Finite Group Databasebeta

← #26104780

Subgroups — #26104780

Order 12032.

530 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20251698C211yesmaximal, characteristic
2C2×#15688878C211yesmaximal, characteristic
3#20241212C212yesmaximal, complemented
4#20241210C212yesmaximal, complemented
5C47×#1749C211yesmaximal, characteristic
6C23×C47⋊C8—21no—
7C2×#13094016C411yescharacteristic
8#15689021C2211yescharacteristic
9#15689021C2211yescharacteristic
10#15689007—21no—
11#15689007—21no—
12#15688996—21no—
13#15688996—21no—
14#15688989C411yescharacteristic
15#15688983C411yescharacteristic
16#15688983C411yescharacteristic
17#15688878—21no—
18#15688878C2212yes—
19C22×C42×C47C2211yescharacteristic
20C2×C22⋊C8×C47—21no—
21C2×#97×C47—21no—
22C47×#336C2212yes—
23C22×C47⋊C8—24no—
24C22×C47⋊C8—22no—
25C22×C47⋊C8—42no—
26C22×C47⋊C8—42no—
27#13094016—42no—
28C2×C4⋊Dic47C2×C411yescharacteristic
29C2×C4⋊Dic47C2×C411yescharacteristic
30C2×C4⋊Dic47C2×C411yescharacteristic
31C2×C4⋊Dic47C2×C411yescharacteristic
32C2×Dic47⋊C4—21no—
33C2×Dic47⋊C4—21no—
34C2×Dic47⋊C4—21no—
35C2×Dic47⋊C4—21no—
36C2×C22⋊Dic47—21no—
37C2×C22⋊Dic47—21no—
38C47⋊#95—21no—
39C47⋊#95—21no—
40C47⋊#95—21no—
41C47⋊#95—21no—
42C47⋊#95—21no—
43C47⋊#95—21no—
44C23×C4×C47C2×C411yescharacteristic
45C23×C4×C47D411yescharacteristic
46C23×C4×C47D411yescharacteristic
47C22×C8×C47—22no—
48C2×C42×C47—42no—
49C2×C42×C47C2×C411yescharacteristic
50C2×C42×C47D411yescharacteristic
51C2×C42×C47D411yescharacteristic
52C2×C42×C47C2311yescharacteristic
53C22⋊C8×C47—24no—
54#97×C47—42no—
55C24×C47C22⋊C411yescharacteristic
56C22×C4×C47—22no—
57C22×C4×C47—22no—
58C22×C4×C47—21no—
59C22×C4×C47—21no—
60C22×C4×C47—22no—
61C22×C4×C47—22no—
62C22×C4×C47—22no—
63C22×C4×C47—22no—
64C22×C4×C47—41no—
65C22×C4×C47—41no—
66C22×C4×C47C22⋊C411yescharacteristic
67C22×C4×C47C22⋊C411yescharacteristic
68C22×C4×C47C22⋊C411yescharacteristic
69C22×C4×C47C22×C411yescharacteristic
70C22×C4×C47C2×D411yescharacteristic
71C22×C4×C47C2×D411yescharacteristic
72C22×Dic47—21no—
73C22×Dic47—21no—
74C22×Dic47—21no—
75C22×Dic47—21no—
76C2×C8×C47—24no—
77C2×C8×C47—42no—
78C2×C8×C47—42no—
79C42×C47—22no—
80C42×C47—22no—
81C42×C47—41no—
82C42×C47—41no—
83C22⋊Dic47—42no—
84C22⋊Dic47—42no—
85C4⋊Dic47—22no—
86C4⋊Dic47—22no—
87C4⋊Dic47—41no—
88C4⋊Dic47—41no—
89Dic47⋊C4—42no—
90Dic47⋊C4—42no—
91Dic47⋊C4—42no—
92Dic47⋊C4—41no—
93Dic47⋊C4—41no—
94C2×C47⋊C8—24no—
95C2×C47⋊C8—48no—
96C2×C47⋊C8—42no—
97C2×C47⋊C8—42no—
98C23×C47—21no—
99C23×C47—21no—
100C23×C47—21no—
101C23×C47—21no—
102C23×C47#10112yes—
103C23×C47#10412yes—
104C23×C47#10412yes—
105C23×C47C2×C22⋊C411yescharacteristic
106C2×C4×C47—24no—
107C2×C4×C47—24no—
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—21no—
117C2×C4×C47—21no—
118C2×C4×C47—21no—
119C2×C4×C47—21no—
120C2×C4×C47—21no—
121C2×C4×C47—21no—
122C2×C4×C47—44no—
123C2×C4×C47—44no—
124C2×C4×C47#9912yes—
125C2×C4×C47#13112yes—
126C2×C4×C47#13512yes—
127C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
128C2×C4×C47#13711yescharacteristic
129C2×Dic47—41no—
130C2×Dic47—41no—
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—
141C8×C47—44no—
142C47⋊C8—44no—
143#7691—471nomaximal, Sylow
144C22×C47—22no—
145C22×C47—22no—
146C22×C47—22no—
147C22×C47—22no—
148C22×C47—22no—
149C22×C47—22no—
150C22×C47#33014yes—
151C22×C47#40811yescharacteristic
152C22×C47C2×#9911yescharacteristic
153C22×C47C2×#10111yescharacteristic
154C22×C47#41611yescharacteristic
155C22×C47#41711yescharacteristic
156C22×C47C2×#10411yescharacteristic
157C22×C47C2×#10411yescharacteristic
158C4×C47—22no—
159C4×C47—22no—
160C4×C47—21no—
161C4×C47—22no—
162C4×C47—21no—
163C4×C47—41no—
164C4×C47—41no—
165Dic47—81no—
166Dic47—81no—
167Dic47—81no—
168Dic47—81no—
169#2851—471no—
170#1749D4711yescharacteristic
171C2×#336—471no—
172#1271—472no—
173#1271—472no—
174C2×C47—24no—
175C2×C47#147211yescharacteristic
176C2×C47#148311yescharacteristic
177C2×C47#149411yescharacteristic
178C2×C47#150111yescharacteristic
179C2×C47#150411yescharacteristic
180C2×C47C2×#33011yescharacteristic
181C2×C47#155211yescharacteristic
182C23×C8—941no—
183C2×#125—471no—
184C22×C42C2×D4711yescharacteristic
185C2×C22⋊C8—21no—
186C2×#97—21no—
187#389—471no—
188#389—471no—
189#387—941no—
190#387—941no—
191#385—941no—
192#385—941no—
193#382—471no—
194#380—471no—
195#380—471no—
196#336—472no—
197#336—941no—
198#336C2×D4712yes—
199C47#769111yescharacteristic, complemented, Sylow
200C23×C4C4×D4711yescharacteristic
201C23×C4D18811yescharacteristic
202C23×C4C22⋊D4711yescharacteristic
203C22×C8—22no—
204C22×C8—944no—
205C22×C8—942no—
206C22×C8—1882no—
207C22×C8—1882no—
208#125—1882no—
209C2×C4⋊C4—471no—
210C2×C4⋊C4—471no—
211C2×C4⋊C4—471no—
212C2×C4⋊C4—471no—
213C2×C4⋊C4—941no—
214C2×C4⋊C4—941no—
215C2×C4⋊C4—941no—
216C2×C4⋊C4—941no—
217C2×C22⋊C4—941no—
218C2×C22⋊C4—941no—
219C2×C42—42no—
220C2×C42C4×D4711yescharacteristic
221C2×C42D18811yescharacteristic
222C2×C42C22⋊D4711yescharacteristic
223C2×C42C22×D4711yescharacteristic
224C22⋊C8—24no—
225#97—42no—
226#95—941no—
227#95—941no—
228#95—941no—
229#95—941no—
230#95—941no—
231#95—941no—
232C24(C2×D47)⋊C411yescharacteristic
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—941no—
244C22×C4—941no—
245C22×C4—941no—
246C22×C4—941no—
247C22×C4(C2×D47)⋊C411yescharacteristic
248C22×C4(C2×D47)⋊C411yescharacteristic
249C22×C4(C2×D47)⋊C411yescharacteristic
250C22×C4C2×C4×D4711yescharacteristic
251C22×C4C2×D18811yescharacteristic
252C22×C4C2×C22⋊D4711yescharacteristic
253C2×C8—24no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—944no—
257C2×C8—1888no—
258C2×C8—1882no—
259C2×C8—1882no—
260C4⋊C4—942no—
261C4⋊C4—942no—
262C4⋊C4—1881no—
263C4⋊C4—1881no—
264C4⋊C4—1882no—
265C4⋊C4—1882no—
266C4⋊C4—1882no—
267C4⋊C4—1881no—
268C4⋊C4—1881no—
269C22⋊C4—1882no—
270C22⋊C4—1882no—
271C42—22no—
272C42—22no—
273C42—41no—
274C42—41no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23#1309394212yes—
280C23#1309395012yes—
281C23#1309395212yes—
282C23C2×(C2×D47)⋊C411yescharacteristic
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—1881no—
302C2×C4—1881no—
303C2×C4—1881no—
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#1309393812yes—
314C2×C4#1309403212yes—
315C2×C4#1309404412yes—
316C2×C4#1309405211yescharacteristic
317C2×C4#1309405711yescharacteristic
318C8—44no—
319C8—1884no—
320C22—22no—
321C22—22no—
322C22—22no—
323C22—22no—
324C22—22no—
325C22—22no—
326C22#1568886414yes—
327C22#1568909111yescharacteristic
328C22C2×#1309393811yescharacteristic
329C22C2×#1309394211yescharacteristic
330C22#1568911711yescharacteristic
331C22#1568912511yescharacteristic
332C22C2×#1309395011yescharacteristic
333C22C2×#1309395211yescharacteristic
334C4—22no—
335C4—22no—
336C4—21no—
337C4—22no—
338C4—21no—
339C4—41no—
340C4—41no—
341C4—3761no—
342C4—3761no—
343C4—3761no—
344C4—3761no—
345C2—24no—
346C2#2024175311yescharacteristic
347C2#2024180211yescharacteristic
348C2#2024184611yescharacteristic
349C2#2024187711yescharacteristic
350C2#2024189311yescharacteristic
351C2C2×#1568886411yescharacteristic
352C2#2024208911yescharacteristic
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#7691
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. #20241212C2semidirect
4. #20241210C2semidirect
199. C47#7691semidirect

Maximal subgroups

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

subgroupindex
#202516982
C2×#156888782
#202412122
#202412102
C47×#17492
#769147