Finite Group Databasebeta

← #26109919

Subgroups — #26109919

Order 12032.

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

normal subgroups only

This group has 466 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×#2357C211yesmaximal, characteristic
6C2×#13093939—21no—
7#15689087C2212yes—
8C4×C22⋊Dic47C2212yes—
9#15688890—21no—
10#15688849—21no—
11#15688847—21no—
12C2×C4×D4×C47C2211yescharacteristic
13C47×#390—21no—
14C47×#388—21no—
15C47×#388—21no—
16C47×#382C411yescharacteristic, complemented
17C47×#381C411yescharacteristic, complemented
18C47×#381C2211yescharacteristic
19C4×C4⋊C4×C47—21no—
20C4×C4⋊C4×C47C411yescharacteristic, complemented
21C4×C22⋊C4×C47C411yescharacteristic, complemented
22C4×C22⋊C4×C47C2211yescharacteristic
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×C47C2×C411yescharacteristic
32C23×C4×C47C2311yescharacteristic
33C4×D4×C47—22no—
34C4×D4×C47—22no—
35C2×C4⋊C4×C47—21no—
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×C47—21no—
42C2×C22⋊C4×C47D411yescharacteristic
43C2×C22⋊C4×C47D411yescharacteristic
44C2×C42×C47—21no—
45C2×C42×C47—21no—
46C2×C42×C47—21no—
47C2×C42×C47C2×C411yescharacteristic
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—81no—
67C22×C4×C47—81no—
68C22×C4×C47C22⋊C411yescharacteristic
69C22×C4×C47C22⋊C411yescharacteristic
70C22×C4×C47C22⋊C411yescharacteristic
71C22×C4×C47C22×C411yescharacteristic
72C22×C4×C47C2×D411yescharacteristic
73C22×C4×C47#4112yes—
74C22×C4×C47#4112yes—
75C22×Dic47—42no—
76C4⋊C4×C47—22no—
77C4⋊C4×C47—42no—
78C4⋊C4×C47—42no—
79C22⋊C4×C47—22no—
80C22⋊C4×C47—22no—
81C22⋊C4×C47—42no—
82C22⋊C4×C47—42no—
83C22⋊C4×C47—81no—
84C22⋊C4×C47—81no—
85C42×C47—21no—
86C42×C47—21no—
87C42×C47—42no—
88C42×C47—81no—
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—21no—
100C23×C47—41no—
101C23×C47—41no—
102C23×C47—41no—
103C23×C47—41no—
104C23×C47C2×C22⋊C411yescharacteristic
105C23×C47#12312yes—
106D4×C47—44no—
107C2×C4×C47—21no—
108C2×C4×C47—21no—
109C2×C4×C47—22no—
110C2×C4×C47—21no—
111C2×C4×C47—21no—
112C2×C4×C47—42no—
113C2×C4×C47—42no—
114C2×C4×C47—41no—
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—81no—
130C2×C4×C47—81no—
131C2×C4×C47#10412yes—
132C2×C4×C47#11711yescharacteristic
133C2×C4×C47#11711yescharacteristic
134C2×C4×C47#12312yes—
135C2×Dic47—44no—
136C2×Dic47—82no—
137C2×Dic47—82no—
138C47⋊C8—82no—
139#8779—471nomaximal, Sylow
140C22×C47—21no—
141C22×C47—21no—
142C22×C47—22no—
143C22×C47—22no—
144C22×C47—22no—
145C22×C47—41no—
146C22×C47—41no—
147C22×C47—41no—
148C22×C47—81no—
149C22×C47—81no—
150C22×C47—81no—
151C22×C47—81no—
152C22×C47#38111yescharacteristic
153C22×C47C2×#10411yescharacteristic
154C22×C47#42111yescharacteristic
155C4×C47—22no—
156C4×C47—22no—
157C4×C47—21no—
158C4×C47—21no—
159C4×C47—42no—
160C4×C47—41no—
161C4×C47—41no—
162C4×C47—81no—
163C4×C47—81no—
164C4×C47—81no—
165Dic47—84no—
166#2357D4711yescharacteristic
167C4×#99—471no—
168#1465—471no—
169#1297—472no—
170#1284—472no—
171C2×C47—21no—
172C2×C47—21no—
173C2×C47—42no—
174C2×C47—81no—
175C2×C47#182611yescharacteristic
176C2×C47#182711yescharacteristic
177C2×C47#182811yescharacteristic
178C2×C4×D4C2×D4711yescharacteristic
179C2×#99—941no—
180#407—472no—
181#390—21no—
182#388—21no—
183#388—21no—
184#382Dic4711yescharacteristic, complemented
185#381Dic4711yescharacteristic, complemented
186#381C2×D4711yescharacteristic
187C4×C4⋊C4—21no—
188C4×C4⋊C4Dic4711yescharacteristic, complemented
189C4×C22⋊C4—472no—
190C4×C22⋊C4Dic4711yescharacteristic, complemented
191C4×C22⋊C4C2×D4711yescharacteristic
192#342—941no—
193#324—941no—
194#323—941no—
195C47#877911yescharacteristic, complemented, Sylow
196C22×D4C22⋊D4711yescharacteristic
197C23×C4C2×Dic4711yescharacteristic
198C23×C4C22×D4711yescharacteristic
199C2×M16—942no—
200C4×D4—22no—
201C4×D4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4C22⋊D4711yescharacteristic
205C2×C22⋊C4—21no—
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—21no—
215C2×C42—942no—
216C2×C42C2×Dic4711yescharacteristic
217#99—944no—
218C22⋊C8—942no—
219C22⋊C8—942no—
220#95—21no—
221#95—21no—
222#95—21no—
223#95—942no—
224#95C2×Dic4711yescharacteristic
225#95C2×Dic4711yescharacteristic
226#95C2×Dic4711yescharacteristic
227#95C2×Dic4711yescharacteristic
228C24C22⋊Dic4711yescharacteristic
229C24C2×C22⋊D4711yescharacteristic
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—41no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—81no—
240C22×C4—81no—
241C22×C4—1882no—
242C22×C4C22⋊Dic4711yescharacteristic
243C22×C4C22⋊Dic4711yescharacteristic
244C22×C4C22⋊Dic4711yescharacteristic
245C22×C4#1062535012yes—
246C22×C4#1062535012yes—
247C22×C4C22×Dic4711yescharacteristic
248C22×C4C2×C22⋊D4711yescharacteristic
249M16—3762no—
250C2×C8—1882no—
251C4⋊C4—22no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C22⋊C4—22no—
255C22⋊C4—22no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—81no—
259C22⋊C4—81no—
260C22⋊C4—944no—
261C22⋊C4—1882no—
262C42—21no—
263C42—21no—
264C42—42no—
265C42—81no—
266C42—81no—
267C42—3762no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23C2×C22⋊Dic4711yescharacteristic
278C23#1309400712yes—
279D4—44no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
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—81no—
303C2×C4—81no—
304C2×C4—3762no—
305C2×C4—3762no—
306C2×C4—1884no—
307C2×C4#1309395112yes—
308C2×C4#1309398311yescharacteristic
309C2×C4#1309398311yescharacteristic
310C2×C4#1309400712yes—
311C8—3762no—
312C22—21no—
313C22—21no—
314C22—22no—
315C22—22no—
316C22—22no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22#1568898611yescharacteristic
325C22C2×#1309395111yescharacteristic
326C22#1568914011yescharacteristic
327C4—22no—
328C4—22no—
329C4—21no—
330C4—21no—
331C4—42no—
332C4—41no—
333C4—41no—
334C4—81no—
335C4—81no—
336C4—81no—
337C4—3764no—
338C2—21no—
339C2—21no—
340C2—42no—
341C2—81no—
342C2#2024348711yescharacteristic
343C2#2024349211yescharacteristic
344C2#2024349611yescharacteristic
subgroupG/H
centerC22#15688986
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2357C2
socleC22×C47#381
prime radicalC22×D4×C47D4
cosocleC23×C4×C47C23
hypercenter#2357D47
hypocenterC47#8779
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2357

Complements

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

NH ≅ G/Ntype
3. #20241274C2semidirect
4. #20241240C2semidirect
16. C47×#382C4semidirect
17. C47×#381C4semidirect
20. C4×C4⋊C4×C47C4semidirect
21. C4×C22⋊C4×C47C4semidirect
184. #382Dic47semidirect
185. #381Dic47semidirect
188. C4×C4⋊C4Dic47semidirect
190. C4×C22⋊C4Dic47semidirect
195. C47#8779semidirect

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×#23572
#877947