Finite Group Databasebeta

← #26197604

Subgroups — #26197604

Order 12032.

938 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 938 conjugacy classes of subgroups — too many to draw. Only its 171 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#20254391C212yesmaximal, complemented
2#20253937C211yesmaximal, characteristic, complemented
3#20242705C218yesmaximal, complemented
4#20242015C211yesmaximal, characteristic, complemented
5#20241986C212yesmaximal, complemented
6C2×C47×#562C211yesmaximal, characteristic
7#15689680C2211yescharacteristic
8#15689533C2214yescomplemented
9#15689522—22no—
10#15689518C2214yescomplemented
11#15689482C2214yescomplemented
12#15689342—24no—
13#15689279—22no—
14#15689274—22no—
15#15689180C2212yes—
16#15689163—22no—
17C4×#10625329C2211yescharacteristic
18#15688865C2214yescomplemented
19#15688863C2218yescomplemented
20C47×#562—22no—
21C47×#562—24no—
22C47×#562C2214yes—
23C2×#125×C47C2211yescharacteristic
24C2×#124×C47C2212yes—
25C2×#123×C47—22no—
26C2×C22⋊Q8×C47—22no—
27C2×#121×C47—22no—
28C2×#10625337—24no—
29C2×#10625335—24no—
30C2×#10625329—21no—
31#13094024C2312yes—
32#13094016—21no—
33#13094003—22no—
34C4×Dic94—22no—
35#13093983—21no—
36#13093957—24no—
37#13093955—24no—
38C4⋊(C47⋊C8)—22no—
39#13093948—24no—
40#13093945—24no—
41C22⋊(C47⋊C8)—22no—
42#13093935C2314yes—
43#13093935C2312yes—
44C4×C47⋊C8—21no—
45C22×Q8×C47D412yes—
46C22×D4×C47—22no—
47C22×D4×C47D412yes—
48C23×C4×C47—22no—
49#125×C47—22no—
50#125×C47D412yes—
51#125×C47C2312yes—
52#124×C47—22no—
53#124×C47—22no—
54#124×C47D414yes—
55#124×C47C2314yes—
56#123×C47—28no—
57#123×C47—28no—
58C22⋊Q8×C47—28no—
59C22⋊Q8×C47—28no—
60#121×C47—28no—
61#121×C47—28no—
62C2×C4⋊C4×C47—22no—
63C2×C4⋊C4×C47—22no—
64C2×C4⋊C4×C47D412yes—
65C2×C22⋊C4×C47—22no—
66C2×C22⋊C4×C47—22no—
67C2×C22⋊C4×C47—22no—
68C2×C42×C47C2311yescharacteristic
69C24×C47—22no—
70C2×Q8×C47—22no—
71C2×Q8×C47—82no—
72C2×Q8×C47C2×D414yes—
73C2×D4×C47—28no—
74C2×D4×C47—22no—
75C2×D4×C47—82no—
76C2×D4×C47—84no—
77C2×D4×C47C2×D414yes—
78C22×C4×C47—22no—
79C22×C4×C47—22no—
80C22×C4×C47—28no—
81C22×C4×C47—84no—
82C22×C4×C47C2×D411yescharacteristic
83C22×C4×C47C2×D412yes—
84C2×Dic94—22no—
85C4⋊C4×C47—28no—
86C4⋊C4×C47—24no—
87C4⋊C4×C47—22no—
88C4⋊C4×C47—24no—
89C4⋊C4×C47C2×D414yes—
90C22⋊C4×C47—28no—
91C22⋊C4×C47—24no—
92C22⋊C4×C47—24no—
93C22⋊C4×C47—24no—
94C22⋊C4×C47—24no—
95C42×C47C2×D411yescharacteristic
96C42×C47C2×D412yes—
97C42×C47C2411yescharacteristic
98C22⋊Dic47—41no—
99#10625337—84no—
100#10625335—84no—
101C4⋊Dic47—22no—
102C4⋊Dic47—22no—
103Dic47⋊C4—42no—
104C4×Dic47—41no—
105#10625329—24no—
106C2×C47⋊C8—22no—
107C2×C47⋊C8—24no—
108C23×C47—24no—
109C23×C47—44no—
110C23×C47—44no—
111C23×C47—44no—
112C23×C47#12011yescharacteristic
113Q8×C47—44no—
114Q8×C47—82no—
115D4×C47—44no—
116D4×C47—82no—
117D4×C47—88no—
118C2×C4×C47—22no—
119C2×C4×C47—24no—
120C2×C4×C47—24no—
121C2×C4×C47—42no—
122C2×C4×C47—44no—
123C2×C4×C47—44no—
124C2×C4×C47—88no—
125C2×C4×C47#12011yescharacteristic
126C2×C4×C47#12012yes—
127C2×C4×C47C22×D411yescharacteristic
128C2×C4×C47C22×D412yes—
129C2×Dic47—42no—
130Dic94—44no—
131C47⋊C8—24no—
132C47⋊C8—44no—
133#19006—471nomaximal, Sylow
134C22×C47—22no—
135C22×C47—44no—
136C22×C47—44no—
137C22×C47—44no—
138C22×C47—88no—
139C22×C47#45612yes—
140C22×C47C2×#12011yescharacteristic
141C4×C47—21no—
142C4×C47—21no—
143C4×C47—22no—
144C4×C47—22no—
145C4×C47—44no—
146C4×C47—44no—
147Dic47—82no—
148C2×#562D4711yescharacteristic
149#3215—472no—
150#3166—471no—
151#1661—478no—
152#1534—471no—
153#1527—472no—
154C2×C47—22no—
155C2×C47—84no—
156C2×C47C2×#45611yescharacteristic
157C2×C47#301912yes—
158#562—22no—
159#562—24no—
160#562C2×D4714yes—
161#533—471no—
162C2×#125C2×D4711yescharacteristic
163C2×#124C2×D4712yes—
164C2×#123—22no—
165C2×C22⋊Q8—22no—
166C2×#121—22no—
167#501—474no—
168#499—942no—
169#497—474no—
170#489—474no—
171#464—944no—
172#451—942no—
173#450—942no—
174#432—472no—
175#428—942no—
176C4×M16—471no—
177#330—474no—
178#329—478no—
179C47#1900611yescharacteristic, complemented, Sylow
180C22×Q8C22⋊D4712yes—
181C22×D4—22no—
182C22×D4C22⋊D4712yes—
183C23×C4—22no—
184C2×Q16—944no—
185C2×SD16—944no—
186C2×M16—941no—
187C4⋊Q8—472no—
188#125—22no—
189#125—941no—
190#125C22⋊D4712yes—
191#125C22×D4712yes—
192#124—22no—
193#124—22no—
194#124C22⋊D4714yes—
195#124C22×D4714yes—
196#123—28no—
197#123—28no—
198C22⋊Q8—28no—
199C22⋊Q8—28no—
200C22⋊Q8—942no—
201#121—28no—
202#121—28no—
203C4×Q8—942no—
204#117—941no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4C22⋊D4712yes—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C42C22×D4711yescharacteristic
212#107—944no—
213#106—944no—
214C4⋊C8—942no—
215#103—944no—
216#102—944no—
217C22⋊C8—942no—
218#97—472no—
219#97—474no—
220C4×C8—941no—
221C24—22no—
222C2×Q8—22no—
223C2×Q8—82no—
224C2×Q8—942no—
225C2×Q8C2×C22⋊D4714yes—
226C2×D4—28no—
227C2×D4—22no—
228C2×D4—82no—
229C2×D4—84no—
230C2×D4C2×C22⋊D4714yes—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—28no—
234C22×C4—84no—
235C22×C4C2×C22⋊D4711yescharacteristic
236C22×C4C2×C22⋊D4712yes—
237Q16—3764no—
238SD16—3764no—
239M16—944no—
240C2×C8—942no—
241C2×C8—944no—
242C4⋊C4—28no—
243C4⋊C4—24no—
244C4⋊C4—22no—
245C4⋊C4—24no—
246C4⋊C4—942no—
247C4⋊C4—942no—
248C4⋊C4—1882no—
249C4⋊C4C2×C22⋊D4714yes—
250C22⋊C4—28no—
251C22⋊C4—24no—
252C22⋊C4—24no—
253C22⋊C4—24no—
254C22⋊C4—24no—
255C22⋊C4—1881no—
256C42—1881no—
257C42C2×C22⋊D4711yescharacteristic
258C42C2×C22⋊D4712yes—
259C42C23×D4711yescharacteristic
260C23—24no—
261C23—44no—
262C23—44no—
263C23—44no—
264C23#1309399311yescharacteristic
265Q8—44no—
266Q8—82no—
267Q8—1884no—
268D4—44no—
269D4—82no—
270D4—88no—
271C2×C4—22no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—42no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—88no—
278C2×C4—1882no—
279C2×C4#1309399311yescharacteristic
280C2×C4#1309399312yes—
281C2×C4C22×C22⋊D4711yescharacteristic
282C2×C4C22×C22⋊D4712yes—
283C8—944no—
284C8—1884no—
285C22—22no—
286C22—44no—
287C22—44no—
288C22—44no—
289C22—88no—
290C22#1568930612yes—
291C22C2×#1309399311yescharacteristic
292C4—21no—
293C4—21no—
294C4—22no—
295C4—22no—
296C4—44no—
297C4—44no—
298C4—3762no—
299C2—22no—
300C2—84no—
301C2C2×#1568930611yescharacteristic
302C2#2025274512yes—
subgroupG/H
centerC22C2×#13093993
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
Fitting, prime radicalC2×C47×#562C2
socleC22×C47C2×#120
hypercenterC2×#562D47
hypocenterC47#19006
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#562

Complements

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

NH ≅ G/Ntype
1. #20254391C2semidirect
2. #20253937C2semidirect
3. #20242705C2semidirect
4. #20242015C2semidirect
5. #20241986C2semidirect
8. #15689533C22semidirect
10. #15689518C22semidirect
11. #15689482C22semidirect
18. #15688865C22semidirect
19. #15688863C22semidirect
179. C47#19006semidirect

Maximal subgroups

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

subgroupindex
#202543912
#202539372
#202427052
#202420152
#202419862
C2×C47×#5622
#1900647