Finite Group Databasebeta

← #26026071

Subgroups — #26026071

Order 11392.

508 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#15667989C211yesmaximal, characteristic
2C22×#13083553C211yesmaximal, characteristic
3#19151539C212yesmaximal, complemented
4#19151532C212yesmaximal, complemented
5#19151529C212yesmaximal, complemented
6#19151518C212yesmaximal, complemented
7#19151471C211yesmaximal, characteristic
8#19151459C211yesmaximal, characteristic
9#19151459C211yesmaximal, characteristic
10#19151456C211yesmaximal, characteristic
11#385×C89C211yesmaximal, characteristic
12C22×Dic178C2211yescharacteristic
13C23×Dic89C2211yescharacteristic
14C23×Dic89C2211yescharacteristic
15#15667989—24no—
16C2×C4⋊Dic89C2211yescharacteristic
17C2×#13083553—24no—
18C2×#13083553C2212yes—
19C2×#13083553C2211yescharacteristic
20C2×#13083553C2212yes—
21C2×#13083553C2211yescharacteristic
22C2×#13083553C2212yes—
23C2×#13083553C2212yes—
24C2×C22⋊Dic89C2211yescharacteristic
25C2×C22⋊Dic89C2211yescharacteristic
26#15667919C2212yes—
27#15667919C2212yes—
28#15667919C2212yes—
29#15667919C2212yes—
30#15667919C2212yes—
31#15667919C2212yes—
32C23×C4×C89C2211yescharacteristic
33C2×C4⋊C4×C89C2212yes—
34C2×C22⋊C4×C89C2211yescharacteristic
35C2×C22⋊C4×C89C2211yescharacteristic
36#95×C89C2212yes—
37C2×Dic178—22no—
38C2×Dic178—42no—
39C22×Dic89—22no—
40C22×Dic89—22no—
41C22×Dic89—42no—
42C22×Dic89D414yes—
43C22×Dic89Q814yescomplemented
44C22×Dic89C2311yescharacteristic
45C22×Dic89C2311yescharacteristic
46C22×Dic89C2312yes—
47C22×Dic89C2311yescharacteristic
48C22×Dic89C2311yescharacteristic
49C22×Dic89C2312yes—
50C4⋊Dic89—22no—
51#13083553—22no—
52#13083553—22no—
53#13083553—24no—
54#13083553—24no—
55#13083553—42no—
56#13083553—42no—
57C22⋊Dic89—22no—
58C22⋊Dic89—22no—
59C24×C89C2311yescharacteristic
60C22×C4×C89—22no—
61C22×C4×C89—42no—
62C22×C4×C89C2311yescharacteristic
63C22×C4×C89C2311yescharacteristic
64C22×C4×C89C2312yes—
65C22×C4×C89C2312yes—
66C4⋊C4×C89—42no—
67C22⋊C4×C89—22no—
68C22⋊C4×C89—22no—
69C23×C89—22no—
70C23×C89—22no—
71C23×C89C2×D412yes—
72C23×C89C2×Q812yes—
73C23×C89#4112yes—
74C23×C89C2411yescharacteristic
75C2×C4×C89—21no—
76C2×C4×C89—21no—
77C2×C4×C89—22no—
78C2×C4×C89—22no—
79C2×C4×C89—22no—
80C2×C4×C89—22no—
81C2×C4×C89—22no—
82C2×C4×C89—22no—
83C2×C4×C89—22no—
84C2×C4×C89—22no—
85C2×C4×C89—44no—
86C2×Dic89—28no—
87C2×Dic89—22no—
88C2×Dic89—22no—
89C2×Dic89—22no—
90C2×Dic89—22no—
91C2×Dic89—21no—
92C2×Dic89—21no—
93C2×Dic89—22no—
94C2×Dic89—22no—
95C2×Dic89—22no—
96C2×Dic89—22no—
97C2×Dic89—22no—
98C2×Dic89—44no—
99C2×Dic89C2×D412yes—
100C2×Dic89C2×D412yes—
101C2×Dic89C2×Q814yes—
102Dic178—44no—
103C22×C89—22no—
104C22×C89—22no—
105C22×C89—24no—
106C22×C89—24no—
107C22×C89C22⋊Q814yes—
108C22×C89C22×D411yescharacteristic
109C22×C89C22×Q811yescharacteristic
110C22×C89C2×#4111yescharacteristic
111C22×C89H(5,𝔽2)12yes—
112C22×C892−1+412yes—
113C4×C89—41no—
114C4×C89—42no—
115C4×C89—41no—
116C4×C89—42no—
117Dic89—24no—
118Dic89—22no—
119Dic89—22no—
120Dic89—41no—
121Dic89—41no—
122Dic89—42no—
123C2×C89—24no—
124C2×C89C2×C22⋊Q811yescharacteristic
125C2×C89#53411yescharacteristic
126C2×C89#53611yescharacteristic
127C2×C89#53912yes—
128C2×C89#54412yes—
129#2627—891nomaximal, Sylow
130C89#262711yescharacteristic, complemented, Sylow
131C2×C22⋊Q8—891no—
132C22×C4⋊C4—891no—
133#400—892no—
134#398—892no—
135#398—892no—
136#395—892no—
137#385—891no—
138#385D8911yescharacteristic
139#381—891no—
140#381—891no—
141#380—891no—
142C22×Q8—891no—
143C23×C4—891no—
144C23×C4—891no—
145C23×C4C2×D8911yescharacteristic
146C22⋊Q8—1784no—
147C2×C4⋊C4—892no—
148C2×C4⋊C4—892no—
149C2×C4⋊C4—892no—
150C2×C4⋊C4—892no—
151C2×C4⋊C4—891no—
152C2×C4⋊C4—891no—
153C2×C4⋊C4—891no—
154C2×C4⋊C4—1784no—
155C2×C4⋊C4C2×D8912yes—
156C2×C22⋊C4—891no—
157C2×C22⋊C4—891no—
158C2×C22⋊C4C2×D8911yescharacteristic
159C2×C22⋊C4C2×D8911yescharacteristic
160#95—892no—
161#95—892no—
162#95—892no—
163#95—892no—
164#95—892no—
165#95—892no—
166#95C2×D8912yes—
167C24C22×D8911yescharacteristic
168C2×Q8—3562no—
169C2×Q8—1782no—
170C22×C4—22no—
171C22×C4—42no—
172C22×C4—894no—
173C22×C4—894no—
174C22×C4—892no—
175C22×C4—891no—
176C22×C4—891no—
177C22×C4—891no—
178C22×C4—892no—
179C22×C4—891no—
180C22×C4—3562no—
181C22×C4—1782no—
182C22×C4—1782no—
183C22×C4C22×D8912yes—
184C22×C4C22×D8911yescharacteristic
185C22×C4C22×D8911yescharacteristic
186C22×C4C22×D8912yes—
187C4⋊C4—42no—
188C4⋊C4—3562no—
189C4⋊C4—3562no—
190C4⋊C4—1784no—
191C4⋊C4—1784no—
192C4⋊C4—1782no—
193C4⋊C4—1782no—
194C4⋊C4—1782no—
195C22⋊C4—22no—
196C22⋊C4—22no—
197C22⋊C4—1782no—
198C22⋊C4—1782no—
199C23—22no—
200C23—22no—
201C23D4×D8912yes—
202C23Q8×D8912yes—
203C23#1308357412yes—
204C23C23×D8911yescharacteristic
205Q8—3564no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—44no—
217C2×C4—892no—
218C2×C4—892no—
219C2×C4—894no—
220C2×C4—3564no—
221C2×C4—1781no—
222C2×C4—1781no—
223C2×C4—1782no—
224C2×C4—1782no—
225C2×C4—1782no—
226C2×C4—1782no—
227C2×C4—1782no—
228C2×C4—1788no—
229C2×C4—1782no—
230C2×C4—1782no—
231C2×C4—1782no—
232C2×C4—1782no—
233C22—22no—
234C22—22no—
235C22—24no—
236C22—24no—
237C22#1566799014yes—
238C22C2×D4×D8911yescharacteristic
239C22C2×Q8×D8911yescharacteristic
240C22C2×#1308357411yescharacteristic
241C22#1566805712yes—
242C22#1566805912yes—
243C4—42no—
244C4—41no—
245C4—41no—
246C4—42no—
247C4—3561no—
248C4—3561no—
249C4—3562no—
250C4—1782no—
251C4—1782no—
252C4—1784no—
253C2—24no—
254C2C2×#1566799011yescharacteristic
255C2#1915215811yescharacteristic
256C2#1915217011yescharacteristic
257C2#1915219012yes—
258C2#1915222712yes—
subgroupG/H
center, FrattiniC23C23×D89
commutator, socle, cosocleC23×C89C24
Fitting#385×C89C2
prime radicalC24×C89C23
hypercenter#385D89
hypocenterC89#2627
LCSG ▷ C23×C89 ▷ C89
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
3. #19151539C2semidirect
4. #19151532C2semidirect
5. #19151529C2semidirect
6. #19151518C2semidirect
43. C22×Dic89Q8semidirect
130. C89#2627semidirect

Maximal subgroups

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

subgroupindex
C2×#156679892
C22×#130835532
#191515392
#191515322
#191515292
#191515182
#191514712
#191514592
#191514562
#385×C892
#262789