Finite Group Databasebeta

← #26129024

Subgroups — #26129024

Order 12032.

388 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 63 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#20244555C211yesmaximal, characteristic
2#20242378C212yesmaximal
3#20241388C212yesmaximal
4#20241318C211yesmaximal, characteristic
5C47×#2817C211yesmaximal, characteristic
6#15689749C2211yescharacteristic
7#15689453—22no—
8#15688986—21no—
9#15688890C2211yescharacteristic
10#15688889—21no—
11#15688859—22no—
12#15688847C2212yes—
13C2×C22⋊Q8×C47—21no—
14C47×#400—22no—
15C47×#397—21no—
16C47×#397C2212yes—
17C47×#395—22no—
18C47×#381C2211yescharacteristic
19#13094003—22no—
20#13094001—42no—
21C2×C22⋊Dic47—21no—
22C2×C22⋊Dic47—21no—
23#13093955—42no—
24#13093947—42no—
25C22⋊(C47⋊C8)—22no—
26C47⋊#95—42no—
27C22×Q8×C47—21no—
28C23×C4×C47D411yescharacteristic
29C22⋊Q8×C47—42no—
30C22⋊Q8×C47—42no—
31C2×C4⋊C4×C47—22no—
32C2×C4⋊C4×C47—21no—
33C2×C4⋊C4×C47—21no—
34C2×C4⋊C4×C47D412yes—
35C2×C22⋊C4×C47—22no—
36C2×C22⋊C4×C47D411yescharacteristic
37C2×C22⋊C4×C47C2311yescharacteristic
38#95×C47—22no—
39#95×C47—21no—
40#95×C47—22no—
41#95×C47—22no—
42#95×C47D412yes—
43C24×C47C2×D411yescharacteristic
44C2×Q8×C47—42no—
45C2×Q8×C47—81no—
46C2×Q8×C47—81no—
47C22×C4×C47—21no—
48C22×C4×C47—21no—
49C22×C4×C47—22no—
50C22×C4×C47—22no—
51C22×C4×C47—21no—
52C22×C4×C47—41no—
53C22×C4×C47—41no—
54C22×C4×C47SD1612yes—
55C22×C4×C47C2×D412yes—
56C22×Dic47—22no—
57C22×Dic47—41no—
58C2×Dic94—22no—
59C4⋊C4×C47—42no—
60C4⋊C4×C47—44no—
61C4⋊C4×C47—42no—
62C4⋊C4×C47—81no—
63C22⋊C4×C47—41no—
64C22⋊C4×C47—42no—
65C22⋊C4×C47—42no—
66C22⋊C4×C47—41no—
67C22⋊Dic47—41no—
68C22⋊Dic47—41no—
69C22⋊Dic47—41no—
70C22⋊Dic47—41no—
71C4⋊Dic47—22no—
72Dic47⋊C4—42no—
73C2×C47⋊C8—42no—
74C23×C47—21no—
75C23×C47—21no—
76C23×C47—42no—
77C23×C47#12011yescharacteristic
78C23×C47C2×SD1611yescharacteristic
79C23×C47AGL(1,ℤ/8)11yescharacteristic
80Q8×C47—84no—
81C2×C4×C47—22no—
82C2×C4×C47—22no—
83C2×C4×C47—41no—
84C2×C4×C47—41no—
85C2×C4×C47—42no—
86C2×C4×C47—42no—
87C2×C4×C47—44no—
88C2×C4×C47—42no—
89C2×C4×C47—42no—
90C2×C4×C47—42no—
91C2×C4×C47—41no—
92C2×C4×C47—42no—
93C2×C4×C47—42no—
94C2×C4×C47—81no—
95C2×Dic47—42no—
96C2×Dic47—82no—
97C2×Dic47—81no—
98C2×Dic47—82no—
99Dic94—82no—
100C47⋊C8—82no—
101#12207—471nomaximal, Sylow
102C22×C47—22no—
103C22×C47—21no—
104C22×C47—21no—
105C22×C47—41no—
106C22×C47—41no—
107C22×C47—42no—
108C22×C47—81no—
109C22×C47#45011yescharacteristic
110C22×C47#45511yescharacteristic
111C22×C47B(4,𝔽2)11yescharacteristic
112C4×C47—42no—
113C4×C47—42no—
114C4×C47—82no—
115C4×C47—81no—
116C4×C47—82no—
117Dic47—82no—
118Dic47—161no—
119#2817D4711yescharacteristic
120#2030—471no—
121#1608—472no—
122#1344—472no—
123#1317—471no—
124C2×C47—21no—
125C2×C47—21no—
126C2×C47—41no—
127C2×C47—41no—
128C2×C47#160111yescharacteristic
129C2×C47#220112yes—
130#543—471no—
131C2×C22⋊Q8—21no—
132#483—942no—
133#400—22no—
134#397—21no—
135#397C2×D4712yes—
136#395—22no—
137#381—941no—
138#381C2×D4711yescharacteristic
139#342—471no—
140#342—941no—
141#328—942no—
142#323—472no—
143C47#1220711yescharacteristic, complemented, Sylow
144C22×Q8—21no—
145C23×C4C22⋊D4711yescharacteristic
146C22⋊Q8—42no—
147C22⋊Q8—42no—
148C22⋊Q8—942no—
149C22⋊Q8—1882no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4C22⋊D4712yes—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—941no—
156C2×C22⋊C4—941no—
157C2×C22⋊C4C22⋊D4711yescharacteristic
158C2×C22⋊C4C22×D4711yescharacteristic
159#106—1882no—
160#103—1882no—
161C22⋊C8—942no—
162#95—22no—
163#95—21no—
164#95—22no—
165#95—22no—
166#95—1882no—
167#95C22⋊D4712yes—
168C24C2×C22⋊D4711yescharacteristic
169C2×Q8—42no—
170C2×Q8—81no—
171C2×Q8—81no—
172C2×Q8—942no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—942no—
181C22×C4—1881no—
182C22×C4#1062533512yes—
183C22×C4C2×C22⋊D4712yes—
184C2×C8—1882no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—44no—
188C4⋊C4—81no—
189C4⋊C4—942no—
190C4⋊C4—1882no—
191C22⋊C4—41no—
192C22⋊C4—42no—
193C22⋊C4—42no—
194C22⋊C4—41no—
195C22⋊C4—1881no—
196C22⋊C4—1881no—
197C22⋊C4—1881no—
198C22⋊C4—1881no—
199C23—21no—
200C23—21no—
201C23—42no—
202C23#1309399311yescharacteristic
203C23C2×#1062533511yescharacteristic
204C23#1309405011yescharacteristic
205Q8—84no—
206Q8—3762no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—44no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—81no—
221C2×C4—3762no—
222C2×C4—3761no—
223C2×C4—3762no—
224C2×C4—1882no—
225C8—3762no—
226C22—22no—
227C22—21no—
228C22—21no—
229C22—41no—
230C22—41no—
231C22—42no—
232C22—81no—
233C22#1568927411yescharacteristic
234C22#1568930111yescharacteristic
235C22#1568931011yescharacteristic
236C4—42no—
237C4—42no—
238C4—82no—
239C4—81no—
240C4—82no—
241C4—3762no—
242C4—7521no—
243C2—21no—
244C2—21no—
245C2—41no—
246C2—41no—
247C2#2024234011yescharacteristic
248C2#2024540112yes—
subgroupG/H
centerC22#15689310
commutator, cosocleC2×C22⋊C4×C47C23
FrattiniC2×C22⋊C4C22×D47
FittingC47×#2817C2
socleC22×C47B(4,𝔽2)
prime radicalC24×C47C2×D4
hypercenter#2817D47
hypocenterC47#12207
LCSG ▷ C2×C22⋊C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2817

Complements

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

NH ≅ G/Ntype
143. C47#12207semidirect

Maximal subgroups

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

subgroupindex
#202445552
#202423782
#202413882
#202413182
C47×#28172
#1220747