Finite Group Databasebeta

← #26028312

Subgroups — #26028312

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×#15668007C211yesmaximal, characteristic
3#19151540C212yesmaximal, complemented
4#19151533C212yesmaximal
5#19151528C211yesmaximal, characteristic, complemented
6#19151525C211yesmaximal, characteristic
7#19151524C211yesmaximal, characteristic
8#19151518C212yesmaximal, complemented
9#19151511C212yesmaximal
10#19151465C211yesmaximal, characteristic, complemented
11#398×C89C211yesmaximal, characteristic, complemented
12C22×Dic178C2212yes—
13#15668011—24no—
14#15668007—24no—
15C2×C4⋊Dic89C2211yescharacteristic
16C2×C4⋊Dic89C2211yescharacteristic
17C2×#13083553C2212yes—
18C2×#13083553C2211yescharacteristic
19C2×#13083553C2212yes—
20C2×#13083551C2211yescharacteristic
21C2×#13083551C2211yescharacteristic
22C2×#13083551C2211yescharacteristic
23C2×#13083551C2212yes—
24C2×#13083551C2212yes—
25C2×C4×Dic89C2211yescharacteristic
26#15667919C2212yes—
27#15667919C2211yescharacteristic
28#15667919C2212yes—
29#15667919C2212yes—
30#15667919C2212yes—
31#15667919C2212yes—
32C2×C4⋊C4×C89C2211yescharacteristic
33C2×C4⋊C4×C89C2211yescharacteristic
34C2×C4⋊C4×C89C2212yes—
35#95×C89C2211yescharacteristic
36#95×C89C2212yes—
37C23×D89C2311yescharacteristic
38C2×Dic178—24no—
39C2×Dic178—42no—
40C2×Dic178—42no—
41C22×Dic89C2312yes—
42C22×Dic89C2312yes—
43C22×Dic89C2311yescharacteristic
44C22×Dic89C2311yescharacteristic
45C22×Dic89C2311yescharacteristic
46C4⋊Dic89—22no—
47C4⋊Dic89—22no—
48#13083553—22no—
49#13083553—24no—
50#13083553—42no—
51#13083551—22no—
52#13083551—22no—
53#13083551—22no—
54#13083551—42no—
55#13083551—42no—
56C4×Dic89D414yes—
57C22×C4×C89C2312yes—
58C22×C4×C89C2312yes—
59C22×C4×C89C2311yescharacteristic
60C22×C4×C89C2311yescharacteristic
61C22×C4×C89C2311yescharacteristic
62C4⋊C4×C89—22no—
63C4⋊C4×C89—22no—
64C4⋊C4×C89—42no—
65C23×C89C2411yescharacteristic
66C22×D89—21no—
67C22×D89—21no—
68C22×D89—21no—
69C22×D89—22no—
70C22×D89—22no—
71C2×C4×C89—22no—
72C2×C4×C89—21no—
73C2×C4×C89—21no—
74C2×C4×C89—22no—
75C2×C4×C89—21no—
76C2×C4×C89—22no—
77C2×C4×C89—22no—
78C2×C4×C89—22no—
79C2×C4×C89—21no—
80C2×C4×C89—22no—
81C2×C4×C89—22no—
82C2×C4×C89—22no—
83C2×C4×C89C2×D412yes—
84C2×Dic89—21no—
85C2×Dic89—21no—
86C2×Dic89—22no—
87C2×Dic89—21no—
88C2×Dic89—22no—
89C2×Dic89—22no—
90C2×Dic89—22no—
91C2×Dic89—22no—
92C2×Dic89—22no—
93C2×Dic89—22no—
94C2×Dic89—22no—
95C2×Dic89C2×D412yes—
96C2×Dic89C2×D412yes—
97Dic178—48no—
98C22×C89C22×D411yescharacteristic
99C22×C89H(5,𝔽2)11yescharacteristic
100C22×C89H(5,𝔽2)11yescharacteristic
101C22×C892−1+412yes—
102C22×C892−1+412yes—
103C2×D89—41no—
104C2×D89—41no—
105C2×D89—41no—
106C2×D89—42no—
107C2×D89—42no—
108C4×C89—22no—
109C4×C89—42no—
110C4×C89—41no—
111C4×C89—41no—
112C4×C89—42no—
113Dic89—22no—
114Dic89—22no—
115Dic89—42no—
116Dic89—41no—
117Dic89—42no—
118C2×C89#53511yescharacteristic
119C2×C89#53611yescharacteristic
120C2×C89#53611yescharacteristic
121C2×C89#56312yes—
122C2×C89#56312yes—
123D89—81no—
124#2821—891nomaximal, Sylow
125C89#282111yescharacteristic, complemented, Sylow
126C2×C4⋊Q8—891no—
127C2×#126—891no—
128#400—892no—
129#399—892no—
130#398—891no—
131#398D8911yescharacteristic, complemented
132#397—891no—
133#397—891no—
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⋊C4C2×D8911yescharacteristic
146C2×C4⋊C4C2×D8911yescharacteristic
147C2×C4⋊C4C2×D8912yes—
148C2×C22⋊C4—892no—
149C2×C22⋊C4—892no—
150C2×C22⋊C4—891no—
151C2×C22⋊C4—891no—
152C2×C22⋊C4—891no—
153C2×C42—891no—
154#95—892no—
155#95—891no—
156#95—892no—
157#95—892no—
158#95—892no—
159#95—892no—
160#95C2×D8911yescharacteristic
161#95C2×D8912yes—
162C24—891no—
163C2×Q8—3562no—
164C2×Q8—3562no—
165C2×Q8—1784no—
166C22×C4—891no—
167C22×C4—891no—
168C22×C4—892no—
169C22×C4—891no—
170C22×C4—892no—
171C22×C4C22×D8912yes—
172C22×C4C22×D8911yescharacteristic
173C22×C4C22×D8912yes—
174C22×C4C22×D8911yescharacteristic
175C22×C4C22×D8911yescharacteristic
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—42no—
179C4⋊C4—3562no—
180C4⋊C4—1782no—
181C4⋊C4—1782no—
182C4⋊C4—1784no—
183C4⋊C4—1782no—
184C22⋊C4—3562no—
185C22⋊C4—3562no—
186C22⋊C4—1782no—
187C22⋊C4—1782no—
188C22⋊C4—1782no—
189C42—894no—
190C23—1781no—
191C23—1781no—
192C23—1781no—
193C23—1782no—
194C23—1782no—
195C23C23×D8911yescharacteristic
196Q8—3568no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—892no—
210C2×C4—892no—
211C2×C4—1781no—
212C2×C4—1782no—
213C2×C4—1782no—
214C2×C4—1782no—
215C2×C4—1782no—
216C2×C4—1782no—
217C2×C4—1782no—
218C2×C4—1782no—
219C2×C4—1781no—
220C2×C4—1781no—
221C2×C4—1782no—
222C2×C4D4×D8912yes—
223C22—3561no—
224C22—3561no—
225C22—3561no—
226C22—3562no—
227C22—3562no—
228C22C2×D4×D8911yescharacteristic
229C22#1566805711yescharacteristic
230C22#1566805811yescharacteristic
231C22#1566805912yes—
232C22#1566806012yes—
233C4—22no—
234C4—42no—
235C4—41no—
236C4—41no—
237C4—42no—
238C4—3562no—
239C4—3561no—
240C4—3562no—
241C4—1782no—
242C4—1782no—
243C2—7121no—
244C2#1915216411yescharacteristic
245C2#1915217011yescharacteristic
246C2#1915217311yescharacteristic
247C2#1915234912yes—
248C2#1915235212yes—
subgroupG/H
center, FrattiniC23C23×D89
commutator, socle, cosocleC23×C89C24
Fitting#398×C89C2
prime radicalC23×D89C23
hypercenter#398D89
hypocenterC89#2821
LCSG ▷ C23×C89 ▷ C89
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#15668011C2semidirect
3. #19151540C2semidirect
5. #19151528C2semidirect
8. #19151518C2semidirect
10. #19151465C2semidirect
11. #398×C89C2semidirect
125. C89#2821semidirect
131. #398D89semidirect

Maximal subgroups

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

subgroupindex
C2×#156680112
C2×#156680072
#191515402
#191515332
#191515282
#191515252
#191515242
#191515182
#191515112
#191514652
#398×C892
#282189