Finite Group Databasebeta

← #26126559

Subgroups — #26126559

Order 12032.

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

normal subgroups only

This group has 468 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#20246656C211yesmaximal, characteristic
2#20243121C211yesmaximal, characteristic
3#20241413C212yesmaximal
4#20241410C212yesmaximal
5C47×#1458C211yesmaximal, characteristic
6C2×C4×Dic94C411yescharacteristic, complemented
7C2×#13093983—21no—
8#15689062C411yescharacteristic, complemented
9#15689059C2211yescharacteristic
10#15689056—21no—
11#15689046—21no—
12#15689043C411yescharacteristic, complemented
13#15689035C2211yescharacteristic
14C4×Dic47⋊C4—21no—
15C4×C4⋊Dic47C411yescharacteristic, complemented
16C4×C22⋊Dic47—21no—
17#15688890—21no—
18#15688890C2212yes—
19C2×#117×C47C2211yescharacteristic
20C47×#431—21no—
21C47×#324—21no—
22C47×#323C2212yes—
23C22×Dic94C2×C411yescharacteristic
24C4×Dic94—41no—
25C4×Dic94—41no—
26#13093983—24no—
27#13093983—42no—
28C2×C4⋊Dic47—21no—
29C2×C4⋊Dic47—41no—
30C2×C4⋊Dic47C2×C411yescharacteristic
31C2×Dic47⋊C4—21no—
32C2×Dic47⋊C4—21no—
33C2×Dic47⋊C4—21no—
34C2×C22⋊Dic47—22no—
35C2×C22⋊Dic47—21no—
36C2×C22⋊Dic47—21no—
37C2×C4×Dic47—21no—
38C2×C4×Dic47—21no—
39C2×C4×Dic47—41no—
40C47⋊#95—21no—
41C47⋊#95—21no—
42C47⋊#95—21no—
43C47⋊#95—21no—
44C47⋊#95C2×C411yescharacteristic
45C47⋊#95C2×C411yescharacteristic
46C23×C4×C47D411yescharacteristic
47#117×C47—41no—
48#117×C47—41no—
49C2×C4⋊C4×C47C2×C411yescharacteristic
50C2×C4⋊C4×C47D411yescharacteristic
51C2×C22⋊C4×C47D411yescharacteristic
52C2×C22⋊C4×C47C2311yescharacteristic
53C2×C42×C47C2×C411yescharacteristic
54C2×C42×C47D411yescharacteristic
55C4⋊C8×C47—41no—
56C22⋊C8×C47—22no—
57C4×C8×C47—41no—
58C24×C47C2×D411yescharacteristic
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47—22no—
62C22×C4×C47C22⋊C411yescharacteristic
63C22×C4×C47C22⋊C411yescharacteristic
64C22×C4×C47C22⋊C411yescharacteristic
65C22×C4×C47C22⋊C411yescharacteristic
66C22×C4×C47SD1612yes—
67C22×C4×C47Q1612yes—
68C22×C4×C47C22×C411yescharacteristic
69C22×C4×C47C2×D411yescharacteristic
70C22×Dic47—21no—
71C22×Dic47—21no—
72C22×Dic47—21no—
73C22×Dic47—21no—
74C22×Dic47—42no—
75C2×Dic94—41no—
76C2×Dic94—42no—
77C2×C8×C47—42no—
78C4⋊C4×C47—41no—
79C4⋊C4×C47—41no—
80C22⋊C4×C47—41no—
81C22⋊C4×C47—41no—
82C42×C47—21no—
83C42×C47—21no—
84C42×C47—21no—
85C42×C47—21no—
86C22⋊Dic47—24no—
87C22⋊Dic47—42no—
88C22⋊Dic47—42no—
89C22⋊Dic47—42no—
90C4⋊Dic47—42no—
91C4⋊Dic47—44no—
92C4⋊Dic47—41no—
93Dic47⋊C4—42no—
94Dic47⋊C4—42no—
95Dic47⋊C4—42no—
96C4×Dic47—44no—
97C4×Dic47—41no—
98C4×Dic47—41no—
99C4×Dic47—42no—
100C23×C47—21no—
101C23×C47—21no—
102C23×C47—41no—
103C23×C47—41no—
104C23×C47C2×C22⋊C411yescharacteristic
105C23×C47C2×SD1611yescharacteristic
106C23×C47C2×Q1611yescharacteristic
107C2×C4×C47—21no—
108C2×C4×C47—21no—
109C2×C4×C47—22no—
110C2×C4×C47—22no—
111C2×C4×C47—22no—
112C2×C4×C47—22no—
113C2×C4×C47—21no—
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—41no—
121C2×C4×C47#9911yescharacteristic
122C2×C4×C47#9911yescharacteristic
123C2×C4×C47#10312yes—
124C2×C4×C47#10312yes—
125C2×C4×C47#10412yes—
126C2×Dic47—22no—
127C2×Dic47—22no—
128C2×Dic47—41no—
129C2×Dic47—41no—
130C2×Dic47—41no—
131C2×Dic47—41no—
132C2×Dic47—44no—
133C2×Dic47—41no—
134C2×Dic47—41no—
135C2×Dic47—41no—
136C2×Dic47—41no—
137C2×Dic47—41no—
138C2×Dic47—41no—
139C2×Dic47—82no—
140C2×Dic47—82no—
141Dic94—42no—
142Dic94—42no—
143C8×C47—82no—
144#11715—471nomaximal, Sylow
145C22×C47—21no—
146C22×C47—21no—
147C22×C47—21no—
148C22×C47—21no—
149C22×C47—41no—
150C22×C47—41no—
151C22×C47—41no—
152C22×C47—41no—
153C22×C47—81no—
154C22×C47C2×#9911yescharacteristic
155C22×C47C2×#10311yescharacteristic
156C22×C47C2×#10411yescharacteristic
157C4×C47—22no—
158C4×C47—41no—
159C4×C47—41no—
160C4×C47—41no—
161C4×C47—41no—
162C4×C47#32812yes—
163C4×C47#32812yes—
164Dic47—42no—
165Dic47—42no—
166Dic47—81no—
167Dic47—84no—
168Dic47—81no—
169#2396—471no—
170#1740—471no—
171#1458D4711yescharacteristic
172#1352—472no—
173#1351—472no—
174C2×C47—21no—
175C2×C47—21no—
176C2×C47—41no—
177C2×C47—41no—
178C2×C47C2×#32811yescharacteristic
179C2×C47#212011yescharacteristic
180C2×C47#212611yescharacteristic
181C2×C4×Q8—471no—
182C2×#117—941no—
183C2×#117C2×D4711yescharacteristic
184#431—21no—
185#400—471no—
186#399—471no—
187#398—941no—
188#396—941no—
189#395—471no—
190#393—471no—
191C4×C4⋊C4—471no—
192C4×C4⋊C4—941no—
193C4×C22⋊C4—941no—
194#342—472no—
195#342—941no—
196#324—21no—
197#323C2×D4712yes—
198C47#1171511yescharacteristic, complemented, Sylow
199C22×Q8—471no—
200C23×C4D18811yescharacteristic
201C4×Q8—1881no—
202C4×Q8—1881no—
203#117—41no—
204#117—41no—
205#117—944no—
206#117—1882no—
207C2×C4⋊C4—471no—
208C2×C4⋊C4—941no—
209C2×C4⋊C4—941no—
210C2×C4⋊C4—941no—
211C2×C4⋊C4—941no—
212C2×C4⋊C4—1881no—
213C2×C4⋊C4C4×D4711yescharacteristic
214C2×C4⋊C4C22⋊D4711yescharacteristic
215C2×C22⋊C4—942no—
216C2×C22⋊C4—941no—
217C2×C22⋊C4—941no—
218C2×C22⋊C4D18811yescharacteristic
219C2×C22⋊C4C22×D4711yescharacteristic
220C2×C42—941no—
221C2×C42—941no—
222C2×C42—1881no—
223C2×C42C4×D4711yescharacteristic
224C2×C42C22⋊D4711yescharacteristic
225C4⋊C8—41no—
226C22⋊C8—22no—
227C4×C8—41no—
228#95—471no—
229#95—471no—
230#95—941no—
231#95—941no—
232#95—941no—
233#95—941no—
234C24C2×D18811yescharacteristic
235C2×Q8—1881no—
236C2×Q8—1882no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—941no—
241C22×C4—941no—
242C22×C4—941no—
243C22×C4—941no—
244C22×C4—1882no—
245C22×C4#1062532512yes—
246C22×C4Dic18812yes—
247C22×C4(C2×D47)⋊C411yescharacteristic
248C22×C4(C2×D47)⋊C411yescharacteristic
249C22×C4(C2×D47)⋊C411yescharacteristic
250C22×C4(C2×D47)⋊C411yescharacteristic
251C22×C4C2×C4×D4711yescharacteristic
252C22×C4C2×C22⋊D4711yescharacteristic
253C2×C8—42no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—1881no—
257C4⋊C4—1882no—
258C4⋊C4—1884no—
259C4⋊C4—1882no—
260C4⋊C4—1882no—
261C4⋊C4—1882no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C22⋊C4—944no—
265C22⋊C4—1882no—
266C22⋊C4—1882no—
267C22⋊C4—1882no—
268C42—21no—
269C42—21no—
270C42—21no—
271C42—21no—
272C42—1884no—
273C42—1881no—
274C42—1881no—
275C42—1882no—
276C23—21no—
277C23—21no—
278C23—41no—
279C23—41no—
280C23C2×(C2×D47)⋊C411yescharacteristic
281C23C2×#1062532511yescharacteristic
282C23C2×Dic18811yescharacteristic
283Q8—1882no—
284Q8—1882no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—942no—
300C2×C4—942no—
301C2×C4—3762no—
302C2×C4—3762no—
303C2×C4—1884no—
304C2×C4—1881no—
305C2×C4—1881no—
306C2×C4—1881no—
307C2×C4—1881no—
308C2×C4—1881no—
309C2×C4—1881no—
310C2×C4—1881no—
311C2×C4—1881no—
312C2×C4—1881no—
313C2×C4—1881no—
314C2×C4#1309393811yescharacteristic
315C2×C4#1309393811yescharacteristic
316C2×C4#1309394912yes—
317C2×C4#1309394912yes—
318C2×C4#1309395012yes—
319C8—82no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—81no—
329C22C2×#1309393811yescharacteristic
330C22C2×#1309394911yescharacteristic
331C22C2×#1309395011yescharacteristic
332C4—22no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—3761no—
338C4—3764no—
339C4—3761no—
340C4—1882no—
341C4—1882no—
342C4#1568886112yes—
343C4#1568886112yes—
344C2—21no—
345C2—21no—
346C2—41no—
347C2—41no—
348C2C2×#1568886111yescharacteristic
349C2#2024503011yescharacteristic
350C2#2024505111yescharacteristic
subgroupG/H
centerC2×C4#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C22⋊C4C22×D47
FittingC47×#1458C2
socleC22×C47C2×#99
prime radicalC24×C47C2×D4
cosocleC2×C22⋊C4×C47C23
hypercenter#1458D47
hypocenterC47#11715
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ #1458

Complements

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

NH ≅ G/Ntype
6. C2×C4×Dic94C4semidirect
8. #15689062C4semidirect
12. #15689043C4semidirect
15. C4×C4⋊Dic47C4semidirect
198. C47#11715semidirect

Maximal subgroups

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

subgroupindex
#202466562
#202431212
#202414132
#202414102
C47×#14582
#1171547