Finite Group Databasebeta

← #26131732

Subgroups — #26131732

Order 12032.

890 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 890 conjugacy classes of subgroups — too many to draw. Only its 155 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#20258099C211yesmaximal, characteristic
2#20245015C212yesmaximal, complemented
3C2×#15688908C212yesmaximal, complemented
4#20245002C211yesmaximal, characteristic, complemented
5C47×#2115C211yesmaximal, characteristic, complemented
6#15689959C411yescharacteristic
7#15689890C411yescharacteristic
8#15689889—22no—
9#15689889C412yes—
10#15689888—24no—
11C2×#13094030—22no—
12C2×#13094030C2212yes—
13#131×D47—24no—
14C2×#13093964C2211yescharacteristic, complemented
15C22×C47⋊C16C2211yescharacteristic, complemented
16#15688908—24no—
17#15688906—24no—
18C22⋊(C47⋊C16)—22no—
19C2×#131×C47C2211yescharacteristic
20C2×M32×C47C2212yescomplemented
21C47×#349—22no—
22#13094072C812yes—
23#13094071—22no—
24#13094070—22no—
25#13094069—22no—
26#13094069C812yes—
27C2×#10625350—22no—
28C2×#10625350C2×C412yes—
29#41×D47—24no—
30#13094032—28no—
31#13094032—24no—
32#13094031—24no—
33#13094030—24no—
34#13094030—24no—
35#13094030—24no—
36#13094030—22no—
37#13094030—22no—
38#13094030D414yes—
39#13094030D414yes—
40C2×#10625329—22no—
41C2×#10625329C2×C412yes—
42C2×#10625324—24no—
43C2×#10625324—22no—
44M16×D47—28no—
45M16×D47—24no—
46C2×C8×D47—24no—
47C2×C8×D47—22no—
48#13093964—22no—
49C2×C47⋊C16—21no—
50C2×C47⋊C16—22no—
51C2×C47⋊C16—21no—
52C2×#41×C47C2×C411yescharacteristic
53#131×C47—22no—
54#131×C47—22no—
55C2×M16×C47C2×C411yescharacteristic
56C2×M16×C47D412yes—
57C22×C8×C47D412yes—
58C22×C8×C47C2311yescharacteristic
59M32×C47—22no—
60M32×C47—22no—
61C2×C16×C47—22no—
62#41×C47—22no—
63#41×C47—22no—
64C2×Q8×C47C2×C811yescharacteristic
65C2×D4×C47C2×C811yescharacteristic
66C2×D4×C47M1612yes—
67C22×C4×C47C22⋊C412yes—
68C22×C4×C47C22×C411yescharacteristic
69C2×C22⋊D47—24no—
70C2×C22⋊D47—22no—
71#10625354—24no—
72Q8×D47—24no—
73#10625352—28no—
74#10625352—24no—
75D4×D47—28no—
76D4×D47—24no—
77#10625350—24no—
78#10625350—24no—
79#10625350—24no—
80#10625350—22no—
81#10625350—22no—
82#10625350C22⋊C414yes—
83#10625350C2×C814yes—
84C2×D188—22no—
85C2×D188M1612yes—
86C2×C4×D47—24no—
87C2×C4×D47—22no—
88C2×Dic94—22no—
89C2×Dic94M1612yes—
90M16×C47—22no—
91M16×C47—24no—
92C2×C8×C47—22no—
93C2×C8×C47—22no—
94C2×C8×C47C22⋊C412yes—
95C2×C8×C47C2×D411yescharacteristic
96C2×C8×C47C2×D412yes—
97C2×C8×C47C2×D412yes—
98C2×C8×C47C2×D411yescharacteristic
99#10625329—24no—
100#10625329—22no—
101#10625329C22⋊C414yes—
102C2×C47⋊C8—24no—
103C2×C47⋊C8—22no—
104#10625324—28no—
105#10625324—24no—
106#10625324—24no—
107C8×D47—28no—
108C8×D47—24no—
109C8×D47—24no—
110C16×C47—42no—
111C47⋊C16—41no—
112C47⋊C16—41no—
113C23×C47C22⋊C812yes—
114C23×C47C2×M1611yescharacteristic
115C22×D47—24no—
116C22×D47—22no—
117Q8×C47—22no—
118D4×C47—24no—
119D4×C47—22no—
120C2×C4×C47—22no—
121C2×C4×C47—22no—
122C2×C4×C47C22⋊C812yes—
123C2×C4×C47C2×C22⋊C412yes—
124C2×C4×C47C2×C22⋊C411yescharacteristic
125C2×C4×C47C22×C811yescharacteristic
126C2×C4×C47C2×M1612yes—
127C22⋊D47—28no—
128C22⋊D47—24no—
129C22⋊D47—24no—
130C2×Dic47—24no—
131C2×Dic47—22no—
132D188—24no—
133D188—22no—
134D188C22⋊C814yes—
135C4×D47—28no—
136C4×D47—24no—
137C4×D47—24no—
138Dic94—24no—
139Dic94—22no—
140Dic94C22⋊C814yes—
141C8×C47—22no—
142C8×C47#12012yes—
143C8×C47#12011yescharacteristic
144C8×C47#12011yescharacteristic
145C47⋊C8—24no—
146#12685—471nomaximal, Sylow
147C22×C47—22no—
148C22×C47—41no—
149C22×C47—42no—
150C22×C47C2×C22⋊C811yescharacteristic
151C22×C47#40712yes—
152C2×D47—24no—
153C2×D47—82no—
154C2×D47—84no—
155C4×C47—22no—
156C4×C47#37911yescharacteristic
157C4×C47C2×C22⋊C812yes—
158C4×C47#40711yescharacteristic
159Dic47—24no—
160#3573—471no—
161#2117—472no—
162C2×#350—472no—
163#2115D4711yescharacteristic, complemented
164#2113—471no—
165C2×C47—22no—
166C2×C47—21no—
167C2×C47—42no—
168C2×C47#177911yescharacteristic
169D47—44no—
170#585—471no—
171#568—471no—
172#568—472no—
173#568—944no—
174#568—942no—
175C2×#131—472no—
176C2×#131—944no—
177C2×#131—942no—
178C2×#131C2×D4711yescharacteristic
179C2×M32—471no—
180C2×M32C2×D4712yescomplemented
181C22×C16—471no—
182#350—944no—
183#350—944no—
184#349—22no—
185C22⋊C16—942no—
186C47#1268511yescharacteristic, complemented, Sylow
1872−1+4—472no—
1882−1+4—942no—
189H(5,𝔽2)—472no—
190H(5,𝔽2)—942no—
191H(5,𝔽2)—942no—
192C2×#41—472no—
193C2×#41—944no—
194C2×#41—942no—
195C2×#41C4×D4711yescharacteristic
196#131—22no—
197#131—22no—
198#131—474no—
199#131—474no—
200#131—944no—
201#131—944no—
202#131—944no—
203#131—948no—
204#131—944no—
205#131—944no—
206#131—942no—
207#131—942no—
208C2×M16—472no—
209C2×M16—944no—
210C2×M16—944no—
211C2×M16—948no—
212C2×M16—942no—
213C2×M16—942no—
214C2×M16C4×D4711yescharacteristic
215C2×M16C22⋊D4712yes—
216C22×C8—944no—
217C22×C8—942no—
218C22×C8D18812yes—
219C22×C8C22×D4711yescharacteristic
220M32—22no—
221M32—22no—
222M32—942no—
223C2×C16—22no—
224C2×C16—941no—
225C2×C16—941no—
226C2×C16—942no—
227#41—22no—
228#41—22no—
229#41—474no—
230#41—474no—
231#41—944no—
232#41—944no—
233#41—944no—
234#41—948no—
235#41—944no—
236#41—944no—
237#41—942no—
238#41—942no—
239C2×Q8—472no—
240C2×Q8—942no—
241C2×Q8—944no—
242C2×Q8C8×D4711yescharacteristic
243C2×D4—472no—
244C2×D4—942no—
245C2×D4—944no—
246C2×D4—948no—
247C2×D4—944no—
248C2×D4—942no—
249C2×D4C8×D4711yescharacteristic
250C2×D4#1062532412yes—
251C22×C4—944no—
252C22×C4—942no—
253C22×C4(C2×D47)⋊C412yes—
254C22×C4C2×C4×D4711yescharacteristic
255M16—22no—
256M16—24no—
257M16—474no—
258M16—948no—
259M16—944no—
260M16—942no—
261M16—944no—
262M16—944no—
263C2×C8—22no—
264C2×C8—22no—
265C2×C8—948no—
266C2×C8—944no—
267C2×C8—942no—
268C2×C8—944no—
269C2×C8—944no—
270C2×C8(C2×D47)⋊C412yes—
271C2×C8C2×D18811yescharacteristic
272C2×C8D4×D4712yes—
273C2×C8D4×D4712yes—
274C2×C8C2×C22⋊D4711yescharacteristic
275C16—42no—
276C16—1881no—
277C16—1881no—
278C23—944no—
279C23—942no—
280C23(C2×D47)⋊C812yes—
281C23C2×#1062532411yescharacteristic
282Q8—22no—
283Q8—474no—
284Q8—944no—
285Q8—942no—
286D4—24no—
287D4—22no—
288D4—474no—
289D4—944no—
290D4—948no—
291D4—944no—
292D4—944no—
293D4—942no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—948no—
297C2×C4—944no—
298C2×C4—942no—
299C2×C4—944no—
300C2×C4—944no—
301C2×C4(C2×D47)⋊C812yes—
302C2×C4C22⋊C4×D4712yes—
303C2×C4C2×(C2×D47)⋊C411yescharacteristic
304C2×C4C2×C8×D4711yescharacteristic
305C2×C4M16×D4712yes—
306C8—22no—
307C8—944no—
308C8#1309399212yes—
309C8#1309399411yescharacteristic
310C8#1309399411yescharacteristic
311C22—22no—
312C22—41no—
313C22—42no—
314C22—944no—
315C22—3762no—
316C22—3764no—
317C22C2×(C2×D47)⋊C811yescharacteristic
318C22#1568908512yes—
319C4—22no—
320C4—944no—
321C4#1568897811yescharacteristic
322C4C22⋊C8×D4712yes—
323C4#1568908611yescharacteristic
324C2—21no—
325C2—22no—
326C2—42no—
327C2—1884no—
328C2#2024329711yescharacteristic
subgroupG/H
centerC8#13093994
commutatorC2×C4×C47C22×C8
FrattiniC22×C8C22×D47
FittingC47×#2115C2
socleC2×C47#1779
prime radical#15689959C4
cosocleC22×C8×C47C23
hypercenter#2115D47
hypocenterC47#12685
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C8 ◁ C22×C8 ◁ #2115

Complements

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

NH ≅ G/Ntype
2. #20245015C2semidirect
3. C2×#15688908C2semidirect
4. #20245002C2semidirect
5. C47×#2115C2semidirect
14. C2×#13093964C22semidirect
15. C22×C47⋊C16C22semidirect
20. C2×M32×C47C22semidirect
163. #2115D47semidirect
180. C2×M32C2×D47semidirect
186. C47#12685semidirect

Maximal subgroups

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

subgroupindex
#202580992
#202450152
C2×#156889082
#202450022
C47×#21152
#1268547