Finite Group Databasebeta

← #26114153

Subgroups — #26114153

Order 12032.

522 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 119 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#20252473C211yesmaximal, characteristic, complemented
2#20243384C211yesmaximal, characteristic, complemented
3#20241813C212yesmaximal
4#20241246C212yesmaximal, complemented
5C47×#2536C211yesmaximal, characteristic
6#15689254—24no—
7#15689180—22no—
8#15689176C2212yes—
9#15689147C2212yes—
10C2×#13093943—21no—
11#15688892—21no—
12D4⋊(C47⋊C8)—24no—
13C2×C22⋊Q8×C47—21no—
14C2×C4×Q8×C47C2211yescharacteristic
15C2×C4×D4×C47C2211yescharacteristic
16C47×#398C412yes—
17C47×#394C412yes—
18C47×#388—22no—
19C47×#382C2211yescharacteristic
20C47×#380—21no—
21C2×#10625329—22no—
22C4⋊(C47⋊C8)—24no—
23C4⋊(C47⋊C8)—22no—
24C4⋊(C47⋊C8)—22no—
25C4⋊(C47⋊C8)—22no—
26#13093943—42no—
27C22⋊(C47⋊C8)—22no—
28#13093935—24no—
29C4×C47⋊C8D414yes—
30C22×Q8×C47D411yescharacteristic
31C22×D4×C47C2×C411yescharacteristic
32C23×C4×C47—21no—
33C22⋊Q8×C47—44no—
34C4×Q8×C47—22no—
35C4×Q8×C47—22no—
36C4×D4×C47—22no—
37C4×D4×C47—22no—
38C2×C4⋊C4×C47—22no—
39C2×C4⋊C4×C47—21no—
40C2×C4⋊C4×C47C2×C411yescharacteristic
41C2×C4⋊C4×C47C2×C412yes—
42C2×C4⋊C4×C47D411yescharacteristic
43C2×C22⋊C4×C47—22no—
44C2×C22⋊C4×C47—22no—
45C2×C22⋊C4×C47—21no—
46C2×C42×C47D412yes—
47C2×C42×C47C2311yescharacteristic
48#95×C47—22no—
49#95×C47—21no—
50#95×C47C2×C412yes—
51C24×C47—21no—
52C2×Q8×C47—22no—
53C2×Q8×C47—22no—
54C2×Q8×C47—21no—
55C2×Q8×C47—21no—
56C2×D4×C47—22no—
57C2×D4×C47—42no—
58C22×C4×C47—22no—
59C22×C4×C47—21no—
60C22×C4×C47—42no—
61C22×C4×C47—42no—
62C22×C4×C47—41no—
63C22×C4×C47—41no—
64C22×C4×C47C22⋊C412yes—
65C22×C4×C47C22⋊C412yes—
66C22×C4×C47C22×C411yescharacteristic
67C22×C4×C47C2×D411yescharacteristic
68C22×C4×C47C2×D411yescharacteristic
69C4⋊C4×C47—24no—
70C4⋊C4×C47—22no—
71C4⋊C4×C47—22no—
72C4⋊C4×C47—44no—
73C4⋊C4×C47—42no—
74C22⋊C4×C47—44no—
75C22⋊C4×C47—44no—
76C22⋊C4×C47—41no—
77C22⋊C4×C47—41no—
78C42×C47—22no—
79C42×C47—22no—
80C42×C47C2×D412yes—
81C42×C47#4112yes—
82#10625329—42no—
83#10625329—42no—
84C2×C47⋊C8—24no—
85C2×C47⋊C8—42no—
86C23×C47—22no—
87C23×C47—41no—
88C23×C47—41no—
89C23×C47—41no—
90C23×C47—41no—
91C23×C47—41no—
92C23×C47—41no—
93C23×C47C2×C22⋊C411yescharacteristic
94Q8×C47—42no—
95Q8×C47—42no—
96D4×C47—24no—
97D4×C47—42no—
98C2×C4×C47—22no—
99C2×C4×C47—22no—
100C2×C4×C47—22no—
101C2×C4×C47—22no—
102C2×C4×C47—41no—
103C2×C4×C47—41no—
104C2×C4×C47—42no—
105C2×C4×C47—42no—
106C2×C4×C47—42no—
107C2×C4×C47—41no—
108C2×C4×C47—41no—
109C2×C4×C47—42no—
110C2×C4×C47—41no—
111C2×C4×C47—82no—
112C2×C4×C47—82no—
113C2×C4×C47C4×D412yes—
114C2×C4×C47#12011yescharacteristic
115C2×C4×C47#12112yes—
116C2×C4×C47#12311yescharacteristic
117C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
118C2×C4×C47AGL(1,ℤ/8)12yes—
119C2×C4×C47#13711yescharacteristic
120C47⋊C8—44no—
121C47⋊C8—82no—
122#9490—471nomaximal, Sylow
123C22×C47—21no—
124C22×C47—21no—
125C22×C47—42no—
126C22×C47—42no—
127C22×C47—42no—
128C22×C47—81no—
129C22×C47—81no—
130C22×C47—81no—
131C22×C47—81no—
132C22×C47#38611yescharacteristic
133C22×C47#41711yescharacteristic
134C22×C47#41811yescharacteristic
135C4×C47—22no—
136C4×C47—21no—
137C4×C47—21no—
138C4×C47—42no—
139C4×C47—42no—
140C4×C47—82no—
141C4×C47—81no—
142C4×C47#46812yes—
143C4×C47#46912yes—
144#2983—471no—
145#2536D4711yescharacteristic
146#1801—471no—
147#1486—472no—
148#1286—472no—
149C2×C47—21no—
150C2×C47—21no—
151C2×C47—82no—
152C2×C47#189111yescharacteristic
153C2×C47#189511yescharacteristic
154C2×C47#191211yescharacteristic
155C2×C22⋊Q8—21no—
156C2×C4×Q8C2×D4711yescharacteristic
157C2×C4×D4C2×D4711yescharacteristic
158#446—944no—
159#432—942no—
160#431—472no—
161#423—472no—
162C2×#101—941no—
163#398Dic4712yes—
164#394Dic4712yes—
165#388—22no—
166#382C2×D4711yescharacteristic
167#380—21no—
168#343—941no—
169D4⋊C8—944no—
170C47#949011yescharacteristic, complemented, Sylow
171C22×Q8C22⋊D4711yescharacteristic
172C22×D4C2×Dic4711yescharacteristic
173C23×C4—21no—
174C2×M16—942no—
175C22⋊Q8—44no—
176C4×Q8—22no—
177C4×Q8—22no—
178C4×D4—22no—
179C4×D4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4C2×Dic4711yescharacteristic
183C2×C4⋊C4C2×Dic4712yes—
184C2×C4⋊C4C22⋊D4711yescharacteristic
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—21no—
188C2×C42C22⋊D4712yes—
189C2×C42C22×D4711yescharacteristic
190C4⋊C8—944no—
191C4⋊C8—942no—
192C4⋊C8—942no—
193C4⋊C8—942no—
194#101—1882no—
195C22⋊C8—942no—
196#97—944no—
197C4×C8—474no—
198#95—22no—
199#95—21no—
200#95C2×Dic4712yes—
201C24—21no—
202C2×Q8—22no—
203C2×Q8—21no—
204C2×Q8—22no—
205C2×Q8—21no—
206C2×D4—22no—
207C2×D4—42no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—42no—
211C22×C4—41no—
212C22×C4—42no—
213C22×C4—41no—
214C22×C4C22⋊Dic4712yes—
215C22×C4C22⋊Dic4712yes—
216C22×C4C22×Dic4711yescharacteristic
217C22×C4C2×C22⋊D4711yescharacteristic
218C22×C4C2×C22⋊D4711yescharacteristic
219M16—1882no—
220M16—1882no—
221C2×C8—944no—
222C2×C8—1882no—
223C4⋊C4—24no—
224C4⋊C4—22no—
225C4⋊C4—22no—
226C4⋊C4—44no—
227C4⋊C4—42no—
228C22⋊C4—44no—
229C22⋊C4—44no—
230C22⋊C4—41no—
231C22⋊C4—41no—
232C42—22no—
233C42—22no—
234C42D4×D4712yes—
235C42#1062535212yes—
236C23—22no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23C2×C22⋊Dic4711yescharacteristic
244Q8—42no—
245Q8—42no—
246D4—24no—
247D4—42no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4D4×Dic4712yes—
264C2×C4#1309399211yescharacteristic
265C2×C4#1309399612yes—
266C2×C4#1309400511yescharacteristic
267C2×C4#1309404911yescharacteristic
268C2×C4#1309405012yes—
269C2×C4#1309405411yescharacteristic
270C8—3762no—
271C8—1884no—
272C22—21no—
273C22—21no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—81no—
278C22—81no—
279C22—81no—
280C22—81no—
281C22#1568900311yescharacteristic
282C22#1568912411yescharacteristic
283C22#1568912811yescharacteristic
284C4—22no—
285C4—21no—
286C4—21no—
287C4—42no—
288C4—42no—
289C4—82no—
290C4—81no—
291C4#1568936712yes—
292C4#1568937412yes—
293C2—21no—
294C2—21no—
295C2—82no—
296C2#2024380711yescharacteristic
297C2#2024382711yescharacteristic
298C2#2024392611yescharacteristic
subgroupG/H
centerC22#15689003
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2536C2
socleC22×C47#386
prime radicalC22×D4×C47C2×C4
cosocleC2×C42×C47C23
hypercenter#2536D47
hypocenterC47#9490
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2536

Complements

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

NH ≅ G/Ntype
1. #20252473C2semidirect
2. #20243384C2semidirect
4. #20241246C2semidirect
170. C47#9490semidirect

Maximal subgroups

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

subgroupindex
#202524732
#202433842
#202418132
#202412462
C47×#25362
#949047