Finite Group Databasebeta

← C47×#11805 #26051503

Subgroups — C47×#11805 #26051503

Order 12032.

448 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 98 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×#2618C211yesmaximal, characteristic
2C47×#1821C211yesmaximal, characteristic, complemented
3C47×#1463C211yesmaximal, characteristic, complemented
4C47×#1346C212yesmaximal, complemented
5C47×#1345C212yesmaximal, complemented
6C2×#123×C47—21no—
7C2×#121×C47C411yescharacteristic
8C22×C4⋊C4×C47C2211yescharacteristic
9C47×#407—21no—
10C47×#407—21no—
11C47×#398C411yescharacteristic
12C47×#398C2211yescharacteristic
13C47×#396—21no—
14C47×#396—21no—
15C47×#394C411yescharacteristic
16C47×#394C2211yescharacteristic
17C47×#386—21no—
18C47×#380C411yescharacteristic
19C47×#343—21no—
20C47×#343C2212yes—
21C47×#324—21no—
22C47×#324C2212yes—
23C22×D4×C47—21no—
24C22×D4×C47C2×C411yescharacteristic
25C23×C4×C47C2×C411yescharacteristic
26C23×C4×C47D411yescharacteristic
27C23×C4×C47D411yescharacteristic
28C2×M16×C47—22no—
29C2×M16×C47—41no—
30#123×C47—44no—
31#121×C47—41no—
32#121×C47—41no—
33C2×C4⋊C4×C47—21no—
34C2×C4⋊C4×C47—21no—
35C2×C4⋊C4×C47—21no—
36C2×C4⋊C4×C47—41no—
37C2×C4⋊C4×C47—41no—
38C2×C4⋊C4×C47C2×C411yescharacteristic
39C2×C4⋊C4×C47C2×C411yescharacteristic
40C2×C4⋊C4×C47D411yescharacteristic
41C2×C4⋊C4×C47D411yescharacteristic
42C2×C4⋊C4×C47C2311yescharacteristic
43C2×C22⋊C4×C47—21no—
44C2×C22⋊C4×C47—21no—
45C2×C22⋊C4×C47—21no—
46C2×C22⋊C4×C47—21no—
47C2×C22⋊C4×C47—21no—
48C2×C22⋊C4×C47—21no—
49C22⋊C8×C47—22no—
50C22⋊C8×C47—41no—
51C22⋊C8×C47—41no—
52C22⋊C8×C47—41no—
53#95×C47—21no—
54#95×C47—21no—
55#95×C47C2×C411yescharacteristic
56#95×C47C2×C411yescharacteristic
57C24×C47—21no—
58C24×C47C22⋊C411yescharacteristic
59C2×D4×C47—41no—
60C2×D4×C47—42no—
61C2×D4×C47—42no—
62C2×D4×C47—41no—
63C2×D4×C47—41no—
64C2×D4×C47—81no—
65C22×C4×C47—21no—
66C22×C4×C47—21no—
67C22×C4×C47—21no—
68C22×C4×C47—21no—
69C22×C4×C47—21no—
70C22×C4×C47—21no—
71C22×C4×C47—22no—
72C22×C4×C47—21no—
73C22×C4×C47—21no—
74C22×C4×C47—21no—
75C22×C4×C47—41no—
76C22×C4×C47—81no—
77C22×C4×C47—81no—
78C22×C4×C47C22⋊C411yescharacteristic
79C22×C4×C47C22⋊C411yescharacteristic
80C22×C4×C47C22⋊C411yescharacteristic
81C22×C4×C47C22×C411yescharacteristic
82C22×C4×C47C2×D411yescharacteristic
83C22×C4×C47C2×D411yescharacteristic
84M16×C47—42no—
85M16×C47—42no—
86M16×C47—82no—
87C2×C8×C47—42no—
88C2×C8×C47—42no—
89C4⋊C4×C47—41no—
90C4⋊C4×C47—41no—
91C4⋊C4×C47—41no—
92C4⋊C4×C47—41no—
93C4⋊C4×C47—42no—
94C4⋊C4×C47—41no—
95C4⋊C4×C47—42no—
96C4⋊C4×C47—81no—
97C22⋊C4×C47—42no—
98C22⋊C4×C47—42no—
99C22⋊C4×C47—42no—
100C22⋊C4×C47—42no—
101C22⋊C4×C47—81no—
102C22⋊C4×C47—81no—
103C23×C47—21no—
104C23×C47—21no—
105C23×C47—41no—
106C23×C47—41no—
107C23×C47—41no—
108C23×C47—41no—
109C23×C47—41no—
110C23×C47—41no—
111C23×C47—41no—
112C23×C47—41no—
113C23×C47—41no—
114C23×C47#9911yescharacteristic
115C23×C47#9911yescharacteristic
116C23×C47C2×C22⋊C411yescharacteristic
117D4×C47—82no—
118D4×C47—82no—
119D4×C47—82no—
120C2×C4×C47—21no—
121C2×C4×C47—21no—
122C2×C4×C47—21no—
123C2×C4×C47—21no—
124C2×C4×C47—21no—
125C2×C4×C47—21no—
126C2×C4×C47—21no—
127C2×C4×C47—21no—
128C2×C4×C47—21no—
129C2×C4×C47—42no—
130C2×C4×C47—42no—
131C2×C4×C47—42no—
132C2×C4×C47—41no—
133C2×C4×C47—41no—
134C2×C4×C47—41no—
135C2×C4×C47—41no—
136C2×C4×C47—41no—
137C2×C4×C47—41no—
138C2×C4×C47—41no—
139C2×C4×C47—41no—
140C2×C4×C47—41no—
141C2×C4×C47—41no—
142C2×C4×C47—41no—
143C2×C4×C47—41no—
144C2×C4×C47—41no—
145C2×C4×C47—81no—
146C2×C4×C47—81no—
147C2×C4×C47—81no—
148C2×C4×C47AGL(1,ℤ/8)12yes—
149C8×C47—82no—
150C8×C47—82no—
151#11805C4711yesmaximal, characteristic, direct product factor, Sylow
152C22×C47—21no—
153C22×C47—21no—
154C22×C47—21no—
155C22×C47—21no—
156C22×C47—21no—
157C22×C47—21no—
158C22×C47—21no—
159C22×C47—21no—
160C22×C47—42no—
161C22×C47—42no—
162C22×C47—81no—
163C22×C47—81no—
164C22×C47—81no—
165C22×C47—81no—
166C22×C47—81no—
167C22×C47—81no—
168C22×C47—81no—
169C22×C47C2×#9911yescharacteristic
170C22×C47#41811yescharacteristic
171C22×C47#42111yescharacteristic
172C4×C47—22no—
173C4×C47—41no—
174C4×C47—41no—
175C4×C47—41no—
176C4×C47—41no—
177C4×C47—41no—
178C4×C47—81no—
179C4×C47—81no—
180C4×C47—81no—
181#2618C2×C4711yescharacteristic
182#1821C2×C4711yescharacteristic, complemented
183#1463C2×C4711yescharacteristic, complemented
184#1346C2×C4712yescomplemented
185#1345C2×C4712yescomplemented
186C2×C47—21no—
187C2×C47—21no—
188C2×C47—41no—
189C2×C47—41no—
190C2×C47—161no—
191C2×C47#151011yescharacteristic
192C2×C47#212111yescharacteristic
193C2×C47#213311yescharacteristic
194C2×#123—21no—
195C2×#121C4×C4711yescharacteristic
196C22×C4⋊C4C22×C4711yescharacteristic
197#407—21no—
198#407—21no—
199#398C4×C4711yescharacteristic
200#398C22×C4711yescharacteristic
201#396—21no—
202#396—21no—
203#394C4×C4711yescharacteristic
204#394C22×C4711yescharacteristic
205#386—21no—
206#380C4×C4711yescharacteristic
207#343—21no—
208#343C22×C4712yes—
209#324—21no—
210#324C22×C4712yes—
211C47#1180511yescharacteristic, direct product factor, Sylow
212C22×D4—21no—
213C22×D4C2×C4×C4711yescharacteristic
214C23×C4C2×C4×C4711yescharacteristic
215C23×C4D4×C4711yescharacteristic
216C23×C4D4×C4711yescharacteristic
217C2×M16—22no—
218C2×M16—41no—
219#123—44no—
220#121—41no—
221#121—41no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4C2×C4×C4711yescharacteristic
228C2×C4⋊C4C2×C4×C4711yescharacteristic
229C2×C4⋊C4D4×C4711yescharacteristic
230C2×C4⋊C4D4×C4711yescharacteristic
231C2×C4⋊C4C23×C4711yescharacteristic
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C22⋊C8—22no—
239C22⋊C8—41no—
240C22⋊C8—41no—
241C22⋊C8—41no—
242#95—21no—
243#95—21no—
244#95C2×C4×C4711yescharacteristic
245#95C2×C4×C4711yescharacteristic
246C24—21no—
247C24C22⋊C4×C4711yescharacteristic
248C2×D4—41no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—81no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—41no—
265C22×C4—81no—
266C22×C4—81no—
267C22×C4C22⋊C4×C4711yescharacteristic
268C22×C4C22⋊C4×C4711yescharacteristic
269C22×C4C22⋊C4×C4711yescharacteristic
270C22×C4C22×C4×C4711yescharacteristic
271C22×C4C2×D4×C4711yescharacteristic
272C22×C4C2×D4×C4711yescharacteristic
273M16—42no—
274M16—42no—
275M16—82no—
276C2×C8—42no—
277C2×C8—42no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—81no—
286C22⋊C4—42no—
287C22⋊C4—42no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—81no—
291C22⋊C4—81no—
292C23—21no—
293C23—21no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23#99×C4711yescharacteristic
304C23#99×C4711yescharacteristic
305C23C2×C22⋊C4×C4711yescharacteristic
306D4—82no—
307D4—82no—
308D4—82no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4AGL(1,ℤ/8)×C4712yes—
338C8—82no—
339C8—82no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—42no—
349C22—42no—
350C22—81no—
351C22—81no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22—81no—
357C22C2×#99×C4711yescharacteristic
358C22C47×#41811yescharacteristic
359C22C47×#42111yescharacteristic
360C4—22no—
361C4—41no—
362C4—41no—
363C4—41no—
364C4—41no—
365C4—41no—
366C4—81no—
367C4—81no—
368C4—81no—
369C2—21no—
370C2—21no—
371C2—41no—
372C2—41no—
373C2—161no—
374C2C47×#151011yescharacteristic
375C2C47×#212111yescharacteristic
376C2C47×#213311yescharacteristic
subgroupG/H
center, socleC22×C47C2×#99
commutatorC22×C4C22×C4×C47
Frattini, cosocleC2×C4⋊C4C23×C47
prime radicalC2×#121×C47C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C47 ◁ C24×C47 ◁ C22×C4⋊C4×C47 ◁ G

Complements

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

NH ≅ G/Ntype
2. C47×#1821C2semidirect
3. C47×#1463C2semidirect
4. C47×#1346C2semidirect
5. C47×#1345C2semidirect
151. #11805C47direct (N × H)
182. #1821C2×C47semidirect
183. #1463C2×C47semidirect
184. #1346C2×C47semidirect
185. #1345C2×C47semidirect
211. C47#11805direct (N × H)

Maximal subgroups

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

subgroupindex
C47×#26182
C47×#18212
C47×#14632
C47×#13462
C47×#13452
#1180547