Finite Group Databasebeta

← #26112064

Subgroups — #26112064

Order 12032.

648 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C2×#15688991C211yesmaximal, characteristic, complemented
2#20243456C211yesmaximal, characteristic, complemented
3C2×#15688857C212yesmaximal
4#20241266C212yesmaximal, complemented
5C47×#2854C211yesmaximal, characteristic
6C2×C22⋊(C47⋊C8)C2212yes—
7#15688991—24no—
8C2×C47⋊#95—21no—
9C2×C47⋊#95—21no—
10C2×C47⋊#95C2212yes—
11#15688878—22no—
12#15688857—24no—
13C2×#123×C47—21no—
14C2×#121×C47C2211yescharacteristic
15C22×C42×C47C2211yescharacteristic
16C47×#390C412yescomplemented
17C47×#389C412yescomplemented
18C47×#388—22no—
19C47×#386—21no—
20C47×#380C2211yescharacteristic
21C23×Dic47—22no—
22C22×C47⋊C8—22no—
23C22⋊(C47⋊C8)—22no—
24C22⋊(C47⋊C8)—22no—
25C47⋊#95—24no—
26C47⋊#95—24no—
27C47⋊#95—22no—
28C47⋊#95D414yes—
29C22×D4×C47—21no—
30C22×D4×C47C2×C411yescharacteristic
31C23×C4×C47D411yescharacteristic
32C23×C4×C47D411yescharacteristic
33C23×C4×C47C2311yescharacteristic
34#123×C47—44no—
35#121×C47—22no—
36#121×C47—22no—
37C2×C4⋊C4×C47—22no—
38C2×C4⋊C4×C47C2×C411yescharacteristic
39C2×C4⋊C4×C47C2×C412yes—
40C2×C22⋊C4×C47—21no—
41C2×C22⋊C4×C47—22no—
42C2×C22⋊C4×C47—22no—
43C2×C22⋊C4×C47—21no—
44C2×C42×C47—22no—
45C2×C42×C47—22no—
46C2×C42×C47C2×C412yes—
47C2×C42×C47D412yes—
48#95×C47—22no—
49#95×C47—21no—
50C24×C47—21no—
51C24×C47C2×D411yescharacteristic
52C2×D4×C47—22no—
53C2×D4×C47—42no—
54C2×D4×C47—41no—
55C2×D4×C47—41no—
56C2×D4×C47—42no—
57C2×D4×C47—42no—
58C22×C4×C47—21no—
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47—21no—
62C22×C4×C47—22no—
63C22×C4×C47—22no—
64C22×C4×C47—22no—
65C22×C4×C47—21no—
66C22×C4×C47—42no—
67C22×C4×C47—42no—
68C22×C4×C47C22⋊C412yes—
69C22×C4×C47C22⋊C412yes—
70C22×C4×C47C22×C411yescharacteristic
71C22×C4×C47C2×D411yescharacteristic
72C22×C4×C47C2×D412yes—
73C22×C4×C47#4112yes—
74C22×Dic47—24no—
75C22×Dic47—44no—
76C22×Dic47—44no—
77C22×Dic47—42no—
78C22×Dic47—42no—
79C4⋊C4×C47—22no—
80C4⋊C4×C47—44no—
81C4⋊C4×C47—42no—
82C22⋊C4×C47—44no—
83C22⋊C4×C47—44no—
84C22⋊C4×C47—42no—
85C22⋊C4×C47—42no—
86C42×C47—24no—
87C42×C47—24no—
88C2×C47⋊C8—44no—
89C2×C47⋊C8—42no—
90C23×C47—22no—
91C23×C47—22no—
92C23×C47—41no—
93C23×C47—42no—
94C23×C47—41no—
95C23×C47—41no—
96C23×C47—42no—
97C23×C47C2×C22⋊C411yescharacteristic
98C23×C47#12011yescharacteristic
99C23×C47#12311yescharacteristic
100C23×C47#13512yes—
101C23×C47AGL(1,ℤ/8)12yes—
102D4×C47—82no—
103D4×C47—82no—
104D4×C47—82no—
105C2×C4×C47—22no—
106C2×C4×C47—22no—
107C2×C4×C47—22no—
108C2×C4×C47—22no—
109C2×C4×C47—22no—
110C2×C4×C47—22no—
111C2×C4×C47—22no—
112C2×C4×C47—22no—
113C2×C4×C47—42no—
114C2×C4×C47—42no—
115C2×C4×C47—42no—
116C2×C4×C47—42no—
117C2×C4×C47—42no—
118C2×C4×C47—42no—
119C2×C4×C47—41no—
120C2×C4×C47—42no—
121C2×C4×C47—41no—
122C2×C4×C47—42no—
123C2×C4×C47—41no—
124C2×C4×C47—42no—
125C2×C4×C47C4×D412yes—
126C2×C4×C47#12112yes—
127C2×Dic47—44no—
128C2×Dic47—44no—
129C2×Dic47—44no—
130C2×Dic47—84no—
131C2×Dic47—84no—
132C47⋊C8—82no—
133#9157—471nomaximal, Sylow
134C22×C47—21no—
135C22×C47—21no—
136C22×C47—21no—
137C22×C47—21no—
138C22×C47—24no—
139C22×C47—24no—
140C22×C47—81no—
141C22×C47—81no—
142C22×C47—81no—
143C22×C47—82no—
144C22×C47—82no—
145C22×C47#38611yescharacteristic
146C22×C47#41012yes—
147C22×C47#41611yescharacteristic
148C22×C47#41811yescharacteristic
149C22×C47#42112yes—
150C22×C47#44912yes—
151C22×C47#48212yes—
152C4×C47—22no—
153C4×C47—22no—
154C4×C47—42no—
155C4×C47—42no—
156C4×C47—81no—
157C4×C47—82no—
158Dic47—84no—
159#2854D4711yescharacteristic
160C2×#383—471no—
161#1817—471no—
162C2×#327—472no—
163#1294—472no—
164C2×C47—22no—
165C2×C47—22no—
166C2×C47—161no—
167C2×C47#151412yes—
168C2×C47#152112yes—
169C2×C47#187511yescharacteristic
170C2×C47#189711yescharacteristic
171C2×C47#189911yescharacteristic
172C2×#123—21no—
173C2×#121C2×D4711yescharacteristic
174C22×C42C2×D4711yescharacteristic
175C2×C22⋊C8—472no—
176#390Dic4712yescomplemented
177#389Dic4712yescomplemented
178#388—22no—
179#386—21no—
180#383—944no—
181#380C2×D4711yescharacteristic
182C2×#95—472no—
183C2×#95—941no—
184C2×#95—941no—
185#336—942no—
186#327—944no—
187C47#915711yescharacteristic, complemented, Sylow
188C22×D4—21no—
189C22×D4C2×Dic4711yescharacteristic
190C23×C4—942no—
191C23×C4C22⋊D4711yescharacteristic
192C23×C4C22⋊D4711yescharacteristic
193C23×C4C22×D4711yescharacteristic
194C22×C8—942no—
195#123—44no—
196#121—22no—
197#121—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4C2×Dic4711yescharacteristic
200C2×C4⋊C4C2×Dic4712yes—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—21no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42C2×Dic4712yes—
208C2×C42C22⋊D4712yes—
209C22⋊C8—942no—
210C22⋊C8—942no—
211#95—22no—
212#95—21no—
213#95—474no—
214#95—944no—
215#95—944no—
216#95—942no—
217C24—21no—
218C24C2×C22⋊D4711yescharacteristic
219C2×D4—22no—
220C2×D4—42no—
221C2×D4—41no—
222C2×D4—41no—
223C2×D4—42no—
224C2×D4—42no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—944no—
236C22×C4—1884no—
237C22×C4—1884no—
238C22×C4—1882no—
239C22×C4—1882no—
240C22×C4C22⋊Dic4712yes—
241C22×C4C22⋊Dic4712yes—
242C22×C4D4×D4712yes—
243C22×C4#1062535212yes—
244C22×C4C22×Dic4711yescharacteristic
245C22×C4C2×C22⋊D4711yescharacteristic
246C2×C8—1884no—
247C2×C8—1882no—
248C4⋊C4—22no—
249C4⋊C4—42no—
250C4⋊C4—44no—
251C22⋊C4—44no—
252C22⋊C4—44no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C42—24no—
256C42—24no—
257C23—22no—
258C23—22no—
259C23—41no—
260C23—42no—
261C23—41no—
262C23—41no—
263C23—42no—
264C23C2×C22⋊Dic4711yescharacteristic
265C23#1309399211yescharacteristic
266C23#1309400511yescharacteristic
267C23#1309404412yes—
268C23#1309405012yes—
269D4—82no—
270D4—82no—
271D4—82no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—3764no—
293C2×C4—3764no—
294C2×C4—1884no—
295C2×C4—1884no—
296C2×C4—1884no—
297C2×C4D4×Dic4712yes—
298C2×C4#1309399612yes—
299C8—3762no—
300C22—24no—
301C22—24no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—82no—
307C22—82no—
308C22—81no—
309C22—81no—
310C22—81no—
311C22#1568900311yescharacteristic
312C22#1568909712yes—
313C22#1568911911yescharacteristic
314C22#1568912811yescharacteristic
315C22#1568914012yes—
316C22#1568926612yes—
317C22#1568944412yes—
318C4—22no—
319C4—22no—
320C4—42no—
321C4—42no—
322C4—82no—
323C4—81no—
324C4—3764no—
325C2—22no—
326C2—22no—
327C2—161no—
328C2#2024193512yes—
329C2#2024196512yes—
330C2#2024371111yescharacteristic
331C2#2024383711yescharacteristic
332C2#2024384711yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic47
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2854C2
socleC23×C47C2×C22⋊C4
prime radicalC2×#121×C47C22
cosocleC23×C4×C47C23
hypercenter#2854D47
hypocenterC47#9157
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2854

Complements

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

NH ≅ G/Ntype
1. C2×#15688991C2semidirect
2. #20243456C2semidirect
4. #20241266C2semidirect
16. C47×#390C4semidirect
17. C47×#389C4semidirect
176. #390Dic47semidirect
177. #389Dic47semidirect
187. C47#9157semidirect

Maximal subgroups

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

subgroupindex
C2×#156889912
#202434562
C2×#156888572
#202412662
C47×#28542
#915747