Finite Group Databasebeta

← #26112061

Subgroups — #26112061

Order 12032.

648 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 127 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#20251719C211yesmaximal, characteristic
2C2×#15688991C211yesmaximal, characteristic, complemented
3C2×#15688858C212yesmaximal
4#20241268C212yesmaximal, complemented
5C47×#1817C211yesmaximal, characteristic, complemented
6C2×#13094007—21no—
7C2×#13093998C2211yescharacteristic
8#15689025C412yescomplemented
9#15689021C412yescomplemented
10#15689014—22no—
11#15689002—21no—
12#15688991—24no—
13#15688983C2211yescharacteristic
14C2×C47⋊#95—21no—
15C2×C47⋊#95—21no—
16C2×C47⋊#95C2212yes—
17#15688858—24no—
18C22×C42×C47C2211yescharacteristic
19C2×C22⋊C8×C47C2212yes—
20C47×#336—22no—
21C22×D188C2×C411yescharacteristic
22C22×C22⋊D47—21no—
23C23×Dic47—22no—
24#13094007—44no—
25#13093998—22no—
26#13093998—22no—
27C2×C4⋊Dic47C2×C411yescharacteristic
28C2×C4⋊Dic47C2×C412yes—
29C2×Dic47⋊C4—22no—
30C2×C22⋊Dic47—21no—
31C2×(C2×D47)⋊C4—22no—
32C2×(C2×D47)⋊C4—22no—
33C2×(C2×D47)⋊C4—21no—
34C47⋊#95—24no—
35C47⋊#95—24no—
36C47⋊#95—22no—
37C47⋊#95—22no—
38C47⋊#95—21no—
39C47⋊#95D414yes—
40C23×C4×C47D411yescharacteristic
41C23×C4×C47D411yescharacteristic
42C23×C4×C47C2311yescharacteristic
43C22×C8×C47—22no—
44C2×C42×C47—22no—
45C2×C42×C47—22no—
46C2×C42×C47C2×C412yes—
47C2×C42×C47D412yes—
48C22⋊C8×C47—22no—
49C22⋊C8×C47—22no—
50C24×C47C2×D411yescharacteristic
51C23×D47—21no—
52C22×C4×C47—21no—
53C22×C4×C47—21no—
54C22×C4×C47—21no—
55C22×C4×C47—21no—
56C22×C4×C47—22no—
57C22×C4×C47—22no—
58C22×C4×C47—42no—
59C22×C4×C47—42no—
60C22×C4×C47C22⋊C412yes—
61C22×C4×C47C22⋊C412yes—
62C22×C4×C47C22×C411yescharacteristic
63C22×C4×C47C2×D411yescharacteristic
64C22×C4×C47C2×D412yes—
65C22×C4×C47#4112yes—
66C2×C22⋊D47—41no—
67C2×C22⋊D47—41no—
68C2×C22⋊D47—42no—
69C2×C22⋊D47—42no—
70C22×Dic47—22no—
71C22×Dic47—21no—
72C22×Dic47—24no—
73C22×Dic47—44no—
74C22×Dic47—44no—
75C22×Dic47—42no—
76C22×Dic47—42no—
77C2×D188—22no—
78C2×D188—42no—
79C2×C8×C47—44no—
80C2×C8×C47—42no—
81C42×C47—24no—
82C42×C47—24no—
83C22⋊Dic47—42no—
84(C2×D47)⋊C4—44no—
85(C2×D47)⋊C4—44no—
86(C2×D47)⋊C4—42no—
87C4⋊Dic47—22no—
88C4⋊Dic47—42no—
89Dic47⋊C4—44no—
90C23×C47—22no—
91C23×C47—22no—
92C23×C47C2×C22⋊C411yescharacteristic
93C23×C47#12011yescharacteristic
94C23×C47#12311yescharacteristic
95C23×C47#13512yes—
96C23×C47AGL(1,ℤ/8)12yes—
97C22×D47—41no—
98C22×D47—42no—
99C22×D47—41no—
100C22×D47—41no—
101C22×D47—42no—
102C2×C4×C47—22no—
103C2×C4×C47—22no—
104C2×C4×C47—22no—
105C2×C4×C47—22no—
106C2×C4×C47—22no—
107C2×C4×C47—22no—
108C2×C4×C47—22no—
109C2×C4×C47—22no—
110C2×C4×C47—42no—
111C2×C4×C47—42no—
112C2×C4×C47—42no—
113C2×C4×C47—42no—
114C2×C4×C47—42no—
115C2×C4×C47—42no—
116C2×C4×C47C4×D412yes—
117C2×C4×C47#12112yes—
118C22⋊D47—82no—
119C22⋊D47—82no—
120C2×Dic47—44no—
121C2×Dic47—44no—
122C2×Dic47—41no—
123C2×Dic47—42no—
124C2×Dic47—41no—
125C2×Dic47—41no—
126C2×Dic47—42no—
127C2×Dic47—42no—
128C2×Dic47—44no—
129C2×Dic47—84no—
130C2×Dic47—84no—
131D188—82no—
132C8×C47—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#38611yescharacteristic
141C22×C47#41012yes—
142C22×C47#41611yescharacteristic
143C22×C47#41811yescharacteristic
144C22×C47#42112yes—
145C22×C47#44912yes—
146C22×C47#48212yes—
147C2×D47—81no—
148C2×D47—81no—
149C2×D47—81no—
150C2×D47—82no—
151C2×D47—82no—
152C4×C47—22no—
153C4×C47—22no—
154C4×C47—42no—
155C4×C47—42no—
156Dic47—84no—
157Dic47—81no—
158Dic47—82no—
159#2854—471no—
160C2×#383—471no—
161#1817D4711yescharacteristic, complemented
162C2×#327—472no—
163#1294—472no—
164C2×C47—22no—
165C2×C47—22no—
166C2×C47#151412yes—
167C2×C47#152112yes—
168C2×C47#187511yescharacteristic
169C2×C47#189711yescharacteristic
170C2×C47#189911yescharacteristic
171D47—161no—
172C2×#123—941no—
173C2×#121—471no—
174C22×C42C2×D4711yescharacteristic
175C2×C22⋊C8C2×D4712yes—
176#390—472no—
177#389—472no—
178#388—942no—
179#386—941no—
180#383—944no—
181#380—471no—
182C2×#95—472no—
183C2×#95—941no—
184C2×#95—941no—
185#336—22no—
186#327—944no—
187C47#915711yescharacteristic, complemented, Sylow
188C22×D4—471no—
189C22×D4—941no—
190C23×C4—942no—
191C23×C4D18811yescharacteristic
192C23×C4D18811yescharacteristic
193C23×C4C22×D4711yescharacteristic
194C22×C8—22no—
195#123—1884no—
196#121—942no—
197#121—942no—
198C2×C4⋊C4—471no—
199C2×C4⋊C4—472no—
200C2×C4⋊C4—942no—
201C2×C22⋊C4—941no—
202C2×C22⋊C4—942no—
203C2×C22⋊C4—942no—
204C2×C22⋊C4—941no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42C4×D4712yes—
208C2×C42C22⋊D4712yes—
209C22⋊C8—22no—
210C22⋊C8—22no—
211#95—474no—
212#95—944no—
213#95—944no—
214#95—942no—
215#95—942no—
216#95—941no—
217C24—941no—
218C24C2×D18811yescharacteristic
219C2×D4—942no—
220C2×D4—1882no—
221C2×D4—1881no—
222C2×D4—1881no—
223C2×D4—1882no—
224C2×D4—1882no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—942no—
234C22×C4—941no—
235C22×C4—944no—
236C22×C4—1884no—
237C22×C4—1884no—
238C22×C4—1882no—
239C22×C4—1882no—
240C22×C4(C2×D47)⋊C412yes—
241C22×C4(C2×D47)⋊C412yes—
242C22×C4C2×C4×D4711yescharacteristic
243C22×C4D4×D4712yes—
244C22×C4#1062535212yes—
245C22×C4C2×C22⋊D4711yescharacteristic
246C2×C8—44no—
247C2×C8—42no—
248C4⋊C4—942no—
249C4⋊C4—1882no—
250C4⋊C4—1884no—
251C22⋊C4—1884no—
252C22⋊C4—1884no—
253C22⋊C4—1882no—
254C22⋊C4—1882no—
255C42—24no—
256C42—24no—
257C23—22no—
258C23—22no—
259C23—1881no—
260C23—1882no—
261C23—1881no—
262C23—1881no—
263C23—1882no—
264C23C2×(C2×D47)⋊C411yescharacteristic
265C23#1309399411yescharacteristic
266C23#1309400811yescharacteristic
267C23#1309404612yes—
268C23#1309405212yes—
269D4—3762no—
270D4—3762no—
271D4—3762no—
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—3764no—
287C2×C4—3764no—
288C2×C4—1884no—
289C2×C4—1884no—
290C2×C4—1881no—
291C2×C4—1884no—
292C2×C4—1882no—
293C2×C4—1881no—
294C2×C4—1882no—
295C2×C4—1881no—
296C2×C4—1882no—
297C2×C4(C22⋊D47)⋊C412yes—
298C2×C4#1309399612yes—
299C8—82no—
300C22—24no—
301C22—24no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—3762no—
307C22—3762no—
308C22—3761no—
309C22—3761no—
310C22—3761no—
311C22#1568900011yescharacteristic
312C22#1568909612yes—
313C22#1568912011yescharacteristic
314C22#1568912911yescharacteristic
315C22#1568913912yes—
316C22#1568927012yes—
317C22#1568944812yes—
318C4—22no—
319C4—22no—
320C4—42no—
321C4—42no—
322C4—3764no—
323C4—3762no—
324C4—3761no—
325C2—22no—
326C2—22no—
327C2—7521no—
328C2#2024193612yes—
329C2#2024196612yes—
330C2#2024370811yescharacteristic
331C2#2024383411yescharacteristic
332C2#2024384411yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1817C2
socleC23×C47C2×C22⋊C4
prime radicalC2×#13093998C22
cosocleC23×C4×C47C23
hypercenter#1817D47
hypocenterC47#9157
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
2. C2×#15688991C2semidirect
4. #20241268C2semidirect
5. C47×#1817C2semidirect
8. #15689025C4semidirect
9. #15689021C4semidirect
161. #1817D47semidirect
187. C47#9157semidirect

Maximal subgroups

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

subgroupindex
#202517192
C2×#156889912
C2×#156888582
#202412682
C47×#18172
#915747