Finite Group Databasebeta

← #26114130

Subgroups — #26114130

Order 12032.

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

normal subgroups only

This group has 542 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#20252469C211yesmaximal, characteristic, complemented
2#20248293C211yesmaximal, characteristic
3#20241812C212yesmaximal
4#20241248C212yesmaximal, complemented
5C47×#1801C211yesmaximal, characteristic, complemented
6C2×#13094000—21no—
7C2×C4×D188C2211yescharacteristic
8Q8×C47⋊C8—24no—
9#15689176—22no—
10#15689176C2212yes—
11#15689053C412yes—
12#15689040C412yes—
13#15689014—22no—
14#15688989C2211yescharacteristic
15#15688980—21no—
16#15688850—24no—
17C2×C4×Q8×C47C2211yescharacteristic
18C47×#423C2212yes—
19C2×#101×C47—21no—
20C47×#343—21no—
21C22×D188C2×C411yescharacteristic
22C22×C4×D47—21no—
23#13094000—42no—
24#13094000—42no—
25C4×D188—22no—
26C4×D188—22no—
27C2×C4⋊Dic47—21no—
28C2×C4⋊Dic47C2×C411yescharacteristic
29C2×Dic47⋊C4—22no—
30C2×(C2×D47)⋊C4—22no—
31C2×(C2×D47)⋊C4—22no—
32C2×(C2×D47)⋊C4—21no—
33C4⋊(C47⋊C8)—24no—
34C4⋊(C47⋊C8)—22no—
35C22⋊(C47⋊C8)—22no—
36C4×C47⋊C8—24no—
37C4×C47⋊C8D414yes—
38C47⋊#95—22no—
39C47⋊#95—21no—
40C47⋊#95C2×C412yes—
41C22×Q8×C47D411yescharacteristic
42C2×M16×C47—22no—
43C4×Q8×C47—22no—
44C4×Q8×C47—22no—
45C2×C4⋊C4×C47C2×C412yes—
46C2×C4⋊C4×C47D411yescharacteristic
47C2×C42×C47D412yes—
48C2×C42×C47C2311yescharacteristic
49C4⋊C8×C47—22no—
50C4⋊C8×C47—22no—
51#101×C47—42no—
52C23×D47—21no—
53C2×Q8×C47—22no—
54C2×Q8×C47—22no—
55C2×Q8×C47D812yes—
56C2×Q8×C47SD1612yes—
57C22×C4×C47C22⋊C412yes—
58C22×C4×C47C22⋊C412yes—
59C22×C4×C47C22×C411yescharacteristic
60C22×C4×C47C2×D411yescharacteristic
61C22×C4×C47C2×D411yescharacteristic
62C22×Dic47—22no—
63C22×Dic47—21no—
64C2×D188—22no—
65C2×D188—42no—
66C2×C4×D47—42no—
67C2×C4×D47—41no—
68C2×C4×D47—41no—
69C2×C4×D47—42no—
70M16×C47—42no—
71M16×C47—42no—
72C2×C8×C47—42no—
73C4⋊C4×C47—24no—
74C4⋊C4×C47—22no—
75C42×C47—22no—
76C42×C47—22no—
77C42×C47C2×D412yes—
78C42×C47#4112yes—
79(C2×D47)⋊C4—44no—
80(C2×D47)⋊C4—44no—
81(C2×D47)⋊C4—41no—
82(C2×D47)⋊C4—41no—
83C4⋊Dic47—22no—
84C4⋊Dic47—41no—
85C4⋊Dic47—41no—
86Dic47⋊C4—44no—
87C2×C47⋊C8—24no—
88C23×C47C2×C22⋊C411yescharacteristic
89C22×D47—22no—
90C22×D47—41no—
91C22×D47—41no—
92C22×D47—41no—
93C22×D47—41no—
94C22×D47—41no—
95C22×D47—41no—
96Q8×C47—24no—
97Q8×C47—24no—
98C2×C4×C47—22no—
99C2×C4×C47—22no—
100C2×C4×C47—22no—
101C2×C4×C47—41no—
102C2×C4×C47—41no—
103C2×C4×C47—42no—
104C2×C4×C47#10214yes—
105C2×C4×C47C4×D412yes—
106C2×C4×C47#12011yescharacteristic
107C2×C4×C47#12112yes—
108C2×C4×C47#12311yescharacteristic
109C2×C4×C47C2×D811yescharacteristic
110C2×C4×C47C2×SD1611yescharacteristic
111C2×C4×C47#13512yes—
112C2×Dic47—42no—
113C2×Dic47—42no—
114C2×Dic47—41no—
115C2×Dic47—41no—
116C2×Dic47—42no—
117C2×Dic47—41no—
118D188—24no—
119D188—42no—
120C4×D47—82no—
121C4×D47—82no—
122C8×C47—82no—
123C47⋊C8—44no—
124#9486—471nomaximal, Sylow
125C22×C47—21no—
126C22×C47—21no—
127C22×C47#38611yescharacteristic
128C22×C47C2×#10211yescharacteristic
129C22×C47#41611yescharacteristic
130C2×D47—42no—
131C2×D47—42no—
132C2×D47—42no—
133C2×D47—81no—
134C2×D47—81no—
135C2×D47—81no—
136C2×D47—81no—
137C4×C47—22no—
138C4×C47—24no—
139C4×C47—21no—
140C4×C47—21no—
141C4×C47—42no—
142C4×C47#46512yes—
143C4×C47#46612yes—
144Dic47—82no—
145Dic47—81no—
146#2982—471no—
147#2536—471no—
148#1801D4711yescharacteristic, complemented
149#1486—472no—
150#1286—472no—
151C2×C47—21no—
152C2×C47—21no—
153C2×C47#189111yescharacteristic
154C2×C47#189311yescharacteristic
155C2×C47#191211yescharacteristic
156D47—82no—
157C2×C22⋊Q8—941no—
158C2×C4×Q8C2×D4711yescharacteristic
159C2×C4×D4—471no—
160C8×Q8—944no—
161#431—472no—
162#431—942no—
163#423C2×D4712yes—
164C2×#101—21no—
165#398—472no—
166#394—472no—
167#388—942no—
168#382—471no—
169#380—941no—
170#343—21no—
171D4⋊C8—944no—
172C47#948611yescharacteristic, complemented, Sylow
173C22×Q8C22⋊D4711yescharacteristic
174C22×D4—471no—
175C23×C4—941no—
176C2×M16—22no—
177C22⋊Q8—1882no—
178C22⋊Q8—1882no—
179C4×Q8—22no—
180C4×Q8—22no—
181C4×D4—942no—
182C4×D4—942no—
183C2×C4⋊C4—471no—
184C2×C4⋊C4—942no—
185C2×C4⋊C4—941no—
186C2×C4⋊C4C4×D4712yes—
187C2×C4⋊C4C22⋊D4711yescharacteristic
188C2×C22⋊C4—942no—
189C2×C22⋊C4—942no—
190C2×C22⋊C4—941no—
191C2×C42D18812yes—
192C2×C42C22×D4711yescharacteristic
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—944no—
196C4⋊C8—942no—
197#101—42no—
198C22⋊C8—942no—
199C4×C8—474no—
200C4×C8—944no—
201#95—472no—
202#95—942no—
203#95—941no—
204C24—941no—
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8#1062533412yes—
208C2×Q8#1062533612yes—
209C2×D4—942no—
210C2×D4—1882no—
211C22×C4—942no—
212C22×C4—941no—
213C22×C4—1882no—
214C22×C4—1881no—
215C22×C4—1881no—
216C22×C4—1882no—
217C22×C4(C2×D47)⋊C412yes—
218C22×C4(C2×D47)⋊C412yes—
219C22×C4C2×C4×D4711yescharacteristic
220C22×C4C2×D18811yescharacteristic
221C22×C4C2×C22⋊D4711yescharacteristic
222M16—42no—
223M16—42no—
224C2×C8—42no—
225C2×C8—944no—
226C4⋊C4—24no—
227C4⋊C4—22no—
228C4⋊C4—942no—
229C4⋊C4—1884no—
230C4⋊C4—1881no—
231C4⋊C4—1881no—
232C22⋊C4—1884no—
233C22⋊C4—1884no—
234C22⋊C4—1881no—
235C22⋊C4—1881no—
236C42—22no—
237C42—22no—
238C42D4×D4712yes—
239C42#1062535212yes—
240C23—942no—
241C23—1881no—
242C23—1881no—
243C23—1881no—
244C23—1881no—
245C23—1881no—
246C23—1881no—
247C23C2×(C2×D47)⋊C411yescharacteristic
248Q8—24no—
249Q8—24no—
250D4—944no—
251D4—1882no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—3762no—
259C2×C4—3762no—
260C2×C4—1882no—
261C2×C4—1882no—
262C2×C4—1881no—
263C2×C4—1882no—
264C2×C4—1881no—
265C2×C4—1881no—
266C2×C4#1309394414yes—
267C2×C4(C22⋊D47)⋊C412yes—
268C2×C4#1309399411yescharacteristic
269C2×C4#1309399612yes—
270C2×C4#1309400811yescharacteristic
271C2×C4C2×#1062533411yescharacteristic
272C2×C4C2×#1062533611yescharacteristic
273C2×C4#1309404412yes—
274C8—82no—
275C8—1884no—
276C22—21no—
277C22—21no—
278C22—3761no—
279C22—3761no—
280C22—3761no—
281C22—3761no—
282C22—1882no—
283C22—1882no—
284C22—1882no—
285C22#1568900011yescharacteristic
286C22C2×#1309394411yescharacteristic
287C22#1568911711yescharacteristic
288C4—22no—
289C4—24no—
290C4—21no—
291C4—21no—
292C4—42no—
293C4—3762no—
294C4—3761no—
295C4#1568935412yes—
296C4#1568935712yes—
297C2—21no—
298C2—21no—
299C2—3762no—
300C2#2024380411yescharacteristic
301C2#2024381411yescharacteristic
302C2#2024392311yescharacteristic
subgroupG/H
centerC22#15689000
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1801C2
socleC22×C47#386
prime radicalC22×D188C2×C4
cosocleC2×C42×C47C23
hypercenter#1801D47
hypocenterC47#9486
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
1. #20252469C2semidirect
4. #20241248C2semidirect
5. C47×#1801C2semidirect
148. #1801D47semidirect
172. C47#9486semidirect

Maximal subgroups

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

subgroupindex
#202524692
#202482932
#202418122
#202412482
C47×#18012
#948647