Finite Group Databasebeta

← #26133116

Subgroups — #26133116

Order 12032.

460 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 95 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#20246478C211yesmaximal, characteristic
2C2×#15688922C211yesmaximal, characteristic
3#20243603C212yesmaximal
4#20241514C212yesmaximal
5C47×#1886C211yesmaximal, characteristic
6#15689166—22no—
7C2×#13093943C2212yes—
8#15689030—21no—
9#15689030C2212yes—
10#15689022—22no—
11C4⋊C4×Dic47—22no—
12C4×C4⋊Dic47—21no—
13#15688922C414yescomplemented
14#15688882—22no—
15C2×C4⋊Q8×C47C2211yescharacteristic
16C47×#519C2211yescharacteristic
17C47×#419—21no—
18C2×#101×C47C2211yescharacteristic
19C47×#339—21no—
20C2×#10625329—22no—
21C2×C4⋊Dic47—21no—
22C2×C4⋊Dic47—21no—
23C2×C4⋊Dic47—21no—
24C2×C4⋊Dic47D412yes—
25C2×C4×Dic47—22no—
26C2×C4×Dic47—21no—
27C2×C4×Dic47D412yes—
28#13093957—42no—
29#13093955—42no—
30#13093943—22no—
31#13093943C2×C414yes—
32C47⋊#95—22no—
33C47⋊#95—22no—
34C47⋊#95—22no—
35C22×Q8×C47D411yescharacteristic
36C22×Q8×C47C2311yescharacteristic
37C2×M16×C47—21no—
38C4⋊Q8×C47—22no—
39C4⋊Q8×C47C2×C412yes—
40C4⋊Q8×C47Q812yes—
41C4×Q8×C47—41no—
42C4×Q8×C47—41no—
43#117×C47—21no—
44#117×C47—22no—
45C2×C4⋊C4×C47—22no—
46C2×C42×C47D411yescharacteristic
47#103×C47—41no—
48#103×C47—41no—
49#101×C47—22no—
50C2×Q8×C47—21no—
51C2×Q8×C47—41no—
52C2×Q8×C47—41no—
53C2×Q8×C47—41no—
54C2×Q8×C47—41no—
55C2×Q8×C47C4⋊C412yes—
56C2×Q8×C47C22×C411yescharacteristic
57C2×Q8×C47C2×Q811yescharacteristic
58C2×Q8×C47#4112yes—
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47C2×D411yescharacteristic
62C22×C4×C47C2×D412yes—
63C22×Dic47—22no—
64C22×Dic47—22no—
65M16×C47—41no—
66M16×C47—41no—
67C2×C8×C47—41no—
68C4⋊C4×C47—24no—
69C4⋊C4×C47—24no—
70C4⋊C4×C47—21no—
71C4⋊C4×C47—21no—
72C4⋊C4×C47—22no—
73C4⋊C4×C47—22no—
74C22⋊C4×C47—41no—
75C42×C47—21no—
76C42×C47—42no—
77C42×C47—41no—
78C42×C47C4⋊C412yes—
79C4⋊Dic47—24no—
80C4⋊Dic47—42no—
81C4⋊Dic47—42no—
82C4⋊Dic47—81no—
83C4×Dic47—44no—
84C4×Dic47—42no—
85C4×Dic47—81no—
86#10625329—24no—
87#10625329—42no—
88C2×C47⋊C8—42no—
89C23×C47#12011yescharacteristic
90Q8×C47—41no—
91Q8×C47—41no—
92Q8×C47—42no—
93Q8×C47—42no—
94Q8×C47—81no—
95C2×C4×C47—22no—
96C2×C4×C47—22no—
97C2×C4×C47—22no—
98C2×C4×C47—22no—
99C2×C4×C47—21no—
100C2×C4×C47—22no—
101C2×C4×C47—21no—
102C2×C4×C47—22no—
103C2×C4×C47—41no—
104C2×C4×C47—41no—
105C2×C4×C47—41no—
106C2×C4×C47—41no—
107C2×C4×C47C2×C4⋊C411yescharacteristic
108C2×C4×C47C4×D412yes—
109C2×C4×C47C22⋊Q812yes—
110C2×C4×C47#12311yescharacteristic
111C2×Dic47—24no—
112C2×Dic47—42no—
113C2×Dic47—42no—
114C2×Dic47—42no—
115C2×Dic47—42no—
116C2×Dic47—42no—
117C8×C47—81no—
118C47⋊C8—44no—
119#12973—471nomaximal, Sylow
120C22×C47—21no—
121C22×C47—21no—
122C22×C47#38511yescharacteristic
123C22×C47B(4,𝔽2)11yescharacteristic
124C22×C47#45811yescharacteristic
125C4×C47—24no—
126C4×C47—22no—
127C4×C47—21no—
128C4×C47—21no—
129C4×C47—22no—
130C4×C47—42no—
131C4×C47—81no—
132C4×C47—81no—
133Dic47—44no—
134Dic47—82no—
135#2369—471no—
136C2×#356—471no—
137#1886D4711yescharacteristic
138#1857—472no—
139#1394—472no—
140C2×C47—21no—
141C2×C47—21no—
142C2×C47#186711yescharacteristic
143C2×C47#220411yescharacteristic
144C2×C47#220511yescharacteristic
145C2×C4⋊Q8C2×D4711yescharacteristic
146#519C2×D4711yescharacteristic
147#428—942no—
148#419—21no—
149C2×#101—472no—
150C2×#101C2×D4711yescharacteristic
151#391—472no—
152#391—941no—
153#389—942no—
154C4×C4⋊C4—942no—
155C4×C4⋊C4—941no—
156#356—474no—
157#339—21no—
158#339—942no—
159C47#1297311yescharacteristic, complemented, Sylow
160C22×Q8C22⋊D4711yescharacteristic
161C22×Q8C22×D4711yescharacteristic
162C2×M16—21no—
163C2×M16—942no—
164C4⋊Q8—22no—
165C4⋊Q8Dic9412yes—
166C4⋊Q8C4×D4712yes—
167C4×Q8—41no—
168C4×Q8—41no—
169#117—22no—
170#117—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—472no—
173C2×C4⋊C4—941no—
174C2×C4⋊C4—941no—
175C2×C4⋊C4—941no—
176C2×C42—472no—
177C2×C42—941no—
178C2×C42—942no—
179C2×C42C22⋊D4711yescharacteristic
180#107—1882no—
181#106—1882no—
182#103—41no—
183#103—41no—
184#101—22no—
185#101—474no—
186#101—942no—
187#95—942no—
188#95—942no—
189#95—942no—
190C2×Q8—21no—
191C2×Q8—41no—
192C2×Q8—41no—
193C2×Q8—41no—
194C2×Q8—41no—
195C2×Q8Dic47⋊C412yes—
196C2×Q8C2×Dic9411yescharacteristic
197C2×Q8C2×C4×D4711yescharacteristic
198C2×Q8#1062535212yes—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—942no—
202C22×C4—942no—
203C22×C4D4×D4712yes—
204C22×C4C2×C22⋊D4711yescharacteristic
205M16—41no—
206M16—41no—
207M16—944no—
208M16—1882no—
209C2×C8—41no—
210C2×C8—1882no—
211C4⋊C4—24no—
212C4⋊C4—24no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—22no—
216C4⋊C4—22no—
217C4⋊C4—944no—
218C4⋊C4—3761no—
219C4⋊C4—1882no—
220C4⋊C4—1882no—
221C22⋊C4—41no—
222C42—21no—
223C42—42no—
224C42—41no—
225C42—3761no—
226C42—1882no—
227C42—1884no—
228C42Dic47⋊C412yes—
229C23#1309399211yescharacteristic
230Q8—41no—
231Q8—41no—
232Q8—42no—
233Q8—42no—
234Q8—81no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—944no—
248C2×C4—1882no—
249C2×C4—1882no—
250C2×C4—1882no—
251C2×C4—1882no—
252C2×C4—1882no—
253C2×C4C2×Dic47⋊C411yescharacteristic
254C2×C4(C22⋊D47)⋊C412yes—
255C2×C4#1309400112yes—
256C2×C4#1309400511yescharacteristic
257C8—81no—
258C8—1884no—
259C22—21no—
260C22—21no—
261C22#1568899511yescharacteristic
262C22#1568930911yescharacteristic
263C22#1568931211yescharacteristic
264C4—24no—
265C4—21no—
266C4—22no—
267C4—21no—
268C4—22no—
269C4—42no—
270C4—81no—
271C4—81no—
272C4—3762no—
273C4—1884no—
274C2—21no—
275C2—21no—
276C2#2024366811yescharacteristic
277C2#2024541311yescharacteristic
278C2#2024541711yescharacteristic
subgroupG/H
centerC22#15689309
commutatorC2×Q8×C47C22×C4
FrattiniC22×Q8C22×D47
FittingC47×#1886C2
socleC22×C47B(4,𝔽2)
prime radicalC23×C47#120
cosocleC22×Q8×C47C23
hypercenter#1886D47
hypocenterC47#12973
LCSG ▷ C2×Q8×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1886

Complements

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

NH ≅ G/Ntype
13. #15688922C4semidirect
159. C47#12973semidirect

Maximal subgroups

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

subgroupindex
#202464782
C2×#156889222
#202436032
#202415142
C47×#18862
#1297347