Finite Group Databasebeta

← #26104398

Subgroups — #26104398

Order 12032.

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

normal subgroups only

This group has 608 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
1C4×#13094021C211yesmaximal, characteristic
2#20243531C211yesmaximal, characteristic, complemented
3#20241208C214yesmaximal, complemented
4C47×#1839C211yesmaximal, characteristic, complemented
5C2×#13094021C411yescharacteristic
6C2×C4×D188—21no—
7C2×C4×D188—21no—
8C2×C4×D188C412yes—
9#15689025—21no—
10#15689025—21no—
11#15689025C2212yes—
12#15688993C411yescharacteristic
13#15688878—21no—
14#15688878—21no—
15#15688878C2212yes—
16C43×C47C2211yescharacteristic
17C47×#336—21no—
18C47×#336—21no—
19C47×#336C2212yes—
20C22×D188—21no—
21C22×D188—21no—
22C22×D188C2×C412yes—
23C22×C4×D47—22no—
24C22×C47⋊C8—22no—
25#13094021—22no—
26#13094021C814yescomplemented
27C4×D188—44no—
28C4×D188—44no—
29C4×D188—44no—
30C2×C4⋊Dic47—21no—
31C2×C4⋊Dic47—21no—
32C2×C4⋊Dic47C2×C412yes—
33C2×(C2×D47)⋊C4—22no—
34C2×(C2×D47)⋊C4—22no—
35C2×(C2×D47)⋊C4—22no—
36C22×C8×C47—22no—
37C2×C42×C47C2×C411yescharacteristic
38C2×C42×C47C2×C411yescharacteristic
39C2×C42×C47D411yescharacteristic
40C2×C42×C47D411yescharacteristic
41C2×C42×C47D411yescharacteristic
42C2×C42×C47D411yescharacteristic
43C2×C42×C47C2311yescharacteristic
44C23×D47—22no—
45C22×C4×C47C22⋊C411yescharacteristic
46C22×C4×C47C22⋊C411yescharacteristic
47C22×C4×C47C22⋊C411yescharacteristic
48C22×C4×C47C22⋊C411yescharacteristic
49C22×C4×C47C22×C411yescharacteristic
50C22×C4×C47C2×D411yescharacteristic
51C22×C4×C47C2×D411yescharacteristic
52C22×Dic47—22no—
53C2×D188—24no—
54C2×D188—24no—
55C2×D188—22no—
56C2×D188—42no—
57C2×D188—42no—
58C2×D188—42no—
59C2×D188—42no—
60C2×D188—42no—
61C2×D188—42no—
62C2×D188C2×C814yes—
63C2×C4×D47—44no—
64C2×C4×D47—44no—
65C2×C4×D47—44no—
66C2×C8×C47—42no—
67C2×C8×C47—42no—
68C2×C8×C47—42no—
69C42×C47—22no—
70C42×C47—22no—
71C42×C47—22no—
72C42×C47—41no—
73C42×C47—41no—
74C42×C47—41no—
75C42×C47C2×C812yes—
76C42×C47M1612yes—
77(C2×D47)⋊C4—44no—
78(C2×D47)⋊C4—44no—
79(C2×D47)⋊C4—44no—
80C4⋊Dic47—42no—
81C4⋊Dic47—42no—
82C4⋊Dic47—42no—
83C2×C47⋊C8—42no—
84C2×C47⋊C8—42no—
85C2×C47⋊C8—42no—
86C23×C47C2×C22⋊C411yescharacteristic
87C22×D47—24no—
88C22×D47—42no—
89C22×D47—42no—
90C22×D47—42no—
91C22×D47—42no—
92C22×D47—42no—
93C22×D47—42no—
94C2×C4×C47—21no—
95C2×C4×C47—21no—
96C2×C4×C47—21no—
97C2×C4×C47—21no—
98C2×C4×C47—21no—
99C2×C4×C47—21no—
100C2×C4×C47—21no—
101C2×C4×C47—21no—
102C2×C4×C47—21no—
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×C47C22⋊C812yes—
110C2×C4×C47C22⋊C812yes—
111C2×C4×C47#10012yes—
112C2×C4×C47C22×C811yescharacteristic
113C2×C4×C47C2×M1611yescharacteristic
114C2×C4×C47AGL(1,ℤ/8)12yes—
115C2×C4×C47AGL(1,ℤ/8)12yes—
116C2×Dic47—42no—
117C2×Dic47—42no—
118C2×Dic47—42no—
119D188—44no—
120D188—44no—
121D188—48no—
122D188—48no—
123C4×D47—88no—
124C8×C47—82no—
125C47⋊C8—82no—
126#7603—471nomaximal, Sylow
127C22×C47C2×C22⋊C811yescharacteristic
128C22×C47#41011yescharacteristic
129C22×C47C2×#10011yescharacteristic
130C22×C47#41811yescharacteristic
131C22×C47#41811yescharacteristic
132C22×C47#42111yescharacteristic
133C22×C47#42111yescharacteristic
134C2×D47—44no—
135C2×D47—44no—
136C2×D47—44no—
137C2×D47—82no—
138C2×D47—82no—
139C2×D47—82no—
140C2×D47—82no—
141C4×C47—22no—
142C4×C47—21no—
143C4×C47—21no—
144C4×C47—22no—
145C4×C47—22no—
146C4×C47—22no—
147C4×C47—22no—
148C4×C47—41no—
149Dic47—82no—
150C4×#127—471no—
151#1839—471no—
152#1839D4711yescharacteristic, complemented
153#1270—474no—
154C2×C47#146011yescharacteristic
155C2×C47#147811yescharacteristic
156C2×C47#147811yescharacteristic
157C2×C47#151411yescharacteristic
158C2×C47#151411yescharacteristic
159C2×C47#153611yescharacteristic
160C2×C47#155511yescharacteristic
161D47—84no—
162C2×#127—471no—
163C2×C4×D4—472no—
164C2×C4×D4—941no—
165C2×C4×D4—941no—
166#390—472no—
167#390—941no—
168#390—941no—
169#384—471no—
170C43C2×D4711yescharacteristic
171#336—21no—
172#336—21no—
173#336—472no—
174#336—941no—
175#336—941no—
176#336C2×D4712yes—
177C47#760311yescharacteristic, complemented, Sylow
178C22×D4—472no—
179C22×D4—941no—
180C22×D4—941no—
181C23×C4—942no—
182C22×C8—22no—
183C22×C8—942no—
184#127—474no—
185#127—942no—
186C4×D4—1884no—
187C4×D4—1884no—
188C4×D4—1884no—
189C2×C4⋊C4—472no—
190C2×C4⋊C4—941no—
191C2×C4⋊C4—941no—
192C2×C22⋊C4—942no—
193C2×C22⋊C4—942no—
194C2×C22⋊C4—942no—
195C2×C42C4×D4711yescharacteristic
196C2×C42C4×D4711yescharacteristic
197C2×C42D18811yescharacteristic
198C2×C42D18811yescharacteristic
199C2×C42C22⋊D4711yescharacteristic
200C2×C42C22⋊D4711yescharacteristic
201C2×C42C22×D4711yescharacteristic
202C24—942no—
203C2×D4—474no—
204C2×D4—944no—
205C2×D4—944no—
206C2×D4—942no—
207C2×D4—1882no—
208C2×D4—1882no—
209C2×D4—1882no—
210C2×D4—1882no—
211C2×D4—1882no—
212C2×D4—1882no—
213C22×C4—942no—
214C22×C4—1884no—
215C22×C4—1884no—
216C22×C4—1884no—
217C22×C4(C2×D47)⋊C411yescharacteristic
218C22×C4(C2×D47)⋊C411yescharacteristic
219C22×C4(C2×D47)⋊C411yescharacteristic
220C22×C4(C2×D47)⋊C411yescharacteristic
221C22×C4C2×C4×D4711yescharacteristic
222C22×C4C2×D18811yescharacteristic
223C22×C4C2×C22⋊D4711yescharacteristic
224C2×C8—42no—
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—1882no—
228C2×C8—1882no—
229C2×C8—1882no—
230C4⋊C4—1882no—
231C4⋊C4—1882no—
232C4⋊C4—1882no—
233C22⋊C4—1884no—
234C22⋊C4—1884no—
235C22⋊C4—1884no—
236C42—22no—
237C42—22no—
238C42—22no—
239C42—41no—
240C42—41no—
241C42—41no—
242C42C8×D4712yes—
243C42#1062532412yes—
244C23—944no—
245C23—1882no—
246C23—1882no—
247C23—1882no—
248C23—1882no—
249C23—1882no—
250C23—1882no—
251C23C2×(C2×D47)⋊C411yescharacteristic
252D4—1884no—
253D4—1884no—
254D4—1888no—
255D4—1888no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—3768no—
272C2×C4—1882no—
273C2×C4—1882no—
274C2×C4—1882no—
275C2×C4(C2×D47)⋊C812yes—
276C2×C4(C2×D47)⋊C812yes—
277C2×C4#1309394012yes—
278C2×C4C2×C8×D4711yescharacteristic
279C2×C4C2×#1062532411yescharacteristic
280C2×C4#1309404912yes—
281C2×C4#1309405212yes—
282C8—82no—
283C8—3762no—
284C22—3762no—
285C22—3762no—
286C22—3762no—
287C22—3762no—
288C22—1884no—
289C22—1884no—
290C22—1884no—
291C22C2×(C2×D47)⋊C811yescharacteristic
292C22#1568909611yescharacteristic
293C22C2×#1309394011yescharacteristic
294C22#1568912711yescharacteristic
295C22#1568912911yescharacteristic
296C22#1568913911yescharacteristic
297C22#1568914111yescharacteristic
298C4—22no—
299C4—22no—
300C4—21no—
301C4—21no—
302C4—22no—
303C4—22no—
304C4—22no—
305C4—41no—
306C4—3762no—
307C2—3764no—
308C2#2024170811yescharacteristic
309C2#2024177811yescharacteristic
310C2#2024178011yescharacteristic
311C2#2024193411yescharacteristic
312C2#2024193611yescharacteristic
313C2#2024202211yescharacteristic
314C2#2024210111yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC2×C4×C47C22×C8
FrattiniC2×C42C22×D47
FittingC47×#1839C2
socleC23×C47C2×C22⋊C4
prime radicalC2×#13094021C4
cosocleC2×C42×C47C23
hypercenter#1839D47
hypocenterC47#7603
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C43 ◁ #1839

Complements

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

NH ≅ G/Ntype
2. #20243531C2semidirect
3. #20241208C2semidirect
4. C47×#1839C2semidirect
26. #13094021C8semidirect
152. #1839D47semidirect
177. C47#7603semidirect

Maximal subgroups

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

subgroupindex
C4×#130940212
#202435312
#202412082
C47×#18392
#760347