Finite Group Databasebeta

← #26126981

Subgroups — #26126981

Order 12032.

388 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 79 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#20249154C211yesmaximal, characteristic
2#20241722C211yesmaximal, characteristic
3#20241393C214yesmaximal, complemented
4C47×#1822C211yesmaximal, characteristic
5C2×#13094024C411yescharacteristic
6C2×#13094003—21no—
7#15689180—21no—
8#15689053—21no—
9#15689053C2212yes—
10#15689048—21no—
11#15689048C412yes—
12#15689029—21no—
13#15688989C411yescharacteristic
14#15688849C2212yes—
15#15688847—21no—
16C2×#117×C47C2211yescharacteristic
17C47×#407—21no—
18C47×#343—21no—
19C47×#343C2212yes—
20C22×Dic94—21no—
21#13094024—41no—
22#13094024—41no—
23#13094003—44no—
24C2×C4⋊Dic47—21no—
25C2×C4⋊Dic47—21no—
26C2×C4⋊Dic47C2×C412yes—
27C2×Dic47⋊C4—22no—
28C2×C22⋊Dic47—22no—
29C4⋊(C47⋊C8)—41no—
30C22⋊(C47⋊C8)—22no—
31#13093935—41no—
32C47⋊#95—21no—
33C47⋊#95—22no—
34C47⋊#95—22no—
35C47⋊#95C2×C412yes—
36C23×C4×C47D411yescharacteristic
37C2×M16×C47—22no—
38#117×C47—41no—
39#117×C47—41no—
40C2×C4⋊C4×C47D411yescharacteristic
41C2×C4⋊C4×C47C2311yescharacteristic
42C2×C22⋊C4×C47C2×C411yescharacteristic
43C2×C22⋊C4×C47D411yescharacteristic
44C2×C42×C47C2×C411yescharacteristic
45C2×C42×C47D411yescharacteristic
46C22⋊C8×C47—41no—
47C22⋊C8×C47—41no—
48C24×C47C22⋊C411yescharacteristic
49C22×C4×C47—21no—
50C22×C4×C47—21no—
51C22×C4×C47—22no—
52C22×C4×C47—41no—
53C22×C4×C47C22⋊C411yescharacteristic
54C22×C4×C47C22⋊C411yescharacteristic
55C22×C4×C47C22⋊C411yescharacteristic
56C22×C4×C47C22×C411yescharacteristic
57C22×C4×C47C2×D411yescharacteristic
58C22×C4×C47C2×D411yescharacteristic
59C22×Dic47—22no—
60C22×Dic47—22no—
61C2×Dic94—42no—
62C2×Dic94—42no—
63C2×Dic94—41no—
64C2×Dic94—41no—
65M16×C47—42no—
66M16×C47—42no—
67C2×C8×C47—42no—
68C4⋊C4×C47—41no—
69C4⋊C4×C47—41no—
70C22⋊C4×C47—41no—
71C22⋊C4×C47—41no—
72C42×C47—21no—
73C42×C47—21no—
74C42×C47—21no—
75C42×C47—21no—
76C22⋊Dic47—44no—
77C4⋊Dic47—42no—
78C4⋊Dic47—42no—
79C4⋊Dic47—42no—
80Dic47⋊C4—44no—
81C2×C47⋊C8—42no—
82C23×C47—21no—
83C23×C47—21no—
84C23×C47—21no—
85C23×C47—41no—
86C23×C47—41no—
87C23×C47C2×C22⋊C411yescharacteristic
88C2×C4×C47—21no—
89C2×C4×C47—21no—
90C2×C4×C47—21no—
91C2×C4×C47—21no—
92C2×C4×C47—21no—
93C2×C4×C47—21no—
94C2×C4×C47—21no—
95C2×C4×C47—21no—
96C2×C4×C47—42no—
97C2×C4×C47—42no—
98C2×C4×C47—41no—
99C2×C4×C47—41no—
100C2×C4×C47—41no—
101C2×C4×C47—41no—
102C2×C4×C47#9911yescharacteristic
103C2×C4×C47#9911yescharacteristic
104C2×C4×C47AGL(1,ℤ/8)12yes—
105C2×Dic47—42no—
106C2×Dic47—42no—
107C2×Dic47—42no—
108C2×Dic47—42no—
109C2×Dic47—42no—
110C2×Dic47—42no—
111Dic94—82no—
112Dic94—82no—
113C8×C47—82no—
114C47⋊C8—82no—
115#11807—471nomaximal, Sylow
116C22×C47—21no—
117C22×C47—21no—
118C22×C47—42no—
119C22×C47—41no—
120C22×C47—41no—
121C22×C47—41no—
122C22×C47—81no—
123C22×C47C2×#9911yescharacteristic
124C22×C47#41811yescharacteristic
125C22×C47#42111yescharacteristic
126C4×C47—22no—
127C4×C47—21no—
128C4×C47—21no—
129C4×C47—41no—
130C4×C47—41no—
131C4×C47—41no—
132C4×C47—41no—
133Dic47—82no—
134Dic47—82no—
135#2621—471no—
136#1822D4711yescharacteristic
137#1464—471no—
138#1346—474no—
139C2×C47—21no—
140C2×C47—21no—
141C2×C47—81no—
142C2×C47#151011yescharacteristic
143C2×C47#212311yescharacteristic
144C2×C47#213511yescharacteristic
145C2×C4⋊Q8—471no—
146C2×C22⋊Q8—941no—
147C2×#117C2×D4711yescharacteristic
148#432—941no—
149#407—21no—
150#398—472no—
151#398—941no—
152#397—472no—
153#397—941no—
154#391—941no—
155#382—471no—
156#343—21no—
157#343C2×D4712yes—
158#324—472no—
159#323—941no—
160C47#1180711yescharacteristic, complemented, Sylow
161C22×Q8—941no—
162C23×C4C22⋊D4711yescharacteristic
163C2×M16—22no—
164C4⋊Q8—1881no—
165C4⋊Q8—1881no—
166C22⋊Q8—1884no—
167#117—41no—
168#117—41no—
169C2×C4⋊C4—472no—
170C2×C4⋊C4—941no—
171C2×C4⋊C4—941no—
172C2×C4⋊C4—942no—
173C2×C4⋊C4C22⋊D4711yescharacteristic
174C2×C4⋊C4C22×D4711yescharacteristic
175C2×C22⋊C4—942no—
176C2×C22⋊C4C4×D4711yescharacteristic
177C2×C22⋊C4D18811yescharacteristic
178C2×C42C4×D4711yescharacteristic
179C2×C42D18811yescharacteristic
180C4⋊C8—1881no—
181C22⋊C8—41no—
182C22⋊C8—41no—
183C22⋊C8—942no—
184#97—1881no—
185#95—472no—
186#95—941no—
187#95—942no—
188#95—942no—
189C24(C2×D47)⋊C411yescharacteristic
190C2×Q8—1882no—
191C2×Q8—1882no—
192C2×Q8—1881no—
193C2×Q8—1881no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—41no—
198C22×C4—942no—
199C22×C4—942no—
200C22×C4(C2×D47)⋊C411yescharacteristic
201C22×C4(C2×D47)⋊C411yescharacteristic
202C22×C4(C2×D47)⋊C411yescharacteristic
203C22×C4C2×C4×D4711yescharacteristic
204C22×C4C2×D18811yescharacteristic
205C22×C4C2×C22⋊D4711yescharacteristic
206M16—42no—
207M16—42no—
208C2×C8—42no—
209C2×C8—1882no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—1882no—
213C4⋊C4—1882no—
214C4⋊C4—1882no—
215C4⋊C4—1884no—
216C22⋊C4—41no—
217C22⋊C4—41no—
218C22⋊C4—1884no—
219C42—21no—
220C42—21no—
221C42—21no—
222C42—21no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—41no—
227C23—41no—
228C23C2×(C2×D47)⋊C411yescharacteristic
229Q8—3762no—
230Q8—3762no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—1882no—
246C2×C4—1882no—
247C2×C4—1882no—
248C2×C4—1882no—
249C2×C4—1882no—
250C2×C4—1882no—
251C2×C4#1309393811yescharacteristic
252C2×C4#1309393811yescharacteristic
253C2×C4#1309404912yes—
254C8—82no—
255C8—3762no—
256C22—21no—
257C22—21no—
258C22—42no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—81no—
263C22C2×#1309393811yescharacteristic
264C22#1568912711yescharacteristic
265C22#1568914111yescharacteristic
266C4—22no—
267C4—21no—
268C4—21no—
269C4—41no—
270C4—41no—
271C4—41no—
272C4—41no—
273C4—3762no—
274C4—3762no—
275C2—21no—
276C2—21no—
277C2—81no—
278C2#2024191611yescharacteristic
279C2#2024504311yescharacteristic
280C2#2024508211yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC47×#1822C2
socleC22×C47C2×#99
prime radicalC24×C47C22⋊C4
cosocleC2×C4⋊C4×C47C23
hypercenter#1822D47
hypocenterC47#11807
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1822

Complements

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

NH ≅ G/Ntype
3. #20241393C2semidirect
160. C47#11807semidirect

Maximal subgroups

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

subgroupindex
#202491542
#202417222
#202413932
C47×#18222
#1180747