Finite Group Databasebeta

← #26191178

Subgroups — #26191178

Order 12032.

906 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 906 conjugacy classes of subgroups — too many to draw. Only its 207 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#20254529C212yesmaximal
2#20248373C211yesmaximal, characteristic
3#20242396C218yesmaximal
4#20241887C212yesmaximal
5#20241712C211yesmaximal, characteristic
6C47×#3502C211yesmaximal, characteristic
7C2×#13094024C2211yescharacteristic
8C2×#13094022—21no—
9#15689519—22no—
10#15689473—22no—
11#15689451C2218yes—
12#15689408—24no—
13#15689176C2212yes—
14#15689053C2214yes—
15#15689028—21no—
16#15689021—22no—
17C4×C4⋊Dic47C2211yescharacteristic
18#15688860C2218yes—
19#15688849C2214yes—
20Q82×C47—22no—
21C47×#557—22no—
22C47×#557—24no—
23C47×#544—21no—
24C47×#544C2214yes—
25C47×#533—22no—
26C47×#533C2212yes—
27C47×#519—21no—
28C2×C4×Q8×C47C2211yescharacteristic
29C22×Dic94—21no—
30#13094024—22no—
31#13094024—22no—
32#13094022—44no—
33C2×C4⋊Dic47—21no—
34C2×C4⋊Dic47—21no—
35C2×C4⋊Dic47C2312yes—
36C2×Dic47⋊C4—24no—
37C2×C4×Dic47—21no—
38#13093955—28no—
39C4⋊(C47⋊C8)—22no—
40#13093948—28no—
41C22⋊(C47⋊C8)C2314yes—
42C4×C47⋊C8—22no—
43C47⋊#95—24no—
44C47⋊#95C2312yes—
45C22×Q8×C47D411yescharacteristic
46C4⋊Q8×C47—22no—
47C4⋊Q8×C47—22no—
48C4⋊Q8×C47—24no—
49C4⋊Q8×C47—28no—
50C4⋊Q8×C47D412yes—
51C4⋊Q8×C47D412yes—
52#125×C47—24no—
53#125×C47—24no—
54#125×C47—28no—
55#125×C47—22no—
56C22⋊Q8×C47—22no—
57C22⋊Q8×C47—24no—
58C22⋊Q8×C47C2314yes—
59C4×Q8×C47—24no—
60C4×Q8×C47—24no—
61C4×Q8×C47—24no—
62C4×Q8×C47—24no—
63C4×Q8×C47—22no—
64C4×Q8×C47—22no—
65C4×Q8×C47—22no—
66C4×Q8×C47D414yes—
67#117×C47—24no—
68#117×C47—22no—
69C2×C4⋊C4×C47D411yescharacteristic
70C2×C4⋊C4×C47C2312yes—
71C2×C42×C47D412yes—
72C2×C42×C47C2311yescharacteristic
73C2×Q8×C47—24no—
74C2×Q8×C47—22no—
75C2×Q8×C47—22no—
76C2×Q8×C47—22no—
77C2×Q8×C47SD1614yes—
78C2×Q8×C47C2×D414yes—
79C22×C4×C47C2×D411yescharacteristic
80C22×C4×C47C2×D412yes—
81C22×C4×C47C2×D412yes—
82C22×C4×C47C2×D411yescharacteristic
83C22×C4×C47C2411yescharacteristic
84C22×Dic47—22no—
85C22×Dic47—22no—
86C2×Dic94—42no—
87C2×Dic94—42no—
88C2×Dic94—42no—
89C4⋊C4×C47—24no—
90C4⋊C4×C47—24no—
91C4⋊C4×C47—24no—
92C4⋊C4×C47—28no—
93C4⋊C4×C47—22no—
94C4⋊C4×C47—22no—
95C4⋊C4×C47—22no—
96C4⋊C4×C47—22no—
97C4⋊C4×C47—24no—
98C4⋊C4×C47—24no—
99C4⋊C4×C47—24no—
100C4⋊C4×C47C2×D414yes—
101C22⋊C4×C47—24no—
102C42×C47—28no—
103C42×C47—22no—
104C42×C47—22no—
105C42×C47—24no—
106C42×C47C2×D412yes—
107C42×C47C2×D412yes—
108C4⋊Dic47—24no—
109C4⋊Dic47—24no—
110C4⋊Dic47—41no—
111C4⋊Dic47—41no—
112Dic47⋊C4—48no—
113C4×Dic47—42no—
114C2×C47⋊C8—24no—
115C23×C47C22×D411yescharacteristic
116Q8×C47—28no—
117Q8×C47—44no—
118Q8×C47—44no—
119Q8×C47—84no—
120C2×C4×C47—22no—
121C2×C4×C47—22no—
122C2×C4×C47—22no—
123C2×C4×C47—24no—
124C2×C4×C47—24no—
125C2×C4×C47—22no—
126C2×C4×C47—41no—
127C2×C4×C47—42no—
128C2×C4×C47#12011yescharacteristic
129C2×C4×C47#12012yes—
130C2×C4×C47#12011yescharacteristic
131C2×C4×C47C2×SD1614yes—
132C2×C4×C47C2×SD1612yes—
133C2×C4×C47#13712yes—
134C2×C4×C47C22×D412yes—
135C2×Dic47—24no—
136C2×Dic47—42no—
137C2×Dic47—42no—
138C2×Dic47—42no—
139C2×Dic47—42no—
140C2×Dic47—42no—
141Dic94—84no—
142C47⋊C8—44no—
143#18410—471nomaximal, Sylow
144C22×C47—21no—
145C22×C47—21no—
146C22×C47C2×#12011yescharacteristic
147C22×C47C22×SD1611yescharacteristic
148C22×C47C2×#13711yescharacteristic
149C4×C47—24no—
150C4×C47—21no—
151C4×C47—22no—
152C4×C47—44no—
153C4×C47—42no—
154C4×C47—44no—
155C4×C47#44814yes—
156C4×C47#45612yes—
157Dic47—44no—
158Dic47—82no—
159#3502D4711yescharacteristic
160#3229—472no—
161#2546—471no—
162#1611—478no—
163#1503—472no—
164#1461—471no—
165C2×C47—21no—
166C2×C47—21no—
167C2×C47C2×#44811yescharacteristic
168C2×C47C2×#45611yescharacteristic
169C2×C47#302811yescharacteristic
170Q82—22no—
171#557—22no—
172#557—24no—
173#544—21no—
174#544C2×D4714yes—
175#533—22no—
176#533C2×D4712yes—
177C2×C4⋊Q8—471no—
178C2×C4⋊Q8—941no—
179#519—21no—
180C2×C4×Q8C2×D4711yescharacteristic
181#498—942no—
182#487—942no—
183#483—478no—
184#475—944no—
185#431—472no—
186#398—474no—
187#391—941no—
188#389—942no—
189C4×C4⋊C4—471no—
190#328—478no—
191#324—474no—
192C47#1841011yescharacteristic, complemented, Sylow
193C22×Q8—941no—
194C22×Q8C22⋊D4711yescharacteristic
195C4⋊Q8—22no—
196C4⋊Q8—22no—
197C4⋊Q8—24no—
198C4⋊Q8—28no—
199C4⋊Q8—942no—
200C4⋊Q8—942no—
201C4⋊Q8—1884no—
202C4⋊Q8C22⋊D4712yes—
203C4⋊Q8C22⋊D4712yes—
204#125—24no—
205#125—24no—
206#125—28no—
207#125—22no—
208C22⋊Q8—22no—
209C22⋊Q8—24no—
210C22⋊Q8C22×D4714yes—
211C4×Q8—24no—
212C4×Q8—24no—
213C4×Q8—24no—
214C4×Q8—24no—
215C4×Q8—22no—
216C4×Q8—22no—
217C4×Q8—22no—
218C4×Q8C22⋊D4714yes—
219#117—24no—
220#117—22no—
221C2×C4⋊C4—472no—
222C2×C4⋊C4—941no—
223C2×C4⋊C4—941no—
224C2×C4⋊C4—944no—
225C2×C4⋊C4C22⋊D4711yescharacteristic
226C2×C4⋊C4C22×D4712yes—
227C2×C42—941no—
228C2×C42C22⋊D4712yes—
229C2×C42C22×D4711yescharacteristic
230#106—948no—
231C4⋊C8—942no—
232#103—948no—
233C22⋊C8—474no—
234C4×C8—942no—
235#95—472no—
236#95—944no—
237C2×Q8—24no—
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—1882no—
242C2×Q8—1882no—
243C2×Q8—1882no—
244C2×Q8#1062533614yes—
245C2×Q8C2×C22⋊D4714yes—
246C22×C4—942no—
247C22×C4—942no—
248C22×C4C2×C22⋊D4711yescharacteristic
249C22×C4C2×C22⋊D4712yes—
250C22×C4C2×C22⋊D4712yes—
251C22×C4C2×C22⋊D4711yescharacteristic
252C22×C4C23×D4711yescharacteristic
253C2×C8—944no—
254C4⋊C4—24no—
255C4⋊C4—24no—
256C4⋊C4—24no—
257C4⋊C4—28no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—944no—
266C4⋊C4—944no—
267C4⋊C4—1881no—
268C4⋊C4—1881no—
269C4⋊C4—1888no—
270C4⋊C4C2×C22⋊D4714yes—
271C22⋊C4—24no—
272C42—28no—
273C42—22no—
274C42—22no—
275C42—24no—
276C42—1882no—
277C42C2×C22⋊D4712yes—
278C42C2×C22⋊D4712yes—
279C23C22×C22⋊D4711yescharacteristic
280Q8—28no—
281Q8—44no—
282Q8—44no—
283Q8—84no—
284Q8—3764no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—24no—
289C2×C4—24no—
290C2×C4—22no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—944no—
294C2×C4—1882no—
295C2×C4—1882no—
296C2×C4—1882no—
297C2×C4—1882no—
298C2×C4—1882no—
299C2×C4#1309399311yescharacteristic
300C2×C4#1309399312yes—
301C2×C4#1309399311yescharacteristic
302C2×C4C2×#1062533614yes—
303C2×C4C2×#1062533612yes—
304C2×C4#1309405512yes—
305C2×C4C22×C22⋊D4712yes—
306C8—1884no—
307C22—21no—
308C22—21no—
309C22C2×#1309399311yescharacteristic
310C22C22×#1062533611yescharacteristic
311C22C2×#1309405511yescharacteristic
312C4—21no—
313C4—22no—
314C4—24no—
315C4—44no—
316C4—42no—
317C4—44no—
318C4—3762no—
319C4—1884no—
320C4#1568926214yes—
321C4#1568930612yes—
322C2—21no—
323C2—21no—
324C2C2×#1568926211yescharacteristic
325C2C2×#1568930611yescharacteristic
326C2#2025279411yescharacteristic
subgroupG/H
centerC22C2×#13093993
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#3502C2
socleC22×C47C2×#120
prime radicalC23×C47C22×D4
hypercenter#3502D47
hypocenterC47#18410
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3502

Complements

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

NH ≅ G/Ntype
192. C47#18410semidirect

Maximal subgroups

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

subgroupindex
#202545292
#202483732
#202423962
#202418872
#202417122
C47×#35022
#1841047