Finite Group Databasebeta

← #26181980

Subgroups — #26181980

Order 12032.

734 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#41×C47⋊C8C211yesmaximal, characteristic
2#20252032C211yesmaximal, characteristic
3#20241688C212yesmaximal, complemented
4#20241685C212yesmaximal, complemented
5#20241681C212yesmaximal, complemented
6#20241678C211yesmaximal, characteristic, complemented
7#20241672C211yesmaximal, characteristic, complemented
8C8×#10625329C211yesmaximal, characteristic, complemented
9M16×C47⋊C8C212yesmaximal, complemented
10C2×#15688844C211yesmaximal, characteristic, complemented
11C47×#2965C211yesmaximal, characteristic
12Q8×C47⋊C8C412yes—
13D4×C47⋊C8C414yes—
14D4×C47⋊C8C412yes—
15#15689184C414yes—
16#15689176C2212yes—
17#15689176C2211yescharacteristic
18#15689176C2212yes—
19#15689176C2212yes—
20#15689150C412yes—
21#15689090C412yes—
22C4×#10625329C2211yescharacteristic
23C2×C4×C47⋊C8C2212yes—
24C2×C4×C47⋊C8C2211yescharacteristic
25C2×C4×C47⋊C8C2211yescharacteristic
26#15688845C2212yes—
27#15688845C2212yes—
28#15688845C2212yes—
29#15688844C2211yescharacteristic
30#15688844C2212yes—
31#15688844C2211yescharacteristic
32#15688844C2211yescharacteristic
33#15688844C2211yescharacteristic
34C8×C47⋊C8C2212yes—
35C8×C47⋊C8C2212yes—
36C4×#41×C47C2211yescharacteristic
37C47×#446C412yes—
38C47×#436C412yes—
39C47×#435C414yes—
40C47×#431C2211yescharacteristic
41C47×#431C2212yes—
42C4×M16×C47C2212yes—
43C2×C4×C8×C47C2211yescharacteristic
44C2×#10625329C2×C412yes—
45C22×C47⋊C8C2×C412yes—
46C22×C47⋊C8C2×C414yes—
47C22×C47⋊C8C2×C412yes—
48C4⋊(C47⋊C8)C2×C412yes—
49C4⋊(C47⋊C8)C2×C414yes—
50C4⋊(C47⋊C8)C2×C414yes—
51C4⋊(C47⋊C8)C2×C414yes—
52C22⋊(C47⋊C8)C2×C414yes—
53C22⋊(C47⋊C8)C2×C414yes—
54C22⋊(C47⋊C8)C2×C412yes—
55C22⋊(C47⋊C8)C2×C414yes—
56#13093935C2×C412yes—
57C4×C47⋊C8C2×C414yes—
58C4×C47⋊C8C2×C412yes—
59C4×C47⋊C8C2×C412yes—
60C4×C47⋊C8C2311yescharacteristic
61C4×C47⋊C8C2311yescharacteristic
62C4×C47⋊C8C2312yes—
63C4×C47⋊C8C2311yescharacteristic
64C4×C47⋊C8C2311yescharacteristic
65C4×C47⋊C8C2312yes—
66C2×#41×C47C2×C411yescharacteristic
67C2×M16×C47C2×C414yes—
68C22×C8×C47C2×C412yes—
69C4×Q8×C47C2×C411yescharacteristic
70C4×Q8×C47C2×C411yescharacteristic
71C4×D4×C47C2×C412yes—
72C4×D4×C47C2×C411yescharacteristic
73C4×D4×C47C2×C412yes—
74C4×D4×C47C2×C411yescharacteristic
75#117×C47C2×C411yescharacteristic
76#117×C47C2×C412yes—
77C2×C42×C47C2311yescharacteristic
78C2×C42×C47C2312yes—
79C4⋊C8×C47C2×C414yes—
80C4⋊C8×C47C2×C412yes—
81C22⋊C8×C47C2×C412yes—
82C22⋊C8×C47C2×C414yes—
83#97×C47C2×C414yes—
84C4×C8×C47C2×C412yes—
85C4×C8×C47C2311yescharacteristic
86C4×C8×C47C2312yes—
87C4×C8×C47C2311yescharacteristic
88#41×C47—24no—
89C2×Q8×C47C4211yescharacteristic
90C2×D4×C47C4212yes—
91C2×D4×C47C4211yescharacteristic
92C22×C4×C47C22×C412yes—
93C22×C4×C47C22×C412yes—
94C22×C4×C47C22×C411yescharacteristic
95C22×C4×C47C22×C411yescharacteristic
96C22×C4×C47C22×C412yes—
97C22×C4×C47C22×C411yescharacteristic
98M16×C47—28no—
99C2×C8×C47—24no—
100C2×C8×C47C22×C412yes—
101C2×C8×C47C22×C414yes—
102C2×C8×C47C22×C412yes—
103C4⋊C4×C47C4212yes—
104C4⋊C4×C47C4212yes—
105C4⋊C4×C47C4211yescharacteristic
106C4⋊C4×C47C4211yescharacteristic
107C22⋊C4×C47C4212yes—
108C22⋊C4×C47C4212yes—
109C22⋊C4×C47C4212yes—
110C42×C47C22×C411yescharacteristic
111C42×C47C22×C412yes—
112C42×C47C22×C412yes—
113C42×C47C22×C411yescharacteristic
114C42×C47C22×C411yescharacteristic
115C42×C47C22×C412yes—
116C42×C47C2411yescharacteristic
117#10625329—24no—
118C2×C47⋊C8—24no—
119C2×C47⋊C8—42no—
120C2×C47⋊C8—44no—
121C2×C47⋊C8C22×C412yes—
122C2×C47⋊C8C22×C412yes—
123C2×C47⋊C8C22×C412yes—
124C2×C47⋊C8C22×C414yes—
125C2×C47⋊C8C22×C412yes—
126C2×C47⋊C8C22×C414yes—
127C23×C47C2×C4212yes—
128C23×C47C2×C4211yescharacteristic
129Q8×C47—22no—
130D4×C47—24no—
131D4×C47—22no—
132C2×C4×C47—22no—
133C2×C4×C47—24no—
134C2×C4×C47—24no—
135C2×C4×C47—22no—
136C2×C4×C47—24no—
137C2×C4×C47C2×C4211yescharacteristic
138C2×C4×C47C2×C4212yes—
139C2×C4×C47C2×C4211yescharacteristic
140C2×C4×C47C2×C4212yes—
141C2×C4×C47C2×C4212yes—
142C2×C4×C47C2×C4211yescharacteristic
143C2×C4×C47#13114yes—
144C2×C4×C47C23×C411yescharacteristic
145C2×C4×C47C23×C411yescharacteristic
146C2×C4×C47C23×C411yescharacteristic
147C8×C47—22no—
148C8×C47—24no—
149C8×C47#13114yes—
150C47⋊C8—22no—
151C47⋊C8—22no—
152C47⋊C8—24no—
153C47⋊C8—22no—
154C47⋊C8—24no—
155C47⋊C8—22no—
156#17331—471nomaximal, Sylow
157C22×C47—21no—
158C22×C47—21no—
159C22×C47—22no—
160C22×C47—22no—
161C22×C47—22no—
162C22×C47#40512yes—
163C22×C47C22×C4211yescharacteristic
164C4×C47—22no—
165C4×C47—21no—
166C4×C47—22no—
167C4×C47—21no—
168C4×C47—22no—
169C4×C47#40512yes—
170C4×C47C2×#13112yes—
171C4×C47C2×#13112yes—
172C4×C47#56812yes—
173#2965D4711yescharacteristic
174C8×#41—471no—
175#2922—471no—
176#1454—472no—
177#1453—472no—
178#1452—472no—
179#1451—471no—
180#1450—471no—
181C8×M16—472no—
182C8×M16—471no—
183C2×#322—471no—
184C2×C47—22no—
185C2×C47—42no—
186C2×C47C2×#40511yescharacteristic
187C2×C47C4×#13111yescharacteristic
188C2×C47#287511yescharacteristic
189C4×#41C2×D4711yescharacteristic
190#446Dic4712yes—
191C8×Q8—472no—
192#436Dic4712yes—
193#435Dic4714yes—
194C8×D4—472no—
195C8×D4—474no—
196#433—474no—
197#431—472no—
198#431—472no—
199#431—471no—
200#431—472no—
201#431C2×D4711yescharacteristic
202#431C2×D4712yes—
203#424—472no—
204#408—472no—
205C4×M16—471no—
206C4×M16C2×D4712yes—
207C2×C4×C8—471no—
208C2×C4×C8—471no—
209C2×C4×C8—472no—
210C2×C4×C8C2×D4711yescharacteristic
211#322—471no—
212#322—471no—
213#322—471no—
214#322—471no—
215#322—472no—
216#322—472no—
217#322—472no—
218#322—472no—
219C82—472no—
220C82—472no—
221C47#1733111yescharacteristic, complemented, Sylow
222C2×#41C4×D4711yescharacteristic
223C2×M16—472no—
224C2×M16C2×Dic4714yes—
225C22×C8—472no—
226C22×C8—472no—
227C22×C8—474no—
228C22×C8C2×Dic4712yes—
229C4×Q8C4×D4711yescharacteristic
230C4×Q8C2×Dic4711yescharacteristic
231C4×D4C4×D4712yes—
232C4×D4C4×D4711yescharacteristic
233C4×D4C2×Dic4712yes—
234C4×D4C2×Dic4711yescharacteristic
235#117C4×D4711yescharacteristic
236#117C4×D4712yes—
237C2×C42C22×D4711yescharacteristic
238C2×C42C22×D4712yes—
239C4⋊C8—474no—
240C4⋊C8—474no—
241C4⋊C8—472no—
242C4⋊C8—474no—
243C4⋊C8C2×Dic4714yes—
244C4⋊C8C2×Dic4712yes—
245C22⋊C8—474no—
246C22⋊C8—474no—
247C22⋊C8—472no—
248C22⋊C8—474no—
249C22⋊C8C2×Dic4712yes—
250C22⋊C8C2×Dic4714yes—
251#97—472no—
252#97C2×Dic4714yes—
253C4×C8—472no—
254C4×C8—472no—
255C4×C8—474no—
256C4×C8—471no—
257C4×C8—472no—
258C4×C8—472no—
259C4×C8—471no—
260C4×C8—471no—
261C4×C8—471no—
262C4×C8C2×Dic4712yes—
263C4×C8C22×D4711yescharacteristic
264C4×C8C22×D4712yes—
265C4×C8C22×D4711yescharacteristic
266#41—24no—
267C2×Q8C4×Dic4711yescharacteristic
268C2×D4C4×Dic4712yes—
269C2×D4C4×Dic4711yescharacteristic
270C22×C4C2×C4×D4712yes—
271C22×C4C2×C4×D4712yes—
272C22×C4C2×C4×D4711yescharacteristic
273C22×C4C2×C4×D4711yescharacteristic
274C22×C4C22×Dic4711yescharacteristic
275C22×C4C22×Dic4712yes—
276M16—28no—
277M16—944no—
278C2×C8—24no—
279C2×C8—472no—
280C2×C8—472no—
281C2×C8—472no—
282C2×C8—474no—
283C2×C8—472no—
284C2×C8—474no—
285C2×C8—944no—
286C2×C8—1882no—
287C2×C8—1884no—
288C2×C8C22×Dic4712yes—
289C2×C8C22×Dic4714yes—
290C2×C8C22×Dic4712yes—
291C4⋊C4C4×Dic4712yes—
292C4⋊C4C4×Dic4711yescharacteristic
293C4⋊C4C4×Dic4712yes—
294C4⋊C4C4×Dic4711yescharacteristic
295C22⋊C4C4×Dic4712yes—
296C22⋊C4C4×Dic4712yes—
297C22⋊C4C4×Dic4712yes—
298C42C2×C4×D4711yescharacteristic
299C42C2×C4×D4712yes—
300C42C2×C4×D4711yescharacteristic
301C42C2×C4×D4712yes—
302C42C22×Dic4711yescharacteristic
303C42C22×Dic4712yes—
304C42C23×D4711yescharacteristic
305C23C2×C4×Dic4712yes—
306C23C2×C4×Dic4711yescharacteristic
307Q8—22no—
308D4—24no—
309D4—22no—
310C2×C4—22no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—22no—
314C2×C4—24no—
315C2×C4C2×C4×Dic4711yescharacteristic
316C2×C4C2×C4×Dic4712yes—
317C2×C4C2×C4×Dic4711yescharacteristic
318C2×C4C2×C4×Dic4712yes—
319C2×C4C2×C4×Dic4712yes—
320C2×C4C2×C4×Dic4711yescharacteristic
321C2×C4#1309403214yes—
322C2×C4C22×C4×D4711yescharacteristic
323C2×C4C22×C4×D4711yescharacteristic
324C2×C4C23×Dic4711yescharacteristic
325C8—22no—
326C8—24no—
327C8—942no—
328C8—942no—
329C8—944no—
330C8—944no—
331C8—942no—
332C8—942no—
333C8#1309403114yes—
334C22—21no—
335C22—21no—
336C22—22no—
337C22—22no—
338C22—22no—
339C22#1568908112yes—
340C22C22×C4×Dic4711yescharacteristic
341C4—22no—
342C4—21no—
343C4—22no—
344C4—21no—
345C4—22no—
346C4#1568908112yes—
347C4C2×#1309403112yes—
348C4C2×#1309403212yes—
349C4#1568988812yes—
350C2—22no—
351C2—42no—
352C2C2×#1568908111yescharacteristic
353C2C4×#1309403111yescharacteristic
354C2#2025181411yescharacteristic
subgroupG/H
center, FrattiniC42C23×D47
commutator, socleC22×C47C22×C42
FittingC47×#2965C2
prime radicalC2×#41×C47C2×C4
cosocleC42×C47C24
hypercenter#2965D47
hypocenterC47#17331
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C42 ◁ #2965

Complements

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

NH ≅ G/Ntype
3. #20241688C2semidirect
4. #20241685C2semidirect
5. #20241681C2semidirect
6. #20241678C2semidirect
7. #20241672C2semidirect
8. C8×#10625329C2semidirect
9. M16×C47⋊C8C2semidirect
10. C2×#15688844C2semidirect
221. C47#17331semidirect

Maximal subgroups

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

subgroupindex
#41×C47⋊C82
#202520322
#202416882
#202416852
#202416812
#202416782
#202416722
C8×#106253292
M16×C47⋊C82
C2×#156888442
C47×#29652
#1733147