Finite Group Databasebeta

← #26111272

Subgroups — #26111272

Order 12032.

756 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 131 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×#15689043C211yesmaximal, characteristic
2#20243322C212yesmaximal
3C2×#15688849C211yesmaximal, characteristic
4#20241266C212yesmaximal
5C47×#2313C211yesmaximal, characteristic
6C23×Dic94C2211yescharacteristic
7C22×Dic47⋊C4—21no—
8C2×#13093947—24no—
9C2×C22⋊(C47⋊C8)C2212yes—
10#15689043—22no—
11#15689043C414yescomplemented
12C2×C47⋊#95—21no—
13C2×C47⋊#95C2211yescharacteristic
14#15688849—22no—
15C22×C4⋊C4×C47C2211yescharacteristic
16C47×#387—24no—
17C47×#381—21no—
18C47×#380C2212yes—
19C4×C22⋊C4×C47—21no—
20C22×Dic94—22no—
21C22×Dic94—22no—
22C22×Dic94—21no—
23C22×Dic94—44no—
24C22×Dic94C2×C412yes—
25C23×Dic47—21no—
26C23×Dic47—21no—
27C22×C47⋊C8—22no—
28C2×Dic47⋊C4—24no—
29C2×Dic47⋊C4—44no—
30#13093947—48no—
31C22⋊(C47⋊C8)—22no—
32C22⋊(C47⋊C8)—22no—
33C47⋊#95—24no—
34C47⋊#95—21no—
35C47⋊#95C2×C412yes—
36C23×C4×C47D411yescharacteristic
37C23×C4×C47D411yescharacteristic
38C23×C4×C47C2311yescharacteristic
39C2×C4⋊C4×C47—22no—
40C2×C4⋊C4×C47—22no—
41C2×C4⋊C4×C47C2×C412yes—
42C2×C4⋊C4×C47D412yes—
43C2×C4⋊C4×C47D414yes—
44C2×C22⋊C4×C47—22no—
45C2×C42×C47—22no—
46#95×C47—22no—
47#95×C47—24no—
48#95×C47—22no—
49C24×C47C2×D411yescharacteristic
50C22×C4×C47—21no—
51C22×C4×C47—21no—
52C22×C4×C47—21no—
53C22×C4×C47—22no—
54C22×C4×C47—22no—
55C22×C4×C47—21no—
56C22×C4×C47—24no—
57C22×C4×C47—42no—
58C22×C4×C47—42no—
59C22×C4×C47C22⋊C412yes—
60C22×C4×C47C22⋊C412yes—
61C22×C4×C47C22×C411yescharacteristic
62C22×C4×C47C2×D412yes—
63C22×C4×C47C2×D411yescharacteristic
64C22×C4×C47#4112yes—
65C22×Dic47—24no—
66C22×Dic47—24no—
67C22×Dic47—22no—
68C22×Dic47—22no—
69C22×Dic47—42no—
70C22×Dic47—42no—
71C22×Dic47—41no—
72C22×Dic47—41no—
73C22×Dic47—42no—
74C2×Dic94—28no—
75C2×Dic94—24no—
76C2×Dic94—44no—
77C2×Dic94—48no—
78C2×Dic94—44no—
79C2×Dic94—44no—
80C2×Dic94—42no—
81C4⋊C4×C47—24no—
82C4⋊C4×C47—24no—
83C4⋊C4×C47—44no—
84C22⋊C4×C47—42no—
85C22⋊C4×C47—42no—
86C42×C47—82no—
87Dic47⋊C4—48no—
88C2×C47⋊C8—44no—
89C2×C47⋊C8—42no—
90C23×C47—22no—
91C23×C47—22no—
92C23×C47C2×C22⋊C411yescharacteristic
93C23×C47#12011yescharacteristic
94C23×C47#12311yescharacteristic
95C23×C47#13512yes—
96C23×C47#13712yes—
97C2×C4×C47—24no—
98C2×C4×C47—22no—
99C2×C4×C47—22no—
100C2×C4×C47—22no—
101C2×C4×C47—22no—
102C2×C4×C47—41no—
103C2×C4×C47—41no—
104C2×C4×C47—42no—
105C2×C4×C47—42no—
106C2×C4×C47—42no—
107C2×C4×C47—42no—
108C2×C4×C47—42no—
109C2×C4×C47—44no—
110C2×C4×C47—44no—
111C2×C4×C47—44no—
112C2×C4×C47—81no—
113C2×C4×C47C4×D412yes—
114C2×C4×C47#12112yes—
115C2×Dic47—24no—
116C2×Dic47—24no—
117C2×Dic47—42no—
118C2×Dic47—42no—
119C2×Dic47—44no—
120C2×Dic47—48no—
121C2×Dic47—42no—
122C2×Dic47—42no—
123C2×Dic47—44no—
124Dic94—28no—
125Dic94—48no—
126Dic94—44no—
127C47⋊C8—82no—
128#9014—471nomaximal, Sylow
129C22×C47—24no—
130C22×C47—24no—
131C22×C47—21no—
132C22×C47—21no—
133C22×C47—21no—
134C22×C47—21no—
135C22×C47#38611yescharacteristic
136C22×C47#41611yescharacteristic
137C22×C47#41911yescharacteristic
138C22×C47#45212yes—
139C22×C47#45612yes—
140C22×C47B(4,𝔽2)12yes—
141C22×C47#48512yes—
142C4×C47—22no—
143C4×C47—22no—
144C4×C47—42no—
145C4×C47—42no—
146C4×C47—84no—
147Dic47—44no—
148Dic47—44no—
149C2×#395—471no—
150#2313D4711yescharacteristic
151#1787—472no—
152C2×#324—471no—
153#1294—472no—
154C2×C47—21no—
155C2×C47—21no—
156C2×C47—22no—
157C2×C47#160314yes—
158C2×C47#187311yescharacteristic
159C2×C47#188611yescharacteristic
160C2×C47#188911yescharacteristic
161C23×Q8—471no—
162C22×C4⋊C4—941no—
163C22×C4⋊C4C2×D4711yescharacteristic
164C2×#103—944no—
165C2×C22⋊C8—472no—
166#395—474no—
167#395—942no—
168#387—24no—
169#381—21no—
170#380C2×D4712yes—
171C4×C22⋊C4—21no—
172C2×#95—471no—
173C2×#95—941no—
174#324—942no—
175C47#901411yescharacteristic, complemented, Sylow
176C22×Q8—472no—
177C22×Q8—941no—
178C22×Q8—942no—
179C22×Q8—942no—
180C22×Q8—1884no—
181C23×C4—941no—
182C23×C4—941no—
183C23×C4C22⋊D4711yescharacteristic
184C23×C4C22⋊D4711yescharacteristic
185C23×C4C22×D4711yescharacteristic
186C22×C8—942no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—944no—
190C2×C4⋊C4—1884no—
191C2×C4⋊C4C4×D4712yes—
192C2×C4⋊C4D18812yes—
193C2×C4⋊C4C22⋊D4714yes—
194C2×C22⋊C4—22no—
195C2×C42—22no—
196#103—1888no—
197C22⋊C8—942no—
198C22⋊C8—942no—
199#95—22no—
200#95—24no—
201#95—22no—
202#95—472no—
203#95—944no—
204#95—941no—
205C24C2×C22⋊D4711yescharacteristic
206C2×Q8—948no—
207C2×Q8—944no—
208C2×Q8—1884no—
209C2×Q8—1884no—
210C2×Q8—1884no—
211C2×Q8—1888no—
212C2×Q8—1882no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—21no—
219C22×C4—24no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—944no—
223C22×C4—944no—
224C22×C4—942no—
225C22×C4—942no—
226C22×C4—1882no—
227C22×C4—1882no—
228C22×C4—1881no—
229C22×C4—1881no—
230C22×C4—1882no—
231C22×C4(C2×D47)⋊C412yes—
232C22×C4(C2×D47)⋊C412yes—
233C22×C4C2×C4×D4711yescharacteristic
234C22×C4C2×D18811yescharacteristic
235C22×C4#1062535012yes—
236C22×C4C2×C22⋊D4712yes—
237C2×C8—1884no—
238C2×C8—1882no—
239C4⋊C4—24no—
240C4⋊C4—24no—
241C4⋊C4—44no—
242C4⋊C4—1888no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C42—82no—
246C23—22no—
247C23—22no—
248C23C2×(C2×D47)⋊C411yescharacteristic
249C23#1309399311yescharacteristic
250C23#1309400711yescharacteristic
251C23#1309404412yes—
252C23#1309405412yes—
253Q8—948no—
254Q8—1888no—
255Q8—1884no—
256C2×C4—24no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—44no—
271C2×C4—81no—
272C2×C4—944no—
273C2×C4—944no—
274C2×C4—1882no—
275C2×C4—1888no—
276C2×C4—1882no—
277C2×C4—1884no—
278C2×C4—1882no—
279C2×C4—1884no—
280C2×C4—1882no—
281C2×C4C4×C22⋊D4712yes—
282C2×C4#1309399812yes—
283C8—3762no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—24no—
289C22—24no—
290C22#1568900211yescharacteristic
291C22#1568911711yescharacteristic
292C22#1568913111yescharacteristic
293C22#1568928512yes—
294C22#1568930612yes—
295C22#1568931012yes—
296C22#1568946412yes—
297C4—22no—
298C4—22no—
299C4—42no—
300C4—42no—
301C4—84no—
302C4—1884no—
303C4—1884no—
304C2—21no—
305C2—21no—
306C2—22no—
307C2#2024235114yes—
308C2#2024370011yescharacteristic
309C2#2024378111yescharacteristic
310C2#2024379611yescharacteristic
subgroupG/H
centerC23#13093993
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2313C2
socleC23×C47#120
prime radicalC24×C47C2×D4
cosocleC23×C4×C47C23
hypercenter#2313D47
hypocenterC47#9014
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2313

Complements

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

NH ≅ G/Ntype
11. #15689043C4semidirect
175. C47#9014semidirect

Maximal subgroups

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

subgroupindex
C2×#156890432
#202433222
C2×#156888492
#202412662
C47×#23132
#901447