Finite Group Databasebeta

← #26113762

Subgroups — #26113762

Order 12032.

380 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 99 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#20252494C211yesmaximal, characteristic
2#20242026C211yesmaximal, characteristic
3#20241939C211yesmaximal, characteristic
4#20241939C211yesmaximal, characteristic
5#20241260C211yesmaximal, characteristic
6#20241253C211yesmaximal, characteristic
7C47×#3185C211yesmaximal, characteristic
8#15689254—22no—
9#15689254—22no—
10#15689184—22no—
11#15689180C2211yescharacteristic
12#15689180C2211yescharacteristic
13#15689180C2211yescharacteristic
14#15688991C2211yescharacteristic
15#15688865—22no—
16#15688860—22no—
17#15688860—22no—
18C47×#533C2211yescharacteristic
19C47×#533C2211yescharacteristic
20C47×#498C411yescharacteristic, complemented
21C47×#494C411yescharacteristic, complemented
22C47×#487C411yescharacteristic, complemented
23C47×#487C411yescharacteristic, complemented
24C47×#470—22no—
25C47×#441—22no—
26C4×M16×C47C2211yescharacteristic
27C4⋊(C47⋊C8)—24no—
28C4⋊(C47⋊C8)—21no—
29C4⋊(C47⋊C8)—21no—
30C4⋊(C47⋊C8)—21no—
31C22⋊(C47⋊C8)—21no—
32C22⋊(C47⋊C8)—21no—
33C22⋊(C47⋊C8)D412yes—
34#13093935—22no—
35#13093935—21no—
36#13093935—21no—
37#13093935D412yes—
38C4×C47⋊C8—22no—
39C47⋊#95—21no—
40C47⋊#95—21no—
41C47⋊#95—21no—
42C2×Q16×C47—22no—
43C2×M16×C47—21no—
44C4⋊Q8×C47C2×C411yescharacteristic
45C4⋊Q8×C47C2×C411yescharacteristic
46C4⋊Q8×C47C2×C411yescharacteristic
47C4⋊Q8×C47C2×C411yescharacteristic
48#125×C47—21no—
49#125×C47—21no—
50C22⋊Q8×C47—21no—
51C22⋊Q8×C47—21no—
52C22⋊Q8×C47—21no—
53C22⋊Q8×C47D412yes—
54C4×Q8×C47—21no—
55C4×Q8×C47—21no—
56C4×Q8×C47—21no—
57C4×Q8×C47D412yes—
58#117×C47—21no—
59#117×C47—21no—
60C2×C42×C47C2311yescharacteristic
61#106×C47—22no—
62#103×C47—22no—
63#103×C47—22no—
64#97×C47—21no—
65C4×C8×C47C2×C411yescharacteristic
66C4×C8×C47C2×C411yescharacteristic
67C2×Q8×C47—21no—
68C2×Q8×C47—21no—
69C2×Q8×C47—21no—
70C2×Q8×C47C22⋊C412yes—
71C22×C4×C47C2×D411yescharacteristic
72C22×C4×C47C2×D411yescharacteristic
73C22×C4×C47#4111yescharacteristic
74C22×Dic47—41no—
75Q16×C47—44no—
76M16×C47—42no—
77C2×C8×C47—21no—
78C2×C8×C47—21no—
79C4⋊C4×C47—21no—
80C4⋊C4×C47—21no—
81C4⋊C4×C47—21no—
82C4⋊C4×C47—22no—
83C4⋊C4×C47—22no—
84C4⋊C4×C47—21no—
85C4⋊C4×C47—21no—
86C4⋊C4×C47—21no—
87C4⋊C4×C47—21no—
88C4⋊C4×C47—41no—
89C4⋊C4×C47—41no—
90C4⋊C4×C47C22⋊C412yes—
91C22⋊C4×C47—22no—
92C22⋊C4×C47—41no—
93C42×C47—22no—
94C42×C47—41no—
95C42×C47C22×C411yescharacteristic
96C42×C47C2×D411yescharacteristic
97C42×C47C2×D411yescharacteristic
98C42×C47#4111yescharacteristic
99C2×C47⋊C8—22no—
100C2×C47⋊C8—22no—
101C2×C47⋊C8—41no—
102C23×C47#12111yescharacteristic
103Q8×C47—42no—
104Q8×C47—42no—
105Q8×C47—81no—
106Q8×C47—81no—
107C2×C4×C47—22no—
108C2×C4×C47—22no—
109C2×C4×C47—41no—
110C2×C4×C47—41no—
111C2×C4×C47—41no—
112C2×C4×C47—41no—
113C2×C4×C47—41no—
114C2×C4×C47C2×C22⋊C411yescharacteristic
115C2×C4×C47C4×D411yescharacteristic
116C2×C4×C47C4×D411yescharacteristic
117C2×C4×C47#12011yescharacteristic
118C2×C4×C47#12111yescharacteristic
119C2×C4×C47#12311yescharacteristic
120C2×Dic47—81no—
121C2×Dic47—81no—
122C2×Dic47—81no—
123C8×C47—22no—
124C8×C47—41no—
125C47⋊C8—42no—
126C47⋊C8—42no—
127C47⋊C8—81no—
128#9416—471nomaximal, Sylow
129C22×C47—21no—
130C22×C47—21no—
131C22×C47—21no—
132C22×C47#38611yescharacteristic
133C4×C47—21no—
134C4×C47—21no—
135C4×C47—21no—
136C4×C47—21no—
137C4×C47—21no—
138C4×C47—21no—
139C4×C47—42no—
140C4×C47—42no—
141C4×C47—81no—
142C4×C47—81no—
143Dic47—161no—
144#3185D4711yescharacteristic
145#2987—471no—
146#1537—471no—
147#1515—471no—
148#1515—471no—
149#1291—471no—
150#1288—471no—
151C2×C47—41no—
152C2×C47#188411yescharacteristic
153C2×C47#190211yescharacteristic
154C2×C47#190511yescharacteristic
155#533C2×D4711yescharacteristic
156#533C2×D4711yescharacteristic
157#498Dic4711yescharacteristic, complemented
158#494Dic4711yescharacteristic, complemented
159#487Dic4711yescharacteristic, complemented
160#487Dic4711yescharacteristic, complemented
161#470—22no—
162#446—942no—
163#446—942no—
164#441—22no—
165#433—942no—
166#432—471no—
167#432—471no—
168#432—471no—
169C4×M16C2×D4711yescharacteristic
170#383—471no—
171#330—942no—
172#328—942no—
173#328—942no—
174C47#941611yescharacteristic, complemented, Sylow
175C2×Q16—22no—
176C2×M16—21no—
177C4⋊Q8C2×Dic4711yescharacteristic
178C4⋊Q8C2×Dic4711yescharacteristic
179C4⋊Q8C2×Dic4711yescharacteristic
180C4⋊Q8C2×Dic4711yescharacteristic
181#125—21no—
182#125—21no—
183C22⋊Q8—21no—
184C22⋊Q8—21no—
185C22⋊Q8—21no—
186C22⋊Q8C22⋊D4712yes—
187C4×Q8—21no—
188C4×Q8—21no—
189C4×Q8—21no—
190C4×Q8C22⋊D4712yes—
191#117—21no—
192#117—21no—
193C2×C42C22×D4711yescharacteristic
194#106—22no—
195C4⋊C8—944no—
196C4⋊C8—941no—
197C4⋊C8—941no—
198C4⋊C8—941no—
199#103—22no—
200#103—22no—
201C22⋊C8—472no—
202C22⋊C8—941no—
203C22⋊C8—941no—
204#97—21no—
205#97—472no—
206#97—942no—
207#97—941no—
208#97—941no—
209C4×C8—942no—
210C4×C8C2×Dic4711yescharacteristic
211C4×C8C2×Dic4711yescharacteristic
212#95—941no—
213#95—941no—
214#95—941no—
215C2×Q8—21no—
216C2×Q8—21no—
217C2×Q8—21no—
218C2×Q8C22⋊Dic4712yes—
219C22×C4—1881no—
220C22×C4D4×D4711yescharacteristic
221C22×C4#1062535211yescharacteristic
222C22×C4C2×C22⋊D4711yescharacteristic
223Q16—44no—
224M16—42no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—942no—
228C2×C8—942no—
229C2×C8—1881no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—22no—
234C4⋊C4—22no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4C22⋊Dic4712yes—
242C22⋊C4—22no—
243C22⋊C4—41no—
244C42—22no—
245C42—41no—
246C42D4×D4711yescharacteristic
247C42#1062535211yescharacteristic
248C42C22×Dic4711yescharacteristic
249C42C2×C22⋊D4711yescharacteristic
250C23#1309399611yescharacteristic
251Q8—42no—
252Q8—42no—
253Q8—81no—
254Q8—81no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—3761no—
263C2×C4—3761no—
264C2×C4—3761no—
265C2×C4C2×C22⋊Dic4711yescharacteristic
266C2×C4D4×Dic4711yescharacteristic
267C2×C4D4×Dic4711yescharacteristic
268C2×C4#1309399211yescharacteristic
269C2×C4#1309399611yescharacteristic
270C2×C4#1309400511yescharacteristic
271C8—22no—
272C8—41no—
273C8—3761no—
274C8—1882no—
275C8—1882no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22#1568900311yescharacteristic
280C4—21no—
281C4—21no—
282C4—21no—
283C4—21no—
284C4—21no—
285C4—21no—
286C4—42no—
287C4—42no—
288C4—81no—
289C4—81no—
290C4—7521no—
291C2—41no—
292C2#2024377211yescharacteristic
293C2#2024386311yescharacteristic
294C2#2024388411yescharacteristic
subgroupG/H
centerC22#15689003
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#3185C2
socleC22×C47#386
prime radicalC23×C47#121
cosocleC2×C42×C47C23
hypercenter#3185D47
hypocenterC47#9416
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3185

Complements

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

NH ≅ G/Ntype
20. C47×#498C4semidirect
21. C47×#494C4semidirect
22. C47×#487C4semidirect
23. C47×#487C4semidirect
157. #498Dic47semidirect
158. #494Dic47semidirect
159. #487Dic47semidirect
160. #487Dic47semidirect
174. C47#9416semidirect

Maximal subgroups

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

subgroupindex
#202524942
#202420262
#202419392
#202412602
#202412532
C47×#31852
#941647