Finite Group Databasebeta

← #37544104

Subgroups — #37544104

Order 23168.

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

normal subgroups only

This group has 400 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#26041835C211yesmaximal, characteristic, complemented
2#124×D181C211yesmaximal, characteristic
3#26041640C212yesmaximal, complemented
4#26041637C212yesmaximal, complemented
5#26041636C211yesmaximal, characteristic, complemented
6#26041634C211yesmaximal, characteristic, complemented
7#26041633C211yesmaximal, characteristic, complemented
8#26041286C212yesmaximal, complemented
9#26041282C212yesmaximal
10C4×#15668432C211yesmaximal, characteristic, complemented
11#488×C181C211yesmaximal, characteristic, complemented
12C2×Q8×D181C2211yescharacteristic
13C2×D4×D181C2211yescharacteristic
14C2×#15668432C2212yes—
15#19153815C2212yescomplemented
16#19153808C2211yescharacteristic, complemented
17#19153805C2211yescharacteristic
18#19153804C2211yescharacteristic
19#19153803C2211yescharacteristic
20#19153802—22no—
21#19153794—22no—
22#19153781—22no—
23#19153774C2212yes—
24C22⋊C4×D181—22no—
25C42×D181C2211yescharacteristic
26#19153741C2212yescomplemented
27#19153740C2212yescomplemented
28#19153739C2212yescomplemented
29#19153738C2212yes—
30#19153737C2212yes—
31#19153736C2212yes—
32#19153726C2212yescomplemented
33C4×#13089678C2211yescharacteristic, complemented
34#125×C181C2211yescharacteristic, complemented
35#124×C181C2211yescharacteristic
36#103×C181C2212yescomplemented
37#102×C181C2212yes—
38C4×C8×C181C2211yescharacteristic, complemented
39C23×D181—21no—
40C2×Dic362—22no—
41C2×Dic362—21no—
42C2×Dic362C2311yescharacteristic
43Q8×D181—24no—
44C2×D724C2311yescharacteristic
45C2×C22⋊D181—21no—
46D4×D181—24no—
47C2×C4×D181—21no—
48C2×C4×D181D412yes—
49C2×C4×D181C2311yescharacteristic
50#15668432—24no—
51C2×#13089678C2312yes—
52C4⋊Dic181C2312yes—
53#15668428—22no—
54#15668428—22no—
55C22⋊Dic181—22no—
56#15668426—22no—
57#15668426—22no—
58#15668426—22no—
59C4×Dic181—22no—
60C4×Dic181D412yes—
61C4×Dic181C2311yescharacteristic
62C2×Q8×C181C2311yescharacteristic
63C2×D4×C181C2311yescharacteristic
64C2×C8×C181C2312yes—
65C4⋊C4×C181—22no—
66C4⋊C4×C181C2312yes—
67C22⋊C4×C181—22no—
68C42×C181C2311yescharacteristic
69C22×D181—21no—
70C22×D181—44no—
71C22×D181—42no—
72C22×D181C2×D411yescharacteristic
73Dic362—22no—
74Dic362—44no—
75Dic362—42no—
76D724—22no—
77C22⋊D181—42no—
78C2×Dic181—22no—
79C2×Dic181—21no—
80C2×Dic181C2×D412yes—
81C2×Dic181C2×D411yescharacteristic
82C4×D181—42no—
83C4×D181—42no—
84C4×D181#4114yes—
85#13089678—22no—
86C23×C181—21no—
87Q8×C181—22no—
88D4×C181—22no—
89C2×C4×C181—21no—
90C2×C4×C181—22no—
91C2×C4×C181C2×D412yes—
92C2×C4×C181C2411yescharacteristic
93C8×C181—22no—
94C22×C181—41no—
95C22×C181—42no—
96C22×C181C22×D411yescharacteristic
97C2×D181—22no—
98C2×D181—41no—
99C2×D181—42no—
100C2×D181—84no—
101C2×D181#12412yes—
102C4×C181—22no—
103C4×C181—41no—
104C4×C181—42no—
105C4×C181C2×#4112yes—
106Dic181—22no—
107Dic181—41no—
108Dic181—42no—
109Dic181#12412yes—
110C2×C181—81no—
111C2×C181C2×#12411yescharacteristic
112C2×C181#57512yes—
113D181—22no—
114D181—81no—
115C181#314011yescharacteristic, complemented, Sylow
116#3140—1811nomaximal, Sylow
117#533—1811no—
118C2×#124—1811no—
119#489—1812no—
120#489—1812no—
121#488—1811no—
122#488—1811no—
123#488—1811no—
124#488D18111yescharacteristic, complemented
125#419—1812no—
126#418—1812no—
127C4×M16—1811no—
128C22×Q8—1811no—
129C22×D4—1811no—
130C2×M16—1812no—
131C4⋊Q8—1812no—
132#125—1811no—
133#125C2×D18111yescharacteristic, complemented
134#124—3622no—
135#124—1811no—
136#124—1811no—
137#124—1811no—
138#124C2×D18111yescharacteristic
139C22⋊Q8—3622no—
140C4×Q8—3622no—
141#117—1812no—
142C2×C22⋊C4—3622no—
143C2×C42—1811no—
144#103—1812no—
145#103—1812no—
146#103—1812no—
147#103C2×D18112yescomplemented
148#102—1812no—
149#102—1812no—
150#102—1812no—
151#102C2×D18112yes—
152#97—1812no—
153C4×C8—1811no—
154C4×C8C2×D18111yescharacteristic, complemented
155C24—3621no—
156C2×Q8—3622no—
157C2×Q8—3621no—
158C2×Q8—3624no—
159C2×Q8—1811no—
160C2×Q8C22×D18111yescharacteristic
161C2×D4—3624no—
162C2×D4—3621no—
163C2×D4—1811no—
164C2×D4C22×D18111yescharacteristic
165C22×C4—3621no—
166C22×C4—1811no—
167C22×C4—1812no—
168M16—3624no—
169C2×C8—1812no—
170C2×C8C22×D18112yes—
171C4⋊C4—22no—
172C4⋊C4—3622no—
173C4⋊C4—3622no—
174C4⋊C4—1812no—
175C4⋊C4C22×D18112yes—
176C22⋊C4—22no—
177C22⋊C4—3622no—
178C22⋊C4—3622no—
179C22⋊C4—3622no—
180C22⋊C4—3622no—
181C42—3622no—
182C42—1812no—
183C42—1811no—
184C42C22×D18111yescharacteristic
185C23—21no—
186C23—3621no—
187C23—1811no—
188C23—7244no—
189C23—7242no—
190Q8—22no—
191Q8—3622no—
192Q8—7244no—
193Q8—7242no—
194D4—22no—
195D4—3622no—
196D4—7242no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—3622no—
200C2×C4—3621no—
201C2×C4—1812no—
202C2×C4—1811no—
203C2×C4—1814no—
204C2×C4—7242no—
205C2×C4—7242no—
206C2×C4D4×D18112yes—
207C2×C4C23×D18111yescharacteristic
208C8—22no—
209C8—3622no—
210C22—41no—
211C22—42no—
212C22—3622no—
213C22—14484no—
214C22—1812no—
215C22—7241no—
216C22—7242no—
217C22C2×D4×D18111yescharacteristic
218C4—22no—
219C4—41no—
220C4—42no—
221C4—3622no—
222C4—1812no—
223C4—7241no—
224C4—7242no—
225C4#41×D18112yes—
226C2—81no—
227C2—3622no—
228C2—14481no—
229C2#124×D18111yescharacteristic
230C2#2604207612yes—
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#488×C181C2
socleC22×C181C22×D4
prime radicalC2×D4×D181C22
hypercenter#488D181
hypocenterC181#3140
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #26041835C2semidirect
3. #26041640C2semidirect
4. #26041637C2semidirect
5. #26041636C2semidirect
6. #26041634C2semidirect
7. #26041633C2semidirect
8. #26041286C2semidirect
10. C4×#15668432C2semidirect
11. #488×C181C2semidirect
15. #19153815C22semidirect
16. #19153808C22semidirect
26. #19153741C22semidirect
27. #19153740C22semidirect
28. #19153739C22semidirect
32. #19153726C22semidirect
33. C4×#13089678C22semidirect
34. #125×C181C22semidirect
36. #103×C181C22semidirect
38. C4×C8×C181C22semidirect
115. C181#3140semidirect
124. #488D181semidirect
133. #125C2×D181semidirect
147. #103C2×D181semidirect
154. C4×C8C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260418352
#124×D1812
#260416402
#260416372
#260416362
#260416342
#260416332
#260412862
#260412822
C4×#156684322
#488×C1812
#3140181