Finite Group Databasebeta

← C2×#30632430 #44564486

Subgroups — C2×#30632430 #44564486

Order 32128.

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#30632430C2116yesmaximal, direct product factor
2C2×#22548702C213yesmaximal, complemented
3C2×#22548701C216yesmaximal, complemented
4C2×#22548684C212yesmaximal, complemented
5C2×#22548683C213yesmaximal, complemented
6C2×#120×C251C211yesmaximal, characteristic, complemented
7C24×D251C2211yescharacteristic, complemented
8C22×Dic502C2213yescomplemented
9C22×C22⋊D251C2216yes—
10#22548702C22124yescomplemented
11#22548701C22148yescomplemented
12#22548684C22116yescomplemented
13#22548683C22124yescomplemented
14C2×C22⋊Dic251C2216yescomplemented
15C2×C22⋊Dic251C2213yescomplemented
16C2×(C2×D251)⋊C4C2216yescomplemented
17C2×C4×Dic251C2213yescomplemented
18C25×C251C2211yescharacteristic, complemented
19C22×D4×C251C2213yes—
20#120×C251C2218yescomplemented
21C2×C22⋊C4×C251C2213yescomplemented
22C23×D251—46no—
23C23×D251C2312yes—
24C23×D251C2314yescomplemented
25C2×Dic502—46no—
26C2×Dic502C23112yescomplemented
27C2×C22⋊D251—412no—
28C2×C22⋊D251C23124yes—
29C22×Dic251C2316yes—
30C22⋊Dic251C23112yescomplemented
31C22⋊Dic251C23124yescomplemented
32(C2×D251)⋊C4C23124yescomplemented
33C4×Dic251C23112yescomplemented
34C24×C251—46no—
35C24×C251C2313yes—
36C24×C251C2311yescharacteristic
37C24×C251C2314yescomplemented
38C2×D4×C251—46no—
39C2×D4×C251C23112yes—
40C22×C4×C251C2313yes—
41C22⋊C4×C251C23112yescomplemented
42C22×D251—212no—
43C22×D251—412no—
44C22×D251—416no—
45C22×D251C2414yes—
46Dic502—412no—
47C22⋊D251—424no—
48C2×Dic251—212no—
49C2×Dic251C24112yes—
50C23×C251—26no—
51C23×C251—26no—
52C23×C251—212no—
53C23×C251—412no—
54C23×C251—416no—
55C23×C251C2416yes—
56C23×C251C2412yes—
57C23×C251C2411yescharacteristic
58D4×C251—412no—
59C2×C4×C251—26no—
60C2×C4×C251C2416yes—
61C22×C251—26no—
62C22×C251—26no—
63C22×C251—212no—
64C22×C251—416no—
65C22×C251—412no—
66C22×C251—44no—
67C22×C251H(5,𝔽2)16yes—
68C22×C251C2511yescharacteristic
69C2×D251—212no—
70C2×D251—416no—
71C2×D251—48no—
72C4×C251—26no—
73Dic251—212no—
74C2×C251—42no—
75C2×C251—46no—
76C2×C251H(3,𝔽4)14yescomplemented
77C2×C251C2×H(5,𝔽2)13yes—
78D251—44no—
79C251C2×H(3,𝔽4)11yescharacteristic, complemented, Sylow
80C2×H(3,𝔽4)—2511nomaximal, Sylow
81H(3,𝔽4)—25116no—
82C2×#124—2513no—
83C2×#124—2516no—
84C2×#120—2512no—
85C2×#120—2513no—
86C2×#120D25111yescharacteristic, complemented
87C25—2511no—
88C25C2×D25111yescharacteristic, complemented
89C22×Q8—2513no—
90C22×D4—2516no—
91C22×D4C2×D25113yes—
92#124—25124no—
93#124—25148no—
94#120—25116no—
95#120—25124no—
96#120C2×D25118yescomplemented
97C2×C22⋊C4—2513no—
98C2×C22⋊C4—2516no—
99C2×C22⋊C4—2516no—
100C2×C22⋊C4C2×D25113yescomplemented
101C2×C42—2513no—
102C24—46no—
103C24—10046no—
104C24—2514no—
105C24—2512no—
106C24C22×D25111yescharacteristic
107C24C22×D25113yes—
108C24C22×D25114yescomplemented
109C2×Q8—10046no—
110C2×Q8—25112no—
111C2×D4—46no—
112C2×D4—100412no—
113C2×D4—25124no—
114C2×D4C22×D251112yes—
115C22×C4—2516no—
116C22×C4C22×D25113yes—
117C22⋊C4—25112no—
118C22⋊C4—25124no—
119C22⋊C4—25124no—
120C22⋊C4C22×D251112yescomplemented
121C42—25112no—
122C23—26no—
123C23—26no—
124C23—212no—
125C23—412no—
126C23—416no—
127C23—100412no—
128C23—100416no—
129C23—50212no—
130C23—2514no—
131C23C23×D25112yes—
132C23C23×D25116yes—
133C23C23×D25111yescharacteristic
134Q8—100412no—
135D4—412no—
136D4—100424no—
137C2×C4—26no—
138C2×C4—50212no—
139C2×C4—25112no—
140C2×C4C23×D25116yes—
141C22—26no—
142C22—26no—
143C22—212no—
144C22—416no—
145C22—412no—
146C22—44no—
147C22—100416no—
148C22—10048no—
149C22—50212no—
150C22#2254876016yes—
151C22C24×D25111yescharacteristic
152C4—26no—
153C4—50212no—
154C2—46no—
155C2—42no—
156C2—10044no—
157C2#3063243014yesdirect product factor
158C2C2×#2254876013yes—
subgroupG/H
centerC23C23×D251
commutator, cosocleC22×C251C25
FrattiniC22C24×D251
FittingC2×#120×C251C2
socleC23×C251C24
hypercenterC2×#120D251
hypocenterC251C2×H(3,𝔽4)
LCSG ▷ C22×C251 ▷ C251
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. #30632430C2direct (N × H)
2. C2×#22548702C2semidirect
3. C2×#22548701C2semidirect
4. C2×#22548684C2semidirect
5. C2×#22548683C2semidirect
6. C2×#120×C251C2semidirect
7. C24×D251C22semidirect
8. C22×Dic502C22semidirect
10. #22548702C22semidirect
11. #22548701C22semidirect
12. #22548684C22semidirect
13. #22548683C22semidirect
14. C2×C22⋊Dic251C22semidirect
15. C2×C22⋊Dic251C22semidirect
16. C2×(C2×D251)⋊C4C22semidirect
17. C2×C4×Dic251C22semidirect
18. C25×C251C22semidirect
20. #120×C251C22semidirect
21. C2×C22⋊C4×C251C22semidirect
24. C23×D251C23semidirect
26. C2×Dic502C23semidirect
30. C22⋊Dic251C23semidirect
31. C22⋊Dic251C23semidirect
32. (C2×D251)⋊C4C23semidirect
33. C4×Dic251C23semidirect
37. C24×C251C23semidirect
41. C22⋊C4×C251C23semidirect
76. C2×C251H(3,𝔽4)semidirect
79. C251C2×H(3,𝔽4)semidirect
86. C2×#120D251semidirect
88. C25C2×D251semidirect
96. #120C2×D251semidirect
100. C2×C22⋊C4C2×D251semidirect
108. C24C22×D251semidirect
120. C22⋊C4C22×D251semidirect
157. C2#30632430direct (N × H)

Maximal subgroups

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

subgroupindex
#306324302
C2×#225487022
C2×#225487012
C2×#225486842
C2×#225486832
C2×#120×C2512
C2×H(3,𝔽4)251