Finite Group Databasebeta

← #26113618

Subgroups — #26113618

Order 12032.

532 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 107 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#20241989C212yesmaximal, complemented
3#20241722C211yesmaximal, characteristic
4#20241282C212yesmaximal, complemented
5C47×#2873C211yesmaximal, characteristic
6C2×#13094016C2211yescharacteristic
7#15689180C2212yes—
8#15689062C411yescharacteristic
9#15689059C411yescharacteristic
10#15689053C411yescharacteristic
11#15689048C411yescharacteristic
12#15689018—21no—
13#15689017—21no—
14#15689007—21no—
15#15688996—21no—
16#15688969C2211yescharacteristic
17#15688864—24no—
18#15688849—21no—
19#15688847—21no—
20C22×M16×C47—21no—
21C2×#117×C47C2211yescharacteristic
22C47×#405—24no—
23C4×M16×C47—21no—
24C4×M16×C47C2212yes—
25#13094016—22no—
26#13094016—22no—
27C2×C4⋊Dic47C2×C411yescharacteristic
28C2×C4⋊Dic47C2×C411yescharacteristic
29C2×Dic47⋊C4—21no—
30C2×Dic47⋊C4—21no—
31C2×Dic47⋊C4—21no—
32C2×Dic47⋊C4—21no—
33C2×C22⋊Dic47—21no—
34C2×C22⋊Dic47—21no—
35C2×C4×Dic47—21no—
36C4⋊(C47⋊C8)—22no—
37C22⋊(C47⋊C8)—22no—
38#13093935—22no—
39C47⋊#95—21no—
40C47⋊#95—21no—
41C47⋊#95—21no—
42C47⋊#95—21no—
43C47⋊#95—21no—
44C47⋊#95C2×C411yescharacteristic
45C47⋊#95C2×C411yescharacteristic
46C23×C4×C47D411yescharacteristic
47C2×M16×C47—24no—
48C2×M16×C47—24no—
49C2×M16×C47—22no—
50C2×M16×C47—22no—
51C22×C8×C47—22no—
52#117×C47—22no—
53#117×C47—22no—
54C2×C4⋊C4×C47C2×C411yescharacteristic
55C2×C4⋊C4×C47D411yescharacteristic
56C2×C22⋊C4×C47C2×C411yescharacteristic
57C2×C22⋊C4×C47D411yescharacteristic
58C2×C42×C47D411yescharacteristic
59C2×C42×C47C2311yescharacteristic
60#97×C47—22no—
61#97×C47D414yes—
62C4×C8×C47—22no—
63C4×C8×C47—22no—
64C24×C47C22⋊C411yescharacteristic
65C22×C4×C47—22no—
66C22×C4×C47—22no—
67C22×C4×C47—21no—
68C22×C4×C47—21no—
69C22×C4×C47C22⋊C411yescharacteristic
70C22×C4×C47C22⋊C411yescharacteristic
71C22×C4×C47C22⋊C411yescharacteristic
72C22×C4×C47C22×C411yescharacteristic
73C22×C4×C47C2×D411yescharacteristic
74C22×C4×C47C2×D411yescharacteristic
75C22×Dic47—21no—
76C22×Dic47—21no—
77C22×Dic47—21no—
78C22×Dic47—21no—
79M16×C47—28no—
80M16×C47—24no—
81M16×C47—24no—
82C2×C8×C47—28no—
83C2×C8×C47—24no—
84C4⋊C4×C47—22no—
85C4⋊C4×C47—22no—
86C22⋊C4×C47—22no—
87C22⋊C4×C47—22no—
88C42×C47—21no—
89C42×C47—21no—
90C42×C47C2×D412yes—
91C42×C47#4112yes—
92C22⋊Dic47—42no—
93C22⋊Dic47—42no—
94C4⋊Dic47—22no—
95C4⋊Dic47—22no—
96Dic47⋊C4—42no—
97Dic47⋊C4—42no—
98Dic47⋊C4—41no—
99Dic47⋊C4—42no—
100Dic47⋊C4—41no—
101C4×Dic47—42no—
102C2×C47⋊C8—42no—
103C23×C47—21no—
104C23×C47—21no—
105C23×C47—41no—
106C23×C47—41no—
107C23×C47#10412yes—
108C23×C47C2×C22⋊C411yescharacteristic
109C2×C4×C47—22no—
110C2×C4×C47—22no—
111C2×C4×C47—24no—
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#10412yes—
123C2×C4×C47C4×D412yes—
124C2×C4×C47#12011yescharacteristic
125C2×C4×C47#12112yes—
126C2×C4×C47#12311yescharacteristic
127C2×Dic47—22no—
128C2×Dic47—41no—
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—
139C8×C47—28no—
140C47⋊C8—82no—
141#9388—471nomaximal, Sylow
142C22×C47—21no—
143C22×C47—21no—
144C22×C47—22no—
145C22×C47—42no—
146C22×C47—41no—
147C22×C47—41no—
148C22×C47—81no—
149C22×C47#38611yescharacteristic
150C22×C47C2×#10411yescharacteristic
151C22×C47C2×#10411yescharacteristic
152C4×C47—22no—
153C4×C47—21no—
154C4×C47—22no—
155C4×C47—21no—
156C4×C47—22no—
157C4×C47—41no—
158C4×C47—41no—
159Dic47—42no—
160Dic47—81no—
161Dic47—81no—
162Dic47—81no—
163#2873D4711yescharacteristic
164#2611—471no—
165#1528—472no—
166#1464—471no—
167#1300—472no—
168C2×C47—21no—
169C2×C47—21no—
170C2×C47—42no—
171C2×C47#188411yescharacteristic
172C2×C47#189611yescharacteristic
173C2×C47#191511yescharacteristic
174C22×M16—21no—
175C2×#125—471no—
176C2×#117C2×D4711yescharacteristic
177#432—472no—
178#405—24no—
179C4×M16—21no—
180C4×M16C2×D4712yes—
181#400—471no—
182#399—471no—
183#398—471no—
184#397—471no—
185#389—941no—
186#388—941no—
187#387—941no—
188#385—941no—
189#376—471no—
190#330—944no—
191#324—941no—
192#323—941no—
193C47#938811yescharacteristic, complemented, Sylow
194C23×C4C22⋊D4711yescharacteristic
195C2×M16—24no—
196C2×M16—24no—
197C2×M16—22no—
198C2×M16—22no—
199C22×C8—22no—
200#125—942no—
201#125—942no—
202#117—22no—
203#117—22no—
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—941no—
213C2×C22⋊C4—941no—
214C2×C22⋊C4C4×D4711yescharacteristic
215C2×C22⋊C4D18811yescharacteristic
216C2×C42—941no—
217C2×C42C22⋊D4711yescharacteristic
218C2×C42C22×D4711yescharacteristic
219C4⋊C8—942no—
220C22⋊C8—942no—
221#97—22no—
222#97—942no—
223#97C22⋊D4714yes—
224C4×C8—22no—
225C4×C8—22no—
226#95—471no—
227#95—471no—
228#95—941no—
229#95—941no—
230#95—941no—
231#95—941no—
232#95—941no—
233C24(C2×D47)⋊C411yescharacteristic
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—21no—
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×C4C2×C4×D4711yescharacteristic
246C22×C4C2×D18811yescharacteristic
247C22×C4C2×C22⋊D4711yescharacteristic
248M16—28no—
249M16—24no—
250M16—24no—
251C2×C8—28no—
252C2×C8—24no—
253C2×C8—1882no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—942no—
257C4⋊C4—942no—
258C4⋊C4—1882no—
259C4⋊C4—1882no—
260C4⋊C4—1881no—
261C4⋊C4—1882no—
262C4⋊C4—1881no—
263C22⋊C4—22no—
264C22⋊C4—22no—
265C22⋊C4—1882no—
266C22⋊C4—1882no—
267C42—21no—
268C42—21no—
269C42—1882no—
270C42#1062535012yes—
271C42C2×C22⋊D4712yes—
272C23—21no—
273C23—21no—
274C23—41no—
275C23—41no—
276C23#1309395212yes—
277C23C2×(C2×D47)⋊C411yescharacteristic
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—24no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—942no—
292C2×C4—1881no—
293C2×C4—1881no—
294C2×C4—1881no—
295C2×C4—1881no—
296C2×C4—1881no—
297C2×C4—1881no—
298C2×C4—1881no—
299C2×C4—1881no—
300C2×C4—1881no—
301C2×C4—1881no—
302C2×C4—1881no—
303C2×C4#1309395212yes—
304C2×C4C4×C22⋊D4712yes—
305C2×C4#1309399311yescharacteristic
306C2×C4#1309399812yes—
307C2×C4#1309400711yescharacteristic
308C8—28no—
309C8—3762no—
310C22—21no—
311C22—21no—
312C22—22no—
313C22—42no—
314C22—41no—
315C22—41no—
316C22—81no—
317C22#1568900211yescharacteristic
318C22C2×#1309395211yescharacteristic
319C22C2×#1309395211yescharacteristic
320C4—22no—
321C4—21no—
322C4—22no—
323C4—21no—
324C4—22no—
325C4—41no—
326C4—41no—
327C4—3761no—
328C4—3761no—
329C4—3761no—
330C4—1882no—
331C2—21no—
332C2—21no—
333C2—42no—
334C2#2024377111yescharacteristic
335C2#2024383111yescharacteristic
336C2#2024394011yescharacteristic
subgroupG/H
centerC22#15689002
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2873C2
socleC22×C47#386
prime radicalC24×C47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#2873D47
hypocenterC47#9388
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2873

Complements

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

NH ≅ G/Ntype
2. #20241989C2semidirect
4. #20241282C2semidirect
193. C47#9388semidirect

Maximal subgroups

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

subgroupindex
#202490862
#202419892
#202417222
#202412822
C47×#28732
#938847