Finite Group Databasebeta

← #26119351

Subgroups — #26119351

Order 12032.

400 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 91 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#20251660C211yesmaximal, characteristic
2#20244362C212yesmaximal
3#20241270C212yesmaximal
4#20241269C211yesmaximal, characteristic
5C47×#1960C211yesmaximal, characteristic
6C2×#13093935C2212yes—
7#15689065—22no—
8#15689062—21no—
9#15689044—22no—
10#15689043—21no—
11#15688989C2212yes—
12#15688885—22no—
13#15688885—22no—
14C2×C4⋊Q8×C47C2211yescharacteristic
15C2×#103×C47—21no—
16C2×#103×C47—21no—
17C2×#97×C47C2211yescharacteristic
18C47×#382C2211yescharacteristic
19C22×Dic94—21no—
20C22×C47⋊C8—22no—
21C2×C4⋊Dic47—21no—
22C2×C4⋊Dic47D412yes—
23C2×C4⋊Dic47D412yes—
24C2×Dic47⋊C4—22no—
25#13093935—22no—
26#13093935—22no—
27C47⋊#95—21no—
28C47⋊#95—22no—
29C47⋊#95—21no—
30C47⋊#95—22no—
31C47⋊#95—22no—
32C22×Q8×C47—21no—
33C22×C8×C47—21no—
34C4⋊Q8×C47—22no—
35C4⋊Q8×C47—21no—
36C4⋊Q8×C47Q812yescomplemented
37C2×C4⋊C4×C47—22no—
38C2×C4⋊C4×C47D411yescharacteristic
39C2×C4⋊C4×C47D411yescharacteristic
40C2×C42×C47C2311yescharacteristic
41#103×C47—42no—
42#103×C47—42no—
43#97×C47—22no—
44#95×C47—22no—
45C2×Q8×C47—22no—
46C2×Q8×C47—41no—
47C2×Q8×C47—41no—
48C2×Q8×C47—41no—
49C2×Q8×C47—41no—
50C2×Q8×C47—41no—
51C22×C4×C47—21no—
52C22×C4×C47—21no—
53C22×C4×C47—41no—
54C22×C4×C47C2×D412yes—
55C22×C4×C47C2×D411yescharacteristic
56C22×Dic47—22no—
57C22×Dic47—22no—
58C2×Dic94—81no—
59C2×Dic94—82no—
60C2×C8×C47—22no—
61C2×C8×C47—41no—
62C2×C8×C47—41no—
63C4⋊C4×C47—24no—
64C4⋊C4×C47—41no—
65C4⋊C4×C47—41no—
66C4⋊C4×C47—42no—
67C42×C47C2×Q811yescharacteristic
68C42×C47#4111yescharacteristic
69C42×C47#4112yes—
70C4⋊Dic47—42no—
71C4⋊Dic47—42no—
72C4⋊Dic47—81no—
73Dic47⋊C4—82no—
74C2×C47⋊C8—24no—
75C2×C47⋊C8—42no—
76C2×C47⋊C8—42no—
77C23×C47#12011yescharacteristic
78Q8×C47—42no—
79Q8×C47—42no—
80Q8×C47—81no—
81Q8×C47—81no—
82C2×C4×C47—21no—
83C2×C4×C47—21no—
84C2×C4×C47—22no—
85C2×C4×C47—22no—
86C2×C4×C47—42no—
87C2×C4×C47—41no—
88C2×C4×C47—41no—
89C2×C4×C47—42no—
90C2×C4×C47—42no—
91C2×C4×C47—81no—
92C2×C4×C47—81no—
93C2×C4×C47C22⋊Q812yes—
94C2×C4×C47C22⋊Q811yescharacteristic
95C2×C4×C47#12412yes—
96C2×C4×C47#12411yescharacteristic
97C2×Dic47—42no—
98C2×Dic47—42no—
99C2×Dic47—42no—
100C2×Dic47—42no—
101C2×Dic47—42no—
102C2×Dic47—42no—
103Dic94—84no—
104C8×C47—42no—
105C47⋊C8—44no—
106#10359—471nomaximal, Sylow
107C22×C47#39311yescharacteristic
108C22×C47#45311yescharacteristic
109C22×C47#45411yescharacteristic
110C22×C47#45512yes—
111C22×C47#45612yes—
112C4×C47—21no—
113C4×C47—21no—
114C4×C47—22no—
115C4×C47—22no—
116C4×C47—81no—
117C4×C47—81no—
118C4×C47—82no—
119Dic47—82no—
120Dic47—82no—
121#2844—471no—
122#1993—472no—
123#1960D4711yescharacteristic
124#1295—472no—
125#1295—471no—
126C2×C47#165911yescharacteristic
127C2×C47#166011yescharacteristic
128C2×C47#166412yes—
129C2×C47#202711yescharacteristic
130C2×C47#202812yes—
131C2×C4⋊Q8C2×D4711yescharacteristic
132C2×#103—21no—
133C2×#103—21no—
134C2×#97—472no—
135C2×#97C2×D4711yescharacteristic
136#400—942no—
137#400—941no—
138#395—942no—
139#395—941no—
140#382—472no—
141#382C2×D4711yescharacteristic
142#340—942no—
143#340—942no—
144C47#1035911yescharacteristic, complemented, Sylow
145C22×Q8—21no—
146C22×Q8—941no—
147C22×C8—21no—
148C22×C8—942no—
149C4⋊Q8—21no—
150C4⋊Q8—22no—
151C4⋊Q8Dic9412yescomplemented
152C2×C4⋊C4—22no—
153C2×C4⋊C4—472no—
154C2×C4⋊C4—472no—
155C2×C4⋊C4—942no—
156C2×C4⋊C4—941no—
157C2×C4⋊C4C22⋊D4711yescharacteristic
158C2×C4⋊C4C22⋊D4711yescharacteristic
159C2×C42C22×D4711yescharacteristic
160#103—42no—
161#103—42no—
162#97—22no—
163#97—942no—
164#97—942no—
165#95—22no—
166#95—942no—
167#95—941no—
168#95—941no—
169#95—942no—
170#95—942no—
171C2×Q8—22no—
172C2×Q8—41no—
173C2×Q8—41no—
174C2×Q8—41no—
175C2×Q8—41no—
176C2×Q8—41no—
177C2×Q8—3761no—
178C2×Q8—3762no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—41no—
182C22×C4—942no—
183C22×C4—942no—
184C22×C4D4×D4712yes—
185C22×C4C2×C22⋊D4711yescharacteristic
186C2×C8—22no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—944no—
190C2×C8—1882no—
191C2×C8—1882no—
192C4⋊C4—24no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—42no—
196C4⋊C4—3762no—
197C4⋊C4—3761no—
198C4⋊C4—1882no—
199C4⋊C4—1882no—
200C42C2×Dic9411yescharacteristic
201C42#1062535011yescharacteristic
202C42#1062535212yes—
203C23#1309399211yescharacteristic
204Q8—42no—
205Q8—42no—
206Q8—81no—
207Q8—81no—
208Q8—3764no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—1882no—
221C2×C4—1882no—
222C2×C4—1882no—
223C2×C4—1882no—
224C2×C4—1882no—
225C2×C4—1882no—
226C2×C4#1309400112yes—
227C2×C4#1309400311yescharacteristic
228C2×C4#1309401012yes—
229C2×C4#1309401111yescharacteristic
230C8—42no—
231C8—1884no—
232C22#1568903511yescharacteristic
233C22#1568929011yescharacteristic
234C22#1568929511yescharacteristic
235C22#1568929912yes—
236C22#1568930412yes—
237C4—21no—
238C4—21no—
239C4—22no—
240C4—22no—
241C4—81no—
242C4—81no—
243C4—82no—
244C4—3762no—
245C4—3762no—
246C2#2024269211yescharacteristic
247C2#2024269711yescharacteristic
248C2#2024271912yes—
249C2#2024454411yescharacteristic
250C2#2024454712yes—
subgroupG/H
centerC23#13093992
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#1960C2
socle, prime radicalC23×C47#120
hypercenter#1960D47
hypocenterC47#10359
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1960

Complements

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

NH ≅ G/Ntype
36. C4⋊Q8×C47Q8semidirect
144. C47#10359semidirect
151. C4⋊Q8Dic94semidirect

Maximal subgroups

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

subgroupindex
#202516602
#202443622
#202412702
#202412692
C47×#19602
#1035947