Finite Group Databasebeta

← #26117568

Subgroups — #26117568

Order 12032.

552 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C2×#15689487C212yesmaximal
2#20251456C211yesmaximal, characteristic, complemented
3C2×#15688864C212yesmaximal, complemented
4#20241269C211yesmaximal, characteristic, complemented
5C47×#1961C211yesmaximal, characteristic
6C2×#13094016C2212yes—
7#15689487—24no—
8#15689487—24no—
9C2×#13093945—22no—
10C2×#13093945—22no—
11C2×#13093935C2212yes—
12#15689063—22no—
13#15689055—22no—
14#15689053—22no—
15#15688864—24no—
16C2×#127×C47C2211yescharacteristic
17C2×#102×C47—22no—
18C2×#97×C47C2211yescharacteristic
19C47×#382C2211yescharacteristic
20C22×C47⋊C8—22no—
21#13094016—22no—
22#13094016—22no—
23#13094016—22no—
24#13094016—22no—
25C2×C4⋊Dic47—22no—
26C2×C4⋊Dic47D412yes—
27C2×C4⋊Dic47D412yes—
28C2×Dic47⋊C4—22no—
29C2×Dic47⋊C4—22no—
30#13093945—28no—
31#13093945—28no—
32#13093935—22no—
33#13093935D414yes—
34C47⋊#95—24no—
35C47⋊#95—22no—
36C22×D4×C47—22no—
37C22×D4×C47—21no—
38C22×C8×C47—21no—
39#127×C47—22no—
40#127×C47—21no—
41#127×C47D412yes—
42C2×C4⋊C4×C47D412yes—
43C2×C42×C47C2311yescharacteristic
44#102×C47—44no—
45#97×C47—22no—
46#95×C47—22no—
47C24×C47—41no—
48C2×D4×C47—24no—
49C2×D4×C47—22no—
50C2×D4×C47—48no—
51C2×D4×C47—42no—
52C2×D4×C47—42no—
53C2×D4×C47—42no—
54C2×D4×C47—41no—
55C22×C4×C47—22no—
56C22×C4×C47C2×D412yes—
57C22×C4×C47C2×D411yescharacteristic
58C22×Dic47—22no—
59C22×Dic47—22no—
60C2×C8×C47—22no—
61C2×C8×C47—41no—
62C2×C8×C47—41no—
63C4⋊C4×C47—42no—
64C42×C47C2×D411yescharacteristic
65C42×C47C2×D412yes—
66C42×C47#4111yescharacteristic
67C4⋊Dic47—22no—
68C4⋊Dic47—22no—
69C4⋊Dic47—22no—
70C4⋊Dic47—22no—
71C4⋊Dic47—82no—
72Dic47⋊C4—42no—
73Dic47⋊C4—42no—
74Dic47⋊C4—42no—
75Dic47⋊C4—42no—
76C2×C47⋊C8—24no—
77C2×C47⋊C8—44no—
78C23×C47—42no—
79C23×C47—81no—
80C23×C47—84no—
81C23×C47—81no—
82C23×C47#12011yescharacteristic
83D4×C47—48no—
84D4×C47—44no—
85D4×C47—84no—
86D4×C47—82no—
87C2×C4×C47—21no—
88C2×C4×C47—24no—
89C2×C4×C47—21no—
90C2×C4×C47—42no—
91C2×C4×C47—44no—
92C2×C4×C47#12012yes—
93C2×C4×C47#12112yes—
94C2×C4×C47#12111yescharacteristic
95C2×C4×C47#12711yescharacteristic
96C2×Dic47—42no—
97C2×Dic47—42no—
98C2×Dic47—42no—
99C2×Dic47—44no—
100C2×Dic47—42no—
101C8×C47—42no—
102C47⋊C8—44no—
103#10080—471nomaximal, Sylow
104C22×C47—84no—
105C22×C47—82no—
106C22×C47—162no—
107C22×C47—162no—
108C22×C47#39211yescharacteristic
109C22×C47#45311yescharacteristic
110C22×C47#45414yes—
111C22×C47#45411yescharacteristic
112C4×C47—22no—
113C4×C47—22no—
114C4×C47—21no—
115C4×C47—21no—
116C4×C47—82no—
117Dic47—82no—
118Dic47—82no—
119C2×#490—472no—
120#2822—471no—
121#1961D4711yescharacteristic
122C2×#330—472no—
123#1295—471no—
124C2×C47—162no—
125C2×C47#165712yes—
126C2×C47#165812yes—
127C2×C47#200312yes—
128C2×C47#200611yescharacteristic
129C2×#127C2×D4711yescharacteristic
130C2×#125—472no—
131#490—944no—
132#490—944no—
133C2×#102—22no—
134C2×#102—942no—
135C2×#102—942no—
136C2×#97—472no—
137C2×#97C2×D4711yescharacteristic
138#400—942no—
139#398—942no—
140#398—942no—
141#382C2×D4711yescharacteristic
142#330—944no—
143C47#1008011yescharacteristic, complemented, Sylow
144C22×D4—22no—
145C22×D4—21no—
146C22×C8—21no—
147C22×C8—942no—
148#127—21no—
149#127—22no—
150#127D18812yes—
151#125—942no—
152#125—942no—
153#125—942no—
154#125—942no—
155C2×C4⋊C4—472no—
156C2×C4⋊C4—472no—
157C2×C4⋊C4—942no—
158C2×C4⋊C4—942no—
159C2×C4⋊C4—942no—
160C2×C4⋊C4C22⋊D4712yes—
161C2×C42C22×D4711yescharacteristic
162#102—44no—
163#102—948no—
164#102—948no—
165#97—22no—
166#97—474no—
167#97—942no—
168#95—22no—
169#95—944no—
170#95—942no—
171C24—41no—
172C2×D4—24no—
173C2×D4—22no—
174C2×D4—42no—
175C2×D4—48no—
176C2×D4—42no—
177C2×D4—42no—
178C2×D4—41no—
179C22×C4—22no—
180C22×C4—942no—
181C22×C4—942no—
182C22×C4D4×D4712yes—
183C22×C4C2×C22⋊D4711yescharacteristic
184C2×C8—22no—
185C2×C8—41no—
186C2×C8—41no—
187C2×C8—944no—
188C2×C8—1884no—
189C4⋊C4—42no—
190C4⋊C4—942no—
191C4⋊C4—942no—
192C4⋊C4—942no—
193C4⋊C4—942no—
194C4⋊C4—3762no—
195C4⋊C4—1882no—
196C4⋊C4—1882no—
197C4⋊C4—1882no—
198C4⋊C4—1882no—
199C42C2×D18811yescharacteristic
200C42#1062535011yescharacteristic
201C42D4×D4712yes—
202C23—42no—
203C23—81no—
204C23—84no—
205C23—81no—
206C23#1309399211yescharacteristic
207D4—48no—
208D4—44no—
209D4—84no—
210D4—82no—
211C2×C4—21no—
212C2×C4—24no—
213C2×C4—21no—
214C2×C4—42no—
215C2×C4—44no—
216C2×C4—1882no—
217C2×C4—1882no—
218C2×C4—1882no—
219C2×C4—1884no—
220C2×C4—1882no—
221C2×C4#1309399412yes—
222C2×C4#1309399712yes—
223C2×C4#1309399811yescharacteristic
224C2×C4#1309402011yescharacteristic
225C8—42no—
226C8—1884no—
227C22—84no—
228C22—82no—
229C22—162no—
230C22—162no—
231C22#1568903211yescharacteristic
232C22#1568929011yescharacteristic
233C22#1568929414yes—
234C22#1568929511yescharacteristic
235C4—22no—
236C4—22no—
237C4—21no—
238C4—21no—
239C4—82no—
240C4—3762no—
241C4—3762no—
242C2—162no—
243C2#2024268412yes—
244C2#2024268912yes—
245C2#2024441112yes—
246C2#2024442911yescharacteristic
subgroupG/H
centerC23#13093992
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#1961C2
socleC23×C47#120
prime radicalC2×#127×C47C22
hypercenter#1961D47
hypocenterC47#10080
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1961

Complements

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

NH ≅ G/Ntype
2. #20251456C2semidirect
3. C2×#15688864C2semidirect
4. #20241269C2semidirect
143. C47#10080semidirect

Maximal subgroups

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

subgroupindex
C2×#156894872
#202514562
C2×#156888642
#202412692
C47×#19612
#1008047