Finite Group Databasebeta

← #26117274

Subgroups — #26117274

Order 12032.

476 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 105 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#20253912C211yesmaximal, characteristic
2#20253428C212yesmaximal
3#20244328C211yesmaximal, characteristic
4#20241792C212yesmaximal
5C47×#1275C211yesmaximal, characteristic
6#15689680C2212yes—
7C2×C4×Dic94C2211yescharacteristic
8#15689532—22no—
9#15689532—22no—
10#15689459—22no—
11#15689419—22no—
12#15689334—22no—
13#15689281—22no—
14C4×Dic188—24no—
15#15689161—22no—
16C2×C4⋊(C47⋊C8)C2211yescharacteristic
17#15688877—22no—
18#15688855—24no—
19C47×#431C2212yes—
20C2×#97×C47C2211yescharacteristic
21C22×Dic94D411yescharacteristic
22C2×Dic188—24no—
23C22×C47⋊C8—21no—
24#13094024D412yes—
25#13094024D412yes—
26#13094016—22no—
27#13094003—22no—
28#13094003—22no—
29C4×Dic94—22no—
30C4×Dic94—22no—
31C4×Dic94—22no—
32C4×Dic94—22no—
33#13093983—22no—
34C2×C4⋊Dic47D411yescharacteristic
35C2×Dic47⋊C4—21no—
36C2×C4×Dic47—21no—
37#13093958—24no—
38#13093958—22no—
39#13093958—22no—
40#13093956—24no—
41C4⋊(C47⋊C8)—22no—
42#13093949—24no—
43#13093949—24no—
44C22×C8×C47D412yes—
45C2×C42×C47C2311yescharacteristic
46C4⋊C8×C47—22no—
47C22⋊C8×C47—22no—
48#97×C47—22no—
49C4×C8×C47D414yes—
50C22×C4×C47C2×D411yescharacteristic
51C22×C4×C47C2×D411yescharacteristic
52C22×C4×C47#4111yescharacteristic
53C22×Dic47—21no—
54C22×Dic47—21no—
55C2×Dic94—22no—
56C2×Dic94—22no—
57C2×Dic94—22no—
58C2×Dic94—42no—
59C2×C8×C47—22no—
60C2×C8×C47—22no—
61C2×C8×C47—24no—
62C2×C8×C47Q1614yescomplemented
63C42×C47C2×D412yes—
64C42×C47C2×D412yes—
65C22⋊Dic47—42no—
66C4⋊Dic47—22no—
67C4⋊Dic47—22no—
68C4⋊Dic47—22no—
69C4⋊Dic47—22no—
70C4⋊Dic47—22no—
71Dic47⋊C4—42no—
72Dic47⋊C4—42no—
73Dic47⋊C4—42no—
74C4×Dic47—41no—
75C4×Dic47—41no—
76C4×Dic47—42no—
77C2×C47⋊C8—41no—
78C2×C47⋊C8—42no—
79Dic188—48no—
80C23×C47#12111yescharacteristic
81C2×C4×C47—41no—
82C2×C4×C47—41no—
83C2×C4×C47#12012yes—
84C2×C4×C47#12011yescharacteristic
85C2×C4×C47#12112yes—
86C2×C4×C47#12711yescharacteristic
87C2×C4×C47C2×Q1612yes—
88C2×C4×C47#13712yes—
89C2×Dic47—22no—
90C2×Dic47—41no—
91C2×Dic47—41no—
92C2×Dic47—42no—
93C2×Dic47—41no—
94C2×Dic47—42no—
95C2×Dic47—42no—
96Dic94—24no—
97Dic94—42no—
98Dic94—44no—
99Dic94—44no—
100C8×C47—24no—
101C8×C47—44no—
102C47⋊C8—81no—
103#10037—471nomaximal, Sylow
104C22×C47—21no—
105C22×C47—21no—
106C22×C47#39211yescharacteristic
107C22×C47#46212yes—
108C4×C47—22no—
109C4×C47—22no—
110C4×C47#45112yes—
111C4×C47#49411yescharacteristic
112C4×C47#49711yescharacteristic
113Dic47—42no—
114Dic47—82no—
115Dic47—82no—
116Dic47—81no—
117#3162—471no—
118#3104—472no—
119#1985—471no—
120#1481—472no—
121#1275D4711yescharacteristic
122C2×C47—22no—
123C2×C47#201011yescharacteristic
124C2×C47#201611yescharacteristic
125C2×C47#201711yescharacteristic
126#533—472no—
127C2×C4×Q8—471no—
128#501—942no—
129#501—942no—
130#484—942no—
131#477—942no—
132#462—942no—
133#451—942no—
134C4×Q16—944no—
135#431C2×D4712yes—
136#427—942no—
137C2×C4⋊C8—471no—
138C2×#97C2×D4711yescharacteristic
139#336—942no—
140Q8⋊C8—944no—
141C47#1003711yescharacteristic, complemented, Sylow
142C22×Q8—471no—
143C2×Q16—944no—
144C22×C8—941no—
145C22×C8D18812yes—
146C4⋊Q8—472no—
147C4⋊Q8—472no—
148#125—942no—
149C22⋊Q8—942no—
150C22⋊Q8—942no—
151C4×Q8—942no—
152C4×Q8—942no—
153C4×Q8—942no—
154C4×Q8—942no—
155#117—942no—
156C2×C4⋊C4—471no—
157C2×C4⋊C4—941no—
158C2×C42—941no—
159C2×C42C22×D4711yescharacteristic
160#107—942no—
161#107—942no—
162#107—944no—
163#106—944no—
164C4⋊C8—22no—
165C4⋊C8—942no—
166#103—944no—
167#103—944no—
168C22⋊C8—22no—
169#97—22no—
170C4×C8D18814yes—
171C2×Q8—942no—
172C2×Q8—942no—
173C2×Q8—942no—
174C2×Q8—1882no—
175C22×C4—941no—
176C22×C4—941no—
177C22×C4C2×D18811yescharacteristic
178C22×C4D4×D4711yescharacteristic
179C22×C4#1062535411yescharacteristic
180Q16—1888no—
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—24no—
184C2×C8—1881no—
185C2×C8—1882no—
186C2×C8Dic18814yescomplemented
187C4⋊C4—942no—
188C4⋊C4—942no—
189C4⋊C4—942no—
190C4⋊C4—942no—
191C4⋊C4—942no—
192C4⋊C4—1882no—
193C4⋊C4—1882no—
194C4⋊C4—1882no—
195C22⋊C4—1882no—
196C42—1881no—
197C42—1881no—
198C42—1882no—
199C42C2×D18812yes—
200C42D4×D4712yes—
201C23#1309399911yescharacteristic
202Q8—944no—
203Q8—1884no—
204Q8—1884no—
205Q8—1882no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—942no—
209C2×C4—1882no—
210C2×C4—1881no—
211C2×C4—1882no—
212C2×C4—1882no—
213C2×C4—1881no—
214C2×C4—1881no—
215C2×C4#1309399412yes—
216C2×C4#1309399411yescharacteristic
217C2×C4#1309399912yes—
218C2×C4#1309402111yescharacteristic
219C2×C4C2×Dic18812yes—
220C2×C4#1309405712yes—
221C8—24no—
222C8—44no—
223C8—3761no—
224C22—21no—
225C22—21no—
226C22#1568903311yescharacteristic
227C22#1568933412yes—
228C4—22no—
229C4—22no—
230C4—3761no—
231C4—3762no—
232C4—3762no—
233C4—1882no—
234C4#1568928112yes—
235C4#1568950311yescharacteristic
236C4#1568951711yescharacteristic
237C2—22no—
238C2#2024445611yescharacteristic
239C2#2024449011yescharacteristic
240C2#2024449311yescharacteristic
subgroupG/H
centerC2×C4#13094021
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#1275C2
socleC22×C47#392
prime radicalC23×C47#121
hypercenter#1275D47
hypocenterC47#10037
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1275

Complements

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

NH ≅ G/Ntype
62. C2×C8×C47Q16semidirect
141. C47#10037semidirect
186. C2×C8Dic188semidirect

Maximal subgroups

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

subgroupindex
#202539122
#202534282
#202443282
#202417922
C47×#12752
#1003747