Finite Group Databasebeta

← #31821185

Subgroups — #31821185

Order 17792.

1464 conjugacy classes of subgroups; only the 545 normal subgroups are shown.

normal subgroups only

This group has 1464 conjugacy classes of subgroups — too many to draw. Only its 545 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#22581789C211yesmaximal, characteristic, complemented
2C2×#17984888C211yesmaximal, characteristic, complemented
3#22581765C218yesmaximal, complemented
4#22581758C218yesmaximal, complemented
5#22581753C216yesmaximal, complemented
6C2×#17984870C216yesmaximal, complemented
7#568×C139C211yesmaximal, characteristic, complemented
8#17984890—21no—
9#17984889C2214yescomplemented
10#17984888—24no—
11#17984888C2218yescomplemented
12#17984888C2214yescomplemented
13#17984887—23no—
14C2×#14555786—21no—
15C2×#14555785C2213yescomplemented
16C2×#14555785C2216yescomplemented
17#41×D139—24no—
18C22×D556C2213yescomplemented
19C2×D4×D139—23no—
20#17984875C2214yescomplemented
21#17984875C22112yescomplemented
22#17984870C22112yescomplemented
23#17984870C22124yescomplemented
24#17984870C22112yescomplemented
25#17984864C22124yescomplemented
26C2×#14555774C22112yescomplemented
27C2×D1112C22112yescomplemented
28C2×#41×C139C2211yescharacteristic
29#131×C139C2218yescomplemented
30C2×M16×C139C2216yescomplemented
31C23×D139—23no—
32#14555786—24no—
33#14555786—28no—
34#14555785—26no—
35#14555785—26no—
36#14555785—26no—
37#14555785C23112yes—
38#14555785C2314yes—
39#14555785C23112yescomplemented
40#14555784—212no—
41C2×Dic278C2311yescharacteristic
42C2×Dic278C2316yescomplemented
43Q8×D139—24no—
44C2×D556—23no—
45C2×D556—26no—
46C2×D556C23112yes—
47C2×D556C2313yes—
48C2×D556C2316yescomplemented
49C2×C22⋊D139—23no—
50C2×C22⋊D139—26no—
51D4×D139—224no—
52D4×D139—212no—
53C2×C4×D139—23no—
54C2×C4×D139—26no—
55Dic556C2318yescomplemented
56#14555774C23124yescomplemented
57#14555774C2318yescomplemented
58D1112C23124yescomplemented
59#41×C139D414yes—
60#41×C139C2314yes—
61C2×Q8×C139D411yescharacteristic
62C2×D4×C139D413yes—
63C22×C4×C139C2313yes—
64M16×C139C2314yes—
65M16×C139C23112yescomplemented
66C2×C8×C139C23112yescomplemented
67C22×D139—23no—
68C22×D139—26no—
69C22×D139—212no—
70C22×D139—86no—
71Dic278—26no—
72Dic278—21no—
73Dic278C2414yes—
74Dic278C2412yes—
75D556—212no—
76D556—26no—
77D556—23no—
78D556C2414yes—
79D556C2416yes—
80C22⋊D139—212no—
81C22⋊D139—212no—
82C22⋊D139—24no—
83C2×Dic139—21no—
84C2×Dic139—26no—
85C4×D139—24no—
86C4×D139—212no—
87C4×D139—212no—
88C23×C139C2×D413yes—
89Q8×C139C2×D414yes—
90D4×C139C2×D4112yes—
91C2×C4×C139C2×D413yes—
92C2×C4×C139C2×D416yes—
93C2×C4×C139C2416yes—
94C2×C4×C139C2411yescharacteristic
95C8×C139C2418yes—
96C22×C139—43no—
97C22×C139C22×D411yescharacteristic
98C22×C139C22×D416yes—
99C2×D139—24no—
100C2×D139—26no—
101C2×D139—46no—
102C2×D139—86no—
103C2×D139—812no—
104C4×C139C22×D411yescharacteristic
105C4×C139C22×D416yes—
106C4×C139C2511yescharacteristic
107Dic139—22no—
108Dic139—24no—
109C2×C139—21no—
110C2×C139—26no—
111C2×C139C23×D411yescharacteristic
112D139—46no—
113D139—44no—
114C139#358511yescharacteristic, complemented, Sylow
115#3585—1391nomaximal, Sylow
116#585—1391no—
117C2×H(5,𝔽2)—1391no—
118#577—1398no—
119#576—1398no—
120#575—1396no—
121C2×AGL(1,ℤ/8)—1396no—
122#568D13911yescharacteristic, complemented
1232−1+4—2781no—
1242−1+4—1394no—
125H(5,𝔽2)—2784no—
126H(5,𝔽2)—2783no—
127H(5,𝔽2)—1394no—
128H(5,𝔽2)—1398no—
129C2×#41—2784no—
130C2×#41—2781no—
131C2×#41—1396no—
132C2×#41—1393no—
133C2×#41C2×D13911yescharacteristic
134C22×D4—2783no—
135C22×D4—1393no—
136#137—1394no—
137#137—13912no—
138AGL(1,ℤ/8)—13912no—
139AGL(1,ℤ/8)—13924no—
140AGL(1,ℤ/8)—13912no—
141#135—13924no—
142C2×SD16—13912no—
143C2×D8—13912no—
144#131C2×D13918yescomplemented
145C2×M16C2×D13916yescomplemented
146C24—2783no—
147#41—2788no—
148#41—2784no—
149#41—27812no—
150#41—2786no—
151#41—2786no—
152#41—2786no—
153#41—13912no—
154#41—13912no—
155#41—1394no—
156#41D55614yes—
157#41C22×D13914yes—
158C2×Q8—2784no—
159C2×Q8—1396no—
160C2×Q8—1391no—
161C2×Q8D55611yescharacteristic
162C2×D4—2783no—
163C2×D4—2786no—
164C2×D4—27824no—
165C2×D4—27812no—
166C2×D4—2783no—
167C2×D4—2786no—
168C2×D4—1396no—
169C2×D4—13912no—
170C2×D4—1393no—
171C2×D4D55613yes—
172C22×C4—2783no—
173C22×C4—2786no—
174C22×C4C22×D13913yes—
175Q16—1398no—
176SD16—13924no—
177SD16—1398no—
178D8—13924no—
179M16C22×D13914yes—
180M16C22×D139112yescomplemented
181C2×C8C22×D139112yescomplemented
182C23—2783no—
183C23—2786no—
184C23—27812no—
185C23—11126no—
186C23C2×D55613yes—
187Q8—2786no—
188Q8—2781no—
189Q8—1394no—
190Q8—1392no—
191Q8C2×D55614yes—
192D4—27812no—
193D4—2784no—
194D4—27812no—
195D4—2783no—
196D4—27812no—
197D4—2786no—
198D4—1394no—
199D4—1396no—
200D4C2×D556112yes—
201C2×C4—2781no—
202C2×C4—2786no—
203C2×C4—2784no—
204C2×C4—27812no—
205C2×C4—27812no—
206C2×C4C2×D55613yes—
207C2×C4C2×D55616yes—
208C2×C4C23×D13916yes—
209C2×C4C23×D13911yescharacteristic
210C8C23×D13918yes—
211C22—43no—
212C22—2784no—
213C22—2786no—
214C22—5566no—
215C22—111212no—
216C22—11126no—
217C22C22×D55611yescharacteristic
218C22C22×D55616yes—
219C4—2782no—
220C4—2784no—
221C4C22×D55611yescharacteristic
222C4C22×D55616yes—
223C4C24×D13911yescharacteristic
224C2—21no—
225C2—26no—
226C2—5566no—
227C2—5564no—
228C2C23×D55611yescharacteristic
subgroupG/H
centerC2C23×D556
commutator, cosocleC4×C139C25
FrattiniC4C24×D139
Fitting#568×C139C2
socleC2×C139C23×D4
hypercenter#568D139
hypocenterC139#3585
LCSG ▷ C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C2 ◁ C2×#41 ◁ #568

Complements

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

NH ≅ G/Ntype
1. #22581789C2semidirect
2. C2×#17984888C2semidirect
3. #22581765C2semidirect
4. #22581758C2semidirect
5. #22581753C2semidirect
6. C2×#17984870C2semidirect
7. #568×C139C2semidirect
9. #17984889C22semidirect
11. #17984888C22semidirect
12. #17984888C22semidirect
15. C2×#14555785C22semidirect
16. C2×#14555785C22semidirect
18. C22×D556C22semidirect
20. #17984875C22semidirect
21. #17984875C22semidirect
22. #17984870C22semidirect
23. #17984870C22semidirect
24. #17984870C22semidirect
25. #17984864C22semidirect
26. C2×#14555774C22semidirect
27. C2×D1112C22semidirect
29. #131×C139C22semidirect
30. C2×M16×C139C22semidirect
39. #14555785C23semidirect
42. C2×Dic278C23semidirect
48. C2×D556C23semidirect
55. Dic556C23semidirect
56. #14555774C23semidirect
57. #14555774C23semidirect
58. D1112C23semidirect
65. M16×C139C23semidirect
66. C2×C8×C139C23semidirect
114. C139#3585semidirect
122. #568D139semidirect
144. #131C2×D139semidirect
145. C2×M16C2×D139semidirect
180. M16C22×D139semidirect
181. C2×C8C22×D139semidirect

Maximal subgroups

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

subgroupindex
#225817892
C2×#179848882
#225817652
#225817582
#225817532
C2×#179848702
#568×C1392
#3585139