
47C4⋊Q8×C47D412yes—
48C4×D4×C47—21no—
49C4×D4×C47—21no—
50C4×D4×C47—21no—
51C4×D4×C47—21no—
52C2×C4⋊C4×C47—21no—
53C2×C4⋊C4×C47—21no—
54C2×C4⋊C4×C47—21no—
55C2×C4⋊C4×C47—21no—
56C2×C4⋊C4×C47D411yescharacteristic
57C2×C22⋊C4×C47—21no—
58C2×C42×C47C2311yescharacteristic
59#106×C47—24no—
60#106×C47—28no—
61C4⋊C8×C47—21no—
62C4⋊C8×C47—21no—
63C4⋊C8×C47—21no—
64C4⋊C8×C47—21no—
65C4⋊C8×C47D412yes—
66C4⋊C8×C47D412yes—
67#103×C47—24no—
68#102×C47—24no—
69C22⋊C8×C47—44no—
70C4×C8×C47—21no—
71C4×C8×C47—21no—
72C24×C47—21no—
73C2×Q8×C47—22no—
74C2×Q8×C47—22no—
75C2×Q8×C47—41no—
76C2×Q8×C47—41no—
77C2×Q8×C47—41no—
78C2×Q8×C47—41no—
79C2×Q8×C47—41no—
80C2×Q8×C47—41no—
81C2×Q8×C47—82no—
82C2×D4×C47—21no—
83C2×D4×C47—21no—
84C2×D4×C47—21no—
85C2×D4×C47—21no—
86C2×D4×C47—21no—
87C2×D4×C47SD1612yes—
88C22×C4×C47—21no—
89C22×C4×C47—41no—
90C22×C4×C47—41no—
91C22×C4×C47—41no—
92C22×C4×C47—41no—
93C22×C4×C47—42no—
94C22×C4×C47—42no—
95C22×C4×C47C2×D411yescharacteristic
96C22×C4×C47C2×D411yescharacteristic
97C22×C4×C47#4111yescharacteristic
98SD16×C47—48no—
99C2×C8×C47—22no—
100C2×C8×C47—22no—
101C2×C8×C47—22no—
102C2×C8×C47—22no—
103C2×C8×C47—41no—
104C2×C8×C47—41no—
105C2×C8×C47—41no—
106C2×C8×C47—41no—
107C2×C8×C47—41no—
108C2×C8×C47—42no—
109C2×C8×C47SD1614yescomplemented
110C4⋊C4×C47—21no—
111C4⋊C4×C47—22no—
112C4⋊C4×C47—22no—
113C4⋊C4×C47—21no—
114C4⋊C4×C47—22no—
115C4⋊C4×C47—22no—
116C4⋊C4×C47—41no—
117C4⋊C4×C47—41no—
118C4⋊C4×C47—41no—
119C4⋊C4×C47—41no—
120C22⋊C4×C47—41no—
121C22⋊C4×C47—41no—
122C42×C47C2×D411yescharacteristic
123C42×C47C2×D411yescharacteristic
124C42×C47C2×D411yescharacteristic
125C42×C47C2×D411yescharacteristic
126C23×C47—22no—
127C23×C47—41no—
128C23×C47—42no—
129C23×C47—41no—
130C23×C47—42no—
131C23×C47#12111yescharacteristic
132Q8×C47—42no—
133Q8×C47—42no—
134Q8×C47—82no—
135Q8×C47—82no—
136Q8×C47—81no—
137Q8×C47—81no—
138D4×C47—22no—
139D4×C47—22no—
140D4×C47—41no—
141D4×C47—41no—
142C2×C4×C47—22no—
143C2×C4×C47—41no—
144C2×C4×C47—42no—
145C2×C4×C47—42no—
146C2×C4×C47—42no—
147C2×C4×C47—81no—
148C2×C4×C47—81no—
149C2×C4×C47—82no—
150C2×C4×C47—82no—
151C2×C4×C47—81no—
152C2×C4×C47—81no—
153C2×C4×C47#12011yescharacteristic
154C2×C4×C47#12011yescharacteristic
155C2×C4×C47#12011yescharacteristic
156C2×C4×C47#12111yescharacteristic
157C2×C4×C47#12111yescharacteristic
158C2×C4×C47#12711yescharacteristic
159C2×C4×C47C2×SD1611yescharacteristic
160C2×C4×C47C2×SD1612yes—
161C2×C4×C47#13511yescharacteristic
162C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
163C2×C4×C47#13711yescharacteristic
164C2×C4×C47#13712yes—
165C8×C47—24no—
166C8×C47—42no—
167C8×C47—42no—
168C8×C47—81no—
169#10271C4711yesmaximal, characteristic, direct product factor, Sylow
170C22×C47—44no—
171C22×C47—42no—
172C22×C47—82no—
173C22×C47—81no—
174C22×C47—81no—
175C22×C47#39211yescharacteristic
176C22×C47#46112yes—
177C22×C47#46511yescharacteristic
178C22×C47#46811yescharacteristic
179C22×C47#47112yes—
180C4×C47—21no—
181C4×C47—21no—
182C4×C47—82no—
183C4×C47—81no—
184C4×C47—82no—
185C4×C47#44811yescharacteristic
186C4×C47#44812yes—
187C4×C47#44911yescharacteristic
188C4×C47#45011yescharacteristic
189C4×C47#45211yescharacteristic
190C4×C47#49211yescharacteristic
191C4×C47#49711yescharacteristic
192C2×#498C2×C4711yescharacteristic, complemented
193C2×#476C2×C4711yescharacteristic
194C2×#461C2×C4711yescharacteristic
195#1954C2×C4711yescharacteristic
196C2×#333C2×C4711yescharacteristic, complemented
197C2×#332C2×C4711yescharacteristic, complemented
198#1277C2×C4711yescharacteristic, complemented
199C2×C47—82no—
200C2×C47#167112yes—
201C2×C47#169112yes—
202C2×C47#200011yescharacteristic
203C2×C47#201311yescharacteristic
204C2×C47#201911yescharacteristic
205C22×SD16—21no—
206C2×C4⋊Q8C22×C4711yescharacteristic
207C2×C4⋊Q8C22×C4711yescharacteristic
208C2×C4×D4C22×C4711yescharacteristic
209#498—22no—
210#498—22no—
211#476—22no—
212#476—22no—
213#461—22no—
214#461—22no—
215C2×#106—22no—
216C2×#106—21no—
217C2×C4⋊C8C22×C4711yescharacteristic
218C2×C4⋊C8C22×C4711yescharacteristic
219C2×C4⋊C8C22×C4711yescharacteristic
220C2×#103—21no—
221C2×#102—21no—
222C2×C22⋊C8—22no—
223C2×C4×C8C22×C4711yescharacteristic
224#333—22no—
225#332—22no—
226C47#1027111yescharacteristic, direct product factor, Sylow
227C22×Q8—21no—
228C22×Q8—21no—
229C22×D4D4×C4711yescharacteristic
230C23×C4—21no—
231C2×SD16—24no—
232C2×SD16—42no—
233C2×SD16—42no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8D4×C4712yes—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8—21no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8D4×C4712yes—
245C4⋊Q8D4×C4712yes—
246C4×D4—21no—
247C4×D4—21no—
248C4×D4—21no—
249C4×D4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4D4×C4711yescharacteristic
255C2×C22⋊C4—21no—
256C2×C42C23×C4711yescharacteristic
257#106—24no—
258#106—28no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8D4×C4712yes—
264C4⋊C8D4×C4712yes—
265#103—24no—
266#102—24no—
267C22⋊C8—44no—
268C4×C8—21no—
269C4×C8—21no—
270C24—21no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—82no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4SD16×C4712yes—
286C22×C4—21no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4C2×D4×C4711yescharacteristic
294C22×C4C2×D4×C4711yescharacteristic
295C22×C4#41×C4711yescharacteristic
296SD16—48no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—42no—
307C2×C8SD16×C4714yescomplemented
308C4⋊C4—22no—
309C4⋊C4—21no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—21no—
313C4⋊C4—22no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C42C2×D4×C4711yescharacteristic
321C42C2×D4×C4711yescharacteristic
322C42C2×D4×C4711yescharacteristic
323C42C2×D4×C4711yescharacteristic
324C23—22no—
325C23—42no—
326C23—41no—
327C23—42no—
328C23—41no—
329C23#121×C4711yescharacteristic
330Q8—42no—
331Q8—42no—
332Q8—82no—
333Q8—82no—
334Q8—81no—
335Q8—81no—
336D4—22no—
337D4—22no—
338D4—41no—
339D4—41no—
340C2×C4—22no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—82no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—82no—
351C2×C4#120×C4711yescharacteristic
352C2×C4#120×C4711yescharacteristic
353C2×C4#120×C4711yescharacteristic
354C2×C4#121×C4711yescharacteristic
355C2×C4#121×C4711yescharacteristic
356C2×C4#127×C4711yescharacteristic
357C2×C4C2×SD16×C4711yescharacteristic
358C2×C4C2×SD16×C4712yes—
359C2×C4#135×C4711yescharacteristic
360C2×C4AGL(1,ℤ/8)×C4711yescharacteristic
361C2×C4#137×C4711yescharacteristic
362C2×C4#137×C4712yes—
363C8—24no—
364C8—42no—
365C8—42no—
366C8—81no—
367C22—44no—
368C22—42no—
369C22—81no—
370C22—81no—
371C22—82no—
372C22C47×#39211yescharacteristic
373C22C47×#46112yes—
374C22C47×#46511yescharacteristic
375C22C47×#46811yescharacteristic
376C22C47×#47112yes—
377C4—21no—
378C4—21no—
379C4—81no—
380C4—82no—
381C4—82no—
382C4C47×#44812yes—
383C4C47×#44811yescharacteristic
384C4C47×#44911yescharacteristic
385C4C47×#45011yescharacteristic
386C4C47×#45211yescharacteristic
387C4C47×#49211yescharacteristic
388C4C47×#49711yescharacteristic
389C2—82no—
390C2C47×#167112yes—
391C2C47×#169112yes—
392C2C47×#200011yescharacteristic
393C2C47×#201311yescharacteristic
394C2C47×#201911yescharacteristic
subgroupG/H
center, socleC23×C47#121
commutator, Frattini, cosocleC2×C42C23×C47
prime radicalC22×D4×C47D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C47 ◁ C2×C42×C47 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C47×#498C2semidirect
5. C2×C47×#333C2semidirect
6. C2×C47×#332C2semidirect
7. C47×#1277C2semidirect
109. C2×C8×C47SD16semidirect
169. #10271C47direct (N × H)
192. C2×#498C2×C47semidirect
196. C2×#333C2×C47semidirect
197. C2×#332C2×C47semidirect
198. #1277C2×C47semidirect
226. C47#10271direct (N × H)
307. C2×C8SD16×C47semidirect

Maximal subgroups

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

subgroupindex
C2×C47×#4982
C2×C47×#4762
C2×C47×#4612
C47×#19542
C2×C47×#3332
C2×C47×#3322
C47×#12772
#1027147