Finite Group Databasebeta

← #26125716

Subgroups — #26125716

Order 12032.

280 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 75 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#20252040C211yesmaximal, characteristic
2#20245200C211yesmaximal, characteristic, complemented
3#20241340C212yesmaximal, complemented
4#20241325C212yesmaximal
5C47×#2165C211yesmaximal, characteristic, complemented
6C4×#10625350C411yescharacteristic
7#15689182—21no—
8#15689182—21no—
9#15689178—21no—
10#15689178—21no—
11#15689151C411yescharacteristic
12#15689151C2211yescharacteristic
13#15689091C411yescharacteristic
14#15689091C2211yescharacteristic
15C2×#13093935C411yescharacteristic
16#15688944—21no—
17#15688905C2212yes—
18#15688901—21no—
19C4×M16×C47C2211yescharacteristic
20C47×#364—21no—
21C47×#349C2212yes—
22C47×#347—21no—
23C2×#10625350C2×C411yescharacteristic
24C22×C47⋊C8C2×C411yescharacteristic
25C22×C47⋊C8C2×C411yescharacteristic
26C4×Dic94—21no—
27C4×D188—21no—
28C4×C22⋊D47—22no—
29#13093983C2×C411yescharacteristic
30#13093981—21no—
31C42×D47—21no—
32#13093964—22no—
33Dic47⋊C8—21no—
34Dic47⋊C8—21no—
35Dic47⋊C8—21no—
36Dic47⋊C8—21no—
37(C2×D47)⋊C8—21no—
38(C2×D47)⋊C8—21no—
39(C2×D47)⋊C8—21no—
40(C2×D47)⋊C8—21no—
41#13093935—21no—
42#13093935—21no—
43C2×M16×C47C2×C411yescharacteristic
44C2×M16×C47C2311yescharacteristic
45C2×C42×C47C2×C411yescharacteristic
46M32×C47—22no—
47#97×C47D411yescharacteristic
48#97×C47D411yescharacteristic
49C4×C8×C47D411yescharacteristic
50C4×C8×C47D411yescharacteristic
51C22×C4×C47—21no—
52C22×C4×C47C22×C411yescharacteristic
53C2×C22⋊D47—21no—
54#10625350—41no—
55#10625350—41no—
56C2×D188M1611yescharacteristic
57C2×C4×D47—21no—
58C2×C4×D47—22no—
59C2×Dic94M1611yescharacteristic
60M16×C47—41no—
61M16×C47—41no—
62C2×C8×C47C22⋊C411yescharacteristic
63C2×C8×C47C22⋊C411yescharacteristic
64C2×C8×C47C2×D411yescharacteristic
65C2×C8×C47C2×D411yescharacteristic
66C42×C47—21no—
67C42×C47C22⋊C411yescharacteristic
68C42×C47C22⋊C411yescharacteristic
69C22⋊Dic47M1612yes—
70(C2×D47)⋊C4—22no—
71C4⋊Dic47—21no—
72Dic47⋊C4—22no—
73C4×Dic47—22no—
74C4×Dic47—21no—
75C2×C47⋊C8—21no—
76C2×C47⋊C8—21no—
77C2×C47⋊C8—41no—
78C2×C47⋊C8—41no—
79C16×C47—42no—
80C47⋊C16—42no—
81C23×C47C2×M1611yescharacteristic
82C22×D47—21no—
83C2×C4×C47—21no—
84C2×C4×C47—21no—
85C2×C4×C47—21no—
86C2×C4×C47—21no—
87C2×C4×C47—81no—
88C2×C4×C47C2×C22⋊C411yescharacteristic
89C2×C4×C47C2×M1611yescharacteristic
90C22⋊D47—42no—
91C2×Dic47—21no—
92C2×Dic47—22no—
93D188—41no—
94C4×D47—41no—
95C4×D47—41no—
96C4×D47—82no—
97Dic94—41no—
98C8×C47—21no—
99C8×C47—21no—
100C8×C47—21no—
101C8×C47—21no—
102C47⋊C8—41no—
103C47⋊C8—41no—
104#11523—471nomaximal, Sylow
105C22×C47—21no—
106C22×C47—41no—
107C22×C47#40711yescharacteristic
108C2×D47—42no—
109C2×D47—41no—
110C4×C47—21no—
111C4×C47—41no—
112C4×C47—41no—
113C4×C47#41011yescharacteristic
114C4×C47#41311yescharacteristic
115Dic47—42no—
116Dic47—41no—
117#2923—471no—
118#2165—471no—
119#2165D4711yescharacteristic, complemented
120#1328—472no—
121#1321—472no—
122C2×C47—21no—
123C2×C47—41no—
124C2×C47#146411yescharacteristic
125D47—81no—
126C4×#41—471no—
127#433—941no—
128#433—941no—
129#432—941no—
130#432—941no—
131#424—471no—
132#424—471no—
133#408—471no—
134#408—471no—
135C4×M16C2×D4711yescharacteristic
136C2×#97—471no—
137#364—21no—
138#364—941no—
139#349—472no—
140#349C2×D4712yes—
141#347—21no—
142#347—941no—
143C47#1152311yescharacteristic, complemented, Sylow
144C2×#41—471no—
145C2×M16C4×D4711yescharacteristic
146C2×M16C22×D4711yescharacteristic
147C22×C8—471no—
148C22×C8—471no—
149C4×Q8—941no—
150C4×D4—941no—
151C4×D4—942no—
152#117—471no—
153#117—941no—
154C2×C42—941no—
155C2×C42C4×D4711yescharacteristic
156M32—22no—
157M32—942no—
158C4⋊C8—941no—
159C4⋊C8—941no—
160C4⋊C8—941no—
161C4⋊C8—941no—
162C22⋊C8—941no—
163C22⋊C8—941no—
164C22⋊C8—941no—
165C22⋊C8—941no—
166#97—941no—
167#97—941no—
168#97D18811yescharacteristic
169#97C22⋊D4711yescharacteristic
170C4×C8D18811yescharacteristic
171C4×C8C22⋊D4711yescharacteristic
172#41—1881no—
173#41—1881no—
174C2×Q8—471no—
175C2×D4—471no—
176C2×D4—941no—
177C22×C4—21no—
178C22×C4—942no—
179C22×C4—941no—
180C22×C4C2×C4×D4711yescharacteristic
181M16—41no—
182M16—41no—
183C2×C8—941no—
184C2×C8—941no—
185C2×C8—1881no—
186C2×C8—1881no—
187C2×C8(C2×D47)⋊C411yescharacteristic
188C2×C8(C2×D47)⋊C411yescharacteristic
189C2×C8C2×D18811yescharacteristic
190C2×C8C2×C22⋊D4711yescharacteristic
191C4⋊C4—942no—
192C4⋊C4—941no—
193C22⋊C4—472no—
194C22⋊C4—942no—
195C42—21no—
196C42—942no—
197C42—941no—
198C42(C2×D47)⋊C411yescharacteristic
199C42(C2×D47)⋊C411yescharacteristic
200C16—42no—
201C16—1882no—
202C23—941no—
203C23M16×D4711yescharacteristic
204Q8—1881no—
205D4—1881no—
206D4—1882no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—81no—
212C2×C4—942no—
213C2×C4—941no—
214C2×C4—3762no—
215C2×C4—1881no—
216C2×C4—1881no—
217C2×C4C2×(C2×D47)⋊C411yescharacteristic
218C2×C4M16×D4711yescharacteristic
219C8—21no—
220C8—21no—
221C8—21no—
222C8—21no—
223C8—1881no—
224C8—1881no—
225C22—21no—
226C22—41no—
227C22—1882no—
228C22—1881no—
229C22#1568908611yescharacteristic
230C4—21no—
231C4—41no—
232C4—41no—
233C4—1882no—
234C4—1881no—
235C4#1568909611yescharacteristic
236C4#1568910511yescharacteristic
237C2—21no—
238C2—41no—
239C2—3761no—
240C2#2024172011yescharacteristic
subgroupG/H
centerC4#15689096
commutatorC22×C4×C47C22×C4
FrattiniC2×M16C22×D47
FittingC47×#2165C2
socleC2×C47#1464
prime radicalC2×#10625350C2×C4
cosocleC2×M16×C47C23
hypercenter#2165D47
hypocenterC47#11523
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C42 ◁ C4×M16 ◁ #2165

Complements

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

NH ≅ G/Ntype
2. #20245200C2semidirect
3. #20241340C2semidirect
5. C47×#2165C2semidirect
119. #2165D47semidirect
143. C47#11523semidirect

Maximal subgroups

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

subgroupindex
#202520402
#202452002
#202413402
#202413252
C47×#21652
#1152347