Finite Group Databasebeta

← #26128877

Subgroups — #26128877

Order 12032.

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

normal subgroups only

This group has 360 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#20242425C212yesmaximal
3#20241414C212yesmaximal, complemented
4#20241332C211yesmaximal, characteristic, complemented
5C47×#2040C211yesmaximal, characteristic
6#15689439—22no—
7#15689065—22no—
8#15689062C2212yes—
9#15689044—22no—
10#15689043—21no—
11#15688989—21no—
12#15688857—22no—
13#15688849C2212yes—
14C47×#537C2211yescharacteristic
15C2×#104×C47—21no—
16C47×#417—21no—
17C47×#382C2211yescharacteristic
18C47×#343C2211yescharacteristic
19C22×Dic94—21no—
20C2×C4⋊Dic47—21no—
21C2×C4⋊Dic47—21no—
22C2×C4⋊Dic47D412yes—
23C2×Dic47⋊C4—22no—
24#13093955—42no—
25#13093945—42no—
26C22⋊(C47⋊C8)—22no—
27C47⋊#95—22no—
28C47⋊#95—21no—
29C47⋊#95—22no—
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—82no—
60C2×Dic94—81no—
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×C47SD1612yes—
70C4⋊Dic47—42no—
71C4⋊Dic47—81no—
72C4⋊Dic47—81no—
73Dic47⋊C4—82no—
74C2×C47⋊C8—42no—
75C23×C47—41no—
76C23×C47#12011yescharacteristic
77Q8×C47—41no—
78Q8×C47—41no—
79D4×C47—41no—
80D4×C47—41no—
81D4×C47—82no—
82D4×C47—81no—
83D4×C47—81no—
84C2×C4×C47—21no—
85C2×C4×C47—21no—
86C2×C4×C47—22no—
87C2×C4×C47—22no—
88C2×C4×C47—41no—
89C2×C4×C47—41no—
90C2×C4×C47—42no—
91C2×C4×C47—42no—
92C2×C4×C47—41no—
93C2×C4×C47—41no—
94C2×C4×C47—81no—
95C2×C4×C47—81no—
96C2×C4×C47C2×SD1611yescharacteristic
97C2×C4×C47#13711yescharacteristic
98C2×Dic47—42no—
99C2×Dic47—42no—
100C2×Dic47—42no—
101C2×Dic47—42no—
102C2×Dic47—42no—
103C2×Dic47—42no—
104Dic94—84no—
105C8×C47—81no—
106C47⋊C8—82no—
107#12185—471nomaximal, Sylow
108C22×C47—21no—
109C22×C47—21no—
110C22×C47—81no—
111C22×C47—81no—
112C22×C47—81no—
113C22×C47#44811yescharacteristic
114C22×C47#45411yescharacteristic
115C22×C47B(4,𝔽2)11yescharacteristic
116C4×C47—21no—
117C4×C47—21no—
118C4×C47—42no—
119C4×C47—42no—
120C4×C47—81no—
121C4×C47—81no—
122Dic47—82no—
123Dic47—82no—
124#2844—471no—
125#2040D4711yescharacteristic
126#1616—472no—
127#1353—472no—
128#1324—471no—
129C2×C47—21no—
130C2×C47—21no—
131C2×C47—161no—
132C2×C47#159911yescharacteristic
133C2×C47#220111yescharacteristic
134C2×C47#220211yescharacteristic
135#537C2×D4711yescharacteristic
136#481—942no—
137C2×#104—21no—
138#417—21no—
139#400—472no—
140#400—942no—
141#395—942no—
142#395—941no—
143#382—941no—
144#382C2×D4711yescharacteristic
145#343C2×D4711yescharacteristic
146#327—942no—
147#324—472no—
148C47#1218511yescharacteristic, complemented, Sylow
149C2×#41—21no—
150C22×Q8—941no—
151C2×M16—21no—
152C22⋊Q8—41no—
153#121—22no—
154#121—41no—
155C2×C4⋊C4—472no—
156C2×C4⋊C4—942no—
157C2×C4⋊C4—941no—
158C2×C4⋊C4—941no—
159C2×C4⋊C4C22⋊D4711yescharacteristic
160C2×C4⋊C4C22×D4711yescharacteristic
161C2×C42C22⋊D4711yescharacteristic
162#106—1882no—
163#104—42no—
164#103—41no—
165#102—41no—
166#102—1882no—
167C22⋊C8—942no—
168#95—22no—
169#95—472no—
170#95—942no—
171#95—941no—
172#95—942no—
173#95—942no—
174#41—41no—
175#41—41no—
176#41—41no—
177#41—41no—
178C2×Q8—21no—
179C2×Q8—3762no—
180C2×Q8—3761no—
181C2×D4—21no—
182C2×D4—22no—
183C2×D4—41no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—41no—
187C22×C4—942no—
188C22×C4—942no—
189C22×C4D4×D4712yes—
190C22×C4C2×C22⋊D4711yescharacteristic
191M16—41no—
192M16—41no—
193C2×C8—41no—
194C2×C8—1882no—
195C4⋊C4—22no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—3762no—
199C4⋊C4—3761no—
200C4⋊C4—3761no—
201C4⋊C4—1882no—
202C22⋊C4—42no—
203C42—21no—
204C42#1062533512yes—
205C23—41no—
206C23#1309399211yescharacteristic
207Q8—41no—
208Q8—41no—
209Q8—3764no—
210D4—41no—
211D4—41no—
212D4—82no—
213D4—81no—
214D4—81no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4—1882no—
228C2×C4—1882no—
229C2×C4—1882no—
230C2×C4—1882no—
231C2×C4—1882no—
232C2×C4—1882no—
233C2×C4C2×#1062533511yescharacteristic
234C2×C4#1309405411yescharacteristic
235C8—81no—
236C8—3762no—
237C22—21no—
238C22—21no—
239C22—81no—
240C22—81no—
241C22—81no—
242C22#1568926111yescharacteristic
243C22#1568929511yescharacteristic
244C22#1568930911yescharacteristic
245C4—21no—
246C4—21no—
247C4—42no—
248C4—42no—
249C4—81no—
250C4—81no—
251C4—3762no—
252C4—3762no—
253C2—21no—
254C2—21no—
255C2—161no—
256C2#2024232711yescharacteristic
257C2#2024540011yescharacteristic
258C2#2024540411yescharacteristic
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#12185
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. #20241414C2semidirect
4. #20241332C2semidirect
148. C47#12185semidirect

Maximal subgroups

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

subgroupindex
#202516602
#202424252
#202414142
#202413322
C47×#20402
#1218547