Finite Group Databasebeta

← #36365182

Subgroups — #36365182

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
3#26012126C211yesmaximal, characteristic, complemented
4#26011741C212yesmaximal, complemented
5#26011688C211yesmaximal, characteristic, complemented
6#26011683C211yesmaximal, characteristic, complemented
7#26011633C212yesmaximal, complemented
8#26011623C212yesmaximal, complemented
9C4×#14583815C211yesmaximal, characteristic, complemented
10#26011570C211yesmaximal, characteristic, complemented
11#493×C173C211yesmaximal, characteristic, complemented
12C2×#14583825C2212yescomplemented
13C2×D4×D173—21no—
14C2×D4×D173C2212yes—
15#19149703—28no—
16C2×#14583815C2211yescharacteristic
17C2×#14583814C2212yescomplemented
18C2×#14583811C2212yes—
19C2×#14583809C2212yescomplemented
20#19149675—21no—
21#19149651—22no—
22#19149650—21no—
23#19149650C2212yes—
24#19149650C2212yescomplemented
25#19149647—22no—
26C4×Dic346C2211yescharacteristic, complemented
27C4×D692C2211yescharacteristic
28C4×D692C2211yescharacteristic
29D4×Dic173—21no—
30#19149611C2211yescharacteristic, complemented
31C4⋊#13063436C2211yescharacteristic, complemented
32#19149604C2211yescharacteristic, complemented
33#19149601C2211yescharacteristic
34#19149600C2212yescomplemented
35#19149600C2212yescomplemented
36#19149591C2212yescomplemented
37C2×D8×C173C2212yescomplemented
38C2×D8×C173C2212yescomplemented
39#127×C173C2211yescharacteristic
40#127×C173C2211yescharacteristic, complemented
41C4×C8×C173C2211yescharacteristic, complemented
42C23×D173—22no—
43#14583825—24no—
44#14583825—42no—
45C2×Dic346C2311yescharacteristic
46C2×D692D411yescharacteristic
47C2×D692C2311yescharacteristic
48C2×C22⋊D173—22no—
49C2×C22⋊D173—22no—
50C2×C22⋊D173—22no—
51C2×C22⋊D173—22no—
52D4×D173—24no—
53D4×D173—42no—
54D4×D173—44no—
55C22×Dic173—22no—
56C2×C4×D173—21no—
57C2×C4×D173C2312yes—
58#14583815D414yescomplemented
59#14583814—24no—
60#14583811—24no—
61#14583809—24no—
62C2×#13063436C2312yes—
63C4⋊Dic173D411yescharacteristic
64C4⋊Dic173C2311yescharacteristic
65C4⋊Dic173C2311yescharacteristic
66#14583805—21no—
67C22⋊Dic173—22no—
68C22⋊Dic173—22no—
69#14583803—21no—
70#14583803D412yes—
71C4×Dic173—21no—
72C2×D4×C173—21no—
73C2×D4×C173—21no—
74C2×D4×C173—21no—
75C2×D4×C173—21no—
76C2×D4×C173C2312yes—
77C2×D4×C173C2312yes—
78D8×C173—24no—
79D8×C173—42no—
80C2×C8×C173C2311yescharacteristic
81C2×C8×C173C2311yescharacteristic
82C42×C173C2311yescharacteristic
83C22×D173—21no—
84C22×D173—44no—
85C22×D173—44no—
86C22×D173—82no—
87C22×D173C2×D412yes—
88Dic346—21no—
89Dic346C2×D412yescomplemented
90D692—22no—
91D692—21no—
92D692C2×D412yes—
93C22⋊D173—44no—
94C22⋊D173—44no—
95C22⋊D173—44no—
96C22⋊D173—82no—
97C2×Dic173—21no—
98C2×Dic173—21no—
99C2×Dic173—21no—
100C2×Dic173—44no—
101C2×Dic173C2×D412yes—
102C4×D173—22no—
103C4×D173—22no—
104C4×D173—42no—
105#13063436—22no—
106C23×C173—22no—
107C23×C173—22no—
108D4×C173—22no—
109D4×C173—22no—
110D4×C173—22no—
111D4×C173—22no—
112D4×C173—42no—
113D4×C173—42no—
114D4×C173—81no—
115D4×C173—81no—
116C2×C4×C173C2×D411yescharacteristic
117C2×C4×C173C2×D411yescharacteristic
118C2×C4×C173C2411yescharacteristic
119C8×C173—21no—
120C8×C173C2×D412yescomplemented
121C22×C173—42no—
122C22×C173—42no—
123C22×C173—42no—
124C22×C173—42no—
125C22×C173—42no—
126C22×C173—42no—
127C22×C173C22×D411yescharacteristic
128C2×D173—22no—
129C2×D173—22no—
130C2×D173—22no—
131C2×D173—22no—
132C2×D173—41no—
133C2×D173—41no—
134C2×D173—88no—
135C4×C173—21no—
136C4×C173AGL(1,ℤ/8)12yescomplemented
137C4×C173C22×D411yescharacteristic
138C4×C173H(5,𝔽2)11yescharacteristic
139Dic173—22no—
140Dic173—22no—
141Dic173—41no—
142Dic173—41no—
143C2×C173—82no—
144C2×C173—82no—
145C2×C173D4211yescharacteristic
146C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
147C2×C173#57611yescharacteristic
148D173—42no—
149D173—42no—
150C173#327411yescharacteristic, complemented, Sylow
151#3274—1731nomaximal, Sylow
152C2×AGL(1,ℤ/8)—1732no—
153#550—1731no—
154D42—1731no—
155#493D17311yescharacteristic, complemented
156#469—1732no—
157#460—1731no—
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—1732no—
168C2×SD16—1731no—
169C2×D8—1732no—
170C2×D8C2×D17312yescomplemented
171C2×D8C2×D17312yescomplemented
172C2×M16—1732no—
173#127—3461no—
174#127C2×D17311yescharacteristic
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×D4—1731no—
185#106—1731no—
186C4⋊C8—1731no—
187#103—1731no—
188#102—1732no—
189#102—1732no—
190#102—1731no—
191C22⋊C8—1732no—
192C4×C8C2×D17311yescharacteristic, complemented
193C24—3462no—
194#41—3464no—
195#41—6922no—
196C2×Q8—1731no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—3464no—
202C2×D4—3462no—
203C2×D4—3462no—
204C2×D4—3462no—
205C2×D4—3462no—
206C2×D4—1731no—
207C2×D4—1731no—
208C2×D4—6922no—
209C2×D4—6924no—
210C2×D4C22×D17312yes—
211C2×D4C22×D17312yes—
212C22×C4—3462no—
213C22×C4—3461no—
214C22×C4—1732no—
215SD16—3464no—
216SD16—1734no—
217D8—24no—
218D8—42no—
219D8—3464no—
220M16—3464no—
221C2×C8—1732no—
222C2×C8C22×D17311yescharacteristic
223C2×C8C22×D17311yescharacteristic
224C4⋊C4—3461no—
225C4⋊C4—1731no—
226C4⋊C4—1731no—
227C4⋊C4—1731no—
228C22⋊C4—3462no—
229C22⋊C4—3462no—
230C22⋊C4—3461no—
231C22⋊C4—1732no—
232C42—3461no—
233C42C22×D17311yescharacteristic
234C23—22no—
235C23—22no—
236C23—3461no—
237C23—13842no—
238C23—1732no—
239C23—6924no—
240C23—6924no—
241Q8—3461no—
242Q8—1732no—
243D4—22no—
244D4—22no—
245D4—22no—
246D4—22no—
247D4—42no—
248D4—42no—
249D4—81no—
250D4—81no—
251D4—3462no—
252D4—3461no—
253D4—13842no—
254D4—1732no—
255D4—6924no—
256D4—6924no—
257D4—6924no—
258C2×C4—3461no—
259C2×C4—3462no—
260C2×C4—3462no—
261C2×C4—3461no—
262C2×C4—3461no—
263C2×C4—1732no—
264C2×C4—6924no—
265C2×C4—6922no—
266C2×C4D4×D17311yescharacteristic
267C2×C4D4×D17311yescharacteristic
268C2×C4C23×D17311yescharacteristic
269C8—21no—
270C8—3462no—
271C8D4×D17312yescomplemented
272C22—42no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—3462no—
279C22—3462no—
280C22—3462no—
281C22—3462no—
282C22—13848no—
283C22—6921no—
284C22—6921no—
285C22C2×D4×D17311yescharacteristic
286C4—21no—
287C4—3462no—
288C4—3462no—
289C4—6921no—
290C4—6921no—
291C4#1914970312yescomplemented
292C4C2×D4×D17311yescharacteristic
293C4#1914972411yescharacteristic
294C2—82no—
295C2—82no—
296C2—6922no—
297C2—6922no—
298C2#2601212611yescharacteristic
299C2C2×#1914970311yescharacteristic
300C2#2601229211yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#493×C173C2
socleC22×C173C22×D4
hypercenter#493D173
hypocenterC173#3274
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #493

Complements

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

NH ≅ G/Ntype
1. C2×#19149703C2semidirect
2. #26012161C2semidirect
3. #26012126C2semidirect
4. #26011741C2semidirect
5. #26011688C2semidirect
6. #26011683C2semidirect
7. #26011633C2semidirect
8. #26011623C2semidirect
9. C4×#14583815C2semidirect
10. #26011570C2semidirect
11. #493×C173C2semidirect
12. C2×#14583825C22semidirect
17. C2×#14583814C22semidirect
19. C2×#14583809C22semidirect
24. #19149650C22semidirect
26. C4×Dic346C22semidirect
30. #19149611C22semidirect
31. C4⋊#13063436C22semidirect
32. #19149604C22semidirect
34. #19149600C22semidirect
35. #19149600C22semidirect
36. #19149591C22semidirect
37. C2×D8×C173C22semidirect
38. C2×D8×C173C22semidirect
40. #127×C173C22semidirect
41. C4×C8×C173C22semidirect
58. #14583815D4semidirect
89. Dic346C2×D4semidirect
120. C8×C173C2×D4semidirect
136. C4×C173AGL(1,ℤ/8)semidirect
150. C173#3274semidirect
155. #493D173semidirect
170. C2×D8C2×D173semidirect
171. C2×D8C2×D173semidirect
175. #127C2×D173semidirect
192. C4×C8C2×D173semidirect
271. C8D4×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
#260121262
#260117412
#260116882
#260116832
#260116332
#260116232
C4×#145838152
#260115702
#493×C1732
#3274173