Finite Group Databasebeta

← #26128853

Subgroups — #26128853

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#20248630C211yesmaximal, characteristic, complemented
2#20242425C212yesmaximal
3#20241414C212yesmaximal, complemented
4#20241334C211yesmaximal, characteristic, complemented
5C47×#2039C211yesmaximal, characteristic
6#15689439—22no—
7#15689062—21no—
8#15689062C2212yes—
9#15689022—22no—
10#15688988—22no—
11C4×C4⋊Dic47—21no—
12#15688857—22no—
13#15688849C2212yes—
14C47×#537C2211yescharacteristic
15C2×#104×C47—21no—
16C47×#417—21no—
17C47×#384C2211yescharacteristic
18C47×#343C2211yescharacteristic
19C2×C4⋊Dic47—21no—
20C2×C4⋊Dic47—21no—
21C2×C4⋊Dic47—21no—
22C2×C4⋊Dic47D412yes—
23C2×C4×Dic47—22no—
24#13093955—42no—
25#13093945—42no—
26C22⋊(C47⋊C8)—22no—
27C47⋊#95—21no—
28C47⋊#95—22no—
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×C47—22no—
38C2×C4⋊C4×C47D411yescharacteristic
39C2×C4⋊C4×C47C2311yescharacteristic
40C2×C42×C47D411yescharacteristic
41#104×C47—42no—
42#103×C47—41no—
43#102×C47—41no—
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—
59M16×C47—41no—
60M16×C47—41no—
61C2×C8×C47—41no—
62C4⋊C4×C47—22no—
63C4⋊C4×C47—41no—
64C4⋊C4×C47—44no—
65C4⋊C4×C47—41no—
66C22⋊C4×C47—42no—
67C42×C47—21no—
68C42×C47SD1612yes—
69C4⋊Dic47—42no—
70C4⋊Dic47—42no—
71C4⋊Dic47—42no—
72C4⋊Dic47—81no—
73C4×Dic47—44no—
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—22no—
85C2×C4×C47—22no—
86C2×C4×C47—22no—
87C2×C4×C47—21no—
88C2×C4×C47—21no—
89C2×C4×C47—22no—
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—
104C8×C47—81no—
105C47⋊C8—82no—
106#12181—471nomaximal, Sylow
107C22×C47—21no—
108C22×C47—21no—
109C22×C47—81no—
110C22×C47—81no—
111C22×C47—81no—
112C22×C47#44811yescharacteristic
113C22×C47#45411yescharacteristic
114C22×C47B(4,𝔽2)11yescharacteristic
115C4×C47—24no—
116C4×C47—21no—
117C4×C47—21no—
118C4×C47—42no—
119C4×C47—81no—
120C4×C47—81no—
121Dic47—82no—
122Dic47—82no—
123#2571—471no—
124#2039D4711yescharacteristic
125#1616—472no—
126#1353—472no—
127#1325—471no—
128C2×C47—21no—
129C2×C47—21no—
130C2×C47—161no—
131C2×C47#159911yescharacteristic
132C2×C47#219811yescharacteristic
133C2×C47#220411yescharacteristic
134#537C2×D4711yescharacteristic
135#481—942no—
136C2×#104—21no—
137#417—21no—
138#400—472no—
139#400—941no—
140#389—942no—
141#384C2×D4711yescharacteristic
142#382—942no—
143C4×C4⋊C4—941no—
144#343C2×D4711yescharacteristic
145#327—942no—
146#324—472no—
147C47#1218111yescharacteristic, complemented, Sylow
148C2×#41—21no—
149C2×M16—21no—
150C22⋊Q8—41no—
151#121—22no—
152#121—41no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—472no—
155C2×C4⋊C4—941no—
156C2×C4⋊C4—941no—
157C2×C4⋊C4—941no—
158C2×C4⋊C4C22⋊D4711yescharacteristic
159C2×C4⋊C4C22×D4711yescharacteristic
160C2×C42—942no—
161C2×C42C22⋊D4711yescharacteristic
162#106—1882no—
163#104—42no—
164#103—41no—
165#102—41no—
166#102—1882no—
167C22⋊C8—942no—
168#95—472no—
169#95—941no—
170#95—942no—
171#95—942no—
172#95—942no—
173#41—41no—
174#41—41no—
175#41—41no—
176#41—41no—
177C2×Q8—21no—
178C2×D4—21no—
179C2×D4—22no—
180C2×D4—41no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—41no—
184C22×C4—942no—
185C22×C4—942no—
186C22×C4D4×D4712yes—
187C22×C4C2×C22⋊D4711yescharacteristic
188M16—41no—
189M16—41no—
190C2×C8—41no—
191C2×C8—1882no—
192C4⋊C4—22no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—44no—
196C4⋊C4—3761no—
197C4⋊C4—1882no—
198C4⋊C4—1882no—
199C4⋊C4—1882no—
200C22⋊C4—42no—
201C42—21no—
202C42—1884no—
203C42#1062533512yes—
204C23—41no—
205C23#1309399211yescharacteristic
206Q8—41no—
207Q8—41no—
208D4—41no—
209D4—41no—
210D4—82no—
211D4—81no—
212D4—81no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4—1882no—
226C2×C4—1882no—
227C2×C4—1882no—
228C2×C4—1882no—
229C2×C4—1882no—
230C2×C4—1882no—
231C2×C4C2×#1062533511yescharacteristic
232C2×C4#1309405411yescharacteristic
233C8—81no—
234C8—3762no—
235C22—21no—
236C22—21no—
237C22—81no—
238C22—81no—
239C22—81no—
240C22#1568926111yescharacteristic
241C22#1568929511yescharacteristic
242C22#1568930911yescharacteristic
243C4—24no—
244C4—21no—
245C4—21no—
246C4—42no—
247C4—81no—
248C4—81no—
249C4—3762no—
250C4—3762no—
251C2—21no—
252C2—21no—
253C2—161no—
254C2#2024232711yescharacteristic
255C2#2024538311yescharacteristic
256C2#2024541311yescharacteristic
subgroupG/H
centerC22#15689309
commutator, cosocleC2×C4⋊C4×C47C23
FrattiniC2×C4⋊C4C22×D47
FittingC47×#2039C2
socleC22×C47B(4,𝔽2)
prime radicalC47×#537C22
hypercenter#2039D47
hypocenterC47#12181
LCSG ▷ C2×C4⋊C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2039

Complements

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

NH ≅ G/Ntype
1. #20248630C2semidirect
3. #20241414C2semidirect
4. #20241334C2semidirect
147. C47#12181semidirect

Maximal subgroups

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

subgroupindex
#202486302
#202424252
#202414142
#202413342
C47×#20392
#1218147