Finite Group Databasebeta

← #26127410

Subgroups — #26127410

Order 12032.

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

normal subgroups only

This group has 452 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#20241409C214yesmaximal, complemented
4C47×#2563C211yesmaximal, 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×C4×D4×C47C2211yescharacteristic
13C47×#393C412yescomplemented
14C47×#393C2212yes—
15C47×#390—21no—
16C47×#382—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×C47C2×C412yes—
28C22×D4×C47D411yescharacteristic
29C23×C4×C47D411yescharacteristic
30C23×C4×C47D411yescharacteristic
31C4⋊Q8×C47—41no—
32C4⋊Q8×C47—41no—
33C4×D4×C47—41no—
34C4×D4×C47—41no—
35C2×C4⋊C4×C47—21no—
36C2×C4⋊C4×C47—21no—
37C2×C4⋊C4×C47—21no—
38C2×C4⋊C4×C47D411yescharacteristic
39C2×C22⋊C4×C47—22no—
40C2×C22⋊C4×C47—22no—
41C2×C22⋊C4×C47C2×C411yescharacteristic
42C2×C22⋊C4×C47C2311yescharacteristic
43C2×C42×C47—21no—
44C2×C42×C47C2×C411yescharacteristic
45#95×C47—22no—
46#95×C47—22no—
47#95×C47C2×C412yes—
48C24×C47C22⋊C411yescharacteristic
49C24×C47C2×D411yescharacteristic
50C2×Q8×C47—42no—
51C2×Q8×C47—82no—
52C2×D4×C47—41no—
53C2×D4×C47—41no—
54C2×D4×C47SD1612yes—
55C2×D4×C47Q1612yes—
56C22×C4×C47—21no—
57C22×C4×C47—21no—
58C22×C4×C47—21no—
59C22×C4×C47—21no—
60C22×C4×C47—22no—
61C22×C4×C47—22no—
62C22×C4×C47—42no—
63C22×C4×C47—42no—
64C22×C4×C47C22⋊C411yescharacteristic
65C22×C4×C47C22⋊C411yescharacteristic
66C22×C4×C47C22⋊C411yescharacteristic
67C22×C4×C47C22×C411yescharacteristic
68C22×C4×C47C2×D411yescharacteristic
69C22×Dic47—42no—
70C4⋊C4×C47—41no—
71C4⋊C4×C47—42no—
72C4⋊C4×C47—42no—
73C4⋊C4×C47—41no—
74C4⋊C4×C47—41no—
75C22⋊C4×C47—41no—
76C22⋊C4×C47—41no—
77C22⋊C4×C47—82no—
78C22⋊C4×C47—82no—
79C42×C47—21no—
80C42×C47—21no—
81C42×C47—42no—
82C22⋊Dic47—42no—
83C22⋊Dic47—42no—
84C4⋊Dic47—82no—
85C2×C47⋊C8—42no—
86C2×C47⋊C8—82no—
87C23×C47—21no—
88C23×C47—21no—
89C23×C47—21no—
90C23×C47—21no—
91C23×C47—41no—
92C23×C47—41no—
93C23×C47—41no—
94C23×C47—41no—
95C23×C47#10312yes—
96C23×C47C2×C22⋊C411yescharacteristic
97C23×C47C2×SD1611yescharacteristic
98C23×C47C2×Q1611yescharacteristic
99Q8×C47—84no—
100D4×C47—42no—
101D4×C47—42no—
102C2×C4×C47—21no—
103C2×C4×C47—21no—
104C2×C4×C47—22no—
105C2×C4×C47—21no—
106C2×C4×C47—21no—
107C2×C4×C47—21no—
108C2×C4×C47—42no—
109C2×C4×C47—42no—
110C2×C4×C47—41no—
111C2×C4×C47—41no—
112C2×C4×C47—41no—
113C2×C4×C47—42no—
114C2×C4×C47—42no—
115C2×C4×C47—42no—
116C2×C4×C47—42no—
117C2×C4×C47—42no—
118C2×C4×C47—42no—
119C2×C4×C47—81no—
120C2×C4×C47—81no—
121C2×C4×C47#9911yescharacteristic
122C2×C4×C47#9911yescharacteristic
123C2×C4×C47#10312yes—
124C2×Dic47—82no—
125C2×Dic47—82no—
126C2×Dic47—82no—
127C47⋊C8—82no—
128#11900—471nomaximal, Sylow
129C22×C47—21no—
130C22×C47—21no—
131C22×C47—21no—
132C22×C47—22no—
133C22×C47—21no—
134C22×C47—42no—
135C22×C47—41no—
136C22×C47—41no—
137C22×C47—41no—
138C22×C47—41no—
139C22×C47—41no—
140C22×C47—41no—
141C22×C47—81no—
142C22×C47—81no—
143C22×C47C2×#9911yescharacteristic
144C22×C47C2×#10311yescharacteristic
145C22×C47#42111yescharacteristic
146C4×C47—21no—
147C4×C47—22no—
148C4×C47—41no—
149C4×C47—41no—
150C4×C47—41no—
151C4×C47—82no—
152C4×C47—82no—
153C4×C47#35412yes—
154Dic47—162no—
155#2563D4711yescharacteristic
156#1799—471no—
157#1459—471no—
158#1351—474no—
159C2×C47—21no—
160C2×C47—21no—
161C2×C47—41no—
162C2×C47—42no—
163C2×C47—41no—
164C2×C47#152011yescharacteristic
165C2×C47#212211yescharacteristic
166C2×C47C2×#35411yescharacteristic
167C2×C4⋊Q8Dic4711yescharacteristic, complemented
168C2×C4×D4C2×D4711yescharacteristic
169C2×#99—941no—
170C2×C22⋊C8—941no—
171#393Dic4712yescomplemented
172#393C2×D4712yes—
173#390—21no—
174#382—21no—
175#380—21no—
176#380—21no—
177#380—941no—
178C4×C4⋊C4Dic4711yescharacteristic, complemented
179#342—472no—
180#324—941no—
181#323—472no—
182C47#1190011yescharacteristic, complemented, Sylow
183C22×Q8C2×Dic4712yes—
184C22×D4C22⋊D4711yescharacteristic
185C23×C4C22⋊D4711yescharacteristic
186C23×C4C22⋊D4711yescharacteristic
187C22×C8—1881no—
188C4⋊Q8—41no—
189C4⋊Q8—41no—
190C4×D4—41no—
191C4×D4—41no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—1881no—
196C2×C4⋊C4C22⋊D4711yescharacteristic
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—942no—
200C2×C22⋊C4C2×Dic4711yescharacteristic
201C2×C22⋊C4C22×D4711yescharacteristic
202C2×C42—21no—
203C2×C42C2×Dic4711yescharacteristic
204#99—1882no—
205C22⋊C8—942no—
206C22⋊C8—1881no—
207#95—22no—
208#95—22no—
209#95—1881no—
210#95C2×Dic4712yes—
211C24C22⋊Dic4711yescharacteristic
212C24C2×C22⋊D4711yescharacteristic
213C2×Q8—42no—
214C2×Q8—82no—
215C2×D4—41no—
216C2×D4—41no—
217C2×D4#1062533612yes—
218C2×D4#1062533712yes—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—1882no—
228C22×C4C22⋊Dic4711yescharacteristic
229C22×C4C22⋊Dic4711yescharacteristic
230C22×C4C22⋊Dic4711yescharacteristic
231C22×C4C22×Dic4711yescharacteristic
232C22×C4C2×C22⋊D4711yescharacteristic
233C2×C8—3762no—
234C2×C8—1882no—
235C4⋊C4—41no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—3762no—
241C22⋊C4—41no—
242C22⋊C4—41no—
243C22⋊C4—82no—
244C22⋊C4—82no—
245C22⋊C4—1882no—
246C22⋊C4—1882no—
247C42—21no—
248C42—21no—
249C42—42no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23#1309394812yes—
259C23C2×C22⋊Dic4711yescharacteristic
260C23C2×#1062533611yescharacteristic
261C23C2×#1062533711yescharacteristic
262Q8—84no—
263D4—42no—
264D4—42no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—3762no—
285C2×C4—3762no—
286C2×C4—3762no—
287C2×C4#1309393911yescharacteristic
288C2×C4#1309393911yescharacteristic
289C2×C4#1309394812yes—
290C8—3762no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—22no—
295C22—21no—
296C22—42no—
297C22—41no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—81no—
304C22—81no—
305C22C2×#1309393911yescharacteristic
306C22C2×#1309394811yescharacteristic
307C22#1568914011yescharacteristic
308C4—22no—
309C4—21no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—82no—
314C4—82no—
315C4—7522no—
316C4#1568891812yes—
317C2—21no—
318C2—21no—
319C2—41no—
320C2—42no—
321C2—41no—
322C2#2024196011yescharacteristic
323C2#2024503811yescharacteristic
324C2C2×#1568891811yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC22×C4×C47C22×C4
FrattiniC2×C22⋊C4C22×D47
FittingC47×#2563C2
socleC22×C47C2×#99
prime radicalC22×D4×C47D4
cosocleC2×C22⋊C4×C47C23
hypercenter#2563D47
hypocenterC47#11900
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2563

Complements

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

NH ≅ G/Ntype
3. #20241409C2semidirect
11. C2×C4⋊Q8×C47C4semidirect
13. C47×#393C4semidirect
19. C4×C4⋊C4×C47C4semidirect
167. C2×C4⋊Q8Dic47semidirect
171. #393Dic47semidirect
178. C4×C4⋊C4Dic47semidirect
182. C47#11900semidirect

Maximal subgroups

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

subgroupindex
#202433762
#202417062
#202414092
C47×#25632
#1190047