Finite Group Databasebeta

← #26112927

Subgroups — #26112927

Order 12032.

410 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 109 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#20254541C211yesmaximal, characteristic
2#20252328C211yesmaximal, characteristic
3#20244361C211yesmaximal, characteristic
4#20242068C211yesmaximal, characteristic
5#20241953C211yesmaximal, characteristic
6#20241247C211yesmaximal, characteristic
7C47×#1275C211yesmaximal, characteristic
8#15689680C2211yescharacteristic
9C2×#13094024C2211yescharacteristic
10#15689528C412yescomplemented
11#15689474C412yescomplemented
12#15689459—22no—
13#15689419—22no—
14#15689251—22no—
15#15689251—22no—
16#15689149—21no—
17#15689147C2211yescharacteristic
18#15689078—21no—
19#15688989C2211yescharacteristic
20#15688886—21no—
21#15688886—21no—
22#15688871—22no—
23#15688861—22no—
24C47×#431C2211yescharacteristic
25C47×#431C2211yescharacteristic
26C2×#97×C47C2211yescharacteristic
27C22×Dic94—21no—
28C2×#10625329—21no—
29#13094024—21no—
30#13094024—21no—
31#13094024—21no—
32#13094024C2×C412yes—
33#13094024C2×C411yescharacteristic
34#13094024C2×C411yescharacteristic
35#13094016—21no—
36#13094003—21no—
37#13094003D412yes—
38C4×Dic94—21no—
39C4×Dic94D412yes—
40#13093983—21no—
41C2×C4⋊Dic47—21no—
42C2×C4⋊Dic47—21no—
43C2×C4⋊Dic47—21no—
44#13093958—24no—
45C4⋊(C47⋊C8)—21no—
46C4⋊(C47⋊C8)D412yes—
47Dic47⋊C8—24no—
48#13093949—24no—
49#13093934—22no—
50C8×Dic47—22no—
51C47⋊#95—21no—
52C47⋊#95—21no—
53C22×C8×C47D411yescharacteristic
54C22×C8×C47D411yescharacteristic
55C2×C42×C47C2311yescharacteristic
56C4⋊C8×C47—21no—
57C4⋊C8×C47—21no—
58C22⋊C8×C47—21no—
59C22⋊C8×C47—21no—
60#97×C47—21no—
61#97×C47—21no—
62C4×C8×C47—21no—
63C4×C8×C47C2×C412yes—
64C22×C4×C47C2×D411yescharacteristic
65C22×C4×C47C2×D411yescharacteristic
66C22×C4×C47#4111yescharacteristic
67C22×Dic47—41no—
68C22×Dic47—41no—
69C2×Dic94—21no—
70C2×Dic94—22no—
71C2×Dic94—41no—
72C2×Dic94—82no—
73C2×Dic94C22⋊C412yes—
74C2×C8×C47—21no—
75C2×C8×C47—21no—
76C2×C8×C47—22no—
77C2×C8×C47—21no—
78C2×C8×C47—21no—
79C2×C8×C47—21no—
80C2×C8×C47—41no—
81C2×C8×C47Q1612yes—
82C42×C47C22×C411yescharacteristic
83C42×C47C2×D411yescharacteristic
84C42×C47C2×D411yescharacteristic
85C42×C47#4111yescharacteristic
86C22⋊Dic47—22no—
87C4⋊Dic47—21no—
88C4⋊Dic47—21no—
89C4⋊Dic47—21no—
90C4⋊Dic47—22no—
91C4⋊Dic47—22no—
92C4⋊Dic47—41no—
93C4⋊Dic47—41no—
94C4⋊Dic47—41no—
95C4⋊Dic47—41no—
96C4⋊Dic47C22⋊C412yes—
97Dic47⋊C4—22no—
98Dic47⋊C4—22no—
99C4×Dic47—22no—
100#10625329—41no—
101#10625329—41no—
102C2×C47⋊C8—22no—
103C23×C47#12111yescharacteristic
104C2×C4×C47—41no—
105C2×C4×C47—41no—
106C2×C4×C47C2×C22⋊C411yescharacteristic
107C2×C4×C47C4×D411yescharacteristic
108C2×C4×C47C4×D411yescharacteristic
109C2×C4×C47#12011yescharacteristic
110C2×C4×C47#12111yescharacteristic
111C2×C4×C47#12311yescharacteristic
112C2×C4×C47C2×Q1611yescharacteristic
113C2×C4×C47#13511yescharacteristic
114C2×C4×C47#13711yescharacteristic
115C2×C4×C47#13711yescharacteristic
116C2×Dic47—22no—
117C2×Dic47—22no—
118C2×Dic47—42no—
119C2×Dic47—82no—
120C2×Dic47—81no—
121C2×Dic47—81no—
122C2×Dic47—81no—
123Dic94—42no—
124Dic94—81no—
125Dic94—82no—
126Dic94—82no—
127C8×C47—22no—
128C8×C47—42no—
129C8×C47—41no—
130C8×C47—81no—
131C47⋊C8—42no—
132#9288—471nomaximal, Sylow
133C22×C47—21no—
134C22×C47—21no—
135C22×C47#38611yescharacteristic
136C22×C47#46211yescharacteristic
137C22×C47#46411yescharacteristic
138C4×C47—21no—
139C4×C47—21no—
140C4×C47—21no—
141C4×C47—21no—
142C4×C47C4×Q1611yescharacteristic
143C4×C47#44111yescharacteristic
144C4×C47#45111yescharacteristic
145C4×C47#45211yescharacteristic
146Dic47—42no—
147Dic47—42no—
148Dic47—82no—
149Dic47—161no—
150#3230—471no—
151#2961—471no—
152#1993—471no—
153#1548—471no—
154#1518—471no—
155#1286—471no—
156#1275D4711yescharacteristic
157C2×C47—21no—
158C2×C47—21no—
159C2×C47#187811yescharacteristic
160C2×C47#190311yescharacteristic
161C2×C47#190911yescharacteristic
162#533—471no—
163C2×C4⋊Q8—471no—
164#500—472no—
165#487—472no—
166#484—942no—
167#477—942no—
168#446—942no—
169#446—942no—
170#431C2×D4711yescharacteristic
171#431C2×D4711yescharacteristic
172#424—941no—
173#423—471no—
174#405—941no—
175C2×#97C2×D4711yescharacteristic
176#382—471no—
177#340—941no—
178#340—941no—
179#333—942no—
180#328—942no—
181C47#928811yescharacteristic, complemented, Sylow
182C22×Q8—941no—
183C2×M16—941no—
184C22×C8D18811yescharacteristic
185C22×C8D18811yescharacteristic
186C4⋊Q8—471no—
187C4⋊Q8—471no—
188C4⋊Q8—472no—
189C4⋊Q8—941no—
190C4⋊Q8—941no—
191C4⋊Q8—941no—
192#125—941no—
193C22⋊Q8—472no—
194C22⋊Q8—941no—
195C4×Q8—472no—
196C4×Q8—941no—
197#117—941no—
198C2×C4⋊C4—941no—
199C2×C4⋊C4—941no—
200C2×C4⋊C4—941no—
201C2×C42C22×D4711yescharacteristic
202#107—944no—
203C4⋊C8—21no—
204C4⋊C8—21no—
205C4⋊C8—472no—
206C4⋊C8—944no—
207C4⋊C8—941no—
208#103—944no—
209C22⋊C8—21no—
210C22⋊C8—21no—
211#97—21no—
212#97—21no—
213#97—942no—
214C4×C8—21no—
215C4×C8—942no—
216C4×C8C4×D4712yes—
217#95—941no—
218#95—941no—
219C2×Q8—472no—
220C2×Q8—941no—
221C2×Q8—942no—
222C2×Q8—3762no—
223C2×Q8—1881no—
224C22×C4—1881no—
225C22×C4—1881no—
226C22×C4C2×D18811yescharacteristic
227C22×C4D4×D4711yescharacteristic
228C22×C4#1062535411yescharacteristic
229M16—1881no—
230M16—1881no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—22no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8—41no—
238C2×C8—942no—
239C2×C8Dic18812yes—
240C4⋊C4—472no—
241C4⋊C4—941no—
242C4⋊C4—941no—
243C4⋊C4—941no—
244C4⋊C4—942no—
245C4⋊C4—942no—
246C4⋊C4—942no—
247C4⋊C4—942no—
248C4⋊C4—1881no—
249C4⋊C4—1881no—
250C4⋊C4—1881no—
251C4⋊C4—1881no—
252C22⋊C4—942no—
253C42—942no—
254C42C2×C4×D4711yescharacteristic
255C42#1062535011yescharacteristic
256C42D4×D4711yescharacteristic
257C42D4×D4711yescharacteristic
258C23#1309399911yescharacteristic
259Q8—3762no—
260Q8—3762no—
261Q8—3761no—
262Q8—1882no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—942no—
266C2×C4—942no—
267C2×C4—3762no—
268C2×C4—3761no—
269C2×C4—3761no—
270C2×C4—3761no—
271C2×C4—1882no—
272C2×C4C22⋊C4×D4711yescharacteristic
273C2×C4#1309398611yescharacteristic
274C2×C4C4×D18811yescharacteristic
275C2×C4#1309399411yescharacteristic
276C2×C4#1309399711yescharacteristic
277C2×C4#1309400911yescharacteristic
278C2×C4C2×Dic18811yescharacteristic
279C2×C4#1309404611yescharacteristic
280C2×C4#1309405711yescharacteristic
281C2×C4#1309405711yescharacteristic
282C8—22no—
283C8—42no—
284C8—41no—
285C8—81no—
286C8—1882no—
287C22—21no—
288C22—21no—
289C22#1568900111yescharacteristic
290C22#1568933411yescharacteristic
291C22#1568934611yescharacteristic
292C4—21no—
293C4—21no—
294C4—21no—
295C4—21no—
296C4—3762no—
297C4—1882no—
298C4—1882no—
299C4—7521no—
300C4C4×Dic18811yescharacteristic
301C4#1568923111yescharacteristic
302C4#1568928111yescharacteristic
303C4#1568928711yescharacteristic
304C2—21no—
305C2—21no—
306C2#2024372811yescharacteristic
307C2#2024386911yescharacteristic
308C2#2024390911yescharacteristic
subgroupG/H
centerC22#15689001
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1275C2
socleC22×C47#386
prime radicalC23×C47#121
cosocleC2×C42×C47C23
hypercenter#1275D47
hypocenterC47#9288
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1275

Complements

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

NH ≅ G/Ntype
10. #15689528C4semidirect
11. #15689474C4semidirect
181. C47#9288semidirect

Maximal subgroups

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

subgroupindex
#202545412
#202523282
#202443612
#202420682
#202419532
#202412472
C47×#12752
#928847