Finite Group Databasebeta

← C47×#9294 #26048992

Subgroups — C47×#9294 #26048992

Order 12032.

490 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C47×#3208C211yesmaximal, characteristic
2C47×#2961C211yesmaximal, characteristic, complemented
3C47×#1961C211yesmaximal, characteristic
4C47×#1545C211yesmaximal, characteristic
5C47×#1518C211yesmaximal, characteristic, complemented
6C47×#1286C211yesmaximal, characteristic, complemented
7C47×#1275C211yesmaximal, characteristic, complemented
8C47×#533C2211yescharacteristic
9C2×#127×C47C2211yescharacteristic
10C47×#492C412yescomplemented
11C47×#486C412yescomplemented
12C47×#460—22no—
13C47×#450—22no—
14C47×#446—22no—
15C47×#446—22no—
16C47×#431C2211yescharacteristic
17C47×#431C2211yescharacteristic
18C47×#424—21no—
19C47×#423C2211yescharacteristic
20C2×#102×C47—21no—
21C2×#102×C47—21no—
22C47×#405—21no—
23C2×#97×C47C2211yescharacteristic
24C47×#382C2211yescharacteristic
25C47×#331—22no—
26C47×#328—22no—
27C22×D4×C47—21no—
28C22×D4×C47—21no—
29C22×D4×C47—21no—
30C2×SD16×C47—24no—
31C2×M16×C47—21no—
32C22×C8×C47D411yescharacteristic
33C22×C8×C47D411yescharacteristic
34C4⋊Q8×C47C2×C411yescharacteristic
35C4⋊Q8×C47C2×C411yescharacteristic
36#127×C47—21no—
37#127×C47—21no—
38#127×C47—21no—
39#127×C47C2×C412yes—
40#125×C47—21no—
41C22⋊Q8×C47—21no—
42C22⋊Q8×C47D412yes—
43C4×Q8×C47—21no—
44C4×Q8×C47D412yes—
45#117×C47—21no—
46C2×C4⋊C4×C47—21no—
47C2×C42×C47C2311yescharacteristic
48C4⋊C8×C47—24no—
49C4⋊C8×C47—21no—
50C4⋊C8×C47—21no—
51C4⋊C8×C47—21no—
52C4⋊C8×C47D412yes—
53#102×C47—24no—
54#102×C47—42no—
55#102×C47—42no—
56C22⋊C8×C47—21no—
57C22⋊C8×C47—21no—
58#97×C47—22no—
59#97×C47—21no—
60#97×C47—21no—
61C4×C8×C47—22no—
62C4×C8×C47—21no—
63C4×C8×C47C2×C412yes—
64#95×C47—21no—
65#95×C47—21no—
66C24×C47—41no—
67C2×Q8×C47—21no—
68C2×Q8×C47C22⋊C412yes—
69C2×D4×C47—22no—
70C2×D4×C47—22no—
71C2×D4×C47—22no—
72C2×D4×C47—41no—
73C2×D4×C47—41no—
74C2×D4×C47—41no—
75C2×D4×C47—41no—
76C2×D4×C47—41no—
77C2×D4×C47—41no—
78C2×D4×C47—41no—
79C2×D4×C47—82no—
80C2×D4×C47—82no—
81C22×C4×C47—41no—
82C22×C4×C47C2×D411yescharacteristic
83C22×C4×C47C2×D411yescharacteristic
84C22×C4×C47#4111yescharacteristic
85SD16×C47—84no—
86M16×C47—41no—
87M16×C47—41no—
88C2×C8×C47—21no—
89C2×C8×C47—21no—
90C2×C8×C47—22no—
91C2×C8×C47—21no—
92C2×C8×C47—22no—
93C2×C8×C47—21no—
94C2×C8×C47—21no—
95C2×C8×C47—41no—
96C2×C8×C47SD1612yescomplemented
97C4⋊C4×C47—21no—
98C4⋊C4×C47—21no—
99C4⋊C4×C47—21no—
100C4⋊C4×C47—22no—
101C4⋊C4×C47—22no—
102C4⋊C4×C47—41no—
103C4⋊C4×C47—41no—
104C4⋊C4×C47C22⋊C412yes—
105C22⋊C4×C47—22no—
106C42×C47—22no—
107C42×C47C22×C411yescharacteristic
108C42×C47C2×D411yescharacteristic
109C42×C47C2×D411yescharacteristic
110C42×C47#4111yescharacteristic
111C23×C47—42no—
112C23×C47—82no—
113C23×C47—82no—
114C23×C47—81no—
115C23×C47—81no—
116C23×C47#12111yescharacteristic
117Q8×C47—42no—
118Q8×C47—81no—
119D4×C47—44no—
120D4×C47—42no—
121D4×C47—42no—
122D4×C47—82no—
123D4×C47—82no—
124D4×C47—82no—
125D4×C47—81no—
126D4×C47—81no—
127C2×C4×C47—22no—
128C2×C4×C47—22no—
129C2×C4×C47—41no—
130C2×C4×C47—41no—
131C2×C4×C47—81no—
132C2×C4×C47—81no—
133C2×C4×C47—81no—
134C2×C4×C47C2×C22⋊C411yescharacteristic
135C2×C4×C47C4×D411yescharacteristic
136C2×C4×C47C4×D411yescharacteristic
137C2×C4×C47#12011yescharacteristic
138C2×C4×C47#12111yescharacteristic
139C2×C4×C47#12311yescharacteristic
140C2×C4×C47C2×SD1611yescharacteristic
141C2×C4×C47#13511yescharacteristic
142C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
143C2×C4×C47#13711yescharacteristic
144C8×C47—22no—
145C8×C47—42no—
146C8×C47—41no—
147C8×C47—42no—
148C8×C47—81no—
149#9294C4711yesmaximal, characteristic, direct product factor, Sylow
150C22×C47—21no—
151C22×C47—21no—
152C22×C47—82no—
153C22×C47—82no—
154C22×C47—82no—
155C22×C47—162no—
156C22×C47—162no—
157C22×C47#38611yescharacteristic
158C22×C47#46011yescharacteristic
159C22×C47#46411yescharacteristic
160C4×C47—21no—
161C4×C47—21no—
162C4×C47—21no—
163C4×C47—21no—
164C4×C47—42no—
165C4×C47—42no—
166C4×C47—161no—
167C4×C47C4×SD1611yescharacteristic
168C4×C47#44011yescharacteristic
169C4×C47#44811yescharacteristic
170C4×C47#44911yescharacteristic
171#3208C2×C4711yescharacteristic
172#2961C2×C4711yescharacteristic, complemented
173#1961C2×C4711yescharacteristic
174#1545C2×C4711yescharacteristic
175#1518C2×C4711yescharacteristic, complemented
176#1286C2×C4711yescharacteristic, complemented
177#1275C2×C4711yescharacteristic, complemented
178C2×C47—21no—
179C2×C47—21no—
180C2×C47—162no—
181C2×C47#188011yescharacteristic
182C2×C47#190311yescharacteristic
183C2×C47#191011yescharacteristic
184#533C22×C4711yescharacteristic
185C2×#127C22×C4711yescharacteristic
186#492C4×C4712yescomplemented
187#486C4×C4712yescomplemented
188#460—22no—
189#450—22no—
190#446—22no—
191#446—22no—
192#431C22×C4711yescharacteristic
193#431C22×C4711yescharacteristic
194#424—21no—
195#423C22×C4711yescharacteristic
196C2×#102—21no—
197C2×#102—21no—
198#405—21no—
199C2×#97C22×C4711yescharacteristic
200#382C22×C4711yescharacteristic
201#331—22no—
202#328—22no—
203C47#929411yescharacteristic, direct product factor, Sylow
204C22×D4—21no—
205C22×D4—21no—
206C22×D4—21no—
207C2×SD16—24no—
208C2×M16—21no—
209C22×C8D4×C4711yescharacteristic
210C22×C8D4×C4711yescharacteristic
211C4⋊Q8C2×C4×C4711yescharacteristic
212C4⋊Q8C2×C4×C4711yescharacteristic
213#127—21no—
214#127—21no—
215#127—21no—
216#127C2×C4×C4712yes—
217#125—21no—
218C22⋊Q8—21no—
219C22⋊Q8D4×C4712yes—
220C4×Q8—21no—
221C4×Q8D4×C4712yes—
222#117—21no—
223C2×C4⋊C4—21no—
224C2×C42C23×C4711yescharacteristic
225C4⋊C8—24no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8D4×C4712yes—
230#102—24no—
231#102—42no—
232#102—42no—
233C22⋊C8—21no—
234C22⋊C8—21no—
235#97—22no—
236#97—21no—
237#97—21no—
238C4×C8—22no—
239C4×C8—21no—
240C4×C8C2×C4×C4712yes—
241#95—21no—
242#95—21no—
243C24—41no—
244C2×Q8—21no—
245C2×Q8C22⋊C4×C4712yes—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—82no—
257C2×D4—82no—
258C22×C4—41no—
259C22×C4C2×D4×C4711yescharacteristic
260C22×C4C2×D4×C4711yescharacteristic
261C22×C4#41×C4711yescharacteristic
262SD16—84no—
263M16—41no—
264M16—41no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—41no—
273C2×C8SD16×C4712yescomplemented
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4C22⋊C4×C4712yes—
282C22⋊C4—22no—
283C42—22no—
284C42C22×C4×C4711yescharacteristic
285C42C2×D4×C4711yescharacteristic
286C42C2×D4×C4711yescharacteristic
287C42#41×C4711yescharacteristic
288C23—42no—
289C23—82no—
290C23—82no—
291C23—81no—
292C23—81no—
293C23#121×C4711yescharacteristic
294Q8—42no—
295Q8—81no—
296D4—44no—
297D4—42no—
298D4—42no—
299D4—82no—
300D4—82no—
301D4—82no—
302D4—81no—
303D4—81no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4C2×C22⋊C4×C4711yescharacteristic
312C2×C4C4×D4×C4711yescharacteristic
313C2×C4C4×D4×C4711yescharacteristic
314C2×C4#120×C4711yescharacteristic
315C2×C4#121×C4711yescharacteristic
316C2×C4#123×C4711yescharacteristic
317C2×C4C2×SD16×C4711yescharacteristic
318C2×C4#135×C4711yescharacteristic
319C2×C4AGL(1,ℤ/8)×C4711yescharacteristic
320C2×C4#137×C4711yescharacteristic
321C8—22no—
322C8—42no—
323C8—41no—
324C8—42no—
325C8—81no—
326C22—21no—
327C22—21no—
328C22—82no—
329C22—82no—
330C22—82no—
331C22—162no—
332C22—162no—
333C22C47×#38611yescharacteristic
334C22C47×#46011yescharacteristic
335C22C47×#46411yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—42no—
341C4—42no—
342C4—161no—
343C4C4×SD16×C4711yescharacteristic
344C4C47×#44011yescharacteristic
345C4C47×#44811yescharacteristic
346C4C47×#44911yescharacteristic
347C2—21no—
348C2—21no—
349C2—162no—
350C2C47×#188011yescharacteristic
351C2C47×#190311yescharacteristic
352C2C47×#191011yescharacteristic
subgroupG/H
center, socleC22×C47#386
commutatorC42C22×C4×C47
Frattini, cosocleC2×C42C23×C47
prime radicalC2×#127×C47C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C47 ◁ C2×C42×C47 ◁ G

Complements

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

NH ≅ G/Ntype
2. C47×#2961C2semidirect
5. C47×#1518C2semidirect
6. C47×#1286C2semidirect
7. C47×#1275C2semidirect
10. C47×#492C4semidirect
11. C47×#486C4semidirect
96. C2×C8×C47SD16semidirect
149. #9294C47direct (N × H)
172. #2961C2×C47semidirect
175. #1518C2×C47semidirect
176. #1286C2×C47semidirect
177. #1275C2×C47semidirect
186. #492C4×C47semidirect
187. #486C4×C47semidirect
203. C47#9294direct (N × H)
273. C2×C8SD16×C47semidirect

Maximal subgroups

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

subgroupindex
C47×#32082
C47×#29612
C47×#19612
C47×#15452
C47×#15182
C47×#12862
C47×#12752
#929447