Finite Group Databasebeta

← C2×#22550711 #30653537

Subgroups — C2×#22550711 #30653537

Order 16256.

5956 conjugacy classes of subgroups; only the 3205 normal subgroups are shown.

normal subgroups only

This group has 5956 conjugacy classes of subgroups — too many to draw. Only its 3205 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#22550711C2132yesmaximal, direct product factor
2C2×#16861256C216yesmaximal, complemented
3C2×#16861254C219yesmaximal, complemented
4C22×#14523247C219yesmaximal, complemented
5C2×#41×D127C216yesmaximal, complemented
6C2×H(5,𝔽2)×C127C211yesmaximal, characteristic, complemented
7#16861256C22196yescomplemented
8#16861254C221144yescomplemented
9C2×#14523249C2212yescomplemented
10C2×#14523248C22118yescomplemented
11C2×#14523247C22118yescomplemented
12C2×#14523247C22136yescomplemented
13C2×#14523247C221144yescomplemented
14#41×D127C22196yescomplemented
15C22×Dic254C2219yescomplemented
16C2×Q8×D127C2216yescomplemented
17C22×C22⋊D127C22118yescomplemented
18C2×D4×D127C22118yescomplemented
19C23×Dic127C2216yescomplemented
20C22×C4×D127C2219yescomplemented
21H(5,𝔽2)×C127C22116yescomplemented
22C2×#41×C127C2216yescomplemented
23C22×D4×C127C2219yescomplemented
24C23×D127C2319yes—
25#14523249C23116yescomplemented
26#14523248C231144yescomplemented
27#14523247C231144yescomplemented
28#14523247C231288yescomplemented
29C2×Dic254C23118yes—
30C2×Dic254C23172yescomplemented
31Q8×D127C23148yescomplemented
32C2×D508C2316yes—
33C2×C22⋊D127C23136yes—
34C2×C22⋊D127C231144yescomplemented
35D4×D127C231144yescomplemented
36C22×Dic127C2319yes—
37C22×Dic127C23118yes—
38C22×Dic127C23148yescomplemented
39C2×C4×D127C2316yes—
40C2×C4×D127C23118yes—
41C2×C4×D127C23172yescomplemented
42C24×C127C2316yes—
43#41×C127C23148yescomplemented
44C2×Q8×C127C2312yes—
45C2×D4×C127C23118yes—
46C2×D4×C127C23172yescomplemented
47C22×C4×C127C2319yes—
48C22×D127—418no—
49C22×D127C2416yes—
50C22×D127C24136yes—
51Dic254C24172yes—
52D508C24124yes—
53C22⋊D127C241144yes—
54C2×Dic127C2411yescharacteristic
55C2×Dic127C2419yes—
56C2×Dic127C24136yes—
57C2×Dic127C24172yes—
58C4×D127C24124yes—
59C4×D127C24172yes—
60C23×C127—412no—
61C23×C127C2419yes—
62C23×C127C24124yes—
63Q8×C127C2418yes—
64D4×C127C24172yes—
65C2×C4×C127C2416yes—
66C2×C4×C127C24136yes—
67C22×C127—218no—
68C22×C127—424no—
69C22×C127C25118yes—
70C22×C127C2511yescharacteristic
71C2×D127—212no—
72C2×D127—436no—
73C2×D127C25112yes—
74C4×C127C25112yes—
75Dic127C2512yes—
76Dic127C25118yes—
77C2×C127—218no—
78C2×C127#58512yescomplemented
79C2×C127C2611yescharacteristic
80D127—212no—
81C2×#585—1271nomaximal, Sylow
82C127C2×#58511yescharacteristic, complemented, Sylow
83#585—12732no—
84C2×2−1+4—1276no—
85C2×H(5,𝔽2)—1279no—
86C2×H(5,𝔽2)D12711yescharacteristic, complemented
87C22×#41—1276no—
88C22×#41—1279no—
892−1+4—12796no—
90H(5,𝔽2)—127144no—
91H(5,𝔽2)C2×D127116yescomplemented
92C2×#41—12796no—
93C2×#41—1272no—
94C2×#41—12718no—
95C2×#41—12736no—
96C2×#41—127144no—
97C2×#41—12718no—
98C2×#41C2×D12716yescomplemented
99C22×Q8—1276no—
100C22×Q8—1279no—
101C22×D4—12718no—
102C22×D4—12718no—
103C22×D4C2×D12719yescomplemented
104C23×C4—1276no—
105C23×C4—1279no—
106C24—1279no—
107C24C22×D12716yes—
108#41—12716no—
109#41—127144no—
110#41—127288no—
111#41—127144no—
112#41C22×D127148yescomplemented
113C2×Q8—12748no—
114C2×Q8—12718no—
115C2×Q8—12772no—
116C2×Q8C22×D12712yes—
117C2×D4—12736no—
118C2×D4—127144no—
119C2×D4—127144no—
120C2×D4—1276no—
121C2×D4C22×D127118yes—
122C2×D4C22×D127172yescomplemented
123C22×C4—1279no—
124C22×C4—12718no—
125C22×C4—12748no—
126C22×C4—1276no—
127C22×C4—12718no—
128C22×C4—12772no—
129C22×C4C22×D12719yes—
130C23—412no—
131C23—12736no—
132C23—1276no—
133C23—50818no—
134C23C23×D12719yes—
135C23C23×D127124yes—
136Q8—12772no—
137Q8C23×D12718yes—
138D4—127144no—
139D4—12724no—
140D4C23×D127172yes—
141C2×C4—12736no—
142C2×C4—12772no—
143C2×C4—1271no—
144C2×C4—1279no—
145C2×C4—12724no—
146C2×C4—12772no—
147C2×C4C23×D127136yes—
148C2×C4C23×D12716yes—
149C22—218no—
150C22—424no—
151C22—12712no—
152C22—50836no—
153C22—25412no—
154C22C24×D127118yes—
155C22C24×D12711yescharacteristic
156C4—1272no—
157C4—12718no—
158C4C24×D127112yes—
159C2—218no—
160C2—25412no—
161C2#2255071112yesdirect product factor
162C2C25×D12711yescharacteristic
subgroupG/H
centerC22C24×D127
commutator, cosocleC2×C127C26
FrattiniC2C25×D127
FittingC2×H(5,𝔽2)×C127C2
socleC22×C127C25
hypercenterC2×H(5,𝔽2)D127
hypocenterC127C2×#585
LCSG ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #22550711C2direct (N × H)
2. C2×#16861256C2semidirect
3. C2×#16861254C2semidirect
4. C22×#14523247C2semidirect
5. C2×#41×D127C2semidirect
6. C2×H(5,𝔽2)×C127C2semidirect
7. #16861256C22semidirect
8. #16861254C22semidirect
9. C2×#14523249C22semidirect
10. C2×#14523248C22semidirect
11. C2×#14523247C22semidirect
12. C2×#14523247C22semidirect
13. C2×#14523247C22semidirect
14. #41×D127C22semidirect
15. C22×Dic254C22semidirect
16. C2×Q8×D127C22semidirect
17. C22×C22⋊D127C22semidirect
18. C2×D4×D127C22semidirect
19. C23×Dic127C22semidirect
20. C22×C4×D127C22semidirect
21. H(5,𝔽2)×C127C22semidirect
22. C2×#41×C127C22semidirect
23. C22×D4×C127C22semidirect
25. #14523249C23semidirect
26. #14523248C23semidirect
27. #14523247C23semidirect
28. #14523247C23semidirect
30. C2×Dic254C23semidirect
31. Q8×D127C23semidirect
34. C2×C22⋊D127C23semidirect
35. D4×D127C23semidirect
38. C22×Dic127C23semidirect
41. C2×C4×D127C23semidirect
43. #41×C127C23semidirect
46. C2×D4×C127C23semidirect
78. C2×C127#585semidirect
82. C127C2×#585semidirect
86. C2×H(5,𝔽2)D127semidirect
91. H(5,𝔽2)C2×D127semidirect
98. C2×#41C2×D127semidirect
103. C22×D4C2×D127semidirect
112. #41C22×D127semidirect
122. C2×D4C22×D127semidirect
161. C2#22550711direct (N × H)

Maximal subgroups

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

subgroupindex
#225507112
C2×#168612562
C2×#168612542
C22×#145232472
C2×#41×D1272
C2×H(5,𝔽2)×C1272
C2×#585127