Finite Group Databasebeta

← #26126769

Subgroups — #26126769

Order 12032.

436 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 91 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#20243121C211yesmaximal, characteristic
2#20241722C211yesmaximal, characteristic
3#20241412C212yesmaximal
4#20241388C212yesmaximal
5C47×#2611C211yesmaximal, characteristic
6C2×#13093983—21no—
7#15689180—21no—
8#15688890—21no—
9#15688890C2212yes—
10#15688849—21no—
11#15688847C2212yes—
12C2×#125×C47C411yescharacteristic, complemented
13C2×#117×C47C2211yescharacteristic
14C47×#400C411yescharacteristic, complemented
15C47×#399C2211yescharacteristic
16C47×#398C411yescharacteristic, complemented
17C47×#397C2211yescharacteristic
18C47×#389—21no—
19C47×#388—21no—
20C47×#387—21no—
21C47×#385—21no—
22C47×#376C411yescharacteristic, complemented
23#13093983—24no—
24#13093983—42no—
25C2×C4⋊Dic47—41no—
26C2×C22⋊Dic47—22no—
27C2×C4×Dic47—41no—
28C4⋊(C47⋊C8)—41no—
29C22⋊(C47⋊C8)—22no—
30#13093935—41no—
31C23×C4×C47D411yescharacteristic
32#125×C47—41no—
33#125×C47—41no—
34#117×C47—41no—
35#117×C47—41no—
36C2×C4⋊C4×C47—21no—
37C2×C4⋊C4×C47—21no—
38C2×C4⋊C4×C47—21no—
39C2×C4⋊C4×C47—21no—
40C2×C4⋊C4×C47C2×C411yescharacteristic
41C2×C4⋊C4×C47C2×C411yescharacteristic
42C2×C4⋊C4×C47C2×C411yescharacteristic
43C2×C4⋊C4×C47D411yescharacteristic
44C2×C22⋊C4×C47—21no—
45C2×C22⋊C4×C47—21no—
46C2×C22⋊C4×C47D411yescharacteristic
47C2×C22⋊C4×C47C2311yescharacteristic
48C2×C42×C47—21no—
49C2×C42×C47C2×C411yescharacteristic
50C2×C42×C47D411yescharacteristic
51#95×C47—21no—
52#95×C47—21no—
53#95×C47—21no—
54#95×C47—21no—
55#95×C47—21no—
56#95×C47C2×C411yescharacteristic
57#95×C47C2×C411yescharacteristic
58C24×C47C2×D411yescharacteristic
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47—21no—
62C22×C4×C47—21no—
63C22×C4×C47—22no—
64C22×C4×C47—21no—
65C22×C4×C47—21no—
66C22×C4×C47—21no—
67C22×C4×C47—21no—
68C22×C4×C47C22⋊C411yescharacteristic
69C22×C4×C47C22⋊C411yescharacteristic
70C22×C4×C47C22⋊C411yescharacteristic
71C22×C4×C47C22⋊C411yescharacteristic
72C22×C4×C47C22×C411yescharacteristic
73C22×C4×C47C2×D411yescharacteristic
74C22×Dic47—42no—
75C4⋊C4×C47—41no—
76C4⋊C4×C47—41no—
77C4⋊C4×C47—42no—
78C4⋊C4×C47—42no—
79C4⋊C4×C47—41no—
80C4⋊C4×C47—41no—
81C4⋊C4×C47—41no—
82C4⋊C4×C47—41no—
83C4⋊C4×C47—42no—
84C22⋊C4×C47—41no—
85C22⋊C4×C47—41no—
86C22⋊C4×C47—42no—
87C22⋊C4×C47—41no—
88C22⋊C4×C47—41no—
89C42×C47—21no—
90C42×C47—21no—
91C42×C47—21no—
92C42×C47—21no—
93C42×C47—42no—
94C22⋊Dic47—24no—
95C22⋊Dic47—42no—
96C4⋊Dic47—44no—
97C4×Dic47—44no—
98C2×C47⋊C8—42no—
99C23×C47—21no—
100C23×C47—21no—
101C23×C47—41no—
102C23×C47—41no—
103C23×C47C2×C22⋊C411yescharacteristic
104C23×C47AGL(1,ℤ/8)11yescharacteristic
105C23×C47#13711yescharacteristic
106C2×C4×C47—22no—
107C2×C4×C47—22no—
108C2×C4×C47—22no—
109C2×C4×C47—22no—
110C2×C4×C47—21no—
111C2×C4×C47—21no—
112C2×C4×C47—21no—
113C2×C4×C47—21no—
114C2×C4×C47—22no—
115C2×C4×C47—21no—
116C2×C4×C47—21no—
117C2×C4×C47—21no—
118C2×C4×C47—21no—
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—41no—
131C2×C4×C47—41no—
132C2×C4×C47—41no—
133C2×C4×C47—41no—
134C2×C4×C47#9911yescharacteristic
135C2×C4×C47#9911yescharacteristic
136C2×C4×C47#10412yes—
137C2×Dic47—44no—
138C2×Dic47—82no—
139C2×Dic47—82no—
140C47⋊C8—82no—
141#11760—471nomaximal, Sylow
142C22×C47—21no—
143C22×C47—21no—
144C22×C47—21no—
145C22×C47—21no—
146C22×C47—41no—
147C22×C47—41no—
148C22×C47—41no—
149C22×C47—41no—
150C22×C47—81no—
151C22×C47C2×#9911yescharacteristic
152C22×C47#41711yescharacteristic
153C22×C47C2×#10411yescharacteristic
154C4×C47—21no—
155C4×C47—21no—
156C4×C47—22no—
157C4×C47—42no—
158C4×C47—41no—
159C4×C47—41no—
160C4×C47—41no—
161C4×C47—41no—
162C4×C47—81no—
163C4×C47—81no—
164C4×C47—81no—
165Dic47—84no—
166#2611D4711yescharacteristic
167#1740—471no—
168#1464—471no—
169#1352—472no—
170#1344—472no—
171C2×C47—21no—
172C2×C47—21no—
173C2×C47—41no—
174C2×C47—41no—
175C2×C47#150411yescharacteristic
176C2×C47#212311yescharacteristic
177C2×C47#212611yescharacteristic
178C2×#125Dic4711yescharacteristic, complemented
179C2×#117—941no—
180C2×#117C2×D4711yescharacteristic
181#432—941no—
182#400Dic4711yescharacteristic, complemented
183#399C2×D4711yescharacteristic
184#398Dic4711yescharacteristic, complemented
185#397C2×D4711yescharacteristic
186#389—21no—
187#388—21no—
188#387—21no—
189#385—21no—
190#376Dic4711yescharacteristic, complemented
191#342—472no—
192#342—941no—
193#324—941no—
194#323—472no—
195C47#1176011yescharacteristic, complemented, Sylow
196C23×C4C22⋊D4711yescharacteristic
197#125—41no—
198#125—41no—
199#117—41no—
200#117—41no—
201#117—944no—
202#117—1882no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—1881no—
208C2×C4⋊C4C2×Dic4711yescharacteristic
209C2×C4⋊C4C2×Dic4711yescharacteristic
210C2×C4⋊C4C2×Dic4711yescharacteristic
211C2×C4⋊C4C22⋊D4711yescharacteristic
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—942no—
215C2×C22⋊C4C22⋊D4711yescharacteristic
216C2×C22⋊C4C22×D4711yescharacteristic
217C2×C42—21no—
218C2×C42—1881no—
219C2×C42C2×Dic4711yescharacteristic
220C2×C42C22⋊D4711yescharacteristic
221C4⋊C8—1881no—
222C22⋊C8—942no—
223#97—1881no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95C2×Dic4711yescharacteristic
230#95C2×Dic4711yescharacteristic
231C24C2×C22⋊D4711yescharacteristic
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—1882no—
242C22×C4C22⋊Dic4711yescharacteristic
243C22×C4C22⋊Dic4711yescharacteristic
244C22×C4C22⋊Dic4711yescharacteristic
245C22×C4C22⋊Dic4711yescharacteristic
246C22×C4C22×Dic4711yescharacteristic
247C22×C4C2×C22⋊D4711yescharacteristic
248C2×C8—1882no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—42no—
258C4⋊C4—1884no—
259C22⋊C4—41no—
260C22⋊C4—41no—
261C22⋊C4—42no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C22⋊C4—944no—
265C22⋊C4—1882no—
266C42—21no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—42no—
271C42—1884no—
272C23—21no—
273C23—21no—
274C23—41no—
275C23—41no—
276C23C2×C22⋊Dic4711yescharacteristic
277C23#1309404911yescharacteristic
278C23#1309405411yescharacteristic
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
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—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—3762no—
308C2×C4—3762no—
309C2×C4—1884no—
310C2×C4#1309393911yescharacteristic
311C2×C4#1309393911yescharacteristic
312C2×C4#1309395112yes—
313C8—3762no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—81no—
323C22C2×#1309393911yescharacteristic
324C22#1568912411yescharacteristic
325C22C2×#1309395111yescharacteristic
326C4—21no—
327C4—21no—
328C4—22no—
329C4—42no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—81no—
335C4—81no—
336C4—81no—
337C4—3764no—
338C2—21no—
339C2—21no—
340C2—41no—
341C2—41no—
342C2#2024189211yescharacteristic
343C2#2024504211yescharacteristic
344C2#2024505311yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC22×C4×C47C22×C4
FrattiniC2×C22⋊C4C22×D47
FittingC47×#2611C2
socleC22×C47C2×#99
prime radicalC24×C47C2×D4
cosocleC2×C22⋊C4×C47C23
hypercenter#2611D47
hypocenterC47#11760
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2611

Complements

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

NH ≅ G/Ntype
12. C2×#125×C47C4semidirect
14. C47×#400C4semidirect
16. C47×#398C4semidirect
22. C47×#376C4semidirect
178. C2×#125Dic47semidirect
182. #400Dic47semidirect
184. #398Dic47semidirect
190. #376Dic47semidirect
195. C47#11760semidirect

Maximal subgroups

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

subgroupindex
#202431212
#202417222
#202414122
#202413882
C47×#26112
#1176047