Finite Group Databasebeta

← #26113626

Subgroups — #26113626

Order 12032.

446 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 95 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#20249086C211yesmaximal, characteristic
3#20241991C212yesmaximal, complemented
4#20241280C212yesmaximal, complemented
5C47×#1464C211yesmaximal, characteristic
6C2×#13094016C2211yescharacteristic
7#15689184—24no—
8#15689147—21no—
9#15689087—21no—
10C4×#10625329C2212yes—
11#15689062C411yescharacteristic
12#15689059C411yescharacteristic
13#15689053C411yescharacteristic
14#15689048C411yescharacteristic
15#15689018—21no—
16#15689017—21no—
17#15689007—21no—
18#15688996—21no—
19#15688969C2211yescharacteristic
20#15688864—24no—
21C2×#117×C47C2211yescharacteristic
22C47×#432C2212yes—
23C47×#324—21no—
24C47×#323—21no—
25C2×#10625329—22no—
26#13094016—22no—
27#13094016—22no—
28C2×C4⋊Dic47C2×C411yescharacteristic
29C2×C4⋊Dic47C2×C411yescharacteristic
30C2×Dic47⋊C4—21no—
31C2×Dic47⋊C4—21no—
32C2×Dic47⋊C4—21no—
33C2×Dic47⋊C4—21no—
34C2×C22⋊Dic47—21no—
35C2×C22⋊Dic47—21no—
36C2×C4×Dic47—21no—
37C4⋊(C47⋊C8)—24no—
38C22⋊(C47⋊C8)—24no—
39#13093935D414yes—
40C4×C47⋊C8—22no—
41C47⋊#95—21no—
42C47⋊#95—21no—
43C47⋊#95—21no—
44C47⋊#95—21no—
45C47⋊#95—21no—
46C47⋊#95C2×C411yescharacteristic
47C47⋊#95C2×C411yescharacteristic
48C23×C4×C47D411yescharacteristic
49#117×C47—22no—
50#117×C47—22no—
51C2×C4⋊C4×C47C2×C411yescharacteristic
52C2×C4⋊C4×C47D411yescharacteristic
53C2×C22⋊C4×C47C2×C411yescharacteristic
54C2×C22⋊C4×C47D411yescharacteristic
55C2×C42×C47D411yescharacteristic
56C2×C42×C47C2311yescharacteristic
57C4⋊C8×C47—22no—
58C22⋊C8×C47—22no—
59#97×C47—22no—
60C24×C47C22⋊C411yescharacteristic
61C22×C4×C47—21no—
62C22×C4×C47—21no—
63C22×C4×C47—41no—
64C22×C4×C47—41no—
65C22×C4×C47C22⋊C411yescharacteristic
66C22×C4×C47C22⋊C411yescharacteristic
67C22×C4×C47C22⋊C411yescharacteristic
68C22×C4×C47C22×C411yescharacteristic
69C22×C4×C47C2×D411yescharacteristic
70C22×C4×C47C2×D411yescharacteristic
71C22×Dic47—21no—
72C22×Dic47—21no—
73C22×Dic47—21no—
74C22×Dic47—21no—
75C2×C8×C47—42no—
76C4⋊C4×C47—22no—
77C4⋊C4×C47—22no—
78C22⋊C4×C47—22no—
79C22⋊C4×C47—22no—
80C42×C47—21no—
81C42×C47—21no—
82C42×C47C2×D412yes—
83C42×C47#4112yes—
84C22⋊Dic47—42no—
85C22⋊Dic47—42no—
86C4⋊Dic47—22no—
87C4⋊Dic47—22no—
88Dic47⋊C4—42no—
89Dic47⋊C4—42no—
90Dic47⋊C4—41no—
91Dic47⋊C4—42no—
92Dic47⋊C4—41no—
93C4×Dic47—42no—
94#10625329—42no—
95#10625329—42no—
96C2×C47⋊C8—24no—
97C23×C47—21no—
98C23×C47—21no—
99C23×C47—21no—
100C23×C47—41no—
101C23×C47—41no—
102C23×C47C2×C22⋊C411yescharacteristic
103C2×C4×C47—21no—
104C2×C4×C47—21no—
105C2×C4×C47—21no—
106C2×C4×C47—21no—
107C2×C4×C47—21no—
108C2×C4×C47—21no—
109C2×C4×C47—21no—
110C2×C4×C47—41no—
111C2×C4×C47—41no—
112C2×C4×C47—42no—
113C2×C4×C47—42no—
114C2×C4×C47—41no—
115C2×C4×C47—41no—
116C2×C4×C47C4×D412yes—
117C2×C4×C47#12011yescharacteristic
118C2×C4×C47#12112yes—
119C2×C4×C47#12311yescharacteristic
120C2×Dic47—22no—
121C2×Dic47—41no—
122C2×Dic47—41no—
123C2×Dic47—41no—
124C2×Dic47—41no—
125C2×Dic47—41no—
126C2×Dic47—41no—
127C2×Dic47—41no—
128C2×Dic47—41no—
129C2×Dic47—41no—
130C2×Dic47—41no—
131C2×Dic47—41no—
132C8×C47—82no—
133C47⋊C8—44no—
134#9390—471nomaximal, Sylow
135C22×C47—21no—
136C22×C47—21no—
137C22×C47—42no—
138C22×C47—41no—
139C22×C47—41no—
140C22×C47—41no—
141C22×C47—81no—
142C22×C47#38611yescharacteristic
143C22×C47#42111yescharacteristic
144C22×C47#42111yescharacteristic
145C4×C47—22no—
146C4×C47—21no—
147C4×C47—21no—
148C4×C47—22no—
149C4×C47—41no—
150C4×C47—41no—
151C4×C47—41no—
152Dic47—42no—
153Dic47—81no—
154Dic47—81no—
155Dic47—81no—
156#2921—471no—
157#2611—471no—
158#1528—472no—
159#1464D4711yescharacteristic
160#1300—472no—
161C2×C47—21no—
162C2×C47—21no—
163C2×C47—81no—
164C2×C47#188411yescharacteristic
165C2×C47#189711yescharacteristic
166C2×C47#191511yescharacteristic
167C2×#125—471no—
168C2×#117C2×D4711yescharacteristic
169#433—944no—
170#432C2×D4712yes—
171#423—941no—
172#407—941no—
173C4×M16—472no—
174#400—471no—
175#399—471no—
176#398—471no—
177#397—471no—
178#389—941no—
179#388—941no—
180#387—941no—
181#385—941no—
182#376—471no—
183#330—944no—
184#324—21no—
185#323—21no—
186C47#939011yescharacteristic, complemented, Sylow
187C23×C4D18811yescharacteristic
188C2×M16—942no—
189#125—942no—
190#125—942no—
191#117—22no—
192#117—22no—
193C2×C4⋊C4—471no—
194C2×C4⋊C4—471no—
195C2×C4⋊C4—941no—
196C2×C4⋊C4—941no—
197C2×C4⋊C4—941no—
198C2×C4⋊C4—941no—
199C2×C4⋊C4C4×D4711yescharacteristic
200C2×C4⋊C4C22⋊D4711yescharacteristic
201C2×C22⋊C4—941no—
202C2×C22⋊C4—941no—
203C2×C22⋊C4C4×D4711yescharacteristic
204C2×C22⋊C4C22⋊D4711yescharacteristic
205C2×C42—941no—
206C2×C42D18811yescharacteristic
207C2×C42C22×D4711yescharacteristic
208C4⋊C8—22no—
209C4⋊C8—944no—
210C22⋊C8—22no—
211C22⋊C8—944no—
212#97—22no—
213#97—474no—
214C4×C8—942no—
215#95—471no—
216#95—471no—
217#95—941no—
218#95—941no—
219#95—941no—
220#95—941no—
221#95—941no—
222C24(C2×D47)⋊C411yescharacteristic
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—941no—
228C22×C4—941no—
229C22×C4—941no—
230C22×C4—941no—
231C22×C4(C2×D47)⋊C411yescharacteristic
232C22×C4(C2×D47)⋊C411yescharacteristic
233C22×C4(C2×D47)⋊C411yescharacteristic
234C22×C4C2×C4×D4711yescharacteristic
235C22×C4C2×D18811yescharacteristic
236C22×C4C2×C22⋊D4711yescharacteristic
237M16—1882no—
238M16—1882no—
239C2×C8—42no—
240C2×C8—944no—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C4⋊C4—942no—
244C4⋊C4—942no—
245C4⋊C4—1882no—
246C4⋊C4—1882no—
247C4⋊C4—1881no—
248C4⋊C4—1882no—
249C4⋊C4—1881no—
250C22⋊C4—22no—
251C22⋊C4—22no—
252C22⋊C4—1882no—
253C22⋊C4—1882no—
254C42—21no—
255C42—21no—
256C42—1882no—
257C42D4×D4712yes—
258C42#1062535212yes—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—41no—
263C23—41no—
264C23C2×(C2×D47)⋊C411yescharacteristic
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—942no—
279C2×C4—1881no—
280C2×C4—1881no—
281C2×C4—1881no—
282C2×C4—1881no—
283C2×C4—1881no—
284C2×C4—1881no—
285C2×C4—1881no—
286C2×C4—1881no—
287C2×C4—1881no—
288C2×C4—1881no—
289C2×C4—1881no—
290C2×C4(C22⋊D47)⋊C412yes—
291C2×C4#1309399411yescharacteristic
292C2×C4#1309399612yes—
293C2×C4#1309400811yescharacteristic
294C8—82no—
295C8—1884no—
296C22—21no—
297C22—21no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—81no—
303C22#1568900011yescharacteristic
304C22#1568913911yescharacteristic
305C22#1568913911yescharacteristic
306C4—22no—
307C4—21no—
308C4—21no—
309C4—22no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—3761no—
314C4—3761no—
315C4—3761no—
316C4—1882no—
317C2—21no—
318C2—21no—
319C2—81no—
320C2#2024376911yescharacteristic
321C2#2024383411yescharacteristic
322C2#2024393811yescharacteristic
subgroupG/H
centerC22#15689000
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1464C2
socleC22×C47#386
prime radicalC24×C47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#1464D47
hypocenterC47#9390
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1464

Complements

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

NH ≅ G/Ntype
3. #20241991C2semidirect
4. #20241280C2semidirect
186. C47#9390semidirect

Maximal subgroups

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

subgroupindex
#202520262
#202490862
#202419912
#202412802
C47×#14642
#939047