Finite Group Databasebeta

← #26118525

Subgroups — #26118525

Order 12032.

542 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 105 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#20254326C212yesmaximal
2#20244212C211yesmaximal, characteristic
3#20241952C212yesmaximal, complemented
4#20241233C211yesmaximal, characteristic, complemented
5C47×#3162C211yesmaximal, characteristic, complemented
6C2×#13094021C2211yescharacteristic
7#15689497—22no—
8#15689471—22no—
9#15689323—24no—
10#15689273—24no—
11#15689176C2212yes—
12C2×#13093946—22no—
13#15688989C2211yescharacteristic
14#15688860—24no—
15C47×#533C2212yes—
16C47×#501—22no—
17C47×#501—22no—
18C47×#427—22no—
19C2×#97×C47C2211yescharacteristic
20C22×D188—21no—
21C22×D188—22no—
22C2×#10625336—28no—
23#13094021—22no—
24#13094021—21no—
25#13094021D412yes—
26C2×C4⋊Dic47—21no—
27C4⋊(C47⋊C8)—22no—
28#13093946—44no—
29#13093944—28no—
30C22⋊(C47⋊C8)D414yes—
31C4×C47⋊C8—22no—
32C47⋊#95—22no—
33C22×C8×C47D412yes—
34C4⋊Q8×C47D412yes—
35C4⋊Q8×C47D412yes—
36#125×C47—22no—
37C22⋊Q8×C47—22no—
38C22⋊Q8×C47—22no—
39C4×Q8×C47—22no—
40C4×Q8×C47—22no—
41#117×C47—22no—
42C2×C42×C47C2311yescharacteristic
43#107×C47—24no—
44#106×C47—22no—
45#106×C47—22no—
46#97×C47—22no—
47C23×D47—41no—
48C2×Q8×C47—22no—
49C2×Q8×C47—22no—
50C22×C4×C47C2×D412yes—
51C22×C4×C47C2×D411yescharacteristic
52C22×Dic47—41no—
53C2×D188—24no—
54C2×D188—22no—
55C2×D188—42no—
56C2×D188—42no—
57C2×D188—41no—
58C2×D188—42no—
59C2×D188—84no—
60C2×C8×C47—22no—
61C2×C8×C47—22no—
62C2×C8×C47SD1614yescomplemented
63C4⋊C4×C47—22no—
64C4⋊C4×C47—22no—
65C4⋊C4×C47—22no—
66C4⋊C4×C47—22no—
67C4⋊C4×C47—42no—
68C4⋊C4×C47—42no—
69C22⋊C4×C47—42no—
70C42×C47—42no—
71C42×C47C2×D411yescharacteristic
72C42×C47C2×D412yes—
73C42×C47#4111yescharacteristic
74#10625336—88no—
75C4⋊Dic47—42no—
76C2×C47⋊C8—24no—
77C23×C47#12711yescharacteristic
78C22×D47—42no—
79C22×D47—84no—
80C22×D47—81no—
81C22×D47—81no—
82Q8×C47—44no—
83Q8×C47—82no—
84C2×C4×C47—42no—
85C2×C4×C47—42no—
86C2×C4×C47—42no—
87C2×C4×C47#12012yes—
88C2×C4×C47#12011yescharacteristic
89C2×C4×C47#12111yescharacteristic
90C2×C4×C47#12112yes—
91C2×C4×C47C2×SD1612yes—
92C2×C4×C47#13712yes—
93C2×Dic47—82no—
94C2×Dic47—81no—
95D188—48no—
96D188—44no—
97D188—84no—
98D188—82no—
99C8×C47—24no—
100C47⋊C8—44no—
101#10220—471nomaximal, Sylow
102C22×C47—22no—
103C22×C47#39211yescharacteristic
104C22×C47#49211yescharacteristic
105C22×C47#49711yescharacteristic
106C2×D47—84no—
107C2×D47—82no—
108C2×D47—162no—
109C2×D47—162no—
110C4×C47—22no—
111C4×C47—22no—
112C4×C47—82no—
113C4×C47—82no—
114C4×C47#44812yes—
115C4×C47#46112yes—
116Dic47—161no—
117#3208—472no—
118#3162D4711yescharacteristic, complemented
119#1961—471no—
120#1518—472no—
121#1281—471no—
122C2×C47—21no—
123C2×C47—21no—
124C2×C47#200812yes—
125C2×C47#201311yescharacteristic
126D47—162no—
127#533C2×D4712yes—
128C2×#127—471no—
129#501—22no—
130#501—22no—
131#492—942no—
132#486—942no—
133#460—944no—
134#450—944no—
135#431—472no—
136#427—22no—
137C2×#102—942no—
138C2×#97C2×D4711yescharacteristic
139#382—471no—
140#328—944no—
141C47#1022011yescharacteristic, complemented, Sylow
142C22×D4—942no—
143C22×D4—941no—
144C2×SD16—948no—
145C22×C8C22⋊D4712yes—
146C4⋊Q8C22⋊D4712yes—
147C4⋊Q8C22⋊D4712yes—
148#127—472no—
149#127—941no—
150#127—942no—
151#125—22no—
152C22⋊Q8—22no—
153C22⋊Q8—22no—
154C4×Q8—22no—
155C4×Q8—22no—
156#117—22no—
157C2×C4⋊C4—941no—
158C2×C42C22×D4711yescharacteristic
159#107—24no—
160#106—22no—
161#106—22no—
162C4⋊C8—942no—
163#102—948no—
164#102—1884no—
165C22⋊C8—474no—
166#97—22no—
167C4×C8—942no—
168#95—942no—
169C24—1881no—
170C2×Q8—22no—
171C2×Q8—22no—
172C2×D4—944no—
173C2×D4—942no—
174C2×D4—3764no—
175C2×D4—1881no—
176C2×D4—1882no—
177C2×D4—1882no—
178C2×D4—1882no—
179C22×C4—1881no—
180C22×C4D4×D4712yes—
181C22×C4C2×C22⋊D4711yescharacteristic
182SD16—3768no—
183C2×C8—22no—
184C2×C8—22no—
185C2×C8—944no—
186C2×C8#1062533614yescomplemented
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—22no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—1882no—
194C22⋊C4—42no—
195C42—42no—
196C42D4×D4711yescharacteristic
197C42#1062535211yescharacteristic
198C42C2×C22⋊D4712yes—
199C23—3764no—
200C23—3761no—
201C23—3761no—
202C23—1882no—
203C23#1309402011yescharacteristic
204Q8—44no—
205Q8—82no—
206D4—3764no—
207D4—3762no—
208D4—1888no—
209D4—1884no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—3762no—
214C2×C4—3761no—
215C2×C4#1309399212yes—
216C2×C4#1309399311yescharacteristic
217C2×C4#1309399511yescharacteristic
218C2×C4#1309399612yes—
219C2×C4C2×#1062533612yes—
220C2×C4#1309405512yes—
221C8—24no—
222C8—1884no—
223C22—22no—
224C22—3764no—
225C22—3762no—
226C22—7522no—
227C22—7522no—
228C22#1568903111yescharacteristic
229C22#1568949711yescharacteristic
230C22#1568951511yescharacteristic
231C4—22no—
232C4—22no—
233C4—82no—
234C4—82no—
235C4—7521no—
236C4#1568926212yes—
237C4#1568932512yes—
238C2—21no—
239C2—21no—
240C2—7522no—
241C2#2024443812yes—
242C2#2024447111yescharacteristic
subgroupG/H
centerC22#15689031
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#3162C2
socleC22×C47#392
prime radicalC2×#13094021C22
hypercenter#3162D47
hypocenterC47#10220
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3162

Complements

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

NH ≅ G/Ntype
3. #20241952C2semidirect
4. #20241233C2semidirect
5. C47×#3162C2semidirect
62. C2×C8×C47SD16semidirect
118. #3162D47semidirect
141. C47#10220semidirect
186. C2×C8#10625336semidirect

Maximal subgroups

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

subgroupindex
#202543262
#202442122
#202419522
#202412332
C47×#31622
#1022047