Finite Group Databasebeta

← C2×#20251664 #26422102

Subgroups — C2×#20251664 #26422102

Order 12032.

32890 conjugacy classes of subgroups; only the 1243 normal subgroups are shown.

normal subgroups only

This group has 32890 conjugacy classes of subgroups — too many to draw. Only its 1243 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C26×D47C211yesmaximal, characteristic, complemented
2C22×#13093992C216yesmaximal, complemented
3#20251664C2116yesmaximal, direct product factor
4C2×#15688979C211yesmaximal, characteristic, complemented
5C2×#15688978C216yesmaximal, complemented
6C22×#120×C47C211yesmaximal, characteristic, complemented
7C25×D47—248no—
8C25×D47—28no—
9C25×D47C2216yes—
10C25×D47C2218yescomplemented
11C23×C22⋊D47C2214yescomplemented
12C2×#13093992—248no—
13C2×#13093992C22148yescomplemented
14C22×C22⋊Dic47C2216yescomplemented
15C22×(C2×D47)⋊C4C22112yescomplemented
16#15688979C2218yescomplemented
17#15688978C22148yescomplemented
18C26×C47C2211yescharacteristic
19C23×D4×C47C2213yescomplemented
20C2×#120×C47—28no—
21C2×#120×C47C2218yescomplemented
22C22×C22⋊C4×C47C2213yescomplemented
23C24×D47—2384no—
24C24×D47—2512no—
25C24×D47—224no—
26C24×D47—224no—
27C24×D47—232no—
28C24×D47—296no—
29C24×D47—2192no—
30C24×D47D4148yescomplemented
31C24×D47C23124yes—
32C24×D47C2314yes—
33C22×C22⋊D47—248no—
34C22×C22⋊D47C23116yescomplemented
35C23×Dic47C2314yescomplemented
36#13093992—2192no—
37C2×C22⋊Dic47—224no—
38C2×C22⋊Dic47C23124yescomplemented
39C2×(C2×D47)⋊C4—248no—
40C2×(C2×D47)⋊C4C23148yescomplemented
41C25×C47—224no—
42C25×C47D418yescomplemented
43C25×C47C2314yes—
44C25×C47C2313yes—
45C25×C47C2311yescharacteristic, complemented
46C22×D4×C47—212no—
47C22×D4×C47—224no—
48C22×D4×C47C23112yescomplemented
49C23×C4×C47C2313yescomplemented
50#120×C47—232no—
51C2×C22⋊C4×C47—212no—
52C2×C22⋊C4×C47C23112yescomplemented
53C24×C47—2128no—
54C24×C47—296no—
55C24×C47—248no—
56C24×C47—224no—
57C24×C47—22no—
58C24×C47—212no—
59C24×C47C2×D4124yes—
60C24×C47C2×D4112yes—
61C24×C47C2×D4116yescomplemented
62C24×C47C2416yes—
63C24×C47C2411yescharacteristic
64C24×C47C2412yescomplemented
65C23×D47—22048no—
66C23×D47—2384no—
67C23×D47—2384no—
68C23×D47—2768no—
69C23×D47—21024no—
70C23×D47—2192no—
71C23×D47—248no—
72C23×D47—248no—
73C23×D47—224no—
74C23×D47—28no—
75C23×D47—2384no—
76C23×D47—2192no—
77C23×D47C2×D4148yes—
78C23×D47C2×D4196yescomplemented
79C23×D47C2418yes—
80C2×D4×C47—224no—
81C2×D4×C47—248no—
82C2×D4×C47—448no—
83C22×C4×C47—26no—
84C22×C4×C47—212no—
85C22×C4×C47C2416yescomplemented
86C2×C22⋊D47—296no—
87C2×C22⋊D47—464no—
88C22×Dic47—224no—
89C22×Dic47C2418yescomplemented
90C22⋊C4×C47—224no—
91C22⋊Dic47—248no—
92(C2×D47)⋊C4—296no—
93C23×C47—2256no—
94C23×C47—2128no—
95C23×C47—296no—
96C23×C47—248no—
97C23×C47—296no—
98C23×C47—224no—
99C23×C47—212no—
100C23×C47—22no—
101C23×C47—416no—
102C23×C47—412no—
103C23×C47#120148yescomplemented
104C23×C47C22×D4124yes—
105C23×C47C22×D4112yes—
106C23×C47C22×D4112yes—
107C23×C47C22×D412yes—
108C23×C47C2511yescharacteristic
109C22×D47—22048no—
110C22×D47—21536no—
111C22×D47—2384no—
112C22×D47—2384no—
113C22×D47—296no—
114C22×D47—296no—
115C22×D47—2192no—
116C22×D47—224no—
117C22×D47—28no—
118C22×D47—2384no—
119C22×D47#120164yescomplemented
120C22×D47C22×D4148yes—
121D4×C47—448no—
122C2×C4×C47—212no—
123C2×C4×C47—26no—
124C2×C4×C47—412no—
125C22⋊D47—464no—
126C2×Dic47—224no—
127C2×Dic47—416no—
128C2×#2846—471nomaximal, Sylow
129C22×C47—2128no—
130C22×C47—296no—
131C22×C47—248no—
132C22×C47—224no—
133C22×C47—48no—
134C22×C47—46no—
135C22×C47—88no—
136C22×C47C2×#120116yes—
137C22×C47C2×#120112yes—
138C22×C47C2×#120124yescomplemented
139C22×C47C23×D416yes—
140C22×C47C23×D411yescharacteristic
141C2×D47—2768no—
142C2×D47—2128no—
143C2×D47—248no—
144C2×D47—248no—
145C2×D47C2×#120132yescomplemented
146C4×C47—46no—
147Dic47—48no—
148C27—471no—
149C22×#120—476no—
150C22×#120D4711yescharacteristic, complemented
151#2846—4716no—
152C2×#379—471no—
153C2×#379—476no—
154C2×C47—224no—
155C2×C47—82no—
156C2×C47#284618yescomplemented
157C2×C47C22×#12014yes—
158C2×C47C22×#12013yes—
159D47—232no—
160C26—476no—
161C26—478no—
162C26—9448no—
163C26—948no—
164C26C2×D4711yescharacteristic
165C23×D4—474no—
166C23×D4C2×D4713yescomplemented
167C2×#120—28no—
168C2×#120—4748no—
169C2×#120—9448no—
170C2×#120C2×D4718yescomplemented
171C22×C22⋊C4—476no—
172C22×C22⋊C4—4712no—
173C22×C22⋊C4C2×D4713yescomplemented
174#379—478no—
175#379—4748no—
176C47C2×#284611yescharacteristic, complemented, Sylow
177C25—224no—
178C25—474no—
179C25—4748no—
180C25—4724no—
181C25—9496no—
182C25—94192no—
183C25—94512no—
184C25—94384no—
185C25—9424no—
186C25—9424no—
187C25—9432no—
188C25C22⋊D4718yescomplemented
189C25C22×D4714yes—
190C25C22×D4713yes—
191C25C22×D4711yescharacteristic, complemented
192C22×D4—224no—
193C22×D4—212no—
194C22×D4—4716no—
195C22×D4—9448no—
196C22×D4C22×D47112yescomplemented
197C23×C4—474no—
198C23×C4C22×D4713yescomplemented
199#120—232no—
200#120—94192no—
201C2×C22⋊C4—212no—
202C2×C22⋊C4—4724no—
203C2×C22⋊C4—4748no—
204C2×C22⋊C4—9448no—
205C2×C22⋊C4—9424no—
206C2×C22⋊C4C22×D47112yescomplemented
207C24—2128no—
208C24—296no—
209C24—248no—
210C24—224no—
211C24—212no—
212C24—22no—
213C24—4748no—
214C24—478no—
215C24—4796no—
216C24—94384no—
217C24—94192no—
218C24—941024no—
219C24—942048no—
220C24—94384no—
221C24—94384no—
222C24—94768no—
223C24—94192no—
224C24—9448no—
225C24—9424no—
226C24—9448no—
227C24—948no—
228C24D4×D47124yes—
229C24C2×C22⋊D47112yes—
230C24C2×C22⋊D47116yescomplemented
231C24C23×D4716yes—
232C24C23×D4711yescharacteristic
233C24C23×D4712yescomplemented
234C2×D4—248no—
235C2×D4—224no—
236C2×D4—448no—
237C2×D4—9496no—
238C2×D4—18864no—
239C22×C4—212no—
240C22×C4—26no—
241C22×C4—478no—
242C22×C4—9424no—
243C22×C4C23×D4716yescomplemented
244C22⋊C4—224no—
245C22⋊C4—9496no—
246C22⋊C4—9448no—
247C23—2256no—
248C23—2128no—
249C23—296no—
250C23—248no—
251C23—296no—
252C23—224no—
253C23—212no—
254C23—22no—
255C23—416no—
256C23—412no—
257C23—4764no—
258C23—4748no—
259C23—94384no—
260C23—942048no—
261C23—941536no—
262C23—94384no—
263C23—94384no—
264C23—9496no—
265C23—9496no—
266C23—94192no—
267C23—9424no—
268C23—948no—
269C23#13093992148yescomplemented
270C23C2×D4×D47124yes—
271C23C2×D4×D47112yes—
272C23C22×C22⋊D47112yes—
273C23C22×C22⋊D4712yes—
274C23C24×D4711yescharacteristic
275D4—448no—
276D4—18864no—
277C2×C4—212no—
278C2×C4—26no—
279C2×C4—412no—
280C2×C4—9424no—
281C2×C4—18816no—
282C22—2128no—
283C22—296no—
284C22—248no—
285C22—224no—
286C22—48no—
287C22—46no—
288C22—88no—
289C22—4732no—
290C22—94768no—
291C22—94128no—
292C22—9448no—
293C22—9448no—
294C22#120×D47116yes—
295C22C2×#13093992112yes—
296C22C2×#13093992124yescomplemented
297C22C22×D4×D4716yes—
298C22C23×C22⋊D4711yescharacteristic
299C4—46no—
300C4—1888no—
301C2—224no—
302C2—82no—
303C2—9432no—
304C2#2025166418yesdirect product factor
305C2C2×#120×D4714yes—
306C2C22×#1309399213yes—
subgroupG/H
centerC24C23×D47
commutator, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC22×#120×C47C2
socleC24×C47C24
hypercenterC22×#120D47
hypocenterC47C2×#2846
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C22×#120

Complements

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

NH ≅ G/Ntype
1. C26×D47C2semidirect
2. C22×#13093992C2semidirect
3. #20251664C2direct (N × H)
4. C2×#15688979C2semidirect
5. C2×#15688978C2semidirect
6. C22×#120×C47C2semidirect
10. C25×D47C22semidirect
11. C23×C22⋊D47C22semidirect
13. C2×#13093992C22semidirect
14. C22×C22⋊Dic47C22semidirect
15. C22×(C2×D47)⋊C4C22semidirect
16. #15688979C22semidirect
17. #15688978C22semidirect
19. C23×D4×C47C22semidirect
21. C2×#120×C47C22semidirect
22. C22×C22⋊C4×C47C22semidirect
30. C24×D47D4semidirect
34. C22×C22⋊D47C23semidirect
35. C23×Dic47C23semidirect
38. C2×C22⋊Dic47C23semidirect
40. C2×(C2×D47)⋊C4C23semidirect
42. C25×C47D4semidirect
45. C25×C47C23semidirect
48. C22×D4×C47C23semidirect
49. C23×C4×C47C23semidirect
52. C2×C22⋊C4×C47C23semidirect
61. C24×C47C2×D4semidirect
64. C24×C47C24semidirect
78. C23×D47C2×D4semidirect
85. C22×C4×C47C24semidirect
89. C22×Dic47C24semidirect
103. C23×C47#120semidirect
119. C22×D47#120semidirect
138. C22×C47C2×#120semidirect
145. C2×D47C2×#120semidirect
150. C22×#120D47semidirect
156. C2×C47#2846semidirect
166. C23×D4C2×D47semidirect
170. C2×#120C2×D47semidirect
173. C22×C22⋊C4C2×D47semidirect
176. C47C2×#2846semidirect
188. C25C22⋊D47semidirect
191. C25C22×D47semidirect
196. C22×D4C22×D47semidirect
198. C23×C4C22×D47semidirect
206. C2×C22⋊C4C22×D47semidirect
230. C24C2×C22⋊D47semidirect
233. C24C23×D47semidirect
243. C22×C4C23×D47semidirect
269. C23#13093992semidirect
296. C22C2×#13093992semidirect
304. C2#20251664direct (N × H)

Maximal subgroups

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

subgroupindex
C26×D472
C22×#130939922
#202516642
C2×#156889792
C2×#156889782
C22×#120×C472
C2×#284647