Finite Group Databasebeta

← #31821184

Subgroups — #31821184

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
2#22581764C218yesmaximal, complemented
3#22581756C218yesmaximal, complemented
4#22581751C216yesmaximal, complemented
5C2×#17984868C216yesmaximal, complemented
6#22581720C211yesmaximal, characteristic, complemented
7C2×H(5,𝔽2)×C139C211yesmaximal, characteristic, complemented
8#17984890—21no—
9#17984889C2214yescomplemented
10#17984888C2214yescomplemented
11#17984887—23no—
12C2×#14555785C2216yescomplemented
13#41×D139—24no—
14#17984873C2214yescomplemented
15#17984872C22112yescomplemented
16#17984868C22112yescomplemented
17#17984868C22124yescomplemented
18#17984867C22112yescomplemented
19#17984862C22124yescomplemented
20C2×#14555773C22112yescomplemented
21C2×#14555770C22112yescomplemented
22#17984849C2218yescomplemented
23C2×#14555769C2216yescomplemented
24H(5,𝔽2)×C139—24no—
25H(5,𝔽2)×C139C2218yescomplemented
26C2×#41×C139—21no—
27C2×#41×C139C2211yescharacteristic
28C2×#41×C139C2213yescomplemented
29C22×D4×C139—23no—
30C22×D4×C139C2213yescomplemented
31#14555786—24no—
32#14555785—26no—
33#14555785—26no—
34#14555785C2314yes—
35#14555785C23112yescomplemented
36#14555784—212no—
37C2×Dic278C2316yescomplemented
38Q8×D139—24no—
39C2×D556C2316yescomplemented
40C2×C22⋊D139—26no—
41D4×D139—212no—
42C2×C4×D139—26no—
43#14555775C2318yescomplemented
44#14555773C23124yescomplemented
45#14555772C2318yescomplemented
46#14555770C23124yescomplemented
47#14555769C2314yes—
48#14555769C23112yescomplemented
49C2×C139⋊C8C23112yescomplemented
50C24×C139—23no—
51#41×C139—28no—
52#41×C139—26no—
53#41×C139D414yes—
54#41×C139C23112yes—
55#41×C139C2314yes—
56C2×Q8×C139D411yescharacteristic
57C2×Q8×C139C2311yescharacteristic
58C2×D4×C139—23no—
59C2×D4×C139—224no—
60C2×D4×C139—23no—
61C2×D4×C139—26no—
62C2×D4×C139D413yes—
63C2×D4×C139C23112yes—
64C2×D4×C139C2313yes—
65C22×C4×C139—23no—
66C22×C4×C139C2313yes—
67C22×D139—26no—
68Dic278—26no—
69Dic278C2414yes—
70D556—26no—
71D556C2414yes—
72C22⋊D139—212no—
73C22⋊D139—24no—
74C2×Dic139—26no—
75C4×D139—24no—
76C4×D139—212no—
77C139⋊C8C2418yes—
78C23×C139—23no—
79C23×C139—212no—
80C23×C139—86no—
81C23×C139C2×D413yes—
82Q8×C139—21no—
83Q8×C139C2×D414yes—
84Q8×C139C2412yes—
85D4×C139—212no—
86D4×C139—212no—
87D4×C139—23no—
88D4×C139C2×D4112yes—
89D4×C139C2416yes—
90C2×C4×C139—21no—
91C2×C4×C139—212no—
92C2×C4×C139C2×D413yes—
93C2×C4×C139C2×D416yes—
94C2×C4×C139C2411yescharacteristic
95C2×C4×C139C2416yes—
96C22×C139—26no—
97C22×C139—43no—
98C22×C139—46no—
99C22×C139—812no—
100C22×C139C22×D411yescharacteristic
101C22×C139C22×D416yes—
102C2×D139—24no—
103C2×D139—86no—
104C4×C139—22no—
105C4×C139C22×D416yes—
106C4×C139C22×D411yescharacteristic
107C4×C139C2511yescharacteristic
108Dic139—24no—
109C2×C139—26no—
110C2×C139—21no—
111C2×C139—46no—
112C2×C139C23×D411yescharacteristic
113D139—44no—
114C139#358511yescharacteristic, complemented, Sylow
115#3585—1391nomaximal, Sylow
116#585—1391no—
117C2×H(5,𝔽2)D13911yescharacteristic, complemented
118#577—1398no—
119#576—1398no—
120#575—1396no—
121C2×AGL(1,ℤ/8)—1396no—
122#568—1391no—
1232−1+4—2781no—
1242−1+4—1394no—
125H(5,𝔽2)—24no—
126H(5,𝔽2)—2783no—
127H(5,𝔽2)—1394no—
128H(5,𝔽2)C2×D13918yescomplemented
129C2×#41—21no—
130C2×#41—2784no—
131C2×#41—1396no—
132C2×#41C2×D13911yescharacteristic
133C2×#41C2×D13913yescomplemented
134C22×D4—23no—
135C22×D4C2×D13913yescomplemented
136#137—13912no—
137#137—1394no—
138AGL(1,ℤ/8)—13912no—
139AGL(1,ℤ/8)—13912no—
140AGL(1,ℤ/8)—13924no—
141#135—13924no—
142C2×SD16—13912no—
143C2×D8—13912no—
144#131—1398no—
145C2×M16—1396no—
146C24—23no—
147#41—28no—
148#41—26no—
149#41—2784no—
150#41—27812no—
151#41—2786no—
152#41—2786no—
153#41—13912no—
154#41—1394no—
155#41C22⋊D13914yes—
156#41C22×D139112yes—
157#41C22×D13914yes—
158C2×Q8—2784no—
159C2×Q8—1396no—
160C2×Q8C22⋊D13911yescharacteristic
161C2×Q8C22×D13911yescharacteristic
162C2×D4—23no—
163C2×D4—224no—
164C2×D4—23no—
165C2×D4—26no—
166C2×D4—2786no—
167C2×D4—27812no—
168C2×D4—1396no—
169C2×D4C22⋊D13913yes—
170C2×D4C22×D139112yes—
171C2×D4C22×D13913yes—
172C22×C4—23no—
173C22×C4—2786no—
174C22×C4C22×D13913yes—
175Q16—1398no—
176SD16—1398no—
177SD16—13924no—
178D8—13924no—
179M16—1394no—
180M16—13912no—
181C2×C8—13912no—
182C23—23no—
183C23—212no—
184C23—86no—
185C23—2786no—
186C23C2×C22⋊D13913yes—
187Q8—21no—
188Q8—2786no—
189Q8—1394no—
190Q8C2×C22⋊D13914yes—
191Q8C23×D13912yes—
192D4—212no—
193D4—212no—
194D4—23no—
195D4—2784no—
196D4—27812no—
197D4—2786no—
198D4—1394no—
199D4C2×C22⋊D139112yes—
200D4C23×D13916yes—
201C2×C4—21no—
202C2×C4—212no—
203C2×C4—2786no—
204C2×C4—2784no—
205C2×C4—27812no—
206C2×C4C2×C22⋊D13913yes—
207C2×C4C2×C22⋊D13916yes—
208C2×C4C23×D13916yes—
209C2×C4C23×D13911yescharacteristic
210C8—1398no—
211C22—26no—
212C22—43no—
213C22—46no—
214C22—812no—
215C22—2784no—
216C22—11126no—
217C22C22×C22⋊D13911yescharacteristic
218C22C22×C22⋊D13916yes—
219C4—22no—
220C4—2784no—
221C4C22×C22⋊D13916yes—
222C4C22×C22⋊D13911yescharacteristic
223C4C24×D13911yescharacteristic
224C2—21no—
225C2—26no—
226C2—46no—
227C2—5564no—
228C2C23×C22⋊D13911yescharacteristic
subgroupG/H
centerC2C23×C22⋊D139
commutator, cosocleC4×C139C25
FrattiniC4C24×D139
FittingC2×H(5,𝔽2)×C139C2
socleC2×C139C23×D4
hypercenterC2×H(5,𝔽2)D139
hypocenterC139#3585
LCSG ▷ C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C2 ◁ C2×#41 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #22581789C2semidirect
2. #22581764C2semidirect
3. #22581756C2semidirect
4. #22581751C2semidirect
5. C2×#17984868C2semidirect
6. #22581720C2semidirect
7. C2×H(5,𝔽2)×C139C2semidirect
9. #17984889C22semidirect
10. #17984888C22semidirect
12. C2×#14555785C22semidirect
14. #17984873C22semidirect
15. #17984872C22semidirect
16. #17984868C22semidirect
17. #17984868C22semidirect
18. #17984867C22semidirect
19. #17984862C22semidirect
20. C2×#14555773C22semidirect
21. C2×#14555770C22semidirect
22. #17984849C22semidirect
23. C2×#14555769C22semidirect
25. H(5,𝔽2)×C139C22semidirect
28. C2×#41×C139C22semidirect
30. C22×D4×C139C22semidirect
35. #14555785C23semidirect
37. C2×Dic278C23semidirect
39. C2×D556C23semidirect
43. #14555775C23semidirect
44. #14555773C23semidirect
45. #14555772C23semidirect
46. #14555770C23semidirect
48. #14555769C23semidirect
49. C2×C139⋊C8C23semidirect
114. C139#3585semidirect
117. C2×H(5,𝔽2)D139semidirect
128. H(5,𝔽2)C2×D139semidirect
133. C2×#41C2×D139semidirect
135. C22×D4C2×D139semidirect

Maximal subgroups

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

subgroupindex
#225817892
#225817642
#225817562
#225817512
C2×#179848682
#225817202
C2×H(5,𝔽2)×C1392
#3585139