Finite Group Databasebeta

← #37544582

Subgroups — #37544582

Order 23168.

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

normal subgroups only

This group has 442 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#26041843C211yesmaximal, characteristic, complemented
2#26041603C212yesmaximal, complemented
3#26041600C212yesmaximal, complemented
4#26041530C212yesmaximal, complemented
5#26041513C212yesmaximal, complemented
6#26041413C212yesmaximal, complemented
7#26041412C212yesmaximal, complemented
8#26041246C211yesmaximal, characteristic, complemented
9#535×C181C211yesmaximal, characteristic, complemented
10C22×D724C2211yescharacteristic, complemented
11C2×#15668434C2212yes—
12C2×#15668434C2212yes—
13C2×#15668433C2211yescharacteristic, complemented
14C22×#13089678C2211yescharacteristic, complemented
15#19153812—21no—
16#19153805—21no—
17#19153790C2212yes—
18#19153790C2212yescomplemented
19#19153784—21no—
20#19153784—21no—
21#19153775C2211yescharacteristic, complemented
22#19153749C2212yescomplemented
23#19153747C2212yescomplemented
24#19153737C2212yescomplemented
25#19153737C2212yescomplemented
26#19153736C2212yescomplemented
27#19153736C2212yescomplemented
28C22⋊#13089678C2212yescomplemented
29C22⋊#13089678C2212yescomplemented
30C2×#41×C181C2211yescharacteristic
31C22×D4×C181C2211yescharacteristic, complemented
32#127×C181—21no—
33#124×C181—21no—
34#121×C181C2212yes—
35#121×C181C2212yescomplemented
36#120×C181—21no—
37#120×C181—21no—
38#117×C181C2211yescharacteristic, complemented
39C23×D181—21no—
40C2×D724—21no—
41C2×D724—41no—
42C2×D724—41no—
43C2×D724C2312yes—
44C2×C22⋊D181—22no—
45C2×C22⋊D181—22no—
46#15668434—42no—
47#15668434—42no—
48#15668433—41no—
49C2×#13089678—41no—
50C2×#13089678C2312yes—
51C2×#13089678C2312yes—
52C4⋊Dic181C2312yes—
53C22⋊Dic181—21no—
54#15668426—22no—
55#15668426—22no—
56C4×Dic181—21no—
57C24×C181—21no—
58#41×C181—41no—
59#41×C181—41no—
60C2×Q8×C181D411yescharacteristic
61C2×D4×C181—22no—
62C2×D4×C181—22no—
63C2×D4×C181—21no—
64C2×D4×C181—41no—
65C2×D4×C181—41no—
66C2×D4×C181D411yescharacteristic
67C2×D4×C181D411yescharacteristic
68C2×D4×C181D411yescharacteristic
69C2×D4×C181C2312yes—
70C22×C4×C181C2311yescharacteristic
71C22×C4×C181C2311yescharacteristic
72C22×C4×C181C2311yescharacteristic
73C4⋊C4×C181C2312yes—
74C22⋊C4×C181—21no—
75C22⋊C4×C181—22no—
76C22⋊C4×C181—22no—
77C42×C181—21no—
78C22×D181—22no—
79C22×D181—42no—
80C22×D181—82no—
81D724—22no—
82D724—22no—
83D724—42no—
84C22⋊D181—82no—
85C22⋊D181—82no—
86C2×Dic181—22no—
87#13089678—22no—
88#13089678—22no—
89C23×C181—22no—
90C23×C181—42no—
91C23×C181—82no—
92C23×C181C2×D411yescharacteristic
93C23×C181C2×D411yescharacteristic
94C23×C181C2×D411yescharacteristic
95Q8×C181—41no—
96D4×C181—22no—
97D4×C181—22no—
98D4×C181—41no—
99D4×C181—41no—
100D4×C181—41no—
101D4×C181—42no—
102D4×C181—82no—
103D4×C181—82no—
104C2×C4×C181—21no—
105C2×C4×C181—21no—
106C2×C4×C181—21no—
107C2×C4×C181—21no—
108C2×C4×C181—21no—
109C2×C4×C181—21no—
110C2×C4×C181—22no—
111C2×C4×C181C2×D411yescharacteristic
112C2×C4×C181C2×D411yescharacteristic
113C2×C4×C181C2×D411yescharacteristic
114C2×C4×C181C2411yescharacteristic
115C22×C181—21no—
116C22×C181—21no—
117C22×C181—21no—
118C22×C181—22no—
119C22×C181—22no—
120C22×C181—22no—
121C22×C181—42no—
122C22×C181—42no—
123C22×C181—42no—
124C22×C181—84no—
125C22×C181C22×D411yescharacteristic
126C2×D181—42no—
127C2×D181—42no—
128C2×D181—42no—
129C2×D181—84no—
130C4×C181—21no—
131C4×C181—21no—
132C4×C181—21no—
133C4×C181—42no—
134C4×C181H(5,𝔽2)11yescharacteristic
135C4×C181H(5,𝔽2)11yescharacteristic
136Dic181—42no—
137C2×C181—41no—
138C2×C181—41no—
139C2×C181—41no—
140C2×C181—82no—
141C2×C181#53411yescharacteristic
142C2×C181#57612yes—
143D181—82no—
144C181#319911yescharacteristic, complemented, Sylow
145#3199—1811nomaximal, Sylow
146#535—1811no—
147#535D18111yescharacteristic, complemented
148#482—1812no—
149#482—1812no—
150#469—1812no—
151#466—1812no—
152#447—1812no—
153#447—1812no—
154#408—1811no—
155C2×#41C2×D18111yescharacteristic
156C22×D4—1811no—
157C22×D4C2×D18111yescharacteristic, complemented
158C2×D8—1812no—
159C2×D8—1812no—
160C2×M16—1811no—
161C22×C8—1811no—
162#127—21no—
163#127—3621no—
164#124—21no—
165#124—3621no—
166#121—1812no—
167#121—1812no—
168#121C2×D18112yes—
169#121C2×D18112yescomplemented
170#120—21no—
171#120—21no—
172#120—3621no—
173#120—3621no—
174#117—1811no—
175#117C2×D18111yescharacteristic, complemented
176#107—1812no—
177#106—1812no—
178#102—1812no—
179#102—1812no—
180#102—1812no—
181#102—1812no—
182C22⋊C8—1812no—
183C22⋊C8—1812no—
184C24—21no—
185C24—3621no—
186#41—41no—
187#41—41no—
188C2×Q8C22⋊D18111yescharacteristic
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—21no—
192C2×D4—41no—
193C2×D4—41no—
194C2×D4—3622no—
195C2×D4—3622no—
196C2×D4—3621no—
197C2×D4—1812no—
198C2×D4—7241no—
199C2×D4—7241no—
200C2×D4C22⋊D18111yescharacteristic
201C2×D4C22⋊D18111yescharacteristic
202C2×D4C22⋊D18111yescharacteristic
203C2×D4C22×D18112yes—
204C22×C4C22×D18111yescharacteristic
205C22×C4C22×D18111yescharacteristic
206C22×C4C22×D18111yescharacteristic
207D8—7242no—
208D8—7242no—
209M16—7241no—
210C2×C8—1812no—
211C2×C8—1812no—
212C2×C8—7241no—
213C4⋊C4—1812no—
214C4⋊C4C22×D18112yes—
215C22⋊C4—21no—
216C22⋊C4—22no—
217C22⋊C4—22no—
218C22⋊C4—3621no—
219C22⋊C4—3622no—
220C22⋊C4—3622no—
221C42—21no—
222C42—3621no—
223C23—22no—
224C23—42no—
225C23—82no—
226C23—3622no—
227C23—14482no—
228C23—7242no—
229C23C2×C22⋊D18111yescharacteristic
230C23C2×C22⋊D18111yescharacteristic
231C23C2×C22⋊D18111yescharacteristic
232Q8—41no—
233D4—22no—
234D4—22no—
235D4—41no—
236D4—41no—
237D4—41no—
238D4—42no—
239D4—82no—
240D4—82no—
241D4—3622no—
242D4—3622no—
243D4—14482no—
244D4—14482no—
245D4—7242no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—3622no—
254C2×C4C2×C22⋊D18111yescharacteristic
255C2×C4C2×C22⋊D18111yescharacteristic
256C2×C4C2×C22⋊D18111yescharacteristic
257C2×C4C23×D18111yescharacteristic
258C8—3622no—
259C8—3622no—
260C22—22no—
261C22—22no—
262C22—22no—
263C22—21no—
264C22—21no—
265C22—21no—
266C22—42no—
267C22—42no—
268C22—42no—
269C22—84no—
270C22—14484no—
271C22—7242no—
272C22—7242no—
273C22—7242no—
274C22C22×C22⋊D18111yescharacteristic
275C4—21no—
276C4—21no—
277C4—21no—
278C4—42no—
279C4—7242no—
280C4#1915386111yescharacteristic
281C4#1915386111yescharacteristic
282C2—41no—
283C2—41no—
284C2—41no—
285C2—82no—
286C2—14482no—
287C2#2604183811yescharacteristic
288C2#2604208212yes—
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#535×C181C2
socleC22×C181C22×D4
hypercenter#535D181
hypocenterC181#3199
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #26041843C2semidirect
2. #26041603C2semidirect
3. #26041600C2semidirect
4. #26041530C2semidirect
5. #26041513C2semidirect
6. #26041413C2semidirect
7. #26041412C2semidirect
8. #26041246C2semidirect
9. #535×C181C2semidirect
10. C22×D724C22semidirect
13. C2×#15668433C22semidirect
14. C22×#13089678C22semidirect
18. #19153790C22semidirect
21. #19153775C22semidirect
22. #19153749C22semidirect
23. #19153747C22semidirect
24. #19153737C22semidirect
25. #19153737C22semidirect
26. #19153736C22semidirect
27. #19153736C22semidirect
28. C22⋊#13089678C22semidirect
29. C22⋊#13089678C22semidirect
31. C22×D4×C181C22semidirect
35. #121×C181C22semidirect
38. #117×C181C22semidirect
144. C181#3199semidirect
147. #535D181semidirect
157. C22×D4C2×D181semidirect
169. #121C2×D181semidirect
175. #117C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260418432
#260416032
#260416002
#260415302
#260415132
#260414132
#260414122
#260412462
#535×C1812
#3199181