Finite Group Databasebeta

← #26112925

Subgroups — #26112925

Order 12032.

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

normal subgroups only

This group has 410 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#20254539C211yesmaximal, characteristic
2#20252325C211yesmaximal, characteristic
3#20244362C211yesmaximal, characteristic
4#20241953C211yesmaximal, characteristic
5#20241248C211yesmaximal, characteristic
6#20241218C211yesmaximal, characteristic
7C47×#1548C211yesmaximal, characteristic
8#15689680C2211yescharacteristic
9#15689526C412yescomplemented
10#15689475C412yescomplemented
11#15689458—22no—
12#15689415—22no—
13#15689252—22no—
14#15689252—22no—
15#15689176C2211yescharacteristic
16#15689151—21no—
17#15689081—21no—
18C2×#13093935C2211yescharacteristic
19#15688989C2211yescharacteristic
20#15688885—21no—
21#15688885—21no—
22#15688861—22no—
23C2×C4⋊Q8×C47C2211yescharacteristic
24C47×#431C2211yescharacteristic
25C47×#423C2211yescharacteristic
26C47×#333—22no—
27C22×C47⋊C8D411yescharacteristic
28C22×C47⋊C8D411yescharacteristic
29#13094024C2×C411yescharacteristic
30#13094024C2×C411yescharacteristic
31#13094016—21no—
32#13094003—21no—
33#13094003D412yes—
34C4×Dic94—21no—
35C4×Dic94D412yes—
36#13093983—21no—
37C2×C4⋊Dic47—21no—
38#13093957—24no—
39C4⋊(C47⋊C8)—21no—
40Dic47⋊C8—24no—
41#13093947—24no—
42C22⋊(C47⋊C8)—21no—
43#13093935—21no—
44#13093935—21no—
45#13093934—22no—
46C4×C47⋊C8C2×C412yes—
47C8×Dic47—22no—
48C47⋊#95—21no—
49C47⋊#95—21no—
50C22×Q8×C47—21no—
51C2×M16×C47—21no—
52C4⋊Q8×C47—21no—
53C4⋊Q8×C47—21no—
54C4⋊Q8×C47—21no—
55C4⋊Q8×C47C2×C412yes—
56C2×C4⋊C4×C47—21no—
57C2×C4⋊C4×C47—21no—
58C2×C42×C47C2311yescharacteristic
59C4⋊C8×C47—21no—
60C4⋊C8×C47—21no—
61C4⋊C8×C47D412yes—
62C22⋊C8×C47—21no—
63C4×C8×C47—21no—
64C2×Q8×C47—22no—
65C2×Q8×C47—41no—
66C2×Q8×C47—82no—
67C22×C4×C47—41no—
68C22×C4×C47C2×D411yescharacteristic
69C22×C4×C47C2×D411yescharacteristic
70C22×C4×C47#4111yescharacteristic
71C22×Dic47—41no—
72C2×Dic94—21no—
73C2×Dic94C22⋊C412yes—
74M16×C47—41no—
75M16×C47—41no—
76C2×C8×C47—22no—
77C2×C8×C47—41no—
78C4⋊C4×C47—22no—
79C4⋊C4×C47—22no—
80C4⋊C4×C47—41no—
81C4⋊C4×C47—41no—
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—41no—
91C4⋊Dic47—41no—
92C4⋊Dic47C22⋊C412yes—
93Dic47⋊C4—22no—
94Dic47⋊C4—22no—
95C4×Dic47—22no—
96C2×C47⋊C8—21no—
97C2×C47⋊C8—21no—
98C2×C47⋊C8—22no—
99C2×C47⋊C8—21no—
100C2×C47⋊C8—21no—
101C2×C47⋊C8—21no—
102C2×C47⋊C8Q1612yes—
103C23×C47#12111yescharacteristic
104Q8×C47—82no—
105Q8×C47—82no—
106C2×C4×C47—41no—
107C2×C4×C47—41no—
108C2×C4×C47—42no—
109C2×C4×C47—82no—
110C2×C4×C47C2×C22⋊C411yescharacteristic
111C2×C4×C47C4×D411yescharacteristic
112C2×C4×C47C4×D411yescharacteristic
113C2×C4×C47#12011yescharacteristic
114C2×C4×C47#12111yescharacteristic
115C2×C4×C47#12311yescharacteristic
116C2×C4×C47C2×Q1611yescharacteristic
117C2×C4×C47#13511yescharacteristic
118C2×C4×C47#13711yescharacteristic
119C2×C4×C47#13711yescharacteristic
120C2×Dic47—22no—
121C2×Dic47—22no—
122C2×Dic47—81no—
123C2×Dic47—81no—
124C2×Dic47—81no—
125Dic94—42no—
126Dic94—81no—
127C8×C47—42no—
128C8×C47—81no—
129C47⋊C8—22no—
130C47⋊C8—42no—
131C47⋊C8—41no—
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×C47—82no—
143C4×C47C4×Q1611yescharacteristic
144C4×C47#44111yescharacteristic
145C4×C47#45111yescharacteristic
146C4×C47#45211yescharacteristic
147Dic47—42no—
148Dic47—42no—
149Dic47—161no—
150#3230—471no—
151#2961—471no—
152#1993—471no—
153#1548D4711yescharacteristic
154#1518—471no—
155#1286—471no—
156#1275—471no—
157C2×C47—21no—
158C2×C47—21no—
159C2×C47#187811yescharacteristic
160C2×C47#190311yescharacteristic
161C2×C47#190911yescharacteristic
162#533—471no—
163C2×C4⋊Q8C2×D4711yescharacteristic
164#500—472no—
165#487—472no—
166#484—942no—
167#477—942no—
168#446—942no—
169#446—942no—
170#431—471no—
171#431C2×D4711yescharacteristic
172#424—941no—
173#423C2×D4711yescharacteristic
174#405—941no—
175C2×#97—471no—
176#382—471no—
177#340—941no—
178#340—941no—
179#333—22no—
180#328—942no—
181C47#928811yescharacteristic, complemented, Sylow
182C22×Q8—21no—
183C2×M16—21no—
184C22×C8—471no—
185C22×C8—471no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—471no—
190C4⋊Q8—471no—
191C4⋊Q8C4×D4712yes—
192#125—941no—
193C22⋊Q8—472no—
194C22⋊Q8—941no—
195C4×Q8—472no—
196C4×Q8—941no—
197#117—941no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—941no—
201C2×C42C22×D4711yescharacteristic
202#107—944no—
203C4⋊C8—21no—
204C4⋊C8—21no—
205C4⋊C8—944no—
206C4⋊C8—941no—
207C4⋊C8C22⋊D4712yes—
208#103—944no—
209C22⋊C8—21no—
210C22⋊C8—941no—
211#97—942no—
212#97—941no—
213#97—941no—
214C4×C8—21no—
215C4×C8—472no—
216C4×C8—942no—
217#95—941no—
218#95—941no—
219C2×Q8—22no—
220C2×Q8—41no—
221C2×Q8—82no—
222C2×Q8—472no—
223C2×Q8—941no—
224C22×C4—41no—
225C22×C4—1881no—
226C22×C4#1062535011yescharacteristic
227C22×C4D4×D4711yescharacteristic
228C22×C4D4×D4711yescharacteristic
229M16—41no—
230M16—41no—
231C2×C8—22no—
232C2×C8—41no—
233C2×C8—472no—
234C2×C8—941no—
235C2×C8—941no—
236C2×C8—942no—
237C2×C8—941no—
238C2×C8—941no—
239C2×C8—941no—
240C4⋊C4—22no—
241C4⋊C4—22no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—472no—
245C4⋊C4—941no—
246C4⋊C4—941no—
247C4⋊C4—941no—
248C4⋊C4—942no—
249C4⋊C4—942no—
250C4⋊C4—1881no—
251C4⋊C4—1881no—
252C22⋊C4—942no—
253C42—942no—
254C42C2×C4×D4711yescharacteristic
255C42D4×D4711yescharacteristic
256C42#1062535211yescharacteristic
257C42C2×C22⋊D4711yescharacteristic
258C23#1309399711yescharacteristic
259Q8—82no—
260Q8—82no—
261Q8—3761no—
262Q8—1882no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—82no—
267C2×C4—942no—
268C2×C4—942no—
269C2×C4—3761no—
270C2×C4—3761no—
271C2×C4—3761no—
272C2×C4C22⋊C4×D4711yescharacteristic
273C2×C4(C22⋊D47)⋊C411yescharacteristic
274C2×C4C4×C22⋊D4711yescharacteristic
275C2×C4#1309399211yescharacteristic
276C2×C4#1309399511yescharacteristic
277C2×C4#1309400611yescharacteristic
278C2×C4Q16×D4711yescharacteristic
279C2×C4#1309404511yescharacteristic
280C2×C4#1309405311yescharacteristic
281C2×C4#1309405611yescharacteristic
282C8—42no—
283C8—81no—
284C8—942no—
285C8—1882no—
286C8—1881no—
287C22—21no—
288C22—21no—
289C22#1568899911yescharacteristic
290C22#1568933211yescharacteristic
291C22#1568934811yescharacteristic
292C4—21no—
293C4—21no—
294C4—21no—
295C4—21no—
296C4—82no—
297C4—1882no—
298C4—1882no—
299C4—7521no—
300C4#1568921511yescharacteristic
301C4#1568922711yescharacteristic
302C4#1568927711yescharacteristic
303C4#1568928211yescharacteristic
304C2—21no—
305C2—21no—
306C2#2024372611yescharacteristic
307C2#2024387111yescharacteristic
308C2#2024390711yescharacteristic
subgroupG/H
centerC22#15688999
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1548C2
socleC22×C47#386
prime radicalC23×C47#121
cosocleC2×C42×C47C23
hypercenter#1548D47
hypocenterC47#9288
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
9. #15689526C4semidirect
10. #15689475C4semidirect
181. C47#9288semidirect

Maximal subgroups

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

subgroupindex
#202545392
#202523252
#202443622
#202419532
#202412482
#202412182
C47×#15482
#928847