Finite Group Databasebeta

← #26131454

Subgroups — #26131454

Order 12032.

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

normal subgroups only

This group has 628 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#20246114C211yesmaximal, characteristic
2C2×#15688916C211yesmaximal, characteristic
3#99×Dic47C212yesmaximal, complemented
4#20241512C212yesmaximal, complemented
5C2×C47×#353C211yesmaximal, characteristic
6C2×D4×Dic47C411yescharacteristic, complemented
7C2×D4×Dic47C2211yescharacteristic
8C2×#13093939C2212yes—
9#15689026C411yescharacteristic, complemented
10#15689026C2211yescharacteristic
11#15689003—22no—
12#15689003—22no—
13#15688989C411yescharacteristic, complemented
14C22⋊C4×Dic47—22no—
15#15688969C411yescharacteristic, complemented
16#15688916C414yescomplemented
17#15688889—22no—
18C2×#127×C47C2211yescharacteristic
19C2×#99×C47C2212yes—
20C47×#353C414yescomplemented
21C23×Dic47—21no—
22C23×Dic47D412yes—
23D4×Dic47—42no—
24D4×Dic47—42no—
25C2×C4⋊Dic47C2×C411yescharacteristic
26C2×C4⋊Dic47C2×C411yescharacteristic
27C2×C22⋊Dic47—22no—
28C2×C22⋊Dic47—21no—
29C2×C22⋊Dic47—22no—
30C2×C22⋊Dic47—22no—
31C2×C22⋊Dic47D412yes—
32C2×C4×Dic47—42no—
33C2×C4×Dic47C2×C411yescharacteristic
34C2×C4×Dic47C2×C411yescharacteristic
35#13093939—22no—
36#13093939C2×C414yescomplemented
37C47⋊#95—21no—
38C47⋊#95—21no—
39C47⋊#95—22no—
40C47⋊#95—42no—
41C22×D4×C47—22no—
42C22×D4×C47C2×C411yescharacteristic
43C22×D4×C47C2311yescharacteristic
44#127×C47—22no—
45#127×C47C2×C412yes—
46#127×C47C2×C412yes—
47C2×C22⋊C4×C47—22no—
48C2×C42×C47C2×C411yescharacteristic
49#99×C47—22no—
50#99×C47C2×C414yescomplemented
51C24×C47—21no—
52C24×C47C2×D412yes—
53C2×D4×C47—24no—
54C2×D4×C47—24no—
55C2×D4×C47—21no—
56C2×D4×C47—41no—
57C2×D4×C47—41no—
58C2×D4×C47—41no—
59C2×D4×C47—41no—
60C2×D4×C47—82no—
61C2×D4×C47—82no—
62C2×D4×C47C4212yescomplemented
63C2×D4×C47C22×C411yescharacteristic
64C2×D4×C47C22×C411yescharacteristic
65C2×D4×C47#4112yes—
66C22×C4×C47—21no—
67C22×C4×C47—42no—
68C22×C4×C47C22×C411yescharacteristic
69C22×Dic47—22no—
70C22×Dic47—22no—
71C22×Dic47—21no—
72C22×Dic47—21no—
73C22×Dic47—42no—
74C22×Dic47—42no—
75C22×Dic47—82no—
76C22×Dic47—82no—
77C22×Dic47C22⋊C411yescharacteristic
78C22×Dic47C22⋊C412yes—
79C22×Dic47C22⋊C411yescharacteristic
80C22⋊C4×C47—24no—
81C22⋊C4×C47—42no—
82C42×C47—21no—
83C42×C47C4212yescomplemented
84C22⋊Dic47—24no—
85C22⋊Dic47—42no—
86C22⋊Dic47—42no—
87C22⋊Dic47—42no—
88C22⋊Dic47—82no—
89C22⋊Dic47—82no—
90C4⋊Dic47—41no—
91C4⋊Dic47—41no—
92C4×Dic47—22no—
93C4×Dic47—41no—
94C4×Dic47—84no—
95C23×C47—22no—
96C23×C47—22no—
97C23×C47—22no—
98C23×C47—42no—
99C23×C47—42no—
100C23×C47—42no—
101C23×C47—42no—
102C23×C47—41no—
103C23×C47—41no—
104C23×C47C2×C22⋊C411yescharacteristic
105C23×C47C4×D412yes—
106C23×C47C4×D412yes—
107D4×C47—41no—
108D4×C47—41no—
109D4×C47—42no—
110D4×C47—42no—
111D4×C47—42no—
112D4×C47—84no—
113D4×C47—84no—
114C2×C4×C47—21no—
115C2×C4×C47—21no—
116C2×C4×C47—22no—
117C2×C4×C47—41no—
118C2×C4×C47—41no—
119C2×C4×C47—44no—
120C2×C4×C47—82no—
121C2×C4×C47—82no—
122C2×C4×C47C2×C4211yescharacteristic
123C2×C4×C47#11711yescharacteristic
124C2×Dic47—21no—
125C2×Dic47—22no—
126C2×Dic47—41no—
127C2×Dic47—44no—
128C2×Dic47—42no—
129C2×Dic47—41no—
130C2×Dic47—44no—
131C2×Dic47—41no—
132C2×Dic47—41no—
133C2×Dic47—42no—
134C2×Dic47—41no—
135C2×Dic47—41no—
136C2×Dic47—82no—
137C2×Dic47—84no—
138C2×Dic47—82no—
139C2×Dic47—82no—
140C2×Dic47#9912yes—
141#12628—471nomaximal, Sylow
142C22×C47—21no—
143C22×C47—21no—
144C22×C47—22no—
145C22×C47—22no—
146C22×C47—42no—
147C22×C47—42no—
148C22×C47—44no—
149C22×C47—42no—
150C22×C47—42no—
151C22×C47—42no—
152C22×C47—82no—
153C22×C47—82no—
154C22×C47—81no—
155C22×C47—81no—
156C22×C47C4×C22⋊C411yescharacteristic
157C22×C47C2×#9911yescharacteristic
158C22×C47#41011yescharacteristic
159C4×C47—21no—
160C4×C47—21no—
161C4×C47—42no—
162C4×C47—84no—
163Dic47—22no—
164Dic47—42no—
165Dic47—41no—
166Dic47—81no—
167Dic47—84no—
168Dic47—81no—
169#2328—471no—
170C2×#353—471no—
171C2×#353D4711yescharacteristic
172C4×#99—472no—
173#1393—472no—
174C2×C47—21no—
175C2×C47—21no—
176C2×C47—42no—
177C2×C47—42no—
178C2×C47—82no—
179C2×C47C4×#9911yescharacteristic
180C2×C47#212711yescharacteristic
181C2×C47#212811yescharacteristic
182C2×#127C2×D4711yescharacteristic
183C2×C4×D4—471no—
184C2×C4×D4—471no—
185C2×#99—472no—
186C2×#99C2×D4712yes—
187#390—471no—
188#390—471no—
189#386—942no—
190#386—942no—
191#382—471no—
192C4×C22⋊C4—942no—
193#376—471no—
194#353—474no—
195#353Dic4714yescomplemented
196#342—942no—
197C47#1262811yescharacteristic, complemented, Sylow
198C22×D4—22no—
199C22×D4C4×D4711yescharacteristic
200C22×D4C22×D4711yescharacteristic
201C23×C4—472no—
202C23×C4—941no—
203#127—22no—
204#127C4×D4712yes—
205#127C2×Dic4712yes—
206C4×D4—1882no—
207C4×D4—1882no—
208C2×C4⋊C4—471no—
209C2×C4⋊C4—471no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—472no—
212C2×C22⋊C4—942no—
213C2×C22⋊C4—942no—
214C2×C22⋊C4—942no—
215C2×C22⋊C4—941no—
216C2×C42—471no—
217C2×C42—471no—
218C2×C42—1882no—
219C2×C42C4×D4711yescharacteristic
220#99—22no—
221#99—474no—
222#99—942no—
223#99C2×Dic4714yescomplemented
224#95—941no—
225#95—941no—
226#95—942no—
227#95—1882no—
228C24—21no—
229C24D4×D4712yes—
230C2×D4—24no—
231C2×D4—24no—
232C2×D4—21no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—82no—
238C2×D4—82no—
239C2×D4C4×Dic4712yescomplemented
240C2×D4C2×C4×D4711yescharacteristic
241C2×D4#1062535212yes—
242C2×D4C22×Dic4711yescharacteristic
243C22×C4—21no—
244C22×C4—42no—
245C22×C4—471no—
246C22×C4—471no—
247C22×C4—472no—
248C22×C4—942no—
249C22×C4—942no—
250C22×C4—941no—
251C22×C4—941no—
252C22×C4—3762no—
253C22×C4—3762no—
254C22×C4—1882no—
255C22×C4—1882no—
256C22×C4C2×C4×D4711yescharacteristic
257C4⋊C4—1881no—
258C4⋊C4—1881no—
259C22⋊C4—24no—
260C22⋊C4—42no—
261C22⋊C4—944no—
262C22⋊C4—3762no—
263C22⋊C4—3762no—
264C22⋊C4—1882no—
265C22⋊C4—1882no—
266C22⋊C4—1882no—
267C42—21no—
268C42—942no—
269C42—3764no—
270C42—1881no—
271C42C4×Dic4712yescomplemented
272C23—22no—
273C23—22no—
274C23—22no—
275C23—42no—
276C23—42no—
277C23—42no—
278C23—42no—
279C23—41no—
280C23—41no—
281C23C22⋊C4×D4711yescharacteristic
282C23(C22⋊D47)⋊C412yes—
283C23D4×Dic4712yes—
284D4—41no—
285D4—42no—
286D4—41no—
287D4—42no—
288D4—42no—
289D4—84no—
290D4—84no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—44no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—472no—
300C2×C4—941no—
301C2×C4—942no—
302C2×C4—3762no—
303C2×C4—3762no—
304C2×C4—3762no—
305C2×C4—3764no—
306C2×C4—1881no—
307C2×C4—1884no—
308C2×C4—1884no—
309C2×C4—1881no—
310C2×C4—1881no—
311C2×C4—1882no—
312C2×C4—1881no—
313C2×C4—1881no—
314C2×C4—1882no—
315C2×C4—1881no—
316C2×C4C2×C4×Dic4711yescharacteristic
317C2×C4#1309398011yescharacteristic
318C22—21no—
319C22—21no—
320C22—22no—
321C22—22no—
322C22—42no—
323C22—42no—
324C22—44no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22—82no—
329C22—82no—
330C22—81no—
331C22—81no—
332C22C22⋊C4×Dic4711yescharacteristic
333C22#99×D4711yescharacteristic
334C22#1568909511yescharacteristic
335C4—21no—
336C4—21no—
337C4—42no—
338C4—84no—
339C4—942no—
340C4—3761no—
341C4—3764no—
342C4—3761no—
343C4—1882no—
344C4—1881no—
345C2—21no—
346C2—21no—
347C2—42no—
348C2—42no—
349C2—82no—
350C2#99×Dic4711yescharacteristic
351C2#2024505411yescharacteristic
352C2#2024505811yescharacteristic
subgroupG/H
centerC22#99×D47
commutatorC2×C4×C47C2×C42
FrattiniC22×D4C22×D47
FittingC2×C47×#353C2
socleC22×C47C2×#99
prime radicalC2×#127×C47C22
cosocleC22×D4×C47C23
hypercenterC2×#353D47
hypocenterC47#12628
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#353

Complements

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

NH ≅ G/Ntype
3. #99×Dic47C2semidirect
4. #20241512C2semidirect
6. C2×D4×Dic47C4semidirect
9. #15689026C4semidirect
13. #15688989C4semidirect
15. #15688969C4semidirect
16. #15688916C4semidirect
20. C47×#353C4semidirect
36. #13093939C2×C4semidirect
50. #99×C47C2×C4semidirect
62. C2×D4×C47C42semidirect
83. C42×C47C42semidirect
195. #353Dic47semidirect
197. C47#12628semidirect
223. #99C2×Dic47semidirect
239. C2×D4C4×Dic47semidirect
271. C42C4×Dic47semidirect

Maximal subgroups

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

subgroupindex
#202461142
C2×#156889162
#99×Dic472
#202415122
C2×C47×#3532
#1262847