Finite Group Databasebeta

← #26134599

Subgroups — #26134599

Order 12032.

540 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 103 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#20248387C211yesmaximal, characteristic
2C2×#15688919C214yesmaximal, complemented
3C4×#13093943C211yesmaximal, characteristic, complemented
4C47×#1801C211yesmaximal, characteristic, complemented
5C2×#13094021C411yescharacteristic
6C2×#13094013C2212yes—
7C2×#13093943C2212yes—
8C4×#10625329—21no—
9#15689040—21no—
10#15689040—21no—
11#15689025—21no—
12#15689025—21no—
13#15689024C412yes—
14C4×C4⋊Dic47C411yescharacteristic
15#15688919—28no—
16#15688892—21no—
17C2×C4×Q8×C47C2211yescharacteristic
18C47×#423—21no—
19C2×#101×C47C2212yes—
20C47×#343—21no—
21C22×D188—21no—
22C22×D188—21no—
23C22×D188C2×C412yes—
24C2×#10625329—22no—
25#13094021—41no—
26#13094021—41no—
27#13094013—24no—
28#13094013—24no—
29C2×C4⋊Dic47—21no—
30C2×(C2×D47)⋊C4—22no—
31C2×(C2×D47)⋊C4—22no—
32C2×(C2×D47)⋊C4—22no—
33C2×(C2×D47)⋊C4—22no—
34C2×C4×Dic47C2×C412yes—
35#13093943—22no—
36#13093943D414yescomplemented
37#13093935—41no—
38C4×C47⋊C8—41no—
39C47⋊#95—21no—
40C22×Q8×C47C2311yescharacteristic
41C2×M16×C47—22no—
42C4×Q8×C47—41no—
43C4×Q8×C47—41no—
44C2×C4⋊C4×C47C2×C411yescharacteristic
45C2×C4⋊C4×C47D411yescharacteristic
46C2×C4⋊C4×C47D411yescharacteristic
47C2×C42×C47C2×C411yescharacteristic
48C2×C42×C47D411yescharacteristic
49C2×C42×C47D411yescharacteristic
50C4⋊C8×C47—41no—
51C4⋊C8×C47—41no—
52#101×C47—24no—
53C23×D47—22no—
54C2×Q8×C47—41no—
55C2×Q8×C47—41no—
56C2×Q8×C47C2×D412yes—
57C2×Q8×C47#4112yes—
58C22×C4×C47C22⋊C411yescharacteristic
59C22×C4×C47C22⋊C411yescharacteristic
60C22×C4×C47C22⋊C411yescharacteristic
61C22×C4×C47C22⋊C411yescharacteristic
62C22×C4×C47C22×C411yescharacteristic
63C22×C4×C47C2×D411yescharacteristic
64C22×C4×C47C2×D411yescharacteristic
65C22×Dic47—22no—
66C2×D188—24no—
67C2×D188—42no—
68C2×D188—42no—
69C2×D188—42no—
70C2×D188—42no—
71C2×D188—42no—
72C2×D188—41no—
73C2×D188—41no—
74C2×D188—82no—
75M16×C47—42no—
76M16×C47—42no—
77C2×C8×C47—42no—
78C4⋊C4×C47—41no—
79C4⋊C4×C47—41no—
80C4⋊C4×C47—41no—
81C42×C47—21no—
82C42×C47—21no—
83C42×C47—21no—
84C42×C47—21no—
85C42×C47—21no—
86C42×C47—21no—
87(C2×D47)⋊C4—44no—
88(C2×D47)⋊C4—42no—
89(C2×D47)⋊C4—42no—
90(C2×D47)⋊C4—82no—
91(C2×D47)⋊C4—82no—
92C4⋊Dic47—42no—
93C4×Dic47—24no—
94#10625329—24no—
95#10625329—42no—
96C2×C47⋊C8—42no—
97C23×C47C2×C22⋊C411yescharacteristic
98C22×D47—42no—
99C22×D47—42no—
100C22×D47—42no—
101C22×D47—42no—
102C22×D47—42no—
103C22×D47—42no—
104C22×D47—42no—
105Q8×C47—42no—
106Q8×C47—42no—
107C2×C4×C47—21no—
108C2×C4×C47—21no—
109C2×C4×C47—21no—
110C2×C4×C47—21no—
111C2×C4×C47—21no—
112C2×C4×C47—21no—
113C2×C4×C47—22no—
114C2×C4×C47—21no—
115C2×C4×C47—21no—
116C2×C4×C47—21no—
117C2×C4×C47—41no—
118C2×C4×C47—41no—
119C2×C4×C47—41no—
120C2×C4×C47#9911yescharacteristic
121C2×C4×C47#9911yescharacteristic
122C2×C4×C47C4×D412yes—
123C2×C4×C47#12112yes—
124C2×C4×C47#12411yescharacteristic
125C2×C4×C47#12711yescharacteristic
126C2×Dic47—42no—
127C2×Dic47—42no—
128C2×Dic47—42no—
129D188—84no—
130D188—82no—
131D188—84no—
132D188—82no—
133C8×C47—82no—
134C47⋊C8—44no—
135#13226—471nomaximal, Sylow
136C22×C47—21no—
137C22×C47—21no—
138C22×C47#39011yescharacteristic
139C22×C47C2×#9911yescharacteristic
140C22×C47#41011yescharacteristic
141C2×D47—82no—
142C2×D47—82no—
143C2×D47—82no—
144C2×D47—82no—
145C2×D47—82no—
146C2×D47—82no—
147C2×D47—82no—
148C4×C47—21no—
149C4×C47—22no—
150C4×C47—21no—
151C4×C47—22no—
152C4×C47—21no—
153C4×C47—41no—
154C4×C47—41no—
155C4×C47—41no—
156C4×C47#35512yescomplemented
157Dic47—82no—
158#2548—471no—
159C2×#355—474no—
160#1801D4711yescharacteristic, complemented
161C4×#101—471no—
162C2×C47—21no—
163C2×C47—21no—
164C2×C47#196411yescharacteristic
165C2×C47C2×#35511yescharacteristic
166C2×C47#213411yescharacteristic
167D47—162no—
168C2×#127—471no—
169C2×#124—472no—
170C2×C4×Q8C2×D4711yescharacteristic
171#423—21no—
172C2×#101—472no—
173C2×#101C2×D4712yes—
174C4×M16—941no—
175#394—941no—
176#394—941no—
177#390—472no—
178#390—941no—
179#390—941no—
180C4×C4⋊C4—471no—
181#355—948no—
182#343—21no—
183#343—941no—
184C47#1322611yescharacteristic, complemented, Sylow
185C22×Q8C22×D4711yescharacteristic
186C22×D4—472no—
187C22×D4—941no—
188C22×D4—941no—
189C2×M16—22no—
190C2×M16—942no—
191#127—1881no—
192#127—1881no—
193#124—944no—
194#124—944no—
195C4×Q8—41no—
196C4×Q8—41no—
197C2×C4⋊C4—941no—
198C2×C4⋊C4C4×D4711yescharacteristic
199C2×C4⋊C4D18811yescharacteristic
200C2×C4⋊C4C22⋊D4711yescharacteristic
201C2×C22⋊C4—942no—
202C2×C22⋊C4—942no—
203C2×C22⋊C4—942no—
204C2×C22⋊C4—942no—
205C2×C42—472no—
206C2×C42C4×D4711yescharacteristic
207C2×C42D18811yescharacteristic
208C2×C42C22⋊D4711yescharacteristic
209C4⋊C8—41no—
210C4⋊C8—41no—
211#101—24no—
212#101—474no—
213#101—942no—
214#97—1881no—
215C4×C8—1881no—
216#95—941no—
217C24—942no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8D4×D4712yes—
221C2×Q8#1062535412yes—
222C2×D4—944no—
223C2×D4—3762no—
224C2×D4—1882no—
225C2×D4—1882no—
226C2×D4—1882no—
227C2×D4—1882no—
228C2×D4—1882no—
229C2×D4—1881no—
230C2×D4—1881no—
231C22×C4—942no—
232C22×C4(C2×D47)⋊C411yescharacteristic
233C22×C4(C2×D47)⋊C411yescharacteristic
234C22×C4(C2×D47)⋊C411yescharacteristic
235C22×C4(C2×D47)⋊C411yescharacteristic
236C22×C4C2×C4×D4711yescharacteristic
237C22×C4C2×D18811yescharacteristic
238C22×C4C2×C22⋊D4711yescharacteristic
239M16—42no—
240M16—42no—
241M16—944no—
242M16—1882no—
243C2×C8—42no—
244C2×C8—1882no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—1882no—
249C22⋊C4—3762no—
250C22⋊C4—3762no—
251C22⋊C4—1884no—
252C22⋊C4—1882no—
253C22⋊C4—1882no—
254C42—21no—
255C42—21no—
256C42—21no—
257C42—21no—
258C42—21no—
259C42—21no—
260C42—944no—
261C23—1882no—
262C23—1882no—
263C23—1882no—
264C23—1882no—
265C23—1882no—
266C23—1882no—
267C23—1882no—
268C23C2×(C2×D47)⋊C411yescharacteristic
269Q8—42no—
270Q8—42no—
271D4—3764no—
272D4—3762no—
273D4—3764no—
274D4—3762no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—1882no—
289C2×C4—1882no—
290C2×C4—1882no—
291C2×C4#1309393811yescharacteristic
292C2×C4#1309393811yescharacteristic
293C2×C4#1309398612yes—
294C2×C4#1309399912yes—
295C2×C4#1309401311yescharacteristic
296C2×C4#1309402011yescharacteristic
297C8—82no—
298C8—1884no—
299C22—21no—
300C22—21no—
301C22—3762no—
302C22—3762no—
303C22—3762no—
304C22—3762no—
305C22—3762no—
306C22—3762no—
307C22—3762no—
308C22#1568902411yescharacteristic
309C22C2×#1309393811yescharacteristic
310C22#1568909611yescharacteristic
311C4—21no—
312C4—21no—
313C4—22no—
314C4—22no—
315C4—21no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—3762no—
320C4#1568891912yescomplemented
321C2—21no—
322C2—21no—
323C2—7522no—
324C2#2024422311yescharacteristic
325C2C2×#1568891911yescharacteristic
326C2#2024507911yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC22×Q8C22×D47
FittingC47×#1801C2
socleC22×C47C2×#99
prime radicalC2×#13094021C4
cosocleC22×Q8×C47C23
hypercenter#1801D47
hypocenterC47#13226
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1801

Complements

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

NH ≅ G/Ntype
2. C2×#15688919C2semidirect
3. C4×#13093943C2semidirect
4. C47×#1801C2semidirect
36. #13093943D4semidirect
156. C4×C47#355semidirect
160. #1801D47semidirect
184. C47#13226semidirect
320. C4#15688919semidirect

Maximal subgroups

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

subgroupindex
#202483872
C2×#156889192
C4×#130939432
C47×#18012
#1322647