Finite Group Databasebeta

← #26105578

Subgroups — #26105578

Order 12032.

394 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 95 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#20244951C212yesmaximal
2#20244926C211yesmaximal, characteristic
3#20241214C211yesmaximal, characteristic
4#20241211C212yesmaximal
5C47×#2734C211yesmaximal, characteristic
6C2×#13093957—21no—
7C2×#13093948—21no—
8#15688993C2211yescharacteristic
9#15688885—22no—
10#15688885—22no—
11#15688878C2211yescharacteristic
12#15688878C2212yes—
13C2×C4⋊Q8×C47—21no—
14C47×#398—21no—
15C47×#395—22no—
16C47×#391C2211yescharacteristic
17C47×#389C2212yes—
18C47×#387—22no—
19C22×C47⋊C8—21no—
20C22×C47⋊C8—22no—
21C2×C4⋊Dic47—21no—
22C2×C4⋊Dic47—22no—
23#13093957—42no—
24#13093948—42no—
25C22×Q8×C47—21no—
26C22×Q8×C47D411yescharacteristic
27C4⋊Q8×C47—42no—
28C4⋊Q8×C47—42no—
29C2×C4⋊C4×C47—22no—
30C2×C4⋊C4×C47—22no—
31C2×C4⋊C4×C47—21no—
32C2×C4⋊C4×C47D412yes—
33C2×C4⋊C4×C47D412yes—
34C2×C4⋊C4×C47D411yescharacteristic
35C2×C42×C47—21no—
36C2×C42×C47C2311yescharacteristic
37#95×C47—21no—
38#95×C47—22no—
39#95×C47—22no—
40#95×C47—22no—
41C2×Q8×C47—21no—
42C2×Q8×C47—42no—
43C2×Q8×C47—41no—
44C2×Q8×C47—41no—
45C2×Q8×C47—81no—
46C2×Q8×C47—81no—
47C2×Q8×C47SD1612yes—
48C22×C4×C47—22no—
49C22×C4×C47—21no—
50C22×C4×C47—22no—
51C22×C4×C47—21no—
52C22×C4×C47C2×D411yescharacteristic
53C22×C4×C47C2×D412yes—
54C22×Dic47—41no—
55C4⋊C4×C47—22no—
56C4⋊C4×C47—44no—
57C4⋊C4×C47—42no—
58C4⋊C4×C47—41no—
59C4⋊C4×C47—42no—
60C4⋊C4×C47—42no—
61C4⋊C4×C47—81no—
62C4⋊C4×C47Q1614yes—
63C42×C47—41no—
64C42×C47—42no—
65C4⋊Dic47—41no—
66C4⋊Dic47—41no—
67C4⋊Dic47—44no—
68C2×C47⋊C8—22no—
69C2×C47⋊C8—41no—
70C2×C47⋊C8—41no—
71C2×C47⋊C8—42no—
72C2×C47⋊C8—42no—
73C2×C47⋊C8—42no—
74C23×C47#12011yescharacteristic
75Q8×C47—42no—
76Q8×C47—81no—
77Q8×C47—84no—
78C2×C4×C47—21no—
79C2×C4×C47—22no—
80C2×C4×C47—22no—
81C2×C4×C47—21no—
82C2×C4×C47—22no—
83C2×C4×C47—24no—
84C2×C4×C47—42no—
85C2×C4×C47—41no—
86C2×C4×C47—42no—
87C2×C4×C47—41no—
88C2×C4×C47—42no—
89C2×C4×C47—41no—
90C2×C4×C47—42no—
91C2×C4×C47—42no—
92C2×C4×C47—42no—
93C2×C4×C47C2×SD1611yescharacteristic
94C2×C4×C47C2×Q1612yes—
95C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
96C2×C4×C47#13712yes—
97C2×Dic47—81no—
98C2×Dic47—82no—
99C47⋊C8—42no—
100C47⋊C8—82no—
101#7829—471nomaximal, Sylow
102C22×C47#45011yescharacteristic
103C22×C47#45112yes—
104C22×C47#45311yescharacteristic
105C22×C47#45412yes—
106C22×C47#45811yescharacteristic
107C4×C47—21no—
108C4×C47—21no—
109C4×C47—22no—
110C4×C47—22no—
111C4×C47—42no—
112C4×C47—44no—
113C4×C47—82no—
114C4×C47—81no—
115Dic47—161no—
116#2734D4711yescharacteristic
117#2100—472no—
118#2095—471no—
119#1272—471no—
120#1271—472no—
121C2×C47#161311yescharacteristic
122C2×C47#161712yes—
123C2×C47#163311yescharacteristic
124C2×C47#163712yes—
125C2×C47#165711yescharacteristic
126C2×C4⋊Q8—21no—
127C2×#107—941no—
128C2×#103—941no—
129#398—21no—
130#395—22no—
131#391C2×D4711yescharacteristic
132#389C2×D4712yes—
133#387—22no—
134#384—471no—
135#340—942no—
136#340—942no—
137#336—471no—
138#336—472no—
139C47#782911yescharacteristic, complemented, Sylow
140C22×Q8—21no—
141C22×Q8C22⋊D4711yescharacteristic
142C22×C8—941no—
143C22×C8—942no—
144C4⋊Q8—42no—
145C4⋊Q8—42no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—942no—
150C2×C4⋊C4—941no—
151C2×C4⋊C4C22⋊D4712yes—
152C2×C4⋊C4C22⋊D4712yes—
153C2×C4⋊C4C22⋊D4711yescharacteristic
154C2×C42—21no—
155C2×C42C22×D4711yescharacteristic
156#107—1882no—
157#103—1882no—
158#95—21no—
159#95—22no—
160#95—22no—
161#95—22no—
162C2×Q8—21no—
163C2×Q8—42no—
164C2×Q8—41no—
165C2×Q8—41no—
166C2×Q8—81no—
167C2×Q8—81no—
168C2×Q8#1062533512yes—
169C22×C4—22no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—1881no—
174C22×C4C2×C22⋊D4711yescharacteristic
175C22×C4C2×C22⋊D4712yes—
176C2×C8—942no—
177C2×C8—1882no—
178C2×C8—1882no—
179C2×C8—1882no—
180C2×C8—1881no—
181C2×C8—1881no—
182C4⋊C4—22no—
183C4⋊C4—41no—
184C4⋊C4—42no—
185C4⋊C4—44no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—81no—
189C4⋊C4—1881no—
190C4⋊C4—1881no—
191C4⋊C4—1884no—
192C4⋊C4#1062533714yes—
193C42—41no—
194C42—42no—
195C23#1309399311yescharacteristic
196Q8—42no—
197Q8—81no—
198Q8—84no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—24no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—41no—
209C2×C4—42no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—3761no—
215C2×C4—3762no—
216C2×C4C2×#1062533511yescharacteristic
217C2×C4C2×#1062533712yes—
218C2×C4#1309405011yescharacteristic
219C2×C4#1309405512yes—
220C8—3762no—
221C8—1882no—
222C22#1568927411yescharacteristic
223C22#1568927912yes—
224C22#1568929111yescharacteristic
225C22#1568929612yes—
226C22#1568931311yescharacteristic
227C4—21no—
228C4—21no—
229C4—22no—
230C4—22no—
231C4—44no—
232C4—42no—
233C4—82no—
234C4—81no—
235C4—7521no—
236C2#2024240811yescharacteristic
237C2#2024243012yes—
238C2#2024252811yescharacteristic
239C2#2024255312yes—
240C2#2024268611yescharacteristic
subgroupG/H
centerC23#13093993
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#2734C2
socle, prime radicalC23×C47#120
hypercenter#2734D47
hypocenterC47#7829
LCSG ▷ C2×C42×C47 ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2734

Complements

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

NH ≅ G/Ntype
139. C47#7829semidirect

Maximal subgroups

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

subgroupindex
#202449512
#202449262
#202412142
#202412112
C47×#27342
#782947