Finite Group Databasebeta

← #26112874

Subgroups — #26112874

Order 12032.

382 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 103 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
1#20252483C211yesmaximal, characteristic
2#20241900C211yesmaximal, characteristic
3#20241900C211yesmaximal, characteristic
4#20241861C211yesmaximal, characteristic
5#20241250C211yesmaximal, characteristic
6#20241216C211yesmaximal, characteristic
7C47×#3231C211yesmaximal, characteristic
8#15689254—21no—
9#15689254—22no—
10Q8×C47⋊C8—21no—
11#15689184—22no—
12#15689180C2211yescharacteristic
13#15689180C2211yescharacteristic
14#15689176C2211yescharacteristic
15#15688969C2211yescharacteristic
16#15688860—22no—
17#15688860—22no—
18Q8⋊(C47⋊C8)—21no—
19Q8⋊(C47⋊C8)—21no—
20C47×#533C2211yescharacteristic
21C47×#533C2211yescharacteristic
22C47×#500C411yescharacteristic, complemented
23C47×#498C411yescharacteristic, complemented
24C47×#487C411yescharacteristic, complemented
25C47×#487C411yescharacteristic, complemented
26C47×#485—22no—
27C47×#479—22no—
28C47×#431C2211yescharacteristic
29C2×C4×Dic47—21no—
30C4⋊(C47⋊C8)—22no—
31C4⋊(C47⋊C8)—22no—
32C4⋊(C47⋊C8)—21no—
33C4⋊(C47⋊C8)—21no—
34C4⋊(C47⋊C8)—21no—
35C22⋊(C47⋊C8)—21no—
36C22⋊(C47⋊C8)—21no—
37C22⋊(C47⋊C8)D412yes—
38#13093935—22no—
39#13093935—21no—
40#13093935—21no—
41C4×C47⋊C8—22no—
42C4×C47⋊C8D411yescharacteristic
43C4×C47⋊C8D411yescharacteristic
44C47⋊#95—21no—
45C47⋊#95—21no—
46C4⋊Q8×C47C2×C411yescharacteristic
47C4⋊Q8×C47C2×C411yescharacteristic
48C4⋊Q8×C47C2×C411yescharacteristic
49C4⋊Q8×C47C2×C411yescharacteristic
50#125×C47—21no—
51#125×C47—21no—
52C22⋊Q8×C47—21no—
53C22⋊Q8×C47—21no—
54C22⋊Q8×C47—21no—
55C22⋊Q8×C47D412yes—
56C4×Q8×C47—21no—
57C4×Q8×C47—21no—
58C4×Q8×C47—21no—
59C4×Q8×C47D412yes—
60#117×C47—21no—
61#117×C47—21no—
62C2×C42×C47C2311yescharacteristic
63#107×C47—22no—
64#106×C47—22no—
65C4⋊C8×C47—21no—
66#103×C47—22no—
67#103×C47—22no—
68C22⋊C8×C47—21no—
69C4×C8×C47C2×C411yescharacteristic
70C4×C8×C47C2×C411yescharacteristic
71C2×Q8×C47—21no—
72C2×Q8×C47—21no—
73C2×Q8×C47—21no—
74C2×Q8×C47C22⋊C412yes—
75C22×C4×C47C2×D411yescharacteristic
76C22×C4×C47C2×D411yescharacteristic
77C22×C4×C47#4111yescharacteristic
78C22×Dic47—41no—
79C2×C8×C47—21no—
80C2×C8×C47—21no—
81C4⋊C4×C47—21no—
82C4⋊C4×C47—21no—
83C4⋊C4×C47—21no—
84C4⋊C4×C47—22no—
85C4⋊C4×C47—22no—
86C4⋊C4×C47—21no—
87C4⋊C4×C47—21no—
88C4⋊C4×C47—21no—
89C4⋊C4×C47—21no—
90C4⋊C4×C47—41no—
91C4⋊C4×C47—41no—
92C4⋊C4×C47C22⋊C412yes—
93C22⋊C4×C47—22no—
94C22⋊C4×C47—41no—
95C42×C47—22no—
96C42×C47—41no—
97C42×C47C22×C411yescharacteristic
98C42×C47C2×D411yescharacteristic
99C42×C47C2×D411yescharacteristic
100C42×C47#4111yescharacteristic
101C4×Dic47—22no—
102C4×Dic47—41no—
103C2×C47⋊C8—21no—
104C2×C47⋊C8—22no—
105C2×C47⋊C8—21no—
106C2×C47⋊C8—41no—
107C23×C47#12111yescharacteristic
108Q8×C47—42no—
109Q8×C47—41no—
110Q8×C47—41no—
111Q8×C47—41no—
112Q8×C47—41no—
113Q8×C47—81no—
114C2×C4×C47—21no—
115C2×C4×C47—21no—
116C2×C4×C47—22no—
117C2×C4×C47—22no—
118C2×C4×C47—41no—
119C2×C4×C47—41no—
120C2×C4×C47—41no—
121C2×C4×C47—41no—
122C2×C4×C47C2×C22⋊C411yescharacteristic
123C2×C4×C47C4×D411yescharacteristic
124C2×C4×C47C4×D411yescharacteristic
125C2×C4×C47#12011yescharacteristic
126C2×C4×C47#12111yescharacteristic
127C2×C4×C47#12311yescharacteristic
128C2×Dic47—42no—
129C2×Dic47—81no—
130C2×Dic47—81no—
131C8×C47—41no—
132C8×C47—41no—
133C47⋊C8—41no—
134C47⋊C8—42no—
135C47⋊C8—41no—
136C47⋊C8—81no—
137#9280—471nomaximal, Sylow
138C22×C47—21no—
139C22×C47—21no—
140C22×C47—21no—
141C22×C47#38611yescharacteristic
142C4×C47—21no—
143C4×C47—21no—
144C4×C47—21no—
145C4×C47—21no—
146C4×C47—21no—
147C4×C47—42no—
148C4×C47—42no—
149C4×C47—81no—
150C4×C47—81no—
151C4×C47#44311yescharacteristic
152C4×C47#44411yescharacteristic
153Dic47—82no—
154#3231D4711yescharacteristic
155#2985—471no—
156#1506—471no—
157#1506—471no—
158#1497—471no—
159#1287—471no—
160#1274—471no—
161C2×C47—41no—
162C2×C47#187611yescharacteristic
163C2×C47#190511yescharacteristic
164C2×C47#190911yescharacteristic
165#533C2×D4711yescharacteristic
166#533C2×D4711yescharacteristic
167#500Dic4711yescharacteristic, complemented
168#498Dic4711yescharacteristic, complemented
169#487Dic4711yescharacteristic, complemented
170#487Dic4711yescharacteristic, complemented
171#485—22no—
172#479—22no—
173#446—941no—
174#446—942no—
175C8×Q8—941no—
176#433—942no—
177#432—471no—
178#432—471no—
179#431—471no—
180#431C2×D4711yescharacteristic
181#376—471no—
182#328—942no—
183#328—942no—
184Q8⋊C8—941no—
185Q8⋊C8—941no—
186C47#928011yescharacteristic, complemented, Sylow
187C4⋊Q8C2×Dic4711yescharacteristic
188C4⋊Q8C2×Dic4711yescharacteristic
189C4⋊Q8C2×Dic4711yescharacteristic
190C4⋊Q8C2×Dic4711yescharacteristic
191#125—21no—
192#125—21no—
193C22⋊Q8—21no—
194C22⋊Q8—21no—
195C22⋊Q8—21no—
196C22⋊Q8C22⋊D4712yes—
197C4×Q8—21no—
198C4×Q8—21no—
199C4×Q8—21no—
200C4×Q8C22⋊D4712yes—
201#117—21no—
202#117—21no—
203C2×C42—941no—
204C2×C42C22×D4711yescharacteristic
205#107—22no—
206#106—22no—
207C4⋊C8—21no—
208C4⋊C8—942no—
209C4⋊C8—942no—
210C4⋊C8—941no—
211C4⋊C8—941no—
212C4⋊C8—941no—
213#103—22no—
214#103—22no—
215C22⋊C8—21no—
216C22⋊C8—472no—
217C22⋊C8—941no—
218C22⋊C8—941no—
219#97—942no—
220#97—941no—
221#97—941no—
222C4×C8—471no—
223C4×C8—471no—
224C4×C8—942no—
225C4×C8C2×Dic4711yescharacteristic
226C4×C8C2×Dic4711yescharacteristic
227#95—941no—
228#95—941no—
229C2×Q8—21no—
230C2×Q8—21no—
231C2×Q8—21no—
232C2×Q8C22⋊Dic4712yes—
233C22×C4—1881no—
234C22×C4D4×D4711yescharacteristic
235C22×C4#1062535211yescharacteristic
236C22×C4C2×C22⋊D4711yescharacteristic
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—941no—
240C2×C8—941no—
241C2×C8—942no—
242C2×C8—1881no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—22no—
247C4⋊C4—22no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4C22⋊Dic4712yes—
255C22⋊C4—22no—
256C22⋊C4—41no—
257C42—22no—
258C42—41no—
259C42—942no—
260C42—1881no—
261C42D4×D4711yescharacteristic
262C42#1062535211yescharacteristic
263C42C22×Dic4711yescharacteristic
264C42C2×C22⋊D4711yescharacteristic
265C23#1309399611yescharacteristic
266Q8—42no—
267Q8—41no—
268Q8—41no—
269Q8—41no—
270Q8—41no—
271Q8—81no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—3761no—
281C2×C4—3761no—
282C2×C4—1882no—
283C2×C4C2×C22⋊Dic4711yescharacteristic
284C2×C4D4×Dic4711yescharacteristic
285C2×C4D4×Dic4711yescharacteristic
286C2×C4#1309399211yescharacteristic
287C2×C4#1309399611yescharacteristic
288C2×C4#1309400511yescharacteristic
289C8—41no—
290C8—41no—
291C8—3761no—
292C8—1881no—
293C8—1881no—
294C8—1882no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22#1568900311yescharacteristic
299C4—21no—
300C4—21no—
301C4—21no—
302C4—21no—
303C4—21no—
304C4—42no—
305C4—42no—
306C4—81no—
307C4—81no—
308C4—3762no—
309C4#1568924011yescharacteristic
310C4#1568924511yescharacteristic
311C2—41no—
312C2#2024371711yescharacteristic
313C2#2024388411yescharacteristic
314C2#2024391111yescharacteristic
subgroupG/H
centerC22#15689003
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#3231C2
socleC22×C47#386
prime radicalC23×C47#121
cosocleC2×C42×C47C23
hypercenter#3231D47
hypocenterC47#9280
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3231

Complements

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

NH ≅ G/Ntype
22. C47×#500C4semidirect
23. C47×#498C4semidirect
24. C47×#487C4semidirect
25. C47×#487C4semidirect
167. #500Dic47semidirect
168. #498Dic47semidirect
169. #487Dic47semidirect
170. #487Dic47semidirect
186. C47#9280semidirect

Maximal subgroups

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

subgroupindex
#202524832
#202419002
#202418612
#202412502
#202412162
C47×#32312
#928047