Finite Group Databasebeta

← #26119458

Subgroups — #26119458

Order 12032.

360 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 77 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#20244331C211yesmaximal, characteristic
2#20241938C211yesmaximal, characteristic
3#20241899C211yesmaximal, characteristic
4#20241281C211yesmaximal, characteristic
5#20241252C211yesmaximal, characteristic
6#20241244C211yesmaximal, characteristic
7C47×#2605C211yesmaximal, characteristic
8#15689680C2211yescharacteristic
9#15689180C2211yescharacteristic
10#15689180C2211yescharacteristic
11C4×#10625329C2211yescharacteristic
12#15688883—21no—
13#15688883—21no—
14#15688859—21no—
15#15688859—21no—
16#15688859—21no—
17#15688859—21no—
18C47×#400—21no—
19C47×#395—21no—
20C47×#391—21no—
21C47×#391—21no—
22C47×#387—21no—
23C47×#387—21no—
24C47×#383C2211yescharacteristic
25C4×C4⋊C4×C47C2211yescharacteristic
26C47×#376C2211yescharacteristic
27C2×#10625329—21no—
28#13094024Q811yescharacteristic, complemented
29#13094024Q811yescharacteristic, complemented
30#13094016—21no—
31#13094003—21no—
32#13094003—21no—
33C4×Dic94—21no—
34C4×Dic94—21no—
35#13093983—21no—
36C4⋊(C47⋊C8)—21no—
37C4⋊(C47⋊C8)—21no—
38C22⋊(C47⋊C8)—21no—
39C22⋊(C47⋊C8)—21no—
40#13093935—21no—
41#13093935—21no—
42#13093935—21no—
43C4×C47⋊C8—21no—
44C22×Q8×C47—21no—
45C2×C4⋊C4×C47—21no—
46C2×C4⋊C4×C47—21no—
47C2×C4⋊C4×C47—21no—
48C2×C4⋊C4×C47—21no—
49C2×C42×C47—21no—
50C2×C42×C47D411yescharacteristic
51C2×C42×C47D411yescharacteristic
52C2×C42×C47C2311yescharacteristic
53#95×C47—21no—
54#95×C47—21no—
55#95×C47—21no—
56#95×C47—21no—
57#95×C47—21no—
58#95×C47—21no—
59#95×C47—21no—
60#95×C47—21no—
61#95×C47D411yescharacteristic
62#95×C47D411yescharacteristic
63#95×C47D411yescharacteristic
64#95×C47D411yescharacteristic
65C2×Q8×C47—42no—
66C2×Q8×C47—42no—
67C2×Q8×C47—81no—
68C22×C4×C47—21no—
69C22×C4×C47—21no—
70C22×C4×C47—21no—
71C22×C4×C47—21no—
72C22×C4×C47—21no—
73C22×C4×C47—21no—
74C22×C4×C47C2×D411yescharacteristic
75C22×C4×C47C2×D411yescharacteristic
76C22×C4×C47C2×D411yescharacteristic
77C2×Dic94—21no—
78C2×Dic94—21no—
79C4⋊C4×C47—42no—
80C4⋊C4×C47—42no—
81C4⋊C4×C47—42no—
82C4⋊C4×C47—81no—
83C42×C47—42no—
84C42×C47—41no—
85C42×C47—41no—
86C42×C47C2×Q811yescharacteristic
87C42×C47#4111yescharacteristic
88C42×C47#4111yescharacteristic
89C42×C47#4111yescharacteristic
90C22⋊Dic47—41no—
91C4⋊Dic47—21no—
92C4⋊Dic47—21no—
93C4⋊Dic47—21no—
94C4⋊Dic47—21no—
95Dic47⋊C4—41no—
96Dic47⋊C4—41no—
97C4×Dic47—41no—
98#10625329—22no—
99#10625329—41no—
100C2×C47⋊C8—41no—
101C2×C47⋊C8—41no—
102C2×C47⋊C8—41no—
103C23×C47#12011yescharacteristic
104Q8×C47—84no—
105C2×C4×C47—21no—
106C2×C4×C47—21no—
107C2×C4×C47—22no—
108C2×C4×C47—22no—
109C2×C4×C47—41no—
110C2×C4×C47—41no—
111C2×C4×C47—41no—
112C2×C4×C47—41no—
113C2×C4×C47—41no—
114C2×C4×C47—41no—
115C2×C4×C47—41no—
116C2×C4×C47—41no—
117C2×C4×C47—41no—
118C2×C4×C47—41no—
119C2×C4×C47—41no—
120C2×C4×C47—41no—
121C2×C4×C47—41no—
122C2×C4×C47—41no—
123C2×C4×C47—41no—
124C2×C4×C47—41no—
125C2×C4×C47—41no—
126C2×C4×C47—41no—
127C2×C4×C47C22⋊Q811yescharacteristic
128C2×C4×C47C22⋊Q811yescharacteristic
129C2×C4×C47C22⋊Q811yescharacteristic
130C2×C4×C47#12411yescharacteristic
131C2×C4×C47#12411yescharacteristic
132C2×C4×C47#12411yescharacteristic
133C2×Dic47—41no—
134C2×Dic47—41no—
135Dic94—81no—
136Dic94—81no—
137C47⋊C8—42no—
138C47⋊C8—81no—
139C47⋊C8—81no—
140#10379—471nomaximal, Sylow
141C22×C47—21no—
142C22×C47—21no—
143C22×C47#39311yescharacteristic
144C22×C47#45511yescharacteristic
145C22×C47#45611yescharacteristic
146C4×C47—21no—
147C4×C47—21no—
148C4×C47—21no—
149C4×C47—21no—
150C4×C47—21no—
151C4×C47—21no—
152C4×C47—42no—
153C4×C47—42no—
154C4×C47—81no—
155C4×C47—81no—
156C4×C47—81no—
157C4×C47—81no—
158Dic47—81no—
159Dic47—81no—
160#2605D4711yescharacteristic
161#1986—471no—
162#1515—471no—
163#1506—471no—
164#1300—471no—
165#1288—471no—
166#1285—471no—
167C2×C47—21no—
168C2×C47—21no—
169C2×C47#202811yescharacteristic
170C2×C47#203111yescharacteristic
171C2×C47#203111yescharacteristic
172#533—471no—
173#432—471no—
174#432—471no—
175C4×M16—471no—
176#400—21no—
177#395—21no—
178#391—21no—
179#391—21no—
180#387—21no—
181#387—21no—
182#383C2×D4711yescharacteristic
183C4×C4⋊C4C2×D4711yescharacteristic
184#376C2×D4711yescharacteristic
185#339—941no—
186#339—941no—
187#328—941no—
188#328—941no—
189#328—941no—
190#328—941no—
191C47#1037911yescharacteristic, complemented, Sylow
192C22×Q8—21no—
193C2×M16—941no—
194C4⋊Q8—471no—
195C4⋊Q8—471no—
196#125—941no—
197C22⋊Q8—941no—
198C22⋊Q8—941no—
199C4×Q8—941no—
200C4×Q8—941no—
201#117—941no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C42—21no—
207C2×C42C22⋊D4711yescharacteristic
208C2×C42C22⋊D4711yescharacteristic
209C2×C42C22×D4711yescharacteristic
210C4⋊C8—941no—
211C4⋊C8—941no—
212C22⋊C8—941no—
213C22⋊C8—941no—
214#97—941no—
215#97—941no—
216#97—941no—
217C4×C8—941no—
218#95—21no—
219#95—21no—
220#95—21no—
221#95—21no—
222#95—21no—
223#95—21no—
224#95—21no—
225#95—21no—
226#95C22⋊D4711yescharacteristic
227#95C22⋊D4711yescharacteristic
228#95C22⋊D4711yescharacteristic
229#95C22⋊D4711yescharacteristic
230C2×Q8—42no—
231C2×Q8—42no—
232C2×Q8—81no—
233C2×Q8—941no—
234C2×Q8—941no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4C2×C22⋊D4711yescharacteristic
242C22×C4C2×C22⋊D4711yescharacteristic
243C22×C4C2×C22⋊D4711yescharacteristic
244M16—942no—
245M16—1881no—
246C2×C8—1881no—
247C2×C8—1881no—
248C2×C8—1881no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—81no—
253C4⋊C4—941no—
254C4⋊C4—941no—
255C4⋊C4—941no—
256C4⋊C4—941no—
257C4⋊C4—1881no—
258C4⋊C4—1881no—
259C22⋊C4—1881no—
260C42—41no—
261C42—41no—
262C42—42no—
263C42—1881no—
264C42Q8×D4711yescharacteristic
265C42#1062535411yescharacteristic
266C42#1062535411yescharacteristic
267C42#1062535411yescharacteristic
268C23#1309399311yescharacteristic
269Q8—84no—
270Q8—3761no—
271Q8—3761no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—1881no—
295C2×C4—1881no—
296C2×C4#1309400011yescharacteristic
297C2×C4#1309400011yescharacteristic
298C2×C4#1309400011yescharacteristic
299C2×C4#1309401311yescharacteristic
300C2×C4#1309401311yescharacteristic
301C2×C4#1309401311yescharacteristic
302C8—3761no—
303C8—3761no—
304C8—1882no—
305C22—21no—
306C22—21no—
307C22#1568903411yescharacteristic
308C22#1568930111yescharacteristic
309C22#1568930611yescharacteristic
310C4—21no—
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—42no—
317C4—42no—
318C4—81no—
319C4—81no—
320C4—81no—
321C4—81no—
322C4—3761no—
323C4—3761no—
324C2—21no—
325C2—21no—
326C2#2024454611yescharacteristic
327C2#2024455711yescharacteristic
328C2#2024455711yescharacteristic
subgroupG/H
centerC22#15689034
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#2605C2
socleC22×C47#393
prime radicalC23×C47#120
hypercenter#2605D47
hypocenterC47#10379
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2605

Complements

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

NH ≅ G/Ntype
28. #13094024Q8semidirect
29. #13094024Q8semidirect
191. C47#10379semidirect

Maximal subgroups

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

subgroupindex
#202443312
#202419382
#202418992
#202412812
#202412522
#202412442
C47×#26052
#1037947