Finite Group Databasebeta

← #26109763

Subgroups — #26109763

Order 12032.

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

normal subgroups only

This group has 450 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#20245872C211yesmaximal, characteristic
2#20241722C211yesmaximal, characteristic
3#20241245C212yesmaximal, complemented
4#20241243C212yesmaximal, complemented
5C47×#1464C211yesmaximal, characteristic
6C2×#13093983C411yescharacteristic
7C22×C4×Dic47C411yescharacteristic
8#15689180C2212yes—
9#15689022—21no—
10#15689022—21no—
11#15688998—21no—
12#15688998—21no—
13C4×C4⋊Dic47C411yescharacteristic
14C4×C4⋊Dic47C411yescharacteristic
15C4×C4⋊Dic47C2211yescharacteristic
16C4×C4⋊Dic47C2211yescharacteristic
17#15688849—21no—
18#15688847—21no—
19C2×#117×C47C2211yescharacteristic
20C47×#432C2212yes—
21C47×#324—21no—
22C47×#323—21no—
23C23×Dic47—21no—
24#13093983—42no—
25C2×C4⋊Dic47—21no—
26C2×C4⋊Dic47—21no—
27C2×C4⋊Dic47—21no—
28C2×C4⋊Dic47—21no—
29C2×C4⋊Dic47—21no—
30C2×C22⋊Dic47—21no—
31C2×C4×Dic47—21no—
32C2×C4×Dic47—42no—
33C2×C4×Dic47C2×C411yescharacteristic
34C2×C4×Dic47C2×C411yescharacteristic
35C2×C4×Dic47C2×C411yescharacteristic
36C2×C4×Dic47C2×C411yescharacteristic
37C4⋊(C47⋊C8)—22no—
38C22⋊(C47⋊C8)—22no—
39#13093935—22no—
40C47⋊#95—21no—
41C47⋊#95—21no—
42C47⋊#95—21no—
43C47⋊#95—21no—
44C23×C4×C47C2×C411yescharacteristic
45#117×C47—22no—
46#117×C47—22no—
47C2×C4⋊C4×C47D411yescharacteristic
48C2×C4⋊C4×C47D411yescharacteristic
49C2×C22⋊C4×C47D411yescharacteristic
50C2×C22⋊C4×C47D411yescharacteristic
51C2×C42×C47C2×C411yescharacteristic
52C2×C42×C47C2311yescharacteristic
53C4⋊C8×C47—22no—
54C22⋊C8×C47—22no—
55#97×C47—22no—
56C24×C47C22⋊C411yescharacteristic
57C22×C4×C47—22no—
58C22×C4×C47—22no—
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47C22⋊C411yescharacteristic
62C22×C4×C47C22⋊C411yescharacteristic
63C22×C4×C47C22⋊C411yescharacteristic
64C22×C4×C47C22×C411yescharacteristic
65C22×C4×C47C2×D411yescharacteristic
66C22×C4×C47C2×D411yescharacteristic
67C22×Dic47—21no—
68C22×Dic47—21no—
69C22×Dic47—21no—
70C22×Dic47—21no—
71C22×Dic47—42no—
72C22×Dic47—42no—
73C22×Dic47—42no—
74C2×C8×C47—42no—
75C4⋊C4×C47—22no—
76C4⋊C4×C47—22no—
77C22⋊C4×C47—22no—
78C22⋊C4×C47—22no—
79C42×C47—21no—
80C42×C47—21no—
81C42×C47#4112yes—
82C42×C47#4112yes—
83C22⋊Dic47—42no—
84C4⋊Dic47—42no—
85C4⋊Dic47—42no—
86C4⋊Dic47—42no—
87C4⋊Dic47—42no—
88C4⋊Dic47—42no—
89C4×Dic47—42no—
90C4×Dic47—41no—
91C4×Dic47—41no—
92C4×Dic47—41no—
93C4×Dic47—41no—
94C2×C47⋊C8—42no—
95C23×C47—21no—
96C23×C47—21no—
97C23×C47—41no—
98C23×C47—41no—
99C23×C47#10012yes—
100C23×C47C2×C22⋊C411yescharacteristic
101C2×C4×C47—24no—
102C2×C4×C47—24no—
103C2×C4×C47—21no—
104C2×C4×C47—21no—
105C2×C4×C47—21no—
106C2×C4×C47—21no—
107C2×C4×C47—21no—
108C2×C4×C47—21no—
109C2×C4×C47—41no—
110C2×C4×C47—41no—
111C2×C4×C47—41no—
112C2×C4×C47—41no—
113C2×C4×C47#10112yes—
114C2×C4×C47#11711yescharacteristic
115C2×C4×C47#11711yescharacteristic
116C2×C4×C47#12312yes—
117C2×C4×C47#12312yes—
118C2×Dic47—22no—
119C2×Dic47—22no—
120C2×Dic47—41no—
121C2×Dic47—41no—
122C2×Dic47—48no—
123C2×Dic47—41no—
124C2×Dic47—41no—
125C2×Dic47—41no—
126C2×Dic47—41no—
127C2×Dic47—41no—
128C2×Dic47—41no—
129C2×Dic47—41no—
130C2×Dic47—41no—
131C8×C47—82no—
132C47⋊C8—82no—
133#8742—471nomaximal, Sylow
134C22×C47—21no—
135C22×C47—21no—
136C22×C47—22no—
137C22×C47—42no—
138C22×C47—42no—
139C22×C47—41no—
140C22×C47—41no—
141C22×C47#38111yescharacteristic
142C22×C47C2×#10011yescharacteristic
143C22×C47C2×#10111yescharacteristic
144C4×C47—22no—
145C4×C47—21no—
146C4×C47—21no—
147C4×C47—22no—
148C4×C47—22no—
149C4×C47—41no—
150C4×C47—41no—
151Dic47—42no—
152Dic47—42no—
153Dic47—81no—
154Dic47—81no—
155#2296—471no—
156#1464—471no—
157#1464D4711yescharacteristic
158#1285—472no—
159#1285—472no—
160C2×C47—21no—
161C2×C47—21no—
162C2×C47—42no—
163C2×C47#182111yescharacteristic
164C2×C47#182711yescharacteristic
165C2×C47#182711yescharacteristic
166C2×#117—471no—
167C2×#117C2×D4711yescharacteristic
168C22×C42—471no—
169#432—472no—
170#432C2×D4712yes—
171#389—941no—
172#389—941no—
173#385—941no—
174#385—941no—
175C4×C4⋊C4—471no—
176C4×C4⋊C4—471no—
177C4×C4⋊C4—471no—
178C4×C4⋊C4—471no—
179#324—21no—
180#324—941no—
181#323—21no—
182#323—941no—
183C47#874211yescharacteristic, complemented, Sylow
184C23×C4—941no—
185C23×C4C4×D4711yescharacteristic
186#117—22no—
187#117—22no—
188#117—1882no—
189C2×C4⋊C4—941no—
190C2×C4⋊C4—941no—
191C2×C4⋊C4—941no—
192C2×C4⋊C4—941no—
193C2×C4⋊C4—941no—
194C2×C4⋊C4D18811yescharacteristic
195C2×C4⋊C4C22⋊D4711yescharacteristic
196C2×C22⋊C4—941no—
197C2×C22⋊C4D18811yescharacteristic
198C2×C22⋊C4C22⋊D4711yescharacteristic
199C2×C42—471no—
200C2×C42—471no—
201C2×C42—471no—
202C2×C42—471no—
203C2×C42—941no—
204C2×C42—1882no—
205C2×C42C4×D4711yescharacteristic
206C2×C42C22×D4711yescharacteristic
207C4⋊C8—22no—
208C4⋊C8—942no—
209C22⋊C8—22no—
210C22⋊C8—942no—
211#97—22no—
212#97—942no—
213#95—941no—
214#95—941no—
215#95—941no—
216#95—941no—
217C24(C2×D47)⋊C411yescharacteristic
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—941no—
223C22×C4—941no—
224C22×C4—941no—
225C22×C4—941no—
226C22×C4—1882no—
227C22×C4—1882no—
228C22×C4—1882no—
229C22×C4(C2×D47)⋊C411yescharacteristic
230C22×C4(C2×D47)⋊C411yescharacteristic
231C22×C4(C2×D47)⋊C411yescharacteristic
232C22×C4C2×C4×D4711yescharacteristic
233C22×C4C2×D18811yescharacteristic
234C22×C4C2×C22⋊D4711yescharacteristic
235C2×C8—42no—
236C2×C8—1882no—
237C4⋊C4—22no—
238C4⋊C4—22no—
239C4⋊C4—1882no—
240C4⋊C4—1882no—
241C4⋊C4—1882no—
242C4⋊C4—1882no—
243C4⋊C4—1882no—
244C22⋊C4—22no—
245C22⋊C4—22no—
246C22⋊C4—1882no—
247C42—21no—
248C42—21no—
249C42—1881no—
250C42—1881no—
251C42—1881no—
252C42—1881no—
253C42—1882no—
254C42#1062535212yes—
255C42#1062535212yes—
256C23—21no—
257C23—21no—
258C23—41no—
259C23—41no—
260C23#1309394012yes—
261C23C2×(C2×D47)⋊C411yescharacteristic
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—942no—
275C2×C4—942no—
276C2×C4—1881no—
277C2×C4—1881no—
278C2×C4—1881no—
279C2×C4—1881no—
280C2×C4—1881no—
281C2×C4—1881no—
282C2×C4—1881no—
283C2×C4—1881no—
284C2×C4—1888no—
285C2×C4—1881no—
286C2×C4—1881no—
287C2×C4#1309394212yes—
288C2×C4#1309398011yescharacteristic
289C2×C4#1309398011yescharacteristic
290C2×C4#1309400512yes—
291C2×C4#1309400812yes—
292C8—82no—
293C8—3762no—
294C22—21no—
295C22—21no—
296C22—22no—
297C22—42no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22#1568898411yescharacteristic
302C22C2×#1309394011yescharacteristic
303C22C2×#1309394211yescharacteristic
304C4—22no—
305C4—21no—
306C4—21no—
307C4—22no—
308C4—22no—
309C4—41no—
310C4—41no—
311C4—3761no—
312C4—3761no—
313C4—1882no—
314C4—1882no—
315C2—21no—
316C2—21no—
317C2—42no—
318C2#2024346611yescharacteristic
319C2#2024348911yescharacteristic
320C2#2024349311yescharacteristic
subgroupG/H
centerC22#15688984
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1464C2
socleC22×C47#381
prime radicalC24×C47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#1464D47
hypocenterC47#8742
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1464

Complements

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

NH ≅ G/Ntype
3. #20241245C2semidirect
4. #20241243C2semidirect
183. C47#8742semidirect

Maximal subgroups

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

subgroupindex
#202458722
#202417222
#202412452
#202412432
C47×#14642
#874247