Finite Group Databasebeta

← #2842×D89 #36760837

Subgroups — #2842×D89 #36760837

Order 22784.

3294 conjugacy classes of subgroups; only the 579 normal subgroups are shown.

normal subgroups only

This group has 3294 conjugacy classes of subgroups — too many to draw. Only its 579 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
1#26028517C214yesmaximal, complemented
2#26028516C213yesmaximal, complemented
3#26028515C218yesmaximal, complemented
4#399×D89C214yesmaximal, complemented
5#396×D89C218yesmaximal, complemented
6#379×D89C213yesmaximal
7C89×#2842C211yesmaximal, characteristic, complemented
8C25×D89C2211yescharacteristic
9C2×C4⋊C4×D89C22112yescomplemented
10C2×C22⋊C4×D89C22112yes—
11C2×C22⋊C4×D89C2216yes—
12#19151535C2214yescomplemented
13#19151534C22112yescomplemented
14#19151533C22112yescomplemented
15#19151522C2218yescomplemented
16#19151521C22124yescomplemented
17#19151520C22124yescomplemented
18#19151454C2213yes—
19#19151453C22112yes—
20#19151453C2216yes—
21#95×D89C2214yescomplemented
22#399×C89C2214yescomplemented
23#396×C89C2218yescomplemented
24#379×C89C2213yes—
25C24×D89—412no—
26C24×D89Q818yes—
27C24×D89C2312yes—
28C24×D89C2311yescharacteristic
29C24×D89C2312yes—
30C24×D89C2311yescharacteristic
31C22×C4×D89C23112yes—
32C2×C4⋊Dic89C23112yescomplemented
33C2×#13083553C23124yescomplemented
34C4⋊C4×D89—424no—
35C2×C22⋊Dic89C23112yes—
36C2×C22⋊Dic89C2316yes—
37C2×#13083551C23124yes—
38C2×#13083551C23112yes—
39C22⋊C4×D89—424no—
40C22⋊C4×D89—412no—
41#15667919C23112yescomplemented
42C25×C89C2311yescharacteristic
43C2×C4⋊C4×C89C23112yescomplemented
44C2×C22⋊C4×C89C23112yes—
45C2×C22⋊C4×C89C2316yes—
46#95×C89C2314yescomplemented
47C23×D89—224no—
48C23×D89—212no—
49C23×D89—448no—
50C23×D89—412no—
51C23×D89—46no—
52C23×D89—464no—
53C23×D89C2×Q818yes—
54C23×D89C2×Q814yes—
55C23×D89C2×Q818yes—
56C23×D89C2×Q814yes—
57C23×D89C2412yes—
58C23×D89C2411yescharacteristic
59C23×D89C2411yescharacteristic
60C22×Dic89C24112yes—
61C2×C4×D89—224no—
62C2×C4×D89—248no—
63C4⋊Dic89—412no—
64#13083553—424no—
65C22⋊Dic89—46no—
66C22⋊Dic89—412no—
67#13083551—412no—
68#13083551—424no—
69C24×C89—46no—
70C24×C89C2×Q814yes—
71C24×C89C2412yes—
72C24×C89C2411yescharacteristic
73C22×C4×C89C24112yes—
74C4⋊C4×C89—412no—
75C22⋊C4×C89—412no—
76C22⋊C4×C89—46no—
77C23×C89—212no—
78C23×C89—26no—
79C23×C89—412no—
80C23×C89—416no—
81C23×C89C22×Q814yes—
82C23×C89C22×Q812yes—
83C23×C89C2511yescharacteristic
84C22×D89—26no—
85C22×D89—212no—
86C22×D89—248no—
87C22×D89—224no—
88C22×D89—424no—
89C22×D89—412no—
90C22×D89—432no—
91C22×D89—416no—
92C22×D89—4128no—
93C22×D89—432no—
94C22×D89—416no—
95C22×D89C22×Q814yes—
96C22×D89C22×Q812yes—
97C22×D89C22×Q812yes—
98C22×D89H(5,𝔽2)112yes—
99C2×C4×C89—224no—
100C2×C4×C89—212no—
101C2×Dic89—212no—
102C2×Dic89—224no—
103C4×D89—448no—
104C22×C89—212no—
105C22×C89—26no—
106C22×C89—416no—
107C22×C89—48no—
108C22×C89—44no—
109C22×C89C23×Q811yescharacteristic
110C22×C89C2×H(5,𝔽2)16yes—
111C2×D89—26no—
112C2×D89—212no—
113C2×D89—432no—
114C2×D89—416no—
115C2×D89—44no—
116C2×D89—48no—
117C2×D89—432no—
118C2×D89—416no—
119C2×D89#543112yes—
120C2×D89#560116yes—
121C4×C89—412no—
122Dic89—412no—
123C2×#2842—891nomaximal, Sylow
124C2×C89—44no—
125C2×C89—42no—
126C2×C89C2×#54313yes—
127C2×C89C2×#56014yes—
128D89—42no—
129D89—44no—
130D89#284218yesdirect product factor
131#2842—894no—
132#2842—898no—
133#2842—893no—
134#2842D8911yescharacteristic, direct product factor
135C2×#399—894no—
136C2×#396—898no—
137C2×#379—893no—
138C89C2×#284211yescharacteristic, complemented, Sylow
139C26—891no—
140C22×C4⋊C4—8912no—
141C22×C22⋊C4—8912no—
142C22×C22⋊C4—896no—
143#399—8912no—
144#399—8912no—
145#399—894no—
146#399C2×D8914yescomplemented
147#396—8924no—
148#396—8924no—
149#396—898no—
150#396C2×D8918yescomplemented
151#379—893no—
152#379—8912no—
153#379—896no—
154#379C2×D8913yes—
155C2×#95—894no—
156C25—898no—
157C25—891no—
158C25—892no—
159C25—892no—
160C25—891no—
161C25—35612no—
162C25C22×D8911yescharacteristic
163C23×C4—8912no—
164C2×C4⋊C4—8924no—
165C2×C4⋊C4—8912no—
166C2×C4⋊C4—35624no—
167C2×C4⋊C4C22×D89112yescomplemented
168C2×C22⋊C4—8912no—
169C2×C22⋊C4—896no—
170C2×C22⋊C4—8924no—
171C2×C22⋊C4—8912no—
172C2×C22⋊C4—35624no—
173C2×C22⋊C4—35612no—
174C2×C22⋊C4C22×D89112yes—
175C2×C22⋊C4C22×D8916yes—
176#95—8912no—
177#95C22×D8914yescomplemented
178C24—46no—
179C24—891no—
180C24—894no—
181C24—898no—
182C24—898no—
183C24—894no—
184C24—892no—
185C24—891no—
186C24—35612no—
187C24—3566no—
188C24—35664no—
189C24—35648no—
190C24—17824no—
191C24—17812no—
192C24Q8×D8914yes—
193C24C23×D8912yes—
194C24C23×D8911yescharacteristic
195C22×C4—8912no—
196C22×C4—17848no—
197C22×C4—17824no—
198C22×C4C23×D89112yes—
199C4⋊C4—412no—
200C4⋊C4—35624no—
201C4⋊C4—35612no—
202C22⋊C4—412no—
203C22⋊C4—46no—
204C22⋊C4—35624no—
205C22⋊C4—35612no—
206C22⋊C4—35612no—
207C22⋊C4—3566no—
208C23—212no—
209C23—26no—
210C23—412no—
211C23—416no—
212C23—892no—
213C23—8912no—
214C23—894no—
215C23—892no—
216C23—35624no—
217C23—35612no—
218C23—35632no—
219C23—35616no—
220C23—356128no—
221C23—35632no—
222C23—35616no—
223C23—17848no—
224C23—17824no—
225C23—17812no—
226C23—1786no—
227C23C2×Q8×D8914yes—
228C23C2×Q8×D8912yes—
229C23C24×D8911yescharacteristic
230C2×C4—224no—
231C2×C4—212no—
232C2×C4—35648no—
233C2×C4—17812no—
234C2×C4—17824no—
235C22—212no—
236C22—26no—
237C22—416no—
238C22—48no—
239C22—44no—
240C22—8916no—
241C22—8912no—
242C22—35632no—
243C22—35616no—
244C22—35632no—
245C22—35616no—
246C22—3568no—
247C22—3564no—
248C22—17812no—
249C22—1786no—
250C22C22×Q8×D8911yescharacteristic
251C22H(5,𝔽2)×D8916yes—
252C4—412no—
253C4—35612no—
254C2—44no—
255C2—42no—
256C2—898no—
257C2—3564no—
258C2—3562no—
259C2#543×D8913yes—
260C2#560×D8914yes—
subgroupG/H
center, FrattiniC23C24×D89
commutator, socle, cosocleC23×C89C25
FittingC89×#2842C2
prime radicalC25×D89C22
hypercenter#2842D89
hypocenterC89C2×#2842
LCSG ▷ C23×C89 ▷ C89
UCSC1 ◁ C23 ◁ #2842

Complements

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

NH ≅ G/Ntype
1. #26028517C2semidirect
2. #26028516C2semidirect
3. #26028515C2semidirect
4. #399×D89C2semidirect
5. #396×D89C2semidirect
7. C89×#2842C2semidirect
9. C2×C4⋊C4×D89C22semidirect
12. #19151535C22semidirect
13. #19151534C22semidirect
14. #19151533C22semidirect
15. #19151522C22semidirect
16. #19151521C22semidirect
17. #19151520C22semidirect
21. #95×D89C22semidirect
22. #399×C89C22semidirect
23. #396×C89C22semidirect
32. C2×C4⋊Dic89C23semidirect
33. C2×#13083553C23semidirect
41. #15667919C23semidirect
43. C2×C4⋊C4×C89C23semidirect
46. #95×C89C23semidirect
130. D89#2842direct (N × H)
134. #2842D89direct (N × H)
138. C89C2×#2842semidirect
146. #399C2×D89semidirect
150. #396C2×D89semidirect
167. C2×C4⋊C4C22×D89semidirect
177. #95C22×D89semidirect

Maximal subgroups

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

subgroupindex
#260285172
#260285162
#260285152
#399×D892
#396×D892
#379×D892
C89×#28422
C2×#284289