Finite Group Databasebeta

← #26109909

Subgroups — #26109909

Order 12032.

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

normal subgroups only

This group has 480 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
1C4×#13093939C211yesmaximal, characteristic
2#20241726C211yesmaximal, characteristic
3#20241274C212yesmaximal, complemented
4#20241240C212yesmaximal, complemented
5C47×#2353C211yesmaximal, characteristic
6C2×#13093939—21no—
7#15689087C2212yes—
8C4×C22⋊Dic47C2212yes—
9#15688890—21no—
10#15688849—21no—
11#15688847—21no—
12C2×C4×D4×C47C2211yescharacteristic
13C47×#388—21no—
14C47×#387—21no—
15C47×#386—21no—
16C47×#383C411yescharacteristic, complemented
17C47×#381C2211yescharacteristic
18C47×#380C411yescharacteristic, complemented
19C4×C22⋊C4×C47—21no—
20C4×C22⋊C4×C47C2211yescharacteristic
21C47×#376C411yescharacteristic, complemented
22C2×#95×C47C411yescharacteristic, complemented
23C2×#10625329—22no—
24C2×C22⋊Dic47—22no—
25C2×C4×Dic47—22no—
26#13093939—24no—
27C22⋊(C47⋊C8)—22no—
28C22⋊(C47⋊C8)—22no—
29C47⋊#95—22no—
30C22×D4×C47D411yescharacteristic
31C23×C4×C47—21no—
32C23×C4×C47C2×C411yescharacteristic
33C23×C4×C47C2311yescharacteristic
34C4×D4×C47—22no—
35C4×D4×C47—22no—
36C2×C4⋊C4×C47—21no—
37C2×C4⋊C4×C47D411yescharacteristic
38C2×C22⋊C4×C47—21no—
39C2×C22⋊C4×C47—21no—
40C2×C22⋊C4×C47—21no—
41C2×C22⋊C4×C47D411yescharacteristic
42C2×C22⋊C4×C47D411yescharacteristic
43C2×C42×C47—21no—
44C2×C42×C47—21no—
45C2×C42×C47C2×C411yescharacteristic
46#95×C47—21no—
47#95×C47—21no—
48#95×C47—21no—
49#95×C47—21no—
50#95×C47—21no—
51#95×C47C2×C411yescharacteristic
52#95×C47C2×C411yescharacteristic
53#95×C47C2×C411yescharacteristic
54#95×C47C2×C411yescharacteristic
55C24×C47C22⋊C411yescharacteristic
56C24×C47C2×D411yescharacteristic
57C2×D4×C47—22no—
58C2×D4×C47—22no—
59C2×D4×C47—41no—
60C22×C4×C47—21no—
61C22×C4×C47—21no—
62C22×C4×C47—21no—
63C22×C4×C47—21no—
64C22×C4×C47—21no—
65C22×C4×C47—21no—
66C22×C4×C47—42no—
67C22×C4×C47—42no—
68C22×C4×C47—81no—
69C22×C4×C47—81no—
70C22×C4×C47—81no—
71C22×C4×C47C22⋊C411yescharacteristic
72C22×C4×C47C22⋊C411yescharacteristic
73C22×C4×C47C22⋊C411yescharacteristic
74C22×C4×C47C22×C411yescharacteristic
75C22×C4×C47C2×D411yescharacteristic
76C22×C4×C47#4112yes—
77C22×C4×C47#4112yes—
78C22×Dic47—42no—
79C4⋊C4×C47—22no—
80C4⋊C4×C47—81no—
81C22⋊C4×C47—22no—
82C22⋊C4×C47—22no—
83C22⋊C4×C47—42no—
84C22⋊C4×C47—42no—
85C22⋊C4×C47—81no—
86C42×C47—21no—
87C42×C47—21no—
88C42×C47—42no—
89C42×C47—81no—
90C22⋊Dic47—24no—
91C22⋊Dic47—42no—
92C4×Dic47—82no—
93#10625329—82no—
94C2×C47⋊C8—42no—
95C23×C47—21no—
96C23×C47—21no—
97C23×C47—21no—
98C23×C47—21no—
99C23×C47—41no—
100C23×C47—41no—
101C23×C47—41no—
102C23×C47—41no—
103C23×C47#10412yes—
104C23×C47C2×C22⋊C411yescharacteristic
105C23×C47#12312yes—
106D4×C47—44no—
107C2×C4×C47—22no—
108C2×C4×C47—21no—
109C2×C4×C47—21no—
110C2×C4×C47—21no—
111C2×C4×C47—21no—
112C2×C4×C47—21no—
113C2×C4×C47—42no—
114C2×C4×C47—42no—
115C2×C4×C47—41no—
116C2×C4×C47—41no—
117C2×C4×C47—41no—
118C2×C4×C47—41no—
119C2×C4×C47—41no—
120C2×C4×C47—41no—
121C2×C4×C47—41no—
122C2×C4×C47—41no—
123C2×C4×C47—41no—
124C2×C4×C47—41no—
125C2×C4×C47—41no—
126C2×C4×C47—41no—
127C2×C4×C47—41no—
128C2×C4×C47—41no—
129C2×C4×C47—41no—
130C2×C4×C47—81no—
131C2×C4×C47—81no—
132C2×C4×C47—84no—
133C2×C4×C47#11711yescharacteristic
134C2×C4×C47#11711yescharacteristic
135C2×C4×C47#12312yes—
136C2×Dic47—44no—
137C2×Dic47—82no—
138C2×Dic47—82no—
139C47⋊C8—82no—
140#8777—471nomaximal, Sylow
141C22×C47—21no—
142C22×C47—21no—
143C22×C47—22no—
144C22×C47—22no—
145C22×C47—22no—
146C22×C47—22no—
147C22×C47—42no—
148C22×C47—42no—
149C22×C47—41no—
150C22×C47—41no—
151C22×C47—81no—
152C22×C47—81no—
153C22×C47#38111yescharacteristic
154C22×C47C2×#10411yescharacteristic
155C22×C47#42111yescharacteristic
156C4×C47—22no—
157C4×C47—21no—
158C4×C47—21no—
159C4×C47—42no—
160C4×C47—41no—
161C4×C47—41no—
162C4×C47—41no—
163C4×C47—81no—
164C4×C47—81no—
165C4×C47—81no—
166Dic47—84no—
167#2353D4711yescharacteristic
168C4×#99—471no—
169#1465—471no—
170#1297—472no—
171#1284—472no—
172C2×C47—21no—
173C2×C47—21no—
174C2×C47—42no—
175C2×C47—42no—
176C2×C47#182611yescharacteristic
177C2×C47#182711yescharacteristic
178C2×C47#182811yescharacteristic
179C2×C4×D4C2×D4711yescharacteristic
180C2×#99—941no—
181#407—472no—
182#388—21no—
183#387—21no—
184#386—21no—
185#383Dic4711yescharacteristic, complemented
186#381C2×D4711yescharacteristic
187#380Dic4711yescharacteristic, complemented
188C4×C22⋊C4—21no—
189C4×C22⋊C4—472no—
190C4×C22⋊C4C2×D4711yescharacteristic
191#376Dic4711yescharacteristic, complemented
192C2×#95Dic4711yescharacteristic, complemented
193#342—941no—
194#324—941no—
195#323—941no—
196C47#877711yescharacteristic, complemented, Sylow
197C22×D4C22⋊D4711yescharacteristic
198C23×C4—21no—
199C23×C4C2×Dic4711yescharacteristic
200C23×C4C22×D4711yescharacteristic
201C2×M16—942no—
202C4×D4—22no—
203C4×D4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4C22⋊D4711yescharacteristic
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—942no—
210C2×C22⋊C4C22⋊D4711yescharacteristic
211C2×C22⋊C4C22⋊D4711yescharacteristic
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—942no—
215C2×C42C2×Dic4711yescharacteristic
216#99—944no—
217C22⋊C8—942no—
218C22⋊C8—942no—
219#95—21no—
220#95—21no—
221#95—21no—
222#95—21no—
223#95—21no—
224#95—942no—
225#95C2×Dic4711yescharacteristic
226#95C2×Dic4711yescharacteristic
227#95C2×Dic4711yescharacteristic
228#95C2×Dic4711yescharacteristic
229C24C22⋊Dic4711yescharacteristic
230C24C2×C22⋊D4711yescharacteristic
231C2×D4—22no—
232C2×D4—22no—
233C2×D4—41no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—81no—
243C22×C4—81no—
244C22×C4—81no—
245C22×C4—1882no—
246C22×C4C22⋊Dic4711yescharacteristic
247C22×C4C22⋊Dic4711yescharacteristic
248C22×C4C22⋊Dic4711yescharacteristic
249C22×C4#1062535012yes—
250C22×C4#1062535012yes—
251C22×C4C22×Dic4711yescharacteristic
252C22×C4C2×C22⋊D4711yescharacteristic
253M16—3762no—
254C2×C8—1882no—
255C4⋊C4—22no—
256C4⋊C4—81no—
257C22⋊C4—22no—
258C22⋊C4—22no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22⋊C4—81no—
262C22⋊C4—944no—
263C22⋊C4—1882no—
264C42—21no—
265C42—21no—
266C42—42no—
267C42—81no—
268C42—3762no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23#1309395112yes—
278C23C2×C22⋊Dic4711yescharacteristic
279C23#1309400712yes—
280D4—44no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—84no—
307C2×C4—3762no—
308C2×C4—3762no—
309C2×C4—1884no—
310C2×C4#1309398311yescharacteristic
311C2×C4#1309398311yescharacteristic
312C2×C4#1309400712yes—
313C8—3762no—
314C22—21no—
315C22—21no—
316C22—22no—
317C22—22no—
318C22—22no—
319C22—22no—
320C22—42no—
321C22—42no—
322C22—41no—
323C22—41no—
324C22—81no—
325C22—81no—
326C22#1568898611yescharacteristic
327C22C2×#1309395111yescharacteristic
328C22#1568914011yescharacteristic
329C4—22no—
330C4—21no—
331C4—21no—
332C4—42no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—81no—
337C4—81no—
338C4—81no—
339C4—3764no—
340C2—21no—
341C2—21no—
342C2—42no—
343C2—42no—
344C2#2024348711yescharacteristic
345C2#2024349211yescharacteristic
346C2#2024349611yescharacteristic
subgroupG/H
centerC22#15688986
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2353C2
socleC22×C47#381
prime radicalC22×D4×C47D4
cosocleC23×C4×C47C23
hypercenter#2353D47
hypocenterC47#8777
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2353

Complements

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

NH ≅ G/Ntype
3. #20241274C2semidirect
4. #20241240C2semidirect
16. C47×#383C4semidirect
18. C47×#380C4semidirect
21. C47×#376C4semidirect
22. C2×#95×C47C4semidirect
185. #383Dic47semidirect
187. #380Dic47semidirect
191. #376Dic47semidirect
192. C2×#95Dic47semidirect
196. C47#8777semidirect

Maximal subgroups

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

subgroupindex
C4×#130939392
#202417262
#202412742
#202412402
C47×#23532
#877747