Finite Group Databasebeta

← C2×#26028528 #36760252

Subgroups — C2×#26028528 #36760252

Order 22784.

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×D89C211yesmaximal, characteristic, complemented
2C22×#15667980C216yesmaximal, complemented
3#26028528C2116yesmaximal, direct product factor
4C2×#19151454C211yesmaximal, characteristic, complemented
5C2×#19151453C216yesmaximal, complemented
6C22×#120×C89C211yesmaximal, characteristic, complemented
7C25×D89—28no—
8C25×D89—248no—
9C25×D89C2216yes—
10C25×D89C2218yescomplemented
11C23×C22⋊D89C2214yescomplemented
12C2×#15667980—248no—
13C2×#15667980C22148yescomplemented
14C22×C22⋊Dic89C2216yescomplemented
15C22×#13083551C22112yescomplemented
16#19151454C2218yescomplemented
17#19151453C22148yescomplemented
18C26×C89C2211yescharacteristic
19C23×D4×C89C2213yescomplemented
20C2×#120×C89—28no—
21C2×#120×C89C2218yescomplemented
22C22×C22⋊C4×C89C2213yescomplemented
23C24×D89—2512no—
24C24×D89—2384no—
25C24×D89—296no—
26C24×D89—2192no—
27C24×D89—224no—
28C24×D89—224no—
29C24×D89—232no—
30C24×D89D4148yescomplemented
31C24×D89C23124yes—
32C24×D89C2314yes—
33C22×C22⋊D89—248no—
34C22×C22⋊D89C23116yescomplemented
35C23×Dic89C2314yescomplemented
36#15667980—2192no—
37C2×C22⋊Dic89—224no—
38C2×C22⋊Dic89C23124yescomplemented
39C2×#13083551—248no—
40C2×#13083551C23148yescomplemented
41C25×C89—224no—
42C25×C89D418yescomplemented
43C25×C89C2314yes—
44C25×C89C2313yes—
45C25×C89C2311yescharacteristic, complemented
46C22×D4×C89—212no—
47C22×D4×C89—224no—
48C22×D4×C89C23112yescomplemented
49C23×C4×C89C2313yescomplemented
50#120×C89—232no—
51C2×C22⋊C4×C89—212no—
52C2×C22⋊C4×C89C23112yescomplemented
53C23×D89—22048no—
54C23×D89—2384no—
55C23×D89—2384no—
56C23×D89—2768no—
57C23×D89—21024no—
58C23×D89—2192no—
59C23×D89—248no—
60C23×D89—248no—
61C23×D89—224no—
62C23×D89—28no—
63C23×D89—2384no—
64C23×D89—2192no—
65C23×D89C2×D4148yes—
66C23×D89C2×D4196yescomplemented
67C23×D89C2418yes—
68C2×C22⋊D89—296no—
69C2×C22⋊D89—464no—
70C22×Dic89—224no—
71C22×Dic89C2418yescomplemented
72C22⋊Dic89—248no—
73#13083551—296no—
74C24×C89—2128no—
75C24×C89—296no—
76C24×C89—248no—
77C24×C89—224no—
78C24×C89—212no—
79C24×C89—22no—
80C24×C89C2×D4124yes—
81C24×C89C2×D4112yes—
82C24×C89C2×D4116yescomplemented
83C24×C89C2416yes—
84C24×C89C2411yescharacteristic
85C24×C89C2412yescomplemented
86C2×D4×C89—248no—
87C2×D4×C89—224no—
88C2×D4×C89—448no—
89C22×C4×C89—26no—
90C22×C4×C89—212no—
91C22×C4×C89C2416yescomplemented
92C22⋊C4×C89—224no—
93C23×C89—2256no—
94C23×C89—2128no—
95C23×C89—296no—
96C23×C89—248no—
97C23×C89—296no—
98C23×C89—224no—
99C23×C89—22no—
100C23×C89—212no—
101C23×C89—412no—
102C23×C89—416no—
103C23×C89#120148yescomplemented
104C23×C89C22×D4124yes—
105C23×C89C22×D4112yes—
106C23×C89C22×D4112yes—
107C23×C89C22×D412yes—
108C23×C89C2511yescharacteristic
109C22×D89—22048no—
110C22×D89—21536no—
111C22×D89—2384no—
112C22×D89—2384no—
113C22×D89—296no—
114C22×D89—296no—
115C22×D89—2192no—
116C22×D89—224no—
117C22×D89—28no—
118C22×D89—2384no—
119C22×D89#120164yescomplemented
120C22×D89C22×D4148yes—
121D4×C89—448no—
122C2×C4×C89—26no—
123C2×C4×C89—212no—
124C2×C4×C89—412no—
125C22⋊D89—464no—
126C2×Dic89—224no—
127C2×Dic89—416no—
128C22×C89—2128no—
129C22×C89—296no—
130C22×C89—248no—
131C22×C89—224no—
132C22×C89—48no—
133C22×C89—46no—
134C22×C89—88no—
135C22×C89C2×#120116yes—
136C22×C89C2×#120112yes—
137C22×C89C2×#120124yescomplemented
138C22×C89C23×D416yes—
139C22×C89C23×D411yescharacteristic
140C2×D89—2768no—
141C2×D89—2128no—
142C2×D89—248no—
143C2×D89—248no—
144C2×D89C2×#120132yescomplemented
145C4×C89—46no—
146Dic89—48no—
147C2×#2846—891nomaximal, Sylow
148C2×C89—224no—
149C2×C89—82no—
150C2×C89#284618yescomplemented
151C2×C89C22×#12014yes—
152C2×C89C22×#12013yes—
153D89—232no—
154C27—891no—
155C22×#120—896no—
156C22×#120D8911yescharacteristic, complemented
157#2846—8916no—
158C2×#379—891no—
159C2×#379—896no—
160C89C2×#284611yescharacteristic, complemented, Sylow
161C26—896no—
162C26—898no—
163C26—17848no—
164C26—1788no—
165C26C2×D8911yescharacteristic
166C23×D4—894no—
167C23×D4C2×D8913yescomplemented
168C2×#120—28no—
169C2×#120—8948no—
170C2×#120—17848no—
171C2×#120C2×D8918yescomplemented
172C22×C22⋊C4—896no—
173C22×C22⋊C4—8912no—
174C22×C22⋊C4C2×D8913yescomplemented
175#379—898no—
176#379—8948no—
177C25—224no—
178C25—894no—
179C25—8948no—
180C25—8924no—
181C25—17896no—
182C25—178192no—
183C25—178512no—
184C25—178384no—
185C25—17824no—
186C25—17824no—
187C25—17832no—
188C25C22⋊D8918yescomplemented
189C25C22×D8914yes—
190C25C22×D8913yes—
191C25C22×D8911yescharacteristic, complemented
192C22×D4—224no—
193C22×D4—212no—
194C22×D4—8916no—
195C22×D4—17848no—
196C22×D4C22×D89112yescomplemented
197C23×C4—894no—
198C23×C4C22×D8913yescomplemented
199#120—232no—
200#120—178192no—
201C2×C22⋊C4—212no—
202C2×C22⋊C4—8924no—
203C2×C22⋊C4—8948no—
204C2×C22⋊C4—17848no—
205C2×C22⋊C4—17824no—
206C2×C22⋊C4C22×D89112yescomplemented
207C24—2128no—
208C24—296no—
209C24—248no—
210C24—224no—
211C24—212no—
212C24—22no—
213C24—8948no—
214C24—898no—
215C24—8996no—
216C24—178192no—
217C24—178384no—
218C24—1781024no—
219C24—1782048no—
220C24—178384no—
221C24—178384no—
222C24—178768no—
223C24—178192no—
224C24—17848no—
225C24—17824no—
226C24—17848no—
227C24—1788no—
228C24D4×D89124yes—
229C24C2×C22⋊D89112yes—
230C24C2×C22⋊D89116yescomplemented
231C24C23×D8916yes—
232C24C23×D8911yescharacteristic
233C24C23×D8912yescomplemented
234C2×D4—248no—
235C2×D4—224no—
236C2×D4—448no—
237C2×D4—35664no—
238C2×D4—17896no—
239C22×C4—212no—
240C22×C4—26no—
241C22×C4—898no—
242C22×C4—17824no—
243C22×C4C23×D8916yescomplemented
244C22⋊C4—224no—
245C22⋊C4—17896no—
246C22⋊C4—17848no—
247C23—2128no—
248C23—2256no—
249C23—224no—
250C23—296no—
251C23—248no—
252C23—296no—
253C23—212no—
254C23—22no—
255C23—416no—
256C23—412no—
257C23—8948no—
258C23—8964no—
259C23—178384no—
260C23—1782048no—
261C23—178384no—
262C23—178384no—
263C23—1781536no—
264C23—17824no—
265C23—1788no—
266C23—17896no—
267C23—17896no—
268C23—178192no—
269C23#15667980148yescomplemented
270C23C2×D4×D89124yes—
271C23C2×D4×D89112yes—
272C23C22×C22⋊D89112yes—
273C23C22×C22⋊D8912yes—
274C23C24×D8911yescharacteristic
275D4—448no—
276D4—35664no—
277C2×C4—212no—
278C2×C4—26no—
279C2×C4—412no—
280C2×C4—35616no—
281C2×C4—17824no—
282C22—2128no—
283C22—248no—
284C22—224no—
285C22—296no—
286C22—48no—
287C22—46no—
288C22—88no—
289C22—8932no—
290C22—178768no—
291C22—17848no—
292C22—17848no—
293C22—178128no—
294C22#120×D89116yes—
295C22C2×#15667980112yes—
296C22C2×#15667980124yescomplemented
297C22C22×D4×D8916yes—
298C22C23×C22⋊D8911yescharacteristic
299C4—46no—
300C4—3568no—
301C2—224no—
302C2—82no—
303C2—17832no—
304C2#2602852818yesdirect product factor
305C2C2×#120×D8914yes—
306C2C22×#1566798013yes—
subgroupG/H
centerC24C23×D89
commutator, cosocleC23×C89C25
FrattiniC23C24×D89
FittingC22×#120×C89C2
socleC24×C89C24
hypercenterC22×#120D89
hypocenterC89C2×#2846
LCSG ▷ C23×C89 ▷ C89
UCSC1 ◁ C24 ◁ C22×#120

Complements

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

NH ≅ G/Ntype
1. C26×D89C2semidirect
2. C22×#15667980C2semidirect
3. #26028528C2direct (N × H)
4. C2×#19151454C2semidirect
5. C2×#19151453C2semidirect
6. C22×#120×C89C2semidirect
10. C25×D89C22semidirect
11. C23×C22⋊D89C22semidirect
13. C2×#15667980C22semidirect
14. C22×C22⋊Dic89C22semidirect
15. C22×#13083551C22semidirect
16. #19151454C22semidirect
17. #19151453C22semidirect
19. C23×D4×C89C22semidirect
21. C2×#120×C89C22semidirect
22. C22×C22⋊C4×C89C22semidirect
30. C24×D89D4semidirect
34. C22×C22⋊D89C23semidirect
35. C23×Dic89C23semidirect
38. C2×C22⋊Dic89C23semidirect
40. C2×#13083551C23semidirect
42. C25×C89D4semidirect
45. C25×C89C23semidirect
48. C22×D4×C89C23semidirect
49. C23×C4×C89C23semidirect
52. C2×C22⋊C4×C89C23semidirect
66. C23×D89C2×D4semidirect
71. C22×Dic89C24semidirect
82. C24×C89C2×D4semidirect
85. C24×C89C24semidirect
91. C22×C4×C89C24semidirect
103. C23×C89#120semidirect
119. C22×D89#120semidirect
137. C22×C89C2×#120semidirect
144. C2×D89C2×#120semidirect
150. C2×C89#2846semidirect
156. C22×#120D89semidirect
160. C89C2×#2846semidirect
167. C23×D4C2×D89semidirect
171. C2×#120C2×D89semidirect
174. C22×C22⋊C4C2×D89semidirect
188. C25C22⋊D89semidirect
191. C25C22×D89semidirect
196. C22×D4C22×D89semidirect
198. C23×C4C22×D89semidirect
206. C2×C22⋊C4C22×D89semidirect
230. C24C2×C22⋊D89semidirect
233. C24C23×D89semidirect
243. C22×C4C23×D89semidirect
269. C23#15667980semidirect
296. C22C2×#15667980semidirect
304. C2#26028528direct (N × H)

Maximal subgroups

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

subgroupindex
C26×D892
C22×#156679802
#260285282
C2×#191514542
C2×#191514532
C22×#120×C892
C2×#284689