Finite Group Databasebeta

← C4×#15688933 #26131526

Subgroups — C4×#15688933 #26131526

Order 12032.

466 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15688933C212yesmaximal
2C4×#13093964C211yesmaximal, characteristic, complemented
3C4×#13093956C211yesmaximal, characteristic, complemented
4#20241493C212yesmaximal, complemented
5C4×#135×C47C211yesmaximal, characteristic
6C2×#13093964C2212yes—
7C2×#13093956C2212yes—
8C4×C4⋊Dic47—21no—
9#15688933C418yesdirect product factor
10#15688899—21no—
11C4×C47⋊C16—21no—
12#15688886—21no—
13C2×#135×C47C2211yescharacteristic
14C4×#41×C47—21no—
15C4×Q16×C47C411yescharacteristic, complemented
16C4×Q16×C47C411yescharacteristic, complemented
17C4×SD16×C47—21no—
18C4×SD16×C47—21no—
19C4×D8×C47C411yescharacteristic, complemented
20C4×D8×C47C411yescharacteristic, complemented
21C47×#427C2211yescharacteristic
22C47×#416—21no—
23C2×C4×C8×C47C2211yescharacteristic
24C2×C4⋊Dic47—22no—
25C2×C4×Dic47—41no—
26#13093964—22no—
27#13093964C2×C414yescomplemented
28C2×C47⋊C16—22no—
29#13093956—22no—
30#13093956C2×C414yescomplemented
31C47⋊#95—41no—
32C2×#41×C47—21no—
33#135×C47—22no—
34#135×C47C2×C414yes—
35C2×Q16×C47C2×C411yescharacteristic
36C2×SD16×C47—21no—
37C2×D8×C47C2×C411yescharacteristic
38C22×C8×C47D411yescharacteristic
39C22×C8×C47C2311yescharacteristic
40C4×Q8×C47—21no—
41C4×Q8×C47—21no—
42C4×D4×C47—21no—
43C4×D4×C47—21no—
44C4×D4×C47—41no—
45C4×D4×C47—41no—
46#117×C47—21no—
47#117×C47—41no—
48C2×C42×C47—41no—
49C2×C42×C47D411yescharacteristic
50#107×C47C2×C411yescharacteristic
51#107×C47C2×C411yescharacteristic
52#106×C47—21no—
53#103×C47—21no—
54#103×C47—21no—
55#102×C47—21no—
56#102×C47—21no—
57C4×C8×C47C2×C411yescharacteristic
58C4×C8×C47C2×C411yescharacteristic
59C4×C8×C47D411yescharacteristic
60C4×C8×C47D411yescharacteristic
61#41×C47—24no—
62#41×C47—42no—
63C2×Q8×C47—21no—
64C2×D4×C47—21no—
65C2×D4×C47—41no—
66C22×C4×C47—41no—
67C22×C4×C47—41no—
68C22×C4×C47—41no—
69C22×C4×C47D811yescharacteristic
70C22×C4×C47D811yescharacteristic
71C22×C4×C47C2×D411yescharacteristic
72C22×Dic47—42no—
73Q16×C47—21no—
74Q16×C47C4212yescomplemented
75SD16×C47—24no—
76D8×C47—21no—
77D8×C47C4212yescomplemented
78C2×C8×C47—22no—
79C2×C8×C47C22⋊C411yescharacteristic
80C2×C8×C47C22⋊C411yescharacteristic
81C2×C8×C47C22×C411yescharacteristic
82C2×C8×C47C22×C412yes—
83C2×C8×C47C2×D411yescharacteristic
84C2×C8×C47#4112yes—
85C4⋊C4×C47—21no—
86C4⋊C4×C47—21no—
87C4⋊C4×C47—41no—
88C4⋊C4×C47—41no—
89C22⋊C4×C47—41no—
90C22⋊C4×C47—41no—
91C22⋊C4×C47—41no—
92C42×C47—41no—
93C42×C47—41no—
94C42×C47—41no—
95C42×C47C22⋊C411yescharacteristic
96C42×C47C22⋊C411yescharacteristic
97C42×C47SD1611yescharacteristic
98C42×C47SD1611yescharacteristic
99C4⋊Dic47—24no—
100C4⋊Dic47—42no—
101C4×Dic47—82no—
102C47⋊C16—24no—
103C23×C47—41no—
104C23×C47C2×D811yescharacteristic
105Q8×C47—22no—
106Q8×C47—41no—
107D4×C47—22no—
108D4×C47—44no—
109D4×C47—41no—
110C2×C4×C47—22no—
111C2×C4×C47—41no—
112C2×C4×C47—41no—
113C2×C4×C47—44no—
114C2×C4×C47—41no—
115C2×C4×C47—41no—
116C2×C4×C47—82no—
117C2×C4×C47—82no—
118C2×C4×C47#10211yescharacteristic
119C2×C4×C47#10211yescharacteristic
120C2×C4×C47#10211yescharacteristic
121C2×C4×C47#10211yescharacteristic
122C2×C4×C47C2×C22⋊C411yescharacteristic
123C2×C4×C47C4×D412yes—
124C2×C4×C47C2×SD1611yescharacteristic
125C2×C4×C47#13512yes—
126C2×Dic47—44no—
127C2×Dic47—82no—
128C2×Dic47—82no—
129C8×C47—21no—
130C8×C47—21no—
131C8×C47C2×C4211yescharacteristic
132C8×C47#11711yescharacteristic
133C8×C47C4×D412yes—
134C4×#360—471nomaximal, Sylow
135C22×C47—21no—
136C22×C47—21no—
137C22×C47—42no—
138C22×C47—81no—
139C22×C47—81no—
140C22×C47C2×#10211yescharacteristic
141C22×C47C4×D812yes—
142C4×C47—21no—
143C4×C47—21no—
144C4×C47—21no—
145C4×C47—42no—
146C4×C47—81no—
147C4×C47—81no—
148C4×C47#36012yescomplemented
149C4×C47C4×C22⋊C411yescharacteristic
150C4×C47#41611yescharacteristic
151C4×C47C4×SD1612yes—
152Dic47—84no—
153C4×#135D4711yescharacteristic
154C2×#360—472no—
155C4×M32—471no—
156C4×#106—471no—
157#1386—472no—
158C2×C47—22no—
159C2×C47—82no—
160C2×C47C4×#10211yescharacteristic
161C2×C47C2×#36011yescharacteristic
162C2×C47#214511yescharacteristic
163C2×#135C2×D4711yescharacteristic
164C4×#41—21no—
165C2×M32—472no—
166C4×Q16Dic4711yescharacteristic, complemented
167C4×Q16Dic4711yescharacteristic, complemented
168C4×SD16—21no—
169C4×SD16—21no—
170C4×D8Dic4711yescharacteristic, complemented
171C4×D8Dic4711yescharacteristic, complemented
172#427C2×D4711yescharacteristic
173C2×#106—472no—
174#416—21no—
175C2×C4×C8C2×D4711yescharacteristic
176C4×C4⋊C4—941no—
177#360—478no—
178#346—941no—
179C4×C16—941no—
180#340—941no—
181C47C4×#36011yescharacteristic, complemented, Sylow
182C2×#41—21no—
183#135—22no—
184#135C4×D4714yes—
185C2×Q16C2×Dic4711yescharacteristic
186C2×SD16—21no—
187C2×D8C2×Dic4711yescharacteristic
188C22×C8C22⋊D4711yescharacteristic
189C22×C8C22×D4711yescharacteristic
190C4×Q8—21no—
191C4×Q8—21no—
192C4×D4—21no—
193C4×D4—21no—
194C4×D4—41no—
195C4×D4—41no—
196#117—21no—
197#117—41no—
198C2×C4⋊C4—942no—
199C2×C42—41no—
200C2×C42—1881no—
201C2×C42C22⋊D4711yescharacteristic
202M32—474no—
203M32—942no—
204C2×C16—942no—
205#107C2×Dic4711yescharacteristic
206#107C2×Dic4711yescharacteristic
207#106—21no—
208#106—474no—
209#106—942no—
210#103—21no—
211#103—21no—
212#102—21no—
213#102—21no—
214C4×C8C2×Dic4711yescharacteristic
215C4×C8C2×Dic4711yescharacteristic
216C4×C8C22⋊D4711yescharacteristic
217C4×C8C22⋊D4711yescharacteristic
218#95—1881no—
219#41—24no—
220#41—42no—
221C2×Q8—21no—
222C2×D4—21no—
223C2×D4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—1882no—
228C22×C4#1062533411yescharacteristic
229C22×C4#1062533411yescharacteristic
230C22×C4C2×C22⋊D4711yescharacteristic
231Q16—21no—
232Q16C4×Dic4712yescomplemented
233SD16—24no—
234D8—21no—
235D8C4×Dic4712yescomplemented
236C2×C8—22no—
237C2×C8C22⋊Dic4711yescharacteristic
238C2×C8C22⋊Dic4711yescharacteristic
239C2×C8C2×C4×D4712yes—
240C2×C8#1062535012yes—
241C2×C8C22×Dic4711yescharacteristic
242C2×C8C2×C22⋊D4711yescharacteristic
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—944no—
248C4⋊C4—1882no—
249C22⋊C4—41no—
250C22⋊C4—41no—
251C22⋊C4—41no—
252C42—41no—
253C42—41no—
254C42—41no—
255C42—3762no—
256C42#1062533511yescharacteristic
257C42#1062533511yescharacteristic
258C42C22⋊Dic4711yescharacteristic
259C42C22⋊Dic4711yescharacteristic
260C16—944no—
261C23—41no—
262C23C2×#1062533411yescharacteristic
263Q8—22no—
264Q8—41no—
265D4—22no—
266D4—44no—
267D4—41no—
268C2×C4—22no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—44no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—3762no—
277C2×C4—3762no—
278C2×C4—1884no—
279C2×C4#1309394511yescharacteristic
280C2×C4#1309394511yescharacteristic
281C2×C4#1309394511yescharacteristic
282C2×C4#1309394511yescharacteristic
283C2×C4C2×C22⋊Dic4711yescharacteristic
284C2×C4C4×C22⋊D4712yes—
285C2×C4C2×#1062533511yescharacteristic
286C2×C4#1309404412yes—
287C8—21no—
288C8—21no—
289C8C2×C4×Dic4711yescharacteristic
290C8#1309398311yescharacteristic
291C8C4×C22⋊D4712yes—
292C22—21no—
293C22—21no—
294C22—42no—
295C22—81no—
296C22—81no—
297C22C2×#1309394511yescharacteristic
298C22C4×#1062533412yes—
299C4—21no—
300C4—21no—
301C4—21no—
302C4—42no—
303C4—81no—
304C4—81no—
305C4—3764no—
306C4#1568893312yesdirect product factor
307C4C4×C22⋊Dic4711yescharacteristic
308C4#1568911911yescharacteristic
309C4C4×#1062533512yes—
310C2—22no—
311C2—82no—
312C2C4×#1309394511yescharacteristic
313C2C2×#1568893311yescharacteristic
314C2#2024512411yescharacteristic
subgroupG/H
centerC2×C4#13093945
commutatorC8×C47C2×C42
FrattiniC22×C8C22×D47
FittingC4×#135×C47C2
socleC22×C47C2×#102
prime radicalC2×#135×C47C22
cosocleC22×C8×C47C23
hypercenterC4×#135D47
hypocenterC47C4×#360
LCSG ▷ C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ C4×#135

Complements

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

NH ≅ G/Ntype
2. C4×#13093964C2semidirect
3. C4×#13093956C2semidirect
4. #20241493C2semidirect
9. #15688933C4direct (N × H)
15. C4×Q16×C47C4semidirect
16. C4×Q16×C47C4semidirect
19. C4×D8×C47C4semidirect
20. C4×D8×C47C4semidirect
27. #13093964C2×C4semidirect
30. #13093956C2×C4semidirect
74. Q16×C47C42semidirect
77. D8×C47C42semidirect
148. C4×C47#360semidirect
166. C4×Q16Dic47semidirect
167. C4×Q16Dic47semidirect
170. C4×D8Dic47semidirect
171. C4×D8Dic47semidirect
181. C47C4×#360semidirect
232. Q16C4×Dic47semidirect
235. D8C4×Dic47semidirect
306. C4#15688933direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156889332
C4×#130939642
C4×#130939562
#202414932
C4×#135×C472
C4×#36047