Finite Group Databasebeta

← #26422997

Subgroups — #26422997

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#20251626C214yesmaximal, complemented
2#20251581C214yesmaximal
3#20251562C214yesmaximal, complemented
4#20251538C214yesmaximal
5#20246695C214yesmaximal, complemented
6#20246693C211yesmaximal, characteristic, complemented
7#20246688C214yesmaximal
8#20246681C211yesmaximal, characteristic, complemented
9#20246656C212yesmaximal
10#20245865C212yesmaximal, complemented
11C47×#2298C211yesmaximal, characteristic
12C2×C4×Dic94C2212yes—
13C47⋊#401C2214yes—
14#15689065C2214yes—
15#15689065C2214yes—
16#15689064C2214yes—
17#15689064C2214yes—
18#15689064C2214yes—
19#15689063C2214yes—
20#15689063C2214yes—
21#15689062C2212yes—
22#15689061C2214yes—
23#15689059C2212yes—
24#15689057C2214yes—
25#15689057C2214yes—
26#15689056C2214yes—
27#15689056C2214yes—
28#15689055C2214yes—
29#15689054C2214yes—
30#15689053C2214yes—
31#15689051C2214yes—
32#15689051C2214yes—
33#15689048C2214yes—
34#15689046C2214yes—
35#15689046C2214yes—
36#15689044C2214yes—
37#15689043C2212yes—
38#15689035C2212yes—
39#15689029C2214yes—
40#15689021C2214yes—
41#15689007C2214yes—
42#15689007C2214yes—
43C4×Dic47⋊C4C2214yes—
44C4×Dic47⋊C4C2214yes—
45C4×Dic47⋊C4C2214yes—
46C4×C4⋊Dic47C2212yes—
47C4×C4⋊Dic47C2214yes—
48C4×C22⋊Dic47C2214yes—
49#15688969C2214yes—
50C2×#117×C47C2211yescharacteristic
51C47×#388C2214yes—
52C47×#387C2214yes—
53C4×C4⋊C4×C47C2212yes—
54C4×C22⋊C4×C47C2212yes—
55C47×#376C2211yescharacteristic
56C43×C47C2211yescharacteristic
57C22×Dic94C2312yes—
58C4×Dic94—28no—
59C2×C4⋊Dic47C2312yes—
60C2×C4⋊Dic47C2314yes—
61C2×C4⋊Dic47C2314yes—
62C2×C4⋊Dic47C2314yes—
63C2×Dic47⋊C4C2314yes—
64C2×Dic47⋊C4C2314yes—
65C2×Dic47⋊C4C2314yes—
66C2×Dic47⋊C4C2314yes—
67C2×Dic47⋊C4C2314yes—
68C2×Dic47⋊C4C2314yes—
69C2×Dic47⋊C4C2314yes—
70C2×C22⋊Dic47C2314yes—
71C2×C22⋊Dic47C2314yes—
72C2×C4×Dic47C2314yes—
73C2×C4×Dic47C2314yes—
74C47⋊#95C2314yes—
75C47⋊#95C2314yes—
76C47⋊#95C2314yes—
77C47⋊#95C2314yes—
78C47⋊#95C2314yes—
79C47⋊#95C2314yes—
80C47⋊#95C2312yes—
81C47⋊#95C2312yes—
82C47⋊#95C2314yes—
83C47⋊#95C2314yes—
84C47⋊#95C2314yes—
85C47⋊#95C2314yes—
86C47⋊#95C2314yes—
87C47⋊#95C2314yes—
88C47⋊#95C2314yes—
89C47⋊#95C2314yes—
90C23×C4×C47C2311yescharacteristic
91#117×C47—24no—
92C2×C4⋊C4×C47C2312yes—
93C2×C4⋊C4×C47C2314yes—
94C2×C22⋊C4×C47C2312yes—
95C2×C22⋊C4×C47C2314yes—
96C2×C42×C47C2312yes—
97C2×C42×C47C2312yes—
98C2×C42×C47C2311yescharacteristic
99C2×C42×C47C2314yes—
100C2×C42×C47C2311yescharacteristic
101#95×C47C2314yes—
102#95×C47C2314yes—
103#95×C47C2314yes—
104C24×C47C2411yescharacteristic
105C22×C4×C47—22no—
106C22×C4×C47—24no—
107C22×C4×C47C2414yes—
108C22×C4×C47C2412yes—
109C22×C4×C47C2414yes—
110C22×C4×C47C2411yescharacteristic
111C22×C4×C47C2412yes—
112C22×C4×C47C2411yescharacteristic
113C22×Dic47C2414yes—
114C22×Dic47C2414yes—
115C22×Dic47C2414yes—
116C22×Dic47C2414yes—
117C2×Dic94—24no—
118C2×Dic94—44no—
119C4⋊C4×C47—24no—
120C4⋊C4×C47—44no—
121C22⋊C4×C47—24no—
122C22⋊C4×C47—44no—
123C42×C47—22no—
124C42×C47—24no—
125C42×C47—24no—
126C42×C47—44no—
127C42×C47#4114yes—
128C22⋊Dic47—44no—
129C22⋊Dic47—44no—
130C4⋊Dic47—24no—
131C4⋊Dic47—44no—
132C4⋊Dic47—44no—
133C4⋊Dic47—44no—
134Dic47⋊C4—28no—
135Dic47⋊C4—44no—
136Dic47⋊C4—44no—
137Dic47⋊C4—44no—
138Dic47⋊C4—44no—
139Dic47⋊C4—44no—
140Dic47⋊C4—44no—
141C4×Dic47—28no—
142C4×Dic47—28no—
143C23×C47—21no—
144C23×C47—22no—
145C23×C47—24no—
146C23×C47C2511yescharacteristic
147C2×C4×C47—24no—
148C2×C4×C47—22no—
149C2×C4×C47—21no—
150C2×C4×C47—24no—
151C2×C4×C47—24no—
152C2×C4×C47—22no—
153C2×C4×C47—24no—
154C2×C4×C47—24no—
155C2×C4×C47—24no—
156C2×C4×C47—24no—
157C2×C4×C47—24no—
158C2×C4×C47—44no—
159C2×C4×C47C2×#4114yes—
160C2×C4×C47C2×#4112yes—
161C2×C4×C472−1+414yes—
162C2×Dic47—24no—
163C2×Dic47—24no—
164C2×Dic47—24no—
165C2×Dic47—24no—
166C2×Dic47—24no—
167C2×Dic47—24no—
168C2×Dic47—24no—
169C2×Dic47—24no—
170C2×Dic47—24no—
171C2×Dic47—24no—
172C2×Dic47—24no—
173C2×Dic47—24no—
174Dic94—48no—
175#36700—471nomaximal, Sylow
176C22×C47—41no—
177C22×C47—42no—
178C22×C47—44no—
179C22×C47C22×#4111yescharacteristic
180C22×C47C2×2−1+412yes—
181C22×C47#58514yes—
182C4×C47—24no—
183C4×C47—41no—
184C4×C47—42no—
185C4×C47—44no—
186C4×C47—44no—
187C4×C47#54114yes—
188Dic47—44no—
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#2399—471no—
200#2396—472no—
201#2298D4711yescharacteristic
202#2295—472no—
203C2×C47—81no—
204C2×C47C2×#54111yescharacteristic
205C2×C47#345512yes—
206C2×C47#353114yes—
207C2×C4×Q8—472no—
208C2×#117C2×D4711yescharacteristic
209#401—474no—
210#400—474no—
211#400—472no—
212#400—474no—
213#400—474no—
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#398—474no—
227#397—474no—
228#397—474no—
229#397—474no—
230#396—474no—
231#396—474no—
232#395—472no—
233#395—474no—
234#393—472no—
235#391—474no—
236#389—474no—
237#388C2×D4714yes—
238#387—474no—
239#387—474no—
240#387C2×D4714yes—
241C4×C4⋊C4—472no—
242C4×C4⋊C4—474no—
243C4×C4⋊C4—474no—
244C4×C4⋊C4—474no—
245C4×C4⋊C4—474no—
246C4×C4⋊C4C2×D4712yes—
247C4×C22⋊C4—474no—
248C4×C22⋊C4C2×D4712yes—
249#376—474no—
250#376C2×D4711yescharacteristic
251C43C2×D4711yescharacteristic
252C47#3670011yescharacteristic, complemented, Sylow
253C22×Q8—472no—
254C23×C4C22×D4711yescharacteristic
255C4×Q8—948no—
256#117—24no—
257C2×C4⋊C4—474no—
258C2×C4⋊C4—472no—
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⋊C4—474no—
267C2×C4⋊C4—474no—
268C2×C4⋊C4C22×D4712yes—
269C2×C4⋊C4C22×D4714yes—
270C2×C22⋊C4—474no—
271C2×C22⋊C4—474no—
272C2×C22⋊C4C22×D4712yes—
273C2×C22⋊C4C22×D4714yes—
274C2×C42—474no—
275C2×C42—474no—
276C2×C42C22×D4711yescharacteristic
277C2×C42C22×D4712yes—
278C2×C42C22×D4712yes—
279C2×C42C22×D4714yes—
280C2×C42C22×D4711yescharacteristic
281#95—474no—
282#95—474no—
283#95—474no—
284#95—474no—
285#95—474no—
286#95—474no—
287#95—472no—
288#95—472no—
289#95—474no—
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—
300C24C23×D4711yescharacteristic
301C2×Q8—944no—
302C2×Q8—1884no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—474no—
306C22×C4—474no—
307C22×C4—474no—
308C22×C4—474no—
309C22×C4C23×D4711yescharacteristic
310C22×C4C23×D4714yes—
311C22×C4C23×D4714yes—
312C22×C4C23×D4712yes—
313C22×C4C23×D4712yes—
314C22×C4C23×D4711yescharacteristic
315C4⋊C4—24no—
316C4⋊C4—44no—
317C4⋊C4—948no—
318C4⋊C4—944no—
319C4⋊C4—1884no—
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—24no—
329C22⋊C4—44no—
330C22⋊C4—1884no—
331C22⋊C4—1884no—
332C42—22no—
333C42—24no—
334C42—24no—
335C42—44no—
336C42—948no—
337C42—948no—
338C42#1062535014yes—
339C23—21no—
340C23—22no—
341C23—24no—
342C23C24×D4711yescharacteristic
343Q8—1888no—
344C2×C4—24no—
345C2×C4—22no—
346C2×C4—21no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—22no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—44no—
356C2×C4—944no—
357C2×C4—944no—
358C2×C4—944no—
359C2×C4—944no—
360C2×C4—944no—
361C2×C4—944no—
362C2×C4—944no—
363C2×C4—944no—
364C2×C4—944no—
365C2×C4—944no—
366C2×C4—944no—
367C2×C4—944no—
368C2×C4C2×#1062535014yes—
369C2×C4C2×#1062535012yes—
370C2×C4#1309407114yes—
371C22—42no—
372C22—41no—
373C22—44no—
374C22C22×#1062535011yescharacteristic
375C22C2×#1309407112yes—
376C22#1568995914yes—
377C4—24no—
378C4—42no—
379C4—41no—
380C4—44no—
381C4—44no—
382C4—1884no—
383C4—1884no—
384C4—1884no—
385C4—1884no—
386C4#1568973514yes—
387C2—81no—
388C2C2×#1568973511yescharacteristic
389C2#2025677312yes—
390C2#2025753314yes—
subgroupG/H
centerC22×C4C23×D47
commutator, socle, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC47×#2298C2
prime radicalC24×C47C24
hypercenter#2298D47
hypocenterC47#36700
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ #2298

Complements

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

NH ≅ G/Ntype
1. #20251626C2semidirect
3. #20251562C2semidirect
5. #20246695C2semidirect
6. #20246693C2semidirect
8. #20246681C2semidirect
10. #20245865C2semidirect
252. C47#36700semidirect

Maximal subgroups

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

subgroupindex
#202516262
#202515812
#202515622
#202515382
#202466952
#202466932
#202466882
#202466812
#202466562
#202458652
C47×#22982
#3670047