Finite Group Databasebeta

← #26111604

Subgroups — #26111604

Order 12032.

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

normal subgroups only

This group has 426 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#20253713C211yesmaximal, characteristic
2#20241983C211yesmaximal, characteristic, complemented
3#20241804C212yesmaximal
4#20241224C212yesmaximal, complemented
5C47×#2986C211yesmaximal, characteristic, complemented
6#15689673C2211yescharacteristic
7#15689480—22no—
8#15689475C412yes—
9#15689471C412yes—
10#15689117—22no—
11C4×#10625329C2212yes—
12C2×C4×C47⋊C8C2211yescharacteristic
13#15688865—22no—
14#15688850—22no—
15#15688850—22no—
16C47×#533C2211yescharacteristic
17C47×#446—22no—
18C8×Q8×C47—22no—
19C47×#433—22no—
20C47×#431C2212yes—
21C2×#10625350—21no—
22C2×#10625329—22no—
23C22×C47⋊C8—21no—
24#13094024C2×C411yescharacteristic
25#13094021C2×C411yescharacteristic
26#13094012—21no—
27#13093998—22no—
28C4×D188—22no—
29#13093947—24no—
30#13093944—24no—
31#13093935—22no—
32C4×C47⋊C8—21no—
33C4×C47⋊C8—22no—
34C4×C47⋊C8C2×C412yes—
35C4⋊Q8×C47C2×C412yes—
36#125×C47D412yes—
37C22⋊Q8×C47—22no—
38C4×Q8×C47—22no—
39#117×C47D412yes—
40C2×C42×C47C2311yescharacteristic
41C4⋊C8×C47—24no—
42C4⋊C8×C47D412yes—
43C4⋊C8×C47D412yes—
44C22⋊C8×C47—22no—
45#97×C47—22no—
46C4×C8×C47—22no—
47C4×C8×C47—22no—
48C2×Q8×C47—22no—
49C22×C4×C47C2×D411yescharacteristic
50C22×C4×C47#4112yes—
51C2×C22⋊D47—41no—
52#10625350—44no—
53C2×D188—21no—
54C2×D188—22no—
55C2×C4×D47—41no—
56C2×Dic94—21no—
57C2×C8×C47—22no—
58C2×C8×C47—22no—
59C4⋊C4×C47—22no—
60C4⋊C4×C47—24no—
61C4⋊C4×C47C22⋊C414yes—
62C22⋊C4×C47—22no—
63C42×C47—22no—
64C42×C47C22×C411yescharacteristic
65C42×C47C2×D411yescharacteristic
66C42×C47C2×D412yes—
67(C2×D47)⋊C4—42no—
68C4⋊Dic47—22no—
69#10625329—24no—
70#10625329—42no—
71C2×C47⋊C8—22no—
72C2×C47⋊C8—42no—
73C2×C47⋊C8—42no—
74C23×C47#12311yescharacteristic
75C22×D47—41no—
76Q8×C47—42no—
77Q8×C47—42no—
78C2×C4×C47—22no—
79C2×C4×C47—22no—
80C2×C4×C47—22no—
81C2×C4×C47—24no—
82C2×C4×C47C2×C22⋊C411yescharacteristic
83C2×C4×C47C4×D412yes—
84C2×C4×C47#12011yescharacteristic
85C2×C4×C47#12112yes—
86C22⋊D47—82no—
87C2×Dic47—41no—
88D188—42no—
89D188—42no—
90D188—42no—
91C4×D47—82no—
92Dic94—42no—
93C8×C47—42no—
94C8×C47—42no—
95C47⋊C8—42no—
96C47⋊C8—44no—
97#9070—471nomaximal, Sylow
98C22×C47—22no—
99C22×C47—21no—
100C22×C47#38611yescharacteristic
101C2×D47—82no—
102C2×D47—81no—
103C4×C47—22no—
104C4×C47—21no—
105C4×C47—44no—
106C4×C47#44312yes—
107C4×C47#44412yes—
108C4×C47#45412yes—
109Dic47—81no—
110#3131—471no—
111#2986D4711yescharacteristic, complemented
112#1526—471no—
113#1484—472no—
114#1278—472no—
115C2×C47—41no—
116C2×C47#187612yes—
117C2×C47#188811yescharacteristic
118D47—161no—
119#533C2×D4711yescharacteristic
120#532—471no—
121#488—942no—
122#487—472no—
123#486—472no—
124#446—22no—
125C8×Q8—22no—
126#433—22no—
127#431C2×D4712yes—
128#416—942no—
129C4×M16—472no—
130C2×C4×C8—471no—
131#330—942no—
132D4⋊C8—942no—
133D4⋊C8—942no—
134C47#907011yescharacteristic, complemented, Sylow
135C2×#41—941no—
136C2×M16—942no—
137C22×C8—941no—
138C4⋊Q8—471no—
139C4⋊Q8C4×D4712yes—
140#127—471no—
141#125D18812yes—
142#124—941no—
143C22⋊Q8—22no—
144#121—942no—
145C4×Q8—22no—
146C4×D4—942no—
147#117C22⋊D4712yes—
148C2×C42C22×D4711yescharacteristic
149C4⋊C8—24no—
150C4⋊C8C22⋊D4712yes—
151C4⋊C8C22⋊D4712yes—
152#103—944no—
153#102—944no—
154C22⋊C8—22no—
155#97—22no—
156#97—942no—
157C4×C8—22no—
158C4×C8—22no—
159C4×C8—472no—
160C4×C8—941no—
161C4×C8—942no—
162#41—1884no—
163C2×Q8—22no—
164C2×Q8—941no—
165C2×D4—941no—
166C2×D4—942no—
167C2×D4—1881no—
168C22×C4—1881no—
169C22×C4#1062535012yes—
170C22×C4C2×C22⋊D4711yescharacteristic
171M16—944no—
172M16—1882no—
173C2×C8—22no—
174C2×C8—22no—
175C2×C8—942no—
176C2×C8—1882no—
177C2×C8—1882no—
178C4⋊C4—22no—
179C4⋊C4—24no—
180C4⋊C4—942no—
181C4⋊C4(C2×D47)⋊C414yes—
182C22⋊C4—22no—
183C22⋊C4—1882no—
184C42—22no—
185C42C2×C4×D4711yescharacteristic
186C42C2×D18811yescharacteristic
187C42C2×C22⋊D4712yes—
188C23—1881no—
189C23#1309400711yescharacteristic
190Q8—42no—
191Q8—42no—
192Q8—1882no—
193D4—3762no—
194D4—1882no—
195D4—1882no—
196D4—1882no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—24no—
201C2×C4—3762no—
202C2×C4—1881no—
203C2×C4C2×(C2×D47)⋊C411yescharacteristic
204C2×C4C4×C22⋊D4712yes—
205C2×C4#1309399311yescharacteristic
206C2×C4#1309399812yes—
207C8—42no—
208C8—42no—
209C8—1882no—
210C8—1884no—
211C22—22no—
212C22—21no—
213C22—3762no—
214C22—3761no—
215C22#1568900211yescharacteristic
216C4—22no—
217C4—21no—
218C4—44no—
219C4—3761no—
220C4#1568923812yes—
221C4#1568924312yes—
222C4#1568929612yes—
223C2—41no—
224C2—7521no—
225C2#2024371512yes—
226C2#2024379111yescharacteristic
subgroupG/H
centerC2×C4#13093993
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2986C2
socleC22×C47#386
prime radical#15689673C22
cosocleC2×C42×C47C23
hypercenter#2986D47
hypocenterC47#9070
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2986

Complements

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

NH ≅ G/Ntype
2. #20241983C2semidirect
4. #20241224C2semidirect
5. C47×#2986C2semidirect
111. #2986D47semidirect
134. C47#9070semidirect

Maximal subgroups

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

subgroupindex
#202537132
#202419832
#202418042
#202412242
C47×#29862
#907047