Finite Group Databasebeta

← #26119000

Subgroups — #26119000

Order 12032.

504 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 109 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#20254536C211yesmaximal, characteristic
2#20251765C211yesmaximal, characteristic
3#20244366C211yesmaximal, characteristic
4#20242073C211yesmaximal, characteristic
5#20242072C211yesmaximal, characteristic
6#20241220C211yesmaximal, characteristic
7C47×#3230C211yesmaximal, characteristic
8C2×#13094024C2211yescharacteristic
9#15689526—21no—
10#15689473—21no—
11#15689456—22no—
12#15689418—22no—
13#15689176C2211yescharacteristic
14#15689147C2211yescharacteristic
15#15689030—22no—
16#15689028—22no—
17#15688988—22no—
18C42×Dic47C2211yescharacteristic
19#15688882—22no—
20#15688870—22no—
21#15688869—22no—
22C47×#533C2211yescharacteristic
23C2×C4⋊Q8×C47C2211yescharacteristic
24C47×#500—21no—
25C47×#487—21no—
26C47×#484—22no—
27C47×#477—22no—
28C47×#431C2211yescharacteristic
29C22×Dic94—21no—
30C2×#10625329—21no—
31#13094024—21no—
32#13094024—21no—
33#13094024—21no—
34#13094024—21no—
35C2×C4⋊Dic47—21no—
36C2×C4⋊Dic47D412yes—
37C2×C4×Dic47—21no—
38C2×C4×Dic47—21no—
39C2×C4×Dic47D412yes—
40#13093957—24no—
41C4⋊(C47⋊C8)—21no—
42C4⋊(C47⋊C8)—21no—
43C4⋊(C47⋊C8)D412yes—
44#13093948—24no—
45C22⋊(C47⋊C8)—21no—
46C4×C47⋊C8—21no—
47C47⋊#95—28no—
48C22×Q8×C47—21no—
49C4⋊Q8×C47—21no—
50C4⋊Q8×C47—21no—
51C4⋊Q8×C47—21no—
52C4⋊Q8×C47—21no—
53C4⋊Q8×C47D411yescharacteristic
54C4⋊Q8×C47D411yescharacteristic
55#125×C47—21no—
56C22⋊Q8×C47—21no—
57C22⋊Q8×C47—21no—
58C4×Q8×C47—21no—
59C4×Q8×C47—21no—
60#117×C47—21no—
61C2×C4⋊C4×C47—21no—
62C2×C4⋊C4×C47D412yes—
63C2×C42×C47C2311yescharacteristic
64#107×C47—24no—
65C4⋊C8×C47D412yes—
66#103×C47—24no—
67C22⋊C8×C47—21no—
68C4×C8×C47—21no—
69C2×Q8×C47—21no—
70C2×Q8×C47—21no—
71C2×Q8×C47—44no—
72C2×Q8×C47—41no—
73C2×Q8×C47—41no—
74C22×C4×C47—22no—
75C22×C4×C47C2×D411yescharacteristic
76C22×C4×C47C2×D411yescharacteristic
77C22×C4×C47C2×D411yescharacteristic
78C22×Dic47—22no—
79C22×Dic47—22no—
80C2×Dic94—44no—
81C2×Dic94—41no—
82C2×Dic94—41no—
83C2×C8×C47—22no—
84C4⋊C4×C47—22no—
85C4⋊C4×C47—21no—
86C4⋊C4×C47—21no—
87C4⋊C4×C47—22no—
88C4⋊C4×C47—21no—
89C4⋊C4×C47—21no—
90C4⋊C4×C47—41no—
91C4⋊C4×C47—41no—
92C4⋊C4×C47—41no—
93C4⋊C4×C47—41no—
94C22⋊C4×C47—41no—
95C42×C47—41no—
96C42×C47C2×D411yescharacteristic
97C42×C47C2×D411yescharacteristic
98C42×C47C2×D411yescharacteristic
99C42×C47#4111yescharacteristic
100C4⋊Dic47—22no—
101C4⋊Dic47—22no—
102C4⋊Dic47—41no—
103C4⋊Dic47—41no—
104C4×Dic47—22no—
105C4×Dic47—42no—
106C4×Dic47—42no—
107C4×Dic47Q1614yes—
108#10625329—42no—
109C2×C47⋊C8—22no—
110C2×C47⋊C8—41no—
111C23×C47#12011yescharacteristic
112Q8×C47—42no—
113Q8×C47—42no—
114Q8×C47—82no—
115Q8×C47—82no—
116C2×C4×C47—22no—
117C2×C4×C47—22no—
118C2×C4×C47—44no—
119C2×C4×C47—41no—
120C2×C4×C47—41no—
121C2×C4×C47—42no—
122C2×C4×C47#12011yescharacteristic
123C2×C4×C47#12011yescharacteristic
124C2×C4×C47#12111yescharacteristic
125C2×C4×C47#12111yescharacteristic
126C2×C4×C47#12111yescharacteristic
127C2×C4×C47#12711yescharacteristic
128C2×C4×C47C2×Q1612yes—
129C2×C4×C47#13712yes—
130C2×Dic47—24no—
131C2×Dic47—42no—
132C2×Dic47—42no—
133C2×Dic47—44no—
134C2×Dic47—42no—
135Dic94—82no—
136Dic94—82no—
137C8×C47—42no—
138C47⋊C8—42no—
139C47⋊C8—81no—
140#10295—471nomaximal, Sylow
141C22×C47—21no—
142C22×C47—21no—
143C22×C47#39211yescharacteristic
144C22×C47#45112yes—
145C4×C47—21no—
146C4×C47—21no—
147C4×C47—82no—
148C4×C47—81no—
149C4×C47—81no—
150C4×C47#45412yes—
151C4×C47#45412yes—
152C4×C47#46212yes—
153C4×C47#49411yescharacteristic
154C4×C47#49711yescharacteristic
155Dic47—44no—
156Dic47—82no—
157#3230—471no—
158#3230D4711yescharacteristic
159#2862—471no—
160#1994—471no—
161#1549—471no—
162#1549—471no—
163#1276—471no—
164C2×C47—22no—
165C2×C47#200311yescharacteristic
166C2×C47#200311yescharacteristic
167C2×C47#201611yescharacteristic
168#533C2×D4711yescharacteristic
169C2×C4⋊Q8—471no—
170C2×C4⋊Q8C2×D4711yescharacteristic
171#500—21no—
172#500—941no—
173#487—21no—
174#487—941no—
175#484—22no—
176#484—942no—
177#477—22no—
178#477—942no—
179#431—471no—
180#431C2×D4711yescharacteristic
181#423—471no—
182#391—942no—
183#391—942no—
184#382—942no—
185C43—471no—
186#339—942no—
187#332—942no—
188#332—942no—
189C47#1029511yescharacteristic, complemented, Sylow
190C22×Q8—21no—
191C22×Q8—941no—
192C2×M16—941no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8—941no—
198C4⋊Q8—941no—
199C4⋊Q8—941no—
200C4⋊Q8—941no—
201C4⋊Q8C22⋊D4711yescharacteristic
202C4⋊Q8C22⋊D4711yescharacteristic
203#125—21no—
204C22⋊Q8—21no—
205C22⋊Q8—21no—
206C4×Q8—21no—
207C4×Q8—21no—
208#117—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—472no—
211C2×C4⋊C4—941no—
212C2×C4⋊C4C22⋊D4712yes—
213C2×C42—472no—
214C2×C42—941no—
215C2×C42—941no—
216C2×C42C22×D4711yescharacteristic
217#107—24no—
218#107—944no—
219C4⋊C8—472no—
220C4⋊C8—941no—
221C4⋊C8—941no—
222C4⋊C8C22⋊D4712yes—
223#103—24no—
224#103—944no—
225C22⋊C8—21no—
226C22⋊C8—941no—
227C4×C8—21no—
228C4×C8—941no—
229#95—948no—
230C2×Q8—21no—
231C2×Q8—21no—
232C2×Q8—44no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—1884no—
236C2×Q8—1881no—
237C2×Q8—1881no—
238C22×C4—22no—
239C22×C4—942no—
240C22×C4—942no—
241C22×C4D4×D4711yescharacteristic
242C22×C4D4×D4711yescharacteristic
243C22×C4C2×C22⋊D4711yescharacteristic
244M16—1882no—
245C2×C8—22no—
246C2×C8—942no—
247C2×C8—1881no—
248C4⋊C4—21no—
249C4⋊C4—22no—
250C4⋊C4—21no—
251C4⋊C4—22no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—942no—
259C4⋊C4—942no—
260C4⋊C4—1881no—
261C4⋊C4—1881no—
262C22⋊C4—41no—
263C42—41no—
264C42—474no—
265C42—942no—
266C42—1882no—
267C42—1882no—
268C42D4×D4711yescharacteristic
269C42#1062535211yescharacteristic
270C42C2×C22⋊D4711yescharacteristic
271C42C2×C22⋊D4711yescharacteristic
272C23#1309399211yescharacteristic
273Q8—42no—
274Q8—42no—
275Q8—82no—
276Q8—82no—
277Q8—3762no—
278Q8—3762no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—44no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—944no—
286C2×C4—1884no—
287C2×C4—1882no—
288C2×C4—1882no—
289C2×C4—1882no—
290C2×C4#1309399211yescharacteristic
291C2×C4#1309399311yescharacteristic
292C2×C4#1309399511yescharacteristic
293C2×C4#1309399611yescharacteristic
294C2×C4#1309399611yescharacteristic
295C2×C4#1309402011yescharacteristic
296C2×C4Q16×D4712yes—
297C2×C4#1309405612yes—
298C8—42no—
299C8—3761no—
300C8—1882no—
301C22—21no—
302C22—21no—
303C22#1568903111yescharacteristic
304C22#1568927712yes—
305C4—21no—
306C4—21no—
307C4—82no—
308C4—81no—
309C4—81no—
310C4—3762no—
311C4—1884no—
312C4#1568929412yes—
313C4#1568929612yes—
314C4#1568933112yes—
315C4#1568950111yescharacteristic
316C4#1568951611yescharacteristic
317C2—22no—
318C2#2024440711yescharacteristic
319C2#2024440811yescharacteristic
320C2#2024448611yescharacteristic
subgroupG/H
centerC22#15689031
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#3230C2
socleC22×C47#392
prime radicalC23×C47#120
hypercenter#3230D47
hypocenterC47#10295
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3230

Complements

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

NH ≅ G/Ntype
189. C47#10295semidirect

Maximal subgroups

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

subgroupindex
#202545362
#202517652
#202443662
#202420732
#202420722
#202412202
C47×#32302
#1029547