Finite Group Databasebeta

← C2×#20249219 #26383346

Subgroups — C2×#20249219 #26383346

Order 12032.

2614 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093981C211yesmaximal, characteristic
2#20249219C2116yesmaximal, direct product factor
3C2×#15689056C214yesmaximal, complemented
4C2×#15689045C214yesmaximal
5C2×#15689019C212yesmaximal, complemented
6C2×#15688993C211yesmaximal, characteristic, complemented
7C2×#15688980C212yesmaximal
8C22×#125×C47C211yesmaximal, characteristic, complemented
9C23×C4×D47C2211yescharacteristic
10C2×#13093981—28no—
11C2×#13093981C2218yes—
12C22×C4⋊Dic47C2212yes—
13C22×C4⋊Dic47C2214yes—
14C22×Dic47⋊C4C2212yes—
15C22×Dic47⋊C4C2214yes—
16C22×(C2×D47)⋊C4C2212yes—
17C22×(C2×D47)⋊C4C2214yes—
18C22×C4×Dic47C2211yescharacteristic
19#15689056C22132yescomplemented
20#15689045C22132yes—
21#15689019C22116yescomplemented
22#15688993C2218yescomplemented
23#15688980C22116yes—
24C2×C47⋊#95C2214yes—
25C2×C47⋊#95C2214yes—
26C2×#125×C47—28no—
27C2×#125×C47C2218yescomplemented
28C22×C4⋊C4×C47C2212yes—
29C22×C4⋊C4×C47C2214yes—
30C22×C42×C47C2211yescharacteristic
31C24×D47C2311yescharacteristic
32C23×Dic47C2312yes—
33C23×Dic47C2314yes—
34C23×Dic47C2311yescharacteristic
35C22×C4×D47—24no—
36C22×C4×D47—24no—
37C22×C4×D47Q818yes—
38C22×C4×D47C2314yes—
39#13093981—232no—
40C2×C4⋊Dic47—28no—
41C2×C4⋊Dic47—48no—
42C2×C4⋊Dic47C2318yes—
43C2×C4⋊Dic47C23116yes—
44C2×Dic47⋊C4—28no—
45C2×Dic47⋊C4—48no—
46C2×Dic47⋊C4C2318yes—
47C2×Dic47⋊C4C23116yes—
48C2×(C2×D47)⋊C4—28no—
49C2×(C2×D47)⋊C4—48no—
50C2×(C2×D47)⋊C4C2318yes—
51C2×(C2×D47)⋊C4C23116yes—
52C2×C4×Dic47—24no—
53C2×C4×Dic47C2314yes—
54C47⋊#95C23116yes—
55C47⋊#95C23116yes—
56C23×C4×C47C2314yes—
57C23×C4×C47C2312yes—
58C23×C4×C47C2311yescharacteristic
59#125×C47—232no—
60C2×C4⋊C4×C47—216no—
61C2×C4⋊C4×C47—28no—
62C2×C4⋊C4×C47C2318yes—
63C2×C4⋊C4×C47C23116yes—
64C2×C42×C47D418yes—
65C2×C42×C47C2314yes—
66C24×C47C2411yescharacteristic
67C23×D47—22no—
68C23×D47—22no—
69C23×D47—24no—
70C23×D47—24no—
71C23×D47C2×Q814yes—
72C23×D47C2412yes—
73C22×C4×C47—24no—
74C22×C4×C47—24no—
75C22×C4×C47—28no—
76C22×C4×C47—28no—
77C22×C4×C47—28no—
78C22×C4×C47C2×D418yes—
79C22×C4×C47C2×D414yes—
80C22×C4×C47C2×Q814yes—
81C22×C4×C47#4114yes—
82C22×C4×C47C2418yes—
83C22×C4×C47C2414yes—
84C22×C4×C47C2412yes—
85C22×Dic47—22no—
86C22×Dic47—22no—
87C22×Dic47—24no—
88C22×Dic47—28no—
89C22×Dic47—28no—
90C22×Dic47—24no—
91C22×Dic47—28no—
92C22×Dic47—24no—
93C22×Dic47C2×Q814yes—
94C22×Dic47C2414yes—
95C22×Dic47C2418yes—
96C22×Dic47C2412yes—
97C2×C4×D47—232no—
98C2×C4×D47—28no—
99C2×C4×D47—28no—
100C2×C4×D47C2×Q8116yes—
101C4⋊C4×C47—232no—
102C4⋊C4×C47—216no—
103C42×C47C2×D4116yes—
104(C2×D47)⋊C4—216no—
105(C2×D47)⋊C4—416no—
106C4⋊Dic47—216no—
107C4⋊Dic47—416no—
108Dic47⋊C4—216no—
109Dic47⋊C4—416no—
110C4×Dic47—28no—
111C23×C47C22×D412yes—
112C23×C47C22×Q812yes—
113C23×C47C2×#4112yes—
114C23×C47H(5,𝔽2)14yes—
115C23×C472−1+414yes—
116C23×C47C2511yescharacteristic
117C22×D47—28no—
118C22×D47—216no—
119C22×D47—22no—
120C22×D47—22no—
121C22×D47—24no—
122C22×D47—24no—
123C22×D47—48no—
124C22×D47—44no—
125C22×D47—44no—
126C22×D47C22×Q814yes—
127C2×C4×C47—216no—
128C2×C4×C47—24no—
129C2×C4×C47—24no—
130C2×C4×C47—28no—
131C2×C4×C47—28no—
132C2×C4×C47—28no—
133C2×C4×C47—416no—
134C2×C4×C47C22⋊Q8116yes—
135C2×C4×C47C22×D418yes—
136C2×C4×C47C22×D414yes—
137C2×C4×C47C22×Q814yes—
138C2×C4×C47C2×#4114yes—
139C2×Dic47—28no—
140C2×Dic47—22no—
141C2×Dic47—22no—
142C2×Dic47—24no—
143C2×Dic47—28no—
144C2×Dic47—24no—
145C2×Dic47—28no—
146C2×Dic47—28no—
147C2×Dic47—24no—
148C2×Dic47—48no—
149C2×Dic47—416no—
150C2×Dic47C22×Q814yes—
151C4×D47—232no—
152C2×#2629—471nomaximal, Sylow
153C22×C47C2×C22⋊Q814yes—
154C22×C47#53718yes—
155C22×C47#54014yes—
156C22×C47#54114yes—
157C22×C47#54418yes—
158C22×C47C23×D411yescharacteristic
159C22×C47C23×Q811yescharacteristic
160C22×C47C22×#4111yescharacteristic
161C22×C47C2×H(5,𝔽2)12yes—
162C22×C47C2×2−1+412yes—
163C2×D47—24no—
164C2×D47—28no—
165C2×D47—416no—
166C2×D47—44no—
167C2×D47—42no—
168C2×D47—42no—
169C4×C47—28no—
170C4×C47—48no—
171C4×C47C2×C22⋊Q818yes—
172Dic47—24no—
173Dic47—44no—
174Dic47—48no—
175C22×#125D4711yescharacteristic, complemented
176C22×#117—471no—
177#2629—4716no—
178C2×#398—474no—
179C2×#396—474no—
180C2×#389—472no—
181C2×#384—471no—
182C2×#380—472no—
183C2×C47#262918yescomplemented
184C2×C47C22×C22⋊Q811yescharacteristic
185C2×C47C2×#53712yes—
186C2×C47C2×#54011yescharacteristic
187C2×C47C2×#54111yescharacteristic
188C2×C47C2×#54412yes—
189D47—44no—
190C24×C4—471no—
191C2×#125—28no—
192C2×#125C2×D4718yescomplemented
193C2×#117—478no—
194C2×#117—948no—
195C22×C4⋊C4—472no—
196C22×C4⋊C4—472no—
197C22×C4⋊C4—474no—
198C22×C4⋊C4—474no—
199C22×C4⋊C4C2×D4714yes—
200C22×C4⋊C4C2×D4712yes—
201C22×C22⋊C4—472no—
202C22×C22⋊C4—474no—
203C22×C42—471no—
204C22×C42C2×D4711yescharacteristic
205#398—4732no—
206#396—4732no—
207#389—4716no—
208#384—478no—
209#380—4716no—
210C2×#95—474no—
211C2×#95—474no—
212C47C2×#262911yescharacteristic, complemented, Sylow
213C25—471no—
214C23×C4—471no—
215C23×C4—478no—
216C23×C4—474no—
217C23×C4—474no—
218C23×C4—472no—
219C23×C4—944no—
220C23×C4—944no—
221C23×C4C22×D4714yes—
222C23×C4C22×D4712yes—
223C23×C4C22×D4711yescharacteristic
224#125—232no—
225#117—9432no—
226C2×C4⋊C4—216no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—478no—
229C2×C4⋊C4—478no—
230C2×C4⋊C4—4716no—
231C2×C4⋊C4—4716no—
232C2×C4⋊C4—948no—
233C2×C4⋊C4—948no—
234C2×C4⋊C4—1888no—
235C2×C4⋊C4—1888no—
236C2×C4⋊C4C22×D4718yes—
237C2×C4⋊C4C22×D47116yes—
238C2×C22⋊C4—478no—
239C2×C22⋊C4—4716no—
240C2×C22⋊C4—948no—
241C2×C22⋊C4—1888no—
242C2×C42—474no—
243C2×C42—944no—
244C2×C42C22⋊D4718yes—
245C2×C42C22×D4714yes—
246#95—4716no—
247#95—4716no—
248C24—474no—
249C24—472no—
250C24—942no—
251C24—942no—
252C24—944no—
253C24—944no—
254C24C23×D4711yescharacteristic
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—28no—
258C22×C4—28no—
259C22×C4—28no—
260C22×C4—474no—
261C22×C4—472no—
262C22×C4—4716no—
263C22×C4—474no—
264C22×C4—478no—
265C22×C4—942no—
266C22×C4—942no—
267C22×C4—9432no—
268C22×C4—948no—
269C22×C4—948no—
270C22×C4—944no—
271C22×C4—948no—
272C22×C4—948no—
273C22×C4—948no—
274C22×C4—944no—
275C22×C4—944no—
276C22×C4Q8×D4714yes—
277C22×C4#1062535414yes—
278C22×C4C2×C22⋊D4718yes—
279C22×C4C2×C22⋊D4714yes—
280C22×C4C23×D4718yes—
281C22×C4C23×D4714yes—
282C22×C4C23×D4712yes—
283C4⋊C4—232no—
284C4⋊C4—216no—
285C4⋊C4—9416no—
286C4⋊C4—9416no—
287C4⋊C4—18816no—
288C4⋊C4—18816no—
289C22⋊C4—9416no—
290C22⋊C4—18816no—
291C42—948no—
292C42C2×C22⋊D47116yes—
293C23—474no—
294C23—9416no—
295C23—948no—
296C23—942no—
297C23—942no—
298C23—944no—
299C23—944no—
300C23—1884no—
301C23—1884no—
302C23—1888no—
303C23C22×C22⋊D4712yes—
304C23C2×Q8×D4712yes—
305C23C2×#1062535412yes—
306C23#1309407014yes—
307C23#1309407114yes—
308C23C24×D4711yescharacteristic
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—216no—
312C2×C4—28no—
313C2×C4—28no—
314C2×C4—28no—
315C2×C4—416no—
316C2×C4—474no—
317C2×C4—948no—
318C2×C4—942no—
319C2×C4—942no—
320C2×C4—9432no—
321C2×C4—944no—
322C2×C4—948no—
323C2×C4—948no—
324C2×C4—944no—
325C2×C4—948no—
326C2×C4—944no—
327C2×C4—18816no—
328C2×C4—1888no—
329C2×C4#13094000116yes—
330C2×C4C22×C22⋊D4718yes—
331C2×C4C22×C22⋊D4714yes—
332C2×C4C2×Q8×D4714yes—
333C2×C4C2×#1062535414yes—
334C22—948no—
335C22—944no—
336C22—18816no—
337C22—1884no—
338C22—1882no—
339C22—1882no—
340C22C2×#1309400014yes—
341C22#1568970118yes—
342C22#1568972914yes—
343C22#1568973614yes—
344C22#1568975318yes—
345C22C23×C22⋊D4711yescharacteristic
346C22C22×Q8×D4711yescharacteristic
347C22C22×#1062535411yescharacteristic
348C22C2×#1309407012yes—
349C22C2×#1309407112yes—
350C4—28no—
351C4—48no—
352C4—944no—
353C4—1884no—
354C4—1888no—
355C4C2×#1309400018yes—
356C2—1884no—
357C2#2024921918yesdirect product factor
358C2C22×#1309400011yescharacteristic
359C2C2×#1568970112yes—
360C2C2×#1568972911yescharacteristic
361C2C2×#1568973611yescharacteristic
362C2C2×#1568975312yes—
subgroupG/H
centerC24C23×D47
commutator, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC22×#125×C47C2
socleC24×C47C24
prime radicalC24×D47C23
hypercenterC22×#125D47
hypocenterC47C2×#2629
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C22×#125

Complements

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

NH ≅ G/Ntype
2. #20249219C2direct (N × H)
3. C2×#15689056C2semidirect
5. C2×#15689019C2semidirect
6. C2×#15688993C2semidirect
8. C22×#125×C47C2semidirect
19. #15689056C22semidirect
21. #15689019C22semidirect
22. #15688993C22semidirect
27. C2×#125×C47C22semidirect
175. C22×#125D47semidirect
183. C2×C47#2629semidirect
192. C2×#125C2×D47semidirect
212. C47C2×#2629semidirect
357. C2#20249219direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130939812
#202492192
C2×#156890562
C2×#156890452
C2×#156890192
C2×#156889932
C2×#156889802
C22×#125×C472
C2×#262947