Finite Group Databasebeta

← #26131470

Subgroups — #26131470

Order 12032.

512 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 125 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#20246101C211yesmaximal, characteristic
2C2×#15688920C211yesmaximal, characteristic
3#101×Dic47C212yesmaximal, complemented
4#20241514C212yesmaximal, complemented
5C2×C47×#355C211yesmaximal, characteristic
6C2×Q8×Dic47C2211yescharacteristic
7C2×D4×Dic47C411yescharacteristic
8C2×#13093943C2212yes—
9M16×Dic47—22no—
10#15689030C2211yescharacteristic
11#15689026C411yescharacteristic
12#15689016—22no—
13#15688998—22no—
14#15688993C411yescharacteristic
15#15688969C411yescharacteristic
16#15688920C414yescomplemented
17#15688891—22no—
18C2×#124×C47C2211yescharacteristic
19C2×#101×C47C2212yes—
20C47×#355C414yescomplemented
21C23×Dic47—21no—
22C2×#10625329—22no—
23Q8×Dic47—42no—
24D4×Dic47—42no—
25C2×C4⋊Dic47—22no—
26C2×C4⋊Dic47C2×C411yescharacteristic
27C2×C4⋊Dic47C2×C411yescharacteristic
28C2×C4⋊Dic47D412yes—
29C2×C22⋊Dic47—21no—
30C2×C22⋊Dic47—22no—
31C2×C4×Dic47C2×C411yescharacteristic
32C2×C4×Dic47C2×C411yescharacteristic
33C2×C4×Dic47D412yes—
34#13093943—22no—
35#13093943C2×C414yescomplemented
36#13093934—42no—
37C8×Dic47—42no—
38C47⋊#95—22no—
39C47⋊#95—21no—
40C47⋊#95—21no—
41C22×Q8×C47C2311yescharacteristic
42C22×D4×C47C2×C411yescharacteristic
43C2×M16×C47—22no—
44#124×C47—22no—
45#124×C47C2×C412yes—
46#124×C47C2×C412yes—
47C2×C22⋊C4×C47—22no—
48C2×C42×C47C2×C411yescharacteristic
49#101×C47—22no—
50#101×C47C2×C414yescomplemented
51C24×C47—21no—
52C2×Q8×C47—41no—
53C2×Q8×C47—41no—
54C2×Q8×C47C22×C411yescharacteristic
55C2×Q8×C47C22×C411yescharacteristic
56C2×Q8×C47#4112yes—
57C2×D4×C47—21no—
58C2×D4×C47—41no—
59C2×D4×C47—41no—
60C2×D4×C47C4212yes—
61C22×C4×C47—21no—
62C22×C4×C47C22×C411yescharacteristic
63C22×C4×C47C2×D412yes—
64C22×Dic47—21no—
65C22×Dic47—21no—
66C22×Dic47—42no—
67C22×Dic47—82no—
68C22×Dic47C22⋊C411yescharacteristic
69C22×Dic47C22⋊C412yes—
70C22×Dic47C22⋊C411yescharacteristic
71M16×C47—24no—
72M16×C47—42no—
73C2×C8×C47—42no—
74C22⋊C4×C47—24no—
75C22⋊C4×C47—24no—
76C42×C47—21no—
77C42×C47C4212yes—
78C22⋊Dic47—42no—
79C22⋊Dic47—82no—
80C4⋊Dic47—41no—
81C4⋊Dic47—42no—
82C4⋊Dic47—41no—
83C4⋊Dic47—82no—
84C4×Dic47—22no—
85C4×Dic47—22no—
86C4×Dic47—22no—
87C4×Dic47—41no—
88#10625329—24no—
89#10625329—42no—
90C2×C47⋊C8—42no—
91C23×C47—22no—
92C23×C47—42no—
93C23×C47—42no—
94C23×C47—41no—
95C23×C47—41no—
96C23×C47C2×C22⋊C411yescharacteristic
97Q8×C47—41no—
98Q8×C47—41no—
99Q8×C47—42no—
100D4×C47—42no—
101D4×C47—42no—
102C2×C4×C47—22no—
103C2×C4×C47—21no—
104C2×C4×C47—22no—
105C2×C4×C47—21no—
106C2×C4×C47—22no—
107C2×C4×C47—41no—
108C2×C4×C47—41no—
109C2×C4×C47C2×C4211yescharacteristic
110C2×C4×C47#11711yescharacteristic
111C2×C4×C47C4×D412yes—
112C2×C4×C47C4×D412yes—
113C2×Dic47—21no—
114C2×Dic47—22no—
115C2×Dic47—41no—
116C2×Dic47—42no—
117C2×Dic47—41no—
118C2×Dic47—42no—
119C2×Dic47—41no—
120C2×Dic47—41no—
121C2×Dic47—41no—
122C2×Dic47—41no—
123C2×Dic47—84no—
124C2×Dic47#9912yes—
125C8×C47—44no—
126C47⋊C8—44no—
127#12632—471nomaximal, Sylow
128C22×C47—21no—
129C22×C47—21no—
130C22×C47—44no—
131C22×C47—42no—
132C22×C47—82no—
133C22×C47—81no—
134C22×C47—81no—
135C22×C47C4×C22⋊C411yescharacteristic
136C22×C47C2×#9911yescharacteristic
137C22×C47#41011yescharacteristic
138C4×C47—22no—
139C4×C47—22no—
140C4×C47—21no—
141C4×C47—21no—
142C4×C47—42no—
143Dic47—22no—
144Dic47—42no—
145Dic47—41no—
146Dic47—81no—
147Dic47—81no—
148#2326—471no—
149C2×#355—471no—
150C2×#355D4711yescharacteristic
151C4×#101—472no—
152#1394—472no—
153C2×C47—21no—
154C2×C47—21no—
155C2×C47—82no—
156C2×C47C4×#9911yescharacteristic
157C2×C47#213311yescharacteristic
158C2×C47#213411yescharacteristic
159C2×#124C2×D4711yescharacteristic
160C2×C4×Q8—471no—
161C2×C4×D4—471no—
162C2×#101—472no—
163C2×#101C2×D4712yes—
164C4×M16—942no—
165#391—471no—
166#390—471no—
167#388—942no—
168#385—942no—
169#384—471no—
170#376—471no—
171#355—474no—
172#355Dic4714yescomplemented
173#343—942no—
174C47#1263211yescharacteristic, complemented, Sylow
175C22×Q8C22×D4711yescharacteristic
176C22×D4C4×D4711yescharacteristic
177C23×C4—941no—
178C2×M16—22no—
179C2×M16—942no—
180#124—22no—
181#124C4×D4712yes—
182#124C2×Dic4712yes—
183C4×Q8—1882no—
184C4×D4—1882no—
185C2×C4⋊C4—471no—
186C2×C4⋊C4—472no—
187C2×C4⋊C4—471no—
188C2×C4⋊C4—942no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—942no—
191C2×C22⋊C4—941no—
192C2×C42—471no—
193C2×C42—472no—
194C2×C42—471no—
195C2×C42C4×D4711yescharacteristic
196#101—22no—
197#101—474no—
198#101—942no—
199#101C2×Dic4714yescomplemented
200#97—1882no—
201C4×C8—1882no—
202#95—941no—
203#95—941no—
204#95—942no—
205C24—21no—
206C2×Q8—41no—
207C2×Q8—41no—
208C2×Q8C2×C4×D4711yescharacteristic
209C2×Q8#1062535212yes—
210C2×Q8C22×Dic4711yescharacteristic
211C2×D4—21no—
212C2×D4—41no—
213C2×D4—41no—
214C2×D4C4×Dic4712yes—
215C22×C4—21no—
216C22×C4—471no—
217C22×C4—471no—
218C22×C4—472no—
219C22×C4—941no—
220C22×C4—941no—
221C22×C4—3762no—
222C22×C4—1882no—
223C22×C4C2×C4×D4711yescharacteristic
224C22×C4D4×D4712yes—
225M16—24no—
226M16—42no—
227M16—944no—
228M16—1882no—
229C2×C8—42no—
230C2×C8—1882no—
231C4⋊C4—3762no—
232C4⋊C4—1881no—
233C4⋊C4—1882no—
234C4⋊C4—1881no—
235C22⋊C4—24no—
236C22⋊C4—24no—
237C22⋊C4—3762no—
238C22⋊C4—1882no—
239C42—21no—
240C42—942no—
241C42—942no—
242C42—942no—
243C42—1881no—
244C42C4×Dic4712yes—
245C23—22no—
246C23—42no—
247C23—42no—
248C23—41no—
249C23—41no—
250C23C22⋊C4×D4711yescharacteristic
251Q8—41no—
252Q8—42no—
253Q8—41no—
254D4—42no—
255D4—42no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—472no—
264C2×C4—942no—
265C2×C4—941no—
266C2×C4—3764no—
267C2×C4—1881no—
268C2×C4—1881no—
269C2×C4—1882no—
270C2×C4—1881no—
271C2×C4—1881no—
272C2×C4—1881no—
273C2×C4—1882no—
274C2×C4—1881no—
275C2×C4C2×C4×Dic4711yescharacteristic
276C2×C4#1309398011yescharacteristic
277C2×C4(C22⋊D47)⋊C412yes—
278C2×C4D4×Dic4712yes—
279C8—44no—
280C8—1884no—
281C22—21no—
282C22—21no—
283C22—44no—
284C22—42no—
285C22—82no—
286C22—81no—
287C22—81no—
288C22C22⋊C4×Dic4711yescharacteristic
289C22#99×D4711yescharacteristic
290C22#1568909511yescharacteristic
291C4—22no—
292C4—22no—
293C4—21no—
294C4—21no—
295C4—42no—
296C4—942no—
297C4—3761no—
298C4—3761no—
299C4—1881no—
300C4—1882no—
301C2—21no—
302C2—21no—
303C2—82no—
304C2#99×Dic4711yescharacteristic
305C2#2024507411yescharacteristic
306C2#2024507811yescharacteristic
subgroupG/H
centerC22#99×D47
commutatorC2×C4×C47C2×C42
FrattiniC22×Q8C22×D47
FittingC2×C47×#355C2
socleC22×C47C2×#99
prime radicalC22×D4×C47C2×C4
cosocleC22×Q8×C47C23
hypercenterC2×#355D47
hypocenterC47#12632
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×#355

Complements

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

NH ≅ G/Ntype
3. #101×Dic47C2semidirect
4. #20241514C2semidirect
16. #15688920C4semidirect
20. C47×#355C4semidirect
35. #13093943C2×C4semidirect
50. #101×C47C2×C4semidirect
172. #355Dic47semidirect
174. C47#12632semidirect
199. #101C2×Dic47semidirect

Maximal subgroups

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

subgroupindex
#202461012
C2×#156889202
#101×Dic472
#202415142
C2×C47×#3552
#1263247