Finite Group Databasebeta

← #26119408

Subgroups — #26119408

Order 12032.

416 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 101 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#20244369C211yesmaximal, characteristic
2#20241886C212yesmaximal
3#20241291C212yesmaximal
4#20241217C211yesmaximal, characteristic
5C47×#2399C211yesmaximal, characteristic
6#15689680C2211yescharacteristic
7#15689176C2212yes—
8#15689147C2211yescharacteristic
9#15688883—21no—
10#15688883—21no—
11#15688859—22no—
12#15688859—22no—
13C2×C4×Q8×C47—21no—
14C47×#398—22no—
15C47×#395—22no—
16C4×C4⋊C4×C47—21no—
17C4×C4⋊C4×C47C2212yes—
18C47×#376C2211yescharacteristic
19C2×#10625329—21no—
20#13094024Q812yescomplemented
21#13094016—21no—
22#13094003—22no—
23C4×Dic94—22no—
24#13093983—21no—
25C4⋊(C47⋊C8)—22no—
26C4⋊(C47⋊C8)—22no—
27C22⋊(C47⋊C8)—22no—
28C4×C47⋊C8—22no—
29C22×Q8×C47—21no—
30C4×Q8×C47—42no—
31C4×Q8×C47—42no—
32C2×C4⋊C4×C47—22no—
33C2×C4⋊C4×C47—21no—
34C2×C4⋊C4×C47—22no—
35C2×C4⋊C4×C47—22no—
36C2×C42×C47—22no—
37C2×C42×C47D411yescharacteristic
38C2×C42×C47D411yescharacteristic
39C2×C42×C47C2311yescharacteristic
40#95×C47—21no—
41#95×C47—21no—
42#95×C47—22no—
43#95×C47D412yes—
44#95×C47D412yes—
45C2×Q8×C47—42no—
46C2×Q8×C47—44no—
47C22×C4×C47—22no—
48C22×C4×C47—21no—
49C22×C4×C47—21no—
50C22×C4×C47—22no—
51C22×C4×C47C2×D412yes—
52C22×C4×C47C2×D411yescharacteristic
53C2×Dic94—22no—
54C4⋊C4×C47—44no—
55C4⋊C4×C47—44no—
56C4⋊C4×C47—44no—
57C4⋊C4×C47—42no—
58C42×C47—21no—
59C42×C47—21no—
60C42×C47—42no—
61C42×C47—42no—
62C42×C47SD1612yes—
63C42×C47Q1612yes—
64C42×C47C2×Q811yescharacteristic
65C42×C47#4112yes—
66C42×C47#4111yescharacteristic
67C22⋊Dic47—41no—
68C4⋊Dic47—22no—
69C4⋊Dic47—22no—
70Dic47⋊C4—42no—
71C4×Dic47—41no—
72#10625329—41no—
73#10625329—41no—
74C2×C47⋊C8—42no—
75C2×C47⋊C8—41no—
76C23×C47#12011yescharacteristic
77Q8×C47—48no—
78C2×C4×C47—24no—
79C2×C4×C47—22no—
80C2×C4×C47—22no—
81C2×C4×C47—42no—
82C2×C4×C47—42no—
83C2×C4×C47—42no—
84C2×C4×C47—42no—
85C2×C4×C47—42no—
86C2×C4×C47—42no—
87C2×C4×C47—42no—
88C2×C4×C47—42no—
89C2×C4×C47C22⋊Q812yes—
90C2×C4×C47C22⋊Q811yescharacteristic
91C2×C4×C47#12412yes—
92C2×C4×C47#12411yescharacteristic
93C2×C4×C47C2×SD1611yescharacteristic
94C2×C4×C47C2×Q1611yescharacteristic
95C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
96C2×C4×C47#13711yescharacteristic
97C2×Dic47—42no—
98Dic94—82no—
99C47⋊C8—82no—
100C47⋊C8—81no—
101#10371—471nomaximal, Sylow
102C22×C47—22no—
103C22×C47#39311yescharacteristic
104C22×C47#45011yescharacteristic
105C22×C47#45111yescharacteristic
106C4×C47—22no—
107C4×C47—22no—
108C4×C47—42no—
109C4×C47—44no—
110C4×C47—42no—
111C4×C47—82no—
112C4×C47#47511yescharacteristic
113C4×C47#47711yescharacteristic
114C4×C47#48711yescharacteristic
115C4×C47#48911yescharacteristic
116Dic47—82no—
117#2399D4711yescharacteristic
118#1995—471no—
119#1503—472no—
120#1303—472no—
121#1274—471no—
122C2×C47—21no—
123C2×C47—21no—
124C2×C47#202411yescharacteristic
125C2×C47#203112yes—
126#533—471no—
127C2×C4×Q8—21no—
128#431—472no—
129#423—471no—
130#398—22no—
131#395—22no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4C2×D4712yes—
134#376C2×D4711yescharacteristic
135#339—941no—
136#339—941no—
137#328—942no—
138#328—942no—
139C47#1037111yescharacteristic, complemented, Sylow
140C22×Q8—21no—
141C2×M16—941no—
142C4⋊Q8—472no—
143#125—941no—
144C22⋊Q8—942no—
145C4×Q8—42no—
146C4×Q8—42no—
147C4×Q8—942no—
148#117—941no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C42—22no—
154C2×C42C22⋊D4711yescharacteristic
155C2×C42C22⋊D4711yescharacteristic
156C2×C42C22×D4711yescharacteristic
157C4⋊C8—942no—
158C4⋊C8—942no—
159C22⋊C8—942no—
160C4×C8—942no—
161#95—21no—
162#95—21no—
163#95—22no—
164#95C22⋊D4712yes—
165#95C22⋊D4712yes—
166C2×Q8—42no—
167C2×Q8—44no—
168C2×Q8—942no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4C2×C22⋊D4712yes—
174C22×C4C2×C22⋊D4711yescharacteristic
175M16—1881no—
176M16—1881no—
177C2×C8—1882no—
178C2×C8—1881no—
179C4⋊C4—44no—
180C4⋊C4—44no—
181C4⋊C4—44no—
182C4⋊C4—42no—
183C4⋊C4—942no—
184C4⋊C4—942no—
185C4⋊C4—1882no—
186C22⋊C4—1881no—
187C42—21no—
188C42—21no—
189C42—42no—
190C42—42no—
191C42—1881no—
192C42#1062533512yes—
193C42#1062533712yes—
194C42Q8×D4711yescharacteristic
195C42#1062535411yescharacteristic
196C42#1062535412yes—
197C23#1309399311yescharacteristic
198Q8—48no—
199Q8—3762no—
200C2×C4—24no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—1882no—
212C2×C4#1309400012yes—
213C2×C4#1309400011yescharacteristic
214C2×C4#1309401312yes—
215C2×C4#1309401311yescharacteristic
216C2×C4C2×#1062533511yescharacteristic
217C2×C4C2×#1062533711yescharacteristic
218C2×C4#1309405011yescharacteristic
219C2×C4#1309405511yescharacteristic
220C8—3762no—
221C8—3761no—
222C22—22no—
223C22#1568903411yescharacteristic
224C22#1568927411yescharacteristic
225C22#1568927911yescharacteristic
226C4—22no—
227C4—22no—
228C4—44no—
229C4—42no—
230C4—42no—
231C4—82no—
232C4—3762no—
233C4#1568940511yescharacteristic
234C4#1568941511yescharacteristic
235C4#1568947511yescharacteristic
236C4#1568948511yescharacteristic
237C2—21no—
238C2—21no—
239C2#2024452611yescharacteristic
240C2#2024455712yes—
subgroupG/H
centerC2×C4#13094013
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#2399C2
socleC22×C47#393
prime radicalC23×C47#120
hypercenter#2399D47
hypocenterC47#10371
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2399

Complements

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

NH ≅ G/Ntype
20. #13094024Q8semidirect
139. C47#10371semidirect

Maximal subgroups

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

subgroupindex
#202443692
#202418862
#202412912
#202412172
C47×#23992
#1037147