Finite Group Databasebeta

← #26216765

Subgroups — #26216765

Order 12032.

1110 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689179C212yesmaximal
2C43×D47C211yesmaximal, characteristic
3#20243540C211yesmaximal, characteristic, complemented
4#20243504C212yesmaximal, complemented
5#20243365C214yesmaximal
6#20243243C214yesmaximal, complemented
7C47×#1842C211yesmaximal, characteristic, complemented
8C2×C42×D47C412yes—
9C2×C42×D47C411yescharacteristic
10C2×C42×D47C411yescharacteristic
11C2×C42×D47C2212yes—
12C2×C42×D47C2211yescharacteristic
13#15689179—28no—
14C2×C4⋊(C47⋊C8)C2214yes—
15C2×C4⋊(C47⋊C8)C2212yes—
16C2×(C2×D47)⋊C8C2214yes—
17C2×#13093934C2214yes—
18C42×Dic47C412yes—
19C42×Dic47C411yescharacteristic
20C42×Dic47C411yescharacteristic
21C42×Dic47C2212yes—
22C42×Dic47C2211yescharacteristic
23#15688877C2218yes—
24C2×C4⋊C8×C47C2214yes—
25C2×C4⋊C8×C47C2212yes—
26C43×C47C2211yescharacteristic
27C22×C4×D47C2×C412yes—
28C22×C4×D47C2×C411yescharacteristic
29C22×C4×D47C2×C411yescharacteristic
30C22×C4×D47C2×C412yes—
31C22×C4×D47C2311yescharacteristic
32C22×C47⋊C8C2314yes—
33C2×C4×Dic47C2×C412yes—
34C2×C4×Dic47C2×C412yes—
35C2×C4×Dic47C2×C412yes—
36C2×C4×Dic47C2×C412yes—
37C2×C4×Dic47C2×C411yescharacteristic
38C2×C4×Dic47C2×C412yes—
39C2×C4×Dic47C2×C412yes—
40C2×C4×Dic47C2×C412yes—
41C2×C4×Dic47C2×C411yescharacteristic
42C2×C4×Dic47C2×C411yescharacteristic
43C2×C4×Dic47C2×C411yescharacteristic
44C2×C4×Dic47C2312yes—
45C2×C4×Dic47C2311yescharacteristic
46C42×D47—28no—
47C42×D47—24no—
48C42×D47—24no—
49C42×D47—24no—
50C42×D47—24no—
51C42×D47D414yes—
52C42×D47Q814yes—
53C4⋊(C47⋊C8)—24no—
54C4⋊(C47⋊C8)—28no—
55(C2×D47)⋊C8—28no—
56#13093934—28no—
57C22×C8×C47C2314yes—
58C2×C42×C47C2×C412yes—
59C2×C42×C47C2×C411yescharacteristic
60C2×C42×C47C2×C411yescharacteristic
61C2×C42×C47C2311yescharacteristic
62C2×C42×C47C2312yes—
63C4⋊C8×C47—24no—
64C4⋊C8×C47—28no—
65C23×D47C22×C411yescharacteristic
66C22×C4×C47C22×C411yescharacteristic
67C22×C4×C47C22×C412yes—
68C22×C4×C47C22×C411yescharacteristic
69C22×C4×C47C22×C412yes—
70C22×C4×C47C2411yescharacteristic
71C22×Dic47C22×C411yescharacteristic
72C22×Dic47C22×C411yescharacteristic
73C22×Dic47C22×C411yescharacteristic
74C22×Dic47C22×C412yes—
75C22×Dic47C22×C412yes—
76C2×C4×D47—24no—
77C2×C4×D47—24no—
78C2×C4×D47—24no—
79C2×C4×D47—24no—
80C2×C4×D47—22no—
81C2×C4×D47—22no—
82C2×C4×D47—24no—
83C2×C4×D47—22no—
84C2×C4×D47—22no—
85C2×C4×D47—24no—
86C2×C4×D47C4⋊C414yes—
87C2×C4×D47C4⋊C414yes—
88C2×C4×D47M1618yes—
89C2×C4×D47C2×D412yes—
90C2×C4×D47C2×Q812yes—
91C2×C8×C47—24no—
92C2×C8×C47—28no—
93C42×C47—22no—
94C42×C47—24no—
95C42×C47M1614yes—
96C42×C47C2×D412yes—
97C42×C47C2×Q812yes—
98C42×C47#4114yes—
99C42×C47#4114yes—
100C4×Dic47—24no—
101C4×Dic47—24no—
102C4×Dic47—24no—
103C4×Dic47—22no—
104C4×Dic47—24no—
105C4×Dic47—22no—
106C4×Dic47—22no—
107C4×Dic47—24no—
108C4×Dic47—24no—
109C4×Dic47—24no—
110C4×Dic47C4⋊C414yes—
111C4×Dic47C4⋊C414yes—
112C4×Dic47M1614yes—
113C4×Dic47C2×D412yes—
114C4×Dic47C2×Q812yes—
115C2×C47⋊C8—24no—
116C2×C47⋊C8—28no—
117C23×C47C23×C411yescharacteristic
118C22×D47—22no—
119C22×D47—21no—
120C22×D47—21no—
121C22×D47C2×C4⋊C412yes—
122C22×D47C2×M1614yes—
123C2×C4×C47—22no—
124C2×C4×C47—22no—
125C2×C4×C47—22no—
126C2×C4×C47C2×C4⋊C412yes—
127C2×C4×C47C2×C4⋊C412yes—
128C2×C4×C47#11714yes—
129C2×C4×C47#11714yes—
130C2×C4×C47C2×M1612yes—
131C2×C4×C47C2×M1614yes—
132C2×C4×C47C2×M1612yes—
133C2×C4×C47C2×M1614yes—
134C2×C4×C47C22×D411yescharacteristic
135C2×C4×C47C22×Q811yescharacteristic
136C2×C4×C47C2×#4112yes—
137C2×C4×C47C2×#4112yes—
138C2×Dic47—22no—
139C2×Dic47—22no—
140C2×Dic47—22no—
141C2×Dic47—22no—
142C2×Dic47—21no—
143C2×Dic47—22no—
144C2×Dic47—21no—
145C2×Dic47—22no—
146C2×Dic47C2×C4⋊C412yes—
147C2×Dic47C2×C4⋊C412yes—
148C2×Dic47C2×C4⋊C412yes—
149C2×Dic47C2×M1614yes—
150C2×Dic47C2×M1614yes—
151C4×D47—216no—
152C4×D47—28no—
153C4×D47—22no—
154C4×D47—22no—
155C4×D47—24no—
156C4×D47—28no—
157C4×D47—216no—
158C8×C47—44no—
159C47⋊C8—44no—
160#20810—471nomaximal, Sylow
161C22×C47C22×C4⋊C411yescharacteristic
162C22×C47C2×#11712yes—
163C22×C47C22×M1612yes—
164C22×C47C22×M1611yescharacteristic
165C22×C47C22×M1611yescharacteristic
166C2×D47—22no—
167C2×D47—24no—
168C2×D47—24no—
169C2×D47—22no—
170C2×D47#42314yes—
171C4×C47—22no—
172C4×C47—24no—
173C4×C47#42314yes—
174C4×C47#43218yescomplemented
175C4×C47#53211yescharacteristic
176C4×C47#53211yescharacteristic
177C4×C47#53312yes—
178Dic47—22no—
179Dic47—22no—
180Dic47—24no—
181Dic47—24no—
182Dic47—22no—
183C2×#432—472no—
184C2×C43—471no—
185#1842—471no—
186#1842D4711yescharacteristic, complemented
187#1831—472no—
188#1797—474no—
189#1767—474no—
190C2×C47#228511yescharacteristic
191C2×C47C2×#42311yescharacteristic
192C2×C47C2×#42311yescharacteristic
193C2×C47C2×#43212yes—
194C2×C47#292012yes—
195D47—24no—
196C22×C42—471no—
197C22×C42—471no—
198C22×C42—472no—
199C22×C42—471no—
200C22×C42—472no—
201#432—948no—
202C2×C4⋊C8—474no—
203C2×C4⋊C8—472no—
204C2×C4⋊C8C2×D4714yes—
205C2×C4⋊C8C2×D4712yes—
206C2×C22⋊C8—474no—
207C2×#97—474no—
208C43—471no—
209C43—472no—
210C43—471no—
211C43—471no—
212C43—472no—
213C43C2×D4711yescharacteristic
214#336—478no—
215C47#2081011yescharacteristic, complemented, Sylow
216C23×C4—471no—
217C23×C4—471no—
218C23×C4—472no—
219C23×C4—471no—
220C23×C4—472no—
221C22×C8—474no—
222C22×C8C22×D4714yes—
223C2×C42—471no—
224C2×C42—471no—
225C2×C42—471no—
226C2×C42—471no—
227C2×C42—474no—
228C2×C42—474no—
229C2×C42—472no—
230C2×C42—472no—
231C2×C42—472no—
232C2×C42—472no—
233C2×C42—472no—
234C2×C42—472no—
235C2×C42—471no—
236C2×C42—472no—
237C2×C42—472no—
238C2×C42—944no—
239C2×C42—948no—
240C2×C42—944no—
241C2×C42—944no—
242C2×C42—944no—
243C2×C42C4×D4712yes—
244C2×C42C4×D4711yescharacteristic
245C2×C42C4×D4711yescharacteristic
246C2×C42C22×D4711yescharacteristic
247C2×C42C22×D4712yes—
248C4⋊C8—24no—
249C4⋊C8—28no—
250C4⋊C8—944no—
251C4⋊C8—948no—
252C22⋊C8—948no—
253#97—948no—
254C24—471no—
255C22×C4—471no—
256C22×C4—471no—
257C22×C4—471no—
258C22×C4—472no—
259C22×C4—472no—
260C22×C4—474no—
261C22×C4—474no—
262C22×C4—478no—
263C22×C4—472no—
264C22×C4—472no—
265C22×C4—944no—
266C22×C4—944no—
267C22×C4—944no—
268C22×C4—944no—
269C22×C4—944no—
270C22×C4—942no—
271C22×C4—942no—
272C22×C4—944no—
273C22×C4—942no—
274C22×C4—942no—
275C22×C4C2×C4×D4711yescharacteristic
276C22×C4C2×C4×D4712yes—
277C22×C4C2×C4×D4711yescharacteristic
278C22×C4C2×C4×D4712yes—
279C22×C4C23×D4711yescharacteristic
280C2×C8—24no—
281C2×C8—28no—
282C2×C8—944no—
283C2×C8—948no—
284C42—22no—
285C42—24no—
286C42—474no—
287C42—474no—
288C42—474no—
289C42—472no—
290C42—472no—
291C42—944no—
292C42—944no—
293C42—944no—
294C42—944no—
295C42—944no—
296C42—944no—
297C42—942no—
298C42—944no—
299C42—942no—
300C42—942no—
301C42#1062532414yes—
302C42D4×D4712yes—
303C42#1062535214yes—
304C42Q8×D4712yes—
305C42#1062535414yes—
306C23—472no—
307C23—474no—
308C23—942no—
309C23—941no—
310C23—941no—
311C23C22×C4×D4711yescharacteristic
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—472no—
316C2×C4—472no—
317C2×C4—472no—
318C2×C4—474no—
319C2×C4—474no—
320C2×C4—948no—
321C2×C4—9416no—
322C2×C4—942no—
323C2×C4—942no—
324C2×C4—942no—
325C2×C4—942no—
326C2×C4—941no—
327C2×C4—942no—
328C2×C4—941no—
329C2×C4—942no—
330C2×C4—948no—
331C2×C4—942no—
332C2×C4—942no—
333C2×C4—944no—
334C2×C4—9416no—
335C2×C4C4⋊C4×D4712yes—
336C2×C4C4⋊C4×D4712yes—
337C2×C4#1309398214yes—
338C2×C4#1309398214yes—
339C2×C4M16×D4712yes—
340C2×C4M16×D4714yes—
341C2×C4C2×#1062532412yes—
342C2×C4C2×#1062532414yes—
343C2×C4C2×D4×D4711yescharacteristic
344C2×C4C2×Q8×D4711yescharacteristic
345C2×C4C2×#1062535212yes—
346C2×C4C2×#1062535412yes—
347C8—44no—
348C8—1884no—
349C22—474no—
350C22—942no—
351C22—942no—
352C22—944no—
353C22—944no—
354C22C2×C4⋊C4×D4711yescharacteristic
355C22C2×#1309398212yes—
356C22C2×M16×D4712yes—
357C22C2×M16×D4711yescharacteristic
358C22C22×#1062532411yescharacteristic
359C4—22no—
360C4—24no—
361C4—942no—
362C4—942no—
363C4—942no—
364C4—944no—
365C4—944no—
366C4#1568914514yes—
367C4#1568917918yescomplemented
368C4#1568967111yescharacteristic
369C4#1568967411yescharacteristic
370C4#1568967812yes—
371C2—944no—
372C2#2024579311yescharacteristic
373C2#423×D4711yescharacteristic
374C2C2×#1568914511yescharacteristic
375C2C2×#1568917912yes—
376C2#2025202012yes—
subgroupG/H
center, FrattiniC22×C4C23×D47
commutator, socleC23×C47C23×C4
FittingC47×#1842C2
prime radicalC23×D47C22×C4
cosocleC22×C4×C47C24
hypercenter#1842D47
hypocenterC47#20810
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ #1842

Complements

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

NH ≅ G/Ntype
3. #20243540C2semidirect
4. #20243504C2semidirect
6. #20243243C2semidirect
7. C47×#1842C2semidirect
174. C4×C47#432semidirect
186. #1842D47semidirect
215. C47#20810semidirect
367. C4#15689179semidirect

Maximal subgroups

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

subgroupindex
C2×#156891792
C43×D472
#202435402
#202435042
#202433652
#202432432
C47×#18422
#2081047