Finite Group Databasebeta

← #26109744

Subgroups — #26109744

Order 12032.

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

normal subgroups only

This group has 418 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#20245991C211yesmaximal, characteristic
2C2×#15688849C211yesmaximal, characteristic
3#20241235C212yesmaximal
4#20241234C212yesmaximal
5C2×C47×#324C211yesmaximal, characteristic
6C2×C22⋊(C47⋊C8)C2212yes—
7#15689009—21no—
8#15689009—21no—
9#15689009—21no—
10#15689009—21no—
11#15688986C411yescharacteristic
12#15688986C411yescharacteristic
13#15688983C411yescharacteristic
14#15688983C411yescharacteristic
15C4×C22⋊Dic47C2211yescharacteristic
16C4×C22⋊Dic47C2211yescharacteristic
17#15688849—21no—
18#15688849—21no—
19C22×C4⋊C4×C47C2211yescharacteristic
20C2×C22⋊C8×C47C2212yes—
21C47×#324—21no—
22C47×#324—21no—
23C22×C47⋊C8—22no—
24C2×C4⋊Dic47—21no—
25C2×C4⋊Dic47—21no—
26C2×C22⋊Dic47C2×C411yescharacteristic
27C2×C22⋊Dic47C2×C411yescharacteristic
28C2×C22⋊Dic47C2×C411yescharacteristic
29C2×C22⋊Dic47C2×C411yescharacteristic
30C2×C4×Dic47—21no—
31C2×C4×Dic47—21no—
32C22⋊(C47⋊C8)—22no—
33C22⋊(C47⋊C8)—22no—
34C47⋊#95—21no—
35C47⋊#95—21no—
36C47⋊#95—21no—
37C47⋊#95—21no—
38C47⋊#95—21no—
39C47⋊#95—21no—
40C47⋊#95—21no—
41C47⋊#95—21no—
42C23×C4×C47C2×C411yescharacteristic
43C23×C4×C47C2×C411yescharacteristic
44C23×C4×C47C2311yescharacteristic
45C22×C8×C47—22no—
46C2×C4⋊C4×C47—22no—
47C2×C4⋊C4×C47—22no—
48C2×C4⋊C4×C47D411yescharacteristic
49C2×C4⋊C4×C47D411yescharacteristic
50C2×C4⋊C4×C47D411yescharacteristic
51C2×C4⋊C4×C47D411yescharacteristic
52C22⋊C8×C47—22no—
53C22⋊C8×C47—22no—
54C24×C47C22×C411yescharacteristic
55C22×C4×C47—21no—
56C22×C4×C47—21no—
57C22×C4×C47—22no—
58C22×C4×C47—22no—
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47—81no—
62C22×C4×C47—81no—
63C22×C4×C47C22⋊C411yescharacteristic
64C22×C4×C47C22⋊C411yescharacteristic
65C22×C4×C47C22⋊C411yescharacteristic
66C22×C4×C47C22⋊C411yescharacteristic
67C22×C4×C47C2×D411yescharacteristic
68C22×C4×C47C2×D411yescharacteristic
69C22×C4×C47#4112yes—
70C22×C4×C47#4112yes—
71C22×Dic47—21no—
72C22×Dic47—21no—
73C22×Dic47—21no—
74C22×Dic47—21no—
75C2×C8×C47—42no—
76C2×C8×C47—82no—
77C4⋊C4×C47—22no—
78C4⋊C4×C47—22no—
79C4⋊C4×C47—22no—
80C4⋊C4×C47—22no—
81C22⋊Dic47—22no—
82C22⋊Dic47—22no—
83C22⋊Dic47—22no—
84C22⋊Dic47—22no—
85C4⋊Dic47—81no—
86C4⋊Dic47—81no—
87C4×Dic47—81no—
88C4×Dic47—81no—
89C2×C47⋊C8—42no—
90C2×C47⋊C8—82no—
91C23×C47—41no—
92C23×C47—41no—
93C23×C47C2×C22⋊C411yescharacteristic
94C23×C47#11711yescharacteristic
95C23×C47#11711yescharacteristic
96C23×C47#13112yes—
97C23×C47#13112yes—
98C2×C4×C47—21no—
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—42no—
107C2×C4×C47—41no—
108C2×C4×C47—41no—
109C2×C4×C47—42no—
110C2×C4×C47—42no—
111C2×C4×C47—42no—
112C2×C4×C47—41no—
113C2×C4×C47—41no—
114C2×C4×C47—81no—
115C2×C4×C47—81no—
116C2×C4×C47#12312yes—
117C2×C4×C47#12312yes—
118C2×Dic47—41no—
119C2×Dic47—41no—
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—
130C8×C47—82no—
131C47⋊C8—82no—
132#8737—471nomaximal, Sylow
133C22×C47—21no—
134C22×C47—21no—
135C22×C47—21no—
136C22×C47—21no—
137C22×C47—21no—
138C22×C47—21no—
139C22×C47—42no—
140C22×C47—42no—
141C22×C47#38111yescharacteristic
142C22×C47#40811yescharacteristic
143C22×C47#40811yescharacteristic
144C22×C47#43312yes—
145C22×C47#43312yes—
146C4×C47—22no—
147C4×C47—22no—
148C4×C47—41no—
149C4×C47—41no—
150C4×C47—41no—
151C4×C47—41no—
152Dic47—81no—
153Dic47—81no—
154Dic47—81no—
155Dic47—81no—
156#2313—471no—
157C2×#324—471no—
158C2×#324D4711yescharacteristic
159#1282—472no—
160#1282—472no—
161C2×C47—21no—
162C2×C47—21no—
163C2×C47—22no—
164C2×C47—22no—
165C2×C47#181811yescharacteristic
166C2×C47#182811yescharacteristic
167C2×C47#183011yescharacteristic
168C22×C4⋊C4C2×D4711yescharacteristic
169C2×C22⋊C8—472no—
170C2×C22⋊C8C2×D4712yes—
171#387—941no—
172#387—941no—
173#387—941no—
174#387—941no—
175#381—471no—
176#381—471no—
177#380—471no—
178#380—471no—
179C4×C22⋊C4—471no—
180C4×C22⋊C4—471no—
181#324—21no—
182#324—21no—
183#324—941no—
184#324—941no—
185C47#873711yescharacteristic, complemented, Sylow
186C23×C4C4×D4711yescharacteristic
187C23×C4C4×D4711yescharacteristic
188C23×C4C22×D4711yescharacteristic
189C22×C8—22no—
190C22×C8—942no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—941no—
194C2×C4⋊C4—941no—
195C2×C4⋊C4D18811yescharacteristic
196C2×C4⋊C4D18811yescharacteristic
197C2×C4⋊C4C22⋊D4711yescharacteristic
198C2×C4⋊C4C22⋊D4711yescharacteristic
199C2×C22⋊C4—471no—
200C2×C22⋊C4—471no—
201C2×C22⋊C4—471no—
202C2×C22⋊C4—471no—
203C2×C42—941no—
204C2×C42—941no—
205C22⋊C8—22no—
206C22⋊C8—22no—
207C22⋊C8—942no—
208C22⋊C8—942no—
209#95—941no—
210#95—941no—
211#95—941no—
212#95—941no—
213#95—941no—
214#95—941no—
215#95—941no—
216#95—941no—
217C24C2×C4×D4711yescharacteristic
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—81no—
225C22×C4—81no—
226C22×C4—941no—
227C22×C4—941no—
228C22×C4—941no—
229C22×C4—941no—
230C22×C4(C2×D47)⋊C411yescharacteristic
231C22×C4(C2×D47)⋊C411yescharacteristic
232C22×C4(C2×D47)⋊C411yescharacteristic
233C22×C4(C2×D47)⋊C411yescharacteristic
234C22×C4C2×D18811yescharacteristic
235C22×C4#1062535212yes—
236C22×C4#1062535212yes—
237C22×C4C2×C22⋊D4711yescharacteristic
238C2×C8—42no—
239C2×C8—82no—
240C2×C8—3762no—
241C2×C8—1882no—
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—3761no—
247C4⋊C4—3761no—
248C22⋊C4—942no—
249C22⋊C4—942no—
250C22⋊C4—942no—
251C22⋊C4—942no—
252C42—3761no—
253C42—3761no—
254C23—41no—
255C23—41no—
256C23C2×(C2×D47)⋊C411yescharacteristic
257C23#1309398011yescharacteristic
258C23#1309398011yescharacteristic
259C23#1309403012yes—
260C23#1309403212yes—
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—41no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—81no—
278C2×C4—81no—
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—1881no—
291C2×C4#1309400512yes—
292C2×C4#1309400812yes—
293C8—82no—
294C8—3762no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—42no—
302C22—42no—
303C22#1568898411yescharacteristic
304C22#1568908911yescharacteristic
305C22#1568909111yescharacteristic
306C22#1568918112yes—
307C22#1568918112yes—
308C4—22no—
309C4—22no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—3761no—
315C4—3761no—
316C4—3761no—
317C4—3761no—
318C2—21no—
319C2—21no—
320C2—22no—
321C2—22no—
322C2#2024345711yescharacteristic
323C2#2024349411yescharacteristic
324C2#2024350011yescharacteristic
subgroupG/H
centerC22#15688984
commutator, prime radicalC24×C47C22×C4
FrattiniC23×C4C22×D47
FittingC2×C47×#324C2
socleC22×C47#381
cosocleC23×C4×C47C23
hypercenterC2×#324D47
hypocenterC47#8737
LCSG ▷ C24×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
185. C47#8737semidirect

Maximal subgroups

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

subgroupindex
#202459912
C2×#156888492
#202412352
#202412342
C2×C47×#3242
#873747