Finite Group Databasebeta

← #26127316

Subgroups — #26127316

Order 12032.

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

normal subgroups only

This group has 472 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#20248380C211yesmaximal, characteristic
2#20243376C211yesmaximal, characteristic
3#20241389C214yesmaximal, complemented
4C47×#1459C211yesmaximal, characteristic
5C2×#13094024C411yescharacteristic, complemented
6C2×#13094011—21no—
7C2×#13093939—21no—
8#15689048C412yescomplemented
9#15689048C2212yes—
10#15689028—21no—
11#15688983—21no—
12#15688983—21no—
13#15688983—21no—
14C4×C4⋊Dic47C411yescharacteristic, complemented
15#15688890C2212yes—
16C2×C4×D4×C47C2211yescharacteristic
17C2×C22⋊C8×C47—21no—
18C47×#324—21no—
19C47×#323C2212yes—
20C22×Dic94—21no—
21#13094024—41no—
22#13094024—41no—
23#13094011—42no—
24#13094011—42no—
25C2×C4⋊Dic47—21no—
26C2×C4⋊Dic47—21no—
27C2×C4⋊Dic47—41no—
28C2×C4⋊Dic47C2×C412yes—
29C2×C22⋊Dic47—22no—
30C2×C22⋊Dic47—22no—
31C2×C22⋊Dic47—22no—
32C2×C4×Dic47—21no—
33#13093939—42no—
34C47⋊#95—22no—
35C47⋊#95—22no—
36C47⋊#95—41no—
37C47⋊#95C2×C412yes—
38C22×D4×C47D411yescharacteristic
39C23×C4×C47D411yescharacteristic
40C23×C4×C47D411yescharacteristic
41C22×C8×C47—41no—
42C4×D4×C47—41no—
43C4×D4×C47—41no—
44C2×C4⋊C4×C47D411yescharacteristic
45C2×C22⋊C4×C47C2×C411yescharacteristic
46C2×C22⋊C4×C47C2311yescharacteristic
47C2×C42×C47C2×C411yescharacteristic
48C22⋊C8×C47—22no—
49C22⋊C8×C47—41no—
50C24×C47C22⋊C411yescharacteristic
51C24×C47C2×D411yescharacteristic
52C2×D4×C47—41no—
53C2×D4×C47—41no—
54C2×D4×C47D812yes—
55C2×D4×C47SD1612yes—
56C22×C4×C47—21no—
57C22×C4×C47—21no—
58C22×C4×C47—21no—
59C22×C4×C47—21no—
60C22×C4×C47—42no—
61C22×C4×C47—42no—
62C22×C4×C47C22⋊C411yescharacteristic
63C22×C4×C47C22⋊C411yescharacteristic
64C22×C4×C47C22⋊C411yescharacteristic
65C22×C4×C47C22×C411yescharacteristic
66C22×C4×C47C2×D411yescharacteristic
67C22×Dic47—22no—
68C22×Dic47—22no—
69C22×Dic47—42no—
70C2×Dic94—42no—
71C2×Dic94—42no—
72C2×Dic94—42no—
73C2×C8×C47—42no—
74C2×C8×C47—82no—
75C4⋊C4×C47—41no—
76C22⋊C4×C47—41no—
77C22⋊C4×C47—41no—
78C42×C47—21no—
79C42×C47—21no—
80C22⋊Dic47—44no—
81C22⋊Dic47—42no—
82C22⋊Dic47—42no—
83C22⋊Dic47—42no—
84C22⋊Dic47—42no—
85C4⋊Dic47—42no—
86C4⋊Dic47—41no—
87C4⋊Dic47—41no—
88C4⋊Dic47—42no—
89C4⋊Dic47—82no—
90C4×Dic47—42no—
91C23×C47—21no—
92C23×C47—21no—
93C23×C47—21no—
94C23×C47—21no—
95C23×C47—41no—
96C23×C47—41no—
97C23×C47—41no—
98C23×C47—41no—
99C23×C47#10212yes—
100C23×C47C2×C22⋊C411yescharacteristic
101C23×C47C2×D811yescharacteristic
102C23×C47C2×SD1611yescharacteristic
103D4×C47—42no—
104D4×C47—42no—
105C2×C4×C47—21no—
106C2×C4×C47—21no—
107C2×C4×C47—22no—
108C2×C4×C47—21no—
109C2×C4×C47—21no—
110C2×C4×C47—21no—
111C2×C4×C47—42no—
112C2×C4×C47—42no—
113C2×C4×C47—41no—
114C2×C4×C47—41no—
115C2×C4×C47—41no—
116C2×C4×C47—81no—
117C2×C4×C47—81no—
118C2×C4×C47#9911yescharacteristic
119C2×C4×C47#9911yescharacteristic
120C2×C4×C47#10212yes—
121C2×Dic47—42no—
122C2×Dic47—42no—
123C2×Dic47—42no—
124C2×Dic47—42no—
125C2×Dic47—42no—
126C2×Dic47—42no—
127C2×Dic47—82no—
128C2×Dic47—82no—
129C2×Dic47—82no—
130Dic94—84no—
131C8×C47—82no—
132#11878—471nomaximal, Sylow
133C22×C47—21no—
134C22×C47—21no—
135C22×C47—21no—
136C22×C47—22no—
137C22×C47—21no—
138C22×C47—42no—
139C22×C47—41no—
140C22×C47—41no—
141C22×C47—41no—
142C22×C47—41no—
143C22×C47—41no—
144C22×C47—41no—
145C22×C47—81no—
146C22×C47—81no—
147C22×C47C2×#9911yescharacteristic
148C22×C47C2×#10211yescharacteristic
149C22×C47#42111yescharacteristic
150C4×C47—21no—
151C4×C47—22no—
152C4×C47—41no—
153C4×C47—41no—
154C4×C47—41no—
155C4×C47#35412yes—
156Dic47—82no—
157Dic47—82no—
158Dic47—162no—
159#2547—471no—
160#1799—471no—
161#1459D4711yescharacteristic
162#1344—474no—
163C2×C47—21no—
164C2×C47—21no—
165C2×C47—41no—
166C2×C47—42no—
167C2×C47—41no—
168C2×C47#151911yescharacteristic
169C2×C47#212311yescharacteristic
170C2×C47C2×#35411yescharacteristic
171C2×C4⋊Q8—471no—
172C2×#124—941no—
173C2×C4×D4C2×D4711yescharacteristic
174C2×#99—941no—
175C2×C22⋊C8—21no—
176#397—472no—
177#397—472no—
178#391—941no—
179#380—941no—
180#380—941no—
181#380—941no—
182C4×C4⋊C4—471no—
183#342—472no—
184#324—21no—
185#323C2×D4712yes—
186C47#1187811yescharacteristic, complemented, Sylow
187C22×Q8—941no—
188C22×D4D18811yescharacteristic
189C23×C4D18811yescharacteristic
190C23×C4C22⋊D4711yescharacteristic
191C22×C8—41no—
192C4⋊Q8—1881no—
193C4⋊Q8—1881no—
194#124—1882no—
195#124—1882no—
196C4×D4—41no—
197C4×D4—41no—
198C2×C4⋊C4—472no—
199C2×C4⋊C4—941no—
200C2×C4⋊C4—941no—
201C2×C4⋊C4—1881no—
202C2×C4⋊C4C22⋊D4711yescharacteristic
203C2×C22⋊C4—942no—
204C2×C22⋊C4—942no—
205C2×C22⋊C4—942no—
206C2×C22⋊C4C4×D4711yescharacteristic
207C2×C22⋊C4C22×D4711yescharacteristic
208C2×C42—941no—
209C2×C42C4×D4711yescharacteristic
210#99—1882no—
211C22⋊C8—22no—
212C22⋊C8—41no—
213#95—472no—
214#95—942no—
215#95—942no—
216#95—1881no—
217C24(C2×D47)⋊C411yescharacteristic
218C24C2×D18811yescharacteristic
219C2×Q8—1882no—
220C2×Q8—1882no—
221C2×Q8—1882no—
222C2×D4—41no—
223C2×D4—41no—
224C2×D4#1062532512yes—
225C2×D4D37612yes—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—942no—
233C22×C4—942no—
234C22×C4—1882no—
235C22×C4(C2×D47)⋊C411yescharacteristic
236C22×C4(C2×D47)⋊C411yescharacteristic
237C22×C4(C2×D47)⋊C411yescharacteristic
238C22×C4C2×C4×D4711yescharacteristic
239C22×C4C2×C22⋊D4711yescharacteristic
240C2×C8—42no—
241C2×C8—82no—
242C4⋊C4—41no—
243C4⋊C4—3762no—
244C4⋊C4—1882no—
245C4⋊C4—1882no—
246C4⋊C4—1881no—
247C4⋊C4—1881no—
248C22⋊C4—41no—
249C22⋊C4—41no—
250C22⋊C4—1882no—
251C22⋊C4—1882no—
252C22⋊C4—1884no—
253C22⋊C4—1882no—
254C22⋊C4—1882no—
255C42—21no—
256C42—21no—
257C42—1882no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23#1309394612yes—
267C23C2×(C2×D47)⋊C411yescharacteristic
268C23C2×D37611yescharacteristic
269C23C2×#1062532511yescharacteristic
270Q8—3764no—
271D4—42no—
272D4—42no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—3762no—
287C2×C4—3762no—
288C2×C4—3762no—
289C2×C4—1882no—
290C2×C4—1882no—
291C2×C4—1882no—
292C2×C4—1882no—
293C2×C4—1882no—
294C2×C4—1882no—
295C2×C4#1309393811yescharacteristic
296C2×C4#1309393811yescharacteristic
297C2×C4#1309394612yes—
298C8—82no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—22no—
303C22—21no—
304C22—42no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—41no—
311C22—81no—
312C22—81no—
313C22C2×#1309393811yescharacteristic
314C22C2×#1309394611yescharacteristic
315C22#1568913911yescharacteristic
316C4—22no—
317C4—21no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—3762no—
322C4—3762no—
323C4—7522no—
324C4#1568891712yes—
325C2—21no—
326C2—21no—
327C2—41no—
328C2—42no—
329C2—41no—
330C2#2024195711yescharacteristic
331C2#2024504311yescharacteristic
332C2C2×#1568891711yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C22⋊C4C22×D47
FittingC47×#1459C2
socleC22×C47C2×#99
prime radicalC22×D4×C47D4
cosocleC2×C22⋊C4×C47C23
hypercenter#1459D47
hypocenterC47#11878
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1459

Complements

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

NH ≅ G/Ntype
3. #20241389C2semidirect
5. C2×#13094024C4semidirect
8. #15689048C4semidirect
14. C4×C4⋊Dic47C4semidirect
186. C47#11878semidirect

Maximal subgroups

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

subgroupindex
#202483802
#202433762
#202413892
C47×#14592
#1187847