Finite Group Databasebeta

← #26104357

Subgroups — #26104357

Order 12032.

566 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689021C211yesmaximal, characteristic
2#20243456C211yesmaximal, characteristic
3#20241210C214yesmaximal, complemented
4C47×#1817C211yesmaximal, characteristic
5C22×C4⋊Dic47C411yescharacteristic
6C22×C4⋊Dic47C411yescharacteristic
7C22×Dic47⋊C4—21no—
8C2×C22⋊(C47⋊C8)—21no—
9#15689021—21no—
10#15689021—21no—
11#15689021C412yes—
12#15689021C2212yes—
13C2×C47⋊#95—21no—
14#15688878—21no—
15#15688878C2212yes—
16C22×C42×C47C2211yescharacteristic
17C2×C22⋊C8×C47—21no—
18C47×#336—21no—
19C47×#336C2212yes—
20C23×Dic47—21no—
21C23×Dic47—21no—
22C22×C47⋊C8—22no—
23C2×C4⋊Dic47—21no—
24C2×C4⋊Dic47—22no—
25C2×C4⋊Dic47—21no—
26C2×C4⋊Dic47—42no—
27C2×C4⋊Dic47C814yescomplemented
28C2×C4⋊Dic47C2×C412yes—
29C2×C4⋊Dic47C2×C412yes—
30C2×Dic47⋊C4—22no—
31C2×Dic47⋊C4—22no—
32C2×Dic47⋊C4—44no—
33C22⋊(C47⋊C8)—42no—
34C47⋊#95—22no—
35C47⋊#95—22no—
36C23×C4×C47C2×C411yescharacteristic
37C23×C4×C47D411yescharacteristic
38C23×C4×C47D411yescharacteristic
39C22×C8×C47—22no—
40C2×C42×C47—42no—
41C2×C42×C47C2×C411yescharacteristic
42C2×C42×C47D411yescharacteristic
43C2×C42×C47D411yescharacteristic
44C2×C42×C47C2311yescharacteristic
45C22⋊C8×C47—42no—
46C24×C47C22⋊C411yescharacteristic
47C22×C4×C47—22no—
48C22×C4×C47—22no—
49C22×C4×C47—22no—
50C22×C4×C47—22no—
51C22×C4×C47—22no—
52C22×C4×C47—22no—
53C22×C4×C47—41no—
54C22×C4×C47—41no—
55C22×C4×C47C22⋊C411yescharacteristic
56C22×C4×C47C22⋊C411yescharacteristic
57C22×C4×C47C22⋊C411yescharacteristic
58C22×C4×C47C2×C812yes—
59C22×C4×C47M1612yes—
60C22×C4×C47C22×C411yescharacteristic
61C22×C4×C47C2×D411yescharacteristic
62C22×C4×C47C2×D411yescharacteristic
63C22×Dic47—24no—
64C22×Dic47—22no—
65C22×Dic47—22no—
66C22×Dic47—42no—
67C22×Dic47—42no—
68C22×Dic47—42no—
69C22×Dic47—42no—
70C22×Dic47—42no—
71C2×C8×C47—42no—
72C2×C8×C47—42no—
73C2×C8×C47—42no—
74C42×C47—41no—
75C42×C47—41no—
76C42×C47—41no—
77C42×C47—41no—
78C4⋊Dic47—24no—
79C4⋊Dic47—22no—
80C4⋊Dic47—42no—
81C4⋊Dic47—42no—
82C4⋊Dic47C2×C814yes—
83Dic47⋊C4—44no—
84Dic47⋊C4—44no—
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×C47C22⋊C812yes—
93C23×C47#9912yes—
94C23×C47#10112yes—
95C23×C47C2×C22⋊C411yescharacteristic
96C2×C4×C47—24no—
97C2×C4×C47—24no—
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—21no—
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×C47C22⋊C812yes—
115C2×C4×C47C22×C811yescharacteristic
116C2×C4×C47C2×M1611yescharacteristic
117C2×C4×C47AGL(1,ℤ/8)12yes—
118C2×C4×C47#13712yes—
119C2×Dic47—24no—
120C2×Dic47—48no—
121C2×Dic47—42no—
122C2×Dic47—42no—
123C2×Dic47—42no—
124C2×Dic47—42no—
125C2×Dic47—42no—
126C2×Dic47—84no—
127C8×C47—82no—
128C47⋊C8—82no—
129#7593—471nomaximal, Sylow
130C22×C47—22no—
131C22×C47—22no—
132C22×C47—22no—
133C22×C47—22no—
134C22×C47—22no—
135C22×C47—22no—
136C22×C47#32414yes—
137C22×C47C2×C22⋊C811yescharacteristic
138C22×C47C2×#9911yescharacteristic
139C22×C47C2×#10111yescharacteristic
140C22×C47#41811yescharacteristic
141C22×C47#41911yescharacteristic
142C22×C47#42111yescharacteristic
143C22×C47#42111yescharacteristic
144C4×C47—22no—
145C4×C47—21no—
146C4×C47—21no—
147C4×C47—22no—
148C4×C47—22no—
149C4×C47—41no—
150C4×C47—41no—
151Dic47—44no—
152Dic47—82no—
153C2×#389—471no—
154#1817—471no—
155#1817D4711yescharacteristic
156#1271—474no—
157C2×C47—24no—
158C2×C47C2×#32411yescharacteristic
159C2×C47#147811yescharacteristic
160C2×C47#147911yescharacteristic
161C2×C47#151011yescharacteristic
162C2×C47#151111yescharacteristic
163C2×C47#153411yescharacteristic
164C2×C47#155611yescharacteristic
165C22×C4⋊C4—471no—
166C22×C4⋊C4—471no—
167C22×C4⋊C4—941no—
168C22×C42C2×D4711yescharacteristic
169C2×C22⋊C8—21no—
170C2×C22⋊C8—941no—
171#389—472no—
172#389—472no—
173#389—941no—
174#389—941no—
175C2×#95—941no—
176#336—21no—
177#336—472no—
178#336—941no—
179#336C2×D4712yes—
180C47#759311yescharacteristic, complemented, Sylow
181C23×C4—941no—
182C23×C4—941no—
183C23×C4C4×D4711yescharacteristic
184C23×C4D18811yescharacteristic
185C23×C4C22⋊D4711yescharacteristic
186C22×C8—22no—
187C22×C8—942no—
188C2×C4⋊C4—474no—
189C2×C4⋊C4—472no—
190C2×C4⋊C4—472no—
191C2×C4⋊C4—942no—
192C2×C4⋊C4—941no—
193C2×C4⋊C4—941no—
194C2×C4⋊C4—942no—
195C2×C4⋊C4—942no—
196C2×C4⋊C4—1882no—
197C2×C4⋊C4—1884no—
198C2×C42—42no—
199C2×C42C4×D4711yescharacteristic
200C2×C42D18811yescharacteristic
201C2×C42C22⋊D4711yescharacteristic
202C2×C42C22×D4711yescharacteristic
203C22⋊C8—42no—
204C22⋊C8—1882no—
205#95—942no—
206#95—942no—
207C24(C2×D47)⋊C411yescharacteristic
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—944no—
217C22×C4—942no—
218C22×C4—942no—
219C22×C4—1882no—
220C22×C4—1882no—
221C22×C4—1882no—
222C22×C4—1882no—
223C22×C4—1882no—
224C22×C4C8×D4712yes—
225C22×C4#1062532412yes—
226C22×C4(C2×D47)⋊C411yescharacteristic
227C22×C4(C2×D47)⋊C411yescharacteristic
228C22×C4(C2×D47)⋊C411yescharacteristic
229C22×C4C2×C4×D4711yescharacteristic
230C22×C4C2×D18811yescharacteristic
231C22×C4C2×C22⋊D4711yescharacteristic
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—1882no—
236C2×C8—1882no—
237C2×C8—1882no—
238C4⋊C4—474no—
239C4⋊C4—944no—
240C4⋊C4—942no—
241C4⋊C4—1882no—
242C4⋊C4—1884no—
243C4⋊C4—1882no—
244C4⋊C4—1884no—
245C42—41no—
246C42—41no—
247C42—41no—
248C42—41no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23(C2×D47)⋊C812yes—
254C23#1309393812yes—
255C23#1309394212yes—
256C23C2×(C2×D47)⋊C411yescharacteristic
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—24no—
264C2×C4—24no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—44no—
274C2×C4—44no—
275C2×C4—944no—
276C2×C4—3764no—
277C2×C4—1882no—
278C2×C4—1882no—
279C2×C4—1882no—
280C2×C4—1888no—
281C2×C4—1882no—
282C2×C4—1882no—
283C2×C4(C2×D47)⋊C812yes—
284C2×C4C2×C8×D4711yescharacteristic
285C2×C4C2×#1062532411yescharacteristic
286C2×C4#1309405212yes—
287C2×C4#1309405412yes—
288C8—82no—
289C8—3762no—
290C22—22no—
291C22—22no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22#1568884814yes—
297C22C2×(C2×D47)⋊C811yescharacteristic
298C22C2×#1309393811yescharacteristic
299C22C2×#1309394211yescharacteristic
300C22#1568912911yescharacteristic
301C22#1568913111yescharacteristic
302C22#1568913911yescharacteristic
303C22#1568914111yescharacteristic
304C4—22no—
305C4—22no—
306C4—21no—
307C4—21no—
308C4—22no—
309C4—41no—
310C4—41no—
311C4—3762no—
312C4—1884no—
313C2—24no—
314C2C2×#1568884811yescharacteristic
315C2#2024178011yescharacteristic
316C2#2024178211yescharacteristic
317C2#2024191811yescharacteristic
318C2#2024192011yescharacteristic
319C2#2024201411yescharacteristic
320C2#2024210411yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC2×C4×C47C22×C8
FrattiniC2×C42C22×D47
FittingC47×#1817C2
socleC23×C47C2×C22⋊C4
prime radicalC24×C47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#1817D47
hypocenterC47#7593
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
3. #20241210C2semidirect
27. C2×C4⋊Dic47C8semidirect
180. C47#7593semidirect

Maximal subgroups

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

subgroupindex
C2×#156890212
#202434562
#202412102
C47×#18172
#759347