Finite Group Databasebeta

← #37544957

Subgroups — #37544957

Order 23168.

328 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#26041836C211yesmaximal, characteristic, complemented
2#26041689C211yesmaximal, characteristic, complemented
3#26041689C211yesmaximal, characteristic, complemented
4#26041638C211yesmaximal, characteristic, complemented
5#26041638C211yesmaximal, characteristic, complemented
6#26041617C212yesmaximal, complemented
7#26041600C212yesmaximal
8#26041586C212yesmaximal, complemented
9#26041579C212yesmaximal, complemented
10#26041336C211yesmaximal, characteristic, complemented
11#529×C181C211yesmaximal, characteristic
12#19153816C2211yescharacteristic
13#19153816C2211yescharacteristic
14#19153808C2212yes—
15#19153795—21no—
16#19153795—21no—
17C4×Dic362—21no—
18C4×Dic362—21no—
19#19153775C2212yes—
20#19153749C2212yes—
21#19153749C2212yes—
22#19153747C2212yes—
23#19153747C2212yes—
24C4⋊#13089678C2212yes—
25#19153740C2212yes—
26#19153740C2212yes—
27#19153737C2212yes—
28#19153737C2212yes—
29C22⋊#13089678C2212yes—
30#19153727C2211yescharacteristic
31#19153727C2211yescharacteristic
32#126×C181—21no—
33#125×C181C2211yescharacteristic
34#124×C181C2211yescharacteristic
35#123×C181—21no—
36C22⋊Q8×C181C2211yescharacteristic
37#121×C181C2211yescharacteristic
38C4×Q8×C181C2211yescharacteristic
39C4×D4×C181—21no—
40C4×D4×C181C2211yescharacteristic
41#117×C181—21no—
42C2×C42×C181C2211yescharacteristic
43C2×Dic362—21no—
44C2×Dic362—21no—
45C2×#13089678C2312yes—
46C2×#13089678C2312yes—
47C4⋊Dic181—21no—
48C4⋊Dic181—21no—
49C4⋊Dic181C2312yes—
50C4⋊Dic181C2312yes—
51#15668428—22no—
52#15668428—22no—
53C22⋊Dic181—22no—
54C4×Dic181—22no—
55C2×Q8×C181C2311yescharacteristic
56C2×D4×C181—21no—
57C2×D4×C181C2311yescharacteristic
58C22×C4×C181—21no—
59C22×C4×C181D411yescharacteristic
60C22×C4×C181D411yescharacteristic
61C22×C4×C181C2311yescharacteristic
62C4⋊C4×C181—21no—
63C4⋊C4×C181—21no—
64C4⋊C4×C181—21no—
65C4⋊C4×C181—21no—
66C4⋊C4×C181C2311yescharacteristic
67C4⋊C4×C181C2311yescharacteristic
68C22⋊C4×C181—21no—
69C22⋊C4×C181—21no—
70C22⋊C4×C181—21no—
71C22⋊C4×C181—21no—
72C22⋊C4×C181—21no—
73C42×C181—21no—
74C42×C181D411yescharacteristic
75C42×C181D411yescharacteristic
76C42×C181C2311yescharacteristic
77C42×C181C2311yescharacteristic
78Dic362—42no—
79Dic362—42no—
80C2×Dic181—22no—
81C2×Dic181—22no—
82#13089678—22no—
83#13089678—22no—
84C23×C181—21no—
85C23×C181C2×D411yescharacteristic
86Q8×C181—22no—
87D4×C181—22no—
88D4×C181—42no—
89C2×C4×C181—22no—
90C2×C4×C181—21no—
91C2×C4×C181—21no—
92C2×C4×C181—21no—
93C2×C4×C181—41no—
94C2×C4×C181—41no—
95C2×C4×C181—42no—
96C2×C4×C181C2×D411yescharacteristic
97C2×C4×C181C2×D411yescharacteristic
98C2×C4×C181C2×D411yescharacteristic
99C2×C4×C181C2×D411yescharacteristic
100C2×C4×C181C2×D411yescharacteristic
101C2×C4×C181C2411yescharacteristic
102C22×C181—21no—
103C22×C181—21no—
104C22×C181—21no—
105C22×C181—41no—
106C22×C181—41no—
107C22×C181—41no—
108C22×C181C22×D411yescharacteristic
109C4×C181—21no—
110C4×C181—21no—
111C4×C181—21no—
112C4×C181—21no—
113C4×C181—21no—
114C4×C181—41no—
115C4×C181—41no—
116C4×C181—41no—
117C4×C1812−1+412yes—
118Dic181—42no—
119Dic181—42no—
120C2×C181—41no—
121C2×C181—81no—
122C2×C181#53611yescharacteristic
123C2×C181#57511yescharacteristic
124C2×C181#57511yescharacteristic
125C181#324011yescharacteristic, complemented, Sylow
126#3240—1811nomaximal, Sylow
127#533—1811no—
128#529D18111yescharacteristic
129#501—1811no—
130#501—1811no—
131#489—1811no—
132#489—1811no—
133#485—1812no—
134#482—1812no—
135#479—1812no—
136#478—1812no—
137#432—1811no—
138C4⋊Q8—1811no—
139C4⋊Q8—1811no—
140#126—21no—
141#125—1812no—
142#125C2×D18111yescharacteristic
143#124C2×D18111yescharacteristic
144#123—21no—
145C22⋊Q8—3621no—
146C22⋊Q8—3621no—
147C22⋊Q8C2×D18111yescharacteristic
148#121C2×D18111yescharacteristic
149C4×Q8—3621no—
150C4×Q8—3621no—
151C4×Q8C2×D18111yescharacteristic
152C4×D4—21no—
153C4×D4C2×D18111yescharacteristic
154#117—21no—
155#117—1812no—
156C2×C42C2×D18111yescharacteristic
157#107—1812no—
158#107—1812no—
159#106—1812no—
160#106—1812no—
161C4⋊C8—1812no—
162#103—1812no—
163#103—1812no—
164#102—1812no—
165#102—1812no—
166C22⋊C8—1812no—
167#97—1811no—
168#97—1811no—
169C2×Q8—3621no—
170C2×Q8—3621no—
171C2×Q8C22×D18111yescharacteristic
172C2×D4—21no—
173C2×D4C22×D18111yescharacteristic
174C22×C4—21no—
175C22×C4C22⋊D18111yescharacteristic
176C22×C4C22⋊D18111yescharacteristic
177C22×C4C22×D18111yescharacteristic
178C2×C8—1812no—
179C2×C8—1812no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—3622no—
185C4⋊C4—3622no—
186C4⋊C4—3621no—
187C4⋊C4—3621no—
188C4⋊C4—1812no—
189C4⋊C4—1812no—
190C4⋊C4C22×D18111yescharacteristic
191C4⋊C4C22×D18111yescharacteristic
192C22⋊C4—21no—
193C22⋊C4—21no—
194C22⋊C4—21no—
195C22⋊C4—21no—
196C22⋊C4—21no—
197C22⋊C4—3622no—
198C42—21no—
199C42—3622no—
200C42C22⋊D18111yescharacteristic
201C42C22⋊D18111yescharacteristic
202C42C22×D18111yescharacteristic
203C42C22×D18111yescharacteristic
204C23—21no—
205C23C2×C22⋊D18111yescharacteristic
206Q8—22no—
207Q8—7242no—
208Q8—7242no—
209D4—22no—
210D4—42no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—3622no—
219C2×C4—3622no—
220C2×C4C2×C22⋊D18111yescharacteristic
221C2×C4C2×C22⋊D18111yescharacteristic
222C2×C4C2×C22⋊D18111yescharacteristic
223C2×C4C2×C22⋊D18111yescharacteristic
224C2×C4C2×C22⋊D18111yescharacteristic
225C2×C4C23×D18111yescharacteristic
226C8—3622no—
227C8—3622no—
228C22—21no—
229C22—21no—
230C22—21no—
231C22—41no—
232C22—41no—
233C22—41no—
234C22C22×C22⋊D18111yescharacteristic
235C4—21no—
236C4—21no—
237C4—21no—
238C4—21no—
239C4—21no—
240C4—41no—
241C4—41no—
242C4—41no—
243C4—7242no—
244C4—7242no—
245C4#1915386312yes—
246C2—41no—
247C2—81no—
248C2#2604185011yescharacteristic
249C2#2604207711yescharacteristic
250C2#2604207711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#529×C181C2
socleC22×C181C22×D4
prime radical#121×C181C22
hypercenter#529D181
hypocenterC181#3240
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. #26041836C2semidirect
2. #26041689C2semidirect
3. #26041689C2semidirect
4. #26041638C2semidirect
5. #26041638C2semidirect
6. #26041617C2semidirect
8. #26041586C2semidirect
9. #26041579C2semidirect
10. #26041336C2semidirect
125. C181#3240semidirect

Maximal subgroups

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

subgroupindex
#260418362
#260416892
#260416382
#260416172
#260416002
#260415862
#260415792
#260413362
#529×C1812
#3240181