Finite Group Databasebeta

← #26117602

Subgroups — #26117602

Order 12032.

600 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×#15689482C212yesmaximal, complemented
2#20244207C211yesmaximal, characteristic, complemented
3C2×#15688863C212yesmaximal, complemented
4#20241269C211yesmaximal, characteristic, complemented
5C47×#2819C211yesmaximal, characteristic
6C2×#13094024C2211yescharacteristic
7#15689482—24no—
8#15689482—24no—
9C2×#13093948—22no—
10C2×#13093947—22no—
11C2×#13093945—22no—
12C2×#13093935C2211yescharacteristic
13C2×#13093935C2212yes—
14#15688863—24no—
15C2×#124×C47C2212yes—
16C47×#394—22no—
17C47×#393—22no—
18C47×#392—22no—
19C47×#382C2211yescharacteristic
20C22×Dic94—21no—
21C22×C47⋊C8—22no—
22C22×C47⋊C8—21no—
23#13094024—22no—
24#13094024—21no—
25#13094024D412yescomplemented
26C2×C4⋊Dic47—22no—
27#13093948—28no—
28#13093947—44no—
29#13093945—28no—
30#13093935—22no—
31#13093935—22no—
32#13093935D414yescomplemented
33C22×Q8×C47D412yes—
34C22×D4×C47—22no—
35C22×D4×C47D412yes—
36#124×C47—22no—
37#124×C47—22no—
38#124×C47—22no—
39#124×C47—22no—
40C2×C4⋊C4×C47D412yes—
41C2×C22⋊C4×C47—22no—
42C2×C22⋊C4×C47—22no—
43C2×C22⋊C4×C47—24no—
44C2×C42×C47C2311yescharacteristic
45#95×C47—22no—
46#95×C47—22no—
47C24×C47—22no—
48C2×Q8×C47—22no—
49C2×Q8×C47—22no—
50C2×Q8×C47—44no—
51C2×D4×C47—22no—
52C2×D4×C47—22no—
53C2×D4×C47—44no—
54C2×D4×C47—82no—
55C2×D4×C47—84no—
56C22×C4×C47—22no—
57C22×C4×C47—22no—
58C22×C4×C47C2×D412yes—
59C22×C4×C47C2×D411yescharacteristic
60C22×Dic47—41no—
61C2×Dic94—22no—
62C2×Dic94—42no—
63C2×Dic94—42no—
64C2×Dic94—41no—
65C4⋊C4×C47—42no—
66C22⋊C4×C47—42no—
67C22⋊C4×C47—42no—
68C22⋊C4×C47—42no—
69C22⋊C4×C47—42no—
70C22⋊C4×C47—84no—
71C42×C47C2×D411yescharacteristic
72C42×C47C2×D412yes—
73C42×C47#4111yescharacteristic
74C4⋊Dic47—24no—
75C4⋊Dic47—42no—
76C2×C47⋊C8—22no—
77C2×C47⋊C8—24no—
78C2×C47⋊C8—41no—
79C2×C47⋊C8—44no—
80C2×C47⋊C8—41no—
81C23×C47—42no—
82C23×C47—42no—
83C23×C47—42no—
84C23×C47—44no—
85C23×C47—44no—
86C23×C47#12011yescharacteristic
87Q8×C47—44no—
88Q8×C47—82no—
89D4×C47—44no—
90D4×C47—82no—
91D4×C47—88no—
92C2×C4×C47—21no—
93C2×C4×C47—21no—
94C2×C4×C47—24no—
95C2×C4×C47—42no—
96C2×C4×C47—42no—
97C2×C4×C47—44no—
98C2×C4×C47—44no—
99C2×C4×C47#12012yes—
100C2×C4×C47#12111yescharacteristic
101C2×C4×C47#12112yes—
102C2×C4×C47#12711yescharacteristic
103C2×Dic47—42no—
104C2×Dic47—81no—
105C2×Dic47—81no—
106Dic94—44no—
107Dic94—82no—
108C47⋊C8—44no—
109C47⋊C8—42no—
110#10085—471nomaximal, Sylow
111C22×C47—84no—
112C22×C47—84no—
113C22×C47—82no—
114C22×C47—82no—
115C22×C47—82no—
116C22×C47#39211yescharacteristic
117C22×C47#45311yescharacteristic
118C22×C47#45411yescharacteristic
119C22×C47#45514yes—
120C4×C47—22no—
121C4×C47—22no—
122C4×C47—21no—
123C4×C47—21no—
124C4×C47—82no—
125C4×C47—82no—
126Dic47—82no—
127C2×#489—472no—
128#2819D4711yescharacteristic
129#1960—471no—
130C2×#329—472no—
131#1295—471no—
132C2×C47—162no—
133C2×C47#165912yes—
134C2×C47#166212yes—
135C2×C47#200412yes—
136C2×C47#200611yescharacteristic
137C2×C4⋊Q8—471no—
138C2×#124C2×D4712yes—
139#489—944no—
140#489—944no—
141C2×#103—942no—
142C2×#103—942no—
143C2×#102—942no—
144C2×#97—472no—
145C2×#97—471no—
146#394—22no—
147#393—22no—
148#392—22no—
149#382C2×D4711yescharacteristic
150#329—944no—
151C47#1008511yescharacteristic, complemented, Sylow
152C22×Q8—941no—
153C22×Q8C22⋊D4712yes—
154C22×D4—22no—
155C22×D4C22⋊D4712yes—
156C22×C8—942no—
157C22×C8—941no—
158C4⋊Q8—472no—
159C4⋊Q8—942no—
160C4⋊Q8—941no—
161#124—22no—
162#124—22no—
163#124—22no—
164#124—22no—
165C2×C4⋊C4—942no—
166C2×C4⋊C4C22⋊D4712yes—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—24no—
170C2×C42C22×D4711yescharacteristic
171#103—948no—
172#103—1884no—
173#102—948no—
174#97—474no—
175#97—942no—
176#97—942no—
177#95—22no—
178#95—22no—
179C24—22no—
180C2×Q8—22no—
181C2×Q8—22no—
182C2×Q8—44no—
183C2×Q8—942no—
184C2×Q8—1882no—
185C2×Q8—1882no—
186C2×Q8—1881no—
187C2×D4—22no—
188C2×D4—22no—
189C2×D4—44no—
190C2×D4—82no—
191C2×D4—84no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—1881no—
195C22×C4C2×C22⋊D4712yes—
196C22×C4C2×C22⋊D4711yescharacteristic
197C2×C8—942no—
198C2×C8—944no—
199C2×C8—1881no—
200C2×C8—1884no—
201C2×C8—1881no—
202C4⋊C4—42no—
203C4⋊C4—944no—
204C4⋊C4—1882no—
205C22⋊C4—42no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C22⋊C4—84no—
210C42D4×D4711yescharacteristic
211C42D4×D4712yes—
212C42#1062535211yescharacteristic
213C23—42no—
214C23—42no—
215C23—42no—
216C23—44no—
217C23—44no—
218C23#1309399311yescharacteristic
219Q8—44no—
220Q8—82no—
221Q8—3762no—
222Q8—1884no—
223D4—44no—
224D4—82no—
225D4—88no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—24no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—44no—
232C2×C4—44no—
233C2×C4—3761no—
234C2×C4—3761no—
235C2×C4—1882no—
236C2×C4#1309399212yes—
237C2×C4#1309399511yescharacteristic
238C2×C4#1309399612yes—
239C2×C4#1309402011yescharacteristic
240C8—1884no—
241C8—1882no—
242C22—84no—
243C22—84no—
244C22—82no—
245C22—82no—
246C22—82no—
247C22#1568903111yescharacteristic
248C22#1568929111yescharacteristic
249C22#1568929611yescharacteristic
250C22#1568930114yes—
251C4—22no—
252C4—22no—
253C4—21no—
254C4—21no—
255C4—82no—
256C4—82no—
257C4—3762no—
258C2—162no—
259C2#2024269412yes—
260C2#2024271112yes—
261C2#2024441512yes—
262C2#2024443011yescharacteristic
subgroupG/H
centerC23#13093993
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
Fitting, prime radicalC47×#2819C2
socleC23×C47#120
hypercenter#2819D47
hypocenterC47#10085
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2819

Complements

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

NH ≅ G/Ntype
1. C2×#15689482C2semidirect
2. #20244207C2semidirect
3. C2×#15688863C2semidirect
4. #20241269C2semidirect
25. #13094024D4semidirect
32. #13093935D4semidirect
151. C47#10085semidirect

Maximal subgroups

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

subgroupindex
C2×#156894822
#202442072
C2×#156888632
#202412692
C47×#28192
#1008547