Finite Group Databasebeta

← #26116842

Subgroups — #26116842

Order 12032.

772 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 772 conjugacy classes of subgroups — too many to draw. Only its 121 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×#15689439C212yesmaximal, complemented
2C2×#15688857C212yesmaximal, complemented
3#20241657C211yesmaximal, characteristic, complemented
4#20241231C211yesmaximal, characteristic, complemented
5C47×#2656C211yesmaximal, characteristic
6C22×C4⋊Dic47C2211yescharacteristic
7#15689439—24no—
8#15689439—24no—
9C2×#13093955—22no—
10C2×#13093945—22no—
11C2×C22⋊(C47⋊C8)C2212yes—
12C2×C47⋊#95—22no—
13C2×C47⋊#95C2211yescharacteristic
14#15688857—24no—
15C2×#121×C47—22no—
16C2×#121×C47C2212yes—
17C47×#399—22no—
18C47×#392—22no—
19C2×#95×C47C2211yescharacteristic
20C23×Dic47—21no—
21C23×Dic47—21no—
22C22×C47⋊C8—22no—
23C2×C4⋊Dic47—22no—
24C2×C4⋊Dic47—22no—
25C2×C4⋊Dic47—21no—
26C2×C4⋊Dic47D412yescomplemented
27#13093955—28no—
28#13093945—28no—
29C22⋊(C47⋊C8)—22no—
30C22⋊(C47⋊C8)D414yescomplemented
31C47⋊#95—22no—
32C47⋊#95—44no—
33C22×D4×C47—22no—
34C22×D4×C47—22no—
35C22×D4×C47D412yes—
36C23×C4×C47D412yes—
37C23×C4×C47C2311yescharacteristic
38#121×C47—22no—
39#121×C47—22no—
40#121×C47—22no—
41#121×C47—22no—
42#121×C47—44no—
43#121×C47—44no—
44C2×C4⋊C4×C47—22no—
45C2×C4⋊C4×C47D412yes—
46C2×C22⋊C4×C47—24no—
47C2×C22⋊C4×C47—22no—
48#95×C47—22no—
49C24×C47—22no—
50C24×C47C2×D411yescharacteristic
51C2×D4×C47—22no—
52C2×D4×C47—22no—
53C2×D4×C47—44no—
54C2×D4×C47—44no—
55C2×D4×C47—42no—
56C2×D4×C47—42no—
57C2×D4×C47—44no—
58C2×D4×C47—44no—
59C2×D4×C47—82no—
60C2×D4×C47—82no—
61C22×C4×C47—22no—
62C22×C4×C47—22no—
63C22×C4×C47—22no—
64C22×C4×C47—42no—
65C22×C4×C47—42no—
66C22×C4×C47—42no—
67C22×C4×C47SD1614yes—
68C22×C4×C47C2×D412yes—
69C22×C4×C47C2×D411yescharacteristic
70C22×C4×C47C2×D412yes—
71C22×C4×C47#4111yescharacteristic
72C22×Dic47—22no—
73C22×Dic47—22no—
74C22×Dic47—42no—
75C22×Dic47—41no—
76C22×Dic47—42no—
77C22×Dic47—42no—
78C22×Dic47—41no—
79C22×Dic47—41no—
80C22×Dic47—41no—
81C22×Dic47—42no—
82C4⋊C4×C47—22no—
83C4⋊C4×C47—22no—
84C4⋊C4×C47—44no—
85C22⋊C4×C47—48no—
86C22⋊C4×C47—42no—
87C22⋊C4×C47—42no—
88C4⋊Dic47—24no—
89C4⋊Dic47—42no—
90C2×C47⋊C8—24no—
91C2×C47⋊C8—44no—
92C23×C47—21no—
93C23×C47—21no—
94C23×C47—22no—
95C23×C47—42no—
96C23×C47—44no—
97C23×C47—42no—
98C23×C47—42no—
99C23×C47—44no—
100C23×C47#12011yescharacteristic
101C23×C47#12111yescharacteristic
102C23×C47#12711yescharacteristic
103C23×C47C2×SD1612yes—
104C23×C47AGL(1,ℤ/8)12yes—
105D4×C47—44no—
106D4×C47—82no—
107D4×C47—88no—
108D4×C47—84no—
109D4×C47—82no—
110D4×C47—82no—
111C2×C4×C47—24no—
112C2×C4×C47—24no—
113C2×C4×C47—41no—
114C2×C4×C47—42no—
115C2×C4×C47—41no—
116C2×C4×C47—42no—
117C2×C4×C47—42no—
118C2×C4×C47—44no—
119C2×C4×C47—42no—
120C2×C4×C47—44no—
121C2×C4×C47—82no—
122C2×C4×C47#12012yes—
123C2×C4×C47#12112yes—
124C2×Dic47—44no—
125C2×Dic47—42no—
126C2×Dic47—44no—
127C2×Dic47—42no—
128C2×Dic47—82no—
129C2×Dic47—81no—
130C2×Dic47—82no—
131C2×Dic47—81no—
132C2×Dic47—81no—
133C2×Dic47—81no—
134C47⋊C8—44no—
135#9968—471nomaximal, Sylow
136C22×C47—22no—
137C22×C47—22no—
138C22×C47—22no—
139C22×C47—22no—
140C22×C47—21no—
141C22×C47—21no—
142C22×C47—21no—
143C22×C47—21no—
144C22×C47—84no—
145C22×C47—84no—
146C22×C47—82no—
147C22×C47—82no—
148C22×C47—82no—
149C22×C47#39211yescharacteristic
150C22×C47#45012yes—
151C22×C47#45311yescharacteristic
152C22×C47#45411yescharacteristic
153C22×C47#45812yes—
154C22×C47#46012yes—
155C22×C47#49211yescharacteristic
156C22×C47#49611yescharacteristic
157C4×C47—22no—
158C4×C47—22no—
159C4×C47—44no—
160C4×C47—82no—
161Dic47—82no—
162Dic47—82no—
163C2×#481—472no—
164#2656D4711yescharacteristic
165C2×#327—472no—
166#1445—471no—
167#1280—471no—
168C2×C47—21no—
169C2×C47—22no—
170C2×C47—21no—
171C2×C47—162no—
172C2×C47#161312yes—
173C2×C47#161612yes—
174C2×C47#200611yescharacteristic
175C2×C47#201411yescharacteristic
176C2×C47#202211yescharacteristic
177C2×#121—22no—
178C2×#121C2×D4712yes—
179C22×C4⋊C4—471no—
180#481—944no—
181#481—944no—
182C2×#106—942no—
183C2×#102—942no—
184C2×C22⋊C8—472no—
185#399—22no—
186#392—22no—
187C2×#95—471no—
188C2×#95—942no—
189C2×#95C2×D4711yescharacteristic
190#327—944no—
191C47#996811yescharacteristic, complemented, Sylow
192C22×D4—22no—
193C22×D4—22no—
194C22×D4C22⋊D4712yes—
195C23×C4—941no—
196C23×C4—941no—
197C23×C4C22⋊D4712yes—
198C23×C4C22×D4711yescharacteristic
199C22×C8—942no—
200#121—22no—
201#121—22no—
202#121—22no—
203#121—22no—
204#121—44no—
205#121—44no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—472no—
208C2×C4⋊C4—942no—
209C2×C4⋊C4—942no—
210C2×C4⋊C4—941no—
211C2×C4⋊C4C22⋊D4712yes—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—22no—
214#106—948no—
215#102—948no—
216C22⋊C8—474no—
217C22⋊C8—942no—
218#95—22no—
219#95—942no—
220#95—1884no—
221C24—22no—
222C24C2×C22⋊D4711yescharacteristic
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—44no—
226C2×D4—44no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—44no—
230C2×D4—44no—
231C2×D4—82no—
232C2×D4—82no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—942no—
240C22×C4—942no—
241C22×C4—1882no—
242C22×C4—1881no—
243C22×C4—1882no—
244C22×C4—1882no—
245C22×C4—1881no—
246C22×C4—1881no—
247C22×C4—1881no—
248C22×C4—1882no—
249C22×C4#1062533514yes—
250C22×C4D4×D4711yescharacteristic
251C22×C4D4×D4712yes—
252C22×C4#1062535211yescharacteristic
253C22×C4C2×C22⋊D4712yes—
254C2×C8—944no—
255C2×C8—1884no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—44no—
259C4⋊C4—944no—
260C4⋊C4—1882no—
261C22⋊C4—48no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C23—21no—
265C23—21no—
266C23—22no—
267C23—44no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—44no—
272C23#1309399311yescharacteristic
273C23#1309399511yescharacteristic
274C23#1309402011yescharacteristic
275C23C2×#1062533512yes—
276C23#1309405012yes—
277D4—44no—
278D4—82no—
279D4—88no—
280D4—82no—
281D4—82no—
282D4—84no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—42no—
293C2×C4—82no—
294C2×C4—3761no—
295C2×C4—3762no—
296C2×C4—3762no—
297C2×C4—3761no—
298C2×C4—3761no—
299C2×C4—3761no—
300C2×C4—1882no—
301C2×C4—1884no—
302C2×C4—1884no—
303C2×C4—1882no—
304C2×C4#1309399212yes—
305C2×C4#1309399612yes—
306C8—1884no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—22no—
312C22—22no—
313C22—22no—
314C22—22no—
315C22—82no—
316C22—82no—
317C22—84no—
318C22—84no—
319C22—82no—
320C22#1568903111yescharacteristic
321C22#1568927412yes—
322C22#1568929111yescharacteristic
323C22#1568929611yescharacteristic
324C22#1568931312yes—
325C22#1568932012yes—
326C22#1568949511yescharacteristic
327C22#1568951011yescharacteristic
328C4—22no—
329C4—22no—
330C4—44no—
331C4—82no—
332C4—3762no—
333C4—3762no—
334C2—21no—
335C2—21no—
336C2—22no—
337C2—162no—
338C2#2024240812yes—
339C2#2024242512yes—
340C2#2024443011yescharacteristic
341C2#2024447611yescharacteristic
342C2#2024451711yescharacteristic
subgroupG/H
centerC23#13093993
commutator, cosocleC23×C4×C47C23
FrattiniC23×C4C22×D47
Fitting, prime radicalC47×#2656C2
socleC23×C47#120
hypercenter#2656D47
hypocenterC47#9968
LCSG ▷ C23×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2656

Complements

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

NH ≅ G/Ntype
1. C2×#15689439C2semidirect
2. C2×#15688857C2semidirect
3. #20241657C2semidirect
4. #20241231C2semidirect
26. C2×C4⋊Dic47D4semidirect
30. C22⋊(C47⋊C8)D4semidirect
191. C47#9968semidirect

Maximal subgroups

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

subgroupindex
C2×#156894392
C2×#156888572
#202416572
#202412312
C47×#26562
#996847