Finite Group Databasebeta

← #26111370

Subgroups — #26111370

Order 12032.

482 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 111 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#20252026C211yesmaximal, characteristic
2#20246656C211yesmaximal, characteristic
3#20241818C212yesmaximal, complemented
4#20241290C212yesmaximal, complemented
5C47×#1458C211yesmaximal, characteristic
6C2×C4×Dic94C2211yescharacteristic
7#15689184—24no—
8#15689147C2212yes—
9#15689087—21no—
10C4×#10625329—21no—
11#15689062C411yescharacteristic
12#15689059C411yescharacteristic
13#15689056—21no—
14#15689046—21no—
15#15689043C411yescharacteristic
16#15689035C411yescharacteristic
17C4×Dic47⋊C4—21no—
18C4×C4⋊Dic47C2211yescharacteristic
19C4×C22⋊Dic47—21no—
20#15688855—24no—
21C2×#117×C47C2211yescharacteristic
22C47×#431C2212yes—
23C47×#324—21no—
24C47×#323—21no—
25C22×Dic94C2×C411yescharacteristic
26C2×#10625329—22no—
27C4×Dic94—22no—
28C4×Dic94—22no—
29C2×C4⋊Dic47—21no—
30C2×C4⋊Dic47C2×C411yescharacteristic
31C2×Dic47⋊C4—21no—
32C2×Dic47⋊C4—21no—
33C2×Dic47⋊C4—21no—
34C2×C22⋊Dic47—21no—
35C2×C22⋊Dic47—21no—
36C2×C4×Dic47—21no—
37C2×C4×Dic47—21no—
38C4⋊(C47⋊C8)—22no—
39C4⋊(C47⋊C8)D414yes—
40C22⋊(C47⋊C8)—24no—
41#13093935—22no—
42C4×C47⋊C8—22no—
43C47⋊#95—21no—
44C47⋊#95—21no—
45C47⋊#95—21no—
46C47⋊#95—21no—
47C47⋊#95C2×C411yescharacteristic
48C47⋊#95C2×C411yescharacteristic
49C23×C4×C47D411yescharacteristic
50#117×C47—22no—
51#117×C47—22no—
52C2×C4⋊C4×C47C2×C411yescharacteristic
53C2×C4⋊C4×C47D411yescharacteristic
54C2×C22⋊C4×C47C2×C411yescharacteristic
55C2×C22⋊C4×C47D411yescharacteristic
56C2×C42×C47D411yescharacteristic
57C2×C42×C47C2311yescharacteristic
58C4⋊C8×C47—22no—
59C22⋊C8×C47—22no—
60C4×C8×C47—22no—
61C24×C47C22⋊C411yescharacteristic
62C22×C4×C47—21no—
63C22×C4×C47—21no—
64C22×C4×C47—22no—
65C22×C4×C47—22no—
66C22×C4×C47C22⋊C411yescharacteristic
67C22×C4×C47C22⋊C411yescharacteristic
68C22×C4×C47C22⋊C411yescharacteristic
69C22×C4×C47C22×C411yescharacteristic
70C22×C4×C47C2×D411yescharacteristic
71C22×C4×C47C2×D411yescharacteristic
72C22×Dic47—21no—
73C22×Dic47—21no—
74C22×Dic47—21no—
75C22×Dic47—21no—
76C2×Dic94—22no—
77C2×Dic94—42no—
78C2×C8×C47—42no—
79C4⋊C4×C47—22no—
80C4⋊C4×C47—22no—
81C22⋊C4×C47—22no—
82C22⋊C4×C47—22no—
83C42×C47—21no—
84C42×C47—21no—
85C42×C47C2×D412yes—
86C42×C47#4112yes—
87C22⋊Dic47—42no—
88C22⋊Dic47—42no—
89C4⋊Dic47—22no—
90C4⋊Dic47—42no—
91Dic47⋊C4—42no—
92Dic47⋊C4—42no—
93Dic47⋊C4—42no—
94C4×Dic47—41no—
95C4×Dic47—41no—
96C4×Dic47—42no—
97#10625329—42no—
98#10625329—42no—
99C2×C47⋊C8—24no—
100C23×C47—21no—
101C23×C47—21no—
102C23×C47—21no—
103C23×C47—41no—
104C23×C47—41no—
105C23×C47C2×C22⋊C411yescharacteristic
106C2×C4×C47—21no—
107C2×C4×C47—21no—
108C2×C4×C47—21no—
109C2×C4×C47—21no—
110C2×C4×C47—21no—
111C2×C4×C47—41no—
112C2×C4×C47—41no—
113C2×C4×C47—42no—
114C2×C4×C47—42no—
115C2×C4×C47—41no—
116C2×C4×C47—41no—
117C2×C4×C47C4×D412yes—
118C2×C4×C47#12011yescharacteristic
119C2×C4×C47#12112yes—
120C2×C4×C47#12311yescharacteristic
121C2×C4×C47#13512yes—
122C2×C4×C47#13712yes—
123C2×Dic47—22no—
124C2×Dic47—22no—
125C2×Dic47—41no—
126C2×Dic47—41no—
127C2×Dic47—41no—
128C2×Dic47—41no—
129C2×Dic47—41no—
130C2×Dic47—41no—
131C2×Dic47—41no—
132C2×Dic47—41no—
133C2×Dic47—41no—
134C2×Dic47—41no—
135Dic94—24no—
136Dic94—42no—
137C8×C47—82no—
138C47⋊C8—44no—
139#9034—471nomaximal, Sylow
140C22×C47—21no—
141C22×C47—21no—
142C22×C47—42no—
143C22×C47—41no—
144C22×C47—41no—
145C22×C47—41no—
146C22×C47—81no—
147C22×C47#38611yescharacteristic
148C22×C47#41611yescharacteristic
149C22×C47#41911yescharacteristic
150C4×C47—22no—
151C4×C47—22no—
152C4×C47—41no—
153C4×C47—41no—
154C4×C47—41no—
155C4×C47#45212yes—
156C4×C47#48512yes—
157Dic47—42no—
158Dic47—42no—
159Dic47—81no—
160Dic47—81no—
161#2921—471no—
162#2396—471no—
163#1487—472no—
164#1458D4711yescharacteristic
165#1303—472no—
166C2×C47—21no—
167C2×C47—21no—
168C2×C47—81no—
169C2×C47#187311yescharacteristic
170C2×C47#188411yescharacteristic
171C2×C47#189111yescharacteristic
172C2×C4×Q8—471no—
173C2×#117C2×D4711yescharacteristic
174#433—944no—
175#431C2×D4712yes—
176#423—472no—
177#407—941no—
178C4×M16—941no—
179#400—471no—
180#399—471no—
181#398—941no—
182#396—941no—
183#395—471no—
184#393—471no—
185C4×C4⋊C4—471no—
186C4×C4⋊C4—941no—
187C4×C22⋊C4—941no—
188Q8⋊C8—944no—
189#324—21no—
190#323—21no—
191C47#903411yescharacteristic, complemented, Sylow
192C22×Q8—471no—
193C23×C4D18811yescharacteristic
194C2×M16—942no—
195C4×Q8—942no—
196C4×Q8—942no—
197#117—22no—
198#117—22no—
199C2×C4⋊C4—471no—
200C2×C4⋊C4—941no—
201C2×C4⋊C4—941no—
202C2×C4⋊C4—941no—
203C2×C4⋊C4—941no—
204C2×C4⋊C4C4×D4711yescharacteristic
205C2×C4⋊C4C22⋊D4711yescharacteristic
206C2×C22⋊C4—941no—
207C2×C22⋊C4—941no—
208C2×C22⋊C4C4×D4711yescharacteristic
209C2×C22⋊C4C22⋊D4711yescharacteristic
210C2×C42—941no—
211C2×C42—941no—
212C2×C42D18811yescharacteristic
213C2×C42C22×D4711yescharacteristic
214C4⋊C8—22no—
215C4⋊C8—474no—
216C4⋊C8—942no—
217C22⋊C8—22no—
218C22⋊C8—944no—
219#97—942no—
220C4×C8—22no—
221C4×C8—942no—
222#95—471no—
223#95—471no—
224#95—941no—
225#95—941no—
226#95—941no—
227#95—941no—
228C24(C2×D47)⋊C411yescharacteristic
229C2×Q8—942no—
230C2×Q8—1882no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—941no—
236C22×C4—941no—
237C22×C4—941no—
238C22×C4—941no—
239C22×C4(C2×D47)⋊C411yescharacteristic
240C22×C4(C2×D47)⋊C411yescharacteristic
241C22×C4(C2×D47)⋊C411yescharacteristic
242C22×C4C2×C4×D4711yescharacteristic
243C22×C4C2×D18811yescharacteristic
244C22×C4C2×C22⋊D4711yescharacteristic
245M16—1882no—
246M16—1882no—
247C2×C8—42no—
248C2×C8—944no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—942no—
252C4⋊C4—1882no—
253C4⋊C4—1882no—
254C4⋊C4—1882no—
255C4⋊C4—1882no—
256C22⋊C4—22no—
257C22⋊C4—22no—
258C22⋊C4—1882no—
259C22⋊C4—1882no—
260C42—21no—
261C42—21no—
262C42—1881no—
263C42—1881no—
264C42—1882no—
265C42D4×D4712yes—
266C42#1062535212yes—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—41no—
271C23—41no—
272C23C2×(C2×D47)⋊C411yescharacteristic
273Q8—944no—
274Q8—1882no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—942no—
287C2×C4—942no—
288C2×C4—1881no—
289C2×C4—1881no—
290C2×C4—1881no—
291C2×C4—1881no—
292C2×C4—1881no—
293C2×C4—1881no—
294C2×C4—1881no—
295C2×C4—1881no—
296C2×C4—1881no—
297C2×C4—1881no—
298C2×C4(C22⋊D47)⋊C412yes—
299C2×C4#1309399411yescharacteristic
300C2×C4#1309399612yes—
301C2×C4#1309400811yescharacteristic
302C2×C4#1309404612yes—
303C2×C4#1309405712yes—
304C8—82no—
305C8—1884no—
306C22—21no—
307C22—21no—
308C22—42no—
309C22—41no—
310C22—41no—
311C22—41no—
312C22—81no—
313C22#1568900011yescharacteristic
314C22#1568912011yescharacteristic
315C22#1568913311yescharacteristic
316C4—22no—
317C4—22no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—3761no—
322C4—3761no—
323C4—1882no—
324C4—1882no—
325C4#1568928712yes—
326C4#1568946512yes—
327C2—21no—
328C2—21no—
329C2—81no—
330C2#2024369811yescharacteristic
331C2#2024376911yescharacteristic
332C2#2024380411yescharacteristic
subgroupG/H
centerC2×C4#13093994
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1458C2
socleC22×C47#386
prime radicalC24×C47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#1458D47
hypocenterC47#9034
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1458

Complements

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

NH ≅ G/Ntype
3. #20241818C2semidirect
4. #20241290C2semidirect
191. C47#9034semidirect

Maximal subgroups

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

subgroupindex
#202520262
#202466562
#202418182
#202412902
C47×#14582
#903447