Finite Group Databasebeta

← #26111283

Subgroups — #26111283

Order 12032.

836 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 135 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#20245986C211yesmaximal, characteristic
2#20243323C212yesmaximal, complemented
3C2×#15688847C211yesmaximal, characteristic
4#20241267C212yesmaximal, complemented
5C2×C47×#393C211yesmaximal, characteristic
6C2×#13093948—24no—
7C2×C22⋊(C47⋊C8)C2212yes—
8#15689016—24no—
9#15688983C2212yes—
10#15688979—21no—
11C4×C22⋊Dic47—21no—
12#15688847—22no—
13C23×Q8×C47C2211yescharacteristic
14C22×C22⋊C4×C47—21no—
15C22×C22⋊C4×C47C2211yescharacteristic
16C47×#393—22no—
17C47×#393C414yescomplemented
18C2×#95×C47—21no—
19C2×#95×C47C2211yescharacteristic
20C22×C47⋊C8—22no—
21C2×C4⋊Dic47D414yes—
22C2×C22⋊Dic47—22no—
23C2×C22⋊Dic47—22no—
24C2×C22⋊Dic47—22no—
25C2×C4×Dic47—22no—
26#13093948—48no—
27C22⋊(C47⋊C8)—22no—
28C22⋊(C47⋊C8)—22no—
29C47⋊#95—22no—
30C47⋊#95—22no—
31C25×C47D411yescharacteristic
32C22×Q8×C47—21no—
33C22×Q8×C47—22no—
34C22×Q8×C47—22no—
35C22×Q8×C47—44no—
36C22×Q8×C47C2×C412yes—
37C23×C4×C47—21no—
38C23×C4×C47—21no—
39C23×C4×C47D411yescharacteristic
40C23×C4×C47C2311yescharacteristic
41C2×C22⋊C4×C47—22no—
42C2×C22⋊C4×C47—22no—
43C2×C22⋊C4×C47—24no—
44C2×C22⋊C4×C47—44no—
45C2×C22⋊C4×C47C2×C412yes—
46C2×C22⋊C4×C47D412yes—
47#95×C47—24no—
48#95×C47—21no—
49#95×C47C2×C412yes—
50C24×C47—21no—
51C24×C47—22no—
52C24×C47—21no—
53C24×C47—22no—
54C24×C47—42no—
55C24×C47—42no—
56C24×C47C22⋊C412yes—
57C24×C47C2×D411yescharacteristic
58C2×Q8×C47—28no—
59C2×Q8×C47—24no—
60C2×Q8×C47—44no—
61C2×Q8×C47—44no—
62C2×Q8×C47—44no—
63C2×Q8×C47—48no—
64C2×Q8×C47—42no—
65C22×C4×C47—24no—
66C22×C4×C47—22no—
67C22×C4×C47—21no—
68C22×C4×C47—21no—
69C22×C4×C47—22no—
70C22×C4×C47—42no—
71C22×C4×C47—42no—
72C22×C4×C47—41no—
73C22×C4×C47—41no—
74C22×C4×C47—42no—
75C22×C4×C47—42no—
76C22×C4×C47—42no—
77C22×C4×C47—42no—
78C22×C4×C47C22⋊C412yes—
79C22×C4×C47C22×C411yescharacteristic
80C22×C4×C47C2×D412yes—
81C22×C4×C47C2×D411yescharacteristic
82C22×C4×C47#4112yes—
83C22×Dic47—24no—
84C22⋊C4×C47—24no—
85C22⋊C4×C47—24no—
86C22⋊C4×C47—48no—
87C22⋊Dic47—42no—
88C22⋊Dic47—42no—
89C22⋊Dic47—82no—
90C22⋊Dic47—82no—
91C4⋊Dic47—44no—
92C4×Dic47—82no—
93C2×C47⋊C8—44no—
94C2×C47⋊C8—42no—
95C23×C47—22no—
96C23×C47—22no—
97C23×C47—22no—
98C23×C47—22no—
99C23×C47—22no—
100C23×C47—42no—
101C23×C47—42no—
102C23×C47—48no—
103C23×C47—42no—
104C23×C47—42no—
105C23×C47—42no—
106C23×C47—42no—
107C23×C47—81no—
108C23×C47—81no—
109C23×C47—84no—
110C23×C47C2×C22⋊C411yescharacteristic
111C23×C47#12011yescharacteristic
112C23×C47#12311yescharacteristic
113C23×C47#13512yes—
114C23×C47#13712yes—
115Q8×C47—28no—
116Q8×C47—48no—
117Q8×C47—44no—
118C2×C4×C47—24no—
119C2×C4×C47—24no—
120C2×C4×C47—22no—
121C2×C4×C47—41no—
122C2×C4×C47—41no—
123C2×C4×C47—42no—
124C2×C4×C47—42no—
125C2×C4×C47—44no—
126C2×C4×C47—42no—
127C2×C4×C47—42no—
128C2×C4×C47—44no—
129C2×C4×C47—42no—
130C2×C4×C47—42no—
131C2×C4×C47—42no—
132C2×C4×C47—42no—
133C2×C4×C47—81no—
134C2×C4×C47—84no—
135C2×C4×C47C4×D412yes—
136C2×C4×C47#12112yes—
137C2×Dic47—44no—
138C2×Dic47—44no—
139C2×Dic47—44no—
140C47⋊C8—82no—
141#9016—471nomaximal, Sylow
142C22×C47—24no—
143C22×C47—24no—
144C22×C47—21no—
145C22×C47—21no—
146C22×C47—21no—
147C22×C47—21no—
148C22×C47—44no—
149C22×C47—42no—
150C22×C47—42no—
151C22×C47—42no—
152C22×C47—82no—
153C22×C47—84no—
154C22×C47—84no—
155C22×C47#38611yescharacteristic
156C22×C47#41611yescharacteristic
157C22×C47#41911yescharacteristic
158C22×C47#45212yes—
159C22×C47#45512yes—
160C22×C47B(4,𝔽2)12yes—
161C22×C47#48512yes—
162C4×C47—22no—
163C4×C47—22no—
164C4×C47—42no—
165C4×C47—44no—
166C4×C47—82no—
167Dic47—84no—
168C2×#393D4711yescharacteristic
169#2312—471no—
170#1787—472no—
171C2×#323—471no—
172#1294—472no—
173C2×C47—21no—
174C2×C47—21no—
175C2×C47—22no—
176C2×C47—82no—
177C2×C47#160614yes—
178C2×C47#187311yescharacteristic
179C2×C47#188511yescharacteristic
180C2×C47#188911yescharacteristic
181C23×Q8C2×D4711yescharacteristic
182C22×C22⋊C4—21no—
183C22×C22⋊C4C2×D4711yescharacteristic
184C2×#103—944no—
185C2×C22⋊C8—472no—
186#393—22no—
187#393Dic4714yescomplemented
188#388—944no—
189#380—472no—
190#379—941no—
191C4×C22⋊C4—941no—
192C2×#95—21no—
193C2×#95C2×D4711yescharacteristic
194#323—942no—
195C47#901611yescharacteristic, complemented, Sylow
196C25C22⋊D4711yescharacteristic
197C22×Q8—21no—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—44no—
201C22×Q8C2×Dic4712yes—
202C23×C4—21no—
203C23×C4—21no—
204C23×C4C22⋊D4711yescharacteristic
205C23×C4C22×D4711yescharacteristic
206C22×C8—942no—
207C2×C4⋊C4—474no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—942no—
213C2×C22⋊C4—942no—
214C2×C22⋊C4—942no—
215C2×C22⋊C4C2×Dic4712yes—
216C2×C22⋊C4C22⋊D4712yes—
217C2×C42—942no—
218#103—1888no—
219C22⋊C8—942no—
220C22⋊C8—942no—
221#95—24no—
222#95—21no—
223#95—942no—
224#95—942no—
225#95C2×Dic4712yes—
226C24—21no—
227C24—22no—
228C24—21no—
229C24—22no—
230C24—42no—
231C24—42no—
232C24C22⋊Dic4712yes—
233C24C2×C22⋊D4711yescharacteristic
234C2×Q8—28no—
235C2×Q8—24no—
236C2×Q8—44no—
237C2×Q8—44no—
238C2×Q8—44no—
239C2×Q8—48no—
240C2×Q8—42no—
241C22×C4—24no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—944no—
255C22×C4C22⋊Dic4712yes—
256C22×C4D4×D4712yes—
257C22×C4#1062535212yes—
258C22×C4C22×Dic4711yescharacteristic
259C22×C4C2×C22⋊D4711yescharacteristic
260C2×C8—1884no—
261C2×C8—1882no—
262C4⋊C4—1884no—
263C22⋊C4—24no—
264C22⋊C4—24no—
265C22⋊C4—48no—
266C22⋊C4—3762no—
267C22⋊C4—3762no—
268C22⋊C4—1882no—
269C22⋊C4—1882no—
270C42—3762no—
271C23—22no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—42no—
277C23—42no—
278C23—48no—
279C23—42no—
280C23—42no—
281C23—42no—
282C23—42no—
283C23—81no—
284C23—81no—
285C23—84no—
286C23C2×C22⋊Dic4711yescharacteristic
287C23#1309399211yescharacteristic
288C23#1309400511yescharacteristic
289C23#1309404412yes—
290C23#1309405512yes—
291Q8—28no—
292Q8—48no—
293Q8—44no—
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—22no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—44no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—44no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—81no—
310C2×C4—84no—
311C2×C4—1884no—
312C2×C4—1884no—
313C2×C4—1884no—
314C2×C4D4×Dic4712yes—
315C2×C4#1309399612yes—
316C8—3762no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—24no—
322C22—24no—
323C22—44no—
324C22—42no—
325C22—42no—
326C22—42no—
327C22—82no—
328C22—84no—
329C22—84no—
330C22#1568900311yescharacteristic
331C22#1568911911yescharacteristic
332C22#1568913211yescharacteristic
333C22#1568928312yes—
334C22#1568930012yes—
335C22#1568930912yes—
336C22#1568946112yes—
337C4—22no—
338C4—22no—
339C4—42no—
340C4—44no—
341C4—82no—
342C4—3764no—
343C2—21no—
344C2—21no—
345C2—22no—
346C2—82no—
347C2#2024236614yes—
348C2#2024370111yescharacteristic
349C2#2024377711yescharacteristic
350C2#2024379711yescharacteristic
subgroupG/H
centerC23#13093992
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC2×C47×#393C2
socleC23×C47#120
prime radicalC25×C47D4
cosocleC23×C4×C47C23
hypercenterC2×#393D47
hypocenterC47#9016
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#393

Complements

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

NH ≅ G/Ntype
2. #20243323C2semidirect
4. #20241267C2semidirect
17. C47×#393C4semidirect
187. #393Dic47semidirect
195. C47#9016semidirect

Maximal subgroups

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

subgroupindex
#202459862
#202433232
C2×#156888472
#202412672
C2×C47×#3932
#901647