Finite Group Databasebeta

← C22×#15688860 #26183360

Subgroups — C22×#15688860 #26183360

Order 12032.

1846 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊(C47⋊C8)C211yesmaximal, characteristic
2C22×C47⋊#95C211yesmaximal, characteristic
3C2×#15688860C2112yesmaximal, direct product factor
4C22×C22⋊Q8×C47C211yesmaximal, characteristic
5C24×Dic47—21no—
6C23×C47⋊C8—21no—
7C2×C22⋊(C47⋊C8)—23no—
8C2×C22⋊(C47⋊C8)C2216yes—
9C2×C47⋊#95—23no—
10C2×C47⋊#95C2216yes—
11#15688860C22116yesdirect product factor
12C23×Q8×C47C411yescharacteristic, complemented
13C24×C4×C47C2211yescharacteristic
14C2×C22⋊Q8×C47—23no—
15C2×C22⋊Q8×C47—23no—
16C2×C22⋊Q8×C47C416yescomplemented
17C2×C22⋊Q8×C47C2216yes—
18C22×C4⋊C4×C47—21no—
19C22×C4⋊C4×C47C411yescharacteristic, complemented
20C22×C22⋊C4×C47—21no—
21C23×Dic47—28no—
22C23×Dic47—26no—
23C23×Dic47—44no—
24C23×Dic47—44no—
25C22×C47⋊C8—26no—
26C22×C47⋊C8—44no—
27C22⋊(C47⋊C8)—26no—
28C22⋊(C47⋊C8)C2314yes—
29C47⋊#95—26no—
30C47⋊#95C2314yes—
31C25×C47D411yescharacteristic
32C22×Q8×C47—23no—
33C22×Q8×C47—44no—
34C22×Q8×C47C2×C416yescomplemented
35C23×C4×C47—21no—
36C23×C4×C47—21no—
37C23×C4×C47—44no—
38C23×C4×C47C2×C411yescharacteristic
39C23×C4×C47C2×C413yes—
40C23×C4×C47D411yescharacteristic
41C23×C4×C47D413yes—
42C23×C4×C47D413yes—
43C23×C4×C47C2313yes—
44C22⋊Q8×C47—212no—
45C22⋊Q8×C47—26no—
46C22⋊Q8×C47—26no—
47C22⋊Q8×C47C2×C4112yescomplemented
48C22⋊Q8×C47C2314yes—
49C2×C4⋊C4×C47—26no—
50C2×C4⋊C4×C47—23no—
51C2×C4⋊C4×C47—26no—
52C2×C4⋊C4×C47C2×C416yescomplemented
53C2×C22⋊C4×C47—26no—
54C2×C22⋊C4×C47—26no—
55C24×C47—24no—
56C24×C47—24no—
57C24×C47C22⋊C413yes—
58C24×C47C22⋊C411yescharacteristic
59C24×C47SD1614yes—
60C24×C47Q1614yes—
61C24×C47C2×D413yes—
62C2×Q8×C47—26no—
63C2×Q8×C47—412no—
64C2×Q8×C47—412no—
65C2×Q8×C47C22×C414yescomplemented
66C22×C4×C47—24no—
67C22×C4×C47—24no—
68C22×C4×C47—26no—
69C22×C4×C47—26no—
70C22×C4×C47—46no—
71C22×C4×C47—46no—
72C22×C4×C47—46no—
73C22×C4×C47—46no—
74C22×C4×C47—44no—
75C22×C4×C47—44no—
76C22×C4×C47C22⋊C413yes—
77C22×C4×C47C22⋊C413yes—
78C22×C4×C47C22⋊C416yes—
79C22×C4×C47C22×C413yes—
80C22×C4×C47C22×C413yes—
81C22×C4×C47C2×D413yes—
82C22×C4×C47C2×D413yes—
83C22×C4×C47C2×D413yes—
84C22×C4×C47C2411yescharacteristic
85C22×Dic47—224no—
86C22×Dic47—24no—
87C22×Dic47—432no—
88C22×Dic47—412no—
89C22×Dic47—412no—
90C4⋊C4×C47—212no—
91C4⋊C4×C47—26no—
92C4⋊C4×C47—24no—
93C4⋊C4×C47C22×C414yescomplemented
94C22⋊C4×C47—212no—
95C22⋊C4×C47—24no—
96C2×C47⋊C8—24no—
97C2×C47⋊C8—412no—
98C23×C47—216no—
99C23×C47—216no—
100C23×C47—26no—
101C23×C47—26no—
102C23×C47—26no—
103C23×C47—26no—
104C23×C47#9914yes—
105C23×C47#103112yes—
106C23×C47#10314yes—
107C23×C47#10414yes—
108C23×C47C2×C22⋊C413yes—
109C23×C47C2×C22⋊C413yes—
110C23×C47C2×SD1616yes—
111C23×C47C2×Q1616yes—
112C23×C47C22×D411yescharacteristic
113Q8×C47—44no—
114Q8×C47—412no—
115C2×C4×C47—26no—
116C2×C4×C47—26no—
117C2×C4×C47—26no—
118C2×C4×C47—26no—
119C2×C4×C47—24no—
120C2×C4×C47—24no—
121C2×C4×C47—46no—
122C2×C4×C47—43no—
123C2×C4×C47—43no—
124C2×C4×C47—43no—
125C2×C4×C47—41no—
126C2×C4×C47—412no—
127C2×C4×C47—412no—
128C2×C4×C47C2×C22⋊C413yes—
129C2×C4×C47C2×C22⋊C413yes—
130C2×C4×C47C23×C411yescharacteristic
131C2×C4×C47C22×D411yescharacteristic
132C2×Dic47—28no—
133C2×Dic47—44no—
134C2×Dic47—448no—
135C2×Dic47—44no—
136C47⋊C8—44no—
137C22×#328—471nomaximal, Sylow
138C22×C47—21no—
139C22×C47—23no—
140C22×C47—21no—
141C22×C47—23no—
142C22×C47—212no—
143C22×C47—224no—
144C22×C47—212no—
145C22×C47#328116yescomplemented
146C22×C47C2×#9916yes—
147C22×C47C2×#10316yes—
148C22×C47C2×#10316yes—
149C22×C47C2×#10416yes—
150C22×C47C22×C22⋊C411yescharacteristic
151C22×C47C22×SD1611yescharacteristic
152C22×C47C22×Q1611yescharacteristic
153C4×C47—21no—
154C4×C47—23no—
155C4×C47—23no—
156C4×C47—21no—
157C4×C47—44no—
158C4×C47—44no—
159Dic47—48no—
160C22×C22⋊Q8D4711yescharacteristic
161C22×C22⋊C8—471no—
162C22×#95—471no—
163C2×#328—4712no—
164C2×C47—21no—
165C2×C47—21no—
166C2×C47—26no—
167C2×C47C2×#328112yescomplemented
168C2×C47C22×#9911yescharacteristic
169C2×C47C22×#10311yescharacteristic
170C2×C47C22×#10411yescharacteristic
171C23×Q8Dic4711yescharacteristic, complemented
172C24×C4—941no—
173C24×C4C2×D4711yescharacteristic
174C23×C8—941no—
175C2×C22⋊Q8—23no—
176C2×C22⋊Q8—23no—
177C2×C22⋊Q8Dic4716yescomplemented
178C2×C22⋊Q8C2×D4716yes—
179C22×C4⋊C4—21no—
180C22×C4⋊C4Dic4711yescharacteristic, complemented
181C22×C22⋊C4—21no—
182C2×C22⋊C8—476no—
183C2×C22⋊C8—943no—
184C2×#95—476no—
185C2×#95—943no—
186#328—4716no—
187C47C22×#32811yescharacteristic, complemented, Sylow
188C25C22⋊D4711yescharacteristic
189C22×Q8—23no—
190C22×Q8—44no—
191C22×Q8C2×Dic4716yescomplemented
192C23×C4—21no—
193C23×C4—21no—
194C23×C4—44no—
195C23×C4—948no—
196C23×C4—946no—
197C23×C4—1884no—
198C23×C4—1884no—
199C23×C4C2×Dic4711yescharacteristic
200C23×C4C2×Dic4713yes—
201C23×C4C22⋊D4711yescharacteristic
202C23×C4C22⋊D4713yes—
203C23×C4C22⋊D4713yes—
204C23×C4C22×D4713yes—
205C22×C8—946no—
206C22×C8—1884no—
207C22⋊Q8—212no—
208C22⋊Q8—26no—
209C22⋊Q8—26no—
210C22⋊Q8C2×Dic47112yescomplemented
211C22⋊Q8C22×D4714yes—
212C2×C4⋊C4—26no—
213C2×C4⋊C4—23no—
214C2×C4⋊C4—26no—
215C2×C4⋊C4C2×Dic4716yescomplemented
216C2×C22⋊C4—26no—
217C2×C22⋊C4—26no—
218C22⋊C8—474no—
219C22⋊C8—946no—
220#95—474no—
221#95—946no—
222C24—24no—
223C24—24no—
224C24#1062533614yes—
225C24#1062533714yes—
226C24C22⋊Dic4713yes—
227C24C22⋊Dic4711yescharacteristic
228C24C2×C22⋊D4713yes—
229C2×Q8—26no—
230C2×Q8—412no—
231C2×Q8—412no—
232C2×Q8C22×Dic4714yescomplemented
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—26no—
236C22×C4—26no—
237C22×C4—46no—
238C22×C4—46no—
239C22×C4—46no—
240C22×C4—46no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—9424no—
244C22×C4—944no—
245C22×C4—18812no—
246C22×C4—18832no—
247C22×C4—18812no—
248C22×C4C22⋊Dic4713yes—
249C22×C4C22⋊Dic4713yes—
250C22×C4C22⋊Dic4716yes—
251C22×C4C22×Dic4713yes—
252C22×C4C22×Dic4713yes—
253C22×C4C2×C22⋊D4713yes—
254C22×C4C2×C22⋊D4713yes—
255C22×C4C2×C22⋊D4713yes—
256C22×C4C23×D4711yescharacteristic
257C2×C8—944no—
258C2×C8—18812no—
259C4⋊C4—212no—
260C4⋊C4—26no—
261C4⋊C4—24no—
262C4⋊C4C22×Dic4714yescomplemented
263C22⋊C4—212no—
264C22⋊C4—24no—
265C23—26no—
266C23—26no—
267C23—216no—
268C23—216no—
269C23—26no—
270C23—26no—
271C23#1309393914yes—
272C23#13093948112yes—
273C23#1309394814yes—
274C23#1309395114yes—
275C23C2×C22⋊Dic4713yes—
276C23C2×C22⋊Dic4713yes—
277C23C2×#1062533616yes—
278C23C2×#1062533716yes—
279C23C22×C22⋊D4711yescharacteristic
280Q8—44no—
281Q8—412no—
282C2×C4—26no—
283C2×C4—26no—
284C2×C4—26no—
285C2×C4—26no—
286C2×C4—24no—
287C2×C4—24no—
288C2×C4—46no—
289C2×C4—43no—
290C2×C4—43no—
291C2×C4—43no—
292C2×C4—41no—
293C2×C4—412no—
294C2×C4—412no—
295C2×C4—948no—
296C2×C4—1884no—
297C2×C4—18848no—
298C2×C4—1884no—
299C2×C4C2×C22⋊Dic4713yes—
300C2×C4C2×C22⋊Dic4713yes—
301C2×C4C23×Dic4711yescharacteristic
302C2×C4C22×C22⋊D4711yescharacteristic
303C8—1884no—
304C22—21no—
305C22—23no—
306C22—21no—
307C22—23no—
308C22—212no—
309C22—224no—
310C22—212no—
311C22#15688860116yesdirect product factor
312C22C2×#1309393916yes—
313C22C2×#1309394816yes—
314C22C2×#1309394816yes—
315C22C2×#1309395116yes—
316C22C22×C22⋊Dic4711yescharacteristic
317C22C22×#1062533611yescharacteristic
318C22C22×#1062533711yescharacteristic
319C4—21no—
320C4—23no—
321C4—23no—
322C4—21no—
323C4—44no—
324C4—44no—
325C4—1888no—
326C2—21no—
327C2—26no—
328C2—21no—
329C2C2×#15688860112yesdirect product factor
330C2C22×#1309393911yescharacteristic
331C2C22×#1309394811yescharacteristic
332C2C22×#1309395111yescharacteristic
subgroupG/H
centerC24C22⋊Dic47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC22×C22⋊Q8×C47C2
socleC24×C47C22⋊C4
prime radicalC25×C47D4
cosocleC22×C4×C47C24
hypercenterC22×C22⋊Q8D47
hypocenterC47C22×#328
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C24 ◁ C24×C4 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. C2×#15688860C2direct (N × H)
11. #15688860C22direct (N × H)
12. C23×Q8×C47C4semidirect
16. C2×C22⋊Q8×C47C4semidirect
19. C22×C4⋊C4×C47C4semidirect
34. C22×Q8×C47C2×C4semidirect
47. C22⋊Q8×C47C2×C4semidirect
52. C2×C4⋊C4×C47C2×C4semidirect
65. C2×Q8×C47C22×C4semidirect
93. C4⋊C4×C47C22×C4semidirect
145. C22×C47#328semidirect
167. C2×C47C2×#328semidirect
171. C23×Q8Dic47semidirect
177. C2×C22⋊Q8Dic47semidirect
180. C22×C4⋊C4Dic47semidirect
187. C47C22×#328semidirect
191. C22×Q8C2×Dic47semidirect
210. C22⋊Q8C2×Dic47semidirect
215. C2×C4⋊C4C2×Dic47semidirect
232. C2×Q8C22×Dic47semidirect
262. C4⋊C4C22×Dic47semidirect
311. C22#15688860direct (N × H)
329. C2C2×#15688860direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C22⋊(C47⋊C8)2
C22×C47⋊#952
C2×#156888602
C22×C22⋊Q8×C472
C22×#32847