Finite Group Databasebeta

← #26119165

Subgroups — #26119165

Order 12032.

602 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 103 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#20254163C211yesmaximal, characteristic
2#20251727C212yesmaximal
3#20241961C212yesmaximal
4#20241256C211yesmaximal, characteristic
5C47×#1817C211yesmaximal, characteristic
6C2×#13094003C2212yes—
7#15689459—22no—
8#15689364—22no—
9#15689281—22no—
10#15689029—22no—
11#15689021—22no—
12#15689007—24no—
13#15688996—22no—
14#15688986—22no—
15#15688983C2212yes—
16#15688861—24no—
17C22×C42×C47C2211yescharacteristic
18C2×C22⋊C8×C47C2211yescharacteristic
19C2×C22⋊C8×C47C2211yescharacteristic
20C47×#336—22no—
21C22×Dic94D412yes—
22C2×Dic188—22no—
23#13094003—22no—
24#13094003—22no—
25#13094003—22no—
26#13094003—22no—
27C2×C4⋊Dic47—22no—
28C2×C4⋊Dic47D412yes—
29C2×Dic47⋊C4—24no—
30C2×Dic47⋊C4—22no—
31C2×C22⋊Dic47—22no—
32C2×C22⋊Dic47—22no—
33#13093958—22no—
34#13093949—24no—
35C47⋊#95—24no—
36C47⋊#95—24no—
37C47⋊#95—24no—
38C47⋊#95D414yes—
39C23×C4×C47D411yescharacteristic
40C23×C4×C47D411yescharacteristic
41C23×C4×C47C2311yescharacteristic
42C22×C8×C47—21no—
43C22×C8×C47D412yes—
44C2×C42×C47—22no—
45C2×C42×C47—22no—
46C2×C42×C47—21no—
47C2×C42×C47—21no—
48C22⋊C8×C47—22no—
49C22⋊C8×C47—22no—
50C24×C47C2×D411yescharacteristic
51C22×C4×C47—22no—
52C22×C4×C47—22no—
53C22×C4×C47—21no—
54C22×C4×C47—21no—
55C22×C4×C47—21no—
56C22×C4×C47—22no—
57C22×C4×C47—42no—
58C22×C4×C47—42no—
59C22×C4×C47Q1612yescomplemented
60C22×C4×C47C2×D412yes—
61C22×C4×C47C2×D411yescharacteristic
62C22×C4×C47C2×D412yes—
63C22×C4×C47#4111yescharacteristic
64C22×Dic47—22no—
65C22×Dic47—24no—
66C22×Dic47—22no—
67C2×Dic94—22no—
68C2×Dic94—22no—
69C2×Dic94—44no—
70C2×C8×C47—22no—
71C2×C8×C47—42no—
72C2×C8×C47—42no—
73C2×C8×C47—41no—
74C42×C47—41no—
75C42×C47—42no—
76C42×C47—41no—
77C22⋊Dic47—42no—
78C22⋊Dic47—42no—
79C22⋊Dic47—42no—
80C22⋊Dic47—42no—
81C4⋊Dic47—22no—
82C4⋊Dic47—22no—
83C4⋊Dic47—44no—
84Dic47⋊C4—48no—
85Dic47⋊C4—42no—
86Dic47⋊C4—42no—
87Dic188—82no—
88C23×C47—41no—
89C23×C47—41no—
90C23×C47#12012yes—
91C23×C47#12111yescharacteristic
92C23×C47C2×Q1611yescharacteristic
93C23×C47#13511yescharacteristic
94C23×C47#13712yes—
95C2×C4×C47—24no—
96C2×C4×C47—22no—
97C2×C4×C47—24no—
98C2×C4×C47—22no—
99C2×C4×C47—24no—
100C2×C4×C47—22no—
101C2×C4×C47—42no—
102C2×C4×C47—41no—
103C2×C4×C47—42no—
104C2×C4×C47—41no—
105C2×C4×C47—41no—
106C2×C4×C47—41no—
107C2×C4×C47—42no—
108C2×C4×C47—41no—
109C2×C4×C47#12011yescharacteristic
110C2×C4×C47#12112yes—
111C2×C4×C47#12711yescharacteristic
112C2×Dic47—44no—
113C2×Dic47—44no—
114C2×Dic47—44no—
115C2×Dic47—44no—
116C2×Dic47—42no—
117C2×Dic47—42no—
118C2×Dic47—44no—
119Dic94—44no—
120Dic94—82no—
121C8×C47—42no—
122C8×C47—81no—
123#10325—471nomaximal, Sylow
124C22×C47—22no—
125C22×C47—21no—
126C22×C47—22no—
127C22×C47—21no—
128C22×C47—41no—
129C22×C47—41no—
130C22×C47—42no—
131C22×C47#39211yescharacteristic
132C22×C47#45112yes—
133C22×C47#45212yes—
134C22×C47#46711yescharacteristic
135C22×C47#47011yescharacteristic
136C4×C47—21no—
137C4×C47—22no—
138C4×C47—42no—
139C4×C47—42no—
140C4×C47#45812yes—
141Dic47—84no—
142Dic47—82no—
143Dic47—82no—
144#3189—471no—
145#2855—472no—
146#1817D4711yescharacteristic
147#1520—472no—
148#1289—471no—
149C2×C47—22no—
150C2×C47—22no—
151C2×C47—21no—
152C2×C47—21no—
153C2×C47#200111yescharacteristic
154C2×C47#201011yescharacteristic
155C2×C47#202211yescharacteristic
156C2×C22⋊Q8—472no—
157C22×C42C2×D4711yescharacteristic
158#484—942no—
159#467—942no—
160#451—942no—
161C2×C22⋊C8C2×D4711yescharacteristic
162C2×C22⋊C8C2×D4711yescharacteristic
163#391—942no—
164#389—942no—
165#387—944no—
166#385—942no—
167#381—942no—
168#380—472no—
169#336—22no—
170#328—944no—
171C47#1032511yescharacteristic, complemented, Sylow
172C22×Q8—472no—
173C23×C4D18811yescharacteristic
174C23×C4D18811yescharacteristic
175C23×C4C22×D4711yescharacteristic
176C2×Q16—942no—
177C22×C8—21no—
178C22×C8C22⋊D4712yes—
179C22⋊Q8—942no—
180C22⋊Q8—942no—
181C22⋊Q8—942no—
182C22⋊Q8—942no—
183C2×C4⋊C4—472no—
184C2×C4⋊C4—944no—
185C2×C4⋊C4—942no—
186C2×C4⋊C4—942no—
187C2×C22⋊C4—942no—
188C2×C22⋊C4—942no—
189C2×C42—21no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—21no—
193#107—942no—
194#103—944no—
195C22⋊C8—22no—
196C22⋊C8—22no—
197#95—474no—
198#95—944no—
199#95—944no—
200#95—944no—
201C24C2×D18811yescharacteristic
202C2×Q8—942no—
203C2×Q8—942no—
204C2×Q8—1884no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—944no—
214C22×C4—942no—
215C22×C4—942no—
216C22×C4Dic18812yescomplemented
217C22×C4#1062535011yescharacteristic
218C22×C4D4×D4712yes—
219C22×C4D4×D4712yes—
220C22×C4C2×C22⋊D4711yescharacteristic
221Q16—3762no—
222C2×C8—22no—
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—41no—
226C4⋊C4—942no—
227C4⋊C4—942no—
228C4⋊C4—1888no—
229C4⋊C4—1884no—
230C4⋊C4—1882no—
231C4⋊C4—1882no—
232C22⋊C4—1882no—
233C22⋊C4—1882no—
234C22⋊C4—1882no—
235C22⋊C4—1882no—
236C42—41no—
237C42—42no—
238C42—41no—
239C23—41no—
240C23—41no—
241C23#1309399412yes—
242C23#1309399811yescharacteristic
243C23C2×Dic18811yescharacteristic
244C23#1309404611yescharacteristic
245C23#1309405712yes—
246Q8—3762no—
247Q8—1884no—
248C2×C4—24no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—24no—
253C2×C4—22no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—1884no—
263C2×C4—1884no—
264C2×C4—1884no—
265C2×C4—1884no—
266C2×C4—1884no—
267C2×C4—1882no—
268C2×C4—1882no—
269C2×C4#1309399211yescharacteristic
270C2×C4#1309399712yes—
271C2×C4#1309402011yescharacteristic
272C8—42no—
273C8—81no—
274C22—22no—
275C22—21no—
276C22—22no—
277C22—21no—
278C22—41no—
279C22—41no—
280C22—42no—
281C22#1568903211yescharacteristic
282C22#1568928112yes—
283C22#1568928712yes—
284C22#1568936411yescharacteristic
285C22#1568938111yescharacteristic
286C4—21no—
287C4—22no—
288C4—42no—
289C4—42no—
290C4—3764no—
291C4—3762no—
292C4—3762no—
293C4#1568931212yes—
294C2—22no—
295C2—22no—
296C2—21no—
297C2—21no—
298C2#2024439811yescharacteristic
299C2#2024445511yescharacteristic
300C2#2024451911yescharacteristic
subgroupG/H
centerC22#15689032
commutator, cosocleC23×C4×C47C23
FrattiniC23×C4C22×D47
FittingC47×#1817C2
socleC22×C47#392
prime radicalC24×C47C2×D4
hypercenter#1817D47
hypocenterC47#10325
LCSG ▷ C23×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1817

Complements

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

NH ≅ G/Ntype
59. C22×C4×C47Q16semidirect
171. C47#10325semidirect
216. C22×C4Dic188semidirect

Maximal subgroups

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

subgroupindex
#202541632
#202517272
#202419612
#202412562
C47×#18172
#1032547