Finite Group Databasebeta

← #26379140

Subgroups — #26379140

Order 12032.

2410 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
1#20246569C212yesmaximal, complemented
2#20246565C212yesmaximal, complemented
3#20246500C214yesmaximal, complemented
4#20246317C218yesmaximal, complemented
5#20246155C218yesmaximal, complemented
6#20245980C214yesmaximal, complemented
7#20245786C212yesmaximal, complemented
8C2×C47×#532C211yesmaximal, characteristic
9C2×#13093983C2214yes—
10C22×C4⋊Dic47C2214yes—
11#15689030C2218yes—
12#15689026C2218yes—
13#15689016C22116yes—
14#15689009C22116yes—
15#15689003C22116yes—
16#15689003C2218yes—
17#15689003C2218yes—
18#15688991C2214yes—
19#15688989C2214yes—
20#15688986C2214yes—
21#15688983C2214yes—
22#15688983C2218yes—
23C4×C4⋊Dic47C2214yes—
24C22⋊C4×Dic47C22116yes—
25C4×C22⋊Dic47C2214yes—
26#15688969C2214yes—
27C22×#41×C47C2212yes—
28C47×#532C4116yescomplemented
29C2×C4⋊Q8×C47C2211yescharacteristic
30C2×#127×C47C2211yescharacteristic
31C2×#124×C47C2212yes—
32C2×#121×C47C2214yes—
33C2×C4×D4×C47C2214yes—
34C22×C42×C47C2211yescharacteristic
35C23×Dic47C2318yes—
36#13093983—48no—
37C2×C4⋊Dic47—48no—
38C2×C4⋊Dic47C2318yes—
39C2×C4⋊Dic47C2318yes—
40C2×C22⋊Dic47C2318yes—
41C2×C22⋊Dic47C23116yes—
42C2×C22⋊Dic47C2318yes—
43C2×C4×Dic47C2318yes—
44C2×C4×Dic47C2318yes—
45C47⋊#95C2318yes—
46C47⋊#95C2318yes—
47C47⋊#95C2318yes—
48C47⋊#95C23116yes—
49C47⋊#95C2318yes—
50C2×#41×C47—44no—
51C2×#41×C47—44no—
52C2×#41×C47C2×C4116yescomplemented
53C22×Q8×C47C2312yes—
54C22×D4×C47C2314yes—
55C22×D4×C47C2312yes—
56C22×D4×C47C2314yes—
57C23×C4×C47C2311yescharacteristic
58C23×C4×C47C2314yes—
59C23×C4×C47C2312yes—
60C4⋊Q8×C47C2×C418yescomplemented
61#127×C47C2×C418yes—
62#124×C47C2×C4116yescomplemented
63#121×C47C2×C4132yescomplemented
64C4×D4×C47C2×C4132yescomplemented
65C2×C4⋊C4×C47C2314yes—
66C2×C22⋊C4×C47C2318yes—
67C2×C42×C47C2×C418yescomplemented
68C2×C42×C47C2311yescharacteristic
69C2×C42×C47C2312yes—
70C2×C42×C47C2311yescharacteristic
71C24×C47C2414yes—
72C24×C47C2411yescharacteristic
73#41×C47—432no—
74C2×Q8×C47—42no—
75C2×Q8×C47—42no—
76C2×Q8×C47C22×C418yes—
77C2×D4×C47—44no—
78C2×D4×C47—44no—
79C2×D4×C47—44no—
80C2×D4×C47—44no—
81C2×D4×C47—42no—
82C2×D4×C47—42no—
83C2×D4×C47C22×C4116yes—
84C2×D4×C47C22×C4116yes—
85C2×D4×C47C22×C418yes—
86C22×C4×C47—24no—
87C22×C4×C47—28no—
88C22×C4×C47—28no—
89C22×C4×C47—44no—
90C22×C4×C47C22×C414yes—
91C22×C4×C47C22×C4116yes—
92C22×C4×C47C22×C418yes—
93C22×C4×C47C2412yes—
94C22×C4×C47C2412yes—
95C22×C4×C47C2411yescharacteristic
96C22×C4×C47C2414yes—
97C22×C4×C47C2411yescharacteristic
98C22×Dic47—48no—
99C22×Dic47—416no—
100C22×Dic47C24116yes—
101C4⋊C4×C47C22×C4116yes—
102C22⋊C4×C47C22×C4132yes—
103C42×C47C22×C418yes—
104C42×C47C22×C414yes—
105C42×C47C22×C414yes—
106C22⋊Dic47—416no—
107C22⋊Dic47—48no—
108C22⋊Dic47—48no—
109C4⋊Dic47—216no—
110C4⋊Dic47—48no—
111C4×Dic47—48no—
112C4×Dic47—48no—
113C23×C47—24no—
114C23×C47—24no—
115C23×C47—22no—
116C23×C47—24no—
117C23×C47—28no—
118C23×C47—28no—
119C23×C47C23×C418yes—
120C23×C47C23×C412yes—
121C23×C47C2511yescharacteristic
122Q8×C47—48no—
123D4×C47—416no—
124D4×C47—48no—
125D4×C47—416no—
126C2×C4×C47—24no—
127C2×C4×C47—24no—
128C2×C4×C47—22no—
129C2×C4×C47—24no—
130C2×C4×C47—24no—
131C2×C4×C47—232no—
132C2×C4×C47—28no—
133C2×C4×C47—48no—
134C2×C4×C47C23×C414yes—
135C2×C4×C47C23×C414yes—
136C2×C4×C47C23×C412yes—
137C2×C4×C47C23×C418yes—
138C2×C4×C47C23×C412yes—
139C2×C4×C47H(5,𝔽2)14yes—
140C2×Dic47—216no—
141C2×Dic47—216no—
142C2×Dic47—216no—
143C2×Dic47—816no—
144#34369—471nomaximal, Sylow
145C22×C47—22no—
146C22×C47—28no—
147C22×C47—24no—
148C22×C47—216no—
149C22×C47—44no—
150C22×C47—48no—
151C22×C47—44no—
152C22×C47—44no—
153C22×C47C24×C411yescharacteristic
154C22×C47C2×H(5,𝔽2)12yes—
155C22×C47#58514yes—
156C4×C47—22no—
157C4×C47—28no—
158C4×C47—24no—
159C4×C47—24no—
160C4×C47#51814yes—
161Dic47—416no—
162C2×#532D4711yescharacteristic
163#2381—472no—
164#2380—472no—
165#2372—474no—
166#2352—478no—
167#2333—478no—
168#2311—474no—
169#2283—472no—
170C2×C47—42no—
171C2×C47—48no—
172C2×C47C2×#51811yescharacteristic
173C2×C47#342812yes—
174C2×C47#355114yes—
175C22×#41C2×D4712yes—
176#532Dic47116yescomplemented
177C2×C4⋊Q8C2×D4711yescharacteristic
178C2×#127C2×D4711yescharacteristic
179C2×#124C2×D4712yes—
180C2×#121C2×D4714yes—
181C2×C4×D4C2×D4714yes—
182C2×#117—474no—
183C22×C4⋊C4—474no—
184C22×C42C2×D4711yescharacteristic
185#391—478no—
186#390—478no—
187#388—4716no—
188#387—4716no—
189#386—478no—
190#386—478no—
191#386—4716no—
192#383—474no—
193#382—474no—
194#381—474no—
195#380—474no—
196#380—478no—
197C4×C4⋊C4—474no—
198C4×C22⋊C4—474no—
199C4×C22⋊C4—4716no—
200#376—474no—
201C47#3436911yescharacteristic, complemented, Sylow
202C2×#41—44no—
203C2×#41—44no—
204C2×#41C2×Dic47116yescomplemented
205C22×Q8C22×D4712yes—
206C22×D4C22×D4714yes—
207C22×D4C22×D4714yes—
208C22×D4C22×D4712yes—
209C23×C4—478no—
210C23×C4C22×D4714yes—
211C23×C4C22×D4711yescharacteristic
212C23×C4C22×D4712yes—
213C4⋊Q8C2×Dic4718yescomplemented
214#127C2×Dic4718yes—
215#124C2×Dic47116yescomplemented
216#121C2×Dic47132yescomplemented
217C4×D4C2×Dic47132yescomplemented
218#117—1888no—
219C2×C4⋊C4—478no—
220C2×C4⋊C4—478no—
221C2×C4⋊C4—1888no—
222C2×C4⋊C4C22×D4714yes—
223C2×C22⋊C4—478no—
224C2×C22⋊C4—478no—
225C2×C22⋊C4—4716no—
226C2×C22⋊C4C22×D4718yes—
227C2×C42—478no—
228C2×C42—478no—
229C2×C42C2×Dic4718yescomplemented
230C2×C42C22×D4712yes—
231C2×C42C22×D4711yescharacteristic
232C2×C42C22×D4711yescharacteristic
233#95—478no—
234#95—478no—
235#95—4716no—
236#95—478no—
237#95—478no—
238C24C23×D4711yescharacteristic
239C24C23×D4714yes—
240#41—432no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8C22×Dic4718yes—
244C2×D4—44no—
245C2×D4—44no—
246C2×D4—44no—
247C2×D4—44no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4C22×Dic47116yes—
251C2×D4C22×Dic47116yes—
252C2×D4C22×Dic4718yes—
253C22×C4—28no—
254C22×C4—28no—
255C22×C4—24no—
256C22×C4—44no—
257C22×C4—4716no—
258C22×C4—1888no—
259C22×C4—18816no—
260C22×C4C22×Dic4714yes—
261C22×C4C22×Dic47116yes—
262C22×C4C22×Dic4718yes—
263C22×C4C23×D4712yes—
264C22×C4C23×D4711yescharacteristic
265C22×C4C23×D4712yes—
266C22×C4C23×D4714yes—
267C22×C4C23×D4711yescharacteristic
268C4⋊C4—9416no—
269C4⋊C4—1888no—
270C4⋊C4C22×Dic47116yes—
271C22⋊C4—18816no—
272C22⋊C4—1888no—
273C22⋊C4—1888no—
274C22⋊C4C22×Dic47132yes—
275C42—1888no—
276C42—1888no—
277C42C22×Dic4718yes—
278C42C22×Dic4714yes—
279C42C22×Dic4714yes—
280C23—24no—
281C23—24no—
282C23—22no—
283C23—24no—
284C23—28no—
285C23—28no—
286C23C23×Dic4718yes—
287C23C23×Dic4712yes—
288C23C24×D4711yescharacteristic
289Q8—48no—
290D4—416no—
291D4—48no—
292D4—416no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—22no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—232no—
299C2×C4—28no—
300C2×C4—48no—
301C2×C4—9416no—
302C2×C4—9416no—
303C2×C4—9416no—
304C2×C4—37616no—
305C2×C4C23×Dic4714yes—
306C2×C4C23×Dic4714yes—
307C2×C4C23×Dic4712yes—
308C2×C4C23×Dic4718yes—
309C2×C4C23×Dic4712yes—
310C2×C4#1309406914yes—
311C22—22no—
312C22—28no—
313C22—24no—
314C22—216no—
315C22—44no—
316C22—48no—
317C22—44no—
318C22—44no—
319C22C24×Dic4711yescharacteristic
320C22C2×#1309406912yes—
321C22#1568995914yes—
322C4—22no—
323C4—28no—
324C4—24no—
325C4—24no—
326C4—18816no—
327C4#1568959814yes—
328C2—42no—
329C2—48no—
330C2C2×#1568959811yescharacteristic
331C2#2025659112yes—
332C2#2025781514yes—
subgroupG/H
center, FrattiniC23C24×D47
commutatorC22×C47C24×C4
Fitting, prime radicalC2×C47×#532C2
socle, cosocleC23×C47C25
hypercenterC2×#532D47
hypocenterC47#34369
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#532

Complements

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

NH ≅ G/Ntype
1. #20246569C2semidirect
2. #20246565C2semidirect
3. #20246500C2semidirect
4. #20246317C2semidirect
5. #20246155C2semidirect
6. #20245980C2semidirect
7. #20245786C2semidirect
28. C47×#532C4semidirect
52. C2×#41×C47C2×C4semidirect
60. C4⋊Q8×C47C2×C4semidirect
62. #124×C47C2×C4semidirect
63. #121×C47C2×C4semidirect
64. C4×D4×C47C2×C4semidirect
67. C2×C42×C47C2×C4semidirect
176. #532Dic47semidirect
201. C47#34369semidirect
204. C2×#41C2×Dic47semidirect
213. C4⋊Q8C2×Dic47semidirect
215. #124C2×Dic47semidirect
216. #121C2×Dic47semidirect
217. C4×D4C2×Dic47semidirect
229. C2×C42C2×Dic47semidirect

Maximal subgroups

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

subgroupindex
#202465692
#202465652
#202465002
#202463172
#202461552
#202459802
#202457862
C2×C47×#5322
#3436947