Finite Group Databasebeta

← #26104504

Subgroups — #26104504

Order 12032.

594 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20243150C211yesmaximal, characteristic
2#20243150C211yesmaximal, characteristic
3#20241211C214yesmaximal, complemented
4C2×C47×#389C211yesmaximal, characteristic
5C2×C22⋊(C47⋊C8)—21no—
6C2×C22⋊(C47⋊C8)—21no—
7C2×#13093935—21no—
8C2×#13093935—21no—
9#15688878C2212yes—
10#15688878C2212yes—
11C22×C4⋊C4×C47—21no—
12C22×C4⋊C4×C47C411yescharacteristic
13C22×C4⋊C4×C47C411yescharacteristic
14C22×C42×C47C2211yescharacteristic
15C47×#389—21no—
16C47×#389—21no—
17C47×#389C412yes—
18C47×#389C2212yes—
19C2×#95×C47—21no—
20C22×C47⋊C8—22no—
21C22×C47⋊C8—22no—
22C22⋊(C47⋊C8)—24no—
23C22⋊(C47⋊C8)—24no—
24#13093935—42no—
25#13093935—42no—
26C23×C4×C47—21no—
27C23×C4×C47—21no—
28C23×C4×C47C2×C411yescharacteristic
29C23×C4×C47D411yescharacteristic
30C23×C4×C47D411yescharacteristic
31C2×C4⋊C4×C47—21no—
32C2×C4⋊C4×C47—24no—
33C2×C4⋊C4×C47—22no—
34C2×C4⋊C4×C47—21no—
35C2×C4⋊C4×C47—22no—
36C2×C4⋊C4×C47—44no—
37C2×C4⋊C4×C47—42no—
38C2×C4⋊C4×C47C2×C412yes—
39C2×C4⋊C4×C47C2×C412yes—
40C2×C42×C47—42no—
41C2×C42×C47C2×C411yescharacteristic
42C2×C42×C47D411yescharacteristic
43C2×C42×C47D411yescharacteristic
44C2×C42×C47C2311yescharacteristic
45#95×C47—22no—
46#95×C47—22no—
47C24×C47C22⋊C411yescharacteristic
48C22×C4×C47—22no—
49C22×C4×C47—22no—
50C22×C4×C47—22no—
51C22×C4×C47—24no—
52C22×C4×C47—22no—
53C22×C4×C47—22no—
54C22×C4×C47—22no—
55C22×C4×C47—22no—
56C22×C4×C47—22no—
57C22×C4×C47—22no—
58C22×C4×C47—41no—
59C22×C4×C47—41no—
60C22×C4×C47—42no—
61C22×C4×C47—42no—
62C22×C4×C47—42no—
63C22×C4×C47—42no—
64C22×C4×C47—42no—
65C22×C4×C47C22⋊C411yescharacteristic
66C22×C4×C47C22⋊C411yescharacteristic
67C22×C4×C47C22⋊C411yescharacteristic
68C22×C4×C47C22×C411yescharacteristic
69C22×C4×C47C2×D411yescharacteristic
70C22×C4×C47C2×D411yescharacteristic
71C4⋊C4×C47—24no—
72C4⋊C4×C47—22no—
73C4⋊C4×C47—44no—
74C4⋊C4×C47—42no—
75C4⋊C4×C47—42no—
76C4⋊C4×C47—44no—
77C4⋊C4×C47M1614yes—
78C42×C47—22no—
79C42×C47—22no—
80C42×C47—41no—
81C42×C47—41no—
82C2×C47⋊C8—24no—
83C2×C47⋊C8—24no—
84C2×C47⋊C8—42no—
85C2×C47⋊C8—42no—
86C2×C47⋊C8—42no—
87C2×C47⋊C8—42no—
88C23×C47—21no—
89C23×C47—21no—
90C23×C47—21no—
91C23×C47—21no—
92C23×C47#10112yes—
93C23×C47#10412yes—
94C23×C47#10412yes—
95C23×C47C2×C22⋊C411yescharacteristic
96C2×C4×C47—28no—
97C2×C4×C47—21no—
98C2×C4×C47—21no—
99C2×C4×C47—21no—
100C2×C4×C47—21no—
101C2×C4×C47—21no—
102C2×C4×C47—21no—
103C2×C4×C47—21no—
104C2×C4×C47—21no—
105C2×C4×C47—24no—
106C2×C4×C47—21no—
107C2×C4×C47—21no—
108C2×C4×C47—21no—
109C2×C4×C47—21no—
110C2×C4×C47—21no—
111C2×C4×C47—21no—
112C2×C4×C47—44no—
113C2×C4×C47—44no—
114C2×C4×C47—48no—
115C2×C4×C47—42no—
116C2×C4×C47—42no—
117C2×C4×C47—42no—
118C2×C4×C47—42no—
119C2×C4×C47—42no—
120C2×C4×C47—84no—
121C2×C4×C47#9912yes—
122C2×C4×C47C2×M1612yes—
123C2×C4×C47AGL(1,ℤ/8)12yes—
124C2×C4×C47#13712yes—
125C47⋊C8—44no—
126C47⋊C8—44no—
127#7627—471nomaximal, Sylow
128C22×C47—22no—
129C22×C47—22no—
130C22×C47—22no—
131C22×C47—22no—
132C22×C47—22no—
133C22×C47—22no—
134C22×C47#33014yes—
135C22×C47#40711yescharacteristic
136C22×C47C2×#9911yescharacteristic
137C22×C47C2×#10111yescharacteristic
138C22×C47#41811yescharacteristic
139C22×C47#41911yescharacteristic
140C22×C47C2×#10411yescharacteristic
141C22×C47C2×#10411yescharacteristic
142C4×C47—22no—
143C4×C47—22no—
144C4×C47—22no—
145C4×C47—22no—
146C4×C47—44no—
147C4×C47—41no—
148C4×C47—41no—
149C4×C47—82no—
150C2×#389D4711yescharacteristic
151#1749—471no—
152#1749—471no—
153#1271—474no—
154C2×C47—24no—
155C2×C47#146311yescharacteristic
156C2×C47#149011yescharacteristic
157C2×C47#149111yescharacteristic
158C2×C47#150711yescharacteristic
159C2×C47#150811yescharacteristic
160C2×C47C2×#33011yescharacteristic
161C2×C47#155611yescharacteristic
162C22×C4⋊C4—21no—
163C22×C4⋊C4Dic4711yescharacteristic
164C22×C4⋊C4Dic4711yescharacteristic
165C22×C42C2×D4711yescharacteristic
166C2×C22⋊C8—941no—
167C2×C22⋊C8—941no—
168C2×#97—941no—
169C2×#97—941no—
170#389—21no—
171#389—21no—
172#389Dic4712yes—
173#389C2×D4712yes—
174C2×#95—21no—
175#336—472no—
176#336—472no—
177C47#762711yescharacteristic, complemented, Sylow
178C23×C4—21no—
179C23×C4—21no—
180C23×C4C2×Dic4711yescharacteristic
181C23×C4C22⋊D4711yescharacteristic
182C23×C4C22⋊D4711yescharacteristic
183C22×C8—942no—
184C22×C8—942no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4C2×Dic4712yes—
193C2×C4⋊C4C2×Dic4712yes—
194C2×C42—42no—
195C2×C42C2×Dic4711yescharacteristic
196C2×C42C22⋊D4711yescharacteristic
197C2×C42C22⋊D4711yescharacteristic
198C2×C42C22×D4711yescharacteristic
199C22⋊C8—944no—
200C22⋊C8—944no—
201#97—1882no—
202#97—1882no—
203#95—22no—
204#95—22no—
205C24C22⋊Dic4711yescharacteristic
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—24no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4C22⋊Dic4711yescharacteristic
224C22×C4C22⋊Dic4711yescharacteristic
225C22×C4C22⋊Dic4711yescharacteristic
226C22×C4C22×Dic4711yescharacteristic
227C22×C4C2×C22⋊D4711yescharacteristic
228C22×C4C2×C22⋊D4711yescharacteristic
229C2×C8—944no—
230C2×C8—944no—
231C2×C8—1882no—
232C2×C8—1882no—
233C2×C8—1882no—
234C2×C8—1882no—
235C4⋊C4—24no—
236C4⋊C4—22no—
237C4⋊C4—42no—
238C4⋊C4—44no—
239C4⋊C4—42no—
240C4⋊C4—44no—
241C4⋊C4#1062532914yes—
242C42—22no—
243C42—22no—
244C42—41no—
245C42—41no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23#1309394312yes—
251C23#1309395112yes—
252C23#1309395112yes—
253C23C2×C22⋊Dic4711yescharacteristic
254C2×C4—28no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—24no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—44no—
271C2×C4—44no—
272C2×C4—48no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—84no—
279C2×C4#1309393912yes—
280C2×C4C2×#1062532912yes—
281C2×C4#1309405012yes—
282C2×C4#1309405512yes—
283C8—1884no—
284C8—1884no—
285C22—22no—
286C22—22no—
287C22—22no—
288C22—22no—
289C22—22no—
290C22—22no—
291C22#1568886514yes—
292C22#1568908711yescharacteristic
293C22C2×#1309393911yescharacteristic
294C22C2×#1309394311yescharacteristic
295C22#1568912811yescharacteristic
296C22#1568913211yescharacteristic
297C22C2×#1309395111yescharacteristic
298C22C2×#1309395111yescharacteristic
299C4—22no—
300C4—22no—
301C4—22no—
302C4—22no—
303C4—44no—
304C4—41no—
305C4—41no—
306C4—82no—
307C2—24no—
308C2#2024171811yescharacteristic
309C2#2024183111yescharacteristic
310C2#2024183511yescharacteristic
311C2#2024190411yescharacteristic
312C2#2024190811yescharacteristic
313C2C2×#1568886511yescharacteristic
314C2#2024210511yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic47
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC2×C47×#389C2
socleC23×C47C2×C22⋊C4
prime radicalC24×C47C22⋊C4
cosocleC2×C42×C47C23
hypercenterC2×#389D47
hypocenterC47#7627
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
3. #20241211C2semidirect
177. C47#7627semidirect

Maximal subgroups

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

subgroupindex
#202431502
#202412112
C2×C47×#3892
#762747