Finite Group Databasebeta

← #26126613

Subgroups — #26126613

Order 12032.

456 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 87 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#20246647C211yesmaximal, characteristic
2C4×#13093939C211yesmaximal, characteristic
3#20241413C212yesmaximal, complemented
4#20241389C212yesmaximal, complemented
5C47×#1460C211yesmaximal, characteristic
6C2×#13093983C411yescharacteristic, complemented
7C2×#13093939—21no—
8#15689062C411yescharacteristic, complemented
9#15689059C2211yescharacteristic
10#15689053C411yescharacteristic, complemented
11#15689051—21no—
12#15689048C2211yescharacteristic
13#15689037—21no—
14C4×Dic47⋊C4—21no—
15C4×C22⋊Dic47—21no—
16C4×C22⋊Dic47—21no—
17C4×C22⋊Dic47C411yescharacteristic, complemented
18#15688890C2212yes—
19C2×C4×D4×C47C2211yescharacteristic
20C47×#431—21no—
21C47×#323—21no—
22C47×#323C2212yes—
23#13093983—41no—
24#13093983—41no—
25C2×C4⋊Dic47C2×C411yescharacteristic
26C2×C4⋊Dic47C2×C411yescharacteristic
27C2×Dic47⋊C4—21no—
28C2×Dic47⋊C4—21no—
29C2×C22⋊Dic47—22no—
30C2×C22⋊Dic47—21no—
31C2×C22⋊Dic47—21no—
32C2×C22⋊Dic47—21no—
33C2×C22⋊Dic47—21no—
34C2×C4×Dic47—21no—
35C2×C4×Dic47—21no—
36C2×C4×Dic47—41no—
37#13093939—24no—
38C47⋊#95—21no—
39C47⋊#95—21no—
40C47⋊#95—21no—
41C47⋊#95—21no—
42C47⋊#95—41no—
43C47⋊#95C2×C411yescharacteristic
44C47⋊#95C2×C411yescharacteristic
45C22×D4×C47D411yescharacteristic
46C23×C4×C47C2×C411yescharacteristic
47C23×C4×C47D411yescharacteristic
48C4×D4×C47—41no—
49C4×D4×C47—41no—
50C2×C4⋊C4×C47C2×C411yescharacteristic
51C2×C22⋊C4×C47D411yescharacteristic
52C2×C22⋊C4×C47C2311yescharacteristic
53C2×C42×C47D411yescharacteristic
54C4⋊C8×C47—41no—
55C22⋊C8×C47—22no—
56C4×C8×C47—41no—
57C24×C47C22⋊C411yescharacteristic
58C24×C47C2×D411yescharacteristic
59C2×D4×C47—22no—
60C2×D4×C47—22no—
61C2×D4×C47—41no—
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×C47C22⋊C411yescharacteristic
69C22×C4×C47C22⋊C411yescharacteristic
70C22×C4×C47C22⋊C411yescharacteristic
71C22×C4×C47C22×C411yescharacteristic
72C22×C4×C47C2×D411yescharacteristic
73C22×Dic47—21no—
74C22×Dic47—21no—
75C22×Dic47—21no—
76C22×Dic47—21no—
77C22×Dic47—42no—
78C2×C8×C47—42no—
79C4⋊C4×C47—41no—
80C22⋊C4×C47—41no—
81C22⋊C4×C47—41no—
82C42×C47—21no—
83C42×C47—21no—
84C22⋊Dic47—24no—
85C22⋊Dic47—42no—
86C22⋊Dic47—42no—
87C22⋊Dic47—42no—
88C22⋊Dic47—42no—
89C22⋊Dic47—42no—
90C4⋊Dic47—41no—
91C4⋊Dic47—41no—
92Dic47⋊C4—42no—
93Dic47⋊C4—42no—
94C4×Dic47—42no—
95C4×Dic47—41no—
96C4×Dic47—41no—
97C4×Dic47—82no—
98C23×C47—21no—
99C23×C47—21no—
100C23×C47—21no—
101C23×C47—21no—
102C23×C47—21no—
103C23×C47—41no—
104C23×C47—41no—
105C23×C47—41no—
106C23×C47—41no—
107C23×C47C2×C22⋊C411yescharacteristic
108C23×C47#13512yes—
109D4×C47—44no—
110C2×C4×C47—21no—
111C2×C4×C47—21no—
112C2×C4×C47—22no—
113C2×C4×C47—21no—
114C2×C4×C47—21no—
115C2×C4×C47—21no—
116C2×C4×C47—42no—
117C2×C4×C47—42no—
118C2×C4×C47—42no—
119C2×C4×C47—41no—
120C2×C4×C47—41no—
121C2×C4×C47—41no—
122C2×C4×C47—81no—
123C2×C4×C47#9911yescharacteristic
124C2×C4×C47#9911yescharacteristic
125C2×C4×C47#10412yes—
126C2×Dic47—22no—
127C2×Dic47—41no—
128C2×Dic47—41no—
129C2×Dic47—41no—
130C2×Dic47—41no—
131C2×Dic47—44no—
132C2×Dic47—41no—
133C2×Dic47—41no—
134C2×Dic47—41no—
135C2×Dic47—41no—
136C2×Dic47—41no—
137C2×Dic47—41no—
138C2×Dic47—41no—
139C2×Dic47—82no—
140C2×Dic47—82no—
141C8×C47—82no—
142#11727—471nomaximal, Sylow
143C22×C47—21no—
144C22×C47—21no—
145C22×C47—22no—
146C22×C47—42no—
147C22×C47—41no—
148C22×C47—41no—
149C22×C47—41no—
150C22×C47—41no—
151C22×C47—41no—
152C22×C47—81no—
153C22×C47—81no—
154C22×C47—81no—
155C22×C47C2×#9911yescharacteristic
156C22×C47#41611yescharacteristic
157C22×C47C2×#10411yescharacteristic
158C4×C47—21no—
159C4×C47—22no—
160C4×C47—41no—
161C4×C47—41no—
162C4×C47—41no—
163C4×C47#35212yes—
164Dic47—42no—
165Dic47—81no—
166Dic47—84no—
167Dic47—81no—
168Dic47—81no—
169#2395—471no—
170C4×#99—471no—
171#1460D4711yescharacteristic
172#1352—472no—
173#1344—472no—
174C2×C47—21no—
175C2×C47—21no—
176C2×C47—42no—
177C2×C47—81no—
178C2×C47#150111yescharacteristic
179C2×C47C2×#35211yescharacteristic
180C2×C47#212611yescharacteristic
181C2×C4×D4C2×D4711yescharacteristic
182C2×#117—471no—
183#431—21no—
184C2×#99—941no—
185#400—471no—
186#399—471no—
187#398—471no—
188#397—471no—
189#397—941no—
190#394—941no—
191C4×C4⋊C4—941no—
192C4×C22⋊C4—471no—
193C4×C22⋊C4—941no—
194C4×C22⋊C4—941no—
195#342—472no—
196#323—21no—
197#323C2×D4712yes—
198C47#1172711yescharacteristic, complemented, Sylow
199C22×D4D18811yescharacteristic
200C23×C4C4×D4711yescharacteristic
201C23×C4D18811yescharacteristic
202C4×D4—41no—
203C4×D4—41no—
204#117—1881no—
205#117—1881no—
206C2×C4⋊C4—471no—
207C2×C4⋊C4—471no—
208C2×C4⋊C4—941no—
209C2×C4⋊C4—941no—
210C2×C4⋊C4C4×D4711yescharacteristic
211C2×C22⋊C4—942no—
212C2×C22⋊C4—941no—
213C2×C22⋊C4—941no—
214C2×C22⋊C4—941no—
215C2×C22⋊C4—941no—
216C2×C22⋊C4C22⋊D4711yescharacteristic
217C2×C22⋊C4C22×D4711yescharacteristic
218C2×C42—941no—
219C2×C42—941no—
220C2×C42—1881no—
221C2×C42C22⋊D4711yescharacteristic
222C4⋊C8—41no—
223#99—944no—
224C22⋊C8—22no—
225C4×C8—41no—
226#95—471no—
227#95—471no—
228#95—941no—
229#95—941no—
230#95—941no—
231#95—941no—
232#95—1881no—
233C24(C2×D47)⋊C411yescharacteristic
234C24C2×D18811yescharacteristic
235C2×D4—22no—
236C2×D4—22no—
237C2×D4—41no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—941no—
245C22×C4—941no—
246C22×C4—941no—
247C22×C4—941no—
248C22×C4—1882no—
249C22×C4(C2×D47)⋊C411yescharacteristic
250C22×C4(C2×D47)⋊C411yescharacteristic
251C22×C4(C2×D47)⋊C411yescharacteristic
252C22×C4C2×C4×D4711yescharacteristic
253C22×C4C2×C22⋊D4711yescharacteristic
254C2×C8—42no—
255C4⋊C4—41no—
256C4⋊C4—1881no—
257C4⋊C4—1881no—
258C4⋊C4—1882no—
259C4⋊C4—1882no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—944no—
263C22⋊C4—1882no—
264C22⋊C4—1882no—
265C22⋊C4—1882no—
266C22⋊C4—1882no—
267C22⋊C4—1882no—
268C42—21no—
269C42—21no—
270C42—3762no—
271C42—1882no—
272C42—1881no—
273C42—1881no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23C2×(C2×D47)⋊C411yescharacteristic
284C23#1309404612yes—
285D4—44no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—81no—
299C2×C4—942no—
300C2×C4—3762no—
301C2×C4—3762no—
302C2×C4—1881no—
303C2×C4—1884no—
304C2×C4—1881no—
305C2×C4—1881no—
306C2×C4—1881no—
307C2×C4—1881no—
308C2×C4—1881no—
309C2×C4—1881no—
310C2×C4—1881no—
311C2×C4—1881no—
312C2×C4—1881no—
313C2×C4—1881no—
314C2×C4#1309393811yescharacteristic
315C2×C4#1309393811yescharacteristic
316C2×C4#1309395012yes—
317C8—82no—
318C22—21no—
319C22—21no—
320C22—22no—
321C22—42no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22C2×#1309393811yescharacteristic
331C22#1568912011yescharacteristic
332C22C2×#1309395011yescharacteristic
333C4—21no—
334C4—22no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—3761no—
339C4—3764no—
340C4—3761no—
341C4—3761no—
342C4—1882no—
343C4#1568891412yes—
344C2—21no—
345C2—21no—
346C2—42no—
347C2—81no—
348C2#2024188011yescharacteristic
349C2C2×#1568891411yescharacteristic
350C2#2024505111yescharacteristic
subgroupG/H
centerC2×C4#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C22⋊C4C22×D47
FittingC47×#1460C2
socleC22×C47C2×#99
prime radicalC22×D4×C47D4
cosocleC2×C22⋊C4×C47C23
hypercenter#1460D47
hypocenterC47#11727
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1460

Complements

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

NH ≅ G/Ntype
3. #20241413C2semidirect
4. #20241389C2semidirect
6. C2×#13093983C4semidirect
8. #15689062C4semidirect
10. #15689053C4semidirect
17. C4×C22⋊Dic47C4semidirect
198. C47#11727semidirect

Maximal subgroups

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

subgroupindex
#202466472
C4×#130939392
#202414132
#202413892
C47×#14602
#1172747