Finite Group Databasebeta

← #26112326

Subgroups — #26112326

Order 12032.

560 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 131 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#20249201C211yesmaximal, characteristic
2#20241959C212yesmaximal
3C2×#15688849C211yesmaximal, characteristic
4#20241255C212yesmaximal
5C47×#2316C211yesmaximal, characteristic
6C2×#13094003C2211yescharacteristic
7C2×C22⋊(C47⋊C8)C2212yes—
8#15689062C412yescomplemented
9#15689056—22no—
10#15689043C412yescomplemented
11#15688996—21no—
12#15688986—21no—
13#15688983C2211yescharacteristic
14#15688859—24no—
15#15688849—22no—
16C2×#117×C47—21no—
17C22×C4⋊C4×C47C2211yescharacteristic
18C47×#389—24no—
19C47×#380—21no—
20C47×#380C2212yes—
21C22×Dic94C2×C411yescharacteristic
22C22×C47⋊C8—22no—
23#13094003—22no—
24#13094003—22no—
25C2×C4⋊Dic47—21no—
26C2×C4⋊Dic47C2×C411yescharacteristic
27C2×Dic47⋊C4—21no—
28C2×Dic47⋊C4—22no—
29C2×C22⋊Dic47—21no—
30C2×C22⋊Dic47—21no—
31C22⋊(C47⋊C8)—22no—
32C22⋊(C47⋊C8)—22no—
33C47⋊#95—22no—
34C47⋊#95—22no—
35C47⋊#95—22no—
36C47⋊#95C2×C412yes—
37C23×C4×C47D411yescharacteristic
38C23×C4×C47D411yescharacteristic
39C23×C4×C47C2311yescharacteristic
40#117×C47—42no—
41#117×C47—42no—
42C2×C4⋊C4×C47—22no—
43C2×C4⋊C4×C47—22no—
44C2×C4⋊C4×C47—22no—
45C2×C4⋊C4×C47—22no—
46C2×C4⋊C4×C47—22no—
47C2×C4⋊C4×C47C2×C412yes—
48C2×C4⋊C4×C47D412yes—
49C2×C22⋊C4×C47—22no—
50C2×C42×C47—22no—
51#95×C47—22no—
52#95×C47D414yes—
53C24×C47C2×D411yescharacteristic
54C22×C4×C47—21no—
55C22×C4×C47—21no—
56C22×C4×C47—21no—
57C22×C4×C47—21no—
58C22×C4×C47—21no—
59C22×C4×C47—21no—
60C22×C4×C47—24no—
61C22×C4×C47—81no—
62C22×C4×C47—81no—
63C22×C4×C47C22⋊C412yes—
64C22×C4×C47C22⋊C412yes—
65C22×C4×C47SD1612yes—
66C22×C4×C47Q1612yes—
67C22×C4×C47C22×C411yescharacteristic
68C22×C4×C47C2×D412yes—
69C22×C4×C47C2×D411yescharacteristic
70C22×C4×C47#4112yes—
71C22×Dic47—21no—
72C22×Dic47—22no—
73C22×Dic47—21no—
74C2×Dic94—22no—
75C2×Dic94—81no—
76C4⋊C4×C47—24no—
77C4⋊C4×C47—24no—
78C4⋊C4×C47—44no—
79C4⋊C4×C47—82no—
80C4⋊C4×C47—82no—
81C22⋊C4×C47—42no—
82C22⋊C4×C47—42no—
83C42×C47—44no—
84C22⋊Dic47—41no—
85C22⋊Dic47—41no—
86C22⋊Dic47—41no—
87C22⋊Dic47—41no—
88C4⋊Dic47—22no—
89C4⋊Dic47—81no—
90Dic47⋊C4—42no—
91Dic47⋊C4—82no—
92C2×C47⋊C8—42no—
93C2×C47⋊C8—82no—
94C23×C47—41no—
95C23×C47—41no—
96C23×C47C2×C22⋊C411yescharacteristic
97C23×C47#12011yescharacteristic
98C23×C47#12311yescharacteristic
99C23×C47C2×SD1611yescharacteristic
100C23×C47C2×Q1611yescharacteristic
101C23×C47AGL(1,ℤ/8)11yescharacteristic
102C23×C47#13711yescharacteristic
103C2×C4×C47—22no—
104C2×C4×C47—22no—
105C2×C4×C47—22no—
106C2×C4×C47—22no—
107C2×C4×C47—22no—
108C2×C4×C47—22no—
109C2×C4×C47—22no—
110C2×C4×C47—42no—
111C2×C4×C47—41no—
112C2×C4×C47—42no—
113C2×C4×C47—41no—
114C2×C4×C47—42no—
115C2×C4×C47—42no—
116C2×C4×C47—44no—
117C2×C4×C47—44no—
118C2×C4×C47—44no—
119C2×C4×C47—81no—
120C2×C4×C47#10314yes—
121C2×C4×C47C4×D412yes—
122C2×C4×C47#12112yes—
123C2×Dic47—42no—
124C2×Dic47—42no—
125C2×Dic47—42no—
126C2×Dic47—42no—
127C2×Dic47—41no—
128C2×Dic47—41no—
129C2×Dic47—42no—
130Dic94—82no—
131C47⋊C8—82no—
132#9200—471nomaximal, Sylow
133C22×C47—21no—
134C22×C47—21no—
135C22×C47—21no—
136C22×C47—21no—
137C22×C47—21no—
138C22×C47—21no—
139C22×C47—41no—
140C22×C47—41no—
141C22×C47—41no—
142C22×C47—41no—
143C22×C47#38611yescharacteristic
144C22×C47C2×#10311yescharacteristic
145C22×C47#41711yescharacteristic
146C22×C47#45011yescharacteristic
147C22×C47#45111yescharacteristic
148C22×C47#48311yescharacteristic
149C22×C47#48411yescharacteristic
150C4×C47—24no—
151C4×C47—22no—
152C4×C47—22no—
153C4×C47—42no—
154C4×C47—84no—
155Dic47—82no—
156Dic47—81no—
157Dic47—81no—
158#2627—471no—
159#2316D4711yescharacteristic
160#1520—472no—
161C2×#324—471no—
162#1289—472no—
163C2×C47—21no—
164C2×C47—21no—
165C2×C47—21no—
166C2×C47—21no—
167C2×C47—21no—
168C2×C47—21no—
169C2×C47#187211yescharacteristic
170C2×C47#191211yescharacteristic
171C2×C47#191311yescharacteristic
172C2×C22⋊Q8—471no—
173C2×#117—21no—
174C22×C4⋊C4C2×D4711yescharacteristic
175C2×C22⋊C8—472no—
176#400—472no—
177#398—942no—
178#395—472no—
179#389—24no—
180#385—941no—
181#381—941no—
182#380—21no—
183#380—471no—
184#380C2×D4712yes—
185#328—944no—
186#324—942no—
187C47#920011yescharacteristic, complemented, Sylow
188C22×Q8—471no—
189C23×C4C22⋊D4711yescharacteristic
190C23×C4C22⋊D4711yescharacteristic
191C23×C4C22×D4711yescharacteristic
192C22×C8—942no—
193C22⋊Q8—942no—
194C22⋊Q8—942no—
195#117—42no—
196#117—42no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—471no—
203C2×C4⋊C4—942no—
204C2×C4⋊C4—941no—
205C2×C4⋊C4—941no—
206C2×C4⋊C4C4×D4712yes—
207C2×C4⋊C4D18812yes—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—941no—
210C2×C22⋊C4—941no—
211C2×C42—22no—
212C22⋊C8—942no—
213C22⋊C8—942no—
214#95—22no—
215#95—472no—
216#95—942no—
217#95—942no—
218#95—942no—
219#95C22⋊D4714yes—
220C24C2×C22⋊D4711yescharacteristic
221C2×Q8—942no—
222C2×Q8—3761no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—24no—
230C22×C4—81no—
231C22×C4—81no—
232C22×C4—942no—
233C22×C4—941no—
234C22×C4—941no—
235C22×C4(C2×D47)⋊C412yes—
236C22×C4(C2×D47)⋊C412yes—
237C22×C4#1062533512yes—
238C22×C4#1062533712yes—
239C22×C4C2×C4×D4711yescharacteristic
240C22×C4C2×D18811yescharacteristic
241C22×C4#1062535012yes—
242C22×C4C2×C22⋊D4712yes—
243C2×C8—3762no—
244C2×C8—1882no—
245C4⋊C4—24no—
246C4⋊C4—24no—
247C4⋊C4—44no—
248C4⋊C4—82no—
249C4⋊C4—82no—
250C4⋊C4—942no—
251C4⋊C4—3762no—
252C4⋊C4—3761no—
253C4⋊C4—1882no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C22⋊C4—1881no—
257C22⋊C4—1881no—
258C22⋊C4—1881no—
259C22⋊C4—1881no—
260C42—44no—
261C23—41no—
262C23—41no—
263C23C2×(C2×D47)⋊C411yescharacteristic
264C23#1309399311yescharacteristic
265C23#1309400711yescharacteristic
266C23C2×#1062533511yescharacteristic
267C23C2×#1062533711yescharacteristic
268C23#1309405011yescharacteristic
269C23#1309405511yescharacteristic
270Q8—3762no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—81no—
288C2×C4—1882no—
289C2×C4—1882no—
290C2×C4—1882no—
291C2×C4—1882no—
292C2×C4—1881no—
293C2×C4—1881no—
294C2×C4—1882no—
295C2×C4#1309394714yes—
296C2×C4C4×C22⋊D4712yes—
297C2×C4#1309399812yes—
298C8—3762no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22#1568900211yescharacteristic
310C22C2×#1309394711yescharacteristic
311C22#1568912211yescharacteristic
312C22#1568927411yescharacteristic
313C22#1568927911yescharacteristic
314C22#1568945311yescharacteristic
315C22#1568945811yescharacteristic
316C4—22no—
317C4—22no—
318C4—24no—
319C4—42no—
320C4—84no—
321C4—3762no—
322C4—3761no—
323C4—3761no—
324C2—21no—
325C2—21no—
326C2—21no—
327C2—21no—
328C2—21no—
329C2—21no—
330C2#2024369511yescharacteristic
331C2#2024392511yescharacteristic
332C2#2024393011yescharacteristic
subgroupG/H
centerC22#15689002
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2316C2
socleC22×C47#386
prime radicalC24×C47C2×D4
cosocleC23×C4×C47C23
hypercenter#2316D47
hypocenterC47#9200
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2316

Complements

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

NH ≅ G/Ntype
8. #15689062C4semidirect
10. #15689043C4semidirect
187. C47#9200semidirect

Maximal subgroups

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

subgroupindex
#202492012
#202419592
C2×#156888492
#202412552
C47×#23162
#920047