Finite Group Databasebeta

← #26127767

Subgroups — #26127767

Order 12032.

306 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 85 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#20253972C211yesmaximal, characteristic
2C4⋊C16×D47C211yesmaximal, characteristic
3#20242150C211yesmaximal, characteristic
4#20241444C211yesmaximal, characteristic, complemented
5#20241442C211yesmaximal, characteristic, complemented
6#20241437C211yesmaximal, characteristic, complemented
7C47×#1362C211yesmaximal, characteristic, complemented
8#15689678—21no—
9C2×C16×D47—21no—
10#15689528C2211yescharacteristic
11#15689473—21no—
12#15689361—21no—
13#15689361C412yes—
14Q16×Dic47—21no—
15Q16×Dic47C412yes—
16#15689179—21no—
17C4×C8×D47C2211yescharacteristic
18C4⋊(C47⋊C16)C2211yescharacteristic
19Dic47⋊C16—21no—
20#15688876C2211yescharacteristic
21#15688873—21no—
22C47×#494C2211yescharacteristic
23C47×C4⋊C16C2211yescharacteristic
24C47×#335C2211yescharacteristic
25C2×C8×D47C2×C411yescharacteristic
26C2×C8×D47D411yescharacteristic
27#13094024—21no—
28#13094015—41no—
29#13094004—41no—
30#13094000—22no—
31Q8×Dic47—22no—
32#13093989—41no—
33#13093982—41no—
34C42×D47D411yescharacteristic
35C2×C47⋊C16—21no—
36C16×D47—42no—
37#13093958—21no—
38#13093958C2×C412yes—
39C4⋊(C47⋊C8)—21no—
40#13093948—22no—
41#13093948—22no—
42(C2×D47)⋊C8—41no—
43#13093934—41no—
44C4×C47⋊C8D411yescharacteristic
45C8×Dic47C2×C411yescharacteristic
46C8×Dic47D411yescharacteristic
47C2×Q16×C47—21no—
48C2×Q16×C47C2×C412yes—
49C4⋊Q8×C47—21no—
50C2×C16×C47—21no—
51C4⋊C8×C47—21no—
52C4×C8×C47C2311yescharacteristic
53C2×Q8×C47—22no—
54C2×C4×D47C22⋊C411yescharacteristic
55C2×C4×D47D811yescharacteristic
56C2×C4×D47SD1611yescharacteristic
57C2×Dic94—41no—
58Q16×C47—42no—
59Q16×C47—41no—
60Q16×C47—41no—
61C2×C8×C47—41no—
62C2×C8×C47C22×C411yescharacteristic
63C2×C8×C47C2×D411yescharacteristic
64C4⋊C4×C47—41no—
65C4⋊C4×C47—41no—
66C42×C47C2×D411yescharacteristic
67(C2×D47)⋊C4—42no—
68C4⋊Dic47—22no—
69C4⋊Dic47—41no—
70C4⋊Dic47—42no—
71Dic47⋊C4—42no—
72C4×Dic47—42no—
73C4×Dic47C22⋊C411yescharacteristic
74C4×Dic47D811yescharacteristic
75C4×Dic47SD1611yescharacteristic
76C2×C47⋊C8—41no—
77C2×C47⋊C8C22⋊C411yescharacteristic
78C2×C47⋊C8C22⋊C411yescharacteristic
79C8×D47—21no—
80C8×D47—21no—
81C8×D47—41no—
82C16×C47—41no—
83C47⋊C16—41no—
84C22×D47#10211yescharacteristic
85Q8×C47—42no—
86Q8×C47—42no—
87C2×C4×C47—42no—
88C2×C4×C47C2×C22⋊C411yescharacteristic
89C2×C4×C47C2×D811yescharacteristic
90C2×C4×C47C2×SD1611yescharacteristic
91C2×Dic47—42no—
92C2×Dic47#10211yescharacteristic
93C2×Dic47#10211yescharacteristic
94C2×Dic47#10211yescharacteristic
95C4×D47—22no—
96C4×D47—21no—
97C4×D47—21no—
98C4×D47—21no—
99C4×D47—21no—
100Dic94—81no—
101Dic94—81no—
102C8×C47—21no—
103C8×C47—81no—
104C8×C47AGL(1,ℤ/8)11yescharacteristic
105C8×C47#13711yescharacteristic
106C47⋊C8—21no—
107C47⋊C8—21no—
108C47⋊C8—81no—
109#11984—471nomaximal, Sylow
110C22×C47C2×#10211yescharacteristic
111C2×D47—21no—
112C2×D47—21no—
113C2×D47#36212yes—
114C4×C47—21no—
115C4×C47—82no—
116C4×C47#41311yescharacteristic
117C4×C47#41711yescharacteristic
118C4×C47#50811yescharacteristic
119C4×C47#50811yescharacteristic
120Dic47—21no—
121Dic47—21no—
122Dic47—21no—
123Dic47—82no—
124#3170—471no—
125C2×C4⋊C16—471no—
126#1568—471no—
127#1364—471no—
128#1364—471no—
129#1362—471no—
130#1362D4711yescharacteristic, complemented
131C2×C47#154711yescharacteristic
132C2×C47#213811yescharacteristic
133C2×C47C2×#36211yescharacteristic
134D47—22no—
135#533—941no—
136C22×C16—941no—
137#500—471no—
138#494C2×D4711yescharacteristic
139#487—941no—
140#467—472no—
141#467—941no—
142C4×Q16—472no—
143C4×Q16—941no—
144#432—941no—
145C2×C4×C8—471no—
146C4⋊C16—471no—
147C4⋊C16—941no—
148C4⋊C16C2×D4711yescharacteristic
149#335—471no—
150#335C2×D4711yescharacteristic
151#334—941no—
152C47#1198411yescharacteristic, complemented, Sylow
153C2×Q16—21no—
154C2×Q16C4×D4712yes—
155C22×C8—471no—
156C22×C8—471no—
157C4⋊Q8—21no—
158C4⋊Q8—941no—
159#125—1881no—
160C22⋊Q8—942no—
161C22⋊Q8—1881no—
162C4×Q8—942no—
163C4×Q8—1881no—
164#117—1881no—
165C2×C42—471no—
166C2×C16—21no—
167C2×C16—941no—
168C2×C16—1882no—
169#107—472no—
170#107—941no—
171C4⋊C8—21no—
172C4⋊C8—941no—
173#103—942no—
174#103—942no—
175C22⋊C8—1881no—
176#97—1881no—
177C4×C8—471no—
178C4×C8—471no—
179C4×C8—471no—
180C4×C8C22×D4711yescharacteristic
181C2×Q8—22no—
182C2×Q8—1881no—
183C22×C4—471no—
184C22×C4—471no—
185C22×C4—471no—
186Q16—42no—
187Q16—41no—
188Q16—41no—
189C2×C8—41no—
190C2×C8—471no—
191C2×C8—471no—
192C2×C8—941no—
193C2×C8—941no—
194C2×C8—1881no—
195C2×C8—1881no—
196C2×C8C2×C4×D4711yescharacteristic
197C2×C8D4×D4711yescharacteristic
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—942no—
201C4⋊C4—1882no—
202C4⋊C4—1882no—
203C4⋊C4—1881no—
204C22⋊C4—1882no—
205C42—471no—
206C42—471no—
207C42—471no—
208C42—1882no—
209C42D4×D4711yescharacteristic
210C16—41no—
211C16—1881no—
212C23—471no—
213Q8—42no—
214Q8—42no—
215Q8—3761no—
216Q8—3761no—
217C2×C4—42no—
218C2×C4—471no—
219C2×C4—471no—
220C2×C4—471no—
221C2×C4—942no—
222C2×C4—941no—
223C2×C4—941no—
224C2×C4—941no—
225C2×C4—941no—
226C2×C4—1882no—
227C2×C4C22⋊C4×D4711yescharacteristic
228C2×C4D8×D4711yescharacteristic
229C2×C4SD16×D4711yescharacteristic
230C8—21no—
231C8—81no—
232C8—941no—
233C8—941no—
234C8—3761no—
235C8#1309404811yescharacteristic
236C8#1309405311yescharacteristic
237C22—472no—
238C22—941no—
239C22—941no—
240C22#102×D4711yescharacteristic
241C4—21no—
242C4—82no—
243C4—941no—
244C4—941no—
245C4—941no—
246C4—3762no—
247C4#1568910411yescharacteristic
248C4#1568912111yescharacteristic
249C4#1568955411yescharacteristic
250C4#1568955611yescharacteristic
251C2—942no—
252C2#2024206211yescharacteristic
253C2#2024509411yescharacteristic
254C2#362×D4711yescharacteristic
subgroupG/H
centerC22#102×D47
commutatorC2×C8×C47C22×C4
FrattiniC4×C8C22×D47
FittingC47×#1362C2
socleC22×C47C2×#102
prime radicalC22×D47#102
cosocleC4×C8×C47C23
hypercenter#1362D47
hypocenterC47#11984
LCSG ▷ C2×C8×C47 ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1362

Complements

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

NH ≅ G/Ntype
4. #20241444C2semidirect
5. #20241442C2semidirect
6. #20241437C2semidirect
7. C47×#1362C2semidirect
130. #1362D47semidirect
152. C47#11984semidirect

Maximal subgroups

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

subgroupindex
#202539722
C4⋊C16×D472
#202421502
#202414442
#202414422
#202414372
C47×#13622
#1198447