Finite Group Databasebeta

← #26108512

Subgroups — #26108512

Order 12032.

834 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 834 conjugacy classes of subgroups — too many to draw. Only its 139 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#20244229C212yesmaximal, complemented
2#20242126C211yesmaximal, characteristic, complemented
3#20241983C211yesmaximal, characteristic, complemented
4#20241827C212yesmaximal, complemented
5C47×#3508C211yesmaximal, characteristic
6#15689147C2212yes—
7C2×#13093941—24no—
8C4×#10625329C2212yes—
9#15688869—22no—
10#15688865—22no—
11Q8⋊(C47⋊C8)—22no—
12D4⋊(C47⋊C8)—22no—
13C47×#559—22no—
14C47×#550—22no—
15C47×#546—24no—
16C47×#540—22no—
17C47×#535C412yes—
18C47×#533C2211yescharacteristic
19C47×#532—22no—
20C2×#127×C47C2211yescharacteristic
21C47×#518C412yes—
22C4×#41×C47C2211yescharacteristic
23C2×#10625329—22no—
24C2×#10625329—22no—
25C4⋊(C47⋊C8)—22no—
26C4⋊(C47⋊C8)—22no—
27#13093941—48no—
28#13093935—22no—
29C4×C47⋊C8—22no—
30C2×#41×C47—24no—
31C2×#41×C47C2×C411yescharacteristic
32C22×D4×C47C2×C412yes—
33C22×D4×C47D414yes—
34C4⋊Q8×C47D412yes—
35#127×C47—22no—
36#127×C47—21no—
37#127×C47—21no—
38#127×C47—22no—
39#125×C47D412yes—
40#124×C47—24no—
41#124×C47—22no—
42#123×C47—28no—
43C22⋊Q8×C47—22no—
44#121×C47—24no—
45#121×C47—28no—
46#121×C47—24no—
47#120×C47—24no—
48C4×Q8×C47—22no—
49C4×Q8×C47D412yes—
50C4×D4×C47—24no—
51C4×D4×C47—22no—
52C4×D4×C47—24no—
53C4×D4×C47—24no—
54C4×D4×C47D412yes—
55#117×C47—21no—
56#117×C47C2×C411yescharacteristic
57#117×C47C2×C412yes—
58C2×C22⋊C4×C47—24no—
59C2×C42×C47—21no—
60C2×C42×C47C2311yescharacteristic
61C24×C47C22⋊C414yes—
62#41×C47—42no—
63#41×C47—84no—
64#41×C47—84no—
65C2×Q8×C47—22no—
66C2×Q8×C47C22⋊C411yescharacteristic
67C2×D4×C47—21no—
68C2×D4×C47—28no—
69C2×D4×C47—22no—
70C2×D4×C47—22no—
71C2×D4×C47—24no—
72C2×D4×C47—24no—
73C2×D4×C47—44no—
74C2×D4×C47—44no—
75C2×D4×C47—44no—
76C2×D4×C47C22⋊C411yescharacteristic
77C22×C4×C47—28no—
78C22×C4×C47—21no—
79C22×C4×C47—22no—
80C22×C4×C47C22×C411yescharacteristic
81C22×C4×C47C2×D412yes—
82C4⋊C4×C47—22no—
83C4⋊C4×C47—22no—
84C4⋊C4×C47—24no—
85C4⋊C4×C47C22⋊C412yes—
86C4⋊C4×C47C22⋊C414yes—
87C22⋊C4×C47—28no—
88C22⋊C4×C47—28no—
89C22⋊C4×C47—21no—
90C22⋊C4×C47—22no—
91C22⋊C4×C47—22no—
92C42×C47—21no—
93C42×C47—22no—
94C42×C47—22no—
95C42×C47C2×D411yescharacteristic
96C42×C47C2×D411yescharacteristic
97C42×C47C2×D412yes—
98#10625329—24no—
99#10625329—42no—
100#10625329—44no—
101C2×C47⋊C8—42no—
102C2×C47⋊C8—42no—
103C23×C47—24no—
104C23×C47—21no—
105C23×C47—48no—
106C23×C47—44no—
107C23×C47C2×C22⋊C411yescharacteristic
108Q8×C47—22no—
109Q8×C47—42no—
110Q8×C47—42no—
111D4×C47—22no—
112D4×C47—44no—
113D4×C47—42no—
114D4×C47—44no—
115D4×C47—44no—
116D4×C47—44no—
117D4×C47—84no—
118D4×C47—88no—
119C2×C4×C47—22no—
120C2×C4×C47—22no—
121C2×C4×C47—22no—
122C2×C4×C47—21no—
123C2×C4×C47—22no—
124C2×C4×C47—24no—
125C2×C4×C47—42no—
126C2×C4×C47—42no—
127C2×C4×C47—42no—
128C2×C4×C47—88no—
129C2×C4×C47C2×C22⋊C411yescharacteristic
130C2×C4×C47C2×C22⋊C411yescharacteristic
131C2×C4×C47#12012yes—
132C2×C4×C47#12012yes—
133C47⋊C8—44no—
134C47⋊C8—82no—
135#8472—471nomaximal, Sylow
136C22×C47—22no—
137C22×C47—21no—
138C22×C47—42no—
139C22×C47—48no—
140C22×C47—41no—
141C22×C47—44no—
142C22×C47—164no—
143C22×C47#37911yescharacteristic
144C4×C47—22no—
145C4×C47—21no—
146C4×C47—42no—
147C4×C47—44no—
148C4×C47—41no—
149C4×C47#41312yes—
150C4×C47#45312yes—
151C4×C47#45512yes—
152#3508D4711yescharacteristic
153#1965—472no—
154#1561—471no—
155#1526—471no—
156#1489—472no—
157C2×C47—41no—
158C2×C47—81no—
159C2×C47—84no—
160C2×C47#178511yescharacteristic
161C2×C47#179212yes—
162#559—22no—
163#550—22no—
164#546—24no—
165#540—22no—
166#535Dic4712yes—
167#533C2×D4711yescharacteristic
168#532—22no—
169C2×#127C2×D4711yescharacteristic
170#518Dic4712yes—
171C4×#41C2×D4711yescharacteristic
172#423—472no—
173C2×#100—944no—
174C4×M16—472no—
175#332—942no—
176#330—942no—
177Q8⋊C8—942no—
178D4⋊C8—942no—
179C47#847211yescharacteristic, complemented, Sylow
180C2×#41—24no—
181C2×#41C2×Dic4711yescharacteristic
182C22×D4C2×Dic4712yes—
183C22×D4C22⋊D4714yes—
184C2×M16—942no—
185C2×M16—942no—
186C4⋊Q8C22⋊D4712yes—
187#127—22no—
188#127—21no—
189#127—21no—
190#127—22no—
191#125C22⋊D4712yes—
192#124—24no—
193#124—22no—
194#123—28no—
195C22⋊Q8—22no—
196#121—24no—
197#121—28no—
198#121—24no—
199#120—24no—
200C4×Q8—22no—
201C4×Q8C22⋊D4712yes—
202C4×D4—24no—
203C4×D4—22no—
204C4×D4—24no—
205C4×D4—24no—
206C4×D4C22⋊D4712yes—
207#117—21no—
208#117C2×Dic4711yescharacteristic
209#117C2×Dic4712yes—
210C2×C22⋊C4—24no—
211C2×C42—21no—
212C2×C42C22×D4711yescharacteristic
213C4⋊C8—942no—
214C4⋊C8—942no—
215#100—1888no—
216#97—942no—
217C4×C8—942no—
218C24C22⋊Dic4714yes—
219#41—42no—
220#41—84no—
221#41—84no—
222C2×Q8—22no—
223C2×Q8C22⋊Dic4711yescharacteristic
224C2×D4—21no—
225C2×D4—28no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—24no—
229C2×D4—24no—
230C2×D4—44no—
231C2×D4—44no—
232C2×D4—44no—
233C2×D4C22⋊Dic4711yescharacteristic
234C22×C4—28no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4C22×Dic4711yescharacteristic
238C22×C4C2×C22⋊D4712yes—
239M16—944no—
240M16—1884no—
241M16—1882no—
242C2×C8—1882no—
243C2×C8—1882no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—24no—
247C4⋊C4C22⋊Dic4712yes—
248C4⋊C4C22⋊Dic4714yes—
249C22⋊C4—28no—
250C22⋊C4—28no—
251C22⋊C4—21no—
252C22⋊C4—22no—
253C22⋊C4—22no—
254C42—21no—
255C42—22no—
256C42—22no—
257C42C2×C22⋊D4712yes—
258C42C2×C22⋊D4711yescharacteristic
259C42C2×C22⋊D4711yescharacteristic
260C23—24no—
261C23—21no—
262C23—44no—
263C23—48no—
264C23C2×C22⋊Dic4711yescharacteristic
265Q8—22no—
266Q8—42no—
267Q8—42no—
268D4—22no—
269D4—44no—
270D4—42no—
271D4—44no—
272D4—44no—
273D4—44no—
274D4—84no—
275D4—88no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—24no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—88no—
286C2×C4C2×C22⋊Dic4711yescharacteristic
287C2×C4C2×C22⋊Dic4711yescharacteristic
288C2×C4#1309399312yes—
289C2×C4#1309399312yes—
290C8—3762no—
291C8—1884no—
292C22—21no—
293C22—22no—
294C22—41no—
295C22—44no—
296C22—42no—
297C22—48no—
298C22—164no—
299C22#1568897911yescharacteristic
300C4—21no—
301C4—22no—
302C4—41no—
303C4—42no—
304C4—44no—
305C4#1568910712yes—
306C4#1568929112yes—
307C4#1568930112yes—
308C2—41no—
309C2—84no—
310C2—81no—
311C2#2024331611yescharacteristic
312C2#2024334212yes—
subgroupG/H
centerC22#15688979
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
Fitting, prime radicalC47×#3508C2
socleC22×C47#379
cosocleC2×C42×C47C23
hypercenter#3508D47
hypocenterC47#8472
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3508

Complements

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

NH ≅ G/Ntype
1. #20244229C2semidirect
2. #20242126C2semidirect
3. #20241983C2semidirect
4. #20241827C2semidirect
179. C47#8472semidirect

Maximal subgroups

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

subgroupindex
#202442292
#202421262
#202419832
#202418272
C47×#35082
#847247