Finite Group Databasebeta

← #37543069

Subgroups — #37543069

Order 23168.

724 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 145 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#26042115C211yesmaximal, characteristic, complemented
2#26042077C212yesmaximal, complemented
3#26041461C212yesmaximal, complemented
4#26041456C212yesmaximal, complemented
5#26041451C212yesmaximal, complemented
6#26041446C212yesmaximal, complemented
7C2×#19153744C212yesmaximal, complemented
8#26041263C211yesmaximal, characteristic, complemented
9#532×C181C211yesmaximal, characteristic, complemented
10#19153864C2212yes—
11#19153863—22no—
12#19153862—22no—
13#19153861—22no—
14#19153861C2212yes—
15C2×#15668449—22no—
16C2×#15668449C2212yes—
17#41×D181—24no—
18#19153847C2214yescomplemented
19#19153846—22no—
20#19153842C2214yescomplemented
21#19153841—22no—
22#19153836—24no—
23C2×#15668433C2212yescomplemented
24#19153744C2214yescomplemented
25#19153744C2214yescomplemented
26#19153735C2212yescomplemented
27#19153733C2212yescomplemented
28C2×#41×C181C2211yescharacteristic
29C2×#41×C181C2211yescharacteristic, complemented
30C4⋊Q8×C181C2211yescharacteristic, complemented
31#127×C181C2211yescharacteristic, complemented
32#124×C181C2211yescharacteristic, complemented
33#124×C181C2211yescharacteristic, complemented
34#121×C181—22no—
35C4×D4×C181—22no—
36C2×C42×C181C2211yescharacteristic, complemented
37#15668450—24no—
38#15668449—24no—
39#15668449—24no—
40#15668449—22no—
41#15668449—24no—
42#15668449—22no—
43#15668449D414yescomplemented
44#15668449C2314yes—
45#15668448—28no—
46#15668448—24no—
47C2×Dic362—22no—
48C2×Dic362D412yescomplemented
49Q8×D181—24no—
50C2×D724—22no—
51C2×D724D412yescomplemented
52C2×C22⋊D181—24no—
53C2×C22⋊D181—22no—
54D4×D181—28no—
55D4×D181—24no—
56C2×C4×D181—24no—
57C2×C4×D181—22no—
58#15668439—24no—
59#15668437—24no—
60#15668436—24no—
61#15668434—24no—
62#15668433—22no—
63#15668433C2314yes—
64C2×#13089678—22no—
65#41×C181—22no—
66#41×C181—22no—
67#41×C181—22no—
68#41×C181—22no—
69C2×Q8×C181C2311yescharacteristic
70C2×Q8×C181C2311yescharacteristic
71C2×D4×C181—22no—
72C2×D4×C181—21no—
73C2×D4×C181D412yes—
74C2×D4×C181C2311yescharacteristic
75C2×D4×C181C2311yescharacteristic
76C22×C4×C181—21no—
77C22×C4×C181—21no—
78C22×C4×C181D412yes—
79C22×C4×C181C2311yescharacteristic
80C4⋊C4×C181—22no—
81C22⋊C4×C181—22no—
82C22⋊C4×C181—22no—
83C42×C181—21no—
84C42×C181C2311yescharacteristic
85C42×C181C2311yescharacteristic
86C22×D181—24no—
87C22×D181—22no—
88Dic362—24no—
89Dic362—22no—
90Dic362C2×D414yes—
91D724—24no—
92D724—22no—
93D724C2×D414yes—
94C22⋊D181—28no—
95C22⋊D181—24no—
96C22⋊D181—24no—
97C2×Dic181—24no—
98C2×Dic181—22no—
99C4×D181—28no—
100C4×D181—24no—
101C4×D181—24no—
102#13089678—24no—
103C23×C181—21no—
104C23×C181C2×D412yes—
105C23×C181C2×D411yescharacteristic
106Q8×C181—22no—
107Q8×C181—22no—
108D4×C181—24no—
109D4×C181—22no—
110D4×C181—22no—
111D4×C181—44no—
112D4×C181—42no—
113C2×C4×C181—21no—
114C2×C4×C181—22no—
115C2×C4×C181—22no—
116C2×C4×C181—21no—
117C2×C4×C181—21no—
118C2×C4×C181—42no—
119C2×C4×C181—41no—
120C2×C4×C181—41no—
121C2×C4×C181C2×D412yes—
122C2×C4×C181C2×D412yes—
123C2×C4×C181C2×D412yes—
124C2×C4×C181C2×D411yescharacteristic
125C2×C4×C181C2411yescharacteristic
126C22×C181—22no—
127C22×C181—41no—
128C22×C181—42no—
129C22×C181—42no—
130C22×C181—41no—
131C22×C181#12012yes—
132C22×C181C22×D411yescharacteristic
133C2×D181—24no—
134C2×D181—82no—
135C2×D181—84no—
136C4×C181—22no—
137C4×C181—22no—
138C4×C181—41no—
139C4×C181—41no—
140C4×C181#12011yescharacteristic
141C4×C181#12011yescharacteristic
142C4×C181C22×D412yes—
143Dic181—24no—
144C2×C181—21no—
145C2×C181—22no—
146C2×C181—42no—
147C2×C181—81no—
148C2×C181C2×#12011yescharacteristic
149D181—44no—
150C181#301811yescharacteristic, complemented, Sylow
151#3018—1811nomaximal, Sylow
152#585—1811no—
153#575—1812no—
154#532D18111yescharacteristic, complemented
155#456—1812no—
156#455—1812no—
157#454—1812no—
158#453—1812no—
159C2×#104—1812no—
160#413—1811no—
1612−1+4—3622no—
1622−1+4—1812no—
163H(5,𝔽2)—3622no—
164H(5,𝔽2)—3622no—
165H(5,𝔽2)—1812no—
166C2×#41—3624no—
167C2×#41—3622no—
168C2×#41—1812no—
169C2×#41C2×D18111yescharacteristic
170C2×#41C2×D18111yescharacteristic, complemented
171#137—3622no—
172#137—1814no—
173AGL(1,ℤ/8)—3622no—
174AGL(1,ℤ/8)—1814no—
175#135—3624no—
176C2×M16—1812no—
177C4⋊Q8C2×D18111yescharacteristic, complemented
178#127C2×D18111yescharacteristic, complemented
179#124C2×D18111yescharacteristic, complemented
180#124C2×D18111yescharacteristic, complemented
181#121—22no—
182C4×D4—22no—
183C2×C42C2×D18111yescharacteristic, complemented
184#104—1814no—
185#104—1814no—
186#101—1812no—
187#100—1812no—
188#41—22no—
189#41—22no—
190#41—22no—
191#41—22no—
192#41—3624no—
193#41—3624no—
194#41—3628no—
195#41—3624no—
196#41—3624no—
197#41—3622no—
198#41—3622no—
199#41—3624no—
200#41—1814no—
201#41—1814no—
202C2×Q8—3622no—
203C2×Q8—3624no—
204C2×Q8—1812no—
205C2×Q8C22×D18111yescharacteristic
206C2×Q8C22×D18111yescharacteristic
207C2×D4—22no—
208C2×D4—21no—
209C2×D4—3622no—
210C2×D4—3628no—
211C2×D4—3624no—
212C2×D4—3624no—
213C2×D4—3622no—
214C2×D4—1812no—
215C2×D4C22⋊D18112yes—
216C2×D4C22×D18111yescharacteristic
217C2×D4C22×D18111yescharacteristic
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—3624no—
221C22×C4—3622no—
222C22×C4C22⋊D18112yes—
223C22×C4C22×D18111yescharacteristic
224Q16—3624no—
225SD16—3624no—
226SD16—3624no—
227D8—3624no—
228M16—3622no—
229M16—1814no—
230C2×C8—3622no—
231C4⋊C4—22no—
232C22⋊C4—22no—
233C22⋊C4—22no—
234C42—21no—
235C42C22×D18111yescharacteristic
236C42C22×D18111yescharacteristic
237C23—21no—
238C23—3624no—
239C23—3622no—
240C23C2×C22⋊D18112yes—
241C23C2×C22⋊D18111yescharacteristic
242Q8—22no—
243Q8—22no—
244Q8—3624no—
245Q8—3622no—
246Q8—1814no—
247D4—24no—
248D4—22no—
249D4—22no—
250D4—44no—
251D4—42no—
252D4—3624no—
253D4—3628no—
254D4—3624no—
255D4—3624no—
256D4—3622no—
257D4—1814no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—3628no—
267C2×C4—3624no—
268C2×C4—3622no—
269C2×C4—3624no—
270C2×C4—3624no—
271C2×C4D4×D18112yes—
272C2×C4D4×D18112yes—
273C2×C4C2×C22⋊D18112yes—
274C2×C4C2×C22⋊D18111yescharacteristic
275C2×C4C23×D18111yescharacteristic
276C8—3624no—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—42no—
281C22—41no—
282C22—3624no—
283C22—14482no—
284C22—14484no—
285C22#1915378412yes—
286C22C22×C22⋊D18111yescharacteristic
287C4—22no—
288C4—22no—
289C4—41no—
290C4—41no—
291C4—3624no—
292C4#1915378411yescharacteristic
293C4#1915378411yescharacteristic
294C4C2×D4×D18112yes—
295C2—21no—
296C2—22no—
297C2—42no—
298C2—81no—
299C2—7244no—
300C2C2×#1915378411yescharacteristic
subgroupG/H
centerC4#19153784
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#532×C181C2
socleC2×C181C2×#120
hypercenter#532D181
hypocenterC181#3018
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C4 ◁ C22×C4 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #26042115C2semidirect
2. #26042077C2semidirect
3. #26041461C2semidirect
4. #26041456C2semidirect
5. #26041451C2semidirect
6. #26041446C2semidirect
7. C2×#19153744C2semidirect
8. #26041263C2semidirect
9. #532×C181C2semidirect
18. #19153847C22semidirect
20. #19153842C22semidirect
23. C2×#15668433C22semidirect
24. #19153744C22semidirect
25. #19153744C22semidirect
26. #19153735C22semidirect
27. #19153733C22semidirect
29. C2×#41×C181C22semidirect
30. C4⋊Q8×C181C22semidirect
31. #127×C181C22semidirect
32. #124×C181C22semidirect
33. #124×C181C22semidirect
36. C2×C42×C181C22semidirect
43. #15668449D4semidirect
48. C2×Dic362D4semidirect
51. C2×D724D4semidirect
150. C181#3018semidirect
154. #532D181semidirect
170. C2×#41C2×D181semidirect
177. C4⋊Q8C2×D181semidirect
178. #127C2×D181semidirect
179. #124C2×D181semidirect
180. #124C2×D181semidirect
183. C2×C42C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260421152
#260420772
#260414612
#260414562
#260414512
#260414462
C2×#191537442
#260412632
#532×C1812
#3018181