Finite Group Databasebeta

← #36365178

Subgroups — #36365178

Order 22144.

556 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 125 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×#19149706C212yesmaximal, complemented
2#26012163C211yesmaximal, characteristic, complemented
3#26012126C211yesmaximal, characteristic, complemented
4#26011867C211yesmaximal, characteristic, complemented
5#26011743C212yesmaximal, complemented
6#26011687C211yesmaximal, characteristic, complemented
7#26011635C212yesmaximal, complemented
8#26011626C212yesmaximal, complemented
9C4×#14583813C211yesmaximal, characteristic, complemented
10#26011568C211yesmaximal, characteristic, complemented
11#460×C173C211yesmaximal, characteristic, complemented
12C2×#14583827C2212yescomplemented
13C2×D4×D173—21no—
14C2×D4×D173C2212yes—
15#19149706—28no—
16C2×#14583813C2211yescharacteristic
17C2×D1384C2212yes—
18C2×#14583811C2212yescomplemented
19C2×#14583811C2212yescomplemented
20C2×#14583809C2212yescomplemented
21#19149676—21no—
22#19149676C2211yescharacteristic, complemented
23#19149654—22no—
24#19149654C2212yescomplemented
25#19149650—21no—
26#19149650C2212yes—
27#19149647—22no—
28C4×D692—21no—
29C4×D692C2211yescharacteristic
30C4×D692C2211yescharacteristic
31#19149610C2211yescharacteristic, complemented
32#19149603C2211yescharacteristic, complemented
33#19149601C2212yescomplemented
34#19149599C2211yescharacteristic
35#19149591C2212yescomplemented
36C4×#13063436C2211yescharacteristic, complemented
37C2×SD16×C173C2212yescomplemented
38#127×C173C2211yescharacteristic
39C4×Q8×C173C2211yescharacteristic, complemented
40C4⋊C8×C173C2211yescharacteristic, complemented
41#102×C173C2212yescomplemented
42C23×D173—22no—
43#14583827—24no—
44#14583827—42no—
45C2×D692—22no—
46C2×D692—22no—
47C2×D692—21no—
48C2×D692—21no—
49C2×D692D411yescharacteristic
50C2×D692C2312yes—
51C2×D692C2311yescharacteristic
52C2×C22⋊D173—22no—
53C2×C22⋊D173—22no—
54D4×D173—24no—
55D4×D173—42no—
56D4×D173—44no—
57C2×C4×D173—22no—
58C2×C4×D173—21no—
59C2×C4×D173C2312yes—
60#14583813D414yescomplemented
61D1384—24no—
62#14583811—24no—
63#14583811—42no—
64#14583809—24no—
65C2×#13063436C2311yescharacteristic
66C2×#13063436C2311yescharacteristic
67C4⋊Dic173D411yescharacteristic
68C4⋊Dic173C2311yescharacteristic
69C22⋊Dic173—22no—
70#14583803—22no—
71#14583803—21no—
72#14583803D412yes—
73C2×Q8×C173C2311yescharacteristic
74C2×D4×C173—21no—
75C2×D4×C173—21no—
76C2×D4×C173C2312yes—
77SD16×C173—24no—
78C2×C8×C173C2312yes—
79C4⋊C4×C173—21no—
80C4⋊C4×C173C2311yescharacteristic
81C42×C173—21no—
82C42×C173C2311yescharacteristic
83C22×D173—21no—
84C22×D173—22no—
85C22×D173—44no—
86C22×D173—44no—
87C22×D173—82no—
88C22×D173C2×D412yes—
89D692—22no—
90D692—22no—
91D692—21no—
92D692—22no—
93D692—44no—
94D692—42no—
95D692—81no—
96D692—82no—
97D692C2×D412yes—
98C22⋊D173—44no—
99C22⋊D173—44no—
100C2×Dic173—21no—
101C2×Dic173C2×D412yes—
102C4×D173—22no—
103C4×D173—22no—
104C4×D173—44no—
105C4×D173—42no—
106#13063436—21no—
107#13063436C2×D412yescomplemented
108C23×C173—22no—
109Q8×C173—21no—
110Q8×C173C2×D412yescomplemented
111D4×C173—22no—
112D4×C173—22no—
113D4×C173—42no—
114D4×C173—81no—
115C2×C4×C173—21no—
116C2×C4×C173—21no—
117C2×C4×C173C2×D411yescharacteristic
118C2×C4×C173C2×D411yescharacteristic
119C2×C4×C173C2411yescharacteristic
120C8×C173—22no—
121C22×C173—42no—
122C22×C173—42no—
123C22×C173—42no—
124C22×C173C22×D411yescharacteristic
125C2×D173—22no—
126C2×D173—22no—
127C2×D173—22no—
128C2×D173—22no—
129C2×D173—41no—
130C2×D173—41no—
131C2×D173—42no—
132C2×D173—42no—
133C2×D173—42no—
134C2×D173—88no—
135C4×C173—22no—
136C4×C173—21no—
137C4×C173—41no—
138C4×C173AGL(1,ℤ/8)12yescomplemented
139C4×C173C22×D411yescharacteristic
140C4×C173H(5,𝔽2)11yescharacteristic
141Dic173—22no—
142Dic173—41no—
143C2×C173—82no—
144C2×C173D4211yescharacteristic
145C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
146C2×C173#57611yescharacteristic
147D173—42no—
148D173—42no—
149D173—82no—
150C173#327411yescharacteristic, complemented, Sylow
151#3274—1731nomaximal, Sylow
152C2×AGL(1,ℤ/8)—1732no—
153#550—1731no—
154D42—1731no—
155#493—1731no—
156#469—1732no—
157#460D17311yescharacteristic, complemented
158#459—1731no—
159#449—1732no—
160#447—1732no—
161C4×SD16—1731no—
162#436—1731no—
163C2×#41—1732no—
164C22×D4—3461no—
165C22×D4—1732no—
166AGL(1,ℤ/8)—3468no—
167C2×SD16—1731no—
168C2×SD16C2×D17312yescomplemented
169C2×D8—1732no—
170C2×D8—1732no—
171C2×D8—1732no—
172C2×M16—1732no—
173#127—3461no—
174#127—1731no—
175#127C2×D17311yescharacteristic
176#121—3461no—
177#121—3462no—
178#121—1732no—
179#121—1732no—
180#120—3462no—
181C4×Q8C2×D17311yescharacteristic, complemented
182C4×D4—3461no—
183C4×D4—1731no—
184C4×D4—1731no—
185#106—1731no—
186C4⋊C8C2×D17311yescharacteristic, complemented
187#103—1731no—
188#102—1732no—
189#102—1731no—
190#102C2×D17312yescomplemented
191C22⋊C8—1732no—
192C4×C8—1731no—
193C24—3462no—
194#41—3464no—
195#41—6922no—
196C2×Q8C22×D17311yescharacteristic
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—3462no—
200C2×D4—3462no—
201C2×D4—3464no—
202C2×D4—3462no—
203C2×D4—3462no—
204C2×D4—3461no—
205C2×D4—3461no—
206C2×D4—1731no—
207C2×D4—1732no—
208C2×D4—1731no—
209C2×D4—6922no—
210C2×D4—6924no—
211C2×D4C22×D17312yes—
212C22×C4—3462no—
213C22×C4—3461no—
214C22×C4—1732no—
215SD16—24no—
216SD16—1734no—
217D8—3464no—
218D8—3464no—
219D8—6922no—
220M16—3464no—
221C2×C8—1731no—
222C2×C8—1731no—
223C2×C8C22×D17312yes—
224C4⋊C4—21no—
225C4⋊C4—1731no—
226C4⋊C4—1731no—
227C4⋊C4C22×D17311yescharacteristic
228C22⋊C4—3462no—
229C22⋊C4—3462no—
230C22⋊C4—3461no—
231C22⋊C4—1732no—
232C42—21no—
233C42C22×D17311yescharacteristic
234C23—22no—
235C23—3461no—
236C23—3462no—
237C23—13842no—
238C23—1732no—
239C23—6924no—
240C23—6924no—
241Q8—21no—
242Q8D4×D17312yescomplemented
243D4—22no—
244D4—22no—
245D4—42no—
246D4—81no—
247D4—3462no—
248D4—3462no—
249D4—3461no—
250D4—3462no—
251D4—13841no—
252D4—13842no—
253D4—1732no—
254D4—6924no—
255D4—6924no—
256D4—6924no—
257D4—6922no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—3462no—
261C2×C4—3462no—
262C2×C4—3461no—
263C2×C4—1732no—
264C2×C4—6924no—
265C2×C4—6922no—
266C2×C4D4×D17311yescharacteristic
267C2×C4D4×D17311yescharacteristic
268C2×C4C23×D17311yescharacteristic
269C8—22no—
270C8—3461no—
271C8—1732no—
272C22—42no—
273C22—42no—
274C22—42no—
275C22—3462no—
276C22—3462no—
277C22—3462no—
278C22—3462no—
279C22—13848no—
280C22—6922no—
281C22—6922no—
282C22—6922no—
283C22—6921no—
284C22—6921no—
285C22C2×D4×D17311yescharacteristic
286C4—21no—
287C4—22no—
288C4—41no—
289C4—3462no—
290C4—6921no—
291C4#1914970612yescomplemented
292C4C2×D4×D17311yescharacteristic
293C4#1914972511yescharacteristic
294C2—82no—
295C2—13842no—
296C2—6922no—
297C2—6922no—
298C2#2601212511yescharacteristic
299C2C2×#1914970611yescharacteristic
300C2#2601229511yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#460×C173C2
socleC22×C173C22×D4
hypercenter#460D173
hypocenterC173#3274
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×#19149706C2semidirect
2. #26012163C2semidirect
3. #26012126C2semidirect
4. #26011867C2semidirect
5. #26011743C2semidirect
6. #26011687C2semidirect
7. #26011635C2semidirect
8. #26011626C2semidirect
9. C4×#14583813C2semidirect
10. #26011568C2semidirect
11. #460×C173C2semidirect
12. C2×#14583827C22semidirect
18. C2×#14583811C22semidirect
19. C2×#14583811C22semidirect
20. C2×#14583809C22semidirect
22. #19149676C22semidirect
24. #19149654C22semidirect
31. #19149610C22semidirect
32. #19149603C22semidirect
33. #19149601C22semidirect
35. #19149591C22semidirect
36. C4×#13063436C22semidirect
37. C2×SD16×C173C22semidirect
39. C4×Q8×C173C22semidirect
40. C4⋊C8×C173C22semidirect
41. #102×C173C22semidirect
60. #14583813D4semidirect
107. #13063436C2×D4semidirect
110. Q8×C173C2×D4semidirect
138. C4×C173AGL(1,ℤ/8)semidirect
150. C173#3274semidirect
157. #460D173semidirect
168. C2×SD16C2×D173semidirect
181. C4×Q8C2×D173semidirect
186. C4⋊C8C2×D173semidirect
190. #102C2×D173semidirect
242. Q8D4×D173semidirect
291. C4#19149706semidirect

Maximal subgroups

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

subgroupindex
C2×#191497062
#260121632
#260121262
#260118672
#260117432
#260116872
#260116352
#260116262
C4×#145838132
#260115682
#460×C1732
#3274173