Finite Group Databasebeta

← #26104708

Subgroups — #26104708

Order 12032.

488 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 111 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#20251727C211yesmaximal, characteristic
2C4×C22⋊(C47⋊C8)C211yesmaximal, characteristic
3#20241213C212yesmaximal, complemented
4#20241212C212yesmaximal, complemented
5C47×#1749C211yesmaximal, characteristic
6C2×#13094003C411yescharacteristic
7C2×C22⋊(C47⋊C8)—21no—
8C2×C4×C47⋊C8—21no—
9#15689029C411yescharacteristic
10#15689029C2211yescharacteristic
11#15689021C411yescharacteristic
12#15689021C2211yescharacteristic
13#15689007—21no—
14#15689007—21no—
15#15688996—21no—
16#15688986—21no—
17#15688983C411yescharacteristic
18#15688878C2212yes—
19C22×C42×C47C2211yescharacteristic
20C2×C22⋊C8×C47—21no—
21C2×#97×C47—21no—
22C47×#336C2212yes—
23C22×Dic94C2×C411yescharacteristic
24C22×C47⋊C8—22no—
25#13094003—42no—
26C2×C4⋊Dic47C2×C411yescharacteristic
27C2×C4⋊Dic47C2×C411yescharacteristic
28C2×C4⋊Dic47C2×C411yescharacteristic
29C2×Dic47⋊C4—21no—
30C2×Dic47⋊C4—21no—
31C2×Dic47⋊C4—21no—
32C2×C22⋊Dic47—21no—
33C2×C22⋊Dic47—21no—
34C22⋊(C47⋊C8)—24no—
35C4×C47⋊C8—42no—
36C47⋊#95—21no—
37C47⋊#95—21no—
38C47⋊#95—21no—
39C47⋊#95—21no—
40C47⋊#95—21no—
41C47⋊#95—21no—
42C47⋊#95—21no—
43C23×C4×C47C2×C411yescharacteristic
44C23×C4×C47D411yescharacteristic
45C23×C4×C47D411yescharacteristic
46C22×C8×C47—22no—
47C2×C42×C47—42no—
48C2×C42×C47C2×C411yescharacteristic
49C2×C42×C47D411yescharacteristic
50C2×C42×C47D411yescharacteristic
51C2×C42×C47C2311yescharacteristic
52C22⋊C8×C47—24no—
53#97×C47—42no—
54C24×C47C22⋊C411yescharacteristic
55C22×C4×C47—22no—
56C22×C4×C47—22no—
57C22×C4×C47—22no—
58C22×C4×C47—22no—
59C22×C4×C47—22no—
60C22×C4×C47—22no—
61C22×C4×C47—41no—
62C22×C4×C47—41no—
63C22×C4×C47—41no—
64C22×C4×C47C22⋊C411yescharacteristic
65C22×C4×C47C22⋊C411yescharacteristic
66C22×C4×C47C22⋊C411yescharacteristic
67C22×C4×C47C22×C411yescharacteristic
68C22×C4×C47C2×D411yescharacteristic
69C22×C4×C47C2×D411yescharacteristic
70C22×Dic47—21no—
71C22×Dic47—21no—
72C22×Dic47—21no—
73C22×Dic47—21no—
74C2×Dic94—22no—
75C2×Dic94—41no—
76C2×Dic94—41no—
77C2×C8×C47—24no—
78C2×C8×C47—42no—
79C2×C8×C47—42no—
80C42×C47—22no—
81C42×C47—22no—
82C42×C47—41no—
83C42×C47—41no—
84C22⋊Dic47—42no—
85C22⋊Dic47—42no—
86C4⋊Dic47—22no—
87C4⋊Dic47—41no—
88C4⋊Dic47—41no—
89Dic47⋊C4—42no—
90Dic47⋊C4—42no—
91Dic47⋊C4—42no—
92C2×C47⋊C8—24no—
93C2×C47⋊C8—42no—
94C2×C47⋊C8—42no—
95C23×C47—21no—
96C23×C47—21no—
97C23×C47—21no—
98C23×C47—21no—
99C23×C47—21no—
100C23×C47—21no—
101C23×C47#10412yes—
102C23×C47C2×C22⋊C411yescharacteristic
103C2×C4×C47—21no—
104C2×C4×C47—21no—
105C2×C4×C47—21no—
106C2×C4×C47—21no—
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—21no—
114C2×C4×C47—21no—
115C2×C4×C47—21no—
116C2×C4×C47—44no—
117C2×C4×C47—44no—
118C2×C4×C47—44no—
119C2×C4×C47#9912yes—
120C2×C4×C47#10412yes—
121C2×C4×C47#13112yes—
122C2×C4×C47#13512yes—
123C2×C4×C47#13712yes—
124C2×Dic47—41no—
125C2×Dic47—41no—
126C2×Dic47—41no—
127C2×Dic47—41no—
128C2×Dic47—41no—
129C2×Dic47—41no—
130C2×Dic47—41no—
131C2×Dic47—41no—
132C2×Dic47—41no—
133C2×Dic47—41no—
134C2×Dic47—41no—
135C2×Dic47—41no—
136Dic94—82no—
137C8×C47—44no—
138C47⋊C8—44no—
139#7675—471nomaximal, Sylow
140C22×C47—22no—
141C22×C47—22no—
142C22×C47—22no—
143C22×C47—41no—
144C22×C47—41no—
145C22×C47—41no—
146C22×C47—41no—
147C22×C47#40811yescharacteristic
148C22×C47C2×#9911yescharacteristic
149C22×C47#41311yescharacteristic
150C22×C47#41611yescharacteristic
151C22×C47#41911yescharacteristic
152C22×C47C2×#10411yescharacteristic
153C22×C47C2×#10411yescharacteristic
154C4×C47—22no—
155C4×C47—22no—
156C4×C47—22no—
157C4×C47—22no—
158C4×C47—22no—
159C4×C47—41no—
160Dic47—81no—
161Dic47—81no—
162Dic47—81no—
163Dic47—81no—
164#2855—471no—
165#1749D4711yescharacteristic
166C4×C22⋊C8—471no—
167#1272—472no—
168#1271—472no—
169C2×C47—42no—
170C2×C47#146811yescharacteristic
171C2×C47#148511yescharacteristic
172C2×C47#149411yescharacteristic
173C2×C47#150111yescharacteristic
174C2×C47#150811yescharacteristic
175C2×C47#152411yescharacteristic
176C2×C47#155411yescharacteristic
177C2×C22⋊Q8—471no—
178C22×C42C2×D4711yescharacteristic
179C2×C22⋊C8—21no—
180C2×C22⋊C8—941no—
181C2×#97—21no—
182C2×C4×C8—941no—
183#391—471no—
184#391—471no—
185#389—471no—
186#389—471no—
187#387—941no—
188#387—941no—
189#385—941no—
190#381—941no—
191#380—471no—
192#336—472no—
193#336C2×D4712yes—
194C47#767511yescharacteristic, complemented, Sylow
195C22×Q8—471no—
196C23×C4C4×D4711yescharacteristic
197C23×C4D18811yescharacteristic
198C23×C4C22⋊D4711yescharacteristic
199C22×C8—22no—
200C22×C8—942no—
201C22⋊Q8—1882no—
202C2×C4⋊C4—471no—
203C2×C4⋊C4—471no—
204C2×C4⋊C4—471no—
205C2×C4⋊C4—941no—
206C2×C4⋊C4—941no—
207C2×C4⋊C4—941no—
208C2×C22⋊C4—941no—
209C2×C22⋊C4—941no—
210C2×C42—42no—
211C2×C42C4×D4711yescharacteristic
212C2×C42D18811yescharacteristic
213C2×C42C22⋊D4711yescharacteristic
214C2×C42C22×D4711yescharacteristic
215C22⋊C8—24no—
216C22⋊C8—944no—
217#97—42no—
218C4×C8—1882no—
219#95—941no—
220#95—941no—
221#95—941no—
222#95—941no—
223#95—941no—
224#95—941no—
225#95—941no—
226C24(C2×D47)⋊C411yescharacteristic
227C2×Q8—942no—
228C2×Q8—1881no—
229C2×Q8—1881no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—941no—
240C22×C4—941no—
241C22×C4—941no—
242C22×C4—941no—
243C22×C4(C2×D47)⋊C411yescharacteristic
244C22×C4(C2×D47)⋊C411yescharacteristic
245C22×C4(C2×D47)⋊C411yescharacteristic
246C22×C4C2×C4×D4711yescharacteristic
247C22×C4C2×D18811yescharacteristic
248C22×C4C2×C22⋊D4711yescharacteristic
249C2×C8—24no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—944no—
253C2×C8—1882no—
254C2×C8—1882no—
255C4⋊C4—942no—
256C4⋊C4—1881no—
257C4⋊C4—1881no—
258C4⋊C4—1882no—
259C4⋊C4—1882no—
260C4⋊C4—1882no—
261C22⋊C4—1882no—
262C22⋊C4—1882no—
263C42—22no—
264C42—22no—
265C42—41no—
266C42—41no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23#1309395012yes—
274C23C2×(C2×D47)⋊C411yescharacteristic
275Q8—3762no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—1881no—
293C2×C4—1881no—
294C2×C4—1881no—
295C2×C4—1881no—
296C2×C4—1881no—
297C2×C4—1881no—
298C2×C4—1881no—
299C2×C4—1881no—
300C2×C4—1881no—
301C2×C4—1881no—
302C2×C4—1881no—
303C2×C4—1881no—
304C2×C4#1309393812yes—
305C2×C4#1309395212yes—
306C2×C4#1309403212yes—
307C2×C4#1309404412yes—
308C2×C4#1309405712yes—
309C8—44no—
310C8—1884no—
311C22—22no—
312C22—22no—
313C22—22no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22—41no—
318C22#1568909111yescharacteristic
319C22C2×#1309393811yescharacteristic
320C22#1568910511yescharacteristic
321C22#1568911711yescharacteristic
322C22#1568913311yescharacteristic
323C22C2×#1309395011yescharacteristic
324C22C2×#1309395211yescharacteristic
325C4—22no—
326C4—22no—
327C4—22no—
328C4—22no—
329C4—22no—
330C4—41no—
331C4—3761no—
332C4—3761no—
333C4—3761no—
334C4—3761no—
335C2—42no—
336C2#2024173711yescharacteristic
337C2#2024181011yescharacteristic
338C2#2024184611yescharacteristic
339C2#2024187711yescharacteristic
340C2#2024190911yescharacteristic
341C2#2024197411yescharacteristic
342C2#2024209911yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1749C2
socleC23×C47C2×C22⋊C4
prime radicalC24×C47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#1749D47
hypocenterC47#7675
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
3. #20241213C2semidirect
4. #20241212C2semidirect
194. C47#7675semidirect

Maximal subgroups

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

subgroupindex
#202517272
C4×C22⋊(C47⋊C8)2
#202412132
#202412122
C47×#17492
#767547