Finite Group Databasebeta

← #26113993

Subgroups — #26113993

Order 12032.

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

normal subgroups only

This group has 536 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#20252473C211yesmaximal, characteristic, complemented
2#20242102C212yesmaximal
3#20241732C211yesmaximal, characteristic, complemented
4#20241244C212yesmaximal, complemented
5C47×#2548C211yesmaximal, characteristic
6#15689254—24no—
7#15689180—21no—
8#15689180C2212yes—
9#15689180C2212yes—
10#15689176—21no—
11#15688867—24no—
12#15688849—21no—
13#15688849—21no—
14C2×#127×C47C2211yescharacteristic
15C2×#124×C47C412yes—
16C2×C4×Q8×C47C2211yescharacteristic
17C47×#394—21no—
18C47×#394—21no—
19C47×#390—21no—
20C47×#390—21no—
21C47×#390C412yes—
22C4×C4⋊C4×C47C2211yescharacteristic
23C4⋊(C47⋊C8)—22no—
24C4⋊(C47⋊C8)—22no—
25C4⋊(C47⋊C8)—22no—
26C4⋊(C47⋊C8)D414yes—
27C22⋊(C47⋊C8)—22no—
28C22⋊(C47⋊C8)—22no—
29#13093935—22no—
30#13093935—22no—
31#13093935—22no—
32C4×C47⋊C8—22no—
33C22×Q8×C47C2×C411yescharacteristic
34C22×D4×C47—21no—
35C22×D4×C47—21no—
36C22×D4×C47C2×C412yes—
37#127×C47—22no—
38#127×C47—22no—
39#124×C47—44no—
40C4×Q8×C47—22no—
41C4×Q8×C47—22no—
42C2×C4⋊C4×C47—21no—
43C2×C4⋊C4×C47C2×C411yescharacteristic
44C2×C4⋊C4×C47D411yescharacteristic
45C2×C4⋊C4×C47D411yescharacteristic
46C2×C22⋊C4×C47—22no—
47C2×C22⋊C4×C47—22no—
48C2×C22⋊C4×C47—22no—
49C2×C22⋊C4×C47—22no—
50C2×C42×C47C2×C412yes—
51C2×C42×C47D411yescharacteristic
52C2×C42×C47D411yescharacteristic
53C2×C42×C47C2311yescharacteristic
54#95×C47—21no—
55C24×C47—22no—
56C2×Q8×C47—22no—
57C2×Q8×C47—22no—
58C2×Q8×C47—41no—
59C2×D4×C47—24no—
60C2×D4×C47—42no—
61C2×D4×C47—42no—
62C2×D4×C47—42no—
63C2×D4×C47—42no—
64C2×D4×C47—41no—
65C2×D4×C47—42no—
66C2×D4×C47—41no—
67C2×D4×C47—82no—
68C22×C4×C47—22no—
69C22×C4×C47C22⋊C411yescharacteristic
70C22×C4×C47C22⋊C411yescharacteristic
71C22×C4×C47C22⋊C411yescharacteristic
72C22×C4×C47C22⋊C411yescharacteristic
73C22×C4×C47C22×C411yescharacteristic
74C22×C4×C47C2×D411yescharacteristic
75C22×C4×C47C2×D411yescharacteristic
76C4⋊C4×C47—22no—
77C4⋊C4×C47—22no—
78C4⋊C4×C47—22no—
79C4⋊C4×C47—42no—
80C22⋊C4×C47—44no—
81C22⋊C4×C47—44no—
82C22⋊C4×C47—82no—
83C22⋊C4×C47—82no—
84C42×C47—21no—
85C42×C47—21no—
86C42×C47—21no—
87C42×C47—21no—
88C42×C47—42no—
89C42×C47C2×D412yes—
90C42×C47#4112yes—
91C2×C47⋊C8—24no—
92C2×C47⋊C8—42no—
93C23×C47—42no—
94C23×C47—42no—
95C23×C47—42no—
96C23×C47—42no—
97C23×C47—42no—
98C23×C47—42no—
99C23×C47—42no—
100C23×C47C2×C22⋊C411yescharacteristic
101Q8×C47—44no—
102D4×C47—84no—
103D4×C47—82no—
104D4×C47—84no—
105D4×C47—82no—
106C2×C4×C47—21no—
107C2×C4×C47—21no—
108C2×C4×C47—21no—
109C2×C4×C47—21no—
110C2×C4×C47—22no—
111C2×C4×C47—21no—
112C2×C4×C47—21no—
113C2×C4×C47—21no—
114C2×C4×C47—21no—
115C2×C4×C47—21no—
116C2×C4×C47—41no—
117C2×C4×C47—41no—
118C2×C4×C47—41no—
119C2×C4×C47—42no—
120C2×C4×C47—42no—
121C2×C4×C47—42no—
122C2×C4×C47#10012yes—
123C2×C4×C47C4×D412yes—
124C2×C4×C47#12011yescharacteristic
125C2×C4×C47#12112yes—
126C2×C4×C47#12311yescharacteristic
127C47⋊C8—44no—
128C47⋊C8—82no—
129#9458—471nomaximal, Sylow
130C22×C47—21no—
131C22×C47—21no—
132C22×C47—82no—
133C22×C47—82no—
134C22×C47—82no—
135C22×C47—82no—
136C22×C47—82no—
137C22×C47—82no—
138C22×C47—82no—
139C22×C47#38611yescharacteristic
140C22×C47C2×#10011yescharacteristic
141C22×C47#41311yescharacteristic
142C4×C47—21no—
143C4×C47—22no—
144C4×C47—22no—
145C4×C47—22no—
146C4×C47—41no—
147C4×C47—41no—
148C4×C47—41no—
149C4×C47—82no—
150C4×C47#45412yes—
151#2983—471no—
152#2548D4711yescharacteristic
153#1555—472no—
154#1467—471no—
155#1285—472no—
156C2×C47—21no—
157C2×C47—21no—
158C2×C47—162no—
159C2×C47#188811yescharacteristic
160C2×C47#190011yescharacteristic
161C2×C47#191011yescharacteristic
162C2×#127C2×D4711yescharacteristic
163C2×#124Dic4712yes—
164C2×C4×Q8C2×D4711yescharacteristic
165#446—944no—
166#432—472no—
167#432—472no—
168#432—941no—
169#431—941no—
170#394—21no—
171#394—21no—
172#390—21no—
173#390—21no—
174#390Dic4712yes—
175C4×C4⋊C4C2×D4711yescharacteristic
176#331—944no—
177#324—941no—
178#324—941no—
179C47#945811yescharacteristic, complemented, Sylow
180C22×Q8C2×Dic4711yescharacteristic
181C22×D4—21no—
182C22×D4—21no—
183C22×D4C2×Dic4712yes—
184#127—22no—
185#127—22no—
186#124—44no—
187C4×Q8—22no—
188C4×Q8—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4C2×Dic4711yescharacteristic
191C2×C4⋊C4C22⋊D4711yescharacteristic
192C2×C4⋊C4C22⋊D4711yescharacteristic
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C42C2×Dic4712yes—
198C2×C42C22⋊D4711yescharacteristic
199C2×C42C22⋊D4711yescharacteristic
200C2×C42C22×D4711yescharacteristic
201C4⋊C8—474no—
202C4⋊C8—942no—
203C4⋊C8—942no—
204C4⋊C8—942no—
205C22⋊C8—942no—
206C22⋊C8—942no—
207#97—942no—
208#97—942no—
209#97—942no—
210C4×C8—942no—
211#95—21no—
212C24—22no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—41no—
216C2×D4—24no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—42no—
220C2×D4—42no—
221C2×D4—41no—
222C2×D4—42no—
223C2×D4—41no—
224C2×D4—82no—
225C22×C4—22no—
226C22×C4C22⋊Dic4711yescharacteristic
227C22×C4C22⋊Dic4711yescharacteristic
228C22×C4C22⋊Dic4711yescharacteristic
229C22×C4C22⋊Dic4711yescharacteristic
230C22×C4C22×Dic4711yescharacteristic
231C22×C4C2×C22⋊D4711yescharacteristic
232C22×C4C2×C22⋊D4711yescharacteristic
233C2×C8—944no—
234C2×C8—1882no—
235C4⋊C4—22no—
236C4⋊C4—22no—
237C4⋊C4—22no—
238C4⋊C4—42no—
239C22⋊C4—44no—
240C22⋊C4—44no—
241C22⋊C4—82no—
242C22⋊C4—82no—
243C42—21no—
244C42—21no—
245C42—21no—
246C42—21no—
247C42—42no—
248C42D4×D4712yes—
249C42#1062535212yes—
250C23—42no—
251C23—42no—
252C23—42no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23—42no—
257C23C2×C22⋊Dic4711yescharacteristic
258Q8—44no—
259D4—84no—
260D4—82no—
261D4—84no—
262D4—82no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4#1309394112yes—
280C2×C4D4×Dic4712yes—
281C2×C4#1309399211yescharacteristic
282C2×C4#1309399612yes—
283C2×C4#1309400511yescharacteristic
284C8—3762no—
285C8—1884no—
286C22—21no—
287C22—21no—
288C22—82no—
289C22—82no—
290C22—82no—
291C22—82no—
292C22—82no—
293C22—82no—
294C22—82no—
295C22#1568900311yescharacteristic
296C22C2×#1309394111yescharacteristic
297C22#1568910711yescharacteristic
298C4—21no—
299C4—22no—
300C4—22no—
301C4—22no—
302C4—41no—
303C4—41no—
304C4—41no—
305C4—82no—
306C4#1568929512yes—
307C2—21no—
308C2—21no—
309C2—162no—
310C2#2024379211yescharacteristic
311C2#2024385211yescharacteristic
312C2#2024391611yescharacteristic
subgroupG/H
centerC22#15689003
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2548C2
socleC22×C47#386
prime radicalC2×#127×C47C22
cosocleC2×C42×C47C23
hypercenter#2548D47
hypocenterC47#9458
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2548

Complements

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

NH ≅ G/Ntype
1. #20252473C2semidirect
3. #20241732C2semidirect
4. #20241244C2semidirect
179. C47#9458semidirect

Maximal subgroups

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

subgroupindex
#202524732
#202421022
#202417322
#202412442
C47×#25482
#945847