Finite Group Databasebeta

← #27080227

Subgroups — #27080227

Order 12032.

2706 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689889C212yesmaximal, complemented
2#20256721C211yesmaximal, characteristic
3#20252112C2116yesmaximal, complemented
4#20252105C214yesmaximal, complemented
5#20252099C214yesmaximal, complemented
6C2×#15689091C212yesmaximal, complemented
7C2×#15689079C211yesmaximal, characteristic, complemented
8C47×#2905C211yesmaximal, characteristic, complemented
9C22×#10625350C2212yes—
10#15689889—216no—
11C2×#13094032C2218yescomplemented
12C2×#13094030C2218yes—
13C22×#10625329C2211yescharacteristic
14C2×M16×D47C2218yescomplemented
15C23×C47⋊C8C2211yescharacteristic
16#15689704C414yes—
17#15689699C414yes—
18#15689692C414yes—
19#15689685C414yes—
20#15689673C2214yes—
21#15689635C2218yes—
22#15689200C22116yescomplemented
23#15689193C22132yescomplemented
24#15689188C22116yescomplemented
25#15689091C2218yes—
26#15689091C2214yescomplemented
27#15689091C2214yescomplemented
28C2×(C2×D47)⋊C8C2218yescomplemented
29#15689079C2218yescomplemented
30C2×#13093935C2212yescomplemented
31C2×C4×C47⋊C8C2212yescomplemented
32C22×M16×C47C2212yes—
33C2×#117×C47C2211yescharacteristic
34C47×#424C2218yescomplemented
35C47×#423C2214yescomplemented
36C2×#10625350—28no—
37C2×#10625350—28no—
38C2×#10625350C2×C414yes—
39C2×#10625350C2×C414yes—
40C2×#10625350C2314yes—
41C2×#10625350C2314yes—
42C22×Dic94C2×C412yes—
43C22×D188C2×C412yes—
44C22×C22⋊D47C2×C414yes—
45C22×C4×D47C2314yes—
46#13094032—232no—
47#13094030—232no—
48C2×#10625329D418yescomplemented
49C2×#10625329C2314yes—
50C2×#10625324C23116yes—
51M16×D47—232no—
52C22×C47⋊C8—24no—
53C22×C47⋊C8C2311yescharacteristic
54C22×C47⋊C8C2311yescharacteristic
55C22×C47⋊C8C2314yes—
56C22×C47⋊C8C2312yes—
57C2×C8×D47C23116yes—
58#13094024C2×C414yes—
59#13094021C2×C414yes—
60#13094012C2×C418yes—
61#13094008C2×C4116yes—
62#13094004C2×C4116yes—
63#13093999C2×C4116yes—
64#13093998C2×C4116yes—
65#13093994C2×C4116yes—
66C4×D188C23116yes—
67(C2×D47)⋊C8C23116yes—
68(C2×D47)⋊C8C23116yes—
69#13093935C2318yes—
70C4×C47⋊C8C2318yes—
71C23×C4×C47C2311yescharacteristic
72C2×M16×C47—28no—
73C2×M16×C47C2314yes—
74C2×M16×C47C2314yes—
75C22×C8×C47C2314yes—
76#117×C47C2×C414yes—
77#117×C47C2314yes—
78C2×C4⋊C4×C47C2×C412yes—
79C2×C22⋊C4×C47C2×C412yes—
80C2×C42×C47C2312yes—
81C4⋊C8×C47C23116yes—
82C24×C47C22×C411yescharacteristic
83C23×D47C22×C414yes—
84C22×C4×C47C22×C414yes—
85C22×C4×C47C22×C411yescharacteristic
86C22×C4×C47C22×C412yes—
87C22×C4×C47C2×D414yes—
88C22×C4×C47#4114yes—
89C22×C4×C47C2411yescharacteristic
90C22×C4×C47C2412yes—
91C2×C22⋊D47—216no—
92C2×C22⋊D47C22×C418yes—
93C2×C22⋊D47C22×C418yes—
94C22×Dic47C22×C414yes—
95#10625350—216no—
96#10625350—216no—
97#10625350—28no—
98#10625350—28no—
99#10625350—28no—
100#10625350—28no—
101C2×D188—28no—
102C2×D188C22×C414yes—
103C2×D188C22×C414yes—
104C2×D188C22×C4116yes—
105C2×C4×D47—28no—
106C2×C4×D47—28no—
107C2×C4×D47C22×C418yes—
108C2×C4×D47C2418yes—
109C2×Dic94—28no—
110C2×Dic94C22×C414yes—
111C2×Dic94C22×C414yes—
112M16×C47—28no—
113M16×C47—28no—
114C2×C8×C47—28no—
115C2×C8×C47C2418yes—
116C4⋊C4×C47C22×C418yes—
117C22⋊C4×C47C22×C418yes—
118C42×C47C22×C414yes—
119C42×C47C2414yes—
120(C2×D47)⋊C4C22×C4132yes—
121C4⋊Dic47C22×C4116yes—
122#10625329C2×D4116yes—
123C2×C47⋊C8—28no—
124C2×C47⋊C8—22no—
125C2×C47⋊C8—22no—
126C2×C47⋊C8C2×D418yes—
127C2×C47⋊C8C2412yes—
128C2×C47⋊C8C2412yes—
129C2×C47⋊C8C2414yes—
130#10625324—232no—
131C8×D47—232no—
132C23×C47—42no—
133C23×C47C4×D414yes—
134C23×C47C23×C412yes—
135C23×C47C23×C411yescharacteristic
136C22×D47—28no—
137C22×D47—48no—
138C22×D47C23×C418yes—
139C2×C4×C47—44no—
140C2×C4×C47C4×D418yes—
141C2×C4×C47C4×D414yes—
142C2×C4×C47C23×C418yes—
143C2×C4×C47C23×C411yescharacteristic
144C2×C4×C47C23×C412yes—
145C2×C4×C47C22×D412yes—
146C2×C4×C47C22×D414yes—
147C2×C4×C47C2×#4114yes—
148C2×C4×C47C2×#4112yes—
149C2×C4×C47C2511yescharacteristic
150C22⋊D47—216no—
151C22⋊D47—216no—
152C2×Dic47—28no—
153C2×Dic47C23×C418yes—
154D188—28no—
155D188—28no—
156D188—416no—
157C4×D47—216no—
158C4×D47—28no—
159C4×D47—28no—
160Dic94—28no—
161Dic94—28no—
162C8×C47—28no—
163C47⋊C8—22no—
164C47⋊C8—22no—
165C47⋊C8C22×D418yes—
166#60143—471nomaximal, Sylow
167C22×C47—22no—
168C22×C47—24no—
169C22×C47—44no—
170C22×C47C2×C4×D414yes—
171C22×C47C2×C4×D412yes—
172C22×C47C24×C411yescharacteristic
173C2×D47—28no—
174C2×D47—216no—
175C2×D47—88no—
176C4×C47—28no—
177C4×C47C2×C4×D412yes—
178C4×C47C2×C4×D414yes—
179C4×C47C23×D411yescharacteristic
180C4×C47C22×#4111yescharacteristic
181Dic47—28no—
182C2×#568—472no—
183#3449—471no—
184#2933—4716no—
185#2932—474no—
186#2931—474no—
187#2905D4711yescharacteristic, complemented
188C2×#408—472no—
189C2×#405—471no—
190C2×C47—24no—
191C2×C47—22no—
192C2×C47C22×C4×D411yescharacteristic
193C2×C47#357312yes—
194D47—48no—
195C22×#41—472no—
196#568—9416no—
197C2×#131—478no—
198C2×#131—478no—
199C22×M16—471no—
200C22×M16—478no—
201C22×M16C2×D4712yes—
202C23×C8—471no—
203#537—474no—
204#536—474no—
205#535—474no—
206#534—474no—
207#532—474no—
208#525—478no—
209C2×#117C2×D4711yescharacteristic
210#436—4716no—
211#435—4732no—
212C8×D4—4716no—
213#424C2×D4718yescomplemented
214#423C2×D4714yescomplemented
215#408—474no—
216#408—474no—
217#408—478no—
218C2×C22⋊C8—478no—
219#405—478no—
220C2×#97—472no—
221C2×C4×C8—472no—
222C47#6014311yescharacteristic, complemented, Sylow
223C2×#41—474no—
224C2×#41—474no—
225C2×#41—474no—
226C2×#41—474no—
227C2×#41—948no—
228C2×#41—948no—
229C22×Q8—472no—
230C22×D4—474no—
231C22×D4—472no—
232C23×C4—474no—
233C23×C4C22×D4711yescharacteristic
234#131—9432no—
235#131—9432no—
236C2×M16—28no—
237C2×M16—478no—
238C2×M16—474no—
239C2×M16—4716no—
240C2×M16—9432no—
241C2×M16C22×D4714yes—
242C2×M16C22×D4714yes—
243C22×C8—472no—
244C22×C8—471no—
245C22×C8—471no—
246C22×C8—474no—
247C22×C8—4716no—
248C22×C8—944no—
249C22×C8C22×D4714yes—
250C4⋊Q8—474no—
251#127—474no—
252#124—478no—
253#123—4716no—
254C22⋊Q8—4716no—
255#121—4716no—
256#121—4716no—
257#120—4716no—
258C4×D4—4716no—
259#117C4×D4714yes—
260#117C22×D4714yes—
261C2×C4⋊C4C4×D4712yes—
262C2×C22⋊C4C4×D4712yes—
263C2×C42C22×D4712yes—
264C4⋊C8C22×D47116yes—
265C22⋊C8—4716no—
266C22⋊C8—4716no—
267#97—478no—
268C4×C8—478no—
269C24—474no—
270C24C2×C4×D4711yescharacteristic
271#41—9416no—
272#41—9416no—
273#41—948no—
274#41—948no—
275#41—948no—
276#41—948no—
277C2×Q8—474no—
278C2×Q8—474no—
279C2×Q8—948no—
280C2×D4—478no—
281C2×D4—478no—
282C2×D4—474no—
283C2×D4—474no—
284C2×D4—4716no—
285C2×D4—9416no—
286C2×D4—948no—
287C22×C4—474no—
288C22×C4—478no—
289C22×C4—478no—
290C22×C4—948no—
291C22×C4—948no—
292C22×C4C2×C4×D4714yes—
293C22×C4C2×C4×D4711yescharacteristic
294C22×C4C2×C4×D4712yes—
295C22×C4D4×D4714yes—
296C22×C4#1062535414yes—
297C22×C4C23×D4711yescharacteristic
298C22×C4C23×D4712yes—
299M16—28no—
300M16—28no—
301M16—4716no—
302M16—9432no—
303C2×C8—28no—
304C2×C8—478no—
305C2×C8—474no—
306C2×C8—472no—
307C2×C8—472no—
308C2×C8—9432no—
309C2×C8—948no—
310C2×C8—942no—
311C2×C8—942no—
312C2×C8C23×D4718yes—
313C4⋊C4—4716no—
314C4⋊C4C2×C4×D4718yes—
315C22⋊C4—4732no—
316C22⋊C4C2×C4×D4718yes—
317C42C2×C4×D4714yes—
318C42C23×D4714yes—
319C23—42no—
320C23—478no—
321C23—948no—
322C23—1888no—
323C23#1309398614yes—
324C23C22×C4×D4712yes—
325C23C22×C4×D4711yescharacteristic
326Q8—948no—
327Q8—948no—
328D4—9416no—
329D4—9416no—
330D4—948no—
331D4—948no—
332D4—18816no—
333C2×C4—44no—
334C2×C4—478no—
335C2×C4—948no—
336C2×C4—9416no—
337C2×C4—948no—
338C2×C4—948no—
339C2×C4#1309398618yes—
340C2×C4#1309398614yes—
341C2×C4C22×C4×D4718yes—
342C2×C4C22×C4×D4711yescharacteristic
343C2×C4C22×C4×D4712yes—
344C2×C4C2×D4×D4714yes—
345C2×C4C2×D4×D4712yes—
346C2×C4C2×#1062535414yes—
347C2×C4C2×#1062535412yes—
348C2×C4C24×D4711yescharacteristic
349C8—28no—
350C8—478no—
351C8—942no—
352C8—942no—
353C22—22no—
354C22—24no—
355C22—44no—
356C22—948no—
357C22—9416no—
358C22—3768no—
359C22C2×#1309398614yes—
360C22C2×#1309398612yes—
361C22C23×C4×D4711yescharacteristic
362C4—28no—
363C4—948no—
364C4C2×#1309398614yes—
365C4C2×#1309398612yes—
366C4C22×D4×D4711yescharacteristic
367C4C22×#1062535411yescharacteristic
368C2—24no—
369C2—22no—
370C2—1888no—
371C2C22×#1309398611yescharacteristic
372C2#2025810112yes—
subgroupG/H
center, FrattiniC2×C4C24×D47
commutator, socleC22×C47C24×C4
FittingC47×#2905C2
prime radical#20256721C2
cosocleC2×C4×C47C25
hypercenter#2905D47
hypocenterC47#60143
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ #2905

Complements

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

NH ≅ G/Ntype
1. C2×#15689889C2semidirect
3. #20252112C2semidirect
4. #20252105C2semidirect
5. #20252099C2semidirect
6. C2×#15689091C2semidirect
7. C2×#15689079C2semidirect
8. C47×#2905C2semidirect
11. C2×#13094032C22semidirect
14. C2×M16×D47C22semidirect
22. #15689200C22semidirect
23. #15689193C22semidirect
24. #15689188C22semidirect
26. #15689091C22semidirect
27. #15689091C22semidirect
28. C2×(C2×D47)⋊C8C22semidirect
29. #15689079C22semidirect
30. C2×#13093935C22semidirect
31. C2×C4×C47⋊C8C22semidirect
34. C47×#424C22semidirect
35. C47×#423C22semidirect
48. C2×#10625329D4semidirect
187. #2905D47semidirect
213. #424C2×D47semidirect
214. #423C2×D47semidirect
222. C47#60143semidirect

Maximal subgroups

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

subgroupindex
C2×#156898892
#202567212
#202521122
#202521052
#202520992
C2×#156890912
C2×#156890792
C47×#29052
#6014347