Finite Group Databasebeta

← #26127414

Subgroups — #26127414

Order 12032.

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

normal subgroups only

This group has 508 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#20241412C214yesmaximal, complemented
4C47×#2542C211yesmaximal, 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×C22⋊Q8×C47—21no—
13C2×C22⋊Q8×C47—21no—
14C2×C4×D4×C47C2211yescharacteristic
15C47×#399C412yescomplemented
16C47×#399C2212yes—
17C47×#390—21no—
18C47×#384—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—
33C22⋊Q8×C47—44no—
34C22⋊Q8×C47—44no—
35C4×D4×C47—41no—
36C4×D4×C47—41no—
37C2×C4⋊C4×C47—21no—
38C2×C4⋊C4×C47—22no—
39C2×C4⋊C4×C47—21no—
40C2×C4⋊C4×C47—22no—
41C2×C4⋊C4×C47C2×C412yes—
42C2×C4⋊C4×C47D411yescharacteristic
43C2×C22⋊C4×C47—22no—
44C2×C22⋊C4×C47—22no—
45C2×C22⋊C4×C47C2×C411yescharacteristic
46C2×C22⋊C4×C47C2311yescharacteristic
47C2×C42×C47—21no—
48C2×C42×C47C2×C411yescharacteristic
49#95×C47C2×C412yes—
50C24×C47C22⋊C411yescharacteristic
51C24×C47C2×D411yescharacteristic
52C2×Q8×C47—42no—
53C2×Q8×C47—42no—
54C2×Q8×C47—41no—
55C2×Q8×C47—41no—
56C2×D4×C47—41no—
57C2×D4×C47—41no—
58C2×D4×C47SD1612yes—
59C2×D4×C47Q1612yes—
60C22×C4×C47—22no—
61C22×C4×C47—21no—
62C22×C4×C47—21no—
63C22×C4×C47—21no—
64C22×C4×C47—21no—
65C22×C4×C47—22no—
66C22×C4×C47—42no—
67C22×C4×C47—42no—
68C22×C4×C47C22⋊C411yescharacteristic
69C22×C4×C47C22⋊C411yescharacteristic
70C22×C4×C47C22⋊C411yescharacteristic
71C22×C4×C47C22×C411yescharacteristic
72C22×C4×C47C2×D411yescharacteristic
73C22×Dic47—42no—
74C4⋊C4×C47—24no—
75C4⋊C4×C47—41no—
76C4⋊C4×C47—44no—
77C4⋊C4×C47—44no—
78C4⋊C4×C47—42no—
79C4⋊C4×C47—42no—
80C22⋊C4×C47—41no—
81C22⋊C4×C47—41no—
82C22⋊C4×C47—44no—
83C22⋊C4×C47—44no—
84C42×C47—21no—
85C42×C47—21no—
86C42×C47—42no—
87C22⋊Dic47—42no—
88C22⋊Dic47—42no—
89C4⋊Dic47—82no—
90C2×C47⋊C8—42no—
91C2×C47⋊C8—82no—
92C23×C47—21no—
93C23×C47—21no—
94C23×C47—21no—
95C23×C47—21no—
96C23×C47—41no—
97C23×C47—41no—
98C23×C47—41no—
99C23×C47—41no—
100C23×C47#10312yes—
101C23×C47C2×C22⋊C411yescharacteristic
102C23×C47C2×SD1611yescharacteristic
103C23×C47C2×Q1611yescharacteristic
104Q8×C47—82no—
105Q8×C47—82no—
106D4×C47—42no—
107D4×C47—42no—
108C2×C4×C47—21no—
109C2×C4×C47—21no—
110C2×C4×C47—22no—
111C2×C4×C47—24no—
112C2×C4×C47—21no—
113C2×C4×C47—21no—
114C2×C4×C47—21no—
115C2×C4×C47—42no—
116C2×C4×C47—42no—
117C2×C4×C47—42no—
118C2×C4×C47—42no—
119C2×C4×C47—41no—
120C2×C4×C47—41no—
121C2×C4×C47—41no—
122C2×C4×C47—42no—
123C2×C4×C47—42no—
124C2×C4×C47—42no—
125C2×C4×C47—81no—
126C2×C4×C47—81no—
127C2×C4×C47#9911yescharacteristic
128C2×C4×C47#9911yescharacteristic
129C2×C4×C47#10312yes—
130C2×Dic47—82no—
131C2×Dic47—82no—
132C2×Dic47—82no—
133C47⋊C8—82no—
134#11901—471nomaximal, Sylow
135C22×C47—21no—
136C22×C47—21no—
137C22×C47—21no—
138C22×C47—22no—
139C22×C47—21no—
140C22×C47—42no—
141C22×C47—41no—
142C22×C47—41no—
143C22×C47—41no—
144C22×C47—41no—
145C22×C47—41no—
146C22×C47—41no—
147C22×C47—81no—
148C22×C47—81no—
149C22×C47C2×#9911yescharacteristic
150C22×C47C2×#10311yescharacteristic
151C22×C47#42111yescharacteristic
152C4×C47—21no—
153C4×C47—22no—
154C4×C47—44no—
155C4×C47—41no—
156C4×C47—41no—
157C4×C47—41no—
158C4×C47—82no—
159C4×C47#35312yes—
160Dic47—162no—
161#2542D4711yescharacteristic
162#1799—471no—
163#1459—471no—
164#1352—474no—
165C2×C47—21no—
166C2×C47—21no—
167C2×C47—41no—
168C2×C47—42no—
169C2×C47—41no—
170C2×C47#152011yescharacteristic
171C2×C47#212311yescharacteristic
172C2×C47C2×#35311yescharacteristic
173C2×C4⋊Q8Dic4711yescharacteristic, complemented
174C2×C22⋊Q8—21no—
175C2×C22⋊Q8—21no—
176C2×C4×D4C2×D4711yescharacteristic
177C2×#99—941no—
178C2×C22⋊C8—941no—
179#399Dic4712yescomplemented
180#399C2×D4712yes—
181#390—21no—
182#384—21no—
183#380—941no—
184C4×C4⋊C4Dic4711yescharacteristic, complemented
185#342—472no—
186#324—941no—
187#323—472no—
188C47#1190111yescharacteristic, complemented, Sylow
189C22×Q8—21no—
190C22×D4C22⋊D4711yescharacteristic
191C23×C4C22⋊D4711yescharacteristic
192C23×C4C22⋊D4711yescharacteristic
193C22×C8—1881no—
194C4⋊Q8—41no—
195C4⋊Q8—41no—
196C22⋊Q8—44no—
197C22⋊Q8—44no—
198C4×D4—41no—
199C4×D4—41no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—1881no—
205C2×C4⋊C4C2×Dic4712yes—
206C2×C4⋊C4C22⋊D4711yescharacteristic
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—942no—
210C2×C22⋊C4C2×Dic4711yescharacteristic
211C2×C22⋊C4C22×D4711yescharacteristic
212C2×C42—21no—
213C2×C42C2×Dic4711yescharacteristic
214#99—1882no—
215C22⋊C8—942no—
216C22⋊C8—1881no—
217#95—1881no—
218#95C2×Dic4712yes—
219C24C22⋊Dic4711yescharacteristic
220C24C2×C22⋊D4711yescharacteristic
221C2×Q8—42no—
222C2×Q8—42no—
223C2×Q8—41no—
224C2×Q8—41no—
225C2×D4—41no—
226C2×D4—41no—
227C2×D4#1062533612yes—
228C2×D4#1062533712yes—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—1882no—
238C22×C4C22⋊Dic4711yescharacteristic
239C22×C4C22⋊Dic4711yescharacteristic
240C22×C4C22⋊Dic4711yescharacteristic
241C22×C4C22×Dic4711yescharacteristic
242C22×C4C2×C22⋊D4711yescharacteristic
243C2×C8—3762no—
244C2×C8—1882no—
245C4⋊C4—24no—
246C4⋊C4—41no—
247C4⋊C4—42no—
248C4⋊C4—44no—
249C4⋊C4—44no—
250C4⋊C4—42no—
251C4⋊C4—3762no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—44no—
255C22⋊C4—44no—
256C22⋊C4—1882no—
257C22⋊C4—1882no—
258C42—21no—
259C42—21no—
260C42—42no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23#1309394812yes—
270C23C2×C22⋊Dic4711yescharacteristic
271C23C2×#1062533611yescharacteristic
272C23C2×#1062533711yescharacteristic
273Q8—82no—
274Q8—82no—
275D4—42no—
276D4—42no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—24no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—3762no—
297C2×C4—3762no—
298C2×C4—3762no—
299C2×C4#1309393911yescharacteristic
300C2×C4#1309393911yescharacteristic
301C2×C4#1309394812yes—
302C8—3762no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—22no—
307C22—21no—
308C22—42no—
309C22—41no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—81no—
316C22—81no—
317C22C2×#1309393911yescharacteristic
318C22C2×#1309394811yescharacteristic
319C22#1568914011yescharacteristic
320C4—22no—
321C4—21no—
322C4—44no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—82no—
327C4—7522no—
328C4#1568891612yes—
329C2—21no—
330C2—21no—
331C2—41no—
332C2—42no—
333C2—41no—
334C2#2024196011yescharacteristic
335C2#2024504211yescharacteristic
336C2C2×#1568891611yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC22×C4×C47C22×C4
FrattiniC2×C22⋊C4C22×D47
FittingC47×#2542C2
socleC22×C47C2×#99
prime radicalC22×D4×C47D4
cosocleC2×C22⋊C4×C47C23
hypercenter#2542D47
hypocenterC47#11901
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2542

Complements

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

NH ≅ G/Ntype
3. #20241412C2semidirect
11. C2×C4⋊Q8×C47C4semidirect
15. C47×#399C4semidirect
19. C4×C4⋊C4×C47C4semidirect
173. C2×C4⋊Q8Dic47semidirect
179. #399Dic47semidirect
184. C4×C4⋊C4Dic47semidirect
188. C47#11901semidirect

Maximal subgroups

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

subgroupindex
#202433762
#202417062
#202414122
C47×#25422
#1190147