Finite Group Databasebeta

← #26197565

Subgroups — #26197565

Order 12032.

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

normal subgroups only

This group has 960 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#20254375C212yesmaximal, complemented
2#20253791C211yesmaximal, characteristic, complemented
3#20242711C214yesmaximal, complemented
4#20242681C214yesmaximal, complemented
5#20242015C211yesmaximal, characteristic, complemented
6#20241986C212yesmaximal, complemented
7C2×C47×#562C211yesmaximal, characteristic, complemented
8#15689673C2211yescharacteristic
9#15689513C2214yescomplemented
10#15689493C2212yescomplemented
11#15689489C2212yescomplemented
12#15689482C2214yescomplemented
13#15689480—22no—
14#15689335—24no—
15#15689262—22no—
16#15689257—22no—
17#15689180C2212yes—
18#15689122—22no—
19C4×#10625329C2211yescharacteristic
20#15688865C2214yescomplemented
21#15688863C2218yescomplemented
22C47×#562—22no—
23C47×#562—24no—
24C47×#562C2214yes—
25C2×#125×C47C2211yescharacteristic
26C2×#124×C47C2212yes—
27C2×#123×C47—22no—
28C2×C22⋊Q8×C47—22no—
29C2×#121×C47—22no—
30C2×#10625350—21no—
31C2×#10625336—24no—
32C2×#10625334—24no—
33C2×#10625329—21no—
34#13094024C2311yescharacteristic
35#13094021C2311yescharacteristic
36#13094012—21no—
37#13093998—22no—
38C4×D188—22no—
39C4⋊(C47⋊C8)—22no—
40#13093948—24no—
41#13093947—24no—
42#13093945—24no—
43#13093944—24no—
44C22⋊(C47⋊C8)—22no—
45#13093935C2314yes—
46#13093935C2312yes—
47C4×C47⋊C8—21no—
48C22×Q8×C47D412yes—
49C22×D4×C47—22no—
50C22×D4×C47D412yes—
51C23×C4×C47—22no—
52#125×C47—22no—
53#125×C47D412yes—
54#125×C47C2312yes—
55#124×C47—22no—
56#124×C47—22no—
57#124×C47D414yes—
58#124×C47C2314yes—
59#123×C47—28no—
60#123×C47—28no—
61C22⋊Q8×C47—28no—
62C22⋊Q8×C47—28no—
63#121×C47—28no—
64#121×C47—28no—
65C2×C4⋊C4×C47—22no—
66C2×C4⋊C4×C47—22no—
67C2×C4⋊C4×C47D412yes—
68C2×C22⋊C4×C47—22no—
69C2×C22⋊C4×C47—22no—
70C2×C22⋊C4×C47—22no—
71C2×C42×C47C2311yescharacteristic
72C24×C47—22no—
73C2×Q8×C47—22no—
74C2×Q8×C47—82no—
75C2×Q8×C47C2×D414yes—
76C2×D4×C47—28no—
77C2×D4×C47—22no—
78C2×D4×C47—82no—
79C2×D4×C47—84no—
80C2×D4×C47C2×D414yes—
81C22×C4×C47—22no—
82C22×C4×C47—22no—
83C22×C4×C47—28no—
84C22×C4×C47—84no—
85C22×C4×C47C2×D411yescharacteristic
86C22×C4×C47C2×D412yes—
87C2×C22⋊D47—41no—
88#10625350—44no—
89C2×D188—21no—
90C2×D188—22no—
91C2×C4×D47—41no—
92C2×Dic94—21no—
93C4⋊C4×C47—28no—
94C4⋊C4×C47—24no—
95C4⋊C4×C47—22no—
96C4⋊C4×C47—24no—
97C4⋊C4×C47C2×D414yes—
98C22⋊C4×C47—28no—
99C22⋊C4×C47—24no—
100C22⋊C4×C47—24no—
101C22⋊C4×C47—24no—
102C22⋊C4×C47—24no—
103C42×C47C2×D411yescharacteristic
104C42×C47C2×D412yes—
105C42×C47C2411yescharacteristic
106#10625336—84no—
107#10625334—84no—
108(C2×D47)⋊C4—42no—
109C4⋊Dic47—22no—
110#10625329—42no—
111C2×C47⋊C8—22no—
112C2×C47⋊C8—24no—
113C23×C47—24no—
114C23×C47—44no—
115C23×C47—44no—
116C23×C47—44no—
117C23×C47#12011yescharacteristic
118C22×D47—41no—
119Q8×C47—44no—
120Q8×C47—82no—
121D4×C47—44no—
122D4×C47—82no—
123D4×C47—88no—
124C2×C4×C47—22no—
125C2×C4×C47—24no—
126C2×C4×C47—24no—
127C2×C4×C47—42no—
128C2×C4×C47—44no—
129C2×C4×C47—44no—
130C2×C4×C47—88no—
131C2×C4×C47#12012yes—
132C2×C4×C47#12011yescharacteristic
133C2×C4×C47C22×D412yes—
134C2×C4×C47C22×D411yescharacteristic
135C22⋊D47—82no—
136C2×Dic47—41no—
137D188—44no—
138D188—42no—
139C4×D47—82no—
140Dic94—42no—
141C47⋊C8—42no—
142C47⋊C8—44no—
143#19001—471nomaximal, Sylow
144C22×C47—22no—
145C22×C47—44no—
146C22×C47—44no—
147C22×C47—44no—
148C22×C47—88no—
149C22×C47#45412yes—
150C22×C47C2×#12011yescharacteristic
151C2×D47—81no—
152C2×D47—82no—
153C4×C47—21no—
154C4×C47—21no—
155C4×C47—22no—
156C4×C47—22no—
157C4×C47—44no—
158C4×C47—44no—
159Dic47—81no—
160C2×#562D4711yescharacteristic, complemented
161#3213—472no—
162#3142—471no—
163#1662—474no—
164#1656—474no—
165#1534—471no—
166#1527—472no—
167C2×C47—22no—
168C2×C47—84no—
169C2×C47C2×#45411yescharacteristic
170C2×C47#301912yes—
171D47—161no—
172#562—22no—
173#562—24no—
174#562C2×D4714yes—
175#532—471no—
176C2×#125C2×D4711yescharacteristic
177C2×#124C2×D4712yes—
178C2×#123—22no—
179C2×C22⋊Q8—22no—
180C2×#121—22no—
181#496—474no—
182#491—472no—
183#490—472no—
184#489—474no—
185#488—942no—
186#463—944no—
187#448—942no—
188#447—942no—
189#432—472no—
190#417—942no—
191C4×M16—471no—
192#330—474no—
193#329—478no—
194C47#1900111yescharacteristic, complemented, Sylow
195C2×#41—941no—
196C22×Q8C22⋊D4712yes—
197C22×D4—22no—
198C22×D4C22⋊D4712yes—
199C23×C4—22no—
200C2×SD16—944no—
201C2×D8—944no—
202C2×M16—941no—
203C4⋊Q8—471no—
204#127—471no—
205#125—22no—
206#125C22⋊D4712yes—
207#125C22×D4712yes—
208#124—22no—
209#124—22no—
210#124—941no—
211#124C22⋊D4714yes—
212#124C22×D4714yes—
213#123—28no—
214#123—28no—
215C22⋊Q8—28no—
216C22⋊Q8—28no—
217#121—28no—
218#121—28no—
219#121—942no—
220C4×D4—942no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4C22⋊D4712yes—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C42C22×D4711yescharacteristic
228C4⋊C8—942no—
229#103—944no—
230#103—944no—
231#102—944no—
232#102—944no—
233C22⋊C8—942no—
234#97—472no—
235#97—474no—
236C4×C8—941no—
237C24—22no—
238#41—1884no—
239C2×Q8—22no—
240C2×Q8—82no—
241C2×Q8—941no—
242C2×Q8C2×C22⋊D4714yes—
243C2×D4—28no—
244C2×D4—22no—
245C2×D4—82no—
246C2×D4—84no—
247C2×D4—941no—
248C2×D4—942no—
249C2×D4—1881no—
250C2×D4C2×C22⋊D4714yes—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—28no—
254C22×C4—84no—
255C22×C4—1881no—
256C22×C4C2×C22⋊D4711yescharacteristic
257C22×C4C2×C22⋊D4712yes—
258SD16—3764no—
259D8—3764no—
260M16—1882no—
261C2×C8—942no—
262C2×C8—944no—
263C4⋊C4—28no—
264C4⋊C4—24no—
265C4⋊C4—22no—
266C4⋊C4—24no—
267C4⋊C4—942no—
268C4⋊C4C2×C22⋊D4714yes—
269C22⋊C4—28no—
270C22⋊C4—24no—
271C22⋊C4—24no—
272C22⋊C4—24no—
273C22⋊C4—24no—
274C22⋊C4—1882no—
275C42C2×C22⋊D4711yescharacteristic
276C42C2×C22⋊D4712yes—
277C42C23×D4711yescharacteristic
278C23—24no—
279C23—44no—
280C23—44no—
281C23—44no—
282C23—1881no—
283C23#1309399311yescharacteristic
284Q8—44no—
285Q8—82no—
286Q8—1882no—
287D4—44no—
288D4—82no—
289D4—88no—
290D4—3762no—
291D4—1884no—
292D4—1882no—
293C2×C4—22no—
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—42no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—88no—
300C2×C4—3762no—
301C2×C4—1881no—
302C2×C4#1309399311yescharacteristic
303C2×C4#1309399312yes—
304C2×C4C22×C22⋊D4711yescharacteristic
305C2×C4C22×C22⋊D4712yes—
306C8—1882no—
307C8—1884no—
308C22—22no—
309C22—44no—
310C22—44no—
311C22—44no—
312C22—88no—
313C22—3761no—
314C22—3762no—
315C22#1568929612yes—
316C22C2×#1309399311yescharacteristic
317C4—21no—
318C4—21no—
319C4—22no—
320C4—22no—
321C4—44no—
322C4—44no—
323C4—3761no—
324C2—22no—
325C2—84no—
326C2—7521no—
327C2C2×#1568929611yescharacteristic
328C2#2025274512yes—
subgroupG/H
centerC22C2×#13093993
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC2×C47×#562C2
socleC22×C47C2×#120
hypercenterC2×#562D47
hypocenterC47#19001
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. #20254375C2semidirect
2. #20253791C2semidirect
3. #20242711C2semidirect
4. #20242681C2semidirect
5. #20242015C2semidirect
6. #20241986C2semidirect
7. C2×C47×#562C2semidirect
9. #15689513C22semidirect
10. #15689493C22semidirect
11. #15689489C22semidirect
12. #15689482C22semidirect
20. #15688865C22semidirect
21. #15688863C22semidirect
160. C2×#562D47semidirect
194. C47#19001semidirect

Maximal subgroups

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

subgroupindex
#202543752
#202537912
#202427112
#202426812
#202420152
#202419862
C2×C47×#5622
#1900147