Finite Group Databasebeta

← #26126767

Subgroups — #26126767

Order 12032.

436 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 83 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#20249086C211yesmaximal, characteristic
2#20241722C211yesmaximal, characteristic
3#20241411C212yesmaximal
4#20241387C212yesmaximal
5C47×#1740C211yesmaximal, characteristic
6C2×#13094016C411yescharacteristic, complemented
7#15689180—21no—
8#15689062C411yescharacteristic, complemented
9#15689059C2211yescharacteristic
10#15689053C411yescharacteristic, complemented
11#15689048C2211yescharacteristic
12#15689018—21no—
13#15689017—21no—
14#15689007—21no—
15#15688996—21no—
16#15688969C411yescharacteristic, complemented
17#15688849—21no—
18#15688847C2212yes—
19C2×#117×C47—21no—
20C2×#117×C47C2211yescharacteristic
21C47×#342—21no—
22C47×#342C2212yes—
23#13094016—41no—
24#13094016—41no—
25C2×C4⋊Dic47C2×C411yescharacteristic
26C2×C4⋊Dic47C2×C411yescharacteristic
27C2×Dic47⋊C4—21no—
28C2×Dic47⋊C4—21no—
29C2×Dic47⋊C4—21no—
30C2×Dic47⋊C4—21no—
31C2×C22⋊Dic47—21no—
32C2×C22⋊Dic47—21no—
33C2×C4×Dic47—21no—
34C4⋊(C47⋊C8)—41no—
35C22⋊(C47⋊C8)—22no—
36#13093935—41no—
37C47⋊#95—21no—
38C47⋊#95—21no—
39C47⋊#95—21no—
40C47⋊#95—21no—
41C47⋊#95—21no—
42C47⋊#95C2×C411yescharacteristic
43C47⋊#95C2×C411yescharacteristic
44C23×C4×C47D411yescharacteristic
45#117×C47—24no—
46#117×C47—41no—
47#117×C47—41no—
48#117×C47—42no—
49C2×C4⋊C4×C47—41no—
50C2×C4⋊C4×C47C2×C411yescharacteristic
51C2×C4⋊C4×C47D411yescharacteristic
52C2×C22⋊C4×C47—22no—
53C2×C22⋊C4×C47D411yescharacteristic
54C2×C22⋊C4×C47C2311yescharacteristic
55C2×C42×C47—41no—
56C2×C42×C47C2×C411yescharacteristic
57C2×C42×C47D411yescharacteristic
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—42no—
65C22×C4×C47C22⋊C411yescharacteristic
66C22×C4×C47C22⋊C411yescharacteristic
67C22×C4×C47C22⋊C411yescharacteristic
68C22×C4×C47C22⋊C411yescharacteristic
69C22×C4×C47C22×C411yescharacteristic
70C22×C4×C47C2×D411yescharacteristic
71C22×Dic47—21no—
72C22×Dic47—21no—
73C22×Dic47—21no—
74C22×Dic47—21no—
75C4⋊C4×C47—41no—
76C4⋊C4×C47—41no—
77C4⋊C4×C47—44no—
78C22⋊C4×C47—24no—
79C22⋊C4×C47—41no—
80C22⋊C4×C47—41no—
81C22⋊C4×C47—42no—
82C42×C47—21no—
83C42×C47—21no—
84C42×C47—21no—
85C42×C47—21no—
86C42×C47—44no—
87C22⋊Dic47—42no—
88C22⋊Dic47—41no—
89C22⋊Dic47—41no—
90C4⋊Dic47—41no—
91C4⋊Dic47—41no—
92Dic47⋊C4—42no—
93Dic47⋊C4—42no—
94Dic47⋊C4—41no—
95Dic47⋊C4—41no—
96Dic47⋊C4—42no—
97C4×Dic47—42no—
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—21no—
115C2×C4×C47—21no—
116C2×C4×C47—21no—
117C2×C4×C47—21no—
118C2×C4×C47—41no—
119C2×C4×C47—41no—
120C2×C4×C47—41no—
121C2×C4×C47—41no—
122C2×C4×C47—44no—
123C2×C4×C47—82no—
124C2×C4×C47—82no—
125C2×C4×C47#9911yescharacteristic
126C2×C4×C47#9911yescharacteristic
127C2×C4×C47#10412yes—
128C2×Dic47—22no—
129C2×Dic47—41no—
130C2×Dic47—41no—
131C2×Dic47—41no—
132C2×Dic47—41no—
133C2×Dic47—41no—
134C2×Dic47—41no—
135C2×Dic47—41no—
136C2×Dic47—41no—
137C2×Dic47—41no—
138C2×Dic47—41no—
139C2×Dic47—41no—
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—41no—
158C4×C47—41no—
159C4×C47—41no—
160C4×C47—41no—
161C4×C47—84no—
162Dic47—42no—
163Dic47—81no—
164Dic47—81no—
165Dic47—81no—
166#2611—471no—
167#1740D4711yescharacteristic
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×#125—471no—
179C2×#117—21no—
180C2×#117C2×D4711yescharacteristic
181#432—941no—
182#400—471no—
183#399—471no—
184#398—471no—
185#397—471no—
186#389—941no—
187#388—941no—
188#387—941no—
189#385—941no—
190#376—471no—
191#342—21no—
192#342C2×D4712yes—
193#324—941no—
194#323—472no—
195C47#1176011yescharacteristic, complemented, Sylow
196C23×C4C22⋊D4711yescharacteristic
197#125—1881no—
198#125—1881no—
199#117—24no—
200#117—41no—
201#117—41no—
202#117—42no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—471no—
205C2×C4⋊C4—471no—
206C2×C4⋊C4—941no—
207C2×C4⋊C4—941no—
208C2×C4⋊C4—941no—
209C2×C4⋊C4—941no—
210C2×C4⋊C4C4×D4711yescharacteristic
211C2×C4⋊C4D18811yescharacteristic
212C2×C22⋊C4—22no—
213C2×C22⋊C4—941no—
214C2×C22⋊C4—941no—
215C2×C22⋊C4C22⋊D4711yescharacteristic
216C2×C22⋊C4C22×D4711yescharacteristic
217C2×C42—41no—
218C2×C42—941no—
219C2×C42C4×D4711yescharacteristic
220C2×C42D18811yescharacteristic
221C4⋊C8—1881no—
222C22⋊C8—942no—
223#97—1881no—
224#95—471no—
225#95—471no—
226#95—941no—
227#95—941no—
228#95—941no—
229#95—941no—
230#95—941no—
231C24C2×C22⋊D4711yescharacteristic
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—42no—
238C22×C4—941no—
239C22×C4—941no—
240C22×C4—941no—
241C22×C4—941no—
242C22×C4(C2×D47)⋊C411yescharacteristic
243C22×C4(C2×D47)⋊C411yescharacteristic
244C22×C4(C2×D47)⋊C411yescharacteristic
245C22×C4(C2×D47)⋊C411yescharacteristic
246C22×C4C2×C4×D4711yescharacteristic
247C22×C4C2×D18811yescharacteristic
248C2×C8—1882no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—44no—
252C4⋊C4—1881no—
253C4⋊C4—1881no—
254C4⋊C4—1882no—
255C4⋊C4—1882no—
256C4⋊C4—1881no—
257C4⋊C4—1881no—
258C4⋊C4—1882no—
259C22⋊C4—24no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—42no—
263C22⋊C4—1882no—
264C22⋊C4—1881no—
265C22⋊C4—1881no—
266C42—21no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—44no—
271C42—1882no—
272C23—21no—
273C23—21no—
274C23—41no—
275C23—41no—
276C23C2×(C2×D47)⋊C411yescharacteristic
277C23#1309405011yescharacteristic
278C23#1309405511yescharacteristic
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—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—44no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—942no—
299C2×C4—1881no—
300C2×C4—1881no—
301C2×C4—1881no—
302C2×C4—1881no—
303C2×C4—1881no—
304C2×C4—1881no—
305C2×C4—1881no—
306C2×C4—1881no—
307C2×C4—1881no—
308C2×C4—1881no—
309C2×C4—1881no—
310C2×C4#1309393811yescharacteristic
311C2×C4#1309393811yescharacteristic
312C2×C4#1309395212yes—
313C8—3762no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—81no—
323C22C2×#1309393811yescharacteristic
324C22#1568912211yescharacteristic
325C22C2×#1309395211yescharacteristic
326C4—21no—
327C4—21no—
328C4—22no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—84no—
334C4—3761no—
335C4—3761no—
336C4—3761no—
337C4—1882no—
338C2—21no—
339C2—21no—
340C2—41no—
341C2—41no—
342C2#2024189011yescharacteristic
343C2#2024504111yescharacteristic
344C2#2024505111yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C22⋊C4C22×D47
FittingC47×#1740C2
socleC22×C47C2×#99
prime radicalC24×C47C2×D4
cosocleC2×C22⋊C4×C47C23
hypercenter#1740D47
hypocenterC47#11760
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1740

Complements

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

NH ≅ G/Ntype
6. C2×#13094016C4semidirect
8. #15689062C4semidirect
10. #15689053C4semidirect
16. #15688969C4semidirect
195. C47#11760semidirect

Maximal subgroups

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

subgroupindex
#202490862
#202417222
#202414112
#202413872
C47×#17402
#1176047