Finite Group Databasebeta

← #26109191

Subgroups — #26109191

Order 12032.

460 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C2×#15689195C211yesmaximal, characteristic
2#20243424C212yesmaximal, complemented
3#20241726C211yesmaximal, characteristic
4#20241264C212yesmaximal, complemented
5C47×#1465C211yesmaximal, characteristic
6C22×#10625329C411yescharacteristic
7C23×C47⋊C8C2211yescharacteristic
8#15689195—22no—
9#15689195—22no—
10C2×#13093959—22no—
11C2×C4⋊(C47⋊C8)C411yescharacteristic
12#15689087C2212yes—
13C2×C22⋊(C47⋊C8)C411yescharacteristic
14C2×C22⋊(C47⋊C8)C2211yescharacteristic
15C2×#13093935C411yescharacteristic
16#15688887—22no—
17#15688849—21no—
18#15688847—21no—
19C2×C4×D4×C47C2211yescharacteristic
20C47×#407C2212yes—
21C47×#324—21no—
22C47×#323—21no—
23C2×#10625329—21no—
24C2×#10625329—21no—
25C2×#10625329—22no—
26C2×#10625329—42no—
27C22×C47⋊C8—42no—
28C22×C47⋊C8C2×C411yescharacteristic
29C22×C47⋊C8C2×C411yescharacteristic
30C22×C47⋊C8C2×C411yescharacteristic
31C22×C47⋊C8C2×C411yescharacteristic
32C22×C47⋊C8D412yes—
33C22×C47⋊C8D412yes—
34#13093959—44no—
35C4⋊(C47⋊C8)—22no—
36C22⋊(C47⋊C8)—22no—
37C22⋊(C47⋊C8)—22no—
38C22⋊(C47⋊C8)—22no—
39C22⋊(C47⋊C8)—22no—
40#13093935—21no—
41#13093935—21no—
42C22×D4×C47Q811yescharacteristic
43C23×C4×C47C2×C411yescharacteristic
44C23×C4×C47C2311yescharacteristic
45C2×M16×C47—22no—
46C4×D4×C47—22no—
47C4×D4×C47—22no—
48C2×C4⋊C4×C47D411yescharacteristic
49C2×C22⋊C4×C47D411yescharacteristic
50C2×C22⋊C4×C47Q811yescharacteristic
51C2×C42×C47C2×C411yescharacteristic
52C22⋊C8×C47—22no—
53C22⋊C8×C47—22no—
54C24×C47C4⋊C411yescharacteristic
55C24×C47C2×Q811yescharacteristic
56C2×D4×C47—22no—
57C2×D4×C47—42no—
58C22×C4×C47—21no—
59C22×C4×C47—21no—
60C22×C4×C47—41no—
61C22×C4×C47—41no—
62C22×C4×C47—41no—
63C22×C4×C47—41no—
64C22×C4×C47C4⋊C411yescharacteristic
65C22×C4×C47C4⋊C411yescharacteristic
66C22×C4×C47C4⋊C411yescharacteristic
67C22×C4×C47C22×C411yescharacteristic
68C22×C4×C47C2×D412yes—
69C22×C4×C47C2×D411yescharacteristic
70C22×C4×C47#4112yes—
71M16×C47—42no—
72M16×C47—42no—
73C2×C8×C47—42no—
74C4⋊C4×C47—22no—
75C22⋊C4×C47—22no—
76C22⋊C4×C47—22no—
77C42×C47—21no—
78C42×C47—21no—
79#10625329—42no—
80#10625329—42no—
81#10625329—42no—
82#10625329—42no—
83C2×C47⋊C8—24no—
84C2×C47⋊C8—22no—
85C2×C47⋊C8—21no—
86C2×C47⋊C8—21no—
87C2×C47⋊C8—41no—
88C2×C47⋊C8—42no—
89C2×C47⋊C8—41no—
90C2×C47⋊C8—41no—
91C2×C47⋊C8—42no—
92C2×C47⋊C8C22⋊C411yescharacteristic
93C2×C47⋊C8C22⋊C412yes—
94C2×C47⋊C8C22⋊C411yescharacteristic
95C23×C47—22no—
96C23×C47—21no—
97C23×C47—22no—
98C23×C47—41no—
99C23×C47—41no—
100C23×C47—41no—
101C23×C47—41no—
102C23×C47C2×C4⋊C411yescharacteristic
103C23×C47C22⋊Q812yes—
104D4×C47—44no—
105C2×C4×C47—21no—
106C2×C4×C47—22no—
107C2×C4×C47—21no—
108C2×C4×C47—22no—
109C2×C4×C47—21no—
110C2×C4×C47—21no—
111C2×C4×C47—21no—
112C2×C4×C47—22no—
113C2×C4×C47—41no—
114C2×C4×C47—42no—
115C2×C4×C47—42no—
116C2×C4×C47—41no—
117C2×C4×C47—41no—
118C2×C4×C47—41no—
119C2×C4×C47—41no—
120C2×C4×C47C2×C22⋊C411yescharacteristic
121C2×C4×C47#11711yescharacteristic
122C2×C4×C47#12112yes—
123C8×C47—82no—
124C47⋊C8—22no—
125C47⋊C8—21no—
126C47⋊C8—41no—
127C47⋊C8—41no—
128C47⋊C8—82no—
129C47⋊C8#9912yes—
130#8613—471nomaximal, Sylow
131C22×C47—21no—
132C22×C47—21no—
133C22×C47—22no—
134C22×C47—22no—
135C22×C47—22no—
136C22×C47—44no—
137C22×C47—41no—
138C22×C47—41no—
139C22×C47—41no—
140C22×C47—82no—
141C22×C47#38011yescharacteristic
142C22×C47#43012yes—
143C4×C47—22no—
144C4×C47—21no—
145C4×C47—41no—
146C4×C47—41no—
147C4×C47—41no—
148C4×C47C2×#9911yescharacteristic
149C4×C47#41011yescharacteristic
150C2×#435—471no—
151#1810—472no—
152#1465—471no—
153#1465D4711yescharacteristic
154#1293—472no—
155C2×C47—22no—
156C2×C47—42no—
157C2×C47—81no—
158C2×C47#179911yescharacteristic
159C2×C47#181111yescharacteristic
160C2×C47#181311yescharacteristic
161C22×M16—471no—
162C23×C8—471no—
163C2×C4×D4C2×D4711yescharacteristic
164#435—942no—
165#435—942no—
166C2×#108—942no—
167C2×C4⋊C8—471no—
168#407—472no—
169#407C2×D4712yes—
170C2×C22⋊C8—471no—
171C2×C22⋊C8—471no—
172C2×#97—471no—
173#341—942no—
174#324—21no—
175#324—941no—
176#323—21no—
177#323—941no—
178C47#861311yescharacteristic, complemented, Sylow
179C22×D4Dic9411yescharacteristic
180C23×C4C4×D4711yescharacteristic
181C23×C4C22×D4711yescharacteristic
182C2×M16—22no—
183C2×M16—941no—
184C2×M16—941no—
185C2×M16—942no—
186C2×M16—1882no—
187C22×C8—471no—
188C22×C8—472no—
189C22×C8—471no—
190C22×C8—472no—
191C22×C8—471no—
192C22×C8—471no—
193C22×C8—1882no—
194C4×D4—22no—
195C4×D4—22no—
196C2×C4⋊C4C22⋊D4711yescharacteristic
197C2×C22⋊C4Dic9411yescharacteristic
198C2×C22⋊C4C22⋊D4711yescharacteristic
199C2×C42C4×D4711yescharacteristic
200#108—1884no—
201C4⋊C8—942no—
202C22⋊C8—22no—
203C22⋊C8—22no—
204C22⋊C8—942no—
205C22⋊C8—942no—
206C22⋊C8—942no—
207C22⋊C8—942no—
208#97—941no—
209#97—941no—
210C24Dic47⋊C411yescharacteristic
211C24C2×Dic9411yescharacteristic
212C2×D4—22no—
213C2×D4—42no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4Dic47⋊C411yescharacteristic
221C22×C4Dic47⋊C411yescharacteristic
222C22×C4Dic47⋊C411yescharacteristic
223C22×C4C2×C4×D4711yescharacteristic
224C22×C4D4×D4712yes—
225C22×C4#1062535212yes—
226C22×C4C2×C22⋊D4711yescharacteristic
227M16—42no—
228M16—42no—
229M16—1882no—
230M16—1882no—
231M16—1882no—
232M16—1882no—
233C2×C8—42no—
234C2×C8—471no—
235C2×C8—471no—
236C2×C8—472no—
237C2×C8—944no—
238C2×C8—942no—
239C2×C8—941no—
240C2×C8—941no—
241C2×C8—1881no—
242C2×C8—1882no—
243C2×C8—1882no—
244C2×C8—1881no—
245C2×C8—1881no—
246C4⋊C4—22no—
247C22⋊C4—22no—
248C22⋊C4—22no—
249C42—21no—
250C42—21no—
251C23—22no—
252C23—21no—
253C23—22no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23C2×Dic47⋊C411yescharacteristic
259C23#1309400112yes—
260D4—44no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4C22⋊C4×D4711yescharacteristic
277C2×C4#1309398011yescharacteristic
278C2×C4#1309399612yes—
279C8—82no—
280C8—472no—
281C8—942no—
282C8—941no—
283C8—3762no—
284C8—1881no—
285C8—1881no—
286C22—21no—
287C22—21no—
288C22—22no—
289C22—22no—
290C22—22no—
291C22—44no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—82no—
296C22#1568898111yescharacteristic
297C22#1568917112yes—
298C4—22no—
299C4—21no—
300C4—41no—
301C4—41no—
302C4—41no—
303C4#99×D4711yescharacteristic
304C4#1568909511yescharacteristic
305C2—22no—
306C2—42no—
307C2—81no—
308C2#2024337411yescharacteristic
309C2#2024342811yescharacteristic
310C2#2024343611yescharacteristic
subgroupG/H
centerC22#15688981
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1465C2
socleC22×C47#380
prime radicalC22×D4×C47Q8
cosocleC23×C4×C47C23
hypercenter#1465D47
hypocenterC47#8613
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1465

Complements

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

NH ≅ G/Ntype
2. #20243424C2semidirect
4. #20241264C2semidirect
178. C47#8613semidirect

Maximal subgroups

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

subgroupindex
C2×#156891952
#202434242
#202417262
#202412642
C47×#14652
#861347