Finite Group Databasebeta

← #26127320

Subgroups — #26127320

Order 12032.

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

normal subgroups only

This group has 440 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#20243384C211yesmaximal, characteristic, complemented
2#20241712C211yesmaximal, characteristic, complemented
3#20241394C212yesmaximal, complemented
4#20241391C212yesmaximal
5C47×#2536C211yesmaximal, characteristic
6#15689176—21no—
7#15689147—21no—
8C2×#13093943—21no—
9#15688892C2212yes—
10#15688849—21no—
11#15688849C2212yes—
12C2×C22⋊Q8×C47—21no—
13C2×C4×Q8×C47C2211yescharacteristic
14C2×C4×D4×C47C411yescharacteristic
15C47×#398C411yescharacteristic
16C47×#398C2211yescharacteristic
17C47×#394C411yescharacteristic
18C47×#394C2211yescharacteristic
19C47×#388—21no—
20C47×#388—21no—
21C47×#382C411yescharacteristic
22C47×#380—21no—
23C2×#10625329—22no—
24C4⋊(C47⋊C8)—41no—
25C4⋊(C47⋊C8)—41no—
26C4⋊(C47⋊C8)—41no—
27#13093943—42no—
28C22⋊(C47⋊C8)—22no—
29C4×C47⋊C8—41no—
30C22×Q8×C47D411yescharacteristic
31C22×D4×C47C2×C411yescharacteristic
32C23×C4×C47—21no—
33C22⋊Q8×C47—42no—
34C22⋊Q8×C47—42no—
35C4×Q8×C47—41no—
36C4×Q8×C47—41no—
37C4×D4×C47—41no—
38C4×D4×C47—41no—
39C2×C4⋊C4×C47—21no—
40C2×C4⋊C4×C47—21no—
41C2×C4⋊C4×C47—21no—
42C2×C4⋊C4×C47C2×C411yescharacteristic
43C2×C4⋊C4×C47C2×C411yescharacteristic
44C2×C4⋊C4×C47D411yescharacteristic
45C2×C4⋊C4×C47C2311yescharacteristic
46C2×C22⋊C4×C47—21no—
47C2×C22⋊C4×C47—21no—
48C2×C22⋊C4×C47—21no—
49C2×C22⋊C4×C47—21no—
50C2×C22⋊C4×C47—21no—
51C2×C42×C47C2×C411yescharacteristic
52C2×C42×C47D411yescharacteristic
53C2×C42×C47D411yescharacteristic
54#95×C47—21no—
55#95×C47—21no—
56#95×C47—21no—
57#95×C47C2×C411yescharacteristic
58#95×C47C2×C411yescharacteristic
59C24×C47—21no—
60C2×Q8×C47—41no—
61C2×Q8×C47—41no—
62C2×Q8×C47D812yes—
63C2×Q8×C47SD1612yes—
64C2×D4×C47—41no—
65C2×D4×C47—42no—
66C22×C4×C47—21no—
67C22×C4×C47—21no—
68C22×C4×C47—21no—
69C22×C4×C47—42no—
70C22×C4×C47—42no—
71C22×C4×C47—42no—
72C22×C4×C47C22⋊C411yescharacteristic
73C22×C4×C47C22⋊C411yescharacteristic
74C22×C4×C47C22⋊C411yescharacteristic
75C22×C4×C47C22⋊C411yescharacteristic
76C22×C4×C47C22×C411yescharacteristic
77C22×C4×C47C2×D411yescharacteristic
78C22×C4×C47C2×D411yescharacteristic
79C4⋊C4×C47—41no—
80C4⋊C4×C47—41no—
81C4⋊C4×C47—41no—
82C4⋊C4×C47—41no—
83C4⋊C4×C47—42no—
84C4⋊C4×C47—42no—
85C4⋊C4×C47—41no—
86C4⋊C4×C47—41no—
87C22⋊C4×C47—42no—
88C22⋊C4×C47—42no—
89C22⋊C4×C47—41no—
90C22⋊C4×C47—41no—
91C22⋊C4×C47—42no—
92C22⋊C4×C47—41no—
93C22⋊C4×C47—41no—
94C42×C47—21no—
95C42×C47—21no—
96C42×C47—21no—
97C42×C47—21no—
98C42×C47—21no—
99C42×C47—21no—
100#10625329—42no—
101#10625329—42no—
102C2×C47⋊C8—42no—
103C2×C47⋊C8—42no—
104C23×C47—22no—
105C23×C47—41no—
106C23×C47—41no—
107C23×C47—41no—
108C23×C47—41no—
109C23×C47—41no—
110C23×C47—41no—
111C23×C47C2×C22⋊C411yescharacteristic
112Q8×C47—42no—
113Q8×C47—42no—
114D4×C47—42no—
115D4×C47—42no—
116C2×C4×C47—21no—
117C2×C4×C47—21no—
118C2×C4×C47—21no—
119C2×C4×C47—21no—
120C2×C4×C47—21no—
121C2×C4×C47—21no—
122C2×C4×C47—21no—
123C2×C4×C47—21no—
124C2×C4×C47—21no—
125C2×C4×C47—41no—
126C2×C4×C47—41no—
127C2×C4×C47—41no—
128C2×C4×C47—41no—
129C2×C4×C47—41no—
130C2×C4×C47—41no—
131C2×C4×C47—41no—
132C2×C4×C47—41no—
133C2×C4×C47—41no—
134C2×C4×C47—41no—
135C2×C4×C47—41no—
136C2×C4×C47—41no—
137C2×C4×C47—41no—
138C2×C4×C47—84no—
139C2×C4×C47#9911yescharacteristic
140C2×C4×C47#9911yescharacteristic
141C2×C4×C47#10212yes—
142C2×C4×C47#10212yes—
143C2×C4×C47C2×D811yescharacteristic
144C2×C4×C47C2×SD1611yescharacteristic
145C47⋊C8—82no—
146C47⋊C8—82no—
147#11879—471nomaximal, Sylow
148C22×C47—21no—
149C22×C47—21no—
150C22×C47—42no—
151C22×C47—42no—
152C22×C47—42no—
153C22×C47—81no—
154C22×C47—81no—
155C22×C47—81no—
156C22×C47—81no—
157C22×C47C2×#9911yescharacteristic
158C22×C47C2×#10211yescharacteristic
159C22×C47#42111yescharacteristic
160C4×C47—21no—
161C4×C47—21no—
162C4×C47—21no—
163C4×C47—22no—
164C4×C47—22no—
165C4×C47—21no—
166C4×C47—41no—
167C4×C47—41no—
168C4×C47—41no—
169C4×C47—81no—
170C4×C47—81no—
171C4×C47—81no—
172#2536D4711yescharacteristic
173#1801—471no—
174#1461—471no—
175#1346—472no—
176#1345—472no—
177C2×C47—21no—
178C2×C47—21no—
179C2×C47—82no—
180C2×C47#151911yescharacteristic
181C2×C47#212011yescharacteristic
182C2×C47#213311yescharacteristic
183C2×C22⋊Q8—21no—
184C2×C4×Q8C2×D4711yescharacteristic
185C2×C4×D4Dic4711yescharacteristic
186#431—941no—
187#423—941no—
188C2×#101—941no—
189#398Dic4711yescharacteristic
190#398C2×D4711yescharacteristic
191#394Dic4711yescharacteristic
192#394C2×D4711yescharacteristic
193#388—21no—
194#388—21no—
195#382Dic4711yescharacteristic
196#380—21no—
197#343—472no—
198#324—472no—
199#324—941no—
200C47#1187911yescharacteristic, complemented, Sylow
201C22×Q8C22⋊D4711yescharacteristic
202C22×D4C2×Dic4711yescharacteristic
203C23×C4—21no—
204C2×M16—942no—
205C22⋊Q8—42no—
206C22⋊Q8—42no—
207C4×Q8—41no—
208C4×Q8—41no—
209C4×D4—41no—
210C4×D4—41no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4C2×Dic4711yescharacteristic
215C2×C4⋊C4C2×Dic4711yescharacteristic
216C2×C4⋊C4C22⋊D4711yescharacteristic
217C2×C4⋊C4C22×D4711yescharacteristic
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C42C2×Dic4711yescharacteristic
224C2×C42C22⋊D4711yescharacteristic
225C2×C42C22⋊D4711yescharacteristic
226C4⋊C8—1881no—
227C4⋊C8—1881no—
228C4⋊C8—1881no—
229#101—1882no—
230C22⋊C8—942no—
231C4×C8—1881no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95C2×Dic4711yescharacteristic
236#95C2×Dic4711yescharacteristic
237C24—21no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8#1062533412yes—
241C2×Q8#1062533512yes—
242C2×D4—41no—
243C2×D4—42no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4C22⋊Dic4711yescharacteristic
251C22×C4C22⋊Dic4711yescharacteristic
252C22×C4C22⋊Dic4711yescharacteristic
253C22×C4C22⋊Dic4711yescharacteristic
254C22×C4C22×Dic4711yescharacteristic
255C22×C4C2×C22⋊D4711yescharacteristic
256C22×C4C2×C22⋊D4711yescharacteristic
257M16—1882no—
258M16—1882no—
259C2×C8—1882no—
260C2×C8—1882no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—42no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C42—21no—
277C42—21no—
278C42—21no—
279C42—21no—
280C42—21no—
281C42—21no—
282C23—22no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23C2×C22⋊Dic4711yescharacteristic
290Q8—42no—
291Q8—42no—
292D4—42no—
293D4—42no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—84no—
317C2×C4#1309393911yescharacteristic
318C2×C4#1309393911yescharacteristic
319C2×C4#1309394512yes—
320C2×C4#1309394512yes—
321C2×C4C2×#1062533411yescharacteristic
322C2×C4C2×#1062533511yescharacteristic
323C8—3762no—
324C8—3762no—
325C22—21no—
326C22—21no—
327C22—42no—
328C22—42no—
329C22—42no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22—81no—
334C22C2×#1309393911yescharacteristic
335C22C2×#1309394511yescharacteristic
336C22#1568914011yescharacteristic
337C4—21no—
338C4—22no—
339C4—22no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—81no—
347C4—81no—
348C4—81no—
349C2—21no—
350C2—21no—
351C2—82no—
352C2#2024195611yescharacteristic
353C2#2024502911yescharacteristic
354C2#2024507711yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC47×#2536C2
socleC22×C47C2×#99
prime radicalC22×D4×C47C2×C4
cosocleC2×C4⋊C4×C47C23
hypercenter#2536D47
hypocenterC47#11879
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2536

Complements

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

NH ≅ G/Ntype
1. #20243384C2semidirect
2. #20241712C2semidirect
3. #20241394C2semidirect
200. C47#11879semidirect

Maximal subgroups

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

subgroupindex
#202433842
#202417122
#202413942
#202413912
C47×#25362
#1187947