Finite Group Databasebeta

← #27185912

Subgroups — #27185912

Order 12928.

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#20266013C211yesmaximal, characteristic, complemented
2#20265988C218yesmaximal, complemented
3#20265980C218yesmaximal, complemented
4#20265975C216yesmaximal, complemented
5C2×#15692281C216yesmaximal, complemented
6#20265944C211yesmaximal, characteristic, complemented
7C2×H(5,𝔽2)×C101C211yesmaximal, characteristic, complemented
8#15692303—21no—
9#15692302C2214yescomplemented
10#15692301C2214yescomplemented
11#15692300—23no—
12C2×#13108074C2216yescomplemented
13#41×D101—24no—
14#15692286C2214yescomplemented
15#15692285C22112yescomplemented
16#15692281C22112yescomplemented
17#15692281C22124yescomplemented
18#15692280C22112yescomplemented
19#15692275C22124yescomplemented
20C2×#13108062C22112yescomplemented
21C2×#13108059C22112yescomplemented
22#15692262C2218yescomplemented
23C2×#13108058C2216yescomplemented
24H(5,𝔽2)×C101—24no—
25H(5,𝔽2)×C101C2218yescomplemented
26C2×#41×C101—21no—
27C2×#41×C101C2211yescharacteristic
28C2×#41×C101C2213yescomplemented
29C22×D4×C101—23no—
30C22×D4×C101C2213yescomplemented
31#13108075—24no—
32#13108074—26no—
33#13108074—26no—
34#13108074C2314yes—
35#13108074C23112yescomplemented
36#13108073—212no—
37C2×Dic202C2316yescomplemented
38Q8×D101—24no—
39C2×D404C2316yescomplemented
40C2×C22⋊D101—26no—
41D4×D101—212no—
42C2×C4×D101—26no—
43#13108064C2318yescomplemented
44#13108062C23124yescomplemented
45#13108061C2318yescomplemented
46#13108059C23124yescomplemented
47#13108058C2314yes—
48#13108058C23112yescomplemented
49C2×#11717579C23112yescomplemented
50C24×C101—23no—
51#41×C101—28no—
52#41×C101—26no—
53#41×C101D414yes—
54#41×C101C23112yes—
55#41×C101C2314yes—
56C2×Q8×C101D411yescharacteristic
57C2×Q8×C101C2311yescharacteristic
58C2×D4×C101—23no—
59C2×D4×C101—224no—
60C2×D4×C101—23no—
61C2×D4×C101—26no—
62C2×D4×C101D413yes—
63C2×D4×C101C2313yes—
64C2×D4×C101C23112yes—
65C22×C4×C101—23no—
66C22×C4×C101C2313yes—
67C23×C101—23no—
68C23×C101—212no—
69C23×C101—86no—
70C23×C101C2×D413yes—
71C22×D101—26no—
72Q8×C101—21no—
73Q8×C101C2×D414yes—
74Q8×C101C2412yes—
75D4×C101—212no—
76D4×C101—212no—
77D4×C101—23no—
78D4×C101C2×D4112yes—
79D4×C101C2416yes—
80C2×C4×C101—21no—
81C2×C4×C101—212no—
82C2×C4×C101C2×D413yes—
83C2×C4×C101C2×D416yes—
84C2×C4×C101C2411yescharacteristic
85C2×C4×C101C2416yes—
86C22⋊D101—212no—
87C22⋊D101—24no—
88C2×Dic101—26no—
89D404—26no—
90D404C2414yes—
91C4×D101—24no—
92C4×D101—212no—
93Dic202—26no—
94Dic202C2414yes—
95#11717579C2418yes—
96C22×C101—26no—
97C22×C101—43no—
98C22×C101—46no—
99C22×C101—812no—
100C22×C101C22×D416yes—
101C22×C101C22×D411yescharacteristic
102C2×D101—24no—
103C2×D101—86no—
104C4×C101—22no—
105C4×C101C22×D416yes—
106C4×C101C22×D411yescharacteristic
107C4×C101C2511yescharacteristic
108Dic101—24no—
109C2×C101—26no—
110C2×C101—21no—
111C2×C101—46no—
112C2×C101C23×D411yescharacteristic
113D101—44no—
114#3585—1011nomaximal, Sylow
115C101#358511yescharacteristic, complemented, Sylow
116#585—1011no—
117C2×H(5,𝔽2)D10111yescharacteristic, complemented
118#577—1018no—
119#576—1018no—
120#575—1016no—
121C2×AGL(1,ℤ/8)—1016no—
122#568—1011no—
1232−1+4—1014no—
1242−1+4—2021no—
125H(5,𝔽2)—24no—
126H(5,𝔽2)—1014no—
127H(5,𝔽2)—2023no—
128H(5,𝔽2)C2×D10118yescomplemented
129C2×#41—21no—
130C2×#41—1016no—
131C2×#41—2024no—
132C2×#41C2×D10111yescharacteristic
133C2×#41C2×D10113yescomplemented
134C22×D4—23no—
135C22×D4C2×D10113yescomplemented
136#137—10112no—
137#137—1014no—
138AGL(1,ℤ/8)—10112no—
139AGL(1,ℤ/8)—10112no—
140AGL(1,ℤ/8)—10124no—
141#135—10124no—
142C2×SD16—10112no—
143C2×D8—10112no—
144#131—1018no—
145C2×M16—1016no—
146C24—23no—
147#41—28no—
148#41—26no—
149#41—1014no—
150#41—10112no—
151#41—2024no—
152#41—20212no—
153#41—2026no—
154#41—2026no—
155#41C22⋊D10114yes—
156#41C22×D101112yes—
157#41C22×D10114yes—
158C2×Q8—1016no—
159C2×Q8—2024no—
160C2×Q8C22⋊D10111yescharacteristic
161C2×Q8C22×D10111yescharacteristic
162C2×D4—23no—
163C2×D4—224no—
164C2×D4—23no—
165C2×D4—26no—
166C2×D4—1016no—
167C2×D4—2026no—
168C2×D4—20212no—
169C2×D4C22⋊D10113yes—
170C2×D4C22×D101112yes—
171C2×D4C22×D10113yes—
172C22×C4—23no—
173C22×C4—2026no—
174C22×C4C22×D10113yes—
175Q16—1018no—
176SD16—1018no—
177SD16—10124no—
178D8—10124no—
179M16—1014no—
180M16—10112no—
181C2×C8—10112no—
182C23—23no—
183C23—212no—
184C23—86no—
185C23—2026no—
186C23C2×C22⋊D10113yes—
187Q8—21no—
188Q8—1014no—
189Q8—2026no—
190Q8C2×C22⋊D10114yes—
191Q8C23×D10112yes—
192D4—212no—
193D4—212no—
194D4—23no—
195D4—1014no—
196D4—2024no—
197D4—20212no—
198D4—2026no—
199D4C2×C22⋊D101112yes—
200D4C23×D10116yes—
201C2×C4—21no—
202C2×C4—212no—
203C2×C4—2026no—
204C2×C4—2024no—
205C2×C4—20212no—
206C2×C4C2×C22⋊D10113yes—
207C2×C4C2×C22⋊D10116yes—
208C2×C4C23×D10116yes—
209C2×C4C23×D10111yescharacteristic
210C8—1018no—
211C22—26no—
212C22—43no—
213C22—46no—
214C22—812no—
215C22—8086no—
216C22—2024no—
217C22C22×C22⋊D10111yescharacteristic
218C22C22×C22⋊D10116yes—
219C4—22no—
220C4—2024no—
221C4C22×C22⋊D10116yes—
222C4C22×C22⋊D10111yescharacteristic
223C4C24×D10111yescharacteristic
224C2—21no—
225C2—26no—
226C2—46no—
227C2—4044no—
228C2C23×C22⋊D10111yescharacteristic
subgroupG/H
centerC2C23×C22⋊D101
commutator, cosocleC4×C101C25
FrattiniC4C24×D101
FittingC2×H(5,𝔽2)×C101C2
socleC2×C101C23×D4
hypercenterC2×H(5,𝔽2)D101
hypocenterC101#3585
LCSG ▷ C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C2×#41 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #20266013C2semidirect
2. #20265988C2semidirect
3. #20265980C2semidirect
4. #20265975C2semidirect
5. C2×#15692281C2semidirect
6. #20265944C2semidirect
7. C2×H(5,𝔽2)×C101C2semidirect
9. #15692302C22semidirect
10. #15692301C22semidirect
12. C2×#13108074C22semidirect
14. #15692286C22semidirect
15. #15692285C22semidirect
16. #15692281C22semidirect
17. #15692281C22semidirect
18. #15692280C22semidirect
19. #15692275C22semidirect
20. C2×#13108062C22semidirect
21. C2×#13108059C22semidirect
22. #15692262C22semidirect
23. C2×#13108058C22semidirect
25. H(5,𝔽2)×C101C22semidirect
28. C2×#41×C101C22semidirect
30. C22×D4×C101C22semidirect
35. #13108074C23semidirect
37. C2×Dic202C23semidirect
39. C2×D404C23semidirect
43. #13108064C23semidirect
44. #13108062C23semidirect
45. #13108061C23semidirect
46. #13108059C23semidirect
48. #13108058C23semidirect
49. C2×#11717579C23semidirect
115. C101#3585semidirect
117. C2×H(5,𝔽2)D101semidirect
128. H(5,𝔽2)C2×D101semidirect
133. C2×#41C2×D101semidirect
135. C22×D4C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202660132
#202659882
#202659802
#202659752
C2×#156922812
#202659442
C2×H(5,𝔽2)×C1012
#3585101