Finite Group Databasebeta

← #26111589

Subgroups — #26111589

Order 12032.

444 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 107 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#20253710C211yesmaximal, characteristic
2#20241980C211yesmaximal, characteristic
3#20241808C212yesmaximal, complemented
4#20241224C212yesmaximal, complemented
5C47×#2979C211yesmaximal, characteristic
6#15689680C2211yescharacteristic
7#15689477—22no—
8#15689473C412yes—
9#15689469C412yes—
10#15689119—22no—
11C4×#10625329C2212yes—
12C2×C4×C47⋊C8C2211yescharacteristic
13#15688863—22no—
14#15688853—22no—
15#15688853—22no—
16C47×#532C2211yescharacteristic
17C47×#436—22no—
18C8×D4×C47—22no—
19C47×#431C2212yes—
20C47×#408—22no—
21C2×#10625329—22no—
22C22×C47⋊C8—21no—
23#13094024C2×C412yes—
24#13094016—21no—
25#13094003—22no—
26C4×Dic94—22no—
27#13093983—21no—
28#13093948—24no—
29#13093945—24no—
30#13093935—22no—
31C4×C47⋊C8—21no—
32C4×C47⋊C8—22no—
33C4×C47⋊C8C2×C412yes—
34C2×#41×C47D412yes—
35C2×M16×C47—22no—
36C22×C8×C47—22no—
37C4⋊Q8×C47C2×C411yescharacteristic
38#127×C47C2×C411yescharacteristic
39#124×C47D412yes—
40#121×C47—22no—
41C4×D4×C47—22no—
42C2×C42×C47C2311yescharacteristic
43C4⋊C8×C47D412yes—
44C4⋊C8×C47D412yes—
45C22⋊C8×C47—24no—
46C22⋊C8×C47—22no—
47C4×C8×C47—22no—
48#41×C47—42no—
49#41×C47—42no—
50C2×Q8×C47C22⋊C412yes—
51C2×D4×C47—22no—
52C2×D4×C47—22no—
53C2×D4×C47C22⋊C412yes—
54C22×C4×C47—22no—
55C22×C4×C47C2×D411yescharacteristic
56C22×C4×C47#4112yes—
57C2×Dic94—22no—
58M16×C47—82no—
59C2×C8×C47—22no—
60C2×C8×C47—22no—
61C2×C8×C47—82no—
62C4⋊C4×C47—22no—
63C22⋊C4×C47—24no—
64C42×C47C22×C411yescharacteristic
65C42×C47C2×D411yescharacteristic
66C42×C47C2×D412yes—
67C22⋊Dic47—41no—
68C4⋊Dic47—22no—
69C4⋊Dic47—22no—
70Dic47⋊C4—42no—
71C4×Dic47—41no—
72#10625329—24no—
73#10625329—42no—
74C2×C47⋊C8—22no—
75C2×C47⋊C8—41no—
76C2×C47⋊C8—41no—
77C2×C47⋊C8—42no—
78C23×C47—22no—
79C23×C47#12311yescharacteristic
80Q8×C47—42no—
81D4×C47—42no—
82D4×C47—42no—
83D4×C47—42no—
84D4×C47—82no—
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×C47C2×C22⋊C411yescharacteristic
93C2×C4×C47C4×D412yes—
94C2×C4×C47#12011yescharacteristic
95C2×C4×C47#12112yes—
96C2×Dic47—42no—
97Dic94—42no—
98Dic94—42no—
99C8×C47—42no—
100C8×C47—42no—
101C47⋊C8—42no—
102C47⋊C8—44no—
103#9067—471nomaximal, Sylow
104C22×C47—22no—
105C22×C47—21no—
106C22×C47—44no—
107C22×C47—42no—
108C22×C47#38611yescharacteristic
109C4×C47—22no—
110C4×C47—21no—
111C4×C47—42no—
112C4×C47#44312yes—
113C4×C47#44412yes—
114C4×C47#45511yescharacteristic
115C4×C47#45511yescharacteristic
116Dic47—82no—
117#3131—471no—
118#2979D4711yescharacteristic
119#1525—471no—
120#1485—472no—
121#1278—472no—
122C2×C47—41no—
123C2×C47—82no—
124C2×C47#187612yes—
125C2×C47#188511yescharacteristic
126#533—471no—
127#532C2×D4711yescharacteristic
128#488—942no—
129#487—472no—
130#486—472no—
131#436—22no—
132C8×D4—22no—
133#431C2×D4712yes—
134#416—942no—
135#408—22no—
136C4×M16—472no—
137C2×C4×C8—471no—
138#329—942no—
139Q8⋊C8—942no—
140Q8⋊C8—942no—
141C47#906711yescharacteristic, complemented, Sylow
142C2×#41C22⋊D4712yes—
143C2×M16—22no—
144C2×M16—942no—
145C22×C8—22no—
146C22×C8—941no—
147C4⋊Q8—472no—
148C4⋊Q8C4×D4711yescharacteristic
149#127C4×D4711yescharacteristic
150#125—941no—
151#124D18812yes—
152C22⋊Q8—942no—
153#121—22no—
154C4×Q8—942no—
155C4×D4—22no—
156#117—941no—
157C2×C42C22×D4711yescharacteristic
158C4⋊C8C22⋊D4712yes—
159C4⋊C8C22⋊D4712yes—
160#103—944no—
161#102—944no—
162C22⋊C8—24no—
163C22⋊C8—22no—
164#97—942no—
165C4×C8—22no—
166C4×C8—472no—
167C4×C8—941no—
168C4×C8—942no—
169#41—42no—
170#41—42no—
171C2×Q8—942no—
172C2×Q8(C2×D47)⋊C412yes—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4(C2×D47)⋊C412yes—
176C22×C4—22no—
177C22×C4#1062535012yes—
178C22×C4C2×C22⋊D4711yescharacteristic
179M16—82no—
180M16—944no—
181M16—1882no—
182C2×C8—22no—
183C2×C8—22no—
184C2×C8—82no—
185C2×C8—942no—
186C2×C8—1881no—
187C2×C8—1881no—
188C2×C8—1882no—
189C4⋊C4—22no—
190C4⋊C4—942no—
191C4⋊C4—942no—
192C4⋊C4—1882no—
193C22⋊C4—24no—
194C22⋊C4—1881no—
195C42—1881no—
196C42C2×C4×D4711yescharacteristic
197C42C2×D18811yescharacteristic
198C42C2×C22⋊D4712yes—
199C23—22no—
200C23#1309400711yescharacteristic
201Q8—42no—
202Q8—1882no—
203Q8—1882no—
204D4—42no—
205D4—42no—
206D4—42no—
207D4—82no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—1882no—
216C2×C4C2×(C2×D47)⋊C411yescharacteristic
217C2×C4C4×C22⋊D4712yes—
218C2×C4#1309399311yescharacteristic
219C2×C4#1309399812yes—
220C8—42no—
221C8—42no—
222C8—1882no—
223C8—1884no—
224C22—22no—
225C22—21no—
226C22—44no—
227C22—42no—
228C22#1568900211yescharacteristic
229C4—22no—
230C4—21no—
231C4—42no—
232C4—3762no—
233C4#1568923812yes—
234C4#1568924312yes—
235C4#1568930111yescharacteristic
236C4#1568930111yescharacteristic
237C2—41no—
238C2—82no—
239C2#2024371512yes—
240C2#2024377611yescharacteristic
subgroupG/H
centerC2×C4#13093993
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2979C2
socleC22×C47#386
prime radicalC47×#532C22
cosocleC2×C42×C47C23
hypercenter#2979D47
hypocenterC47#9067
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2979

Complements

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

NH ≅ G/Ntype
3. #20241808C2semidirect
4. #20241224C2semidirect
141. C47#9067semidirect

Maximal subgroups

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

subgroupindex
#202537102
#202419802
#202418082
#202412242
C47×#29792
#906747