Finite Group Databasebeta

← #26126556

Subgroups — #26126556

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#20241702C211yesmaximal, characteristic
3#20241411C212yesmaximal
4#20241408C212yesmaximal
5C47×#1740C211yesmaximal, characteristic
6C2×C4×Dic94C411yescharacteristic, complemented
7#15689176—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#15688849—21no—
18#15688847C2212yes—
19C2×#117×C47—21no—
20C2×#117×C47C2211yescharacteristic
21C47×#342—21no—
22C47×#342C2212yes—
23C22×Dic94C2×C411yescharacteristic
24C4×Dic94—41no—
25C4×Dic94—41no—
26C2×C4⋊Dic47—21no—
27C2×C4⋊Dic47C2×C411yescharacteristic
28C2×Dic47⋊C4—21no—
29C2×Dic47⋊C4—21no—
30C2×Dic47⋊C4—21no—
31C2×C22⋊Dic47—21no—
32C2×C22⋊Dic47—21no—
33C2×C4×Dic47—21no—
34C2×C4×Dic47—21no—
35C4⋊(C47⋊C8)—41no—
36C22⋊(C47⋊C8)—22no—
37C4×C47⋊C8—41no—
38C47⋊#95—21no—
39C47⋊#95—21no—
40C47⋊#95—21no—
41C47⋊#95—21no—
42C47⋊#95C2×C411yescharacteristic
43C47⋊#95C2×C411yescharacteristic
44C23×C4×C47D411yescharacteristic
45#117×C47—24no—
46#117×C47—41no—
47#117×C47—41no—
48#117×C47—42no—
49C2×C4⋊C4×C47—41no—
50C2×C4⋊C4×C47C2×C411yescharacteristic
51C2×C4⋊C4×C47D411yescharacteristic
52C2×C22⋊C4×C47—22no—
53C2×C22⋊C4×C47D411yescharacteristic
54C2×C22⋊C4×C47C2311yescharacteristic
55C2×C42×C47—41no—
56C2×C42×C47C2×C411yescharacteristic
57C2×C42×C47D411yescharacteristic
58C24×C47C2×D411yescharacteristic
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47—22no—
62C22×C4×C47—42no—
63C22×C4×C47C22⋊C411yescharacteristic
64C22×C4×C47C22⋊C411yescharacteristic
65C22×C4×C47C22⋊C411yescharacteristic
66C22×C4×C47C22⋊C411yescharacteristic
67C22×C4×C47SD1612yes—
68C22×C4×C47Q1612yes—
69C22×C4×C47C22×C411yescharacteristic
70C22×C4×C47C2×D411yescharacteristic
71C22×Dic47—21no—
72C22×Dic47—21no—
73C22×Dic47—21no—
74C22×Dic47—21no—
75C2×Dic94—41no—
76C2×Dic94—42no—
77C4⋊C4×C47—41no—
78C4⋊C4×C47—41no—
79C4⋊C4×C47—44no—
80C22⋊C4×C47—24no—
81C22⋊C4×C47—41no—
82C22⋊C4×C47—41no—
83C22⋊C4×C47—42no—
84C42×C47—21no—
85C42×C47—21no—
86C42×C47—21no—
87C42×C47—21no—
88C42×C47—44no—
89C22⋊Dic47—42no—
90C22⋊Dic47—42no—
91C4⋊Dic47—42no—
92C4⋊Dic47—41no—
93Dic47⋊C4—42no—
94Dic47⋊C4—42no—
95Dic47⋊C4—42no—
96C4×Dic47—41no—
97C4×Dic47—41no—
98C4×Dic47—42no—
99C2×C47⋊C8—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—44no—
122C2×C4×C47—82no—
123C2×C4×C47—82no—
124C2×C4×C47#9911yescharacteristic
125C2×C4×C47#9911yescharacteristic
126C2×C4×C47#10312yes—
127C2×C4×C47#10312yes—
128C2×C4×C47#10412yes—
129C2×Dic47—22no—
130C2×Dic47—22no—
131C2×Dic47—41no—
132C2×Dic47—41no—
133C2×Dic47—41no—
134C2×Dic47—41no—
135C2×Dic47—41no—
136C2×Dic47—41no—
137C2×Dic47—41no—
138C2×Dic47—41no—
139C2×Dic47—41no—
140C2×Dic47—41no—
141Dic94—42no—
142Dic94—42no—
143C47⋊C8—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—84no—
163C4×C47#32812yes—
164C4×C47#32812yes—
165Dic47—42no—
166Dic47—42no—
167Dic47—81no—
168Dic47—81no—
169#2396—471no—
170#1740D4711yescharacteristic
171#1458—471no—
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—21no—
183C2×#117C2×D4711yescharacteristic
184#431—941no—
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—21no—
195#342C2×D4712yes—
196#324—941no—
197#323—472no—
198C47#1171511yescharacteristic, complemented, Sylow
199C22×Q8—471no—
200C23×C4C22⋊D4711yescharacteristic
201C4×Q8—1881no—
202C4×Q8—1881no—
203#117—24no—
204#117—41no—
205#117—41no—
206#117—42no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—471no—
209C2×C4⋊C4—941no—
210C2×C4⋊C4—941no—
211C2×C4⋊C4—941no—
212C2×C4⋊C4—941no—
213C2×C4⋊C4C4×D4711yescharacteristic
214C2×C4⋊C4D18811yescharacteristic
215C2×C22⋊C4—22no—
216C2×C22⋊C4—941no—
217C2×C22⋊C4—941no—
218C2×C22⋊C4C22⋊D4711yescharacteristic
219C2×C22⋊C4C22×D4711yescharacteristic
220C2×C42—41no—
221C2×C42—941no—
222C2×C42—941no—
223C2×C42C4×D4711yescharacteristic
224C2×C42D18811yescharacteristic
225C4⋊C8—1881no—
226C22⋊C8—942no—
227C4×C8—1881no—
228#95—471no—
229#95—471no—
230#95—941no—
231#95—941no—
232#95—941no—
233#95—941no—
234C24C2×C22⋊D4711yescharacteristic
235C2×Q8—1881no—
236C2×Q8—1882no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—42no—
241C22×C4—941no—
242C22×C4—941no—
243C22×C4—941no—
244C22×C4—941no—
245C22×C4(C2×D47)⋊C411yescharacteristic
246C22×C4(C2×D47)⋊C411yescharacteristic
247C22×C4(C2×D47)⋊C411yescharacteristic
248C22×C4(C2×D47)⋊C411yescharacteristic
249C22×C4#1062533512yes—
250C22×C4#1062533712yes—
251C22×C4C2×C4×D4711yescharacteristic
252C22×C4C2×D18811yescharacteristic
253C2×C8—1882no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—44no—
257C4⋊C4—1881no—
258C4⋊C4—1882no—
259C4⋊C4—1882no—
260C4⋊C4—1882no—
261C4⋊C4—1882no—
262C22⋊C4—24no—
263C22⋊C4—41no—
264C22⋊C4—41no—
265C22⋊C4—42no—
266C22⋊C4—1882no—
267C22⋊C4—1882no—
268C42—21no—
269C42—21no—
270C42—21no—
271C42—21no—
272C42—44no—
273C42—1881no—
274C42—1881no—
275C42—1882no—
276C23—21no—
277C23—21no—
278C23—41no—
279C23—41no—
280C23C2×(C2×D47)⋊C411yescharacteristic
281C23C2×#1062533511yescharacteristic
282C23C2×#1062533711yescharacteristic
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—44no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—942no—
303C2×C4—942no—
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#1309394712yes—
317C2×C4#1309394712yes—
318C2×C4#1309395212yes—
319C8—3762no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—81no—
329C22C2×#1309393811yescharacteristic
330C22C2×#1309394711yescharacteristic
331C22C2×#1309395211yescharacteristic
332C4—22no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—84no—
338C4—3761no—
339C4—3761no—
340C4—1882no—
341C4—1882no—
342C4#1568885912yes—
343C4#1568885912yes—
344C2—21no—
345C2—21no—
346C2—41no—
347C2—41no—
348C2C2×#1568885911yescharacteristic
349C2#2024502711yescharacteristic
350C2#2024505111yescharacteristic
subgroupG/H
centerC2×C4#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C22⋊C4C22×D47
FittingC47×#1740C2
socleC22×C47C2×#99
prime radicalC24×C47C2×D4
cosocleC2×C22⋊C4×C47C23
hypercenter#1740D47
hypocenterC47#11715
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ #1740

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
#202417022
#202414112
#202414082
C47×#17402
#1171547