Finite Group Databasebeta

← #26026409

Subgroups — #26026409

Order 11392.

498 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 129 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×#15667997C211yesmaximal, characteristic
2C2×#15667992C211yesmaximal, characteristic, complemented
3#19151533C211yesmaximal, characteristic, complemented
4#19151527C212yesmaximal, complemented
5#19151518C211yesmaximal, characteristic, complemented
6#19151516C212yesmaximal
7#19151511C211yesmaximal, characteristic, complemented
8#19151495C212yesmaximal, complemented
9#19151490C212yesmaximal, complemented
10#19151445C211yesmaximal, characteristic, complemented
11#400×C89C211yesmaximal, characteristic, complemented
12C22×Dic178C2211yescharacteristic
13C22×D356C2211yescharacteristic
14C22×C4×D89C2211yescharacteristic
15#15667997—24no—
16#15667992—24no—
17C2×C4⋊Dic89C2212yes—
18C2×#13083553C2211yescharacteristic
19C2×#13083553C2212yes—
20C2×#13083553C2212yes—
21C2×#13083551C2212yes—
22C2×#13083551C2211yescharacteristic
23C2×#13083551C2212yes—
24C2×#13083551C2211yescharacteristic
25C2×#13083551C2212yes—
26C2×#13083551C2211yescharacteristic
27C2×#13083551C2212yes—
28C2×C4×Dic89C2212yes—
29#15667919C2211yescharacteristic
30#15667919C2212yes—
31#15667919C2212yes—
32C2×C4⋊C4×C89C2211yescharacteristic
33C2×C4⋊C4×C89C2211yescharacteristic
34#95×C89C2211yescharacteristic
35#95×C89C2212yes—
36#95×C89C2212yes—
37C23×D89C2311yescharacteristic
38C23×D89C2311yescharacteristic
39C2×Dic178—22no—
40C2×Dic178—42no—
41C2×D356—22no—
42C2×D356—42no—
43C22×Dic89C2312yes—
44C22×Dic89C2311yescharacteristic
45C22×Dic89C2311yescharacteristic
46C22×Dic89C2312yes—
47C2×C4×D89—42no—
48C2×C4×D89D414yes—
49C4⋊Dic89—24no—
50#13083553—22no—
51#13083553—24no—
52#13083553—42no—
53#13083551—24no—
54#13083551—22no—
55#13083551—22no—
56#13083551—22no—
57#13083551—24no—
58#13083551—24no—
59#13083551—42no—
60C4×Dic89—42no—
61C22×C4×C89C2311yescharacteristic
62C22×C4×C89C2312yes—
63C22×C4×C89C2311yescharacteristic
64C22×C4×C89C2312yes—
65C22×C4×C89C2311yescharacteristic
66C4⋊C4×C89—22no—
67C4⋊C4×C89—22no—
68C23×C89C2411yescharacteristic
69C22×D89—22no—
70C22×D89—22no—
71C22×D89—21no—
72C22×D89—21no—
73C22×D89—21no—
74C22×D89—22no—
75C22×D89—22no—
76C22×D89—21no—
77C22×D89—21no—
78C22×D89C2×D412yes—
79C2×C4×C89—22no—
80C2×C4×C89—21no—
81C2×C4×C89—22no—
82C2×C4×C89—21no—
83C2×C4×C89—22no—
84C2×C4×C89—22no—
85C2×C4×C89—22no—
86C2×C4×C89—22no—
87C2×C4×C89—22no—
88C2×C4×C89—21no—
89C2×C4×C89—22no—
90C2×C4×C89—21no—
91C2×C4×C89C2×D412yes—
92C2×Dic89—22no—
93C2×Dic89—22no—
94C2×Dic89—22no—
95C2×Dic89—21no—
96C2×Dic89—22no—
97C2×Dic89—22no—
98C2×Dic89—22no—
99C2×Dic89—22no—
100C2×Dic89C2×D412yes—
101C2×Dic89#4114yes—
102D356—44no—
103C4×D89—44no—
104Dic178—44no—
105C22×C89C22×D411yescharacteristic
106C22×C89C2×#4112yes—
107C22×C89H(5,𝔽2)12yes—
108C22×C892−1+411yescharacteristic
109C22×C892−1+411yescharacteristic
110C2×D89—22no—
111C2×D89—22no—
112C2×D89—22no—
113C2×D89—42no—
114C2×D89—42no—
115C2×D89—41no—
116C2×D89—41no—
117C2×D89—41no—
118C2×D89—42no—
119C2×D89—42no—
120C4×C89—22no—
121C4×C89—42no—
122C4×C89—41no—
123C4×C89—41no—
124C4×C89—42no—
125Dic89—22no—
126Dic89—24no—
127Dic89—41no—
128Dic89—42no—
129C2×C89#53211yescharacteristic
130C2×C89#53411yescharacteristic
131C2×C89#53611yescharacteristic
132C2×C89#53912yes—
133C2×C89#54212yes—
134D89—42no—
135D89—81no—
136#2663—891nomaximal, Sylow
137C89#266311yescharacteristic, complemented, Sylow
138C2×#123—891no—
139C2×C22⋊Q8—891no—
140#400D8911yescharacteristic, complemented
141#399—891no—
142#397—892no—
143#395—891no—
144#394—892no—
145#393—891no—
146#389—892no—
147#388—892no—
148C4×C22⋊C4—891no—
149C22×Q8—891no—
150C22×D4—891no—
151C23×C4—891no—
152#123—1784no—
153C22⋊Q8—1784no—
154C2×C4⋊C4—892no—
155C2×C4⋊C4—892no—
156C2×C4⋊C4—891no—
157C2×C4⋊C4—892no—
158C2×C4⋊C4C2×D8911yescharacteristic
159C2×C4⋊C4C2×D8911yescharacteristic
160C2×C22⋊C4—892no—
161C2×C22⋊C4—892no—
162C2×C22⋊C4—891no—
163C2×C22⋊C4—891no—
164C2×C22⋊C4—891no—
165C2×C22⋊C4—892no—
166C2×C22⋊C4—892no—
167C2×C42—892no—
168#95—891no—
169#95—892no—
170#95—892no—
171#95C2×D8911yescharacteristic
172#95C2×D8912yes—
173#95C2×D8912yes—
174C24—891no—
175C24—891no—
176C2×Q8—3562no—
177C2×Q8—1782no—
178C2×D4—3562no—
179C2×D4—1782no—
180C22×C4—892no—
181C22×C4—891no—
182C22×C4—894no—
183C22×C4—891no—
184C22×C4—892no—
185C22×C4—3562no—
186C22×C4C22×D8911yescharacteristic
187C22×C4C22×D8912yes—
188C22×C4C22×D8912yes—
189C22×C4C22×D8911yescharacteristic
190C22×C4C22×D8911yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—3562no—
194C4⋊C4—1784no—
195C4⋊C4—1782no—
196C4⋊C4—1784no—
197C22⋊C4—3562no—
198C22⋊C4—1782no—
199C22⋊C4—1784no—
200C22⋊C4—1784no—
201C22⋊C4—1784no—
202C22⋊C4—1782no—
203C22⋊C4—1782no—
204C42—3562no—
205C23—892no—
206C23—1781no—
207C23—1782no—
208C23—1782no—
209C23—1781no—
210C23—1781no—
211C23—1782no—
212C23—1782no—
213C23—1781no—
214C23—1781no—
215C23C23×D8911yescharacteristic
216Q8—3564no—
217D4—3564no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—892no—
231C2×C4—894no—
232C2×C4—3564no—
233C2×C4—1781no—
234C2×C4—1782no—
235C2×C4—1782no—
236C2×C4—1782no—
237C2×C4—1782no—
238C2×C4—1782no—
239C2×C4—1782no—
240C2×C4—1782no—
241C2×C4D4×D8912yes—
242C22—3562no—
243C22—3562no—
244C22—3561no—
245C22—3561no—
246C22—3561no—
247C22—3562no—
248C22—3562no—
249C22—1782no—
250C22—1782no—
251C22—1782no—
252C22C2×D4×D8911yescharacteristic
253C22#41×D8912yes—
254C22#1566805812yes—
255C22#1566805911yescharacteristic
256C22#1566806011yescharacteristic
257C4—22no—
258C4—42no—
259C4—41no—
260C4—41no—
261C4—42no—
262C4—3561no—
263C4—3562no—
264C4—1782no—
265C4—1784no—
266C2—3562no—
267C2—7121no—
268C2#1915214711yescharacteristic
269C2#1915215911yescharacteristic
270C2#1915217111yescharacteristic
271C2#1915219612yes—
272C2#1915221412yes—
subgroupG/H
center, FrattiniC23C23×D89
commutator, socle, cosocleC23×C89C24
Fitting#400×C89C2
prime radicalC22×D356C22
hypercenter#400D89
hypocenterC89#2663
LCSG ▷ C23×C89 ▷ C89
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
2. C2×#15667992C2semidirect
3. #19151533C2semidirect
4. #19151527C2semidirect
5. #19151518C2semidirect
7. #19151511C2semidirect
8. #19151495C2semidirect
9. #19151490C2semidirect
10. #19151445C2semidirect
11. #400×C89C2semidirect
137. C89#2663semidirect
140. #400D89semidirect

Maximal subgroups

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

subgroupindex
C2×#156679972
C2×#156679922
#191515332
#191515272
#191515182
#191515162
#191515112
#191514952
#191514902
#191514452
#400×C892
#266389