Finite Group Databasebeta

← #26028315

Subgroups — #26028315

Order 11392.

430 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 125 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×#15668011C211yesmaximal, characteristic, complemented
2C2×#15668005C211yesmaximal, characteristic
3#19151538C211yesmaximal, characteristic, complemented
4#19151536C211yesmaximal, characteristic, complemented
5#19151534C211yesmaximal, characteristic
6#19151531C212yesmaximal, complemented
7#19151526C212yesmaximal
8#19151518C212yesmaximal, complemented
9#19151510C212yesmaximal
10#19151465C211yesmaximal, characteristic, complemented
11#399×C89C211yesmaximal, characteristic
12C22×Dic178C2212yes—
13#15668011—24no—
14#15668005—24no—
15C2×C4⋊Dic89C2211yescharacteristic
16C2×C4⋊Dic89C2211yescharacteristic
17C2×#13083553C2212yes—
18C2×#13083553C2211yescharacteristic
19C2×#13083553C2211yescharacteristic
20C2×#13083553C2212yes—
21C2×C22⋊Dic89C2211yescharacteristic
22C2×C22⋊Dic89C2211yescharacteristic
23C2×C22⋊Dic89C2212yes—
24C2×C4×Dic89C2211yescharacteristic
25#15667919C2212yes—
26#15667919C2212yes—
27#15667919C2211yescharacteristic
28#15667919C2212yes—
29#15667919C2212yes—
30#15667919C2212yes—
31#15667919C2212yes—
32C2×C4⋊C4×C89C2211yescharacteristic
33C2×C4⋊C4×C89C2212yes—
34C2×C22⋊C4×C89C2211yescharacteristic
35C2×C22⋊C4×C89C2212yes—
36#95×C89C2211yescharacteristic
37C2×Dic178—24no—
38C2×Dic178—42no—
39C2×Dic178—42no—
40C22×Dic89C2312yes—
41C22×Dic89C2312yes—
42C22×Dic89C2311yescharacteristic
43C22×Dic89C2311yescharacteristic
44C22×Dic89C2311yescharacteristic
45C22×Dic89C2311yescharacteristic
46C4⋊Dic89—22no—
47C4⋊Dic89—22no—
48#13083553—22no—
49#13083553—22no—
50#13083553—24no—
51#13083553—42no—
52C22⋊Dic89—22no—
53C22⋊Dic89—22no—
54C22⋊Dic89—42no—
55C4×Dic89D414yes—
56C24×C89C2311yescharacteristic
57C22×C4×C89C2312yes—
58C22×C4×C89C2312yes—
59C22×C4×C89C2311yescharacteristic
60C22×C4×C89C2311yescharacteristic
61C4⋊C4×C89—22no—
62C4⋊C4×C89—42no—
63C22⋊C4×C89—22no—
64C22⋊C4×C89—42no—
65C23×C89—21no—
66C23×C89—22no—
67C23×C89—22no—
68C23×C89—21no—
69C23×C89—21no—
70C23×C89C2411yescharacteristic
71C2×C4×C89—22no—
72C2×C4×C89—21no—
73C2×C4×C89—22no—
74C2×C4×C89—22no—
75C2×C4×C89—22no—
76C2×C4×C89—22no—
77C2×C4×C89—22no—
78C2×C4×C89—22no—
79C2×C4×C89—22no—
80C2×C4×C89C2×D412yes—
81C2×Dic89—22no—
82C2×Dic89—21no—
83C2×Dic89—21no—
84C2×Dic89—21no—
85C2×Dic89—21no—
86C2×Dic89—22no—
87C2×Dic89—22no—
88C2×Dic89—22no—
89C2×Dic89—22no—
90C2×Dic89—22no—
91C2×Dic89—21no—
92C2×Dic89—22no—
93C2×Dic89—21no—
94C2×Dic89—22no—
95C2×Dic89C2×D412yes—
96C2×Dic89C2×D412yes—
97Dic178—48no—
98C22×C89—41no—
99C22×C89—41no—
100C22×C89—41no—
101C22×C89—42no—
102C22×C89—42no—
103C22×C89C22×D411yescharacteristic
104C22×C89H(5,𝔽2)12yes—
105C22×C892−1+412yes—
106C22×C892−1+411yescharacteristic
107C22×C892−1+411yescharacteristic
108C4×C89—22no—
109C4×C89—42no—
110C4×C89—41no—
111C4×C89—42no—
112Dic89—22no—
113Dic89—22no—
114Dic89—42no—
115Dic89—41no—
116Dic89—41no—
117Dic89—42no—
118C2×C89—81no—
119C2×C89#53511yescharacteristic
120C2×C89#53611yescharacteristic
121C2×C89#53611yescharacteristic
122C2×C89#56312yes—
123C2×C89#56312yes—
124#2821—891nomaximal, Sylow
125C89#282111yescharacteristic, complemented, Sylow
126C2×C4⋊Q8—891no—
127C2×#126—891no—
128#400—891no—
129#400—891no—
130#399—891no—
131#399D8911yescharacteristic
132#398—892no—
133#397—892no—
134#395—892no—
135#393—892no—
136#383—891no—
137C22×Q8—892no—
138C4⋊Q8—1784no—
139#126—1784no—
140C2×C4⋊C4—892no—
141C2×C4⋊C4—891no—
142C2×C4⋊C4—891no—
143C2×C4⋊C4—891no—
144C2×C4⋊C4—892no—
145C2×C4⋊C4—891no—
146C2×C4⋊C4C2×D8911yescharacteristic
147C2×C4⋊C4C2×D8912yes—
148C2×C22⋊C4—892no—
149C2×C22⋊C4—891no—
150C2×C22⋊C4—891no—
151C2×C22⋊C4C2×D8911yescharacteristic
152C2×C22⋊C4C2×D8912yes—
153C2×C42—891no—
154#95—892no—
155#95—892no—
156#95—891no—
157#95—892no—
158#95—892no—
159#95—892no—
160#95—892no—
161#95C2×D8911yescharacteristic
162C24C22×D8911yescharacteristic
163C2×Q8—3562no—
164C2×Q8—3562no—
165C2×Q8—1784no—
166C22×C4—891no—
167C22×C4—891no—
168C22×C4—892no—
169C22×C4—891no—
170C22×C4—891no—
171C22×C4—892no—
172C22×C4C22×D8912yes—
173C22×C4C22×D8912yes—
174C22×C4C22×D8911yescharacteristic
175C22×C4C22×D8911yescharacteristic
176C4⋊C4—22no—
177C4⋊C4—42no—
178C4⋊C4—3562no—
179C4⋊C4—1782no—
180C4⋊C4—1782no—
181C4⋊C4—1784no—
182C4⋊C4—1782no—
183C4⋊C4—1782no—
184C22⋊C4—22no—
185C22⋊C4—42no—
186C22⋊C4—3562no—
187C22⋊C4—1782no—
188C22⋊C4—1782no—
189C42—894no—
190C23—21no—
191C23—22no—
192C23—22no—
193C23—21no—
194C23—21no—
195C23C23×D8911yescharacteristic
196Q8—3568no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—892no—
207C2×C4—892no—
208C2×C4—1781no—
209C2×C4—1781no—
210C2×C4—1782no—
211C2×C4—1782no—
212C2×C4—1782no—
213C2×C4—1782no—
214C2×C4—1782no—
215C2×C4—1781no—
216C2×C4—1782no—
217C2×C4—1781no—
218C2×C4—1782no—
219C2×C4—1782no—
220C2×C4—1781no—
221C2×C4—1781no—
222C2×C4D4×D8912yes—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—42no—
227C22—42no—
228C22C2×D4×D8911yescharacteristic
229C22#1566805712yes—
230C22#1566805912yes—
231C22#1566805911yescharacteristic
232C22#1566806011yescharacteristic
233C4—22no—
234C4—42no—
235C4—41no—
236C4—42no—
237C4—3562no—
238C4—3561no—
239C4—3561no—
240C4—3562no—
241C4—1782no—
242C4—1782no—
243C2—81no—
244C2#1915216111yescharacteristic
245C2#1915217011yescharacteristic
246C2#1915217111yescharacteristic
247C2#1915234912yes—
248C2#1915235112yes—
subgroupG/H
center, FrattiniC23C23×D89
commutator, socle, cosocleC23×C89C24
Fitting#399×C89C2
prime radicalC24×C89C23
hypercenter#399D89
hypocenterC89#2821
LCSG ▷ C23×C89 ▷ C89
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#15668011C2semidirect
3. #19151538C2semidirect
4. #19151536C2semidirect
6. #19151531C2semidirect
8. #19151518C2semidirect
10. #19151465C2semidirect
125. C89#2821semidirect

Maximal subgroups

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

subgroupindex
C2×#156680112
C2×#156680052
#191515382
#191515362
#191515342
#191515312
#191515262
#191515182
#191515102
#191514652
#399×C892
#282189