Finite Group Databasebeta

← #26110651

Subgroups — #26110651

Order 12032.

428 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 111 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#20246574C211yesmaximal, characteristic
2#20243554C212yesmaximal
3C2×#15688865C211yesmaximal, characteristic
4#20241270C212yesmaximal
5C47×#1992C211yesmaximal, characteristic
6C2×#13093957—22no—
7C2×#13093955—22no—
8C2×#13093935C2212yes—
9#15689022—22no—
10#15689009—22no—
11#15689009—22no—
12#15688989C2212yes—
13#15688865C414yescomplemented
14C2×#125×C47C2211yescharacteristic
15C2×#97×C47C2211yescharacteristic
16C47×#384C2211yescharacteristic
17C47×#340—22no—
18C22×C47⋊C8—22no—
19C2×C4⋊Dic47—22no—
20C2×C4⋊Dic47D412yes—
21C2×C4⋊Dic47D412yes—
22C2×C4×Dic47—22no—
23#13093957—44no—
24#13093955—44no—
25#13093935—22no—
26#13093935C2×C414yes—
27C47⋊#95—22no—
28C47⋊#95—22no—
29C47⋊#95—22no—
30C47⋊#95—22no—
31C22×C8×C47—21no—
32#125×C47—22no—
33#125×C47C2×C412yes—
34#125×C47Q812yes—
35C2×C4⋊C4×C47—22no—
36C2×C4⋊C4×C47—21no—
37C2×C4⋊C4×C47—22no—
38C2×C4⋊C4×C47D412yes—
39C2×C42×C47C2311yescharacteristic
40#97×C47—22no—
41C22×C4×C47—22no—
42C22×C4×C47—41no—
43C22×C4×C47C2×D411yescharacteristic
44C22×C4×C47C2×D412yes—
45C22×Dic47—22no—
46C22×Dic47—22no—
47C2×C8×C47—22no—
48C2×C8×C47—41no—
49C2×C8×C47—41no—
50C4⋊C4×C47—24no—
51C4⋊C4×C47—24no—
52C4⋊C4×C47—22no—
53C4⋊C4×C47—42no—
54C4⋊C4×C47—42no—
55C4⋊C4×C47—42no—
56C4⋊C4×C47C4⋊C414yes—
57C42×C47C22×C411yescharacteristic
58C42×C47C2×Q811yescharacteristic
59C42×C47#4112yes—
60C4⋊Dic47—42no—
61C4⋊Dic47—42no—
62C4⋊Dic47—82no—
63C4×Dic47—82no—
64C2×C47⋊C8—24no—
65C2×C47⋊C8—24no—
66C2×C47⋊C8—24no—
67C23×C47#12011yescharacteristic
68C2×C4×C47—22no—
69C2×C4×C47—21no—
70C2×C4×C47—22no—
71C2×C4×C47—21no—
72C2×C4×C47—24no—
73C2×C4×C47—42no—
74C2×C4×C47—42no—
75C2×C4×C47—82no—
76C2×C4×C47—81no—
77C2×C4×C47C2×C4⋊C411yescharacteristic
78C2×C4×C47C4×D412yes—
79C2×C4×C47C22⋊Q812yes—
80C2×C4×C47#12311yescharacteristic
81C2×Dic47—42no—
82C2×Dic47—42no—
83C2×Dic47—42no—
84C2×Dic47—42no—
85C2×Dic47—42no—
86C2×Dic47—42no—
87C8×C47—42no—
88C47⋊C8—28no—
89#8913—471nomaximal, Sylow
90C22×C47#38511yescharacteristic
91C22×C47#45312yes—
92C22×C47#45411yescharacteristic
93C22×C47#45612yes—
94C22×C47#45611yescharacteristic
95C4×C47—21no—
96C4×C47—21no—
97C4×C47—22no—
98C4×C47—22no—
99C4×C47—44no—
100C4×C47—161no—
101Dic47—82no—
102Dic47—82no—
103#2382—471no—
104#1992D4711yescharacteristic
105#1847—472no—
106C2×#330—471no—
107#1295—472no—
108C2×C47#165611yescharacteristic
109C2×C47#166012yes—
110C2×C47#166311yescharacteristic
111C2×C47#185711yescharacteristic
112C2×C47#185912yes—
113C2×#125C2×D4711yescharacteristic
114C2×#107—942no—
115C2×#106—942no—
116C2×#97—472no—
117C2×#97C2×D4711yescharacteristic
118#389—942no—
119#387—942no—
120#387—942no—
121#384C2×D4711yescharacteristic
122#382—472no—
123#340—22no—
124#330—474no—
125C47#891311yescharacteristic, complemented, Sylow
126C22×C8—21no—
127C22×C8—942no—
128#125—22no—
129#125Dic9412yes—
130#125C4×D4712yes—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—472no—
135C2×C4⋊C4—472no—
136C2×C4⋊C4—942no—
137C2×C4⋊C4C22⋊D4712yes—
138C2×C42—942no—
139C2×C42C22×D4711yescharacteristic
140#107—1884no—
141#106—1884no—
142#97—22no—
143#97—474no—
144#97—942no—
145#95—942no—
146#95—942no—
147#95—942no—
148#95—942no—
149C22×C4—22no—
150C22×C4—41no—
151C22×C4—942no—
152C22×C4—942no—
153C22×C4D4×D4712yes—
154C22×C4C2×C22⋊D4711yescharacteristic
155C2×C8—22no—
156C2×C8—41no—
157C2×C8—41no—
158C2×C8—944no—
159C2×C8—944no—
160C2×C8—944no—
161C4⋊C4—24no—
162C4⋊C4—24no—
163C4⋊C4—22no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—3762no—
168C4⋊C4—1882no—
169C4⋊C4—1882no—
170C4⋊C4Dic47⋊C414yes—
171C42—3762no—
172C42C2×Dic9411yescharacteristic
173C42C2×C4×D4711yescharacteristic
174C42#1062535212yes—
175C23#1309399211yescharacteristic
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—21no—
180C2×C4—24no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—82no—
184C2×C4—81no—
185C2×C4—1882no—
186C2×C4—1882no—
187C2×C4—1882no—
188C2×C4—1882no—
189C2×C4—1882no—
190C2×C4—1882no—
191C2×C4C2×Dic47⋊C411yescharacteristic
192C2×C4(C22⋊D47)⋊C412yes—
193C2×C4#1309400112yes—
194C2×C4#1309400511yescharacteristic
195C8—42no—
196C8—948no—
197C22#1568899511yescharacteristic
198C22#1568928912yes—
199C22#1568929511yescharacteristic
200C22#1568930412yes—
201C22#1568930511yescharacteristic
202C4—21no—
203C4—21no—
204C4—22no—
205C4—22no—
206C4—44no—
207C4—161no—
208C4—3762no—
209C4—3762no—
210C2#2024268011yescharacteristic
211C2#2024269812yes—
212C2#2024271411yescharacteristic
213C2#2024360211yescharacteristic
214C2#2024361212yes—
subgroupG/H
centerC23#13093992
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1992C2
socle, prime radicalC23×C47#120
cosocleC2×C42×C47C23
hypercenter#1992D47
hypocenterC47#8913
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1992

Complements

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

NH ≅ G/Ntype
13. #15688865C4semidirect
125. C47#8913semidirect

Maximal subgroups

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

subgroupindex
#202465742
#202435542
C2×#156888652
#202412702
C47×#19922
#891347