Finite Group Databasebeta

← #26112317

Subgroups — #26112317

Order 12032.

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

normal subgroups only

This group has 578 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#20252026C211yesmaximal, characteristic, complemented
2#20242056C212yesmaximal, complemented
3#20241702C211yesmaximal, characteristic, complemented
4#20241246C212yesmaximal, complemented
5C47×#2619C211yesmaximal, characteristic
6#15689184—24no—
7#15689176C2212yes—
8#15689147C2212yes—
9#15689087—21no—
10C4×#10625329—21no—
11#15688867—24no—
12#15688849—21no—
13#15688847—21no—
14C2×#127×C47C2211yescharacteristic
15C2×#121×C47—21no—
16C2×#117×C47C2211yescharacteristic
17C47×#394—21no—
18C47×#394C412yes—
19C47×#392—21no—
20C47×#392C412yes—
21C47×#390—21no—
22C47×#382C2211yescharacteristic
23C2×#10625329—22no—
24C4⋊(C47⋊C8)—22no—
25C4⋊(C47⋊C8)—22no—
26C4⋊(C47⋊C8)D414yescomplemented
27C22⋊(C47⋊C8)—24no—
28C22⋊(C47⋊C8)—22no—
29#13093935—22no—
30C4×C47⋊C8—22no—
31C4×C47⋊C8—22no—
32C22×D4×C47—22no—
33C22×D4×C47—21no—
34C22×D4×C47—21no—
35C22×D4×C47C2×C412yes—
36C23×C4×C47D411yescharacteristic
37#127×C47—22no—
38#127×C47—22no—
39#121×C47—44no—
40#117×C47—22no—
41#117×C47—22no—
42C2×C4⋊C4×C47—21no—
43C2×C4⋊C4×C47C2×C411yescharacteristic
44C2×C4⋊C4×C47D411yescharacteristic
45C2×C22⋊C4×C47—22no—
46C2×C22⋊C4×C47—22no—
47C2×C22⋊C4×C47—22no—
48C2×C22⋊C4×C47C2×C411yescharacteristic
49C2×C22⋊C4×C47D411yescharacteristic
50C2×C42×C47D411yescharacteristic
51C2×C42×C47C2311yescharacteristic
52#95×C47—21no—
53#95×C47C2×C412yes—
54C24×C47—22no—
55C24×C47C22⋊C411yescharacteristic
56C2×D4×C47—24no—
57C2×D4×C47—44no—
58C2×D4×C47—41no—
59C2×D4×C47—42no—
60C2×D4×C47—41no—
61C2×D4×C47—42no—
62C2×D4×C47—42no—
63C2×D4×C47—82no—
64C2×D4×C47—82no—
65C2×D4×C47—82no—
66C2×D4×C47—82no—
67C22×C4×C47—21no—
68C22×C4×C47—21no—
69C22×C4×C47—22no—
70C22×C4×C47—22no—
71C22×C4×C47—22no—
72C22×C4×C47C22⋊C411yescharacteristic
73C22×C4×C47C22⋊C411yescharacteristic
74C22×C4×C47C22⋊C411yescharacteristic
75C22×C4×C47C22×C411yescharacteristic
76C22×C4×C47C2×D411yescharacteristic
77C22×C4×C47C2×D411yescharacteristic
78C4⋊C4×C47—22no—
79C4⋊C4×C47—22no—
80C4⋊C4×C47—42no—
81C22⋊C4×C47—22no—
82C22⋊C4×C47—22no—
83C22⋊C4×C47—44no—
84C22⋊C4×C47—82no—
85C22⋊C4×C47—82no—
86C42×C47—21no—
87C42×C47—21no—
88C42×C47C2×D412yes—
89C42×C47#4112yes—
90#10625329—42no—
91#10625329—42no—
92C2×C47⋊C8—24no—
93C2×C47⋊C8—42no—
94C23×C47—21no—
95C23×C47—21no—
96C23×C47—21no—
97C23×C47—41no—
98C23×C47—41no—
99C23×C47—42no—
100C23×C47—42no—
101C23×C47—42no—
102C23×C47—42no—
103C23×C47—42no—
104C23×C47—42no—
105C23×C47—42no—
106C23×C47C2×C22⋊C411yescharacteristic
107D4×C47—84no—
108D4×C47—84no—
109D4×C47—88no—
110D4×C47—82no—
111D4×C47—82no—
112C2×C4×C47—21no—
113C2×C4×C47—21no—
114C2×C4×C47—21no—
115C2×C4×C47—21no—
116C2×C4×C47—21no—
117C2×C4×C47—41no—
118C2×C4×C47—41no—
119C2×C4×C47—42no—
120C2×C4×C47—42no—
121C2×C4×C47—41no—
122C2×C4×C47—41no—
123C2×C4×C47—42no—
124C2×C4×C47—42no—
125C2×C4×C47—42no—
126C2×C4×C47C4×D412yes—
127C2×C4×C47#12011yescharacteristic
128C2×C4×C47#12112yes—
129C2×C4×C47#12311yescharacteristic
130C2×C4×C47#13512yes—
131C2×C4×C47AGL(1,ℤ/8)12yes—
132C47⋊C8—44no—
133C47⋊C8—82no—
134#9198—471nomaximal, Sylow
135C22×C47—21no—
136C22×C47—21no—
137C22×C47—42no—
138C22×C47—41no—
139C22×C47—41no—
140C22×C47—41no—
141C22×C47—81no—
142C22×C47—82no—
143C22×C47—82no—
144C22×C47—82no—
145C22×C47—82no—
146C22×C47—82no—
147C22×C47—82no—
148C22×C47—82no—
149C22×C47#38611yescharacteristic
150C22×C47#41611yescharacteristic
151C22×C47#41811yescharacteristic
152C4×C47—22no—
153C4×C47—22no—
154C4×C47—41no—
155C4×C47—41no—
156C4×C47—41no—
157C4×C47—82no—
158C4×C47#44912yes—
159C4×C47#48212yes—
160#2921—471no—
161#2619D4711yescharacteristic
162#1545—472no—
163#1458—471no—
164#1286—472no—
165C2×C47—21no—
166C2×C47—21no—
167C2×C47—81no—
168C2×C47—162no—
169C2×C47#187511yescharacteristic
170C2×C47#191011yescharacteristic
171C2×C47#191411yescharacteristic
172C2×#127C2×D4711yescharacteristic
173C2×#121—21no—
174C2×#117C2×D4711yescharacteristic
175#433—944no—
176#431—472no—
177#423—472no—
178#407—941no—
179C4×M16—941no—
180#394—21no—
181#394Dic4712yes—
182#392—21no—
183#392Dic4712yes—
184#390—21no—
185#382C2×D4711yescharacteristic
186#331—944no—
187#324—941no—
188#323—941no—
189C47#919811yescharacteristic, complemented, Sylow
190C22×D4—22no—
191C22×D4—21no—
192C22×D4—21no—
193C22×D4C2×Dic4712yes—
194C23×C4C22⋊D4711yescharacteristic
195C2×M16—942no—
196#127—22no—
197#127—22no—
198#121—44no—
199#117—22no—
200#117—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4C2×Dic4711yescharacteristic
203C2×C4⋊C4C22⋊D4711yescharacteristic
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4C2×Dic4711yescharacteristic
208C2×C22⋊C4C22⋊D4711yescharacteristic
209C2×C42C22⋊D4711yescharacteristic
210C2×C42C22×D4711yescharacteristic
211C4⋊C8—474no—
212C4⋊C8—942no—
213C4⋊C8—942no—
214C22⋊C8—944no—
215C22⋊C8—942no—
216#97—942no—
217C4×C8—942no—
218C4×C8—942no—
219#95—21no—
220#95C2×Dic4712yes—
221C24—22no—
222C24C22⋊Dic4711yescharacteristic
223C2×D4—24no—
224C2×D4—44no—
225C2×D4—41no—
226C2×D4—42no—
227C2×D4—41no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—82no—
231C2×D4—82no—
232C2×D4—82no—
233C2×D4—82no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4C22⋊Dic4711yescharacteristic
240C22×C4C22⋊Dic4711yescharacteristic
241C22×C4C22⋊Dic4711yescharacteristic
242C22×C4C22×Dic4711yescharacteristic
243C22×C4C2×C22⋊D4711yescharacteristic
244C22×C4C2×C22⋊D4711yescharacteristic
245M16—1882no—
246M16—1882no—
247C2×C8—944no—
248C2×C8—1882no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—42no—
252C22⋊C4—22no—
253C22⋊C4—22no—
254C22⋊C4—44no—
255C22⋊C4—82no—
256C22⋊C4—82no—
257C42—21no—
258C42—21no—
259C42D4×D4712yes—
260C42#1062535212yes—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—41no—
265C23—41no—
266C23—42no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23C2×C22⋊Dic4711yescharacteristic
274D4—84no—
275D4—84no—
276D4—88no—
277D4—82no—
278D4—82no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4D4×Dic4712yes—
294C2×C4#1309399211yescharacteristic
295C2×C4#1309399612yes—
296C2×C4#1309400511yescharacteristic
297C2×C4#1309404412yes—
298C2×C4#1309405012yes—
299C8—3762no—
300C8—1884no—
301C22—21no—
302C22—21no—
303C22—42no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—81no—
308C22—82no—
309C22—82no—
310C22—82no—
311C22—82no—
312C22—82no—
313C22—82no—
314C22—82no—
315C22#1568900311yescharacteristic
316C22#1568911911yescharacteristic
317C22#1568912811yescharacteristic
318C4—22no—
319C4—22no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—82no—
324C4#1568926612yes—
325C4#1568944412yes—
326C2—21no—
327C2—21no—
328C2—81no—
329C2—162no—
330C2#2024371111yescharacteristic
331C2#2024391611yescharacteristic
332C2#2024393611yescharacteristic
subgroupG/H
centerC22#15689003
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
Fitting, prime radicalC47×#2619C2
socleC22×C47#386
cosocleC2×C42×C47C23
hypercenter#2619D47
hypocenterC47#9198
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2619

Complements

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

NH ≅ G/Ntype
1. #20252026C2semidirect
2. #20242056C2semidirect
3. #20241702C2semidirect
4. #20241246C2semidirect
26. C4⋊(C47⋊C8)D4semidirect
189. C47#9198semidirect

Maximal subgroups

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

subgroupindex
#202520262
#202420562
#202417022
#202412462
C47×#26192
#919847