Finite Group Databasebeta

← #26184025

Subgroups — #26184025

Order 12032.

1202 conjugacy classes of subgroups; only the 395 normal subgroups are shown.

normal subgroups only

This group has 1202 conjugacy classes of subgroups — too many to draw. Only its 395 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#20252469C211yesmaximal, characteristic, complemented
2#20246309C211yesmaximal, characteristic, complemented
3#20241948C216yesmaximal, complemented
4#20241883C216yesmaximal, complemented
5C47×#3497C211yesmaximal, characteristic
6C2×Q8×Dic47—21no—
7Q8×C47⋊C8—24no—
8#15689176C2216yes—
9#15689022—23no—
10#15688989C2213yes—
11C4×C4⋊Dic47C2213yes—
12#15688857C22112yes—
13#15688857C2214yes—
14C2×2−1+4×C47—21no—
15C47×#550C414yescomplemented
16D4×Q8×C47C414yescomplemented
17C47×#548—24no—
18C47×#537—23no—
19C47×#532C2216yes—
20C2×C4×Q8×C47C2211yescharacteristic
21Q8×Dic47—44no—
22C2×C4⋊Dic47—23no—
23C2×C4⋊Dic47—23no—
24C2×C4×Dic47—23no—
25C4⋊(C47⋊C8)—26no—
26C22⋊(C47⋊C8)C2314yes—
27C4×C47⋊C8—26no—
28C47⋊#95—23no—
29C47⋊#95C2311yescharacteristic
30C47⋊#95C2313yes—
312−1+4×C47—44no—
322−1+4×C47—44no—
33C2×#41×C47—24no—
34C2×#41×C47—26no—
35C22×Q8×C47—24no—
36C22×Q8×C47D411yescharacteristic
37C4⋊Q8×C47C2×C416yes—
38#127×C47C2×C416yes—
39#124×C47—26no—
40C22⋊Q8×C47—212no—
41#121×C47—26no—
42#121×C47—212no—
43#121×C47C2314yes—
44C4×Q8×C47C2×C414yes—
45C4×Q8×C47D414yes—
46C4×D4×C47—26no—
47C4×D4×C47C2×C4112yes—
48C2×C4⋊C4×C47D413yes—
49C2×C42×C47C2313yes—
50#41×C47—416no—
51#41×C47—412no—
52#41×C47—412no—
53C2×Q8×C47—26no—
54C2×Q8×C47—416no—
55C2×Q8×C47C22⋊C414yes—
56C2×Q8×C47D814yes—
57C2×Q8×C47SD1614yes—
58C2×D4×C47—24no—
59C2×D4×C47—26no—
60C2×D4×C47—26no—
61C2×D4×C47C22×C414yes—
62C22×C4×C47—212no—
63C22×C4×C47C2×D413yes—
64C22×C4×C47C2×D413yes—
65C22×C4×C47C2411yescharacteristic
66C22×Dic47—21no—
67C22×Dic47—23no—
68C4⋊C4×C47—26no—
69C4⋊C4×C47C22⋊C4112yes—
70C4⋊C4×C47C22×C414yes—
71C22⋊C4×C47—212no—
72C22⋊C4×C47—24no—
73C42×C47C22×C416yes—
74C42×C47C2×D416yes—
75C4⋊Dic47—46no—
76C4⋊Dic47—46no—
77C4×Dic47—46no—
78C2×C47⋊C8—24no—
79C23×C47—24no—
80C23×C47C22×D411yescharacteristic
81Q8×C47—412no—
82Q8×C47#102116yes—
83D4×C47—44no—
84D4×C47—44no—
85D4×C47—44no—
86D4×C47—412no—
87D4×C47—86no—
88C2×C4×C47—24no—
89C2×C4×C47—43no—
90C2×C4×C47—41no—
91C2×C4×C47—424no—
92C2×C4×C47C2×C22⋊C416yes—
93C2×C4×C47C2×C22⋊C416yes—
94C2×C4×C47C2×D816yes—
95C2×C4×C47C2×SD1616yes—
96C2×C4×C47C23×C411yescharacteristic
97C2×C4×C47C22×D411yescharacteristic
98C2×Dic47—22no—
99C2×Dic47—41no—
100C2×Dic47—41no—
101C2×Dic47—43no—
102C2×Dic47—43no—
103C2×Dic47—43no—
104C47⋊C8—44no—
105#17618—471nomaximal, Sylow
106C22×C47—21no—
107C22×C47—21no—
108C22×C47—44no—
109C22×C47—44no—
110C22×C47—44no—
111C22×C47C22×C22⋊C411yescharacteristic
112C22×C47C22×D811yescharacteristic
113C22×C47C22×SD1611yescharacteristic
114C4×C47—21no—
115C4×C47—21no—
116C4×C47—26no—
117C4×C47—44no—
118C4×C47C2×#102112yes—
119Dic47—42no—
120Dic47—83no—
121#3497D4711yescharacteristic
122#2982—471no—
123#2351—471no—
124#1517—476no—
125#1502—476no—
126C2×C47—21no—
127C2×C47—21no—
128C2×C47—84no—
129C2×C47#288411yescharacteristic
130C2×C47C22×#10211yescharacteristic
131C2×C47#290111yescharacteristic
132C2×2−1+4—21no—
133#550Dic4714yescomplemented
134D4×Q8Dic4714yescomplemented
135#548—24no—
136#537—23no—
137#532C2×D4716yes—
138C2×C4×Q8—941no—
139C2×C4×Q8C2×D4711yescharacteristic
140C8×Q8—944no—
141#431—476no—
142#389—943no—
143#382—473no—
144C4×C4⋊C4—473no—
145#327—474no—
146#327—4712no—
147C47#1761811yescharacteristic, complemented, Sylow
1482−1+4—44no—
1492−1+4—44no—
150C2×#41—24no—
151C2×#41—26no—
152C22×Q8—24no—
153C22×Q8C22⋊D4711yescharacteristic
154C4⋊Q8C2×Dic4716yes—
155#127C2×Dic4716yes—
156#124—26no—
157C22⋊Q8—212no—
158#121—26no—
159#121—212no—
160#121C22×D4714yes—
161C4×Q8—1884no—
162C4×Q8C2×Dic4714yes—
163C4×Q8C22⋊D4714yes—
164C4×D4—26no—
165C4×D4C2×Dic47112yes—
166C2×C4⋊C4—943no—
167C2×C4⋊C4—943no—
168C2×C4⋊C4C22⋊D4713yes—
169C2×C42—943no—
170C2×C42C22×D4713yes—
171C4⋊C8—946no—
172C22⋊C8—474no—
173C4×C8—946no—
174#95—471no—
175#95—473no—
176#95—943no—
177#41—416no—
178#41—412no—
179#41—412no—
180C2×Q8—26no—
181C2×Q8—416no—
182C2×Q8#1062533414yes—
183C2×Q8#1062533514yes—
184C2×Q8C22⋊Dic4714yes—
185C2×D4—24no—
186C2×D4—26no—
187C2×D4—26no—
188C2×D4C22×Dic4714yes—
189C22×C4—212no—
190C22×C4—941no—
191C22×C4—943no—
192C22×C4C2×C22⋊D4713yes—
193C22×C4C2×C22⋊D4713yes—
194C22×C4C23×D4711yescharacteristic
195C2×C8—944no—
196C4⋊C4—26no—
197C4⋊C4—1886no—
198C4⋊C4—1886no—
199C4⋊C4C22⋊Dic47112yes—
200C4⋊C4C22×Dic4714yes—
201C22⋊C4—212no—
202C22⋊C4—24no—
203C42—1886no—
204C42C22×Dic4716yes—
205C42C2×C22⋊D4716yes—
206C23—24no—
207C23C22×C22⋊D4711yescharacteristic
208Q8—412no—
209Q8#13093945116yes—
210D4—44no—
211D4—44no—
212D4—44no—
213D4—412no—
214D4—86no—
215C2×C4—24no—
216C2×C4—43no—
217C2×C4—41no—
218C2×C4—424no—
219C2×C4—942no—
220C2×C4—1883no—
221C2×C4—1883no—
222C2×C4—1883no—
223C2×C4—1881no—
224C2×C4—1881no—
225C2×C4C2×C22⋊Dic4716yes—
226C2×C4C2×C22⋊Dic4716yes—
227C2×C4C2×#1062533416yes—
228C2×C4C2×#1062533516yes—
229C2×C4C23×Dic4711yescharacteristic
230C2×C4C22×C22⋊D4711yescharacteristic
231C8—1884no—
232C22—21no—
233C22—21no—
234C22—44no—
235C22—44no—
236C22—44no—
237C22C22×C22⋊Dic4711yescharacteristic
238C22C22×#1062533411yescharacteristic
239C22C22×#1062533511yescharacteristic
240C4—26no—
241C4—21no—
242C4—21no—
243C4—44no—
244C4—3763no—
245C4—1882no—
246C4C2×#13093945112yes—
247C2—21no—
248C2—21no—
249C2—84no—
250C2#2025185211yescharacteristic
251C2C22×#1309394511yescharacteristic
252C2#2025193511yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
Fitting, prime radicalC47×#3497C2
socleC22×C47C22×C22⋊C4
cosocleC22×C4×C47C24
hypercenter#3497D47
hypocenterC47#17618
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #3497

Complements

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

NH ≅ G/Ntype
1. #20252469C2semidirect
2. #20246309C2semidirect
3. #20241948C2semidirect
4. #20241883C2semidirect
15. C47×#550C4semidirect
16. D4×Q8×C47C4semidirect
133. #550Dic47semidirect
134. D4×Q8Dic47semidirect
147. C47#17618semidirect

Maximal subgroups

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

subgroupindex
#202524692
#202463092
#202419482
#202418832
C47×#34972
#1761847