Finite Group Databasebeta

← #26110287

Subgroups — #26110287

Order 12032.

460 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 103 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#20252417C211yesmaximal, characteristic
2#20243593C212yesmaximal, complemented
3#20241706C211yesmaximal, characteristic
4#20241276C212yesmaximal, complemented
5C47×#1870C211yesmaximal, characteristic
6#15689195—22no—
7#15689195—22no—
8#15689180—21no—
9C2×#13093959—22no—
10#15689087C2212yes—
11C2×C22⋊(C47⋊C8)—21no—
12C2×C22⋊(C47⋊C8)C2212yes—
13#15688887—22no—
14#15688849C412yes—
15#15688847C412yes—
16C22×M16×C47C2211yescharacteristic
17C2×C4×D4×C47C2211yescharacteristic
18C2×C22⋊C8×C47—21no—
19C2×C22⋊C8×C47C2211yescharacteristic
20C47×#336—21no—
21C2×#10625329D412yes—
22C2×#10625329D412yes—
23C22×C47⋊C8—22no—
24C22×C47⋊C8—21no—
25C22×C47⋊C8—21no—
26#13093959—44no—
27C4⋊(C47⋊C8)—21no—
28C4⋊(C47⋊C8)—21no—
29C22⋊(C47⋊C8)—22no—
30C22⋊(C47⋊C8)—22no—
31C22⋊(C47⋊C8)—22no—
32C22⋊(C47⋊C8)—22no—
33C22⋊(C47⋊C8)C2×C414yes—
34#13093935—22no—
35C22×D4×C47Q811yescharacteristic
36C23×C4×C47D411yescharacteristic
37C23×C4×C47C2311yescharacteristic
38C2×M16×C47—22no—
39C2×M16×C47—42no—
40C22×C8×C47—21no—
41C22×C8×C47—21no—
42C4×D4×C47—22no—
43C4×D4×C47—22no—
44C2×C4⋊C4×C47C2×C411yescharacteristic
45C2×C22⋊C4×C47C2×C411yescharacteristic
46C2×C22⋊C4×C47Q811yescharacteristic
47C2×C42×C47D411yescharacteristic
48C22⋊C8×C47—22no—
49C22⋊C8×C47—42no—
50C24×C47C4⋊C411yescharacteristic
51C24×C47C2×Q811yescharacteristic
52C2×D4×C47—22no—
53C2×D4×C47—22no—
54C2×D4×C47—41no—
55C22×C4×C47—21no—
56C22×C4×C47—21no—
57C22×C4×C47—41no—
58C22×C4×C47—41no—
59C22×C4×C47—41no—
60C22×C4×C47—41no—
61C22×C4×C47C4⋊C411yescharacteristic
62C22×C4×C47C4⋊C411yescharacteristic
63C22×C4×C47C4⋊C411yescharacteristic
64C22×C4×C47C22×C411yescharacteristic
65C22×C4×C47C2×D412yes—
66C22×C4×C47C2×D411yescharacteristic
67C22×C4×C47#4112yes—
68M16×C47—24no—
69M16×C47—42no—
70C2×C8×C47—22no—
71C2×C8×C47—41no—
72C2×C8×C47—41no—
73C2×C8×C47—41no—
74C2×C8×C47—41no—
75C2×C8×C47—41no—
76C4⋊C4×C47—22no—
77C22⋊C4×C47—22no—
78C22⋊C4×C47—22no—
79C42×C47—21no—
80C42×C47—21no—
81#10625329—42no—
82#10625329—42no—
83C2×C47⋊C8—24no—
84C2×C47⋊C8—22no—
85C2×C47⋊C8—22no—
86C2×C47⋊C8—42no—
87C2×C47⋊C8—42no—
88C2×C47⋊C8—81no—
89C2×C47⋊C8—81no—
90C23×C47—21no—
91C23×C47—21no—
92C23×C47—21no—
93C23×C47—21no—
94C23×C47—41no—
95C23×C47—41no—
96C23×C47—41no—
97C23×C47—41no—
98C23×C47#10812yescomplemented
99C23×C47C2×C4⋊C411yescharacteristic
100C23×C47C22⋊Q812yes—
101D4×C47—44no—
102C2×C4×C47—22no—
103C2×C4×C47—21no—
104C2×C4×C47—22no—
105C2×C4×C47—21no—
106C2×C4×C47—21no—
107C2×C4×C47—21no—
108C2×C4×C47—21no—
109C2×C4×C47—21no—
110C2×C4×C47—21no—
111C2×C4×C47—41no—
112C2×C4×C47—42no—
113C2×C4×C47—41no—
114C2×C4×C47—41no—
115C2×C4×C47—41no—
116C2×C4×C47—41no—
117C2×C4×C47—81no—
118C2×C4×C47C4×D412yes—
119C2×C4×C47#12011yescharacteristic
120C2×C4×C47#12311yescharacteristic
121C8×C47—42no—
122C8×C47—81no—
123C47⋊C8—44no—
124C47⋊C8—42no—
125C47⋊C8—42no—
126#8859—471nomaximal, Sylow
127C22×C47—21no—
128C22×C47—21no—
129C22×C47—22no—
130C22×C47—22no—
131C22×C47—22no—
132C22×C47—22no—
133C22×C47—42no—
134C22×C47—41no—
135C22×C47—41no—
136C22×C47—81no—
137C22×C47—81no—
138C22×C47—81no—
139C22×C47#38511yescharacteristic
140C22×C47C2×#10811yescharacteristic
141C22×C47#43011yescharacteristic
142C4×C47—22no—
143C4×C47—21no—
144C4×C47—41no—
145C4×C47—41no—
146C4×C47—41no—
147C4×C47B(4,𝔽2)11yescharacteristic
148C4×C47#45811yescharacteristic
149#2973—471no—
150#1870D4711yescharacteristic
151#1855—472no—
152#1459—471no—
153#1298—472no—
154C2×C47—21no—
155C2×C47—21no—
156C2×C47—42no—
157C2×C47—42no—
158C2×C47#185511yescharacteristic
159C2×C47#185811yescharacteristic
160C2×C47#186711yescharacteristic
161C22×M16C2×D4711yescharacteristic
162C2×C4×D4C2×D4711yescharacteristic
163#435—942no—
164#435—942no—
165#432—941no—
166C2×#108—942no—
167#407—472no—
168C2×C22⋊C8—21no—
169C2×C22⋊C8—472no—
170C2×C22⋊C8—941no—
171C2×C22⋊C8C2×D4711yescharacteristic
172#341—942no—
173#336—21no—
174#324—472no—
175#323—472no—
176C47#885911yescharacteristic, complemented, Sylow
177C22×D4Dic9411yescharacteristic
178C23×C4C22⋊D4711yescharacteristic
179C23×C4C22×D4711yescharacteristic
180C2×M16—22no—
181C2×M16—42no—
182C2×M16—472no—
183C2×M16—472no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—941no—
187C22×C8—941no—
188C22×C8—942no—
189C4×D4—22no—
190C4×D4—22no—
191C2×C4⋊C4C4×D4711yescharacteristic
192C2×C22⋊C4Dic9411yescharacteristic
193C2×C22⋊C4C4×D4711yescharacteristic
194C2×C42C22⋊D4711yescharacteristic
195#108—1884no—
196C4⋊C8—941no—
197C4⋊C8—941no—
198C22⋊C8—22no—
199C22⋊C8—42no—
200C22⋊C8—474no—
201C22⋊C8—942no—
202C22⋊C8—942no—
203C22⋊C8—942no—
204C22⋊C8—942no—
205#97—942no—
206C24Dic47⋊C411yescharacteristic
207C24C2×Dic9411yescharacteristic
208C2×D4—22no—
209C2×D4—22no—
210C2×D4—41no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4Dic47⋊C411yescharacteristic
218C22×C4Dic47⋊C411yescharacteristic
219C22×C4Dic47⋊C411yescharacteristic
220C22×C4C2×C4×D4711yescharacteristic
221C22×C4D4×D4712yes—
222C22×C4#1062535212yes—
223C22×C4C2×C22⋊D4711yescharacteristic
224M16—24no—
225M16—42no—
226M16—1882no—
227M16—1882no—
228C2×C8—22no—
229C2×C8—41no—
230C2×C8—41no—
231C2×C8—41no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—944no—
235C2×C8—942no—
236C2×C8—942no—
237C2×C8—3761no—
238C2×C8—3761no—
239C2×C8—1882no—
240C2×C8—1882no—
241C4⋊C4—22no—
242C22⋊C4—22no—
243C22⋊C4—22no—
244C42—21no—
245C42—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23#1309395912yescomplemented
255C23C2×Dic47⋊C411yescharacteristic
256C23#1309400112yes—
257D4—44no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—81no—
274C2×C4(C22⋊D47)⋊C412yes—
275C2×C4#1309399211yescharacteristic
276C2×C4#1309400511yescharacteristic
277C8—42no—
278C8—81no—
279C8—1884no—
280C8—1882no—
281C8—1882no—
282C22—21no—
283C22—21no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—22no—
288C22—42no—
289C22—41no—
290C22—41no—
291C22—81no—
292C22—81no—
293C22—81no—
294C22#1568899511yescharacteristic
295C22C2×#1309395911yescharacteristic
296C22#1568917111yescharacteristic
297C4—21no—
298C4—22no—
299C4—41no—
300C4—41no—
301C4—41no—
302C4#1568930911yescharacteristic
303C4#1568931211yescharacteristic
304C2—21no—
305C2—21no—
306C2—42no—
307C2—42no—
308C2#2024359211yescharacteristic
309C2#2024360711yescharacteristic
310C2#2024366811yescharacteristic
subgroupG/H
centerC2×C4#13093992
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1870C2
socleC22×C47#385
prime radicalC22×D4×C47Q8
cosocleC23×C4×C47C23
hypercenter#1870D47
hypocenterC47#8859
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1870

Complements

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

NH ≅ G/Ntype
2. #20243593C2semidirect
4. #20241276C2semidirect
98. C23×C47#108semidirect
176. C47#8859semidirect
254. C23#13093959semidirect

Maximal subgroups

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

subgroupindex
#202524172
#202435932
#202417062
#202412762
C47×#18702
#885947