Finite Group Databasebeta

← #26120212

Subgroups — #26120212

Order 12032.

372 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 85 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#20244233C211yesmaximal, characteristic
2#20242176C211yesmaximal, characteristic
3#20242126C211yesmaximal, characteristic
4#20242069C211yesmaximal, characteristic
5#20241809C211yesmaximal, characteristic
6#20241279C211yesmaximal, characteristic
7C47×#3209C211yesmaximal, characteristic
8#15689176C2211yescharacteristic
9#15689147C2211yescharacteristic
10#15689147C2211yescharacteristic
11C2×#13093943—22no—
12C4×#10625329C2211yescharacteristic
13#15688875—21no—
14#15688873—21no—
15#15688872—22no—
16#15688869—21no—
17#15688869—21no—
18Q8⋊(C47⋊C8)—22no—
19C47×#533C2211yescharacteristic
20C2×C4⋊Q8×C47C2211yescharacteristic
21C47×#494—21no—
22C47×#487—21no—
23C47×#462—21no—
24C47×#462—21no—
25C47×#451—22no—
26C47×#431C2211yescharacteristic
27C2×#10625329—21no—
28C2×#10625329—21no—
29C2×#10625329—21no—
30C4⋊(C47⋊C8)—21no—
31C4⋊(C47⋊C8)—21no—
32C4⋊(C47⋊C8)—21no—
33C4⋊(C47⋊C8)—21no—
34C4⋊(C47⋊C8)D412yes—
35#13093943—44no—
36C22⋊(C47⋊C8)—21no—
37#13093935—21no—
38C4×C47⋊C8—21no—
39C4×C47⋊C8—21no—
40C22×Q8×C47D412yes—
41C2×Q16×C47—22no—
42C2×Q16×C47—22no—
43C4⋊Q8×C47—21no—
44C4⋊Q8×C47—21no—
45C4⋊Q8×C47—21no—
46C4⋊Q8×C47—21no—
47C4⋊Q8×C47D411yescharacteristic
48C4⋊Q8×C47D411yescharacteristic
49#125×C47—21no—
50C22⋊Q8×C47—21no—
51C22⋊Q8×C47—21no—
52C4×Q8×C47—21no—
53C4×Q8×C47—21no—
54#117×C47—21no—
55C2×C4⋊C4×C47—21no—
56C2×C4⋊C4×C47—21no—
57C2×C42×C47C2311yescharacteristic
58C4⋊C8×C47D411yescharacteristic
59C4⋊C8×C47D411yescharacteristic
60#103×C47—22no—
61#103×C47—22no—
62C22⋊C8×C47—21no—
63C4×C8×C47—21no—
64C2×Q8×C47—22no—
65C2×Q8×C47—21no—
66C2×Q8×C47—21no—
67C2×Q8×C47—22no—
68C2×Q8×C47—42no—
69C2×Q8×C47—42no—
70C22×C4×C47—22no—
71C22×C4×C47C2×D411yescharacteristic
72C22×C4×C47#4111yescharacteristic
73C22×C4×C47#4111yescharacteristic
74Q16×C47—82no—
75Q16×C47—82no—
76C2×C8×C47—21no—
77C2×C8×C47—21no—
78C4⋊C4×C47—21no—
79C4⋊C4×C47—21no—
80C4⋊C4×C47—21no—
81C4⋊C4×C47—21no—
82C4⋊C4×C47—41no—
83C4⋊C4×C47—41no—
84C4⋊C4×C47—41no—
85C4⋊C4×C47—41no—
86C4⋊C4×C47—41no—
87C4⋊C4×C47—41no—
88C22⋊C4×C47—41no—
89C42×C47—41no—
90C42×C47C2×D411yescharacteristic
91C42×C47C2×D411yescharacteristic
92C42×C47C2×D411yescharacteristic
93C42×C47#4111yescharacteristic
94#10625329—22no—
95#10625329—41no—
96#10625329—42no—
97#10625329—81no—
98C2×C47⋊C8—22no—
99C2×C47⋊C8—41no—
100C2×C47⋊C8—41no—
101C2×C47⋊C8—41no—
102C23×C47#12311yescharacteristic
103Q8×C47—41no—
104Q8×C47—42no—
105Q8×C47—41no—
106Q8×C47—42no—
107Q8×C47—41no—
108Q8×C47—41no—
109Q8×C47—82no—
110C2×C4×C47—21no—
111C2×C4×C47—21no—
112C2×C4×C47—21no—
113C2×C4×C47—22no—
114C2×C4×C47—21no—
115C2×C4×C47—44no—
116C2×C4×C47—41no—
117C2×C4×C47—41no—
118C2×C4×C47—42no—
119C2×C4×C47#12011yescharacteristic
120C2×C4×C47#12111yescharacteristic
121C2×C4×C47#12111yescharacteristic
122C2×C4×C47#12111yescharacteristic
123C2×C4×C47#12311yescharacteristic
124C2×C4×C47#12411yescharacteristic
125C8×C47—41no—
126C8×C47—41no—
127C47⋊C8—42no—
128C47⋊C8—42no—
129C47⋊C8—81no—
130C47⋊C8—81no—
131#10493—471nomaximal, Sylow
132C22×C47—21no—
133C22×C47—21no—
134C22×C47—21no—
135C22×C47#39411yescharacteristic
136C4×C47—21no—
137C4×C47—21no—
138C4×C47—21no—
139C4×C47—81no—
140C4×C47—82no—
141C4×C47—81no—
142C4×C47#45311yescharacteristic
143C4×C47#45511yescharacteristic
144C4×C47#47111yescharacteristic
145C4×C47#47312yes—
146C4×C47#47311yescharacteristic
147#3209D4711yescharacteristic
148#1966—471no—
149#1574—471no—
150#1561—471no—
151#1548—471no—
152#1485—471no—
153#1299—471no—
154C2×C47—41no—
155C2×C47#203711yescharacteristic
156C2×C47#204711yescharacteristic
157C2×C47#205211yescharacteristic
158#533C2×D4711yescharacteristic
159C2×C4⋊Q8C2×D4711yescharacteristic
160#494—21no—
161#487—21no—
162#462—21no—
163#462—21no—
164#451—22no—
165#431—471no—
166#431C2×D4711yescharacteristic
167#423—471no—
168#423—471no—
169C2×#101—942no—
170C4×M16—471no—
171#335—941no—
172#334—941no—
173#333—942no—
174#332—941no—
175#332—941no—
176Q8⋊C8—942no—
177C47#1049311yescharacteristic, complemented, Sylow
178C22×Q8C22⋊D4712yes—
179C2×Q16—22no—
180C2×Q16—22no—
181C2×M16—941no—
182C2×M16—941no—
183C2×M16—941no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8C22⋊D4711yescharacteristic
189C4⋊Q8C22⋊D4711yescharacteristic
190#125—21no—
191C22⋊Q8—21no—
192C22⋊Q8—21no—
193C4×Q8—21no—
194C4×Q8—21no—
195#117—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C42C22×D4711yescharacteristic
199C4⋊C8—472no—
200C4⋊C8—941no—
201C4⋊C8—941no—
202C4⋊C8—941no—
203C4⋊C8—941no—
204C4⋊C8C22⋊D4711yescharacteristic
205C4⋊C8C22⋊D4711yescharacteristic
206#103—22no—
207#103—22no—
208#101—1884no—
209C22⋊C8—21no—
210C22⋊C8—941no—
211#97—941no—
212C4×C8—21no—
213C4×C8—941no—
214C4×C8—941no—
215C2×Q8—22no—
216C2×Q8—21no—
217C2×Q8—21no—
218C2×Q8—22no—
219C2×Q8—42no—
220C2×Q8—42no—
221C22×C4—22no—
222C22×C4#1062535211yescharacteristic
223C22×C4#1062535211yescharacteristic
224C22×C4C2×C22⋊D4711yescharacteristic
225Q16—82no—
226Q16—82no—
227M16—942no—
228M16—3761no—
229M16—1882no—
230M16—1881no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—942no—
234C2×C8—1881no—
235C2×C8—1881no—
236C2×C8—1881no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C22⋊C4—41no—
248C42—41no—
249C42D4×D4711yescharacteristic
250C42#1062535211yescharacteristic
251C42C2×C22⋊D4711yescharacteristic
252C42C2×C22⋊D4711yescharacteristic
253C23#1309400511yescharacteristic
254Q8—41no—
255Q8—42no—
256Q8—41no—
257Q8—42no—
258Q8—41no—
259Q8—41no—
260Q8—82no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—44no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4#1309399311yescharacteristic
271C2×C4#1309399511yescharacteristic
272C2×C4#1309399611yescharacteristic
273C2×C4#1309399611yescharacteristic
274C2×C4#1309400511yescharacteristic
275C2×C4#1309401111yescharacteristic
276C8—41no—
277C8—41no—
278C8—3761no—
279C8—3761no—
280C8—1882no—
281C8—1882no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22#1568903711yescharacteristic
286C4—21no—
287C4—21no—
288C4—21no—
289C4—82no—
290C4—81no—
291C4—81no—
292C4#1568929111yescharacteristic
293C4#1568930111yescharacteristic
294C4#1568938811yescharacteristic
295C4#1568939512yes—
296C4#1568939611yescharacteristic
297C2—41no—
298C2#2024459511yescharacteristic
299C2#2024465311yescharacteristic
300C2#2024468411yescharacteristic
subgroupG/H
centerC22#15689037
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#3209C2
socleC22×C47#394
prime radicalC23×C47#123
hypercenter#3209D47
hypocenterC47#10493
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3209

Complements

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

NH ≅ G/Ntype
177. C47#10493semidirect

Maximal subgroups

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

subgroupindex
#202442332
#202421762
#202421262
#202420692
#202418092
#202412792
C47×#32092
#1049347