Finite Group Databasebeta

← C4×#15689120 #26214030

Subgroups — C4×#15689120 #26214030

Order 12032.

1670 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#10625350C211yesmaximal, characteristic
2C2×#15689120C212yesmaximal
3C4×#13093983C211yesmaximal, characteristic, complemented
4C4×#13093949C214yesmaximal, complemented
5C4×#13093946C214yesmaximal, complemented
6#20243105C212yesmaximal, complemented
7C22×C4×C8×C47C211yesmaximal, characteristic, complemented
8C22×#10625350C2211yescharacteristic
9C4×#10625350—24no—
10C4×#10625350C418yescomplemented
11C2×C4×Dic94C2212yes—
12C2×C4×D188C2212yes—
13C2×C4×C22⋊D47—22no—
14C2×#13093983C2211yescharacteristic
15C2×#13093983C2212yes—
16C2×#13093981—21no—
17C2×C42×D47—21no—
18#15689120C4116yesdirect product factor
19C2×#13093949C2214yes—
20C2×#13093946C2214yes—
21C4×C4⋊Dic47C2214yes—
22C4×C22⋊Dic47—22no—
23#15688969—21no—
24C42×Dic47—21no—
25#15688886C2218yes—
26C23×C8×C47C2212yes—
27C22×C42×C47C2211yescharacteristic
28C2×C4×C8×C47—24no—
29C2×C4×C8×C47C2214yes—
30C2×#10625350—22no—
31C2×#10625350—24no—
32C2×#10625350C2×C414yes—
33C2×#10625350C2×C418yes—
34C22×Dic94C2311yescharacteristic
35C22×D188C2311yescharacteristic
36C22×C22⋊D47—21no—
37C22×C4×D47—21no—
38C22×C4×D47—22no—
39C4×Dic94—24no—
40C4×Dic94C2×C418yes—
41C4×D188—24no—
42C4×D188C2×C418yes—
43C4×C22⋊D47—216no—
44#13093983—22no—
45#13093983—24no—
46#13093983C2×C414yes—
47#13093983C2×C418yescomplemented
48#13093981—28no—
49C2×C4⋊Dic47C2314yes—
50C2×C4⋊Dic47C2312yes—
51C2×Dic47⋊C4—22no—
52C2×C22⋊Dic47—22no—
53C2×C22⋊Dic47—21no—
54C2×(C2×D47)⋊C4—22no—
55C2×C4×Dic47—22no—
56C2×C4×Dic47—21no—
57C2×C4×Dic47—22no—
58C2×C4×Dic47—24no—
59C42×D47—28no—
60#13093949C2×C4116yescomplemented
61#13093946C2×C4116yescomplemented
62C47⋊#95—24no—
63C23×C4×C47D412yes—
64C23×C4×C47C2311yescharacteristic
65C22×C8×C47—24no—
66C22×C8×C47—24no—
67C22×C8×C47C2×C418yescomplemented
68C22×C8×C47C2314yes—
69C2×C42×C47C2×C414yes—
70C2×C42×C47D411yescharacteristic
71C2×C42×C47D411yescharacteristic
72C2×C42×C47D414yes—
73C2×C42×C47C2312yes—
74C4×C8×C47—28no—
75C24×C47C2×D411yescharacteristic
76C23×D47—21no—
77C22×C4×C47—28no—
78C22×C4×C47C22⋊C418yes—
79C22×C4×C47C22×C412yes—
80C22×C4×C47C22×C414yes—
81C22×C4×C47C2×D412yes—
82C22×C4×C47C2×D411yescharacteristic
83C22×C4×C47C2×D412yes—
84C22×C4×C47C2×D412yes—
85C22×C4×C47#4114yes—
86C22×C4×C47C2411yescharacteristic
87C2×C22⋊D47—24no—
88C2×C22⋊D47—28no—
89C22×Dic47—21no—
90C22×Dic47—24no—
91C22×Dic47—22no—
92#10625350—28no—
93#10625350—216no—
94#10625350C42116yescomplemented
95C2×D188—21no—
96C2×D188—22no—
97C2×D188C22×C414yes—
98C2×D188C22×C412yes—
99C2×C4×D47—24no—
100C2×C4×D47—28no—
101C2×C4×D47—28no—
102C2×C4×D47—48no—
103C2×Dic94—21no—
104C2×Dic94—22no—
105C2×Dic94C22×C414yes—
106C2×Dic94C22×C412yes—
107C2×C8×C47—216no—
108C2×C8×C47—24no—
109C2×C8×C47—24no—
110C2×C8×C47C22×C418yes—
111C42×C47C22⋊C414yes—
112C42×C47C22⋊C414yes—
113C42×C47C22×C414yes—
114C42×C47C2×D414yes—
115C22⋊Dic47—24no—
116C22⋊Dic47—28no—
117(C2×D47)⋊C4—28no—
118C4⋊Dic47—22no—
119C4⋊Dic47—24no—
120C4⋊Dic47C22×C418yes—
121C4⋊Dic47C22×C414yes—
122Dic47⋊C4—28no—
123C4×Dic47—24no—
124C4×Dic47—28no—
125C4×Dic47—28no—
126C4×Dic47—48no—
127C23×C47—24no—
128C23×C47C2×C22⋊C412yes—
129C23×C47C4×D414yes—
130C23×C47C22×D411yescharacteristic
131C22×D47—22no—
132C22×D47—24no—
133C22×D47—44no—
134C2×C4×C47—24no—
135C2×C4×C47—216no—
136C2×C4×C47C2×C4214yes—
137C2×C4×C47C2×C22⋊C414yes—
138C2×C4×C47C2×C22⋊C412yes—
139C2×C4×C47C2×C22⋊C414yes—
140C2×C4×C47#11714yes—
141C2×C4×C47C4×D418yes—
142C2×C4×C47C4×D414yes—
143C2×C4×C47#13518yes—
144C2×C4×C47C23×C411yescharacteristic
145C2×C4×C47C23×C412yes—
146C2×C4×C47C22×D411yescharacteristic
147C2×C4×C47C2×#4112yes—
148C22⋊D47—216no—
149C2×Dic47—22no—
150C2×Dic47—24no—
151C2×Dic47—24no—
152C2×Dic47—28no—
153C2×Dic47—48no—
154C2×Dic47—44no—
155D188—22no—
156D188—24no—
157D188C2×C4214yes—
158C4×D47—216no—
159C4×D47—416no—
160Dic94—22no—
161Dic94—24no—
162Dic94C2×C4214yes—
163C8×C47—28no—
164C4×#416—471nomaximal, Sylow
165C22×C47—24no—
166C22×C47—28no—
167C22×C47C4×C22⋊C414yes—
168C22×C47C22×C22⋊C411yescharacteristic
169C22×C47C2×C4×D412yes—
170C22×C47C2×#13514yes—
171C2×D47—24no—
172C2×D47—44no—
173C2×D47—48no—
174C4×C47—24no—
175C4×C47C4×C22⋊C414yes—
176C4×C47#41618yescomplemented
177C4×C47C22×C4211yescharacteristic
178C4×C47C2×#11711yescharacteristic
179C4×C47C2×C4×D412yes—
180Dic47—24no—
181Dic47—44no—
182Dic47—48no—
183C2×C4×#41—471no—
184C2×#416—472no—
185C22×C4×C8D4711yescharacteristic, complemented
186C4×#117—471no—
187C4×#103—474no—
188C4×#102—474no—
189#1735—472no—
190C2×C47—24no—
191C2×C47C2×C4×C22⋊C411yescharacteristic
192C2×C47C2×#41612yes—
193C2×C47C4×#13514yes—
194D47—44no—
195C22×#41—471no—
196C23×C8C2×D4712yes—
197C4×#41—478no—
198C4×#41—944no—
199C2×C4×Q8—472no—
200C2×C4×D4—472no—
201C2×C4×D4—942no—
202C2×#117—472no—
203C2×#117—471no—
204C2×#117—941no—
205C22×C42—941no—
206C22×C42C2×D4711yescharacteristic
207#416—4716no—
208C2×#103—474no—
209C2×#102—474no—
210C2×C4×C8—24no—
211C2×C4×C8C2×D4714yes—
212C4×C4⋊C4—474no—
213C4×C22⋊C4—942no—
214#376—941no—
215C43—941no—
216#340—478no—
217C47C4×#41611yescharacteristic, complemented, Sylow
218C2×#41—474no—
219C2×#41—478no—
220C2×#41—944no—
221C2×#41—942no—
222C22×Q8—471no—
223C22×D4—471no—
224C22×D4—941no—
225C23×C4—941no—
226C23×C4—942no—
227C23×C4D18812yes—
228C23×C4C22×D4711yescharacteristic
229C22×C8—24no—
230C22×C8—24no—
231C22×C8C4×D4718yescomplemented
232C22×C8C22×D4714yes—
233C4×Q8—478no—
234C4×Q8—944no—
235C4×D4—478no—
236C4×D4—9416no—
237C4×D4—944no—
238#117—474no—
239#117—478no—
240#117—944no—
241#117—942no—
242#117—948no—
243C2×C4⋊C4—472no—
244C2×C4⋊C4—474no—
245C2×C4⋊C4—942no—
246C2×C22⋊C4—941no—
247C2×C22⋊C4—942no—
248C2×C22⋊C4—942no—
249C2×C42—941no—
250C2×C42—942no—
251C2×C42—942no—
252C2×C42—948no—
253C2×C42—944no—
254C2×C42C4×D4714yes—
255C2×C42D18811yescharacteristic
256C2×C42D18811yescharacteristic
257C2×C42C22⋊D4714yes—
258C2×C42C22×D4712yes—
259#103—4716no—
260#102—4716no—
261C4×C8—28no—
262#95—944no—
263C24—941no—
264C24C2×D18811yescharacteristic
265#41—4716no—
266#41—948no—
267#41—9416no—
268C2×Q8—474no—
269C2×Q8—472no—
270C2×Q8—941no—
271C2×Q8—942no—
272C2×D4—474no—
273C2×D4—472no—
274C2×D4—944no—
275C2×D4—948no—
276C2×D4—941no—
277C2×D4—942no—
278C22×C4—28no—
279C22×C4—941no—
280C22×C4—944no—
281C22×C4—948no—
282C22×C4—944no—
283C22×C4—942no—
284C22×C4—948no—
285C22×C4—1888no—
286C22×C4(C2×D47)⋊C418yes—
287C22×C4C2×C4×D4712yes—
288C22×C4C2×C4×D4714yes—
289C22×C4C2×D18812yes—
290C22×C4C2×D18811yescharacteristic
291C22×C4C2×D18812yes—
292C22×C4#1062535014yes—
293C22×C4C2×C22⋊D4712yes—
294C22×C4C23×D4711yescharacteristic
295C2×C8—216no—
296C2×C8—24no—
297C2×C8—24no—
298C2×C8C2×C4×D4718yes—
299C4⋊C4—478no—
300C4⋊C4—474no—
301C4⋊C4—948no—
302C4⋊C4—942no—
303C4⋊C4—944no—
304C22⋊C4—944no—
305C22⋊C4—948no—
306C22⋊C4—948no—
307C42—944no—
308C42—948no—
309C42—948no—
310C42—1888no—
311C42(C2×D47)⋊C414yes—
312C42(C2×D47)⋊C414yes—
313C42C2×C4×D4714yes—
314C42C2×C22⋊D4714yes—
315C23—24no—
316C23—942no—
317C23—944no—
318C23—1884no—
319C23C2×(C2×D47)⋊C412yes—
320C23C4×D18814yes—
321C23C22×D18811yescharacteristic
322Q8—474no—
323Q8—942no—
324Q8—944no—
325D4—474no—
326D4—9416no—
327D4—942no—
328D4—944no—
329C2×C4—24no—
330C2×C4—216no—
331C2×C4—942no—
332C2×C4—944no—
333C2×C4—9416no—
334C2×C4—944no—
335C2×C4—948no—
336C2×C4—1888no—
337C2×C4—1884no—
338C2×C4—18816no—
339C2×C4C42×D4714yes—
340C2×C4C2×(C2×D47)⋊C414yes—
341C2×C4C2×(C2×D47)⋊C412yes—
342C2×C4C2×(C2×D47)⋊C414yes—
343C2×C4#1309398114yes—
344C2×C4C4×C22⋊D4718yes—
345C2×C4C4×D18814yes—
346C2×C4#1309404618yes—
347C2×C4C22×C4×D4711yescharacteristic
348C2×C4C22×C4×D4712yes—
349C2×C4C22×C22⋊D4711yescharacteristic
350C2×C4C2×#1062535012yes—
351C8—28no—
352C22—28no—
353C22—24no—
354C22—944no—
355C22—1888no—
356C22—1884no—
357C22C4×(C2×D47)⋊C414yes—
358C22C22×(C2×D47)⋊C411yescharacteristic
359C22C2×C4×D18812yes—
360C22C2×#1309404614yes—
361C4—24no—
362C4—944no—
363C4—1888no—
364C4—1884no—
365C4C4×(C2×D47)⋊C414yes—
366C4#1568912018yesdirect product factor
367C4C2×C42×D4711yescharacteristic
368C4C2×#1309398111yescharacteristic
369C4C2×C4×C22⋊D4712yes—
370C2—24no—
371C2—1884no—
372C2C2×C4×(C2×D47)⋊C411yescharacteristic
373C2C2×#1568912012yes—
374C2C4×#1309404614yes—
subgroupG/H
centerC2×C42D188
commutatorC4×C47C22×C42
FrattiniC22×C4C23×D47
FittingC22×C4×C8×C47C2
socleC23×C47C22×D4
prime radicalC22×#10625350C22
cosocleC22×C4×C47C24
hypercenterC22×C4×C8D47
hypocenterC47C4×#416
LCSG ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C42 ◁ C22×C42 ◁ C22×C4×C8

Complements

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

NH ≅ G/Ntype
3. C4×#13093983C2semidirect
4. C4×#13093949C2semidirect
5. C4×#13093946C2semidirect
6. #20243105C2semidirect
7. C22×C4×C8×C47C2semidirect
10. C4×#10625350C4semidirect
18. #15689120C4direct (N × H)
47. #13093983C2×C4semidirect
60. #13093949C2×C4semidirect
61. #13093946C2×C4semidirect
67. C22×C8×C47C2×C4semidirect
94. #10625350C42semidirect
176. C4×C47#416semidirect
185. C22×C4×C8D47semidirect
217. C47C4×#416semidirect
231. C22×C8C4×D47semidirect
366. C4#15689120direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×#106253502
C2×#156891202
C4×#130939832
C4×#130939492
C4×#130939462
#202431052
C22×C4×C8×C472
C4×#41647