Finite Group Databasebeta

← #37542991

Subgroups — #37542991

Order 23168.

784 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 149 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×#19153862C211yesmaximal, characteristic, complemented
2C2×#19153844C211yesmaximal, characteristic, complemented
3C22×D1448C211yesmaximal, characteristic, complemented
4#26041843C211yesmaximal, characteristic, complemented
5#26041426C214yesmaximal, complemented
6#26041415C214yesmaximal, complemented
7#26041285C211yesmaximal, characteristic, complemented
8#26041276C211yesmaximal, characteristic, complemented
9#408×C181C211yesmaximal, characteristic, complemented
10#19153862—24no—
11#19153862—24no—
12C2×#15668450—21no—
13C2×#15668449C2211yescharacteristic
14C2×#15668449C2212yescomplemented
15C22×D724C2211yescharacteristic
16C22×D724C2212yescomplemented
17C22×D724C2211yescharacteristic, complemented
18C2×D4×D181—22no—
19C2×D4×D181—21no—
20#19153844—24no—
21C2×#15668438C2212yescomplemented
22C2×D1448—24no—
23C2×D1448C2212yescomplemented
24C2×D1448C2214yescomplemented
25#19153812—21no—
26#19153805—21no—
27#19153790C2214yescomplemented
28#19153784—22no—
29#19153775C2211yescharacteristic, complemented
30#19153741C2212yescomplemented
31#19153738C2212yescomplemented
32#19153738C2214yescomplemented
33C2×#41×C181C2211yescharacteristic
34C2×M16×C181C2211yescharacteristic, complemented
35C22×C8×C181C2211yescharacteristic, complemented
36C22⋊C8×C181C2214yescomplemented
37C23×D181—21no—
38C23×D181—22no—
39C23×D181—21no—
40#15668450—28no—
41#15668449—24no—
42#15668449—24no—
43#15668449—22no—
44#15668449D414yescomplemented
45C2×Dic362C2311yescharacteristic
46C2×D724—24no—
47C2×D724—24no—
48C2×D724—21no—
49C2×D724—22no—
50C2×D724—22no—
51C2×D724—22no—
52C2×D724—22no—
53C2×D724—21no—
54C2×D724D414yescomplemented
55C2×D724C2312yes—
56C2×D724C2312yes—
57C2×D724C2311yescharacteristic
58C2×C22⋊D181—21no—
59C2×C22⋊D181—22no—
60C2×C22⋊D181—24no—
61D4×D181—216no—
62D4×D181—28no—
63C2×C4×D181—21no—
64C2×C4×D181—22no—
65#15668438—24no—
66D1448—28no—
67D1448—24no—
68C4⋊Dic181C2312yes—
69C22⋊Dic181—21no—
70#15668426—24no—
71C4×Dic181—21no—
72#41×C181—22no—
73#41×C181—22no—
74C2×Q8×C181D411yescharacteristic
75C2×D4×C181D412yes—
76C2×D4×C181D411yescharacteristic
77C22×C4×C181C2312yes—
78C22×C4×C181C2311yescharacteristic
79M16×C181—22no—
80C2×C8×C181—22no—
81C2×C8×C181C2312yes—
82C2×C8×C181C2312yes—
83C22×D181—21no—
84C22×D181—22no—
85C22×D181—28no—
86C22×D181—24no—
87C22×D181—22no—
88C22×D181—42no—
89C22×D181—82no—
90C22×D181—84no—
91C22×D181—82no—
92Dic362—21no—
93Dic362C2×D412yes—
94D724—28no—
95D724—24no—
96D724—22no—
97D724—21no—
98D724—22no—
99D724—22no—
100D724—42no—
101D724C2×D412yes—
102D724C2×D414yes—
103C22⋊D181—28no—
104C22⋊D181—24no—
105C22⋊D181—84no—
106C2×Dic181—21no—
107C2×Dic181—22no—
108C4×D181—28no—
109C4×D181—24no—
110C23×C181C2×D412yes—
111C23×C181C2×D411yescharacteristic
112Q8×C181—22no—
113D4×C181—24no—
114D4×C181—22no—
115C2×C4×C181—22no—
116C2×C4×C181—22no—
117C2×C4×C181C2×D412yes—
118C2×C4×C181C2×D412yes—
119C2×C4×C181C2×D412yes—
120C2×C4×C181C2×D411yescharacteristic
121C2×C4×C181C2411yescharacteristic
122C8×C181—22no—
123C8×C181—22no—
124C22×C181—22no—
125C22×C181—22no—
126C22×C181—24no—
127C22×C181#12012yes—
128C22×C181C22×D411yescharacteristic
129C2×D181—24no—
130C2×D181—22no—
131C2×D181—44no—
132C2×D181—42no—
133C2×D181—42no—
134C2×D181—44no—
135C2×D181—84no—
136C2×D181—88no—
137C2×D181—84no—
138C4×C181—22no—
139C4×C181#12012yes—
140C4×C181C22×D411yescharacteristic
141C4×C181C22×D411yescharacteristic
142Dic181—22no—
143Dic181—42no—
144C2×C181—22no—
145C2×C181—42no—
146C2×C181C2×#12011yescharacteristic
147C2×C181#57612yes—
148D181—44no—
149D181—42no—
150D181—82no—
151C181#300811yescharacteristic, complemented, Sylow
152#3008—1811nomaximal, Sylow
153C2×H(5,𝔽2)—1811no—
154C2×AGL(1,ℤ/8)—1811no—
155C22×D8—1811no—
156#535—1811no—
157#449—1814no—
158#447—1814no—
159#418—1811no—
160#416—1811no—
161#408D18111yescharacteristic, complemented
162H(5,𝔽2)—3624no—
163H(5,𝔽2)—3624no—
164C2×#41—3621no—
165C2×#41—1812no—
166C2×#41—1811no—
167C2×#41C2×D18111yescharacteristic
168C22×D4—3622no—
169C22×D4—3621no—
170C22×D4—1811no—
171C22×D4—1812no—
172C22×D4—1811no—
173AGL(1,ℤ/8)—3624no—
174C2×SD16—1812no—
175C2×D8—3624no—
176C2×D8—1814no—
177C2×D8—1812no—
178C2×M16C2×D18111yescharacteristic, complemented
179C22×C8C2×D18111yescharacteristic, complemented
180#127—3621no—
181#124—3621no—
182#121—1814no—
183#120—3622no—
184#117—1811no—
185#103—1812no—
186#102—1812no—
187#102—1814no—
188C22⋊C8C2×D18114yescomplemented
189C24—3622no—
190C24—3621no—
191C24—3621no—
192#41—22no—
193#41—22no—
194#41—3624no—
195#41—3624no—
196#41—3628no—
197#41—3622no—
198#41—1814no—
199C2×Q8—1811no—
200C2×Q8D72411yescharacteristic
201C2×D4—3624no—
202C2×D4—3624no—
203C2×D4—36216no—
204C2×D4—3628no—
205C2×D4—3622no—
206C2×D4—3621no—
207C2×D4—3624no—
208C2×D4—3622no—
209C2×D4—3622no—
210C2×D4—3622no—
211C2×D4—3622no—
212C2×D4—3621no—
213C2×D4—3621no—
214C2×D4—1814no—
215C2×D4—1812no—
216C2×D4—1811no—
217C2×D4—1812no—
218C2×D4D72412yes—
219C2×D4D72411yescharacteristic
220C22×C4—3622no—
221C22×C4—3621no—
222C22×C4C22×D18112yes—
223C22×C4C22×D18111yescharacteristic
224SD16—3624no—
225D8—3628no—
226D8—3624no—
227M16—22no—
228C2×C8—22no—
229C2×C8C22×D18112yes—
230C2×C8C22×D18112yes—
231C4⋊C4—1812no—
232C22⋊C4—3621no—
233C22⋊C4—3624no—
234C42—3621no—
235C23—3628no—
236C23—3624no—
237C23—3622no—
238C23—3621no—
239C23—3622no—
240C23—14482no—
241C23—14482no—
242C23—14484no—
243C23—7242no—
244C23C2×D72412yes—
245C23C2×D72411yescharacteristic
246Q8—22no—
247Q8—3621no—
248Q8—1812no—
249D4—24no—
250D4—22no—
251D4—3628no—
252D4—3628no—
253D4—3624no—
254D4—3624no—
255D4—3622no—
256D4—3621no—
257D4—3622no—
258D4—3622no—
259D4—14484no—
260D4—1812no—
261D4—1814no—
262D4—7242no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—3628no—
266C2×C4—3621no—
267C2×C4—3624no—
268C2×C4—3622no—
269C2×C4D4×D18112yes—
270C2×C4D4×D18112yes—
271C2×C4C2×D72412yes—
272C2×C4C2×D72411yescharacteristic
273C2×C4C23×D18111yescharacteristic
274C8—22no—
275C8—22no—
276C22—22no—
277C22—22no—
278C22—24no—
279C22—3624no—
280C22—3622no—
281C22—14484no—
282C22—14484no—
283C22—14488no—
284C22—7242no—
285C22—7244no—
286C22—7244no—
287C22—7242no—
288C22#1915378612yes—
289C22C22×D72411yescharacteristic
290C4—22no—
291C4—3622no—
292C4—7242no—
293C4#1915378612yes—
294C4C2×D4×D18111yescharacteristic
295C4C2×D4×D18111yescharacteristic
296C2—22no—
297C2—42no—
298C2—14482no—
299C2—7244no—
300C2—7242no—
301C2C2×#1915378611yescharacteristic
302C2#2604208412yes—
subgroupG/H
centerC22C22×D724
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#408×C181C2
socleC22×C181C22×D4
hypercenter#408D181
hypocenterC181#3008
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
1. C2×#19153862C2semidirect
2. C2×#19153844C2semidirect
3. C22×D1448C2semidirect
4. #26041843C2semidirect
5. #26041426C2semidirect
6. #26041415C2semidirect
7. #26041285C2semidirect
8. #26041276C2semidirect
9. #408×C181C2semidirect
14. C2×#15668449C22semidirect
16. C22×D724C22semidirect
17. C22×D724C22semidirect
21. C2×#15668438C22semidirect
23. C2×D1448C22semidirect
24. C2×D1448C22semidirect
27. #19153790C22semidirect
29. #19153775C22semidirect
30. #19153741C22semidirect
31. #19153738C22semidirect
32. #19153738C22semidirect
34. C2×M16×C181C22semidirect
35. C22×C8×C181C22semidirect
36. C22⋊C8×C181C22semidirect
44. #15668449D4semidirect
54. C2×D724D4semidirect
151. C181#3008semidirect
161. #408D181semidirect
178. C2×M16C2×D181semidirect
179. C22×C8C2×D181semidirect
188. C22⋊C8C2×D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538622
C2×#191538442
C22×D14482
#260418432
#260414262
#260414152
#260412852
#260412762
#408×C1812
#3008181