Finite Group Databasebeta

← #26112039

Subgroups — #26112039

Order 12032.

630 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15688989C211yesmaximal, characteristic
2#20243456C211yesmaximal, characteristic
3C2×#15688860C212yesmaximal
4#20241266C212yesmaximal
5C47×#2855C211yesmaximal, characteristic
6C22×C4⋊Dic47—21no—
7C2×C22⋊(C47⋊C8)C2212yes—
8#15688989—24no—
9C2×C47⋊#95—21no—
10C2×C47⋊#95C2212yes—
11#15688878—22no—
12#15688860—24no—
13C2×C22⋊Q8×C47C2211yescharacteristic
14C22×C42×C47C2211yescharacteristic
15C47×#391C412yescomplemented
16C47×#389C412yescomplemented
17C47×#387—22no—
18C47×#385—21no—
19C47×#381—21no—
20C47×#380C2211yescharacteristic
21C23×Dic47—22no—
22C22×C47⋊C8—22no—
23C2×C4⋊Dic47—24no—
24C2×C4⋊Dic47—42no—
25C2×C4⋊Dic47—42no—
26C22⋊(C47⋊C8)—22no—
27C22⋊(C47⋊C8)—22no—
28C47⋊#95—24no—
29C47⋊#95—22no—
30C47⋊#95D414yes—
31C22×Q8×C47C2×C411yescharacteristic
32C23×C4×C47D411yescharacteristic
33C23×C4×C47D411yescharacteristic
34C23×C4×C47C2311yescharacteristic
35C22⋊Q8×C47—22no—
36C22⋊Q8×C47—22no—
37C2×C4⋊C4×C47—22no—
38C2×C4⋊C4×C47—21no—
39C2×C4⋊C4×C47C2×C411yescharacteristic
40C2×C4⋊C4×C47C2×C412yes—
41C2×C22⋊C4×C47—21no—
42C2×C22⋊C4×C47—21no—
43C2×C42×C47—22no—
44C2×C42×C47—22no—
45C2×C42×C47C2×C412yes—
46C2×C42×C47D412yes—
47#95×C47—22no—
48#95×C47—22no—
49#95×C47—22no—
50#95×C47—21no—
51C24×C47C2×D411yescharacteristic
52C2×Q8×C47—22no—
53C2×Q8×C47—42no—
54C22×C4×C47—21no—
55C22×C4×C47—21no—
56C22×C4×C47—21no—
57C22×C4×C47—21no—
58C22×C4×C47—21no—
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47—22no—
62C22×C4×C47—21no—
63C22×C4×C47—42no—
64C22×C4×C47—42no—
65C22×C4×C47C22⋊C412yes—
66C22×C4×C47C22⋊C412yes—
67C22×C4×C47SD1612yes—
68C22×C4×C47Q1612yes—
69C22×C4×C47C22×C411yescharacteristic
70C22×C4×C47C2×D411yescharacteristic
71C22×C4×C47C2×D412yes—
72C22×C4×C47#4112yes—
73C22×Dic47—24no—
74C22×Dic47—44no—
75C22×Dic47—44no—
76C22×Dic47—42no—
77C22×Dic47—42no—
78C4⋊C4×C47—22no—
79C4⋊C4×C47—44no—
80C4⋊C4×C47—42no—
81C4⋊C4×C47—42no—
82C22⋊C4×C47—41no—
83C22⋊C4×C47—41no—
84C22⋊C4×C47—41no—
85C22⋊C4×C47—41no—
86C42×C47—24no—
87C42×C47—24no—
88C4⋊Dic47—48no—
89C2×C47⋊C8—44no—
90C2×C47⋊C8—42no—
91C23×C47—22no—
92C23×C47—22no—
93C23×C47C2×C22⋊C411yescharacteristic
94C23×C47#12011yescharacteristic
95C23×C47#12311yescharacteristic
96C23×C47C2×SD1611yescharacteristic
97C23×C47C2×Q1611yescharacteristic
98C23×C47AGL(1,ℤ/8)11yescharacteristic
99C23×C47#13711yescharacteristic
100Q8×C47—82no—
101C2×C4×C47—22no—
102C2×C4×C47—22no—
103C2×C4×C47—22no—
104C2×C4×C47—22no—
105C2×C4×C47—22no—
106C2×C4×C47—22no—
107C2×C4×C47—22no—
108C2×C4×C47—22no—
109C2×C4×C47—22no—
110C2×C4×C47—22no—
111C2×C4×C47—22no—
112C2×C4×C47—42no—
113C2×C4×C47—42no—
114C2×C4×C47—42no—
115C2×C4×C47—42no—
116C2×C4×C47—41no—
117C2×C4×C47—41no—
118C2×C4×C47—42no—
119C2×C4×C47—42no—
120C2×C4×C47—42no—
121C2×C4×C47—42no—
122C2×C4×C47—42no—
123C2×C4×C47#10314yes—
124C2×C4×C47C4×D412yes—
125C2×C4×C47#12112yes—
126C2×Dic47—44no—
127C2×Dic47—44no—
128C2×Dic47—44no—
129C2×Dic47—84no—
130C2×Dic47—84no—
131C47⋊C8—82no—
132#9152—471nomaximal, Sylow
133C22×C47—21no—
134C22×C47—21no—
135C22×C47—21no—
136C22×C47—21no—
137C22×C47—22no—
138C22×C47—22no—
139C22×C47—22no—
140C22×C47—22no—
141C22×C47#38611yescharacteristic
142C22×C47#41012yes—
143C22×C47C2×#10311yescharacteristic
144C22×C47#41711yescharacteristic
145C22×C47#42112yes—
146C22×C47#45011yescharacteristic
147C22×C47#45111yescharacteristic
148C22×C47#48311yescharacteristic
149C22×C47#48411yescharacteristic
150C4×C47—24no—
151C4×C47—22no—
152C4×C47—22no—
153C4×C47—42no—
154C4×C47—81no—
155C4×C47—81no—
156C4×C47—82no—
157Dic47—84no—
158#2855D4711yescharacteristic
159C2×#382—471no—
160#1817—471no—
161C2×#328—472no—
162#1294—472no—
163C2×C47—21no—
164C2×C47—21no—
165C2×C47—21no—
166C2×C47—21no—
167C2×C47#151212yes—
168C2×C47#151812yes—
169C2×C47#187211yescharacteristic
170C2×C47#189711yescharacteristic
171C2×C47#189911yescharacteristic
172C2×C22⋊Q8C2×D4711yescharacteristic
173C22×C4⋊C4—941no—
174C22×C42C2×D4711yescharacteristic
175C2×C22⋊C8—472no—
176#391Dic4712yescomplemented
177#389Dic4712yescomplemented
178#387—22no—
179#385—21no—
180#382—944no—
181#381—21no—
182#380C2×D4711yescharacteristic
183C2×#95—472no—
184C2×#95—941no—
185#336—942no—
186#328—944no—
187C47#915211yescharacteristic, complemented, Sylow
188C22×Q8C2×Dic4711yescharacteristic
189C23×C4—942no—
190C23×C4C22⋊D4711yescharacteristic
191C23×C4C22⋊D4711yescharacteristic
192C23×C4C22×D4711yescharacteristic
193C22×C8—942no—
194C22⋊Q8—22no—
195C22⋊Q8—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—944no—
199C2×C4⋊C4—1882no—
200C2×C4⋊C4—1882no—
201C2×C4⋊C4C2×Dic4711yescharacteristic
202C2×C4⋊C4C2×Dic4712yes—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42C2×Dic4712yes—
208C2×C42C22⋊D4712yes—
209C22⋊C8—942no—
210C22⋊C8—942no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—21no—
215#95—474no—
216#95—944no—
217#95—942no—
218C24C2×C22⋊D4711yescharacteristic
219C2×Q8—22no—
220C2×Q8—42no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—944no—
233C22×C4—1884no—
234C22×C4—1884no—
235C22×C4—1882no—
236C22×C4—1882no—
237C22×C4#1062533612yes—
238C22×C4#1062533712yes—
239C22×C4C22⋊Dic4712yes—
240C22×C4C22⋊Dic4712yes—
241C22×C4D4×D4712yes—
242C22×C4#1062535212yes—
243C22×C4C22×Dic4711yescharacteristic
244C22×C4C2×C22⋊D4711yescharacteristic
245C2×C8—1884no—
246C2×C8—1882no—
247C4⋊C4—22no—
248C4⋊C4—42no—
249C4⋊C4—44no—
250C4⋊C4—42no—
251C4⋊C4—1888no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—41no—
255C22⋊C4—41no—
256C42—24no—
257C42—24no—
258C23—22no—
259C23—22no—
260C23C2×C22⋊Dic4711yescharacteristic
261C23#1309399211yescharacteristic
262C23#1309400511yescharacteristic
263C23C2×#1062533611yescharacteristic
264C23C2×#1062533711yescharacteristic
265C23#1309404911yescharacteristic
266C23#1309405411yescharacteristic
267Q8—82no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—3764no—
291C2×C4—3764no—
292C2×C4—1884no—
293C2×C4—1884no—
294C2×C4—1884no—
295C2×C4#1309394814yes—
296C2×C4D4×Dic4712yes—
297C2×C4#1309399612yes—
298C8—3762no—
299C22—22no—
300C22—22no—
301C22—22no—
302C22—22no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22#1568900311yescharacteristic
308C22#1568909712yes—
309C22C2×#1309394811yescharacteristic
310C22#1568912411yescharacteristic
311C22#1568914012yes—
312C22#1568927311yescharacteristic
313C22#1568927811yescharacteristic
314C22#1568945111yescharacteristic
315C22#1568945611yescharacteristic
316C4—22no—
317C4—22no—
318C4—24no—
319C4—42no—
320C4—82no—
321C4—81no—
322C4—81no—
323C4—3764no—
324C2—21no—
325C2—21no—
326C2—21no—
327C2—21no—
328C2#2024192612yes—
329C2#2024195212yes—
330C2#2024369611yescharacteristic
331C2#2024383711yescharacteristic
332C2#2024384711yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic47
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2855C2
socleC23×C47C2×C22⋊C4
prime radicalC24×C47C2×D4
cosocleC23×C4×C47C23
hypercenter#2855D47
hypocenterC47#9152
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2855

Complements

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

NH ≅ G/Ntype
15. C47×#391C4semidirect
16. C47×#389C4semidirect
176. #391Dic47semidirect
177. #389Dic47semidirect
187. C47#9152semidirect

Maximal subgroups

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

subgroupindex
C2×#156889892
#202434562
C2×#156888602
#202412662
C47×#28552
#915247