Finite Group Databasebeta

← #26025736

Subgroups — #26025736

Order 11392.

520 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 137 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×#15667998C211yesmaximal, characteristic
2C2×#15667977C211yesmaximal, characteristic, complemented
3#19151524C211yesmaximal, characteristic, complemented
4#19151511C211yesmaximal, characteristic, complemented
5#19151510C211yesmaximal, characteristic, complemented
6#19151502C212yesmaximal, complemented
7#19151498C212yesmaximal
8#19151487C212yesmaximal, complemented
9#19151486C212yesmaximal, complemented
10#19151449C211yesmaximal, characteristic, complemented
11#397×C89C211yesmaximal, characteristic, complemented
12C22×Dic178C2211yescharacteristic
13C22×Dic178C2211yescharacteristic
14C22×C22⋊D89C2211yescharacteristic
15#15667998—24no—
16#15667977—24no—
17C2×#13083553C2212yes—
18C2×#13083553C2212yes—
19C2×C22⋊Dic89C2211yescharacteristic
20C2×C22⋊Dic89C2212yes—
21C2×#13083551C2211yescharacteristic
22C2×#13083551C2212yes—
23C2×#13083551C2211yescharacteristic
24C2×#13083551C2212yes—
25C2×C4×Dic89C2212yes—
26C2×C4×Dic89C2212yes—
27C2×C4×Dic89C2211yescharacteristic
28#15667919C2211yescharacteristic
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×Dic178—22no—
39C2×Dic178—24no—
40C2×Dic178—22no—
41C2×Dic178—24no—
42C2×C22⋊D89—22no—
43C2×C22⋊D89—24no—
44C22×Dic89C2311yescharacteristic
45C22×Dic89C2312yes—
46C22×Dic89C2311yescharacteristic
47C22×Dic89C2312yes—
48C22×Dic89C2311yescharacteristic
49#13083553—24no—
50#13083553—24no—
51C22⋊Dic89—22no—
52C22⋊Dic89—24no—
53#13083551—22no—
54#13083551—22no—
55#13083551—24no—
56#13083551—42no—
57C4×Dic89—24no—
58C4×Dic89—42no—
59C4×Dic89D414yes—
60C24×C89C2311yescharacteristic
61C22×C4×C89C2311yescharacteristic
62C22×C4×C89C2312yes—
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—22no—
71C23×C89—21no—
72C23×C89—21no—
73C23×C89C2411yescharacteristic
74C22×D89—21no—
75C22×D89—22no—
76C22×D89—22no—
77C22×D89—21no—
78C22×D89—21no—
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—22no—
92C2×Dic89—21no—
93C2×Dic89—21no—
94C2×Dic89—21no—
95C2×Dic89—22no—
96C2×Dic89—22no—
97C2×Dic89—22no—
98C2×Dic89—22no—
99C2×Dic89C2×D412yes—
100C2×Dic89C2×D412yes—
101C2×Dic89#4114yes—
102C2×Dic89#4114yes—
103Dic178—28no—
104Dic178—44no—
105C22×C89—41no—
106C22×C89—41no—
107C22×C89—41no—
108C22×C89—42no—
109C22×C89—42no—
110C22×C89C22×D411yescharacteristic
111C22×C89C2×#4112yes—
112C22×C89C2×#4112yes—
113C22×C89H(5,𝔽2)11yescharacteristic
114C22×C892−1+411yescharacteristic
115C2×D89—41no—
116C2×D89—41no—
117C2×D89—41no—
118C2×D89—42no—
119C2×D89—42no—
120C4×C89—22no—
121C4×C89—41no—
122C4×C89—42no—
123C4×C89—42no—
124Dic89—24no—
125Dic89—22no—
126Dic89—41no—
127Dic89—42no—
128Dic89#12414yes—
129C2×C89—81no—
130C2×C89#52511yescharacteristic
131C2×C89C2×#12411yescharacteristic
132C2×C89#53711yescharacteristic
133C2×C89#55612yes—
134C2×C89#55912yes—
135D89—81no—
136#2592—891nomaximal, Sylow
137C89#259211yescharacteristic, complemented, Sylow
138C2×#124—891no—
139C2×C4×Q8—891no—
140#397—891no—
141#397D8911yescharacteristic, complemented
142#393—891no—
143#393—891no—
144#391—892no—
145#390—892no—
146#388—892no—
147#388—892no—
148C4×C4⋊C4—891no—
149C22×Q8—891no—
150C22×Q8—891no—
151C22×D4—891no—
152#124—1784no—
153C4×Q8—1784no—
154C2×C4⋊C4—892no—
155C2×C4⋊C4—892no—
156C2×C4⋊C4C2×D8911yescharacteristic
157C2×C22⋊C4—892no—
158C2×C22⋊C4—891no—
159C2×C22⋊C4—892no—
160C2×C22⋊C4—891no—
161C2×C22⋊C4—891no—
162C2×C22⋊C4—892no—
163C2×C22⋊C4C2×D8911yescharacteristic
164C2×C22⋊C4C2×D8912yes—
165C2×C42—891no—
166C2×C42—892no—
167C2×C42—892no—
168#95—891no—
169#95—892no—
170#95—892no—
171#95—892no—
172#95C2×D8911yescharacteristic
173#95C2×D8912yes—
174C24—891no—
175C24C22×D8911yescharacteristic
176C2×Q8—1782no—
177C2×Q8—1784no—
178C2×Q8—1784no—
179C2×Q8—1782no—
180C2×D4—1782no—
181C2×D4—1784no—
182C22×C4—891no—
183C22×C4—892no—
184C22×C4—891no—
185C22×C4—891no—
186C22×C4—892no—
187C22×C4C22×D8911yescharacteristic
188C22×C4C22×D8912yes—
189C22×C4C22×D8912yes—
190C22×C4C22×D8911yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—1784no—
193C4⋊C4—1784no—
194C22⋊C4—22no—
195C22⋊C4—42no—
196C22⋊C4—3562no—
197C22⋊C4—1782no—
198C22⋊C4—1784no—
199C22⋊C4—1782no—
200C22⋊C4—1784no—
201C22⋊C4—1782no—
202C42—894no—
203C42—3562no—
204C42—1784no—
205C23—21no—
206C23—22no—
207C23—22no—
208C23—21no—
209C23—21no—
210C23—1781no—
211C23—1782no—
212C23—1782no—
213C23—1781no—
214C23—1781no—
215C23C23×D8911yescharacteristic
216Q8—3564no—
217Q8—1788no—
218D4—3564no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—892no—
229C2×C4—894no—
230C2×C4—892no—
231C2×C4—894no—
232C2×C4—1781no—
233C2×C4—1782no—
234C2×C4—1782no—
235C2×C4—1782no—
236C2×C4—1782no—
237C2×C4—1782no—
238C2×C4—1782no—
239C2×C4—1781no—
240C2×C4—1781no—
241C2×C4D4×D8912yes—
242C22—41no—
243C22—42no—
244C22—42no—
245C22—41no—
246C22—41no—
247C22—3561no—
248C22—3562no—
249C22—3562no—
250C22—3561no—
251C22—3561no—
252C22C2×D4×D8911yescharacteristic
253C22#41×D8912yes—
254C22#41×D8912yes—
255C22#1566805711yescharacteristic
256C22#1566805911yescharacteristic
257C4—22no—
258C4—42no—
259C4—42no—
260C4—41no—
261C4—894no—
262C4—3562no—
263C4—3561no—
264C4—1782no—
265C4—1784no—
266C2—81no—
267C2—7121no—
268C2#1915210611yescharacteristic
269C2#124×D8911yescharacteristic
270C2#1915217711yescharacteristic
271C2#1915231512yes—
272C2#1915233212yes—
subgroupG/H
center, FrattiniC23C23×D89
commutator, socle, cosocleC23×C89C24
Fitting#397×C89C2
prime radicalC22×C22⋊D89C22
hypercenter#397D89
hypocenterC89#2592
LCSG ▷ C23×C89 ▷ C89
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
2. C2×#15667977C2semidirect
3. #19151524C2semidirect
4. #19151511C2semidirect
5. #19151510C2semidirect
6. #19151502C2semidirect
8. #19151487C2semidirect
9. #19151486C2semidirect
10. #19151449C2semidirect
11. #397×C89C2semidirect
137. C89#2592semidirect
141. #397D89semidirect

Maximal subgroups

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

subgroupindex
C2×#156679982
C2×#156679772
#191515242
#191515112
#191515102
#191515022
#191514982
#191514872
#191514862
#191514492
#397×C892
#259289