Finite Group Databasebeta

← #26120403

Subgroups — #26120403

Order 12032.

402 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 99 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#20244366C212yesmaximal
2#20243539C211yesmaximal, characteristic
3#20242078C212yesmaximal
4#20241263C211yesmaximal, characteristic
5C47×#3249C211yesmaximal, characteristic
6#15689147C2212yes—
7C2×C4⋊(C47⋊C8)C2211yescharacteristic
8C2×Dic47⋊C8—22no—
9C42×Dic47C2211yescharacteristic
10#15688882—22no—
11#15688882—22no—
12#15688872—22no—
13#15688869—22no—
14C47×#533C2212yes—
15C47×#477—22no—
16C47×#467—22no—
17C47×#462—22no—
18C2×C4⋊C8×C47C2211yescharacteristic
19C2×#10625329—22no—
20C22×C47⋊C8—21no—
21C2×C4×Dic47—21no—
22C2×C4×Dic47—21no—
23C2×C4×Dic47D411yescharacteristic
24C2×C4×Dic47D411yescharacteristic
25C4⋊(C47⋊C8)—22no—
26C4⋊(C47⋊C8)—22no—
27C4⋊(C47⋊C8)—22no—
28Dic47⋊C8—44no—
29C2×Q16×C47—22no—
30C22×C8×C47D412yes—
31C4⋊Q8×C47D412yes—
32C4⋊Q8×C47D412yes—
33#125×C47—22no—
34C22⋊Q8×C47—22no—
35C22⋊Q8×C47—22no—
36C4×Q8×C47—22no—
37C4×Q8×C47—22no—
38#117×C47—22no—
39C2×C42×C47C2311yescharacteristic
40#107×C47—22no—
41C4⋊C8×C47—22no—
42#103×C47—24no—
43C2×Q8×C47—22no—
44C2×Q8×C47—22no—
45C22×C4×C47C2×D411yescharacteristic
46C22×C4×C47C2×D411yescharacteristic
47C22×C4×C47#4111yescharacteristic
48C22×Dic47—21no—
49C22×Dic47—21no—
50Q16×C47—82no—
51C2×C8×C47—22no—
52C2×C8×C47—42no—
53C4⋊C4×C47—22no—
54C4⋊C4×C47—22no—
55C4⋊C4×C47—22no—
56C4⋊C4×C47—22no—
57C4⋊C4×C47—42no—
58C4⋊C4×C47—42no—
59C22⋊C4×C47—42no—
60C42×C47—42no—
61C42×C47C2×D412yes—
62C42×C47#4112yes—
63C4×Dic47—22no—
64C4×Dic47—22no—
65C4×Dic47—21no—
66C4×Dic47—22no—
67C4×Dic47—42no—
68C4×Dic47Q1612yescomplemented
69#10625329—42no—
70#10625329—42no—
71C2×C47⋊C8—42no—
72C2×C47⋊C8—41no—
73C2×C47⋊C8—42no—
74C23×C47#12111yescharacteristic
75Q8×C47—42no—
76Q8×C47—42no—
77Q8×C47—82no—
78C2×C4×C47—21no—
79C2×C4×C47—21no—
80C2×C4×C47—41no—
81C2×C4×C47—42no—
82C2×C4×C47—42no—
83C2×C4×C47#12011yescharacteristic
84C2×C4×C47#12112yes—
85C2×C4×C47#12312yes—
86C2×C4×C47#12411yescharacteristic
87C2×C4×C47C2×Q1611yescharacteristic
88C2×C4×C47#13511yescharacteristic
89C2×C4×C47#13712yes—
90C2×Dic47—22no—
91C2×Dic47—22no—
92C2×Dic47—22no—
93C2×Dic47—22no—
94C2×Dic47—41no—
95C2×Dic47—41no—
96C8×C47—42no—
97C47⋊C8—81no—
98C47⋊C8—82no—
99#10522—471nomaximal, Sylow
100C22×C47—21no—
101C22×C47—21no—
102C22×C47#39411yescharacteristic
103C22×C47#46711yescharacteristic
104C22×C47#47011yescharacteristic
105C4×C47—22no—
106C4×C47—21no—
107C4×C47—82no—
108C4×C47—82no—
109C4×C47#45311yescharacteristic
110C4×C47#45611yescharacteristic
111C4×C47#46212yes—
112C4×C47#46412yes—
113Dic47—24no—
114Dic47—42no—
115#3249D4711yescharacteristic
116#1994—472no—
117#1842—471no—
118#1550—472no—
119#1292—471no—
120C2×C47—21no—
121C2×C47—21no—
122C2×C47#203911yescharacteristic
123C2×C47#205011yescharacteristic
124C2×C47#205611yescharacteristic
125#533C2×D4712yes—
126#477—22no—
127#467—22no—
128#462—22no—
129#423—472no—
130C2×C4⋊C8—471no—
131C2×C4⋊C8—942no—
132C2×C4⋊C8C2×D4711yescharacteristic
133C43—471no—
134#339—942no—
135#339—942no—
136#333—942no—
137#332—942no—
138C47#1052211yescharacteristic, complemented, Sylow
139C2×Q16—22no—
140C2×M16—942no—
141C22×C8—941no—
142C22×C8C22⋊D4712yes—
143C4⋊Q8C22⋊D4712yes—
144C4⋊Q8C22⋊D4712yes—
145#125—22no—
146C22⋊Q8—22no—
147C22⋊Q8—22no—
148C4×Q8—22no—
149C4×Q8—22no—
150#117—22no—
151C2×C42—471no—
152C2×C42—471no—
153C2×C42—941no—
154C2×C42—941no—
155C2×C42C22×D4711yescharacteristic
156#107—22no—
157C4⋊C8—22no—
158C4⋊C8—942no—
159C4⋊C8—942no—
160C4⋊C8—942no—
161C4⋊C8—1884no—
162#103—24no—
163C2×Q8—22no—
164C2×Q8—22no—
165C22×C4—941no—
166C22×C4—941no—
167C22×C4D4×D4711yescharacteristic
168C22×C4#1062535211yescharacteristic
169C22×C4C2×C22⋊D4711yescharacteristic
170Q16—82no—
171M16—1882no—
172M16—1882no—
173C2×C8—22no—
174C2×C8—42no—
175C2×C8—1882no—
176C2×C8—1881no—
177C2×C8—1882no—
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C22⋊C4—42no—
185C42—42no—
186C42—472no—
187C42—942no—
188C42—942no—
189C42—941no—
190C42—942no—
191C42—1882no—
192C42#1062535212yes—
193C42C2×C22⋊D4712yes—
194C23#1309399511yescharacteristic
195Q8—42no—
196Q8—42no—
197Q8—82no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—41no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—942no—
204C2×C4—942no—
205C2×C4—942no—
206C2×C4—942no—
207C2×C4—1881no—
208C2×C4—1881no—
209C2×C4#1309399311yescharacteristic
210C2×C4#1309399612yes—
211C2×C4#1309400512yes—
212C2×C4#1309401111yescharacteristic
213C2×C4Q16×D4711yescharacteristic
214C2×C4#1309404711yescharacteristic
215C2×C4#1309405612yes—
216C8—42no—
217C8—3761no—
218C8—3762no—
219C22—21no—
220C22—21no—
221C22#1568903711yescharacteristic
222C22#1568936111yescharacteristic
223C22#1568937811yescharacteristic
224C4—21no—
225C4—22no—
226C4—82no—
227C4—82no—
228C4—944no—
229C4—1882no—
230C4#1568929111yescharacteristic
231C4#1568930611yescharacteristic
232C4#1568933112yes—
233C4#1568934712yes—
234C2—21no—
235C2—21no—
236C2#2024460911yescharacteristic
237C2#2024467211yescharacteristic
238C2#2024470811yescharacteristic
subgroupG/H
centerC22#15689037
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#3249C2
socleC22×C47#394
prime radicalC23×C47#121
hypercenter#3249D47
hypocenterC47#10522
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3249

Complements

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

NH ≅ G/Ntype
68. C4×Dic47Q16semidirect
138. C47#10522semidirect

Maximal subgroups

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

subgroupindex
#202443662
#202435392
#202420782
#202412632
C47×#32492
#1052247