Finite Group Databasebeta

← #26126822

Subgroups — #26126822

Order 12032.

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

normal subgroups only

This group has 316 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#20254038C211yesmaximal, characteristic
2#20243143C211yesmaximal, characteristic, complemented
3#20241380C212yesmaximal, complemented
4#20241374C212yesmaximal, complemented
5C47×#2166C211yesmaximal, characteristic, complemented
6#15689728—21no—
7#15689525C411yescharacteristic, complemented
8#15689500C411yescharacteristic, complemented
9#15689488—21no—
10#15689376C2212yes—
11#15689359—21no—
12#15689236C412yescomplemented
13C4×D376—21no—
14#15689078—21no—
15C47⋊#338C2211yescharacteristic
16#15688880C2211yescharacteristic
17#15688879—21no—
18C47×#405C2211yescharacteristic
19C47×#364—21no—
20C47×#349C2212yes—
21C47×#347—21no—
22C2×D376—21no—
23C2×D376C2×C412yes—
24C2×#10625329—21no—
25#13094021—21no—
26#13094016—21no—
27#13094009—42no—
28#13093998—22no—
29#13093997—42no—
30C4×D188—22no—
31#13093983—21no—
32#13093958—21no—
33#13093956C2×C412yes—
34#13093946—22no—
35#13093946—22no—
36#13093934—22no—
37C8×Dic47—22no—
38C2×M16×C47C2311yescharacteristic
39C22×C8×C47D411yescharacteristic
40#117×C47D411yescharacteristic
41M32×C47—22no—
42#97×C47C2×C411yescharacteristic
43#97×C47D411yescharacteristic
44C4×C8×C47C2×C411yescharacteristic
45C4×C8×C47D411yescharacteristic
46C22×C4×C47C2×D411yescharacteristic
47C2×C22⋊D47—42no—
48C2×D188—22no—
49C2×D188—42no—
50C2×C4×D47—42no—
51M16×C47—41no—
52C2×C8×C47—21no—
53C2×C8×C47—21no—
54C2×C8×C47C22⋊C411yescharacteristic
55C2×C8×C47C22⋊C411yescharacteristic
56C2×C8×C47C22×C411yescharacteristic
57C2×C8×C47C2×D411yescharacteristic
58C4⋊C4×C47—21no—
59C22⋊C4×C47—21no—
60C42×C47C22⋊C411yescharacteristic
61C42×C47C22⋊C411yescharacteristic
62C22⋊Dic47—41no—
63(C2×D47)⋊C4—42no—
64(C2×D47)⋊C4—42no—
65C4⋊Dic47—22no—
66C4⋊Dic47—41no—
67Dic47⋊C4—42no—
68C4×Dic47—22no—
69#10625329—41no—
70#10625329—41no—
71C2×C47⋊C8—22no—
72D376—42no—
73D376—41no—
74D376—41no—
75C16×C47—42no—
76C23×C47AGL(1,ℤ/8)11yescharacteristic
77C22×D47—42no—
78C2×C4×C47—21no—
79C2×C4×C47—21no—
80C2×C4×C47—21no—
81C2×C4×C47—21no—
82C2×C4×C47C2×C22⋊C411yescharacteristic
83C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
84C22⋊D47—162no—
85C2×Dic47—42no—
86C2×Dic47—42no—
87D188—42no—
88D188—42no—
89D188—162no—
90C4×D47—162no—
91C8×C47—21no—
92C8×C47—21no—
93C8×C47—21no—
94C8×C47#10411yescharacteristic
95C8×C47#10411yescharacteristic
96C47⋊C8—42no—
97#11770—471nomaximal, Sylow
98C22×C47—21no—
99C22×C47—41no—
100C22×C47#41811yescharacteristic
101C2×D47—82no—
102C2×D47—82no—
103C2×D47—82no—
104C4×C47—21no—
105C4×C47—41no—
106C4×C47—41no—
107C4×C47#41011yescharacteristic
108C4×C47C2×#10411yescharacteristic
109Dic47—82no—
110Dic47—82no—
111#3176—471no—
112#2166D4711yescharacteristic, complemented
113#1747—471no—
114#1341—472no—
115#1339—472no—
116C2×C47—21no—
117C2×C47—41no—
118C2×C47#150511yescharacteristic
119D47—162no—
120#540—941no—
121#499—471no—
122#493—471no—
123#490—941no—
124#469—472no—
125#466—941no—
126#442—472no—
127C4×D8—941no—
128#405—941no—
129#405C2×D4711yescharacteristic
130#364—21no—
131#349C2×D4712yes—
132#347—21no—
133#338—471no—
134H(3,ℤ/4)—471no—
135H(3,ℤ/4)—941no—
136C47#1177011yescharacteristic, complemented, Sylow
137C2×D8—472no—
138C2×D8—941no—
139C2×M16—941no—
140C2×M16C22×D4711yescharacteristic
141C22×C8D18811yescharacteristic
142#127—941no—
143#125—941no—
144#123—1882no—
145#121—942no—
146#121—1882no—
147C4×D4—942no—
148#117—941no—
149#117D18811yescharacteristic
150M32—22no—
151#107—941no—
152#106—472no—
153#102—942no—
154#102—942no—
155#97—942no—
156#97C4×D4711yescharacteristic
157#97C22⋊D4711yescharacteristic
158C4×C8—942no—
159C4×C8C4×D4711yescharacteristic
160C4×C8C22⋊D4711yescharacteristic
161C2×D4—942no—
162C2×D4—1882no—
163C2×D4—1882no—
164C22×C4—1882no—
165C22×C4C2×D18811yescharacteristic
166D8—1882no—
167D8—1881no—
168D8—1881no—
169M16—41no—
170M16—1881no—
171M16—1881no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—942no—
175C2×C8(C2×D47)⋊C411yescharacteristic
176C2×C8(C2×D47)⋊C411yescharacteristic
177C2×C8C2×C4×D4711yescharacteristic
178C2×C8C2×C22⋊D4711yescharacteristic
179C4⋊C4—21no—
180C4⋊C4—942no—
181C4⋊C4—1882no—
182C4⋊C4—1881no—
183C22⋊C4—21no—
184C22⋊C4—1882no—
185C22⋊C4—1881no—
186C22⋊C4—1882no—
187C42—942no—
188C42(C2×D47)⋊C411yescharacteristic
189C42(C2×D47)⋊C411yescharacteristic
190C16—42no—
191C23—1882no—
192C23#1309405211yescharacteristic
193D4—1882no—
194D4—1882no—
195D4—7522no—
196D4—7522no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—1882no—
202C2×C4—1882no—
203C2×C4—7522no—
204C2×C4C2×(C2×D47)⋊C411yescharacteristic
205C2×C4#1309405211yescharacteristic
206C8—21no—
207C8—21no—
208C8—21no—
209C8—1882no—
210C8#1309395011yescharacteristic
211C8#1309395011yescharacteristic
212C22—21no—
213C22—41no—
214C22—3762no—
215C22—3762no—
216C22—3762no—
217C22#1568912911yescharacteristic
218C4—21no—
219C4—41no—
220C4—41no—
221C4—3762no—
222C4—3762no—
223C4#1568909611yescharacteristic
224C4C2×#1309395011yescharacteristic
225C2—21no—
226C2—41no—
227C2—7522no—
228C2#2024189711yescharacteristic
subgroupG/H
centerC2#20241897
commutatorC2×C8×C47C22×C4
FrattiniC2×M16C22×D47
FittingC47×#2166C2
socleC2×C47#1505
prime radical#20254038C2
cosocleC2×M16×C47C23
hypercenter#2166D47
hypocenterC47#11770
LCSG ▷ C2×C8×C47 ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #2166

Complements

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

NH ≅ G/Ntype
2. #20243143C2semidirect
3. #20241380C2semidirect
4. #20241374C2semidirect
5. C47×#2166C2semidirect
7. #15689525C4semidirect
8. #15689500C4semidirect
12. #15689236C4semidirect
112. #2166D47semidirect
136. C47#11770semidirect

Maximal subgroups

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

subgroupindex
#202540382
#202431432
#202413802
#202413742
C47×#21662
#1177047