Finite Group Databasebeta

← #26127754

Subgroups — #26127754

Order 12032.

358 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 91 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#20253968C211yesmaximal, characteristic
2C4⋊C16×D47C211yesmaximal, characteristic, complemented
3#20242145C211yesmaximal, characteristic, complemented
4#20241443C211yesmaximal, characteristic, complemented
5#20241435C211yesmaximal, characteristic, complemented
6#20241433C211yesmaximal, characteristic, complemented
7C47×#1364C211yesmaximal, characteristic, complemented
8#15689674—21no—
9C2×C16×D47—21no—
10#15689500C2211yescharacteristic
11#15689471—21no—
12#15689360—21no—
13#15689360C412yes—
14#15689205—21no—
15#15689205C412yes—
16#15689145—21no—
17C4×C8×D47C2211yescharacteristic
18C4⋊(C47⋊C16)C2211yescharacteristic
19Dic47⋊C16—21no—
20#15688876C2211yescharacteristic
21#15688875—21no—
22C47×#500C2211yescharacteristic
23C47×C4⋊C16C2211yescharacteristic
24C47×#335C2211yescharacteristic
25C2×#10625354—41no—
26C2×D376—21no—
27C2×D376C2×C412yes—
28C2×#10625324—41no—
29C2×C8×D47C2×C411yescharacteristic
30C2×C8×D47D411yescharacteristic
31#13094021—21no—
32#13094013—41no—
33#13093999—22no—
34#13093999—41no—
35#13093986—22no—
36#13093986—41no—
37C42×D47D411yescharacteristic
38C2×C47⋊C16—21no—
39C16×D47—42no—
40C4⋊(C47⋊C8)—21no—
41Dic47⋊C8—41no—
42#13093944—22no—
43#13093944—22no—
44C4×C47⋊C8D411yescharacteristic
45C8×Dic47C2×C411yescharacteristic
46C8×Dic47D411yescharacteristic
47C4⋊Q8×C47—21no—
48C2×C16×C47—21no—
49#107×C47—21no—
50#107×C47C2×C412yes—
51C4⋊C8×C47—21no—
52C4×C8×C47C2311yescharacteristic
53C2×Q8×C47—41no—
54#10625354—82no—
55#10625354—82no—
56C2×D188—22no—
57C2×D188—42no—
58C2×D188—41no—
59C2×D188—41no—
60C2×C4×D47—42no—
61C2×C4×D47C22⋊C411yescharacteristic
62C2×C4×D47D811yescharacteristic
63C2×C4×D47SD1611yescharacteristic
64C2×C8×C47—41no—
65C2×C8×C47C22×C411yescharacteristic
66C2×C8×C47C2×D411yescharacteristic
67C4⋊C4×C47—22no—
68C4⋊C4×C47—41no—
69C42×C47C2×D411yescharacteristic
70(C2×D47)⋊C4—42no—
71(C2×D47)⋊C4—42no—
72C4×Dic47C22⋊C411yescharacteristic
73C4×Dic47D811yescharacteristic
74C4×Dic47SD1611yescharacteristic
75C2×C47⋊C8—41no—
76C2×C47⋊C8C22⋊C411yescharacteristic
77C2×C47⋊C8C22⋊C411yescharacteristic
78D376—42no—
79D376—41no—
80D376—41no—
81#10625324—82no—
82C8×D47—21no—
83C8×D47—21no—
84C8×D47—41no—
85C16×C47—41no—
86C47⋊C16—41no—
87C22×D47—42no—
88C22×D47#10211yescharacteristic
89Q8×C47—81no—
90Q8×C47—81no—
91C2×C4×C47—42no—
92C2×C4×C47C2×C22⋊C411yescharacteristic
93C2×C4×C47C2×D811yescharacteristic
94C2×C4×C47C2×SD1611yescharacteristic
95C2×Dic47#10211yescharacteristic
96C2×Dic47#10211yescharacteristic
97C2×Dic47#10211yescharacteristic
98D188—42no—
99D188—42no—
100D188—82no—
101D188—84no—
102D188—81no—
103D188—81no—
104C4×D47—21no—
105C4×D47—21no—
106C4×D47—41no—
107C4×D47—84no—
108C4×D47D1612yes—
109C4×D47SD3212yes—
110C8×C47—21no—
111C8×C47—81no—
112C8×C47AGL(1,ℤ/8)11yescharacteristic
113C8×C47#13711yescharacteristic
114C47⋊C8—21no—
115C47⋊C8—21no—
116C47⋊C8—81no—
117#11981—471nomaximal, Sylow
118C22×C47C2×#10211yescharacteristic
119C2×D47—21no—
120C2×D47—21no—
121C2×D47—82no—
122C2×D47—82no—
123C2×D47—82no—
124C2×D47#35712yes—
125C4×C47—21no—
126C4×C47—82no—
127C4×C47#41311yescharacteristic
128C4×C47#41711yescharacteristic
129C4×C47C2×D1611yescharacteristic
130C4×C47C2×SD3211yescharacteristic
131Dic47—21no—
132Dic47—21no—
133Dic47#35712yes—
134#3169—471no—
135C2×C4⋊C16—471no—
136#1567—471no—
137#1364—471no—
138#1364D4711yescharacteristic, complemented
139#1361—471no—
140#1361—471no—
141C2×C47#154711yescharacteristic
142C2×C47C2×#35711yescharacteristic
143C2×C47#214811yescharacteristic
144D47—22no—
145D47—162no—
146#532—941no—
147C22×C16—941no—
148#500C2×D4711yescharacteristic
149#493—471no—
150#486—941no—
151#466—472no—
152#466—941no—
153C4×D8—472no—
154C4×D8—941no—
155#423—941no—
156C2×C4×C8—471no—
157C4⋊C16—471no—
158C4⋊C16—941no—
159C4⋊C16C2×D4711yescharacteristic
160#335—471no—
161#335—941no—
162#335C2×D4711yescharacteristic
163C47#1198111yescharacteristic, complemented, Sylow
164C2×#41—1881no—
165C2×D8—472no—
166C2×D8—941no—
167C2×M16—1881no—
168C22×C8—471no—
169C22×C8—471no—
170C4⋊Q8—21no—
171#127—941no—
172#124—1881no—
173#121—942no—
174#121—1881no—
175C4×D4—942no—
176C4×D4—1881no—
177C2×C42—471no—
178C2×C16—21no—
179C2×C16—941no—
180C2×C16—1882no—
181#107—21no—
182#107C4×D4712yes—
183C4⋊C8—21no—
184C4⋊C8—941no—
185C4⋊C8—1881no—
186#102—942no—
187#102—942no—
188C4×C8—471no—
189C4×C8—471no—
190C4×C8—471no—
191C4×C8C22×D4711yescharacteristic
192#41—3762no—
193#41—3762no—
194C2×Q8—41no—
195C2×D4—942no—
196C2×D4—1882no—
197C2×D4—1881no—
198C2×D4—1881no—
199C22×C4—471no—
200C22×C4—471no—
201C22×C4—471no—
202C22×C4—1882no—
203D8—1882no—
204D8—1881no—
205D8—1881no—
206M16—3762no—
207C2×C8—41no—
208C2×C8—471no—
209C2×C8—471no—
210C2×C8—941no—
211C2×C8—941no—
212C2×C8—1881no—
213C2×C8—1881no—
214C2×C8C2×C4×D4711yescharacteristic
215C2×C8D4×D4711yescharacteristic
216C4⋊C4—22no—
217C4⋊C4—41no—
218C22⋊C4—1882no—
219C22⋊C4—1882no—
220C42—471no—
221C42—471no—
222C42—471no—
223C42D4×D4711yescharacteristic
224C16—41no—
225C16—1881no—
226C23—471no—
227C23—1882no—
228Q8—81no—
229Q8—81no—
230D4—3762no—
231D4—3764no—
232D4—3761no—
233D4—3761no—
234D4—1882no—
235D4—1882no—
236C2×C4—42no—
237C2×C4—471no—
238C2×C4—472no—
239C2×C4—472no—
240C2×C4—471no—
241C2×C4—471no—
242C2×C4—941no—
243C2×C4—941no—
244C2×C4—3764no—
245C2×C4—1881no—
246C2×C4C22⋊C4×D4711yescharacteristic
247C2×C4D8×D4711yescharacteristic
248C2×C4SD16×D4711yescharacteristic
249C8—21no—
250C8—81no—
251C8—941no—
252C8—941no—
253C8—3761no—
254C8#1309405111yescharacteristic
255C8#1309405611yescharacteristic
256C22—472no—
257C22—941no—
258C22—941no—
259C22—3762no—
260C22—3762no—
261C22—3762no—
262C22#102×D4711yescharacteristic
263C4—21no—
264C4—82no—
265C4—472no—
266C4—941no—
267C4—941no—
268C4#1568910611yescharacteristic
269C4#1568912311yescharacteristic
270C4D16×D4711yescharacteristic
271C4SD32×D4711yescharacteristic
272C2—942no—
273C2—7522no—
274C2#2024206411yescharacteristic
275C2#357×D4711yescharacteristic
276C2#2024513611yescharacteristic
subgroupG/H
centerC22#102×D47
commutatorC2×C8×C47C22×C4
FrattiniC4×C8C22×D47
FittingC47×#1364C2
socleC22×C47C2×#102
prime radical#20253968C2
cosocleC4×C8×C47C23
hypercenter#1364D47
hypocenterC47#11981
LCSG ▷ C2×C8×C47 ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1364

Complements

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

NH ≅ G/Ntype
2. C4⋊C16×D47C2semidirect
3. #20242145C2semidirect
4. #20241443C2semidirect
5. #20241435C2semidirect
6. #20241433C2semidirect
7. C47×#1364C2semidirect
138. #1364D47semidirect
163. C47#11981semidirect

Maximal subgroups

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

subgroupindex
#202539682
C4⋊C16×D472
#202421452
#202414432
#202414352
#202414332
C47×#13642
#1198147