Finite Group Databasebeta

← #26112992

Subgroups — #26112992

Order 12032.

454 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 107 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#20254338C211yesmaximal, characteristic
2#20246134C211yesmaximal, characteristic
3#20244368C211yesmaximal, characteristic
4#20242070C211yesmaximal, characteristic
5#20241951C211yesmaximal, characteristic
6#20241216C211yesmaximal, characteristic
7C47×#1286C211yesmaximal, characteristic
8#15689680C2211yescharacteristic
9C2×#13094024C2211yescharacteristic
10#15689503C412yescomplemented
11#15689474C412yescomplemented
12#15689334—22no—
13#15689281—22no—
14#15689176C2211yescharacteristic
15#15689027—22no—
16#15689019—21no—
17#15689008—22no—
18C4⋊C4×Dic47—21no—
19#15688969C2211yescharacteristic
20#15688884—21no—
21#15688884—21no—
22#15688871—22no—
23#15688859—22no—
24C47×#431C2211yescharacteristic
25C47×#423C2211yescharacteristic
26C47×#382C2211yescharacteristic
27C22×Dic94D412yes—
28C2×Dic188—24no—
29#13094024—21no—
30#13094024—21no—
31#13094024C2×C411yescharacteristic
32#13094024C2×C411yescharacteristic
33#13094024C2×C412yes—
34#13094024D412yes—
35#13094016—21no—
36#13094003—21no—
37#13094003—21no—
38C4×Dic94—21no—
39C4×Dic94—21no—
40#13093983—21no—
41C2×C4⋊Dic47—21no—
42C2×C4⋊Dic47—21no—
43C2×Dic47⋊C4—22no—
44C2×C4×Dic47—22no—
45C2×C4×Dic47D411yescharacteristic
46C2×C4×Dic47D411yescharacteristic
47C4⋊(C47⋊C8)—21no—
48#13093949—24no—
49C22⋊(C47⋊C8)—21no—
50C4×C47⋊C8—21no—
51C47⋊#95—22no—
52C47⋊#95—22no—
53C47⋊#95—21no—
54C47⋊#95—21no—
55C2×M16×C47—21no—
56C2×C4⋊C4×C47—21no—
57C2×C42×C47C2311yescharacteristic
58C4⋊C8×C47—21no—
59C4⋊C8×C47—21no—
60C4⋊C8×C47—21no—
61C22⋊C8×C47—21no—
62C4×C8×C47C2×C412yes—
63#95×C47—21no—
64#95×C47D412yes—
65C22×C4×C47—22no—
66C22×C4×C47C2×D411yescharacteristic
67C22×C4×C47C2×D411yescharacteristic
68C22×C4×C47C2×D411yescharacteristic
69C22×Dic47—21no—
70C22×Dic47—22no—
71C22×Dic47—21no—
72C2×Dic94—21no—
73C2×Dic94—22no—
74C2×Dic94—21no—
75C2×Dic94—82no—
76C2×Dic94C22⋊C414yes—
77M16×C47—41no—
78M16×C47—41no—
79C2×C8×C47—22no—
80C2×C8×C47—41no—
81C4⋊C4×C47—41no—
82C4⋊C4×C47—41no—
83C42×C47C22×C411yescharacteristic
84C42×C47C2×D411yescharacteristic
85C42×C47#4111yescharacteristic
86C42×C47#4111yescharacteristic
87C22⋊Dic47—41no—
88C4⋊Dic47—22no—
89C4⋊Dic47—22no—
90C4⋊Dic47—22no—
91C4⋊Dic47—22no—
92C4⋊Dic47—21no—
93C4⋊Dic47—21no—
94C4⋊Dic47—21no—
95C4⋊Dic47—21no—
96Dic47⋊C4—41no—
97Dic47⋊C4—41no—
98Dic47⋊C4—82no—
99C4×Dic47—21no—
100C4×Dic47—21no—
101C4×Dic47—21no—
102C4×Dic47—41no—
103C4×Dic47—82no—
104C4×Dic47Q1612yescomplemented
105C2×C47⋊C8—41no—
106Dic188—84no—
107C23×C47#12011yescharacteristic
108C2×C4×C47—41no—
109C2×C4×C47—41no—
110C2×C4×C47—42no—
111C2×C4×C47—42no—
112C2×C4×C47—42no—
113C2×C4×C47C2×C22⋊C411yescharacteristic
114C2×C4×C47C4×D411yescharacteristic
115C2×C4×C47C4×D411yescharacteristic
116C2×C4×C47#12111yescharacteristic
117C2×C4×C47#12111yescharacteristic
118C2×C4×C47#12311yescharacteristic
119C2×C4×C47C2×Q1611yescharacteristic
120C2×C4×C47#13511yescharacteristic
121C2×C4×C47#13711yescharacteristic
122C2×C4×C47#13711yescharacteristic
123C2×Dic47—24no—
124C2×Dic47—22no—
125C2×Dic47—41no—
126C2×Dic47—42no—
127C2×Dic47—42no—
128C2×Dic47—41no—
129C2×Dic47—41no—
130C2×Dic47—41no—
131C2×Dic47—41no—
132C2×Dic47—41no—
133C2×Dic47—41no—
134Dic94—44no—
135Dic94—42no—
136Dic94—82no—
137Dic94—81no—
138C8×C47—42no—
139C8×C47—81no—
140C47⋊C8—81no—
141#9297—471nomaximal, Sylow
142C22×C47—21no—
143C22×C47—21no—
144C22×C47#38611yescharacteristic
145C22×C47#45111yescharacteristic
146C22×C47#45211yescharacteristic
147C4×C47—21no—
148C4×C47—21no—
149C4×C47—21no—
150C4×C47—21no—
151C4×C47—82no—
152C4×C47C4×Q1611yescharacteristic
153C4×C47#44111yescharacteristic
154C4×C47#46211yescharacteristic
155C4×C47#46411yescharacteristic
156Dic47—42no—
157Dic47—44no—
158Dic47—81no—
159Dic47—81no—
160Dic47—81no—
161#3209—471no—
162#2331—471no—
163#1995—471no—
164#1548—471no—
165#1518—471no—
166#1286D4711yescharacteristic
167#1274—471no—
168C2×C47—21no—
169C2×C47—21no—
170C2×C47#187811yescharacteristic
171C2×C47#190511yescharacteristic
172C2×C47#191111yescharacteristic
173#533—471no—
174C2×C4⋊Q8—471no—
175#494—472no—
176#487—472no—
177#462—942no—
178#451—942no—
179#431—471no—
180#431C2×D4711yescharacteristic
181#423C2×D4711yescharacteristic
182#391—942no—
183#389—941no—
184#387—942no—
185#382C2×D4711yescharacteristic
186C4×C4⋊C4—941no—
187#376—471no—
188#339—941no—
189#339—941no—
190#333—942no—
191#328—942no—
192C47#929711yescharacteristic, complemented, Sylow
193C22×Q8—472no—
194C2×Q16—944no—
195C2×M16—21no—
196C4⋊Q8—472no—
197C4⋊Q8—471no—
198C4⋊Q8—471no—
199C4⋊Q8—472no—
200C4⋊Q8—941no—
201C4⋊Q8—941no—
202#125—941no—
203C22⋊Q8—941no—
204C22⋊Q8—941no—
205C4×Q8—941no—
206C4×Q8—941no—
207#117—941no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—941no—
210C2×C4⋊C4—941no—
211C2×C4⋊C4—942no—
212C2×C42—471no—
213C2×C42—471no—
214C2×C42—942no—
215C2×C42C22×D4711yescharacteristic
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—941no—
220#103—944no—
221C22⋊C8—21no—
222C22⋊C8—941no—
223C4×C8—941no—
224C4×C8C4×D4712yes—
225#95—21no—
226#95—942no—
227#95—942no—
228#95—941no—
229#95—941no—
230#95D18812yes—
231C2×Q8—474no—
232C2×Q8—942no—
233C2×Q8—941no—
234C2×Q8—941no—
235C2×Q8—3762no—
236C22×C4—22no—
237C22×C4—941no—
238C22×C4—942no—
239C22×C4—941no—
240C22×C4C2×D18811yescharacteristic
241C22×C4D4×D4711yescharacteristic
242C22×C4D4×D4711yescharacteristic
243Q16—3764no—
244M16—41no—
245M16—41no—
246C2×C8—22no—
247C2×C8—41no—
248C2×C8—1881no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—942no—
252C4⋊C4—942no—
253C4⋊C4—942no—
254C4⋊C4—942no—
255C4⋊C4—941no—
256C4⋊C4—941no—
257C4⋊C4—941no—
258C4⋊C4—941no—
259C4⋊C4—3762no—
260C4⋊C4—1881no—
261C4⋊C4—1881no—
262C22⋊C4—1881no—
263C42—472no—
264C42—941no—
265C42—941no—
266C42—941no—
267C42—3762no—
268C42—1881no—
269C42C2×C4×D4711yescharacteristic
270C42#1062535011yescharacteristic
271C42D4×D4711yescharacteristic
272C42#1062535411yescharacteristic
273C23#1309399411yescharacteristic
274Q8—3761no—
275Q8—3762no—
276Q8—1882no—
277Q8—1884no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—944no—
284C2×C4—942no—
285C2×C4—1881no—
286C2×C4—1881no—
287C2×C4—1881no—
288C2×C4—1881no—
289C2×C4—1881no—
290C2×C4—1881no—
291C2×C4—1882no—
292C2×C4—1882no—
293C2×C4—1881no—
294C2×C4C22⋊C4×D4711yescharacteristic
295C2×C4#1309398611yescharacteristic
296C2×C4C4×D18811yescharacteristic
297C2×C4#1309399711yescharacteristic
298C2×C4#1309399911yescharacteristic
299C2×C4#1309400911yescharacteristic
300C2×C4Q16×D4711yescharacteristic
301C2×C4#1309404311yescharacteristic
302C2×C4#1309405311yescharacteristic
303C2×C4#1309405311yescharacteristic
304C8—42no—
305C8—81no—
306C8—3761no—
307C22—21no—
308C22—21no—
309C22#1568900111yescharacteristic
310C22#1568928011yescharacteristic
311C22#1568928611yescharacteristic
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—82no—
317C4—3761no—
318C4—3761no—
319C4—3761no—
320C4—1882no—
321C4—1884no—
322C4#1568921711yescharacteristic
323C4#1568922911yescharacteristic
324C4#1568933211yescharacteristic
325C4#1568934411yescharacteristic
326C2—21no—
327C2—21no—
328C2#2024373011yescharacteristic
329C2#2024388111yescharacteristic
330C2#2024391911yescharacteristic
subgroupG/H
centerC22#15689001
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1286C2
socleC22×C47#386
prime radicalC23×C47#120
cosocleC2×C42×C47C23
hypercenter#1286D47
hypocenterC47#9297
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1286

Complements

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

NH ≅ G/Ntype
10. #15689503C4semidirect
11. #15689474C4semidirect
104. C4×Dic47Q16semidirect
192. C47#9297semidirect

Maximal subgroups

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

subgroupindex
#202543382
#202461342
#202443682
#202420702
#202419512
#202412162
C47×#12862
#929747