Finite Group Databasebeta

← #26023890

Subgroups — #26023890

Order 11392.

1348 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 1348 conjugacy classes of subgroups — too many to draw. Only its 197 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
1C23×D356C211yesmaximal, characteristic, complemented
2C2×#15668008C212yesmaximal, complemented
3C2×#15667980C214yesmaximal, complemented
4#19151507C214yesmaximal, complemented
5#19151499C212yesmaximal, complemented
6C4×C22⋊Dic89C211yesmaximal, characteristic, complemented
7C2×#121×C89C211yesmaximal, characteristic, complemented
8C24×D89C2212yes—
9C22×D356—28no—
10C22×D356—24no—
11C22×D356C2212yes—
12C22×D356C2212yes—
13C22×C22⋊D89C2218yescomplemented
14#15668008—28no—
15#15667980—216no—
16C2×C22⋊Dic89C2212yes—
17C2×#13083551C2218yescomplemented
18C2×C4×Dic89C2212yescomplemented
19#15667919C2212yescomplemented
20C22×D4×C89C2212yes—
21C22×D4×C89C2211yescharacteristic, complemented
22C23×C4×C89C2211yescharacteristic
23#121×C89—24no—
24C2×C4⋊C4×C89C2211yescharacteristic, complemented
25C2×C22⋊C4×C89C2212yescomplemented
26C23×D89—24no—
27C23×D89—28no—
28C23×D89—28no—
29C23×D89—44no—
30C23×D89C2314yes—
31C2×D356—232no—
32C2×D356—28no—
33C2×D356—24no—
34C2×D356—24no—
35C2×D356—24no—
36C2×D356D418yescomplemented
37C2×C22⋊D89—216no—
38C2×C22⋊D89—216no—
39C2×C22⋊D89—48no—
40C22×Dic89C2314yes—
41C22⋊Dic89D418yescomplemented
42#13083551—216no—
43C4×Dic89—24no—
44C24×C89C2312yes—
45C24×C89C2311yescharacteristic
46C2×D4×C89—28no—
47C2×D4×C89—24no—
48C2×D4×C89—24no—
49C2×D4×C89—22no—
50C22×C4×C89—24no—
51C22×C4×C89D414yescomplemented
52C22×C4×C89C2312yes—
53C22×C4×C89C2311yescharacteristic
54C22×C4×C89C2311yescharacteristic
55C4⋊C4×C89—22no—
56C22⋊C4×C89—24no—
57C23×C89—22no—
58C23×C89—22no—
59C23×C89—22no—
60C23×C89—24no—
61C23×C89—24no—
62C23×C89—24no—
63C23×C89C2×D414yes—
64C23×C89C2×D412yes—
65C23×C89C2411yescharacteristic
66C22×D89—216no—
67C22×D89—28no—
68C22×D89—24no—
69C22×D89—24no—
70C22×D89—28no—
71C22×D89—432no—
72C22×D89—48no—
73C22×D89—48no—
74C22×D89C2×D418yes—
75D4×C89—48no—
76D4×C89—44no—
77C2×C4×C89—22no—
78C2×C4×C89—28no—
79C2×C4×C89—22no—
80C2×C4×C89—24no—
81C2×C4×C89C2×D414yes—
82C2×C4×C89C2×D412yes—
83C2×C4×C89C2×D412yes—
84C22⋊D89—432no—
85C2×Dic89—24no—
86C2×Dic89—24no—
87C2×Dic89C2×D418yescomplemented
88D356—216no—
89D356—216no—
90C22×C89—22no—
91C22×C89—22no—
92C22×C89—28no—
93C22×C89—42no—
94C22×C89—42no—
95C22×C89—42no—
96C22×C89—48no—
97C22×C89#12714yescomplemented
98C22×C89C22×D411yescharacteristic
99C22×C89C22×D412yes—
100C22×C89C22×D412yes—
101C22×C89H(5,𝔽2)12yes—
102C2×D89—28no—
103C2×D89—216no—
104C2×D89—432no—
105C2×D89—44no—
106C2×D89—44no—
107C2×D89—48no—
108C4×C89—22no—
109C4×C89—42no—
110C4×C89#12014yescomplemented
111Dic89—28no—
112C2×C89—24no—
113C2×C89—82no—
114C2×C89C2×#12011yescharacteristic
115C2×C89C2×#12711yescharacteristic
116C2×C89#53511yescharacteristic
117C2×C89D4214yes—
118D89—48no—
119#2433—891nomaximal, Sylow
120C89#243311yescharacteristic, complemented, Sylow
121C23×D4—891no—
122C2×#127—892no—
123C2×#121D8911yescharacteristic, complemented
124C2×#120—894no—
125#392—894no—
126#390—892no—
127C4×C22⋊C4—891no—
128C25—892no—
129C22×D4—898no—
130C22×D4—892no—
131C22×D4—892no—
132C22×D4—1788no—
133C22×D4—1784no—
134C22×D4C2×D8912yes—
135C22×D4C2×D8911yescharacteristic, complemented
136C23×C4C2×D8911yescharacteristic
137#127—1788no—
138#121—24no—
139#120—17816no—
140C2×C4⋊C4C2×D8911yescharacteristic, complemented
141C2×C22⋊C4—892no—
142C2×C22⋊C4—898no—
143C2×C22⋊C4C2×D8912yescomplemented
144C2×C42—892no—
145#95—892no—
146C24—894no—
147C24—3564no—
148C24—1788no—
149C24—1788no—
150C24—1784no—
151C24C22×D8912yes—
152C24C22×D8911yescharacteristic
153C2×D4—28no—
154C2×D4—24no—
155C2×D4—24no—
156C2×D4—22no—
157C2×D4—898no—
158C2×D4—3568no—
159C2×D4—1788no—
160C2×D4—17832no—
161C2×D4—17816no—
162C2×D4—17816no—
163C2×D4—1784no—
164C2×D4—1784no—
165C2×D4—1784no—
166C22×C4—24no—
167C22×C4—894no—
168C22×C4C22⋊D8914yescomplemented
169C22×C4C22×D8912yes—
170C22×C4C22×D8911yescharacteristic
171C22×C4C22×D8911yescharacteristic
172C4⋊C4—22no—
173C22⋊C4—24no—
174C22⋊C4—898no—
175C22⋊C4—17816no—
176C42—1784no—
177C23—22no—
178C23—22no—
179C23—24no—
180C23—24no—
181C23—22no—
182C23—24no—
183C23—898no—
184C23—3568no—
185C23—35632no—
186C23—3568no—
187C23—17816no—
188C23—1788no—
189C23—1784no—
190C23—1784no—
191C23—1788no—
192C23D4×D8914yes—
193C23C2×C22⋊D8912yes—
194C23C23×D8911yescharacteristic
195D4—48no—
196D4—44no—
197D4—35632no—
198D4—17816no—
199D4—17816no—
200C2×C4—22no—
201C2×C4—28no—
202C2×C4—24no—
203C2×C4—22no—
204C2×C4—898no—
205C2×C4—1784no—
206C2×C4—1784no—
207C2×C4D4×D8914yes—
208C2×C4C2×C22⋊D8912yes—
209C2×C4C2×C22⋊D8912yes—
210C22—22no—
211C22—28no—
212C22—22no—
213C22—42no—
214C22—42no—
215C22—48no—
216C22—42no—
217C22—35632no—
218C22—3564no—
219C22—3564no—
220C22—3568no—
221C22—17816no—
222C22—1788no—
223C22#1566800814yescomplemented
224C22C2×D4×D8912yes—
225C22C2×D4×D8912yes—
226C22C22×C22⋊D8911yescharacteristic
227C22#1566805812yes—
228C4—22no—
229C4—42no—
230C4—1788no—
231C4#1566798014yescomplemented
232C2—24no—
233C2—82no—
234C2—3568no—
235C2C2×#1566798011yescharacteristic
236C2C2×#1566800811yescharacteristic
237C2#1915216211yescharacteristic
238C2#1915223214yes—
subgroupG/H
center, FrattiniC23C23×D89
commutator, socle, cosocleC23×C89C24
FittingC2×#121×C89C2
hypercenterC2×#121D89
hypocenterC89#2433
LCSG ▷ C23×C89 ▷ C89
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C23×D356C2semidirect
2. C2×#15668008C2semidirect
3. C2×#15667980C2semidirect
4. #19151507C2semidirect
5. #19151499C2semidirect
6. C4×C22⋊Dic89C2semidirect
7. C2×#121×C89C2semidirect
13. C22×C22⋊D89C22semidirect
17. C2×#13083551C22semidirect
18. C2×C4×Dic89C22semidirect
19. #15667919C22semidirect
21. C22×D4×C89C22semidirect
24. C2×C4⋊C4×C89C22semidirect
25. C2×C22⋊C4×C89C22semidirect
36. C2×D356D4semidirect
41. C22⋊Dic89D4semidirect
51. C22×C4×C89D4semidirect
87. C2×Dic89C2×D4semidirect
97. C22×C89#127semidirect
110. C4×C89#120semidirect
120. C89#2433semidirect
123. C2×#121D89semidirect
135. C22×D4C2×D89semidirect
140. C2×C4⋊C4C2×D89semidirect
143. C2×C22⋊C4C2×D89semidirect
168. C22×C4C22⋊D89semidirect
223. C22#15668008semidirect
231. C4#15667980semidirect

Maximal subgroups

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

subgroupindex
C23×D3562
C2×#156680082
C2×#156679802
#191515072
#191514992
C4×C22⋊Dic892
C2×#121×C892
#243389