Finite Group Databasebeta

← #26026031

Subgroups — #26026031

Order 11392.

526 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 133 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×#15667994C211yesmaximal, characteristic
2C2×C4⋊C4×D89C211yesmaximal, characteristic, complemented
3#19151540C212yesmaximal, complemented
4#19151534C212yesmaximal, complemented
5#19151530C211yesmaximal, characteristic, complemented
6#19151529C211yesmaximal, characteristic, complemented
7#19151523C211yesmaximal, characteristic, complemented
8#19151474C212yesmaximal
9#19151472C212yesmaximal, complemented
10#19151460C211yesmaximal, characteristic, complemented
11#397×C89C211yesmaximal, characteristic, complemented
12C22×C22⋊D89C2211yescharacteristic
13C22×C4×D89C2211yescharacteristic
14C22×C4×D89C2212yes—
15#15667994—24no—
16C2×C4⋊Dic89C2211yescharacteristic
17C2×C4⋊Dic89C2211yescharacteristic
18C2×C4⋊Dic89C2212yes—
19C2×#13083553C2211yescharacteristic
20C2×#13083553C2212yes—
21C2×#13083553C2212yes—
22C4⋊C4×D89—24no—
23C2×C22⋊Dic89C2211yescharacteristic
24C2×C22⋊Dic89C2212yes—
25C2×#13083551C2212yes—
26C2×#13083551C2211yescharacteristic
27#15667919C2211yescharacteristic
28#15667919C2212yes—
29#15667919C2212yes—
30#15667919C2212yes—
31#15667919C2212yes—
32C2×C4⋊C4×C89C2211yescharacteristic
33C2×C22⋊C4×C89C2211yescharacteristic
34C2×C22⋊C4×C89C2212yes—
35#95×C89C2211yescharacteristic
36#95×C89C2212yes—
37C23×D89C2311yescharacteristic
38C2×C22⋊D89—22no—
39C2×C22⋊D89—24no—
40C22×Dic89C2312yes—
41C22×Dic89C2311yescharacteristic
42C22×Dic89C2312yes—
43C22×Dic89C2311yescharacteristic
44C22×Dic89C2311yescharacteristic
45C2×C4×D89—24no—
46C2×C4×D89—24no—
47C2×C4×D89—42no—
48C2×C4×D89—42no—
49C2×C4×D89D414yes—
50C4⋊Dic89—22no—
51C4⋊Dic89—22no—
52C4⋊Dic89—42no—
53#13083553—22no—
54#13083553—24no—
55#13083553—42no—
56C22⋊Dic89—22no—
57C22⋊Dic89—42no—
58#13083551—22no—
59#13083551—24no—
60C24×C89C2311yescharacteristic
61C22×C4×C89C2312yes—
62C22×C4×C89C2311yescharacteristic
63C22×C4×C89C2312yes—
64C22×C4×C89C2311yescharacteristic
65C4⋊C4×C89—22no—
66C22⋊C4×C89—22no—
67C22⋊C4×C89—42no—
68C23×C89—21no—
69C23×C89—22no—
70C23×C89—21no—
71C23×C89—21no—
72C23×C89—22no—
73C23×C89C2411yescharacteristic
74C22×D89—21no—
75C22×D89—21no—
76C22×D89—22no—
77C22×D89C2×D412yes—
78C22×D89#4114yes—
79C2×C4×C89—22no—
80C2×C4×C89—21no—
81C2×C4×C89—22no—
82C2×C4×C89—22no—
83C2×C4×C89—22no—
84C2×C4×C89—22no—
85C2×C4×C89—22no—
86C2×C4×C89—22no—
87C2×C4×C89—22no—
88C2×C4×C89C2×D412yes—
89C22⋊D89—44no—
90C2×Dic89—22no—
91C2×Dic89—21no—
92C2×Dic89—22no—
93C2×Dic89—21no—
94C2×Dic89—22no—
95C2×Dic89—22no—
96C2×Dic89—22no—
97C2×Dic89—22no—
98C2×Dic89—21no—
99C2×Dic89—21no—
100C2×Dic89—22no—
101C2×Dic89—22no—
102C2×Dic89C2×D412yes—
103C4×D89—28no—
104C4×D89—48no—
105C22×C89—41no—
106C22×C89—41no—
107C22×C89—41no—
108C22×C89—42no—
109C22×C89—42no—
110C22×C89C22×D411yescharacteristic
111C22×C89C2×#4112yes—
112C22×C89H(5,𝔽2)11yescharacteristic
113C22×C89H(5,𝔽2)11yescharacteristic
114C22×C892−1+412yes—
115C2×D89—22no—
116C2×D89—22no—
117C2×D89—24no—
118C2×D89—24no—
119C2×D89#12314yes—
120C4×C89—22no—
121C4×C89—42no—
122C4×C89—41no—
123C4×C89—42no—
124Dic89—22no—
125Dic89—41no—
126Dic89—42no—
127Dic89—41no—
128Dic89—42no—
129C2×C89—81no—
130C2×C89C2×#12311yescharacteristic
131C2×C89#53511yescharacteristic
132C2×C89#53611yescharacteristic
133C2×C89#53912yes—
134C2×C89#53912yes—
135D89—24no—
136#2623—891nomaximal, Sylow
137C89#262311yescharacteristic, complemented, Sylow
138C2×#123—891no—
139C22×C4⋊C4—891no—
140#400—892no—
141#399—892no—
142#398—891no—
143#398—891no—
144#397—891no—
145#397D8911yescharacteristic, complemented
146#386—892no—
147#385—892no—
148#381—891no—
149C22×D4—891no—
150C23×C4—891no—
151C23×C4—892no—
152#123—1784no—
153C2×C4⋊C4—892no—
154C2×C4⋊C4—892no—
155C2×C4⋊C4—891no—
156C2×C4⋊C4—892no—
157C2×C4⋊C4—891no—
158C2×C4⋊C4—891no—
159C2×C4⋊C4—1784no—
160C2×C4⋊C4C2×D8911yescharacteristic
161C2×C22⋊C4—892no—
162C2×C22⋊C4—891no—
163C2×C22⋊C4—891no—
164C2×C22⋊C4—892no—
165C2×C22⋊C4C2×D8911yescharacteristic
166C2×C22⋊C4C2×D8912yes—
167#95—891no—
168#95—892no—
169#95—892no—
170#95—892no—
171#95—892no—
172#95C2×D8911yescharacteristic
173#95C2×D8912yes—
174C24—891no—
175C24C22×D8911yescharacteristic
176C2×D4—1782no—
177C2×D4—1784no—
178C22×C4—891no—
179C22×C4—894no—
180C22×C4—892no—
181C22×C4—891no—
182C22×C4—892no—
183C22×C4—891no—
184C22×C4—3562no—
185C22×C4—3562no—
186C22×C4—1784no—
187C22×C4—1784no—
188C22×C4C22×D8911yescharacteristic
189C22×C4C22×D8912yes—
190C22×C4C22×D8912yes—
191C22×C4C22×D8911yescharacteristic
192C4⋊C4—22no—
193C4⋊C4—3562no—
194C4⋊C4—3562no—
195C4⋊C4—1784no—
196C4⋊C4—1782no—
197C4⋊C4—1782no—
198C4⋊C4—1782no—
199C22⋊C4—22no—
200C22⋊C4—42no—
201C22⋊C4—3562no—
202C22⋊C4—1782no—
203C22⋊C4—1784no—
204C22⋊C4—1782no—
205C23—21no—
206C23—22no—
207C23—21no—
208C23—21no—
209C23—22no—
210C23—892no—
211C23—894no—
212C23—1781no—
213C23—1781no—
214C23—1782no—
215C23C23×D8911yescharacteristic
216D4—3564no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—892no—
227C2×C4—3568no—
228C2×C4—1788no—
229C2×C4—1781no—
230C2×C4—1782no—
231C2×C4—1781no—
232C2×C4—1782no—
233C2×C4—1782no—
234C2×C4—1781no—
235C2×C4—1782no—
236C2×C4—1782no—
237C2×C4—1782no—
238C2×C4—1781no—
239C2×C4—1782no—
240C2×C4—1782no—
241C2×C4D4×D8912yes—
242C22—41no—
243C22—41no—
244C22—42no—
245C22—42no—
246C22—41no—
247C22—894no—
248C22—1782no—
249C22—1782no—
250C22—1784no—
251C22—1784no—
252C22C2×D4×D8911yescharacteristic
253C22#41×D8912yes—
254C22#1566805711yescharacteristic
255C22#1566805811yescharacteristic
256C22#1566805912yes—
257C4—22no—
258C4—42no—
259C4—41no—
260C4—42no—
261C4—3562no—
262C4—3561no—
263C4—3562no—
264C4—3561no—
265C4—1782no—
266C2—81no—
267C2—1784no—
268C2#123×D8911yescharacteristic
269C2#1915216411yescharacteristic
270C2#1915216811yescharacteristic
271C2#1915218912yes—
272C2#1915219112yes—
subgroupG/H
center, FrattiniC23C23×D89
commutator, socle, cosocleC23×C89C24
Fitting#397×C89C2
prime radicalC22×C22⋊D89C22
hypercenter#397D89
hypocenterC89#2623
LCSG ▷ C23×C89 ▷ C89
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D89C2semidirect
3. #19151540C2semidirect
4. #19151534C2semidirect
5. #19151530C2semidirect
6. #19151529C2semidirect
7. #19151523C2semidirect
9. #19151472C2semidirect
10. #19151460C2semidirect
11. #397×C89C2semidirect
137. C89#2623semidirect
145. #397D89semidirect

Maximal subgroups

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

subgroupindex
C2×#156679942
C2×C4⋊C4×D892
#191515402
#191515342
#191515302
#191515292
#191515232
#191514742
#191514722
#191514602
#397×C892
#262389