Finite Group Databasebeta

← #36365186

Subgroups — #36365186

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×#19149703C212yesmaximal, complemented
2#26012161C211yesmaximal, characteristic, complemented
3D4×D692C211yesmaximal, characteristic, complemented
4#26011867C211yesmaximal, characteristic, complemented
5#26011743C212yesmaximal, complemented
6#26011692C211yesmaximal, characteristic, complemented
7#26011637C212yesmaximal, complemented
8#26011623C212yesmaximal, complemented
9C4×#14583814C211yesmaximal, characteristic, complemented
10#26011568C211yesmaximal, characteristic, complemented
11#459×C173C211yesmaximal, characteristic, complemented
12C2×#14583825C2212yescomplemented
13C22×D692—21no—
14C2×D4×D173C2212yes—
15#19149703—28no—
16C2×#14583815C2212yescomplemented
17C2×#14583814C2211yescharacteristic
18C2×#14583811C2212yescomplemented
19C2×#14583811C2212yescomplemented
20C2×#14583809C2212yescomplemented
21#19149676C2211yescharacteristic
22#19149675—21no—
23#19149654C2212yes—
24#19149653—21no—
25#19149651—22no—
26#19149650C2212yescomplemented
27#19149649—22no—
28C4×Dic346C2211yescharacteristic, complemented
29C4×D692C2211yescharacteristic
30D4×Dic173—21no—
31#19149610C2211yescharacteristic, complemented
32#19149602C2211yescharacteristic, complemented
33#19149601C2212yescomplemented
34#19149600C2211yescharacteristic
35#19149591C2212yescomplemented
36C4×#13063436C2211yescharacteristic, complemented
37C2×D8×C173C2212yes—
38#127×C173C2211yescharacteristic, complemented
39C4×D4×C173C2211yescharacteristic
40C4⋊C8×C173C2211yescharacteristic, complemented
41#102×C173C2212yescomplemented
42C23×D173—22no—
43#14583825—24no—
44#14583825—42no—
45C2×Dic346C2311yescharacteristic
46C2×D692—22no—
47C2×D692—21no—
48C2×D692—21no—
49C2×D692—44no—
50C2×D692D411yescharacteristic
51C2×D692C2312yes—
52C2×C22⋊D173—22no—
53C2×C22⋊D173—22no—
54C2×C22⋊D173—22no—
55D4×D173—24no—
56D4×D173—42no—
57C22×Dic173—22no—
58C2×C4×D173C2312yes—
59#14583815—24no—
60#14583814D414yescomplemented
61#14583811—24no—
62#14583811—42no—
63#14583809—24no—
64C2×#13063436C2311yescharacteristic
65C2×#13063436C2311yescharacteristic
66C4⋊Dic173D411yescharacteristic
67C4⋊Dic173C2311yescharacteristic
68#14583805—21no—
69C22⋊Dic173—22no—
70#14583803—22no—
71#14583803D412yes—
72C4×Dic173—21no—
73C2×D4×C173—21no—
74C2×D4×C173—21no—
75C2×D4×C173C2312yes—
76C2×D4×C173C2311yescharacteristic
77C22×C4×C173—21no—
78D8×C173—24no—
79C2×C8×C173C2312yes—
80C4⋊C4×C173C2311yescharacteristic
81C22⋊C4×C173—21no—
82C42×C173C2311yescharacteristic
83C22×D173—22no—
84C22×D173—44no—
85C22×D173—44no—
86C22×D173—82no—
87C22×D173C2×D412yes—
88Dic346—21no—
89Dic346C2×D412yescomplemented
90D692—22no—
91D692—22no—
92D692—22no—
93D692—44no—
94D692—42no—
95D692—81no—
96C22⋊D173—44no—
97C22⋊D173—44no—
98C22⋊D173—82no—
99C2×Dic173—21no—
100C2×Dic173—21no—
101C2×Dic173—44no—
102C2×Dic173C2×D412yes—
103C4×D173—22no—
104C4×D173—22no—
105#13063436—21no—
106#13063436C2×D412yescomplemented
107C23×C173—21no—
108C23×C173—22no—
109D4×C173—22no—
110D4×C173—21no—
111D4×C173—22no—
112D4×C173—42no—
113D4×C173—81no—
114D4×C173C2×D412yes—
115C2×C4×C173—21no—
116C2×C4×C173—42no—
117C2×C4×C173C2×D411yescharacteristic
118C2×C4×C173C2×D411yescharacteristic
119C2×C4×C173C2411yescharacteristic
120C8×C173—22no—
121C22×C173—22no—
122C22×C173—41no—
123C22×C173—41no—
124C22×C173—42no—
125C22×C173—42no—
126C22×C173—42no—
127C22×C173C22×D411yescharacteristic
128C2×D173—22no—
129C2×D173—22no—
130C2×D173—22no—
131C2×D173—42no—
132C2×D173—42no—
133C2×D173—42no—
134C2×D173—88no—
135C4×C173—21no—
136C4×C173—41no—
137C4×C173AGL(1,ℤ/8)12yescomplemented
138C4×C173C22×D411yescharacteristic
139C4×C173H(5,𝔽2)11yescharacteristic
140Dic173—22no—
141Dic173—22no—
142Dic173—41no—
143C2×C173—42no—
144C2×C173—82no—
145C2×C173D4211yescharacteristic
146C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
147C2×C173#57611yescharacteristic
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#460—1731no—
158#459D17311yescharacteristic, complemented
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×SD16—1732no—
169C2×D8—1732no—
170C2×D8—1732no—
171C2×D8C2×D17312yes—
172C2×M16—1732no—
173#127—3461no—
174#127—1731no—
175#127C2×D17311yescharacteristic, complemented
176#121—3461no—
177#121—3462no—
178#121—1732no—
179#121—1732no—
180#120—3462no—
181C4×Q8—1731no—
182C4×D4—3461no—
183C4×D4—1731no—
184C4×D4C2×D17311yescharacteristic
185#106—1731no—
186C4⋊C8C2×D17311yescharacteristic, complemented
187#103—1731no—
188#102—1731no—
189#102—1732no—
190#102C2×D17312yescomplemented
191C22⋊C8—1732no—
192C4×C8—1731no—
193C24—3462no—
194#41—3464no—
195#41—6922no—
196C2×Q8—1731no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—3462no—
200C2×D4—3464no—
201C2×D4—3462no—
202C2×D4—3462no—
203C2×D4—3462no—
204C2×D4—3461no—
205C2×D4—3461no—
206C2×D4—1731no—
207C2×D4—1732no—
208C2×D4—6924no—
209C2×D4—6922no—
210C2×D4C22×D17312yes—
211C2×D4C22×D17311yescharacteristic
212C22×C4—21no—
213C22×C4—3462no—
214C22×C4—1732no—
215SD16—3464no—
216SD16—1734no—
217D8—24no—
218D8—3464no—
219D8—6922no—
220M16—3464no—
221C2×C8—1731no—
222C2×C8—1731no—
223C2×C8C22×D17312yes—
224C4⋊C4—3461no—
225C4⋊C4—1731no—
226C4⋊C4—1731no—
227C4⋊C4C22×D17311yescharacteristic
228C22⋊C4—21no—
229C22⋊C4—3462no—
230C22⋊C4—3462no—
231C22⋊C4—1732no—
232C42—3461no—
233C42C22×D17311yescharacteristic
234C23—21no—
235C23—22no—
236C23—3462no—
237C23—13842no—
238C23—1732no—
239C23—6924no—
240C23—6924no—
241Q8—3461no—
242Q8—1732no—
243D4—22no—
244D4—21no—
245D4—22no—
246D4—42no—
247D4—81no—
248D4—3462no—
249D4—3462no—
250D4—3462no—
251D4—13841no—
252D4—13842no—
253D4—6924no—
254D4—6924no—
255D4—6924no—
256D4—6922no—
257D4D4×D17312yes—
258C2×C4—21no—
259C2×C4—42no—
260C2×C4—3461no—
261C2×C4—3462no—
262C2×C4—3462no—
263C2×C4—3461no—
264C2×C4—1732no—
265C2×C4—6924no—
266C2×C4D4×D17311yescharacteristic
267C2×C4D4×D17311yescharacteristic
268C2×C4C23×D17311yescharacteristic
269C8—22no—
270C8—3461no—
271C8—1732no—
272C22—22no—
273C22—41no—
274C22—41no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—3462no—
279C22—3462no—
280C22—3462no—
281C22—13848no—
282C22—6922no—
283C22—6922no—
284C22—6922no—
285C22C2×D4×D17311yescharacteristic
286C4—21no—
287C4—41no—
288C4—3462no—
289C4—3462no—
290C4—6921no—
291C4#1914970312yescomplemented
292C4C2×D4×D17311yescharacteristic
293C4#1914972511yescharacteristic
294C2—42no—
295C2—82no—
296C2—13842no—
297C2—6922no—
298C2#2601212511yescharacteristic
299C2C2×#1914970311yescharacteristic
300C2#2601229511yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#459×C173C2
socleC22×C173C22×D4
hypercenter#459D173
hypocenterC173#3274
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#19149703C2semidirect
2. #26012161C2semidirect
3. D4×D692C2semidirect
4. #26011867C2semidirect
5. #26011743C2semidirect
6. #26011692C2semidirect
7. #26011637C2semidirect
8. #26011623C2semidirect
9. C4×#14583814C2semidirect
10. #26011568C2semidirect
11. #459×C173C2semidirect
12. C2×#14583825C22semidirect
16. C2×#14583815C22semidirect
18. C2×#14583811C22semidirect
19. C2×#14583811C22semidirect
20. C2×#14583809C22semidirect
26. #19149650C22semidirect
28. C4×Dic346C22semidirect
31. #19149610C22semidirect
32. #19149602C22semidirect
33. #19149601C22semidirect
35. #19149591C22semidirect
36. C4×#13063436C22semidirect
38. #127×C173C22semidirect
40. C4⋊C8×C173C22semidirect
41. #102×C173C22semidirect
60. #14583814D4semidirect
89. Dic346C2×D4semidirect
106. #13063436C2×D4semidirect
137. C4×C173AGL(1,ℤ/8)semidirect
150. C173#3274semidirect
158. #459D173semidirect
175. #127C2×D173semidirect
186. C4⋊C8C2×D173semidirect
190. #102C2×D173semidirect
291. C4#19149703semidirect

Maximal subgroups

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

subgroupindex
C2×#191497032
#260121612
D4×D6922
#260118672
#260117432
#260116922
#260116372
#260116232
C4×#145838142
#260115682
#459×C1732
#3274173