Finite Group Databasebeta

← #26127275

Subgroups — #26127275

Order 12032.

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

normal subgroups only

This group has 364 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#20253043C211yesmaximal, characteristic
2#20245009C211yesmaximal, characteristic
3#20241375C214yesmaximal, complemented
4C47×#1812C211yesmaximal, characteristic
5C22×Dic188C411yescharacteristic, complemented
6#15689694—21no—
7C2×#13093964—21no—
8#15689381C412yescomplemented
9#15689381C2212yes—
10#15689364—21no—
11#15689364—21no—
12#15689161C411yescharacteristic, complemented
13#15689133—21no—
14#15688905—21no—
15#15688905C2212yes—
16C2×#131×C47C2211yescharacteristic
17C47×#427—21no—
18C47×#410—21no—
19C47×H(3,ℤ/4)C2212yes—
20C22×Dic94—21no—
21C2×Dic188—21no—
22C2×Dic188—41no—
23C2×Dic188—41no—
24C2×Dic188C2×C412yes—
25#13094022—41no—
26#13094011—41no—
27#13094005—41no—
28#13094005—41no—
29#13094003—22no—
30#13094003—22no—
31#13093983—21no—
32#13093964—22no—
33#13093964—41no—
34C2×C47⋊C16—41no—
35#13093958—21no—
36#13093956C2×C412yes—
37#13093949—22no—
38#13093949—22no—
39C2×#41×C47D411yescharacteristic
40#131×C47—41no—
41#131×C47—41no—
42C2×M16×C47C2×C411yescharacteristic
43C2×M16×C47D411yescharacteristic
44C2×M16×C47C2311yescharacteristic
45C22×C8×C47C2×C411yescharacteristic
46C22×C8×C47D411yescharacteristic
47C22×C8×C47D411yescharacteristic
48#117×C47—22no—
49#107×C47—41no—
50#106×C47—41no—
51#99×C47—41no—
52#99×C47—41no—
53#41×C47—41no—
54#41×C47—41no—
55C2×Q8×C47SD1611yescharacteristic
56C2×D4×C47D811yescharacteristic
57C2×D4×C47D811yescharacteristic
58C2×D4×C47SD1611yescharacteristic
59C22×C4×C47C22⋊C411yescharacteristic
60C22×C4×C47C22⋊C411yescharacteristic
61C22×C4×C47C2×D411yescharacteristic
62C22×Dic47—41no—
63C2×Dic94—22no—
64C2×Dic94—42no—
65C2×Dic94—42no—
66C2×Dic94—41no—
67M16×C47—41no—
68M16×C47—41no—
69M16×C47—41no—
70C2×C8×C47—21no—
71C2×C8×C47—21no—
72C2×C8×C47—21no—
73C2×C8×C47—21no—
74C2×C8×C47—21no—
75C2×C8×C47—21no—
76C2×C8×C47C22⋊C411yescharacteristic
77C2×C8×C47C22⋊C411yescharacteristic
78C2×C8×C47C22×C411yescharacteristic
79C2×C8×C47C2×D411yescharacteristic
80C4⋊C4×C47—42no—
81C22⋊C4×C47—42no—
82C42×C47—42no—
83C22⋊Dic47—41no—
84C22⋊Dic47—42no—
85C22⋊Dic47—42no—
86C4⋊Dic47—22no—
87Dic47⋊C4—42no—
88Dic47⋊C4—42no—
89C4×Dic47—41no—
90Dic188—42no—
91Dic188—41no—
92Dic188—41no—
93C47⋊C16—42no—
94C23×C47#10211yescharacteristic
95C23×C47#10211yescharacteristic
96C23×C47C2×D811yescharacteristic
97Q8×C47—41no—
98D4×C47—41no—
99D4×C47—41no—
100D4×C47—41no—
101C2×C4×C47—21no—
102C2×C4×C47—21no—
103C2×C4×C47—21no—
104C2×C4×C47—21no—
105C2×C4×C47—21no—
106C2×C4×C47—21no—
107C2×C4×C47—82no—
108C2×C4×C47#10211yescharacteristic
109C2×C4×C47#10211yescharacteristic
110C2×C4×C47C2×C22⋊C411yescharacteristic
111C2×C4×C47C2×SD1611yescharacteristic
112C2×Dic47—42no—
113C2×Dic47—42no—
114C2×Dic47—82no—
115Dic94—42no—
116Dic94—42no—
117Dic94—82no—
118C8×C47—21no—
119C8×C47—21no—
120C8×C47—21no—
121C8×C47#9911yescharacteristic
122C8×C47#9911yescharacteristic
123#11869—471nomaximal, Sylow
124C22×C47—21no—
125C22×C47—21no—
126C22×C47—21no—
127C22×C47—41no—
128C22×C47—41no—
129C22×C47—41no—
130C22×C47C2×#10211yescharacteristic
131C4×C47—21no—
132C4×C47—21no—
133C4×C47—21no—
134C4×C47—162no—
135C4×C47C2×#9911yescharacteristic
136C4×C47#42111yescharacteristic
137Dic47—82no—
138Dic47—82no—
139#3059—471no—
140#2115—471no—
141#1812D4711yescharacteristic
142#1340—474no—
143C2×C47—21no—
144C2×C47—41no—
145C2×C47—41no—
146C2×C47—41no—
147C2×C47#151911yescharacteristic
148C22×Q16—471no—
149C2×#131C2×D4711yescharacteristic
150#536—941no—
151C2×M32—941no—
152#470—472no—
153#470—472no—
154#467—941no—
155#467—941no—
156#427—21no—
157#427—471no—
158#419—941no—
159#410—21no—
160#349—472no—
161#349—941no—
162H(3,ℤ/4)C2×D4712yes—
163C47#1186911yescharacteristic, complemented, Sylow
164C2×#41C22⋊D4711yescharacteristic
165C22×Q8—941no—
166C2×Q16—472no—
167C2×Q16—941no—
168C2×Q16—1881no—
169C2×Q16—1881no—
170#131—41no—
171#131—41no—
172C2×M16C4×D4711yescharacteristic
173C2×M16D18811yescharacteristic
174C2×M16C22×D4711yescharacteristic
175C22×C8C4×D4711yescharacteristic
176C22×C8D18811yescharacteristic
177C22×C8C22⋊D4711yescharacteristic
178C4⋊Q8—1881no—
179#124—1881no—
180#123—1881no—
181#123—1881no—
182C22⋊Q8—942no—
183C22⋊Q8—942no—
184#117—22no—
185#117—941no—
186M32—942no—
187M32—1881no—
188C2×C16—1881no—
189#107—41no—
190#107—941no—
191#106—41no—
192#106—472no—
193#103—942no—
194#103—942no—
195#99—41no—
196#99—41no—
197#41—41no—
198#41—41no—
199C2×Q8—942no—
200C2×Q8—1882no—
201C2×Q8—1882no—
202C2×Q8—1881no—
203C2×Q8#1062533611yescharacteristic
204C2×D4#1062533411yescharacteristic
205C2×D4#1062533411yescharacteristic
206C2×D4#1062533611yescharacteristic
207C22×C4—1881no—
208C22×C4(C2×D47)⋊C411yescharacteristic
209C22×C4(C2×D47)⋊C411yescharacteristic
210C22×C4C2×C22⋊D4711yescharacteristic
211Q16—1882no—
212Q16—1881no—
213Q16—1881no—
214M16—41no—
215M16—41no—
216M16—41no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8(C2×D47)⋊C411yescharacteristic
224C2×C8(C2×D47)⋊C411yescharacteristic
225C2×C8C2×C4×D4711yescharacteristic
226C2×C8C2×D18811yescharacteristic
227C4⋊C4—42no—
228C4⋊C4—942no—
229C4⋊C4—1882no—
230C4⋊C4—1882no—
231C22⋊C4—42no—
232C22⋊C4—1881no—
233C22⋊C4—1882no—
234C22⋊C4—1882no—
235C42—42no—
236C42—1881no—
237C16—1882no—
238C23#1309394411yescharacteristic
239C23#1309394411yescharacteristic
240C23C2×#1062533411yescharacteristic
241Q8—41no—
242Q8—3762no—
243Q8—1882no—
244Q8—1882no—
245D4—41no—
246D4—41no—
247D4—41no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—82no—
255C2×C4—3762no—
256C2×C4—1882no—
257C2×C4—1882no—
258C2×C4#1309394411yescharacteristic
259C2×C4#1309394411yescharacteristic
260C2×C4C2×(C2×D47)⋊C411yescharacteristic
261C2×C4C2×#1062533611yescharacteristic
262C8—21no—
263C8—21no—
264C8—21no—
265C8#1309393811yescharacteristic
266C8#1309393811yescharacteristic
267C22—21no—
268C22—21no—
269C22—21no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22C2×#1309394411yescharacteristic
274C4—21no—
275C4—21no—
276C4—21no—
277C4—162no—
278C4—3762no—
279C4—3762no—
280C4C2×#1309393811yescharacteristic
281C4#1568914111yescharacteristic
282C2—21no—
283C2—41no—
284C2—41no—
285C2—41no—
286C2#2024195511yescharacteristic
subgroupG/H
centerC2#20241955
commutatorC2×C8×C47C22×C4
FrattiniC2×M16C22×D47
FittingC47×#1812C2
socleC2×C47#1519
prime radicalC2×#41×C47D4
cosocleC2×M16×C47C23
hypercenter#1812D47
hypocenterC47#11869
LCSG ▷ C2×C8×C47 ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ #1812

Complements

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

NH ≅ G/Ntype
3. #20241375C2semidirect
5. C22×Dic188C4semidirect
8. #15689381C4semidirect
12. #15689161C4semidirect
163. C47#11869semidirect

Maximal subgroups

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

subgroupindex
#202530432
#202450092
#202413752
C47×#18122
#1186947