Finite Group Databasebeta

← #26127315

Subgroups — #26127315

Order 12032.

472 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 107 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#20243376C211yesmaximal, characteristic
2#20241706C211yesmaximal, characteristic
3#20241388C214yesmaximal, complemented
4C47×#2547C211yesmaximal, characteristic
5C2×#13093939—21no—
6C2×C22⋊(C47⋊C8)—21no—
7#15688983—21no—
8#15688890C2212yes—
9#15688849—21no—
10#15688847C2212yes—
11C2×C4⋊Q8×C47C411yescharacteristic, complemented
12C2×#124×C47—21no—
13C2×C4×D4×C47C2211yescharacteristic
14C47×#397C412yescomplemented
15C47×#397C2212yes—
16C47×#391—21no—
17C47×#380—21no—
18C47×#380—21no—
19C4×C4⋊C4×C47C411yescharacteristic, complemented
20C22×C47⋊C8—41no—
21C2×C4⋊Dic47—41no—
22C2×C22⋊Dic47—22no—
23#13093939—42no—
24C22⋊(C47⋊C8)—22no—
25C22⋊(C47⋊C8)—41no—
26C47⋊#95—41no—
27C22×Q8×C47—21no—
28C22×D4×C47D411yescharacteristic
29C23×C4×C47D411yescharacteristic
30C23×C4×C47D411yescharacteristic
31C4⋊Q8×C47—41no—
32C4⋊Q8×C47—41no—
33#124×C47—42no—
34#124×C47—42no—
35C4×D4×C47—41no—
36C4×D4×C47—41no—
37C2×C4⋊C4×C47—21no—
38C2×C4⋊C4×C47—21no—
39C2×C4⋊C4×C47C2×C412yes—
40C2×C4⋊C4×C47D411yescharacteristic
41C2×C22⋊C4×C47—22no—
42C2×C22⋊C4×C47—22no—
43C2×C22⋊C4×C47C2×C411yescharacteristic
44C2×C22⋊C4×C47C2311yescharacteristic
45C2×C42×C47—21no—
46C2×C42×C47C2×C411yescharacteristic
47#95×C47—22no—
48#95×C47—22no—
49#95×C47C2×C412yes—
50C24×C47C22⋊C411yescharacteristic
51C24×C47C2×D411yescharacteristic
52C2×Q8×C47—42no—
53C2×Q8×C47—42no—
54C2×Q8×C47—42no—
55C2×D4×C47—41no—
56C2×D4×C47—41no—
57C2×D4×C47D812yes—
58C2×D4×C47SD1612yes—
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47—21no—
62C22×C4×C47—21no—
63C22×C4×C47—22no—
64C22×C4×C47—22no—
65C22×C4×C47—42no—
66C22×C4×C47—42no—
67C22×C4×C47C22⋊C411yescharacteristic
68C22×C4×C47C22⋊C411yescharacteristic
69C22×C4×C47C22⋊C411yescharacteristic
70C22×C4×C47C22×C411yescharacteristic
71C22×C4×C47C2×D411yescharacteristic
72C22×Dic47—42no—
73C4⋊C4×C47—41no—
74C4⋊C4×C47—42no—
75C4⋊C4×C47—42no—
76C4⋊C4×C47—41no—
77C4⋊C4×C47—41no—
78C22⋊C4×C47—41no—
79C22⋊C4×C47—41no—
80C22⋊C4×C47—44no—
81C22⋊C4×C47—42no—
82C22⋊C4×C47—42no—
83C42×C47—21no—
84C42×C47—21no—
85C42×C47—42no—
86C22⋊Dic47—42no—
87C22⋊Dic47—42no—
88C4⋊Dic47—82no—
89C2×C47⋊C8—42no—
90C2×C47⋊C8—82no—
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
103Q8×C47—84no—
104D4×C47—42no—
105D4×C47—42no—
106C2×C4×C47—21no—
107C2×C4×C47—21no—
108C2×C4×C47—22no—
109C2×C4×C47—21no—
110C2×C4×C47—21no—
111C2×C4×C47—21no—
112C2×C4×C47—42no—
113C2×C4×C47—42no—
114C2×C4×C47—41no—
115C2×C4×C47—41no—
116C2×C4×C47—41no—
117C2×C4×C47—42no—
118C2×C4×C47—42no—
119C2×C4×C47—42no—
120C2×C4×C47—42no—
121C2×C4×C47—42no—
122C2×C4×C47—42no—
123C2×C4×C47—81no—
124C2×C4×C47—81no—
125C2×C4×C47#9911yescharacteristic
126C2×C4×C47#9911yescharacteristic
127C2×C4×C47#10212yes—
128C2×Dic47—82no—
129C2×Dic47—82no—
130C2×Dic47—82no—
131C47⋊C8—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—82no—
156C4×C47—82no—
157C4×C47#35412yes—
158Dic47—162no—
159#2547D4711yescharacteristic
160#1799—471no—
161#1459—471no—
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⋊Q8Dic4711yescharacteristic, complemented
172C2×#124—21no—
173C2×C4×D4C2×D4711yescharacteristic
174C2×#99—941no—
175C2×C22⋊C8—941no—
176#397Dic4712yescomplemented
177#397C2×D4712yes—
178#391—21no—
179#380—21no—
180#380—21no—
181#380—941no—
182C4×C4⋊C4Dic4711yescharacteristic, complemented
183#342—472no—
184#324—941no—
185#323—472no—
186C47#1187811yescharacteristic, complemented, Sylow
187C22×Q8—21no—
188C22×D4C22⋊D4711yescharacteristic
189C23×C4C22⋊D4711yescharacteristic
190C23×C4C22⋊D4711yescharacteristic
191C22×C8—1881no—
192C4⋊Q8—41no—
193C4⋊Q8—41no—
194#124—42no—
195#124—42no—
196C4×D4—41no—
197C4×D4—41no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—1881no—
201C2×C4⋊C4C2×Dic4712yes—
202C2×C4⋊C4C22⋊D4711yescharacteristic
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—942no—
206C2×C22⋊C4C2×Dic4711yescharacteristic
207C2×C22⋊C4C22×D4711yescharacteristic
208C2×C42—21no—
209C2×C42C2×Dic4711yescharacteristic
210#99—1882no—
211C22⋊C8—942no—
212C22⋊C8—1881no—
213#95—22no—
214#95—22no—
215#95—1881no—
216#95C2×Dic4712yes—
217C24C22⋊Dic4711yescharacteristic
218C24C2×C22⋊D4711yescharacteristic
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×D4—41no—
223C2×D4—41no—
224C2×D4#1062533412yes—
225C2×D4#1062533512yes—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—1882no—
235C22×C4C22⋊Dic4711yescharacteristic
236C22×C4C22⋊Dic4711yescharacteristic
237C22×C4C22⋊Dic4711yescharacteristic
238C22×C4C22×Dic4711yescharacteristic
239C22×C4C2×C22⋊D4711yescharacteristic
240C2×C8—3762no—
241C2×C8—1882no—
242C4⋊C4—41no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—3762no—
248C22⋊C4—41no—
249C22⋊C4—41no—
250C22⋊C4—44no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—1882no—
254C22⋊C4—1882no—
255C42—21no—
256C42—21no—
257C42—42no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23#1309394512yes—
267C23C2×C22⋊Dic4711yescharacteristic
268C23C2×#1062533411yescharacteristic
269C23C2×#1062533511yescharacteristic
270Q8—84no—
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—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—3762no—
293C2×C4—3762no—
294C2×C4—3762no—
295C2×C4#1309393911yescharacteristic
296C2×C4#1309393911yescharacteristic
297C2×C4#1309394512yes—
298C8—3762no—
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×#1309393911yescharacteristic
314C22C2×#1309394511yescharacteristic
315C22#1568914011yescharacteristic
316C4—22no—
317C4—21no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—82no—
322C4—82no—
323C4—7522no—
324C4#1568891812yes—
325C2—21no—
326C2—21no—
327C2—41no—
328C2—42no—
329C2—41no—
330C2#2024195611yescharacteristic
331C2#2024504211yescharacteristic
332C2C2×#1568891811yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC22×C4×C47C22×C4
FrattiniC2×C22⋊C4C22×D47
FittingC47×#2547C2
socleC22×C47C2×#99
prime radicalC22×D4×C47D4
cosocleC2×C22⋊C4×C47C23
hypercenter#2547D47
hypocenterC47#11878
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2547

Complements

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

NH ≅ G/Ntype
3. #20241388C2semidirect
11. C2×C4⋊Q8×C47C4semidirect
14. C47×#397C4semidirect
19. C4×C4⋊C4×C47C4semidirect
171. C2×C4⋊Q8Dic47semidirect
176. #397Dic47semidirect
182. C4×C4⋊C4Dic47semidirect
186. C47#11878semidirect

Maximal subgroups

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

subgroupindex
#202433762
#202417062
#202413882
C47×#25472
#1187847