Finite Group Databasebeta

← #26025607

Subgroups — #26025607

Order 11392.

564 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 147 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×#15667977C211yesmaximal, characteristic, complemented
2C2×#15667972C211yesmaximal, characteristic
3#19151533C211yesmaximal, characteristic, complemented
4#19151511C211yesmaximal, characteristic, complemented
5#19151510C211yesmaximal, characteristic, complemented
6#19151498C212yesmaximal
7#19151493C212yesmaximal, complemented
8#19151470C212yesmaximal, complemented
9#19151469C212yesmaximal, complemented
10#19151462C211yesmaximal, characteristic, complemented
11#399×C89C211yesmaximal, characteristic, complemented
12C22×Dic178C2211yescharacteristic
13C22×C22⋊D89C2211yescharacteristic
14C23×Dic89C2211yescharacteristic
15C22×C4×D89C2211yescharacteristic
16#15667977—24no—
17#15667972—24no—
18C2×#13083553C2212yes—
19C2×#13083553C2212yes—
20C2×#13083553C2211yescharacteristic
21C2×#13083553C2212yes—
22C2×C22⋊Dic89C2212yes—
23C2×#13083551C2212yes—
24C2×#13083551C2211yescharacteristic
25C2×#13083551C2212yes—
26C2×C4×Dic89C2212yes—
27C2×C4×Dic89C2211yescharacteristic
28#15667919C2212yes—
29#15667919C2211yescharacteristic
30#15667919C2212yes—
31#15667919C2212yes—
32C2×C4⋊C4×C89C2211yescharacteristic
33C2×C4⋊C4×C89C2212yes—
34C2×C22⋊C4×C89C2211yescharacteristic
35C2×C22⋊C4×C89C2212yes—
36#95×C89C2211yescharacteristic
37C23×D89C2311yescharacteristic
38C2×Dic178—22no—
39C2×Dic178—24no—
40C2×C22⋊D89—22no—
41C2×C22⋊D89—22no—
42C2×C22⋊D89Q814yescomplemented
43C22×Dic89—22no—
44C22×Dic89—42no—
45C22×Dic89C2311yescharacteristic
46C22×Dic89C2312yes—
47C22×Dic89C2311yescharacteristic
48C22×Dic89C2312yes—
49C22×Dic89C2311yescharacteristic
50C2×C4×D89—22no—
51C2×C4×D89—42no—
52#13083553—22no—
53#13083553—24no—
54#13083553—24no—
55#13083553—42no—
56C22⋊Dic89—24no—
57#13083551—22no—
58#13083551—24no—
59#13083551—42no—
60C4×Dic89—24no—
61C4×Dic89D414yes—
62C24×C89C2311yescharacteristic
63C22×C4×C89C2311yescharacteristic
64C22×C4×C89C2312yes—
65C22×C4×C89C2312yes—
66C22×C4×C89C2311yescharacteristic
67C4⋊C4×C89—22no—
68C4⋊C4×C89—42no—
69C22⋊C4×C89—22no—
70C22⋊C4×C89—42no—
71C23×C89—21no—
72C23×C89—22no—
73C23×C89—21no—
74C23×C89—22no—
75C23×C89C2×Q812yes—
76C23×C89C2411yescharacteristic
77C22×D89—21no—
78C22×D89—22no—
79C22×D89—21no—
80C22×D89—22no—
81C22×D89C2×Q812yes—
82C2×C4×C89—22no—
83C2×C4×C89—21no—
84C2×C4×C89—22no—
85C2×C4×C89—22no—
86C2×C4×C89—22no—
87C2×C4×C89—22no—
88C2×C4×C89—22no—
89C2×C4×C89—22no—
90C2×C4×C89—22no—
91C2×C4×C89C2×D412yes—
92C22⋊D89—28no—
93C2×Dic89—22no—
94C2×Dic89—22no—
95C2×Dic89—22no—
96C2×Dic89—21no—
97C2×Dic89—22no—
98C2×Dic89—22no—
99C2×Dic89—22no—
100C2×Dic89—22no—
101C2×Dic89—44no—
102C2×Dic89C2×D412yes—
103C2×Dic89C2×D412yes—
104C2×Dic89C2×Q812yes—
105C2×Dic89#4114yes—
106C2×Dic89#4112yes—
107C4×D89—44no—
108Dic178—28no—
109C22×C89—22no—
110C22×C89—24no—
111C22×C89—41no—
112C22×C89—41no—
113C22×C89—42no—
114C22×C89C22×D411yescharacteristic
115C22×C89C22×Q811yescharacteristic
116C22×C89C2×#4111yescharacteristic
117C22×C89C2×#4112yes—
118C22×C89H(5,𝔽2)12yes—
119C2×D89—22no—
120C2×D89—24no—
121C2×D89—41no—
122C2×D89—41no—
123C2×D89—42no—
124C4×C89—22no—
125C4×C89—41no—
126C4×C89—42no—
127C4×C89—42no—
128Dic89—24no—
129Dic89—22no—
130Dic89—41no—
131Dic89—42no—
132Dic89C22⋊Q814yes—
133C2×C89—42no—
134C2×C89C2×C22⋊Q811yescharacteristic
135C2×C89#52511yescharacteristic
136C2×C89#53511yescharacteristic
137C2×C89#55412yes—
138C2×C89#55912yes—
139D89—42no—
140#2580—891nomaximal, Sylow
141C89#258011yescharacteristic, complemented, Sylow
142C2×C4×Q8—891no—
143C2×C4×D4—891no—
144#399—891no—
145#399D8911yescharacteristic, complemented
146#393—891no—
147#393—891no—
148#390—892no—
149#389—892no—
150#385—892no—
151#385—892no—
152#382—891no—
153C22×Q8—891no—
154C22×D4—891no—
155C23×C4—891no—
156C23×C4—891no—
157C4×Q8—1784no—
158C4×D4—1784no—
159C2×C4⋊C4—892no—
160C2×C4⋊C4—891no—
161C2×C4⋊C4—892no—
162C2×C4⋊C4—892no—
163C2×C4⋊C4C2×D8911yescharacteristic
164C2×C4⋊C4C2×D8912yes—
165C2×C22⋊C4—892no—
166C2×C22⋊C4—892no—
167C2×C22⋊C4—891no—
168C2×C22⋊C4—892no—
169C2×C22⋊C4C2×D8911yescharacteristic
170C2×C22⋊C4C2×D8912yes—
171C2×C42—891no—
172C2×C42—892no—
173#95—892no—
174#95—891no—
175#95—892no—
176#95—892no—
177#95C2×D8911yescharacteristic
178C24—891no—
179C24C22×D8911yescharacteristic
180C2×Q8—1784no—
181C2×Q8—1782no—
182C2×D4—894no—
183C2×D4—1782no—
184C2×D4—1782no—
185C22×C4—891no—
186C22×C4—892no—
187C22×C4—891no—
188C22×C4—892no—
189C22×C4—891no—
190C22×C4—3562no—
191C22×C4—3562no—
192C22×C4—1782no—
193C22×C4—1782no—
194C22×C4C22×D8912yes—
195C22×C4C22×D8911yescharacteristic
196C22×C4C22×D8912yes—
197C22×C4C22×D8911yescharacteristic
198C4⋊C4—22no—
199C4⋊C4—42no—
200C4⋊C4—3562no—
201C4⋊C4—1782no—
202C4⋊C4—1784no—
203C4⋊C4—1784no—
204C22⋊C4—22no—
205C22⋊C4—42no—
206C22⋊C4—3562no—
207C22⋊C4—1784no—
208C22⋊C4—1784no—
209C22⋊C4—1782no—
210C42—894no—
211C42—1784no—
212C23—21no—
213C23—22no—
214C23—21no—
215C23—22no—
216C23—892no—
217C23—1781no—
218C23—1782no—
219C23—1781no—
220C23—1782no—
221C23Q8×D8912yes—
222C23C23×D8911yescharacteristic
223Q8—1788no—
224D4—1788no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—892no—
235C2×C4—892no—
236C2×C4—892no—
237C2×C4—892no—
238C2×C4—894no—
239C2×C4—3564no—
240C2×C4—3564no—
241C2×C4—1781no—
242C2×C4—1782no—
243C2×C4—1782no—
244C2×C4—1782no—
245C2×C4—1782no—
246C2×C4—1782no—
247C2×C4—1782no—
248C2×C4—1782no—
249C2×C4D4×D8912yes—
250C22—24no—
251C22—22no—
252C22—41no—
253C22—42no—
254C22—41no—
255C22—3561no—
256C22—3562no—
257C22—3561no—
258C22—1784no—
259C22—1782no—
260C22C2×D4×D8911yescharacteristic
261C22C2×Q8×D8911yescharacteristic
262C22#41×D8912yes—
263C22#41×D8911yescharacteristic
264C22#1566805712yes—
265C4—22no—
266C4—41no—
267C4—42no—
268C4—42no—
269C4—894no—
270C4—3561no—
271C4—3562no—
272C4—1782no—
273C4—1784no—
274C2—42no—
275C2—3562no—
276C2C22⋊Q8×D8911yescharacteristic
277C2#1915210611yescharacteristic
278C2#1915216111yescharacteristic
279C2#1915230212yes—
280C2#1915233212yes—
subgroupG/H
center, FrattiniC23C23×D89
commutator, socle, cosocleC23×C89C24
Fitting#399×C89C2
prime radicalC22×C22⋊D89C22
hypercenter#399D89
hypocenterC89#2580
LCSG ▷ C23×C89 ▷ C89
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#15667977C2semidirect
3. #19151533C2semidirect
4. #19151511C2semidirect
5. #19151510C2semidirect
7. #19151493C2semidirect
8. #19151470C2semidirect
9. #19151469C2semidirect
10. #19151462C2semidirect
11. #399×C89C2semidirect
42. C2×C22⋊D89Q8semidirect
141. C89#2580semidirect
145. #399D89semidirect

Maximal subgroups

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

subgroupindex
C2×#156679772
C2×#156679722
#191515332
#191515112
#191515102
#191514982
#191514932
#191514702
#191514692
#191514622
#399×C892
#258089