Finite Group Databasebeta

← #26422996

Subgroups — #26422996

Order 12032.

1356 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20251636C214yesmaximal, complemented
2#20251591C214yesmaximal
3#20251567C214yesmaximal, complemented
4#20251548C214yesmaximal
5#20246695C214yesmaximal, complemented
6#20246693C211yesmaximal, characteristic, complemented
7#20246689C214yesmaximal
8#20246658C212yesmaximal
9#20245883C211yesmaximal, characteristic
10#20245866C212yesmaximal, complemented
11C47×#2399C211yesmaximal, characteristic, complemented
12C2×#13093981C2211yescharacteristic
13C47⋊#401C2214yes—
14#15689065C2214yes—
15#15689065C2214yes—
16#15689064C2214yes—
17#15689064C2214yes—
18#15689063C2214yes—
19#15689062C2214yes—
20#15689061C2212yes—
21#15689061C2214yes—
22#15689061C2214yes—
23#15689058C2212yes—
24#15689057C2214yes—
25#15689057C2214yes—
26#15689056C2214yes—
27#15689056C2214yes—
28#15689054C2214yes—
29#15689053C2214yes—
30#15689052C2214yes—
31#15689050C2214yes—
32#15689049C2214yes—
33#15689045C2214yes—
34#15689045C2214yes—
35#15689042C2212yes—
36#15689034C2212yes—
37#15689030C2214yes—
38#15689022C2214yes—
39#15689012C2214yes—
40#15689009C2214yes—
41#15689009C2214yes—
42#15689005C2214yes—
43C4×Dic47⋊C4C2212yes—
44C4×Dic47⋊C4C2214yes—
45C4×Dic47⋊C4C2214yes—
46C4×C4⋊Dic47C2212yes—
47C4×C4⋊Dic47C2214yes—
48C4×(C2×D47)⋊C4C2212yes—
49C4×(C2×D47)⋊C4C2214yes—
50#15688969C2214yes—
51C42×Dic47C2211yescharacteristic
52C2×C4×Q8×C47C2212yes—
53C47×#398C2214yes—
54C47×#395C2214yes—
55C4×C4⋊C4×C47C2214yes—
56C47×#376C2211yescharacteristic
57C22×C4×D47C2311yescharacteristic
58#13093981—24no—
59C2×C4⋊Dic47C2314yes—
60C2×C4⋊Dic47C2314yes—
61C2×C4⋊Dic47C2314yes—
62C2×Dic47⋊C4C2312yes—
63C2×Dic47⋊C4C2314yes—
64C2×Dic47⋊C4C2314yes—
65C2×Dic47⋊C4C2314yes—
66C2×Dic47⋊C4C2314yes—
67C2×Dic47⋊C4C2314yes—
68C2×(C2×D47)⋊C4C2312yes—
69C2×(C2×D47)⋊C4C2314yes—
70C2×(C2×D47)⋊C4C2314yes—
71C2×(C2×D47)⋊C4C2314yes—
72C2×C4×Dic47C2312yes—
73C2×C4×Dic47C2314yes—
74C2×C4×Dic47C2311yescharacteristic
75C2×C4×Dic47C2314yes—
76C47⋊#95C2314yes—
77C47⋊#95C2314yes—
78C47⋊#95C2314yes—
79C47⋊#95C2314yes—
80C47⋊#95C2314yes—
81C47⋊#95C2314yes—
82C47⋊#95C2314yes—
83C47⋊#95C2312yes—
84C47⋊#95C2312yes—
85C47⋊#95C2314yes—
86C47⋊#95C2314yes—
87C47⋊#95C2314yes—
88C47⋊#95C2314yes—
89C47⋊#95C2314yes—
90C47⋊#95C2314yes—
91C47⋊#95C2314yes—
92C22×Q8×C47C2312yes—
93C4×Q8×C47—28no—
94C2×C4⋊C4×C47C2314yes—
95C2×C4⋊C4×C47C2312yes—
96C2×C4⋊C4×C47C2314yes—
97C2×C4⋊C4×C47C2314yes—
98C2×C42×C47C2314yes—
99C2×C42×C47C2312yes—
100C2×C42×C47C2311yescharacteristic
101#95×C47C2314yes—
102#95×C47C2314yes—
103#95×C47C2314yes—
104C23×D47C2411yescharacteristic
105C2×Q8×C47—24no—
106C2×Q8×C47—44no—
107C22×C4×C47C2414yes—
108C22×C4×C47C2414yes—
109C22×C4×C47C2414yes—
110C22×C4×C47C2412yes—
111C22×C4×C47C2411yescharacteristic
112C22×Dic47C2411yescharacteristic
113C22×Dic47C2414yes—
114C22×Dic47C2414yes—
115C22×Dic47C2414yes—
116C22×Dic47C2412yes—
117C2×C4×D47—22no—
118C2×C4×D47—24no—
119C4⋊C4×C47—24no—
120C4⋊C4×C47—28no—
121C4⋊C4×C47—44no—
122C4⋊C4×C47—44no—
123C42×C47—28no—
124C42×C47—24no—
125C42×C47#4114yes—
126(C2×D47)⋊C4—24no—
127(C2×D47)⋊C4—44no—
128(C2×D47)⋊C4—44no—
129(C2×D47)⋊C4—44no—
130C4⋊Dic47—44no—
131C4⋊Dic47—44no—
132C4⋊Dic47—44no—
133Dic47⋊C4—24no—
134Dic47⋊C4—44no—
135Dic47⋊C4—44no—
136Dic47⋊C4—44no—
137Dic47⋊C4—44no—
138Dic47⋊C4—44no—
139C4×Dic47—22no—
140C4×Dic47—24no—
141C4×Dic47—28no—
142C4×Dic47—44no—
143C23×C47C2511yescharacteristic
144C22×D47—21no—
145C22×D47—22no—
146C22×D47—24no—
147Q8×C47—48no—
148C2×C4×C47—24no—
149C2×C4×C47—24no—
150C2×C4×C47—24no—
151C2×C4×C47—24no—
152C2×C4×C47—24no—
153C2×C4×C47—24no—
154C2×C4×C47—24no—
155C2×C4×C47—24no—
156C2×C4×C47—24no—
157C2×C4×C47—24no—
158C2×C4×C47C2×#4114yes—
159C2×C4×C47C2×#4112yes—
160C2×C4×C472−1+414yes—
161C2×Dic47—22no—
162C2×Dic47—21no—
163C2×Dic47—24no—
164C2×Dic47—24no—
165C2×Dic47—24no—
166C2×Dic47—22no—
167C2×Dic47—24no—
168C2×Dic47—24no—
169C2×Dic47—24no—
170C2×Dic47—24no—
171C2×Dic47—24no—
172C2×Dic47—24no—
173C2×Dic47—24no—
174C4×D47—44no—
175#36700—471nomaximal, Sylow
176C22×C47C22×#4111yescharacteristic
177C22×C47C2×2−1+412yes—
178C22×C47#58514yes—
179C2×D47—41no—
180C2×D47—42no—
181C2×D47—44no—
182C4×C47—24no—
183C4×C47—44no—
184C4×C47—44no—
185C4×C47—44no—
186C4×C47#54114yes—
187Dic47—41no—
188Dic47—42no—
189Dic47—44no—
190Dic47—44no—
191Dic47—44no—
192#2839—474no—
193#2836—474no—
194#2833—474no—
195#2831—474no—
196#2402—474no—
197#2402—471no—
198#2401—474no—
199#2399D4711yescharacteristic, complemented
200#2396—472no—
201#2298—471no—
202#2295—472no—
203C2×C47C2×#54111yescharacteristic
204C2×C47#345512yes—
205C2×C47#353114yes—
206D47—81no—
207C2×C4×Q8C2×D4712yes—
208C2×#117—471no—
209#401—474no—
210#400—474no—
211#400—474no—
212#400—474no—
213#400—472no—
214#400—474no—
215#400—474no—
216#400—474no—
217#400—474no—
218#400—474no—
219#399—472no—
220#398—474no—
221#398—474no—
222#398—474no—
223#398—474no—
224#398—474no—
225#398—474no—
226#398C2×D4714yes—
227#397—474no—
228#397—474no—
229#397—474no—
230#396—474no—
231#396—474no—
232#395—472no—
233#395C2×D4714yes—
234#393—472no—
235#391—474no—
236#389—474no—
237#388—474no—
238#387—474no—
239#387—474no—
240#387—474no—
241C4×C4⋊C4—472no—
242C4×C4⋊C4—474no—
243C4×C4⋊C4—474no—
244C4×C4⋊C4—474no—
245C4×C4⋊C4—472no—
246C4×C4⋊C4C2×D4714yes—
247C4×C22⋊C4—474no—
248C4×C22⋊C4—472no—
249#376—474no—
250#376C2×D4711yescharacteristic
251C43—471no—
252C47#3670011yescharacteristic, complemented, Sylow
253C22×Q8C22×D4712yes—
254C23×C4—471no—
255C4×Q8—28no—
256#117—944no—
257C2×C4⋊C4—472no—
258C2×C4⋊C4—474no—
259C2×C4⋊C4—474no—
260C2×C4⋊C4—474no—
261C2×C4⋊C4—474no—
262C2×C4⋊C4—474no—
263C2×C4⋊C4—474no—
264C2×C4⋊C4—474no—
265C2×C4⋊C4—474no—
266C2×C4⋊C4C22×D4714yes—
267C2×C4⋊C4C22×D4712yes—
268C2×C4⋊C4C22×D4714yes—
269C2×C4⋊C4C22×D4714yes—
270C2×C22⋊C4—472no—
271C2×C22⋊C4—474no—
272C2×C22⋊C4—474no—
273C2×C22⋊C4—474no—
274C2×C42—471no—
275C2×C42—472no—
276C2×C42—474no—
277C2×C42—474no—
278C2×C42C22×D4714yes—
279C2×C42C22×D4712yes—
280C2×C42C22×D4711yescharacteristic
281#95—474no—
282#95—474no—
283#95—474no—
284#95—474no—
285#95—474no—
286#95—474no—
287#95—474no—
288#95—472no—
289#95—472no—
290#95—474no—
291#95—474no—
292#95—474no—
293#95—474no—
294#95—474no—
295#95—474no—
296#95—474no—
297#95C22×D4714yes—
298#95C22×D4714yes—
299#95C22×D4714yes—
300C24—471no—
301C2×Q8—24no—
302C2×Q8—44no—
303C22×C4—471no—
304C22×C4—474no—
305C22×C4—474no—
306C22×C4—474no—
307C22×C4—472no—
308C22×C4—942no—
309C22×C4—944no—
310C22×C4C23×D4714yes—
311C22×C4C23×D4714yes—
312C22×C4C23×D4714yes—
313C22×C4C23×D4712yes—
314C22×C4C23×D4711yescharacteristic
315C4⋊C4—24no—
316C4⋊C4—28no—
317C4⋊C4—44no—
318C4⋊C4—44no—
319C4⋊C4—944no—
320C4⋊C4—1884no—
321C4⋊C4—1884no—
322C4⋊C4—1884no—
323C4⋊C4—1884no—
324C4⋊C4—1884no—
325C4⋊C4—1884no—
326C4⋊C4—1884no—
327C4⋊C4—1884no—
328C22⋊C4—944no—
329C22⋊C4—1884no—
330C22⋊C4—1884no—
331C22⋊C4—1884no—
332C42—28no—
333C42—24no—
334C42—942no—
335C42—944no—
336C42—948no—
337C42—1884no—
338C42#1062535214yes—
339C23—941no—
340C23—942no—
341C23—944no—
342C23C24×D4711yescharacteristic
343Q8—48no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—942no—
355C2×C4—941no—
356C2×C4—944no—
357C2×C4—944no—
358C2×C4—944no—
359C2×C4—942no—
360C2×C4—944no—
361C2×C4—944no—
362C2×C4—944no—
363C2×C4—944no—
364C2×C4—944no—
365C2×C4—944no—
366C2×C4—944no—
367C2×C4—1884no—
368C2×C4C2×#1062535214yes—
369C2×C4C2×#1062535212yes—
370C2×C4#1309407214yes—
371C22—1882no—
372C22—1881no—
373C22—1884no—
374C22C22×#1062535211yescharacteristic
375C22C2×#1309407212yes—
376C22#1568995914yes—
377C4—24no—
378C4—44no—
379C4—44no—
380C4—44no—
381C4—1882no—
382C4—1881no—
383C4—1884no—
384C4—1884no—
385C4—1884no—
386C4#1568973414yes—
387C2—3761no—
388C2C2×#1568973411yescharacteristic
389C2#2025677612yes—
390C2#2025753514yes—
subgroupG/H
centerC22×C4C23×D47
commutator, socle, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC47×#2399C2
prime radicalC23×D47C24
hypercenter#2399D47
hypocenterC47#36700
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ #2399

Complements

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

NH ≅ G/Ntype
1. #20251636C2semidirect
3. #20251567C2semidirect
5. #20246695C2semidirect
6. #20246693C2semidirect
10. #20245866C2semidirect
11. C47×#2399C2semidirect
199. #2399D47semidirect
252. C47#36700semidirect

Maximal subgroups

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

subgroupindex
#202516362
#202515912
#202515672
#202515482
#202466952
#202466932
#202466892
#202466582
#202458832
#202458662
C47×#23992
#3670047