Finite Group Databasebeta

← #26128872

Subgroups — #26128872

Order 12032.

372 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 59 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#20251660C211yesmaximal, characteristic, complemented
2#20242357C212yesmaximal
3#20241417C212yesmaximal, complemented
4#20241332C211yesmaximal, characteristic, complemented
5C47×#2040C211yesmaximal, characteristic
6#15689261—22no—
7#15689065—22no—
8#15689062—21no—
9#15689044—22no—
10#15689043C2212yes—
11#15688989—21no—
12#15688857—22no—
13#15688849C2212yes—
14C47×#537C2211yescharacteristic
15C2×#104×C47—21no—
16C47×#417—21no—
17C47×#382C2211yescharacteristic
18C47×#343C2211yescharacteristic
19C22×Dic94D412yes—
20C2×#10625335—42no—
21C2×C4⋊Dic47—21no—
22C2×C4⋊Dic47—21no—
23C2×C4⋊Dic47—21no—
24C2×Dic47⋊C4—22no—
25#13093947—42no—
26C22⋊(C47⋊C8)—22no—
27C47⋊#95—22no—
28C47⋊#95—22no—
29C47⋊#95—21no—
30C47⋊#95—22no—
31C47⋊#95D412yes—
32C2×#41×C47—21no—
33C2×M16×C47—21no—
34C22⋊Q8×C47—41no—
35#121×C47—22no—
36#121×C47—41no—
37C2×C4⋊C4×C47D411yescharacteristic
38C2×C4⋊C4×C47C2311yescharacteristic
39C2×C42×C47D411yescharacteristic
40#104×C47—42no—
41#103×C47—41no—
42#102×C47—41no—
43#95×C47—22no—
44#41×C47—41no—
45#41×C47—41no—
46#41×C47—41no—
47#41×C47—41no—
48C2×Q8×C47—21no—
49C2×D4×C47—21no—
50C2×D4×C47—22no—
51C2×D4×C47—41no—
52C22×C4×C47—21no—
53C22×C4×C47—21no—
54C22×C4×C47—41no—
55C22×C4×C47C2×D411yescharacteristic
56C22×C4×C47C2×D412yes—
57C22×Dic47—22no—
58C22×Dic47—22no—
59C2×Dic94—42no—
60C2×Dic94—82no—
61M16×C47—41no—
62M16×C47—41no—
63C2×C8×C47—41no—
64C4⋊C4×C47—22no—
65C4⋊C4×C47—41no—
66C4⋊C4×C47—41no—
67C22⋊C4×C47—42no—
68C42×C47—21no—
69C42×C47SD1612yescomplemented
70#10625335—84no—
71C4⋊Dic47—81no—
72C4⋊Dic47—81no—
73C4⋊Dic47—81no—
74Dic47⋊C4—82no—
75C2×C47⋊C8—42no—
76C23×C47—41no—
77C23×C47#12011yescharacteristic
78Q8×C47—41no—
79Q8×C47—41no—
80D4×C47—41no—
81D4×C47—41no—
82D4×C47—82no—
83D4×C47—81no—
84D4×C47—81no—
85C2×C4×C47—21no—
86C2×C4×C47—21no—
87C2×C4×C47—22no—
88C2×C4×C47—22no—
89C2×C4×C47—41no—
90C2×C4×C47—41no—
91C2×C4×C47—42no—
92C2×C4×C47—42no—
93C2×C4×C47—41no—
94C2×C4×C47—41no—
95C2×C4×C47—81no—
96C2×C4×C47—81no—
97C2×C4×C47C2×SD1611yescharacteristic
98C2×C4×C47#13711yescharacteristic
99C2×Dic47—42no—
100C2×Dic47—42no—
101C2×Dic47—42no—
102C2×Dic47—42no—
103C2×Dic47—42no—
104C2×Dic47—42no—
105Dic94—44no—
106Dic94—82no—
107C8×C47—81no—
108C47⋊C8—82no—
109#12184—471nomaximal, Sylow
110C22×C47—21no—
111C22×C47—21no—
112C22×C47—81no—
113C22×C47—81no—
114C22×C47—81no—
115C22×C47#44811yescharacteristic
116C22×C47#45411yescharacteristic
117C22×C47B(4,𝔽2)11yescharacteristic
118C4×C47—21no—
119C4×C47—21no—
120C4×C47—42no—
121C4×C47—42no—
122C4×C47—81no—
123C4×C47—81no—
124Dic47—82no—
125Dic47—82no—
126#2844—471no—
127#2040D4711yescharacteristic
128#1604—472no—
129#1354—472no—
130#1324—471no—
131C2×C47—21no—
132C2×C47—21no—
133C2×C47—161no—
134C2×C47#159911yescharacteristic
135C2×C47#219911yescharacteristic
136C2×C47#220311yescharacteristic
137#537C2×D4711yescharacteristic
138#448—942no—
139C2×#104—21no—
140#417—21no—
141#400—942no—
142#400—941no—
143#395—472no—
144#395—942no—
145#382—941no—
146#382C2×D4711yescharacteristic
147#343C2×D4711yescharacteristic
148#327—942no—
149#324—472no—
150C47#1218411yescharacteristic, complemented, Sylow
151C2×#41—21no—
152C22×Q8—472no—
153C2×SD16—1882no—
154C2×M16—21no—
155C22⋊Q8—41no—
156#121—22no—
157#121—41no—
158C2×C4⋊C4—941no—
159C2×C4⋊C4—941no—
160C2×C4⋊C4—941no—
161C2×C4⋊C4—942no—
162C2×C4⋊C4C22⋊D4711yescharacteristic
163C2×C4⋊C4C22×D4711yescharacteristic
164C2×C42C22⋊D4711yescharacteristic
165#104—42no—
166#103—41no—
167#103—1882no—
168#102—41no—
169C22⋊C8—942no—
170#95—22no—
171#95—472no—
172#95—942no—
173#95—942no—
174#95—941no—
175#95—942no—
176#41—41no—
177#41—41no—
178#41—41no—
179#41—41no—
180C2×Q8—21no—
181C2×Q8—3762no—
182C2×Q8—1882no—
183C2×D4—21no—
184C2×D4—22no—
185C2×D4—41no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—41no—
189C22×C4—942no—
190C22×C4—942no—
191C22×C4D4×D4712yes—
192C22×C4C2×C22⋊D4711yescharacteristic
193SD16—3764no—
194M16—41no—
195M16—41no—
196C2×C8—41no—
197C2×C8—1882no—
198C4⋊C4—22no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4—3762no—
202C4⋊C4—3761no—
203C4⋊C4—3761no—
204C4⋊C4—3761no—
205C22⋊C4—42no—
206C42—21no—
207C42#1062533512yescomplemented
208C23—41no—
209C23#1309399211yescharacteristic
210Q8—41no—
211Q8—41no—
212Q8—3762no—
213Q8—1884no—
214D4—41no—
215D4—41no—
216D4—82no—
217D4—81no—
218D4—81no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—81no—
230C2×C4—81no—
231C2×C4—1882no—
232C2×C4—1882no—
233C2×C4—1882no—
234C2×C4—1882no—
235C2×C4—1882no—
236C2×C4—1882no—
237C2×C4C2×#1062533511yescharacteristic
238C2×C4#1309405411yescharacteristic
239C8—81no—
240C8—3762no—
241C22—21no—
242C22—21no—
243C22—81no—
244C22—81no—
245C22—81no—
246C22#1568926111yescharacteristic
247C22#1568929511yescharacteristic
248C22#1568930911yescharacteristic
249C4—21no—
250C4—21no—
251C4—42no—
252C4—42no—
253C4—81no—
254C4—81no—
255C4—3762no—
256C4—3762no—
257C2—21no—
258C2—21no—
259C2—161no—
260C2#2024232711yescharacteristic
261C2#2024538811yescharacteristic
262C2#2024540811yescharacteristic
subgroupG/H
centerC22#15689309
commutator, cosocleC2×C4⋊C4×C47C23
FrattiniC2×C4⋊C4C22×D47
FittingC47×#2040C2
socleC22×C47B(4,𝔽2)
prime radicalC47×#537C22
hypercenter#2040D47
hypocenterC47#12184
LCSG ▷ C2×C4⋊C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2040

Complements

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

NH ≅ G/Ntype
1. #20251660C2semidirect
3. #20241417C2semidirect
4. #20241332C2semidirect
69. C42×C47SD16semidirect
150. C47#12184semidirect
207. C42#10625335semidirect

Maximal subgroups

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

subgroupindex
#202516602
#202423572
#202414172
#202413322
C47×#20402
#1218447