Finite Group Databasebeta

← #26133077

Subgroups — #26133077

Order 12032.

536 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C2×#15688918C211yesmaximal, characteristic
2#20243777C211yesmaximal, characteristic
3#20243668C212yesmaximal
4#20241506C212yesmaximal
5C47×#2372C211yesmaximal, characteristic
6#15689598C2211yescharacteristic
7#15689132—21no—
8C2×#13093941C2211yescharacteristic
9C2×#13093939C2212yes—
10#15688996—22no—
11#15688918C414yescomplemented
12#15688889—22no—
13#15688883—21no—
14C2×#124×C47C2211yescharacteristic
15C47×#391—21no—
16C47×#388—22no—
17C47×#386C2212yes—
18C4×C22⋊C4×C47—22no—
19C47×#376—21no—
20C2×#10625329—21no—
21D4×Dic47—42no—
22#13093983—21no—
23C2×Dic47⋊C4—42no—
24C2×C22⋊Dic47—22no—
25C2×C22⋊Dic47—22no—
26#13093948—42no—
27#13093941—22no—
28#13093939—22no—
29#13093939C2×C414yes—
30C47⋊#95—42no—
31C22×Q8×C47D411yescharacteristic
32C22×D4×C47C2311yescharacteristic
33C23×C4×C47D412yes—
34#124×C47—22no—
35#124×C47C2×C412yes—
36#124×C47Q812yes—
37C2×C4⋊C4×C47—21no—
38C2×C22⋊C4×C47—22no—
39C2×C22⋊C4×C47—22no—
40C2×C22⋊C4×C47D412yes—
41C2×C42×C47—22no—
42C2×C42×C47—21no—
43C2×C42×C47D411yescharacteristic
44#95×C47—21no—
45#95×C47—21no—
46#95×C47—22no—
47#95×C47—22no—
48C24×C47C2×D412yes—
49C2×Q8×C47—21no—
50C2×Q8×C47—41no—
51C2×Q8×C47—41no—
52C2×Q8×C47C4⋊C412yescomplemented
53C2×D4×C47—41no—
54C2×D4×C47—41no—
55C2×D4×C47C22×C411yescharacteristic
56C2×D4×C47C2×Q811yescharacteristic
57C2×D4×C47#4112yes—
58C22×C4×C47—21no—
59C22×C4×C47—22no—
60C22×C4×C47—21no—
61C22×C4×C47—22no—
62C22×C4×C47—44no—
63C22×C4×C47—42no—
64C22×C4×C47C2×D411yescharacteristic
65C22×Dic47—22no—
66C22×Dic47—22no—
67C22×Dic47—42no—
68C4⋊C4×C47—81no—
69C22⋊C4×C47—24no—
70C22⋊C4×C47—24no—
71C22⋊C4×C47—44no—
72C22⋊C4×C47—42no—
73C42×C47—21no—
74C42×C47—44no—
75C42×C47—81no—
76C42×C47C4⋊C412yescomplemented
77C22⋊Dic47—24no—
78C22⋊Dic47—42no—
79C22⋊Dic47—42no—
80C22⋊Dic47—41no—
81C4⋊Dic47—22no—
82Dic47⋊C4—84no—
83C4×Dic47—41no—
84#10625329—41no—
85#10625329—41no—
86C2×C47⋊C8—41no—
87C23×C47—22no—
88C23×C47—22no—
89C23×C47—42no—
90C23×C47—42no—
91C23×C47C4×D412yes—
92C23×C47#12011yescharacteristic
93C23×C47C22⋊Q812yes—
94Q8×C47—42no—
95Q8×C47—81no—
96D4×C47—41no—
97D4×C47—41no—
98D4×C47—42no—
99C2×C4×C47—21no—
100C2×C4×C47—21no—
101C2×C4×C47—24no—
102C2×C4×C47—22no—
103C2×C4×C47—22no—
104C2×C4×C47—42no—
105C2×C4×C47—42no—
106C2×C4×C47—42no—
107C2×C4×C47—41no—
108C2×C4×C47—41no—
109C2×C4×C47—44no—
110C2×C4×C47—42no—
111C2×C4×C47—42no—
112C2×C4×C47—42no—
113C2×C4×C47—82no—
114C2×C4×C47C2×C4⋊C411yescharacteristic
115C2×C4×C47#12311yescharacteristic
116C2×Dic47—44no—
117C2×Dic47—42no—
118C2×Dic47—82no—
119C2×Dic47—82no—
120C2×Dic47—82no—
121C2×Dic47—82no—
122C47⋊C8—81no—
123#12965—471nomaximal, Sylow
124C22×C47—22no—
125C22×C47—22no—
126C22×C47—22no—
127C22×C47—42no—
128C22×C47—42no—
129C22×C47—42no—
130C22×C47—42no—
131C22×C47—82no—
132C22×C47#38511yescharacteristic
133C22×C47B(4,𝔽2)11yescharacteristic
134C22×C47#45811yescharacteristic
135C4×C47—21no—
136C4×C47—21no—
137C4×C47—42no—
138C4×C47—42no—
139C4×C47—44no—
140C4×C47—82no—
141Dic47—84no—
142Dic47—82no—
143#2372D4711yescharacteristic
144C2×#354—471no—
145#1885—471no—
146#1867—472no—
147#1390—472no—
148C2×C47—21no—
149C2×C47—21no—
150C2×C47—42no—
151C2×C47—42no—
152C2×C47#186711yescharacteristic
153C2×C47#219712yes—
154C2×#124C2×D4711yescharacteristic
155#518—471no—
156#419—941no—
157C2×#100—471no—
158C2×#99—472no—
159#391—21no—
160#388—22no—
161#386C2×D4712yes—
162#385—942no—
163C4×C22⋊C4—22no—
164#376—21no—
165#354—474no—
166#342—942no—
167#339—941no—
168C47#1296511yescharacteristic, complemented, Sylow
169C22×Q8C22⋊D4711yescharacteristic
170C22×D4C22×D4711yescharacteristic
171C23×C4C22⋊D4712yes—
172C2×M16—941no—
173#124—22no—
174#124Dic9412yes—
175#124C4×D4712yes—
176C4×D4—1882no—
177#117—941no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—1882no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—942no—
183C2×C22⋊C4—942no—
184C2×C22⋊C4C22⋊D4712yes—
185C2×C42—22no—
186C2×C42—21no—
187C2×C42C22⋊D4711yescharacteristic
188#103—1882no—
189#100—942no—
190#99—474no—
191#99—942no—
192#95—21no—
193#95—21no—
194#95—22no—
195#95—22no—
196#95—1882no—
197C24C2×C22⋊D4712yes—
198C2×Q8—21no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×Q8Dic47⋊C412yescomplemented
202C2×D4—41no—
203C2×D4—41no—
204C2×D4C2×Dic9411yescharacteristic
205C2×D4C2×C4×D4711yescharacteristic
206C2×D4#1062535012yes—
207C22×C4—22no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—44no—
212C22×C4—42no—
213C22×C4—942no—
214C22×C4—942no—
215C22×C4—1882no—
216C22×C4C2×C22⋊D4711yescharacteristic
217M16—1881no—
218M16—1881no—
219C2×C8—1881no—
220C4⋊C4—81no—
221C4⋊C4—942no—
222C4⋊C4—3764no—
223C22⋊C4—24no—
224C22⋊C4—24no—
225C22⋊C4—44no—
226C22⋊C4—42no—
227C22⋊C4—944no—
228C22⋊C4—1882no—
229C22⋊C4—1882no—
230C22⋊C4—1881no—
231C42—21no—
232C42—44no—
233C42—81no—
234C42—1881no—
235C42Dic47⋊C412yescomplemented
236C23—22no—
237C23—22no—
238C23—42no—
239C23—42no—
240C23C4×C22⋊D4712yes—
241C23#1309399311yescharacteristic
242C23#1309400312yes—
243Q8—42no—
244Q8—81no—
245D4—41no—
246D4—41no—
247D4—42no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—44no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—82no—
263C2×C4—3762no—
264C2×C4—3762no—
265C2×C4—3762no—
266C2×C4—3762no—
267C2×C4—1884no—
268C2×C4—1882no—
269C2×C4C2×Dic47⋊C411yescharacteristic
270C2×C4#1309400711yescharacteristic
271C8—3761no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—82no—
280C22#1568899611yescharacteristic
281C22#1568931011yescharacteristic
282C22#1568931311yescharacteristic
283C4—21no—
284C4—21no—
285C4—42no—
286C4—44no—
287C4—42no—
288C4—82no—
289C4—3764no—
290C4—3762no—
291C2—21no—
292C2—21no—
293C2—42no—
294C2—42no—
295C2#2024366911yescharacteristic
296C2#2024537812yes—
subgroupG/H
centerC22#15689310
commutatorC2×D4×C47C22×C4
FrattiniC22×D4C22×D47
FittingC47×#2372C2
socleC22×C47B(4,𝔽2)
prime radical, cosocleC22×D4×C47C23
hypercenter#2372D47
hypocenterC47#12965
LCSG ▷ C2×D4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #2372

Complements

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

NH ≅ G/Ntype
11. #15688918C4semidirect
52. C2×Q8×C47C4⋊C4semidirect
76. C42×C47C4⋊C4semidirect
168. C47#12965semidirect
201. C2×Q8Dic47⋊C4semidirect
235. C42Dic47⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#156889182
#202437772
#202436682
#202415062
C47×#23722
#1296547