Finite Group Databasebeta

← #26126872

Subgroups — #26126872

Order 12032.

414 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 87 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#20248373C211yesmaximal, characteristic
2#20241732C211yesmaximal, characteristic
3#20241393C214yesmaximal
4C4×#101×C47C211yesmaximal, characteristic
5C2×#13094024C411yescharacteristic
6C2×#13094022—21no—
7#15689180—21no—
8#15689053C412yes—
9#15689053C2212yes—
10#15689028—21no—
11#15689021—21no—
12#15689021—21no—
13C4×C4⋊Dic47C411yescharacteristic
14#15688849—21no—
15#15688849C2212yes—
16C2×C4×Q8×C47C2211yescharacteristic
17C2×#101×C47—21no—
18C4×M16×C47—21no—
19C47×#343C2212yes—
20C22×Dic94—21no—
21#13094024—41no—
22#13094024—41no—
23#13094022—44no—
24C2×C4⋊Dic47—21no—
25C2×C4⋊Dic47—21no—
26C2×C4⋊Dic47C2×C412yes—
27C2×Dic47⋊C4—22no—
28C2×Dic47⋊C4—22no—
29C2×C4×Dic47—21no—
30C4⋊(C47⋊C8)—41no—
31C22⋊(C47⋊C8)—22no—
32#13093935—41no—
33C47⋊#95—22no—
34C47⋊#95—22no—
35C47⋊#95C2×C412yes—
36C22×Q8×C47D411yescharacteristic
37C2×M16×C47—22no—
38C4×Q8×C47—41no—
39C4×Q8×C47—41no—
40C2×C4⋊C4×C47C2×C411yescharacteristic
41C2×C4⋊C4×C47D411yescharacteristic
42C2×C4⋊C4×C47C2311yescharacteristic
43C2×C42×C47C2×C411yescharacteristic
44C2×C42×C47D411yescharacteristic
45C2×C42×C47D411yescharacteristic
46#101×C47—24no—
47#97×C47—41no—
48C4×C8×C47—41no—
49C2×Q8×C47—22no—
50C2×Q8×C47—22no—
51C2×Q8×C47—41no—
52C22×C4×C47C22⋊C411yescharacteristic
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—42no—
64M16×C47—24no—
65M16×C47—42no—
66C2×C8×C47—42no—
67C4⋊C4×C47—41no—
68C4⋊C4×C47—41no—
69C4⋊C4×C47—41no—
70C42×C47—21no—
71C42×C47—21no—
72C42×C47—21no—
73C42×C47—21no—
74C42×C47—21no—
75C42×C47—21no—
76C4⋊Dic47—24no—
77C4⋊Dic47—41no—
78C4⋊Dic47—41no—
79C4⋊Dic47—42no—
80Dic47⋊C4—44no—
81Dic47⋊C4—44no—
82C4×Dic47—42no—
83C2×C47⋊C8—42no—
84C23×C47C2×C22⋊C411yescharacteristic
85Q8×C47—44no—
86C2×C4×C47—21no—
87C2×C4×C47—21no—
88C2×C4×C47—21no—
89C2×C4×C47—21no—
90C2×C4×C47—22no—
91C2×C4×C47—21no—
92C2×C4×C47—21no—
93C2×C4×C47—21no—
94C2×C4×C47—21no—
95C2×C4×C47—21no—
96C2×C4×C47—21no—
97C2×C4×C47—41no—
98C2×C4×C47—41no—
99C2×C4×C47—41no—
100C2×C4×C47#9911yescharacteristic
101C2×C4×C47#9911yescharacteristic
102C2×C4×C47#10412yes—
103C2×C4×C47AGL(1,ℤ/8)12yes—
104C2×Dic47—24no—
105C2×Dic47—42no—
106C2×Dic47—42no—
107C2×Dic47—42no—
108C2×Dic47—42no—
109C2×Dic47—42no—
110Dic94—84no—
111C8×C47—44no—
112C47⋊C8—82no—
113#11782—471nomaximal, Sylow
114C22×C47—21no—
115C22×C47—21no—
116C22×C47C2×#9911yescharacteristic
117C22×C47#41811yescharacteristic
118C22×C47C2×#10411yescharacteristic
119C4×C47—21no—
120C4×C47—22no—
121C4×C47—22no—
122C4×C47—41no—
123C4×C47—41no—
124C4×C47—41no—
125C4×C47—41no—
126C4×C47#35612yes—
127Dic47—44no—
128Dic47—82no—
129#2546—471no—
130C4×#101D4711yescharacteristic
131#1467—471no—
132#1346—474no—
133C2×C47—21no—
134C2×C47—21no—
135C2×C47#150711yescharacteristic
136C2×C47#212311yescharacteristic
137C2×C47C2×#35611yescharacteristic
138C2×C4⋊Q8—471no—
139C2×C4⋊Q8—941no—
140C2×C4×Q8C2×D4711yescharacteristic
141#432—941no—
142C2×#101—21no—
143C4×M16—21no—
144#398—472no—
145#398—472no—
146#391—941no—
147#389—941no—
148#389—941no—
149C4×C4⋊C4—471no—
150#343C2×D4712yes—
151#324—472no—
152#324—941no—
153C47#1178211yescharacteristic, complemented, Sylow
154C22×Q8—941no—
155C22×Q8C22⋊D4711yescharacteristic
156C2×M16—22no—
157C4⋊Q8—1881no—
158C4⋊Q8—1881no—
159C4⋊Q8—1884no—
160C4×Q8—41no—
161C4×Q8—41no—
162C2×C4⋊C4—472no—
163C2×C4⋊C4—941no—
164C2×C4⋊C4—942no—
165C2×C4⋊C4—941no—
166C2×C4⋊C4—942no—
167C2×C4⋊C4C4×D4711yescharacteristic
168C2×C4⋊C4D18811yescharacteristic
169C2×C4⋊C4C22×D4711yescharacteristic
170C2×C42—941no—
171C2×C42C4×D4711yescharacteristic
172C2×C42D18811yescharacteristic
173C2×C42C22⋊D4711yescharacteristic
174C4⋊C8—1881no—
175#101—24no—
176C22⋊C8—942no—
177#97—41no—
178#97—1881no—
179C4×C8—41no—
180#95—472no—
181#95—942no—
182#95—942no—
183C2×Q8—22no—
184C2×Q8—22no—
185C2×Q8—41no—
186C2×Q8—1882no—
187C2×Q8—1882no—
188C2×Q8—1882no—
189C22×C4—942no—
190C22×C4—942no—
191C22×C4(C2×D47)⋊C411yescharacteristic
192C22×C4(C2×D47)⋊C411yescharacteristic
193C22×C4(C2×D47)⋊C411yescharacteristic
194C22×C4(C2×D47)⋊C411yescharacteristic
195C22×C4C2×C4×D4711yescharacteristic
196C22×C4C2×D18811yescharacteristic
197C22×C4C2×C22⋊D4711yescharacteristic
198M16—24no—
199M16—42no—
200C2×C8—42no—
201C2×C8—1882no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—944no—
206C4⋊C4—1882no—
207C4⋊C4—1884no—
208C4⋊C4—1881no—
209C4⋊C4—1881no—
210C4⋊C4—1884no—
211C42—21no—
212C42—21no—
213C42—21no—
214C42—21no—
215C42—21no—
216C42—21no—
217C42—1882no—
218C23C2×(C2×D47)⋊C411yescharacteristic
219Q8—44no—
220Q8—3764no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—944no—
236C2×C4—1882no—
237C2×C4—1882no—
238C2×C4—1882no—
239C2×C4—1882no—
240C2×C4—1882no—
241C2×C4#1309393811yescharacteristic
242C2×C4#1309393811yescharacteristic
243C2×C4#1309395212yes—
244C2×C4#1309404912yes—
245C8—44no—
246C8—3762no—
247C22—21no—
248C22—21no—
249C22C2×#1309393811yescharacteristic
250C22#1568912711yescharacteristic
251C22C2×#1309395211yescharacteristic
252C4—21no—
253C4—22no—
254C4—22no—
255C4—41no—
256C4—41no—
257C4—41no—
258C4—41no—
259C4—3762no—
260C4—1884no—
261C4#1568892112yes—
262C2—21no—
263C2—21no—
264C2#2024190311yescharacteristic
265C2#2024504311yescharacteristic
266C2C2×#1568892111yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC4×#101×C47C2
socleC22×C47C2×#99
prime radicalC23×C47C2×C22⋊C4
cosocleC2×C4⋊C4×C47C23
hypercenterC4×#101D47
hypocenterC47#11782
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ C4×#101

Complements

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

NH ≅ G/Ntype
153. C47#11782semidirect

Maximal subgroups

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

subgroupindex
#202483732
#202417322
#202413932
C4×#101×C472
#1178247