Finite Group Databasebeta

← #26127169

Subgroups — #26127169

Order 12032.

382 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 79 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#20249086C211yesmaximal, characteristic
2#20241702C211yesmaximal, characteristic
3#20241411C212yesmaximal
4#20241387C212yesmaximal
5C47×#1820C211yesmaximal, characteristic
6C2×#13094016C411yescharacteristic, complemented
7#15689176—21no—
8#15689062C411yescharacteristic, complemented
9#15689059C2211yescharacteristic
10#15689053C411yescharacteristic, complemented
11#15689048C2211yescharacteristic
12#15689018—21no—
13#15689017—21no—
14#15689007—21no—
15#15688996—21no—
16#15688969C411yescharacteristic, complemented
17#15688849—21no—
18#15688847C2212yes—
19C2×#117×C47C2211yescharacteristic
20C47×#381—21no—
21C47×#342—21no—
22C47×#342C2212yes—
23#13094016—41no—
24#13094016—41no—
25C2×C4⋊Dic47C2×C411yescharacteristic
26C2×C4⋊Dic47C2×C411yescharacteristic
27C2×Dic47⋊C4—21no—
28C2×Dic47⋊C4—21no—
29C2×Dic47⋊C4—21no—
30C2×Dic47⋊C4—21no—
31C2×C22⋊Dic47—21no—
32C2×C22⋊Dic47—21no—
33C2×C4×Dic47—21no—
34C4⋊(C47⋊C8)—41no—
35C22⋊(C47⋊C8)—22no—
36C4×C47⋊C8—41no—
37C47⋊#95—21no—
38C47⋊#95—21no—
39C47⋊#95—21no—
40C47⋊#95—21no—
41C47⋊#95—21no—
42C47⋊#95C2×C411yescharacteristic
43C47⋊#95C2×C411yescharacteristic
44C23×C4×C47D411yescharacteristic
45#117×C47—41no—
46#117×C47—41no—
47C2×C4⋊C4×C47C2×C411yescharacteristic
48C2×C4⋊C4×C47D411yescharacteristic
49C2×C22⋊C4×C47—22no—
50C2×C22⋊C4×C47D411yescharacteristic
51C2×C22⋊C4×C47C2311yescharacteristic
52C2×C42×C47C2×C411yescharacteristic
53C2×C42×C47D411yescharacteristic
54#95×C47—41no—
55#95×C47—41no—
56C24×C47C2×D411yescharacteristic
57C22×C4×C47—22no—
58C22×C4×C47—21no—
59C22×C4×C47—21no—
60C22×C4×C47—41no—
61C22×C4×C47—42no—
62C22×C4×C47C22⋊C411yescharacteristic
63C22×C4×C47C22⋊C411yescharacteristic
64C22×C4×C47C22⋊C411yescharacteristic
65C22×C4×C47C22⋊C411yescharacteristic
66C22×C4×C47C22×C411yescharacteristic
67C22×C4×C47C2×D411yescharacteristic
68C22×Dic47—21no—
69C22×Dic47—21no—
70C22×Dic47—21no—
71C22×Dic47—21no—
72C4⋊C4×C47—41no—
73C4⋊C4×C47—41no—
74C22⋊C4×C47—41no—
75C22⋊C4×C47—41no—
76C22⋊C4×C47—42no—
77C22⋊C4×C47—42no—
78C42×C47—21no—
79C42×C47—21no—
80C42×C47—21no—
81C42×C47—21no—
82C22⋊Dic47—42no—
83C22⋊Dic47—42no—
84C4⋊Dic47—41no—
85C4⋊Dic47—41no—
86Dic47⋊C4—42no—
87Dic47⋊C4—42no—
88Dic47⋊C4—42no—
89Dic47⋊C4—41no—
90Dic47⋊C4—41no—
91C4×Dic47—42no—
92C2×C47⋊C8—42no—
93C23×C47—21no—
94C23×C47—21no—
95C23×C47—41no—
96C23×C47—41no—
97C23×C47C2×C22⋊C411yescharacteristic
98C23×C47#13512yes—
99C2×C4×C47—21no—
100C2×C4×C47—21no—
101C2×C4×C47—21no—
102C2×C4×C47—21no—
103C2×C4×C47—21no—
104C2×C4×C47—21no—
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—41no—
114C2×C4×C47—82no—
115C2×C4×C47—82no—
116C2×C4×C47—82no—
117C2×C4×C47#9911yescharacteristic
118C2×C4×C47#9911yescharacteristic
119C2×Dic47—22no—
120C2×Dic47—41no—
121C2×Dic47—41no—
122C2×Dic47—41no—
123C2×Dic47—41no—
124C2×Dic47—41no—
125C2×Dic47—41no—
126C2×Dic47—41no—
127C2×Dic47—41no—
128C2×Dic47—41no—
129C2×Dic47—41no—
130C2×Dic47—41no—
131C47⋊C8—82no—
132#11846—471nomaximal, Sylow
133C22×C47—21no—
134C22×C47—21no—
135C22×C47—22no—
136C22×C47—42no—
137C22×C47—41no—
138C22×C47—41no—
139C22×C47—81no—
140C22×C47C2×#9911yescharacteristic
141C22×C47#41611yescharacteristic
142C22×C47#42111yescharacteristic
143C4×C47—21no—
144C4×C47—21no—
145C4×C47—41no—
146C4×C47—41no—
147C4×C47—41no—
148C4×C47—41no—
149C4×C47—41no—
150C4×C47—162no—
151Dic47—42no—
152Dic47—81no—
153Dic47—81no—
154Dic47—81no—
155#2611—471no—
156#1820D4711yescharacteristic
157#1458—471no—
158#1352—472no—
159#1344—472no—
160C2×C47—21no—
161C2×C47—21no—
162C2×C47—42no—
163C2×C47#151611yescharacteristic
164C2×C47#212311yescharacteristic
165C2×C47#212611yescharacteristic
166C2×#125—471no—
167C2×#117C2×D4711yescharacteristic
168#431—941no—
169#400—471no—
170#399—471no—
171#398—471no—
172#397—471no—
173#389—941no—
174#388—941no—
175#387—941no—
176#385—941no—
177#381—21no—
178#376—471no—
179#342—21no—
180#342C2×D4712yes—
181#324—941no—
182#323—472no—
183C47#1184611yescharacteristic, complemented, Sylow
184C23×C4C22⋊D4711yescharacteristic
185#125—1881no—
186#125—1881no—
187#117—41no—
188#117—41no—
189C2×C4⋊C4—471no—
190C2×C4⋊C4—471no—
191C2×C4⋊C4—941no—
192C2×C4⋊C4—941no—
193C2×C4⋊C4—941no—
194C2×C4⋊C4—941no—
195C2×C4⋊C4C4×D4711yescharacteristic
196C2×C4⋊C4D18811yescharacteristic
197C2×C22⋊C4—22no—
198C2×C22⋊C4—941no—
199C2×C22⋊C4—941no—
200C2×C22⋊C4C22⋊D4711yescharacteristic
201C2×C22⋊C4C22×D4711yescharacteristic
202C2×C42—941no—
203C2×C42C4×D4711yescharacteristic
204C2×C42D18811yescharacteristic
205C4⋊C8—1881no—
206C22⋊C8—942no—
207C4×C8—1881no—
208#95—41no—
209#95—41no—
210#95—471no—
211#95—471no—
212#95—941no—
213#95—941no—
214#95—941no—
215#95—941no—
216#95—941no—
217C24C2×C22⋊D4711yescharacteristic
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—41no—
222C22×C4—42no—
223C22×C4—941no—
224C22×C4—941no—
225C22×C4—941no—
226C22×C4—941no—
227C22×C4(C2×D47)⋊C411yescharacteristic
228C22×C4(C2×D47)⋊C411yescharacteristic
229C22×C4(C2×D47)⋊C411yescharacteristic
230C22×C4(C2×D47)⋊C411yescharacteristic
231C22×C4C2×C4×D4711yescharacteristic
232C22×C4C2×D18811yescharacteristic
233C2×C8—1882no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—1881no—
237C4⋊C4—1881no—
238C4⋊C4—1882no—
239C4⋊C4—1882no—
240C4⋊C4—1882no—
241C4⋊C4—1881no—
242C4⋊C4—1881no—
243C22⋊C4—41no—
244C22⋊C4—41no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C22⋊C4—1882no—
248C22⋊C4—1882no—
249C42—21no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—1882no—
254C23—21no—
255C23—21no—
256C23—41no—
257C23—41no—
258C23C2×(C2×D47)⋊C411yescharacteristic
259C23#1309404412yes—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—942no—
279C2×C4—1881no—
280C2×C4—1881no—
281C2×C4—1881no—
282C2×C4—1881no—
283C2×C4—1881no—
284C2×C4—1881no—
285C2×C4—1881no—
286C2×C4—1881no—
287C2×C4—1881no—
288C2×C4—1881no—
289C2×C4—1881no—
290C2×C4#1309393811yescharacteristic
291C2×C4#1309393811yescharacteristic
292C8—3762no—
293C22—21no—
294C22—21no—
295C22—22no—
296C22—42no—
297C22—41no—
298C22—41no—
299C22—81no—
300C22C2×#1309393811yescharacteristic
301C22#1568911711yescharacteristic
302C22#1568914111yescharacteristic
303C4—21no—
304C4—21no—
305C4—41no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—41no—
310C4—162no—
311C4—3761no—
312C4—3761no—
313C4—3761no—
314C4—1882no—
315C2—21no—
316C2—21no—
317C2—42no—
318C2#2024194211yescharacteristic
319C2#2024504111yescharacteristic
320C2#2024505111yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C22⋊C4C22×D47
FittingC47×#1820C2
socleC22×C47C2×#99
prime radicalC24×C47C2×D4
cosocleC2×C22⋊C4×C47C23
hypercenter#1820D47
hypocenterC47#11846
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1820

Complements

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

NH ≅ G/Ntype
6. C2×#13094016C4semidirect
8. #15689062C4semidirect
10. #15689053C4semidirect
16. #15688969C4semidirect
183. C47#11846semidirect

Maximal subgroups

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

subgroupindex
#202490862
#202417022
#202414112
#202413872
C47×#18202
#1184647