Finite Group Databasebeta

← #26110635

Subgroups — #26110635

Order 12032.

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

normal subgroups only

This group has 500 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#20246562C211yesmaximal, characteristic
2#20244205C211yesmaximal, characteristic
3#20243556C212yesmaximal, complemented
4#20241270C212yesmaximal, complemented
5C2×C47×#329C211yesmaximal, characteristic
6C2×#13093958—22no—
7C2×#13093956—22no—
8C2×#13093948—21no—
9C2×#13093945—21no—
10C2×#13093935C2211yescharacteristic
11#15689030—21no—
12#15689016—22no—
13#15689003—21no—
14#15688998—22no—
15#15688993C2211yescharacteristic
16#15688989C2212yes—
17C2×#124×C47C2211yescharacteristic
18C2×#97×C47C2212yes—
19C47×#329C414yescomplemented
20C23×Dic47—21no—
21C22×C47⋊C8—21no—
22C2×C4⋊Dic47—21no—
23C2×C4⋊Dic47—22no—
24C2×C4⋊Dic47—21no—
25C2×C4⋊Dic47D412yes—
26C2×C4⋊Dic47D412yes—
27C2×C22⋊Dic47—22no—
28C2×C22⋊Dic47—21no—
29C2×C4×Dic47—21no—
30#13093958—44no—
31#13093956—44no—
32#13093948—42no—
33#13093945—42no—
34#13093935—22no—
35C47⋊#95—22no—
36C47⋊#95—22no—
37C47⋊#95—22no—
38C22×Q8×C47D411yescharacteristic
39C22×D4×C47D411yescharacteristic
40C22×C8×C47—22no—
41#124×C47—22no—
42#124×C47C2×C412yes—
43#124×C47Q812yes—
44C2×C22⋊C4×C47—22no—
45C2×C42×C47C2311yescharacteristic
46#97×C47—22no—
47#97×C47C2×C414yescomplemented
48C24×C47—21no—
49C2×Q8×C47—21no—
50C2×Q8×C47—41no—
51C2×Q8×C47—41no—
52C2×Q8×C47C4⋊C412yes—
53C2×D4×C47—21no—
54C2×D4×C47—41no—
55C2×D4×C47—41no—
56C2×D4×C47C4⋊C412yes—
57C22×C4×C47—21no—
58C22×C4×C47C2×D411yescharacteristic
59C22×C4×C47C2×D412yes—
60C22×Dic47—22no—
61C22×Dic47—22no—
62C22×Dic47—41no—
63C22×Dic47—82no—
64C22×Dic47—81no—
65C2×C8×C47—24no—
66C2×C8×C47—24no—
67C2×C8×C47—24no—
68C22⋊C4×C47—24no—
69C22⋊C4×C47—24no—
70C42×C47C22×C411yescharacteristic
71C42×C47C2×Q811yescharacteristic
72C42×C47#4112yes—
73C22⋊Dic47—82no—
74C22⋊Dic47—81no—
75C4⋊Dic47—42no—
76C4⋊Dic47—42no—
77C4⋊Dic47—41no—
78C4⋊Dic47—41no—
79C4⋊Dic47—42no—
80C4⋊Dic47—42no—
81C4⋊Dic47—81no—
82C4×Dic47—81no—
83C2×C47⋊C8—22no—
84C2×C47⋊C8—41no—
85C2×C47⋊C8—41no—
86C23×C47—22no—
87C23×C47—42no—
88C23×C47—41no—
89C23×C47—41no—
90C23×C47—42no—
91C23×C47#12011yescharacteristic
92Q8×C47—42no—
93Q8×C47—81no—
94D4×C47—42no—
95D4×C47—81no—
96C2×C4×C47—21no—
97C2×C4×C47—22no—
98C2×C4×C47—21no—
99C2×C4×C47—22no—
100C2×C4×C47—22no—
101C2×C4×C47—41no—
102C2×C4×C47—41no—
103C2×C4×C47C2×C4⋊C411yescharacteristic
104C2×C4×C47C4×D412yes—
105C2×C4×C47C22⋊Q812yes—
106C2×C4×C47#12311yescharacteristic
107C2×Dic47—42no—
108C2×Dic47—42no—
109C2×Dic47—42no—
110C2×Dic47—42no—
111C2×Dic47—42no—
112C2×Dic47—42no—
113C2×Dic47—84no—
114C2×Dic47—82no—
115C2×Dic47—81no—
116C8×C47—28no—
117C47⋊C8—42no—
118#8910—471nomaximal, Sylow
119C22×C47—42no—
120C22×C47—44no—
121C22×C47—81no—
122C22×C47—81no—
123C22×C47—82no—
124C22×C47#38511yescharacteristic
125C22×C47#45312yes—
126C22×C47#45311yescharacteristic
127C22×C47#45511yescharacteristic
128C22×C47#45612yes—
129C4×C47—21no—
130C4×C47—21no—
131C4×C47—22no—
132C4×C47—22no—
133C4×C47—42no—
134Dic47—82no—
135Dic47—82no—
136Dic47—161no—
137#2379—471no—
138#1959—471no—
139#1847—472no—
140C2×#329D4711yescharacteristic
141#1295—472no—
142C2×C47—82no—
143C2×C47#165511yescharacteristic
144C2×C47#166011yescharacteristic
145C2×C47#166112yes—
146C2×C47#185611yescharacteristic
147C2×C47#185912yes—
148C2×#124C2×D4711yescharacteristic
149C2×#107—942no—
150C2×#106—942no—
151C2×#103—941no—
152C2×#102—941no—
153C2×#97—471no—
154C2×#97C2×D4712yes—
155#391—941no—
156#388—942no—
157#386—941no—
158#385—942no—
159#384—471no—
160#382—472no—
161#329Dic4714yescomplemented
162C47#891011yescharacteristic, complemented, Sylow
163C22×Q8D18811yescharacteristic
164C22×D4D18811yescharacteristic
165C23×C4—941no—
166C22×C8—22no—
167C22×C8—941no—
168#124—22no—
169#124Dic9412yes—
170#124C2×Dic4712yes—
171C2×C4⋊C4—472no—
172C2×C4⋊C4—472no—
173C2×C4⋊C4—941no—
174C2×C4⋊C4—941no—
175C2×C4⋊C4—942no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—942no—
178C2×C22⋊C4—941no—
179C2×C42—941no—
180C2×C42C22×D4711yescharacteristic
181#107—1884no—
182#106—1884no—
183#103—1882no—
184#102—1882no—
185#97—22no—
186#97—942no—
187#97C2×Dic4714yescomplemented
188#95—942no—
189#95—942no—
190#95—942no—
191C24—21no—
192C2×Q8—21no—
193C2×Q8—41no—
194C2×Q8—41no—
195C2×Q8C4⋊Dic4712yes—
196C2×D4—21no—
197C2×D4—41no—
198C2×D4—41no—
199C2×D4C4⋊Dic4712yes—
200C22×C4—21no—
201C22×C4—942no—
202C22×C4—942no—
203C22×C4—3762no—
204C22×C4—3761no—
205C22×C4—1881no—
206C22×C4C2×D18811yescharacteristic
207C22×C4D4×D4712yes—
208C2×C8—24no—
209C2×C8—24no—
210C2×C8—24no—
211C2×C8—942no—
212C2×C8—1881no—
213C2×C8—1881no—
214C4⋊C4—3761no—
215C4⋊C4—1881no—
216C4⋊C4—1881no—
217C4⋊C4—1882no—
218C4⋊C4—1882no—
219C4⋊C4—1882no—
220C4⋊C4—1882no—
221C22⋊C4—24no—
222C22⋊C4—24no—
223C22⋊C4—3762no—
224C22⋊C4—3761no—
225C42—3761no—
226C42C2×Dic9411yescharacteristic
227C42#1062535212yes—
228C42C22×Dic4711yescharacteristic
229C23—22no—
230C23—42no—
231C23—41no—
232C23—41no—
233C23—42no—
234C23#1309399411yescharacteristic
235Q8—42no—
236Q8—81no—
237D4—42no—
238D4—81no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—3764no—
247C2×C4—3762no—
248C2×C4—3761no—
249C2×C4—1882no—
250C2×C4—1882no—
251C2×C4—1882no—
252C2×C4—1882no—
253C2×C4—1882no—
254C2×C4—1882no—
255C2×C4C2×C4⋊Dic4711yescharacteristic
256C2×C4D4×Dic4712yes—
257C2×C4#1309400112yes—
258C2×C4#1309400811yescharacteristic
259C8—28no—
260C8—1882no—
261C22—42no—
262C22—44no—
263C22—81no—
264C22—82no—
265C22—81no—
266C22#1568899811yescharacteristic
267C22#1568929212yes—
268C22#1568929311yescharacteristic
269C22#1568930311yescharacteristic
270C22#1568930712yes—
271C4—21no—
272C4—21no—
273C4—22no—
274C4—22no—
275C4—42no—
276C4—3762no—
277C4—3762no—
278C4—7521no—
279C2—82no—
280C2#2024267811yescharacteristic
281C2#2024270011yescharacteristic
282C2#2024270812yes—
283C2#2024360011yescharacteristic
284C2#2024361612yes—
subgroupG/H
centerC23#13093994
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC2×C47×#329C2
socleC23×C47#120
prime radicalC22×D4×C47D4
cosocleC2×C42×C47C23
hypercenterC2×#329D47
hypocenterC47#8910
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#329

Complements

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

NH ≅ G/Ntype
3. #20243556C2semidirect
4. #20241270C2semidirect
19. C47×#329C4semidirect
47. #97×C47C2×C4semidirect
161. #329Dic47semidirect
162. C47#8910semidirect
187. #97C2×Dic47semidirect

Maximal subgroups

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

subgroupindex
#202465622
#202442052
#202435562
#202412702
C2×C47×#3292
#891047