Finite Group Databasebeta

← C2×#20265920 #27185253

Subgroups — C2×#20265920 #27185253

Order 12928.

1408 conjugacy classes of subgroups; only the 469 normal subgroups are shown.

normal subgroups only

This group has 1408 conjugacy classes of subgroups — too many to draw. Only its 469 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#20265920C2116yesmaximal, direct product factor
2C2×#15692242C213yesmaximal, complemented
3C2×#15692241C216yesmaximal, complemented
4C2×#15692224C212yesmaximal, complemented
5C2×#15692223C213yesmaximal, complemented
6C2×#120×C101C211yesmaximal, characteristic, complemented
7C24×D101C2211yescharacteristic, complemented
8C22×Dic202C2213yescomplemented
9C22×C22⋊D101C2216yes—
10#15692242C22124yescomplemented
11#15692241C22148yescomplemented
12#15692224C22116yescomplemented
13#15692223C22124yescomplemented
14C2×C22⋊Dic101C2213yescomplemented
15C2×C22⋊Dic101C2216yescomplemented
16C2×#13108051C2216yescomplemented
17C2×C4×Dic101C2213yescomplemented
18C25×C101C2211yescharacteristic, complemented
19C22×D4×C101C2213yes—
20#120×C101C2218yescomplemented
21C2×C22⋊C4×C101C2213yescomplemented
22C23×D101—46no—
23C23×D101C2312yes—
24C23×D101C2314yescomplemented
25C2×Dic202—46no—
26C2×Dic202C23112yescomplemented
27C2×C22⋊D101—412no—
28C2×C22⋊D101C23124yes—
29C22×Dic101C2316yes—
30C22⋊Dic101C23112yescomplemented
31C22⋊Dic101C23124yescomplemented
32#13108051C23124yescomplemented
33C4×Dic101C23112yescomplemented
34C24×C101—46no—
35C24×C101C2313yes—
36C24×C101C2311yescharacteristic
37C24×C101C2314yescomplemented
38C2×D4×C101—46no—
39C2×D4×C101C23112yes—
40C22×C4×C101C2313yes—
41C22⋊C4×C101C23112yescomplemented
42C23×C101—26no—
43C23×C101—26no—
44C23×C101—212no—
45C23×C101—412no—
46C23×C101—416no—
47C23×C101C2416yes—
48C23×C101C2412yes—
49C23×C101C2411yescharacteristic
50C22×D101—212no—
51C22×D101—412no—
52C22×D101—416no—
53C22×D101C2414yes—
54D4×C101—412no—
55C2×C4×C101—26no—
56C2×C4×C101C2416yes—
57C22⋊D101—424no—
58C2×Dic101—212no—
59C2×Dic101C24112yes—
60Dic202—412no—
61C22×C101—212no—
62C22×C101—26no—
63C22×C101—26no—
64C22×C101—416no—
65C22×C101—44no—
66C22×C101—412no—
67C22×C101H(5,𝔽2)16yes—
68C22×C101C2511yescharacteristic
69C2×D101—212no—
70C2×D101—416no—
71C2×D101—48no—
72C4×C101—26no—
73Dic101—212no—
74C2×C101—42no—
75C2×C101—46no—
76C2×C101H(3,𝔽4)14yescomplemented
77C2×C101C2×H(5,𝔽2)13yes—
78D101—44no—
79C2×H(3,𝔽4)—1011nomaximal, Sylow
80C101C2×H(3,𝔽4)11yescharacteristic, complemented, Sylow
81H(3,𝔽4)—10116no—
82C2×#124—1013no—
83C2×#124—1016no—
84C2×#120—1012no—
85C2×#120—1013no—
86C2×#120D10111yescharacteristic, complemented
87C25—1011no—
88C25C2×D10111yescharacteristic, complemented
89C22×Q8—1013no—
90C22×D4—1016no—
91C22×D4C2×D10113yes—
92#124—10124no—
93#124—10148no—
94#120—10116no—
95#120—10124no—
96#120C2×D10118yescomplemented
97C2×C22⋊C4—1013no—
98C2×C22⋊C4—1016no—
99C2×C22⋊C4—1016no—
100C2×C22⋊C4C2×D10113yescomplemented
101C2×C42—1013no—
102C24—46no—
103C24—1014no—
104C24—1012no—
105C24—4046no—
106C24C22×D10111yescharacteristic
107C24C22×D10113yes—
108C24C22×D10114yescomplemented
109C2×Q8—10112no—
110C2×Q8—4046no—
111C2×D4—46no—
112C2×D4—10124no—
113C2×D4—40412no—
114C2×D4C22×D101112yes—
115C22×C4—1016no—
116C22×C4C22×D10113yes—
117C22⋊C4—10112no—
118C22⋊C4—10124no—
119C22⋊C4—10124no—
120C22⋊C4C22×D101112yescomplemented
121C42—10112no—
122C23—26no—
123C23—26no—
124C23—212no—
125C23—412no—
126C23—416no—
127C23—1014no—
128C23—40412no—
129C23—40416no—
130C23—20212no—
131C23C23×D10112yes—
132C23C23×D10116yes—
133C23C23×D10111yescharacteristic
134Q8—40412no—
135D4—412no—
136D4—40424no—
137C2×C4—26no—
138C2×C4—10112no—
139C2×C4—20212no—
140C2×C4C23×D10116yes—
141C22—26no—
142C22—26no—
143C22—212no—
144C22—416no—
145C22—412no—
146C22—44no—
147C22—40416no—
148C22—4048no—
149C22—20212no—
150C22#1569230016yes—
151C22C24×D10111yescharacteristic
152C4—26no—
153C4—20212no—
154C2—46no—
155C2—42no—
156C2—4044no—
157C2#2026592014yesdirect product factor
158C2C2×#1569230013yes—
subgroupG/H
centerC23C23×D101
commutator, cosocleC22×C101C25
FrattiniC22C24×D101
FittingC2×#120×C101C2
socleC23×C101C24
hypercenterC2×#120D101
hypocenterC101C2×H(3,𝔽4)
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. #20265920C2direct (N × H)
2. C2×#15692242C2semidirect
3. C2×#15692241C2semidirect
4. C2×#15692224C2semidirect
5. C2×#15692223C2semidirect
6. C2×#120×C101C2semidirect
7. C24×D101C22semidirect
8. C22×Dic202C22semidirect
10. #15692242C22semidirect
11. #15692241C22semidirect
12. #15692224C22semidirect
13. #15692223C22semidirect
14. C2×C22⋊Dic101C22semidirect
15. C2×C22⋊Dic101C22semidirect
16. C2×#13108051C22semidirect
17. C2×C4×Dic101C22semidirect
18. C25×C101C22semidirect
20. #120×C101C22semidirect
21. C2×C22⋊C4×C101C22semidirect
24. C23×D101C23semidirect
26. C2×Dic202C23semidirect
30. C22⋊Dic101C23semidirect
31. C22⋊Dic101C23semidirect
32. #13108051C23semidirect
33. C4×Dic101C23semidirect
37. C24×C101C23semidirect
41. C22⋊C4×C101C23semidirect
76. C2×C101H(3,𝔽4)semidirect
80. C101C2×H(3,𝔽4)semidirect
86. C2×#120D101semidirect
88. C25C2×D101semidirect
96. #120C2×D101semidirect
100. C2×C22⋊C4C2×D101semidirect
108. C24C22×D101semidirect
120. C22⋊C4C22×D101semidirect
157. C2#20265920direct (N × H)

Maximal subgroups

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

subgroupindex
#202659202
C2×#156922422
C2×#156922412
C2×#156922242
C2×#156922232
C2×#120×C1012
C2×H(3,𝔽4)101