Finite Group Databasebeta

← #26374962

Subgroups — #26374962

Order 12032.

2858 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093986C211yesmaximal, characteristic
2#20246027C218yesmaximal, complemented
3#20246006C211yesmaximal, characteristic, complemented
4#20245985C214yesmaximal, complemented
5#20245968C212yesmaximal
6#20245846C218yesmaximal, complemented
7#20245770C214yesmaximal, complemented
8#117×Dic47C212yesmaximal, complemented
9C47×#2316C211yesmaximal, characteristic, complemented
10C23×D188C2211yescharacteristic
11C23×C4×D47C2212yes—
12C2×#13093986C4116yescomplemented
13C2×#13093986C2214yes—
14C2×#13093986C2214yes—
15C2×#13093983C2212yes—
16C2×#13093982C2218yescomplemented
17C22×(C2×D47)⋊C4C2212yes—
18C22×C4×Dic47C2211yescharacteristic
19#15689025C2218yes—
20#15689024C2218yes—
21#15689022C2218yescomplemented
22#15689015C22116yescomplemented
23#15689013C22116yescomplemented
24#15688985C2218yescomplemented
25#15688983C2214yes—
26#15688980C2218yescomplemented
27#15688978C2218yes—
28C4⋊C4×Dic47C2214yescomplemented
29C4⋊C4×Dic47C2214yescomplemented
30C22⋊C4×Dic47C2214yes—
31C22⋊C4×Dic47C2214yes—
32#15688970C2218yescomplemented
33#15688969C2214yescomplemented
34C42×Dic47C2214yescomplemented
35C2×#117×C47C2212yes—
36C22×C4⋊C4×C47C2211yescharacteristic
37C47×#389C2218yescomplemented
38C47×#380C2214yes—
39C24×D47C2312yes—
40C22×D188—24no—
41C22×D188—24no—
42C22×D188C2×C418yes—
43C22×D188C2314yes—
44C23×Dic47C2311yescharacteristic
45C23×Dic47C2311yescharacteristic
46C22×C4×D47—48no—
47C22×C4×D47C2×C4116yes—
48C22×C4×D47C2314yes—
49C22×C4×D47C2314yes—
50#13093986C2×C4132yes—
51#13093986C2×C4132yes—
52#13093983—28no—
53#13093982—416no—
54C2×C4⋊Dic47C2314yes—
55C2×C4⋊Dic47C2318yes—
56C2×C22⋊Dic47C2314yes—
57C2×(C2×D47)⋊C4C2×C4116yes—
58C2×(C2×D47)⋊C4C23116yes—
59C2×(C2×D47)⋊C4C23116yes—
60C2×(C2×D47)⋊C4C2318yes—
61C2×C4×Dic47C2×C418yes—
62C2×C4×Dic47C2314yes—
63C2×C4×Dic47C2318yes—
64C2×C4×Dic47C2318yes—
65C2×C4×Dic47C2312yes—
66C2×C4×Dic47C2312yes—
67C47⋊#95C2318yes—
68C47⋊#95C2318yes—
69C47⋊#95C2318yes—
70C23×C4×C47C2312yes—
71C23×C4×C47C2311yescharacteristic
72#117×C47—28no—
73C2×C4⋊C4×C47C2×C418yes—
74C2×C4⋊C4×C47C2314yes—
75C2×C4⋊C4×C47C2318yes—
76C2×C4⋊C4×C47C2314yes—
77C2×C22⋊C4×C47C2314yes—
78C2×C42×C47C2314yes—
79#95×C47C2318yes—
80C24×C47C2411yescharacteristic
81C23×D47—24no—
82C23×D47—24no—
83C23×D47—44no—
84C23×D47—44no—
85C23×D47C22×C418yes—
86C23×D47C2414yes—
87C22×C4×C47—44no—
88C22×C4×C47C22×C418yes—
89C22×C4×C47C22×C414yes—
90C22×C4×C47#4114yes—
91C22×C4×C47#4114yes—
92C22×C4×C47C2418yes—
93C22×C4×C47C2414yes—
94C22×C4×C47C2412yes—
95C22×Dic47—24no—
96C22×Dic47—42no—
97C22×Dic47C22×C414yes—
98C22×Dic47C22×C414yes—
99C22×Dic47C2418yes—
100C22×Dic47C2411yescharacteristic
101C22×Dic47C2412yes—
102C22×Dic47C2411yescharacteristic
103C2×D188—232no—
104C2×D188—28no—
105C2×D188—28no—
106C2×D188C22×C4116yes—
107C2×C4×D47—216no—
108C2×C4×D47—432no—
109C2×C4×D47—48no—
110C2×C4×D47C22×C4116yes—
111C2×C4×D47C22×C4116yes—
112C4⋊C4×C47—28no—
113C4⋊C4×C47—48no—
114C4⋊C4×C47C22×C4116yes—
115C22⋊C4×C47—28no—
116C42×C47—28no—
117C22⋊Dic47—28no—
118(C2×D47)⋊C4—416no—
119(C2×D47)⋊C4—416no—
120(C2×D47)⋊C4C22×C4132yes—
121C4⋊Dic47—28no—
122C4⋊Dic47—48no—
123C4×Dic47—28no—
124C4×Dic47—216no—
125C4×Dic47—48no—
126C4×Dic47C22×C418yes—
127C4×Dic47C22×C418yes—
128C23×C47—22no—
129C23×C47—22no—
130C23×C47C23×C412yes—
131C23×C47C2×#4112yes—
132C23×C47C2×#4112yes—
133C23×C47C2511yescharacteristic
134C22×D47—216no—
135C22×D47—24no—
136C22×D47—24no—
137C22×D47—28no—
138C22×D47—28no—
139C22×D47—48no—
140C22×D47—48no—
141C22×D47—416no—
142C22×D47—416no—
143C22×D47C23×C418yes—
144C2×C4×C47—28no—
145C2×C4×C47—28no—
146C2×C4×C47—28no—
147C2×C4×C47—28no—
148C2×C4×C47—48no—
149C2×C4×C47#117116yescomplemented
150C2×C4×C47C23×C418yes—
151C2×C4×C47C23×C414yes—
152C2×C4×C47C2×#4114yes—
153C2×C4×C47C2×#4114yes—
154C2×Dic47—22no—
155C2×Dic47—24no—
156C2×Dic47—28no—
157C2×Dic47—28no—
158C2×Dic47—28no—
159C2×Dic47—28no—
160C2×Dic47—44no—
161C2×Dic47C23×C414yes—
162C2×Dic47C23×C412yes—
163C2×Dic47C23×C412yes—
164C2×Dic47H(5,𝔽2)14yes—
165D188—232no—
166C4×D47—232no—
167C4×D47—416no—
168#34058—471nomaximal, Sylow
169C22×C47—22no—
170C22×C47—22no—
171C22×C47—24no—
172C22×C47—24no—
173C22×C47C2×#11714yes—
174C22×C47C24×C411yescharacteristic
175C22×C47C22×#4111yescharacteristic
176C22×C47C22×#4111yescharacteristic
177C22×C47C2×H(5,𝔽2)12yes—
178C22×C47#58512yes—
179C2×D47—28no—
180C2×D47—28no—
181C2×D47—28no—
182C2×D47—432no—
183C2×D47—48no—
184C2×D47—48no—
185C4×C47—28no—
186C4×C47—48no—
187C4×C47C2×#11718yes—
188Dic47—22no—
189Dic47—24no—
190Dic47—48no—
191Dic47#51814yes—
192C22×C4×D4—471no—
193#2319—478no—
194#2316—471no—
195#2316D4711yescharacteristic, complemented
196#2312—474no—
197#2309—472no—
198#2293—478no—
199#2280—474no—
200C4×#117—472no—
201C2×C47—24no—
202C2×C47C22×#11711yescharacteristic
203C2×C47C2×#51811yescharacteristic
204C2×C47#342811yescharacteristic
205C2×C47#346012yes—
206C2×C47#346012yes—
207D47—48no—
208C23×D4—471no—
209C24×C4—472no—
210C2×C4×D4—4716no—
211C2×C4×D4—474no—
212C2×C4×D4—474no—
213C2×#117—472no—
214C2×#117—478no—
215C2×#117C2×D4712yes—
216C22×C4⋊C4C2×D4711yescharacteristic
217C22×C22⋊C4—472no—
218C22×C42—471no—
219#390—478no—
220#390—478no—
221#389—478no—
222#389C2×D4718yescomplemented
223#388—4716no—
224#388—4716no—
225#381—478no—
226#380—474no—
227#380—478no—
228#380C2×D4714yes—
229#379—478no—
230C4×C4⋊C4—474no—
231C4×C4⋊C4—474no—
232C4×C22⋊C4—474no—
233C4×C22⋊C4—474no—
234C4×C22⋊C4—478no—
235#376—474no—
236C43—474no—
237C47#3405811yescharacteristic, complemented, Sylow
238C25—472no—
239C22×D4—478no—
240C22×D4—474no—
241C22×D4—944no—
242C22×D4—944no—
243C23×C4—471no—
244C23×C4—471no—
245C23×C4—474no—
246C23×C4—4716no—
247C23×C4—474no—
248C23×C4—1888no—
249C23×C4C22×D4712yes—
250C23×C4C22×D4711yescharacteristic
251C4×D4—4732no—
252C4×D4—4732no—
253#117—28no—
254#117—948no—
255#117—18816no—
256C2×C4⋊C4—474no—
257C2×C4⋊C4—478no—
258C2×C4⋊C4C4×D4718yes—
259C2×C4⋊C4C22×D4714yes—
260C2×C4⋊C4C22×D4718yes—
261C2×C4⋊C4C22×D4714yes—
262C2×C22⋊C4—4716no—
263C2×C22⋊C4—478no—
264C2×C22⋊C4—474no—
265C2×C22⋊C4—4716no—
266C2×C22⋊C4—4716no—
267C2×C22⋊C4C22×D4714yes—
268C2×C42—478no—
269C2×C42—472no—
270C2×C42—472no—
271C2×C42—478no—
272C2×C42—474no—
273C2×C42—478no—
274C2×C42C22×D4714yes—
275#95—478no—
276#95—478no—
277#95—478no—
278#95C22×D4718yes—
279C24—474no—
280C24—478no—
281C24—944no—
282C24—944no—
283C24—1884no—
284C24—1884no—
285C24C23×D4711yescharacteristic
286C2×D4—4716no—
287C2×D4—9432no—
288C2×D4—948no—
289C2×D4—948no—
290C22×C4—44no—
291C22×C4—471no—
292C22×C4—474no—
293C22×C4—471no—
294C22×C4—472no—
295C22×C4—474no—
296C22×C4—4716no—
297C22×C4—4716no—
298C22×C4—478no—
299C22×C4—9416no—
300C22×C4—944no—
301C22×C4—1888no—
302C22×C4—18832no—
303C22×C4—1882no—
304C22×C4C2×C4×D4718yes—
305C22×C4C2×C4×D4714yes—
306C22×C4#1062535214yes—
307C22×C4#1062535414yes—
308C22×C4C23×D4718yes—
309C22×C4C23×D4714yes—
310C22×C4C23×D4712yes—
311C4⋊C4—28no—
312C4⋊C4—48no—
313C4⋊C4—948no—
314C4⋊C4—1888no—
315C4⋊C4C2×C4×D47116yes—
316C22⋊C4—28no—
317C22⋊C4—4732no—
318C22⋊C4—948no—
319C22⋊C4—18816no—
320C22⋊C4—18816no—
321C42—28no—
322C42—478no—
323C42—478no—
324C42—9416no—
325C42—948no—
326C42—1888no—
327C23—22no—
328C23—22no—
329C23—478no—
330C23—9416no—
331C23—944no—
332C23—944no—
333C23—948no—
334C23—948no—
335C23—1888no—
336C23—1888no—
337C23—18816no—
338C23—18816no—
339C23C22×C4×D4712yes—
340C23C2×#1062535212yes—
341C23C2×#1062535412yes—
342C23C24×D4711yescharacteristic
343D4—9432no—
344C2×C4—28no—
345C2×C4—28no—
346C2×C4—28no—
347C2×C4—28no—
348C2×C4—48no—
349C2×C4—472no—
350C2×C4—474no—
351C2×C4—474no—
352C2×C4—472no—
353C2×C4—9432no—
354C2×C4—948no—
355C2×C4—942no—
356C2×C4—944no—
357C2×C4—948no—
358C2×C4—948no—
359C2×C4—948no—
360C2×C4—18816no—
361C2×C4—1884no—
362C2×C4#13093982116yescomplemented
363C2×C4C22×C4×D4718yes—
364C2×C4C22×C4×D4714yes—
365C2×C4C2×#1062535214yes—
366C2×C4C2×#1062535414yes—
367C22—24no—
368C22—24no—
369C22—22no—
370C22—22no—
371C22—948no—
372C22—948no—
373C22—948no—
374C22—18832no—
375C22—1888no—
376C22—1888no—
377C22C2×#1309398214yes—
378C22C23×C4×D4711yescharacteristic
379C22C22×#1062535211yescharacteristic
380C22C22×#1062535411yescharacteristic
381C22H(5,𝔽2)×D4712yes—
382C22#1568996012yes—
383C4—28no—
384C4—48no—
385C4—474no—
386C4—942no—
387C4—944no—
388C4—1888no—
389C4C2×#1309398218yes—
390C2—24no—
391C2—1888no—
392C2C22×#1309398211yescharacteristic
393C2#518×D4711yescharacteristic
394C2#2025659011yescharacteristic
395C2#2025680512yes—
396C2#2025681112yes—
subgroupG/H
center, FrattiniC23C24×D47
commutatorC22×C47C24×C4
FittingC47×#2316C2
socle, cosocleC23×C47C25
prime radicalC23×D188C22
hypercenter#2316D47
hypocenterC47#34058
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ #2316

Complements

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

NH ≅ G/Ntype
2. #20246027C2semidirect
3. #20246006C2semidirect
4. #20245985C2semidirect
6. #20245846C2semidirect
7. #20245770C2semidirect
8. #117×Dic47C2semidirect
9. C47×#2316C2semidirect
12. C2×#13093986C4semidirect
16. C2×#13093982C22semidirect
21. #15689022C22semidirect
22. #15689015C22semidirect
23. #15689013C22semidirect
24. #15688985C22semidirect
26. #15688980C22semidirect
28. C4⋊C4×Dic47C22semidirect
29. C4⋊C4×Dic47C22semidirect
32. #15688970C22semidirect
33. #15688969C22semidirect
34. C42×Dic47C22semidirect
37. C47×#389C22semidirect
149. C2×C4×C47#117semidirect
195. #2316D47semidirect
222. #389C2×D47semidirect
237. C47#34058semidirect
362. C2×C4#13093982semidirect

Maximal subgroups

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

subgroupindex
C22×#130939862
#202460272
#202460062
#202459852
#202459682
#202458462
#202457702
#117×Dic472
C47×#23162
#3405847