Finite Group Databasebeta

← #26111605

Subgroups — #26111605

Order 12032.

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

normal subgroups only

This group has 426 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#20252490C211yesmaximal, characteristic, complemented
2#20241983C211yesmaximal, characteristic, complemented
3#20241805C212yesmaximal
4#20241225C212yesmaximal, complemented
5C47×#3131C211yesmaximal, characteristic
6#15689254—22no—
7Q8×C47⋊C8—22no—
8#15689184—22no—
9#15689176C2212yes—
10C4×#10625329C2212yes—
11#15688865—22no—
12D4⋊(C47⋊C8)—22no—
13D4⋊(C47⋊C8)—22no—
14C47×#533C2211yescharacteristic
15C47×#532C2211yescharacteristic
16C47×#488—22no—
17C47×#487C412yes—
18C47×#486C412yes—
19C47×#416—22no—
20C2×C4×C8×C47C2211yescharacteristic
21C2×#10625329—22no—
22C4⋊(C47⋊C8)—24no—
23C4⋊(C47⋊C8)D412yes—
24C4⋊(C47⋊C8)D412yes—
25C22⋊(C47⋊C8)—22no—
26#13093935—22no—
27#13093935—22no—
28C4×C47⋊C8—22no—
29C4×C47⋊C8—22no—
30C4×C47⋊C8—22no—
31C2×#41×C47—21no—
32C22×C8×C47—21no—
33C4⋊Q8×C47C2×C411yescharacteristic
34C4⋊Q8×C47C2×C412yes—
35#127×C47C2×C411yescharacteristic
36#125×C47D412yes—
37#124×C47—21no—
38C22⋊Q8×C47—22no—
39#121×C47—22no—
40C4×Q8×C47—22no—
41C4×D4×C47—22no—
42#117×C47D412yes—
43C2×C42×C47C2311yescharacteristic
44#103×C47—24no—
45#102×C47—24no—
46C4×C8×C47—21no—
47C4×C8×C47C2×C412yes—
48#41×C47—44no—
49C2×Q8×C47—22no—
50C2×Q8×C47—21no—
51C2×D4×C47—21no—
52C2×D4×C47—22no—
53C2×D4×C47—41no—
54C22×C4×C47—41no—
55C22×C4×C47C2×D411yescharacteristic
56C22×C4×C47#4112yes—
57C2×C8×C47—22no—
58C2×C8×C47—42no—
59C4⋊C4×C47—22no—
60C4⋊C4×C47—24no—
61C4⋊C4×C47—22no—
62C4⋊C4×C47C22⋊C414yes—
63C22⋊C4×C47—22no—
64C22⋊C4×C47—42no—
65C42×C47—22no—
66C42×C47C22×C411yescharacteristic
67C42×C47C2×D411yescharacteristic
68C42×C47C2×D412yes—
69#10625329—24no—
70#10625329—42no—
71C2×C47⋊C8—22no—
72C2×C47⋊C8—22no—
73C2×C47⋊C8—42no—
74C23×C47—41no—
75C23×C47#12311yescharacteristic
76Q8×C47—42no—
77Q8×C47—42no—
78Q8×C47—42no—
79D4×C47—42no—
80D4×C47—42no—
81D4×C47—42no—
82D4×C47—82no—
83C2×C4×C47—22no—
84C2×C4×C47—22no—
85C2×C4×C47—22no—
86C2×C4×C47—24no—
87C2×C4×C47—41no—
88C2×C4×C47—82no—
89C2×C4×C47C2×C22⋊C411yescharacteristic
90C2×C4×C47C4×D412yes—
91C2×C4×C47#12011yescharacteristic
92C2×C4×C47#12112yes—
93C8×C47—42no—
94C47⋊C8—44no—
95C47⋊C8—42no—
96C47⋊C8—42no—
97#9070—471nomaximal, Sylow
98C22×C47—22no—
99C22×C47—21no—
100C22×C47—82no—
101C22×C47—81no—
102C22×C47#38611yescharacteristic
103C4×C47—22no—
104C4×C47—21no—
105C4×C47—44no—
106C4×C47—81no—
107C4×C47#44312yes—
108C4×C47#44412yes—
109C4×C47#45412yes—
110#3131D4711yescharacteristic
111#2986—471no—
112#1526—471no—
113#1484—472no—
114#1278—472no—
115C2×C47—41no—
116C2×C47—161no—
117C2×C47#187612yes—
118C2×C47#188811yescharacteristic
119#533C2×D4711yescharacteristic
120#532C2×D4711yescharacteristic
121#488—22no—
122#487Dic4712yes—
123#486Dic4712yes—
124#446—942no—
125C8×Q8—942no—
126#433—942no—
127#431—472no—
128#416—22no—
129C4×M16—472no—
130C2×C4×C8C2×D4711yescharacteristic
131#330—942no—
132D4⋊C8—942no—
133D4⋊C8—942no—
134C47#907011yescharacteristic, complemented, Sylow
135C2×#41—21no—
136C2×M16—942no—
137C22×C8—21no—
138C4⋊Q8C2×Dic4711yescharacteristic
139C4⋊Q8C2×Dic4712yes—
140#127C2×Dic4711yescharacteristic
141#125C22⋊D4712yes—
142#124—21no—
143C22⋊Q8—22no—
144#121—22no—
145C4×Q8—22no—
146C4×D4—22no—
147#117C22⋊D4712yes—
148C2×C42C22×D4711yescharacteristic
149C4⋊C8—472no—
150C4⋊C8—472no—
151C4⋊C8—944no—
152#103—24no—
153#102—24no—
154C22⋊C8—942no—
155#97—942no—
156#97—942no—
157C4×C8—21no—
158C4×C8—942no—
159C4×C8—942no—
160C4×C8—942no—
161C4×C8C2×Dic4712yes—
162#41—44no—
163C2×Q8—22no—
164C2×Q8—21no—
165C2×D4—21no—
166C2×D4—22no—
167C2×D4—41no—
168C22×C4—41no—
169C22×C4#1062535212yes—
170C22×C4C2×C22⋊D4711yescharacteristic
171M16—944no—
172M16—1882no—
173C2×C8—22no—
174C2×C8—42no—
175C2×C8—942no—
176C2×C8—942no—
177C2×C8—1882no—
178C4⋊C4—22no—
179C4⋊C4—24no—
180C4⋊C4—22no—
181C4⋊C4C22⋊Dic4714yes—
182C22⋊C4—22no—
183C22⋊C4—42no—
184C42—22no—
185C42D4×D4712yes—
186C42C22×Dic4711yescharacteristic
187C42C2×C22⋊D4711yescharacteristic
188C23—41no—
189C23#1309400511yescharacteristic
190Q8—42no—
191Q8—42no—
192Q8—42no—
193D4—42no—
194D4—42no—
195D4—42no—
196D4—82no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—24no—
201C2×C4—41no—
202C2×C4—82no—
203C2×C4C2×C22⋊Dic4711yescharacteristic
204C2×C4D4×Dic4712yes—
205C2×C4#1309399211yescharacteristic
206C2×C4#1309399612yes—
207C8—42no—
208C8—1884no—
209C8—1882no—
210C8—1882no—
211C22—22no—
212C22—21no—
213C22—82no—
214C22—81no—
215C22#1568900311yescharacteristic
216C4—22no—
217C4—21no—
218C4—44no—
219C4—81no—
220C4#1568924012yes—
221C4#1568924512yes—
222C4#1568929512yes—
223C2—41no—
224C2—161no—
225C2#2024371712yes—
226C2#2024379211yescharacteristic
subgroupG/H
centerC2×C4#13093992
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#3131C2
socleC22×C47#386
prime radicalC47×#532C22
cosocleC2×C42×C47C23
hypercenter#3131D47
hypocenterC47#9070
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3131

Complements

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

NH ≅ G/Ntype
1. #20252490C2semidirect
2. #20241983C2semidirect
4. #20241225C2semidirect
134. C47#9070semidirect

Maximal subgroups

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

subgroupindex
#202524902
#202419832
#202418052
#202412252
C47×#31312
#907047