Finite Group Databasebeta

← C47×#8776 #26048474

Subgroups — C47×#8776 #26048474

Order 12032.

480 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C47×#2353C211yesmaximal, characteristic
2C4×#99×C47C211yesmaximal, characteristic
3C47×#1465C211yesmaximal, characteristic
4C47×#1297C212yesmaximal, complemented
5C47×#1284C212yesmaximal, complemented
6C2×C4×D4×C47C2211yescharacteristic
7C2×#99×C47—21no—
8C47×#407C2212yes—
9C47×#388—21no—
10C47×#387—21no—
11C47×#386—21no—
12C47×#383C411yescharacteristic, complemented
13C47×#381C2211yescharacteristic
14C47×#380C411yescharacteristic, complemented
15C4×C22⋊C4×C47—21no—
16C4×C22⋊C4×C47C2212yes—
17C4×C22⋊C4×C47C2211yescharacteristic
18C47×#376C411yescharacteristic, complemented
19C2×#95×C47C411yescharacteristic, complemented
20C47×#342—21no—
21C47×#324—21no—
22C47×#323—21no—
23C22×D4×C47D411yescharacteristic
24C23×C4×C47—21no—
25C23×C4×C47C2×C411yescharacteristic
26C23×C4×C47C2311yescharacteristic
27C2×M16×C47—22no—
28C4×D4×C47—22no—
29C4×D4×C47—22no—
30C2×C4⋊C4×C47—21no—
31C2×C4⋊C4×C47D411yescharacteristic
32C2×C22⋊C4×C47—22no—
33C2×C22⋊C4×C47—21no—
34C2×C22⋊C4×C47—21no—
35C2×C22⋊C4×C47—21no—
36C2×C22⋊C4×C47D411yescharacteristic
37C2×C22⋊C4×C47D411yescharacteristic
38C2×C42×C47—21no—
39C2×C42×C47—21no—
40C2×C42×C47—22no—
41C2×C42×C47C2×C411yescharacteristic
42#99×C47—24no—
43C22⋊C8×C47—22no—
44C22⋊C8×C47—22no—
45#95×C47—21no—
46#95×C47—21no—
47#95×C47—21no—
48#95×C47—21no—
49#95×C47—21no—
50#95×C47—22no—
51#95×C47C2×C411yescharacteristic
52#95×C47C2×C411yescharacteristic
53#95×C47C2×C411yescharacteristic
54#95×C47C2×C411yescharacteristic
55C24×C47C22⋊C411yescharacteristic
56C24×C47C2×D411yescharacteristic
57C2×D4×C47—22no—
58C2×D4×C47—22no—
59C2×D4×C47—41no—
60C22×C4×C47—21no—
61C22×C4×C47—21no—
62C22×C4×C47—21no—
63C22×C4×C47—21no—
64C22×C4×C47—21no—
65C22×C4×C47—21no—
66C22×C4×C47—42no—
67C22×C4×C47—42no—
68C22×C4×C47—42no—
69C22×C4×C47—81no—
70C22×C4×C47—81no—
71C22×C4×C47—81no—
72C22×C4×C47C22⋊C411yescharacteristic
73C22×C4×C47C22⋊C411yescharacteristic
74C22×C4×C47C22⋊C411yescharacteristic
75C22×C4×C47C22×C411yescharacteristic
76C22×C4×C47C2×D411yescharacteristic
77C22×C4×C47#4112yes—
78C22×C4×C47#4112yes—
79M16×C47—82no—
80C2×C8×C47—42no—
81C4⋊C4×C47—22no—
82C4⋊C4×C47—81no—
83C22⋊C4×C47—22no—
84C22⋊C4×C47—22no—
85C22⋊C4×C47—24no—
86C22⋊C4×C47—42no—
87C22⋊C4×C47—42no—
88C22⋊C4×C47—42no—
89C22⋊C4×C47—81no—
90C42×C47—21no—
91C42×C47—21no—
92C42×C47—42no—
93C42×C47—81no—
94C42×C47—82no—
95C23×C47—21no—
96C23×C47—21no—
97C23×C47—21no—
98C23×C47—21no—
99C23×C47—41no—
100C23×C47—41no—
101C23×C47—41no—
102C23×C47—41no—
103C23×C47#10412yes—
104C23×C47C2×C22⋊C411yescharacteristic
105C23×C47#12312yes—
106D4×C47—44no—
107C2×C4×C47—22no—
108C2×C4×C47—21no—
109C2×C4×C47—21no—
110C2×C4×C47—21no—
111C2×C4×C47—21no—
112C2×C4×C47—21no—
113C2×C4×C47—41no—
114C2×C4×C47—42no—
115C2×C4×C47—42no—
116C2×C4×C47—41no—
117C2×C4×C47—41no—
118C2×C4×C47—41no—
119C2×C4×C47—41no—
120C2×C4×C47—41no—
121C2×C4×C47—41no—
122C2×C4×C47—41no—
123C2×C4×C47—44no—
124C2×C4×C47—41no—
125C2×C4×C47—41no—
126C2×C4×C47—41no—
127C2×C4×C47—41no—
128C2×C4×C47—41no—
129C2×C4×C47—41no—
130C2×C4×C47—41no—
131C2×C4×C47—81no—
132C2×C4×C47—81no—
133C2×C4×C47—84no—
134C2×C4×C47—82no—
135C2×C4×C47—82no—
136C2×C4×C47#11711yescharacteristic
137C2×C4×C47#11711yescharacteristic
138C2×C4×C47#12312yes—
139C8×C47—82no—
140#8776C4711yesmaximal, characteristic, direct product factor, Sylow
141C22×C47—21no—
142C22×C47—21no—
143C22×C47—22no—
144C22×C47—22no—
145C22×C47—22no—
146C22×C47—22no—
147C22×C47—42no—
148C22×C47—42no—
149C22×C47—41no—
150C22×C47—41no—
151C22×C47—81no—
152C22×C47—81no—
153C22×C47#38111yescharacteristic
154C22×C47C2×#10411yescharacteristic
155C22×C47#42111yescharacteristic
156C4×C47—22no—
157C4×C47—21no—
158C4×C47—21no—
159C4×C47—42no—
160C4×C47—41no—
161C4×C47—41no—
162C4×C47—41no—
163C4×C47—81no—
164C4×C47—81no—
165C4×C47—81no—
166C4×C47—84no—
167#2353C2×C4711yescharacteristic
168C4×#99C2×C4711yescharacteristic
169#1465C2×C4711yescharacteristic
170#1297C2×C4712yescomplemented
171#1284C2×C4712yescomplemented
172C2×C47—21no—
173C2×C47—21no—
174C2×C47—42no—
175C2×C47—42no—
176C2×C47#182611yescharacteristic
177C2×C47#182711yescharacteristic
178C2×C47#182811yescharacteristic
179C2×C4×D4C22×C4711yescharacteristic
180C2×#99—21no—
181#407C22×C4712yes—
182#388—21no—
183#387—21no—
184#386—21no—
185#383C4×C4711yescharacteristic, complemented
186#381C22×C4711yescharacteristic
187#380C4×C4711yescharacteristic, complemented
188C4×C22⋊C4—21no—
189C4×C22⋊C4C22×C4712yes—
190C4×C22⋊C4C22×C4711yescharacteristic
191#376C4×C4711yescharacteristic, complemented
192C2×#95C4×C4711yescharacteristic, complemented
193#342—21no—
194#324—21no—
195#323—21no—
196C47#877611yescharacteristic, direct product factor, Sylow
197C22×D4D4×C4711yescharacteristic
198C23×C4—21no—
199C23×C4C2×C4×C4711yescharacteristic
200C23×C4C23×C4711yescharacteristic
201C2×M16—22no—
202C4×D4—22no—
203C4×D4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4D4×C4711yescharacteristic
206C2×C22⋊C4—22no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4D4×C4711yescharacteristic
211C2×C22⋊C4D4×C4711yescharacteristic
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—22no—
215C2×C42C2×C4×C4711yescharacteristic
216#99—24no—
217C22⋊C8—22no—
218C22⋊C8—22no—
219#95—21no—
220#95—21no—
221#95—21no—
222#95—21no—
223#95—21no—
224#95—22no—
225#95C2×C4×C4711yescharacteristic
226#95C2×C4×C4711yescharacteristic
227#95C2×C4×C4711yescharacteristic
228#95C2×C4×C4711yescharacteristic
229C24C22⋊C4×C4711yescharacteristic
230C24C2×D4×C4711yescharacteristic
231C2×D4—22no—
232C2×D4—22no—
233C2×D4—41no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—81no—
244C22×C4—81no—
245C22×C4—81no—
246C22×C4C22⋊C4×C4711yescharacteristic
247C22×C4C22⋊C4×C4711yescharacteristic
248C22×C4C22⋊C4×C4711yescharacteristic
249C22×C4C22×C4×C4711yescharacteristic
250C22×C4C2×D4×C4711yescharacteristic
251C22×C4#41×C4712yes—
252C22×C4#41×C4712yes—
253M16—82no—
254C2×C8—42no—
255C4⋊C4—22no—
256C4⋊C4—81no—
257C22⋊C4—22no—
258C22⋊C4—22no—
259C22⋊C4—24no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—42no—
263C22⋊C4—81no—
264C42—21no—
265C42—21no—
266C42—42no—
267C42—81no—
268C42—82no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23#104×C4712yes—
278C23C2×C22⋊C4×C4711yescharacteristic
279C23#123×C4712yes—
280D4—44no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—44no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—84no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4#117×C4711yescharacteristic
311C2×C4#117×C4711yescharacteristic
312C2×C4#123×C4712yes—
313C8—82no—
314C22—21no—
315C22—21no—
316C22—22no—
317C22—22no—
318C22—22no—
319C22—22no—
320C22—42no—
321C22—42no—
322C22—41no—
323C22—41no—
324C22—81no—
325C22—81no—
326C22C47×#38111yescharacteristic
327C22C2×#104×C4711yescharacteristic
328C22C47×#42111yescharacteristic
329C4—22no—
330C4—21no—
331C4—21no—
332C4—42no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—81no—
337C4—81no—
338C4—81no—
339C4—84no—
340C2—21no—
341C2—21no—
342C2—42no—
343C2—42no—
344C2C47×#182611yescharacteristic
345C2C47×#182711yescharacteristic
346C2C47×#182811yescharacteristic
subgroupG/H
center, socleC22×C47#381
commutatorC22×C4C22×C4×C47
Frattini, cosocleC23×C4C23×C47
prime radicalC22×D4×C47D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C47 ◁ C23×C4×C47 ◁ G

Complements

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

NH ≅ G/Ntype
4. C47×#1297C2semidirect
5. C47×#1284C2semidirect
12. C47×#383C4semidirect
14. C47×#380C4semidirect
18. C47×#376C4semidirect
19. C2×#95×C47C4semidirect
140. #8776C47direct (N × H)
170. #1297C2×C47semidirect
171. #1284C2×C47semidirect
185. #383C4×C47semidirect
187. #380C4×C47semidirect
191. #376C4×C47semidirect
192. C2×#95C4×C47semidirect
196. C47#8776direct (N × H)

Maximal subgroups

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

subgroupindex
C47×#23532
C4×#99×C472
C47×#14652
C47×#12972
C47×#12842
#877647