Finite Group Databasebeta

← #26369314

Subgroups — #26369314

Order 12032.

674 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×M32×D47C211yesmaximal, characteristic
2C2×#15689160C211yesmaximal, characteristic
3#427×D47C211yesmaximal, characteristic
4#20245177C211yesmaximal, characteristic, complemented
5#20245176C211yesmaximal, characteristic, complemented
6#20245175C212yesmaximal, complemented
7#20245174C211yesmaximal, characteristic, complemented
8#20245173C212yesmaximal, complemented
9#20245171C212yesmaximal, complemented
10#20245169C212yesmaximal, complemented
11C47×#2159C211yesmaximal, characteristic, complemented
12C22×C8×D47C2211yescharacteristic
13C2×#13093982—21no—
14#117×D47—21no—
15C2×#13093964C2211yescharacteristic
16C2×#13093963C2212yes—
17M32×D47C418yescomplemented
18C2×C16×D47C2212yes—
19#15689161C2211yescharacteristic
20#15689160—21no—
21#15689160—21no—
22#15689160C2211yescharacteristic
23#15689160C2211yescharacteristic
24#15689160C2212yes—
25#15689160C2212yes—
26#15689159—21no—
27#15689158—21no—
28C2×#13093958C2211yescharacteristic
29#107×D47C2211yescharacteristic
30#107×D47C2211yescharacteristic
31C2×#13093955—21no—
32#106×D47—21no—
33#15688951C2212yescomplemented
34#15688950C2212yescomplemented
35#15688950C2212yescomplemented
36#15688949C2212yescomplemented
37#15688948C2212yescomplemented
38#15688948C2212yescomplemented
39C2×M32×C47C2211yescharacteristic
40C47×#427C2211yescharacteristic
41C2×#107×C47C2211yescharacteristic
42C47×#367C2212yescomplemented
43C47×#366C2212yescomplemented
44C22×C4×D47D411yescharacteristic
45C22×C47⋊C8D411yescharacteristic
46C2×C8×D47C2×C414yes—
47C2×C8×D47D411yescharacteristic
48C2×C8×D47D411yescharacteristic
49C2×C8×D47Q814yes—
50C2×C8×D47C2311yescharacteristic
51C2×C8×D47C2311yescharacteristic
52#13093983—21no—
53#13093982—21no—
54#13093982—21no—
55#13093982—22no—
56#13093982—22no—
57#13093982—41no—
58#13093982—41no—
59#13093981—41no—
60#13093980—41no—
61C2×C4⋊Dic47—21no—
62C4⋊C4×D47—21no—
63C4⋊C4×D47—21no—
64C2×(C2×D47)⋊C4—41no—
65C22⋊C4×D47—41no—
66C2×C4×Dic47—41no—
67C42×D47—41no—
68#13093964C2×C414yescomplemented
69#13093963C2×C418yescomplemented
70C2×C47⋊C16C2312yes—
71C16×D47C2×C418yescomplemented
72#13093958—21no—
73#13093958C2311yescharacteristic
74#13093958C2311yescharacteristic
75#13093958C2312yes—
76#13093957—21no—
77#13093957—21no—
78#13093956—21no—
79#13093955—21no—
80#13093955—21no—
81#13093955—22no—
82#13093955—22no—
83C22×C8×C47C2311yescharacteristic
84#117×C47—21no—
85C2×C4⋊C4×C47—21no—
86M32×C47C2×C414yescomplemented
87C2×C16×C47C2312yes—
88#107×C47—21no—
89#107×C47C2311yescharacteristic
90#107×C47C2312yes—
91#107×C47C2311yescharacteristic
92#106×C47—21no—
93C23×D47D811yescharacteristic
94C22×C4×C47—41no—
95C22×C4×C47C2×D411yescharacteristic
96C22×Dic47—41no—
97C22×Dic47D811yescharacteristic
98C2×C4×D47—41no—
99C2×C4×D47—41no—
100C2×C4×D47C4⋊C414yes—
101C2×C4×D47D811yescharacteristic
102C2×C4×D47D811yescharacteristic
103C2×C4×D47Q1614yes—
104C2×C4×D47C2×D411yescharacteristic
105C2×C4×D47C2×D411yescharacteristic
106C2×C8×C47C22×C412yes—
107C2×C8×C47C2×D411yescharacteristic
108C2×C8×C47C2×Q812yes—
109C2×C8×C47C2411yescharacteristic
110C4⋊C4×C47—21no—
111C4⋊C4×C47—21no—
112C4⋊C4×C47—22no—
113C4⋊C4×C47—41no—
114C22⋊C4×C47—41no—
115C42×C47—41no—
116C22⋊Dic47—41no—
117(C2×D47)⋊C4—41no—
118(C2×D47)⋊C4—41no—
119(C2×D47)⋊C4—42no—
120(C2×D47)⋊C4—42no—
121C4⋊Dic47—22no—
122C4⋊Dic47—21no—
123C4⋊Dic47—21no—
124C4⋊Dic47—41no—
125Dic47⋊C4—41no—
126Dic47⋊C4—41no—
127C4×Dic47—41no—
128C4×Dic47—41no—
129C4×Dic47—41no—
130C4×Dic47—42no—
131C4×Dic47—42no—
132C2×C47⋊C8C4⋊C414yes—
133C2×C47⋊C8C2×D411yescharacteristic
134C2×C47⋊C8C2×D411yescharacteristic
135C8×D47C4⋊C414yes—
136C8×D47C4⋊C414yes—
137C8×D47C22×C412yes—
138C8×D47C22×C412yes—
139C8×D47C2×Q812yes—
140C8×D47C2×Q812yes—
141C16×C47C22×C414yes—
142C47⋊C16C22×C414yes—
143C23×C47C2×D811yescharacteristic
144C22×D47—41no—
145C22×D47—41no—
146C22×D47#10714yes—
147C22×D47C2×D811yescharacteristic
148C22×D47C2×D811yescharacteristic
149C2×C4×C47—42no—
150C2×C4×C47—41no—
151C2×C4×C47—41no—
152C2×C4×C47—82no—
153C2×C4×C47C2×C4⋊C412yes—
154C2×C4×C47C2×D811yescharacteristic
155C2×C4×C47C2×Q1612yes—
156C2×C4×C47C22×D411yescharacteristic
157C2×Dic47—41no—
158C2×Dic47—41no—
159C2×Dic47—42no—
160C2×Dic47—82no—
161C2×Dic47#10714yes—
162C2×Dic47C2×D811yescharacteristic
163C2×Dic47C2×D811yescharacteristic
164C4×D47—82no—
165C4×D47—82no—
166C4×D47#10714yes—
167C4×D47#10714yes—
168C4×D47C2×C4⋊C412yes—
169C4×D47C2×C4⋊C412yes—
170C4×D47C2×Q1612yes—
171C4×D47C2×Q1612yes—
172C8×C47C2×C4⋊C412yes—
173C8×C47C23×C411yescharacteristic
174C8×C47C22×Q811yescharacteristic
175C47⋊C8C2×C4⋊C412yes—
176C47⋊C8C2×C4⋊C412yes—
177#33409—471nomaximal, Sylow
178C22×C47—21no—
179C22×C47—21no—
180C22×C47C2×#10712yes—
181C22×C47C22×D811yescharacteristic
182C2×D47—21no—
183C2×D47—21no—
184C2×D47—21no—
185C2×D47—21no—
186C2×D47—44no—
187C2×D47C2×#10712yes—
188C2×D47C2×#10712yes—
189C4×C47—81no—
190C4×C47—81no—
191C4×C47—82no—
192C4×C47C2×#10712yes—
193C4×C47C22×C4⋊C411yescharacteristic
194C4×C47C22×Q1611yescharacteristic
195Dic47—81no—
196Dic47—81no—
197Dic47—82no—
198Dic47C2×#10712yes—
199Dic47C2×#10712yes—
200C22×M32—471no—
201C2×#427—471no—
202C2×#427—471no—
203#2159—471no—
204#2159—471no—
205#2159—471no—
206#2159—472no—
207#2159—472no—
208#2159D4711yescharacteristic, complemented
209#2158—472no—
210#2158—472no—
211C2×C47—22no—
212C2×C47C22×#10711yescharacteristic
213C2×C47#341411yescharacteristic
214C2×C47#341411yescharacteristic
215D47—22no—
216D47—22no—
217C23×C8—471no—
218C2×#117—941no—
219C2×#117—941no—
220C2×M32—471no—
221C2×M32—478no—
222C2×M32—472no—
223C2×M32C2×D4711yescharacteristic
224C22×C16—472no—
225#427—471no—
226#427—471no—
227#427—472no—
228#427—471no—
229#427—472no—
230#427—941no—
231#427—941no—
232#427—941no—
233#427—941no—
234#427C2×D4711yescharacteristic
235C2×#107—471no—
236C2×#107—471no—
237C2×#107—471no—
238C2×#107C2×D4711yescharacteristic
239C2×#106—941no—
240C2×#106—941no—
241#367—472no—
242#367—472no—
243#367—472no—
244#367C2×D4712yescomplemented
245#366—472no—
246#366—472no—
247#366—472no—
248#366C2×D4712yescomplemented
249C47#3340911yescharacteristic, complemented, Sylow
250C23×C4—471no—
251C22×C8—471no—
252C22×C8—474no—
253C22×C8—471no—
254C22×C8—471no—
255C22×C8—474no—
256C22×C8—471no—
257C22×C8—471no—
258C22×C8C22×D4711yescharacteristic
259#117—21no—
260#117—941no—
261#117—941no—
262#117—942no—
263#117—942no—
264#117—941no—
265#117—1881no—
266#117—1881no—
267#117—1881no—
268#117—1881no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—941no—
271C2×C4⋊C4—941no—
272C2×C4⋊C4—941no—
273C2×C22⋊C4—1881no—
274C2×C22⋊C4—1881no—
275C2×C42—1881no—
276C2×C42—1881no—
277M32—474no—
278M32—478no—
279M32C4×D4714yescomplemented
280C2×C16—472no—
281C2×C16—478no—
282C2×C16C22×D4712yes—
283#107—21no—
284#107—471no—
285#107—471no—
286#107—472no—
287#107—941no—
288#107—941no—
289#107—941no—
290#107C22×D4711yescharacteristic
291#107C22×D4712yes—
292#107C22×D4711yescharacteristic
293#106—21no—
294#106—941no—
295#106—941no—
296#106—942no—
297#106—942no—
298#106—941no—
299C24—471no—
300C22×C4—41no—
301C22×C4—471no—
302C22×C4—474no—
303C22×C4—471no—
304C22×C4—471no—
305C22×C4—474no—
306C22×C4—471no—
307C22×C4—471no—
308C22×C4—1881no—
309C22×C4—1881no—
310C22×C4—1881no—
311C22×C4D4×D4711yescharacteristic
312C2×C8—474no—
313C2×C8—471no—
314C2×C8—471no—
315C2×C8—474no—
316C2×C8—474no—
317C2×C8—472no—
318C2×C8—472no—
319C2×C8—472no—
320C2×C8—472no—
321C2×C8C2×C4×D4712yes—
322C2×C8D4×D4711yescharacteristic
323C2×C8Q8×D4712yes—
324C2×C8C23×D4711yescharacteristic
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—22no—
328C4⋊C4—41no—
329C4⋊C4—942no—
330C4⋊C4—941no—
331C4⋊C4—941no—
332C4⋊C4—1881no—
333C4⋊C4—1881no—
334C4⋊C4—1881no—
335C22⋊C4—41no—
336C22⋊C4—1881no—
337C22⋊C4—1881no—
338C22⋊C4—1882no—
339C22⋊C4—1882no—
340C22⋊C4—1881no—
341C42—41no—
342C42—1881no—
343C42—1881no—
344C42—1882no—
345C42—1882no—
346C42—1881no—
347C16—474no—
348C16C2×C4×D4714yes—
349C23—471no—
350C23—471no—
351C23—474no—
352C23—1881no—
353C23—1881no—
354C23D8×D4711yescharacteristic
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—82no—
359C2×C4—471no—
360C2×C4—471no—
361C2×C4—474no—
362C2×C4—472no—
363C2×C4—472no—
364C2×C4—472no—
365C2×C4—472no—
366C2×C4—474no—
367C2×C4—474no—
368C2×C4—3762no—
369C2×C4—3762no—
370C2×C4—3762no—
371C2×C4—1881no—
372C2×C4—1881no—
373C2×C4—1882no—
374C2×C4C4⋊C4×D4712yes—
375C2×C4D8×D4711yescharacteristic
376C2×C4Q16×D4712yes—
377C2×C4C2×D4×D4711yescharacteristic
378C8—472no—
379C8—472no—
380C8C4⋊C4×D4712yes—
381C8C22×C4×D4711yescharacteristic
382C8C2×Q8×D4711yescharacteristic
383C22—21no—
384C22—21no—
385C22—472no—
386C22—472no—
387C22—941no—
388C22—941no—
389C22—941no—
390C22—941no—
391C22—1884no—
392C22#107×D4712yes—
393C22C2×D8×D4711yescharacteristic
394C4—82no—
395C4—81no—
396C4—81no—
397C4—472no—
398C4—472no—
399C4—3761no—
400C4—3761no—
401C4—3762no—
402C4#107×D4712yes—
403C4C2×C4⋊C4×D4711yescharacteristic
404C4C2×Q16×D4711yescharacteristic
405C2—22no—
406C2—942no—
407C2—942no—
408C2C2×#107×D4711yescharacteristic
409C2#2025652011yescharacteristic
410C2#2025652311yescharacteristic
subgroupG/H
centerC22C2×D8×D47
commutatorC8×C47C23×C4
FrattiniC2×C8C23×D47
FittingC47×#2159C2
socleC22×C47C22×D8
prime radicalC23×D47D8
cosocleC2×C8×C47C24
hypercenter#2159D47
hypocenterC47#33409
LCSG ▷ C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2159

Complements

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

NH ≅ G/Ntype
4. #20245177C2semidirect
5. #20245176C2semidirect
6. #20245175C2semidirect
7. #20245174C2semidirect
8. #20245173C2semidirect
9. #20245171C2semidirect
10. #20245169C2semidirect
11. C47×#2159C2semidirect
17. M32×D47C4semidirect
33. #15688951C22semidirect
34. #15688950C22semidirect
35. #15688950C22semidirect
36. #15688949C22semidirect
37. #15688948C22semidirect
38. #15688948C22semidirect
42. C47×#367C22semidirect
43. C47×#366C22semidirect
68. #13093964C2×C4semidirect
69. #13093963C2×C4semidirect
71. C16×D47C2×C4semidirect
86. M32×C47C2×C4semidirect
208. #2159D47semidirect
244. #367C2×D47semidirect
248. #366C2×D47semidirect
249. C47#33409semidirect
279. M32C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×M32×D472
C2×#156891602
#427×D472
#202451772
#202451762
#202451752
#202451742
#202451732
#202451712
#202451692
C47×#21592
#3340947