Finite Group Databasebeta

← #37545393

Subgroups — #37545393

Order 23168.

620 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 133 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×D8×D181C212yesmaximal, complemented
2D4×D724C211yesmaximal, characteristic, complemented
3#26041914C211yesmaximal, characteristic, complemented
4#26041655C211yesmaximal, characteristic, complemented
5#26041514C212yesmaximal, complemented
6#26041475C211yesmaximal, characteristic, complemented
7#26041414C212yesmaximal, complemented
8#26041411C212yesmaximal, complemented
9C4×#15668434C211yesmaximal, characteristic, complemented
10#26041344C211yesmaximal, characteristic, complemented
11#459×C181C211yesmaximal, characteristic, complemented
12C22×D724—21no—
13C2×D4×D181—21no—
14C2×D4×D181C2212yescomplemented
15C2×D4×D181C2212yescomplemented
16C2×D1448C2212yescomplemented
17C2×#15668434C2211yescharacteristic
18C2×#15668434C2212yescomplemented
19C2×#15668434C2212yescomplemented
20D8×D181—28no—
21C2×C8×D181C2212yescomplemented
22#19153813C2211yescharacteristic, complemented
23#19153791C2212yescomplemented
24#19153790—21no—
25#19153787—21no—
26#19153787C2212yescomplemented
27#19153786—22no—
28#19153784—22no—
29C4×D724C2211yescharacteristic
30C4×D724C2211yescharacteristic, complemented
31#19153749C2211yescharacteristic, complemented
32#19153738C2212yescomplemented
33#19153737C2211yescharacteristic, complemented
34#19153736C2211yescharacteristic, complemented
35#19153728C2212yescomplemented
36C4×#13089678C2211yescharacteristic, complemented
37C2×D8×C181C2212yescomplemented
38#127×C181C2211yescharacteristic, complemented
39C4×D4×C181C2211yescharacteristic, complemented
40C4⋊C8×C181C2211yescharacteristic, complemented
41#102×C181C2212yescomplemented
42C23×D181—22no—
43C23×D181—22no—
44C2×D724—22no—
45C2×D724—21no—
46C2×D724—21no—
47C2×D724—44no—
48C2×D724D411yescharacteristic
49C2×D724C2312yes—
50C2×D724C2311yescharacteristic
51C2×C22⋊D181—22no—
52C2×C22⋊D181—22no—
53C2×C22⋊D181—22no—
54D4×D181—24no—
55D4×D181—24no—
56D4×D181—42no—
57D4×D181—42no—
58D4×D181—44no—
59C2×C4×D181—21no—
60C2×C4×D181C2312yes—
61D1448—24no—
62#15668434—24no—
63#15668434—42no—
64#15668434D414yescomplemented
65C2×#13089678C2311yescharacteristic
66C2×#13089678C2311yescharacteristic
67C8×D181—24no—
68C4⋊Dic181D411yescharacteristic
69C4⋊Dic181C2311yescharacteristic
70C22⋊Dic181—22no—
71#15668426—22no—
72#15668426—21no—
73#15668426D412yes—
74C2×D4×C181—21no—
75C2×D4×C181—21no—
76C2×D4×C181C2312yes—
77C2×D4×C181C2311yescharacteristic
78C22×C4×C181—21no—
79D8×C181—24no—
80C2×C8×C181C2312yes—
81C4⋊C4×C181C2311yescharacteristic
82C22⋊C4×C181—21no—
83C42×C181C2311yescharacteristic
84C22×D181—21no—
85C22×D181—22no—
86C22×D181—44no—
87C22×D181—44no—
88C22×D181—44no—
89C22×D181—44no—
90C22×D181—82no—
91C22×D181—82no—
92C22×D181C2×D412yes—
93D724—22no—
94D724—21no—
95D724—22no—
96D724—44no—
97D724—42no—
98D724—81no—
99D724D814yescomplemented
100D724C2×D412yescomplemented
101C22⋊D181—44no—
102C22⋊D181—44no—
103C22⋊D181—44no—
104C2×Dic181—21no—
105C2×Dic181C2×D412yes—
106C4×D181—22no—
107C4×D181—22no—
108C4×D181—42no—
109#13089678—21no—
110#13089678C2×D412yescomplemented
111C23×C181—21no—
112C23×C181—22no—
113D4×C181—22no—
114D4×C181—21no—
115D4×C181—22no—
116D4×C181—42no—
117D4×C181—81no—
118D4×C181C2×D412yescomplemented
119C2×C4×C181—21no—
120C2×C4×C181—42no—
121C2×C4×C181C2×D411yescharacteristic
122C2×C4×C181C2×D411yescharacteristic
123C2×C4×C181C2411yescharacteristic
124C8×C181—22no—
125C22×C181—22no—
126C22×C181—41no—
127C22×C181—41no—
128C22×C181—42no—
129C22×C181—42no—
130C22×C181—42no—
131C22×C181C22×D411yescharacteristic
132C2×D181—22no—
133C2×D181—22no—
134C2×D181—22no—
135C2×D181—41no—
136C2×D181—41no—
137C2×D181—42no—
138C2×D181—42no—
139C2×D181—42no—
140C2×D181—88no—
141C2×D181—88no—
142C2×D181C2×D814yescomplemented
143C4×C181—21no—
144C4×C181—41no—
145C4×C181C2×D812yescomplemented
146C4×C181C22×D411yescharacteristic
147C4×C181H(5,𝔽2)11yescharacteristic
148Dic181—22no—
149Dic181—41no—
150C2×C181—42no—
151C2×C181—82no—
152C2×C181D4211yescharacteristic
153C2×C181C22×D811yescharacteristic
154C2×C181#57611yescharacteristic
155D181—24no—
156D181—42no—
157D181—82no—
158C181D4×D811yescharacteristic, complemented, Sylow
159D4×D8—1811nomaximal, Sylow
160C22×D8—1812no—
161D42—1811no—
162D42—1811no—
163#493—1811no—
164#466—1812no—
165#459—1811no—
166#459D18111yescharacteristic, complemented
167#447—1812no—
168#447—1812no—
169C4×D8—1811no—
170C8×D4—1811no—
171C22×D4—3621no—
172C22×D4—3621no—
173C22×D4—1812no—
174C22×D4—1812no—
175C2×D8—3628no—
176C2×D8—1812no—
177C2×D8—1811no—
178C2×D8—1812no—
179C2×D8—1812no—
180C2×D8C2×D18112yescomplemented
181C22×C8—1812no—
182#127—1811no—
183#127C2×D18111yescharacteristic, complemented
184#121—3621no—
185#121—3621no—
186#121—1812no—
187#121—1812no—
188#120—3622no—
189#120—3622no—
190C4×D4—1811no—
191C4×D4—1811no—
192C4×D4C2×D18111yescharacteristic, complemented
193#107—1811no—
194C4⋊C8C2×D18111yescharacteristic, complemented
195#102—1811no—
196#102—1812no—
197#102—1811no—
198#102C2×D18112yescomplemented
199C22⋊C8—1812no—
200C4×C8—1811no—
201C24—3622no—
202C24—3622no—
203C2×D4—21no—
204C2×D4—21no—
205C2×D4—3622no—
206C2×D4—3624no—
207C2×D4—3624no—
208C2×D4—3622no—
209C2×D4—3622no—
210C2×D4—3622no—
211C2×D4—3621no—
212C2×D4—3621no—
213C2×D4—1811no—
214C2×D4—1812no—
215C2×D4—1811no—
216C2×D4—7244no—
217C2×D4—7242no—
218C2×D4—7242no—
219C2×D4—7244no—
220C2×D4C22×D18112yes—
221C2×D4C22×D18111yescharacteristic
222C22×C4—21no—
223C22×C4—3621no—
224C22×C4—1812no—
225D8—24no—
226D8—3624no—
227D8—3624no—
228D8—1814no—
229D8—7242no—
230C2×C8—3624no—
231C2×C8—1811no—
232C2×C8—1811no—
233C2×C8C22×D18112yes—
234C4⋊C4—1811no—
235C4⋊C4—1811no—
236C4⋊C4C22×D18111yescharacteristic
237C22⋊C4—21no—
238C22⋊C4—3622no—
239C22⋊C4—3622no—
240C22⋊C4—3621no—
241C22⋊C4—1812no—
242C42C22×D18111yescharacteristic
243C23—21no—
244C23—22no—
245C23—3621no—
246C23—3622no—
247C23—14482no—
248C23—14482no—
249C23—1812no—
250C23—7244no—
251C23—7244no—
252C23—7244no—
253C23—7244no—
254D4—22no—
255D4—21no—
256D4—22no—
257D4—42no—
258D4—81no—
259D4—3622no—
260D4—3621no—
261D4—3622no—
262D4—14481no—
263D4—1814no—
264D4—1812no—
265D4—7244no—
266D4—7244no—
267D4—7244no—
268D4—7244no—
269D4—7242no—
270D4D4×D18112yescomplemented
271C2×C4—21no—
272C2×C4—42no—
273C2×C4—3622no—
274C2×C4—3622no—
275C2×C4—3621no—
276C2×C4—1812no—
277C2×C4—7242no—
278C2×C4D4×D18111yescharacteristic
279C2×C4D4×D18111yescharacteristic
280C2×C4C23×D18111yescharacteristic
281C8—22no—
282C8—3621no—
283C8—1812no—
284C22—22no—
285C22—41no—
286C22—41no—
287C22—42no—
288C22—42no—
289C22—42no—
290C22—3622no—
291C22—3622no—
292C22—3622no—
293C22—14488no—
294C22—14488no—
295C22—1814no—
296C22—7242no—
297C22—7242no—
298C22—7242no—
299C22—7241no—
300C22—7241no—
301C22C2×D4×D18111yescharacteristic
302C4—21no—
303C4—41no—
304C4—3622no—
305C4—7241no—
306C4D8×D18112yescomplemented
307C4C2×D4×D18111yescharacteristic
308C4#1915386211yescharacteristic
309C2—42no—
310C2—82no—
311C2—3624no—
312C2—14482no—
313C2—7242no—
314C2#2604191311yescharacteristic
315C2C2×D8×D18111yescharacteristic
316C2#2604208311yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#459×C181C2
socleC22×C181C22×D4
hypercenter#459D181
hypocenterC181D4×D8
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×D8×D181C2semidirect
2. D4×D724C2semidirect
3. #26041914C2semidirect
4. #26041655C2semidirect
5. #26041514C2semidirect
6. #26041475C2semidirect
7. #26041414C2semidirect
8. #26041411C2semidirect
9. C4×#15668434C2semidirect
10. #26041344C2semidirect
11. #459×C181C2semidirect
14. C2×D4×D181C22semidirect
15. C2×D4×D181C22semidirect
16. C2×D1448C22semidirect
18. C2×#15668434C22semidirect
19. C2×#15668434C22semidirect
21. C2×C8×D181C22semidirect
22. #19153813C22semidirect
23. #19153791C22semidirect
26. #19153787C22semidirect
30. C4×D724C22semidirect
31. #19153749C22semidirect
32. #19153738C22semidirect
33. #19153737C22semidirect
34. #19153736C22semidirect
35. #19153728C22semidirect
36. C4×#13089678C22semidirect
37. C2×D8×C181C22semidirect
38. #127×C181C22semidirect
39. C4×D4×C181C22semidirect
40. C4⋊C8×C181C22semidirect
41. #102×C181C22semidirect
64. #15668434D4semidirect
99. D724D8semidirect
100. D724C2×D4semidirect
110. #13089678C2×D4semidirect
118. D4×C181C2×D4semidirect
142. C2×D181C2×D8semidirect
145. C4×C181C2×D8semidirect
158. C181D4×D8semidirect
166. #459D181semidirect
180. C2×D8C2×D181semidirect
183. #127C2×D181semidirect
192. C4×D4C2×D181semidirect
194. C4⋊C8C2×D181semidirect
198. #102C2×D181semidirect
270. D4D4×D181semidirect
306. C4D8×D181semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D1812
D4×D7242
#260419142
#260416552
#260415142
#260414752
#260414142
#260414112
C4×#156684342
#260413442
#459×C1812
D4×D8181