Finite Group Databasebeta

← C2×#19128536 #24922270

Subgroups — C2×#19128536 #24922270

Order 10624.

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#19128536C2132yesmaximal, direct product factor
2C2×#14581839C216yesmaximal, complemented
3C2×#14581837C219yesmaximal, complemented
4C22×#13050589C219yesmaximal, complemented
5C2×#41×D83C216yesmaximal, complemented
6C2×H(5,𝔽2)×C83C211yesmaximal, characteristic, complemented
7#14581839C22196yescomplemented
8#14581837C221144yescomplemented
9C2×#13050591C2212yescomplemented
10C2×#13050590C22118yescomplemented
11C2×#13050589C22118yescomplemented
12C2×#13050589C22136yescomplemented
13C2×#13050589C221144yescomplemented
14#41×D83C22196yescomplemented
15C22×Dic166C2219yescomplemented
16C2×Q8×D83C2216yescomplemented
17C22×C22⋊D83C22118yescomplemented
18C2×D4×D83C22118yescomplemented
19C23×Dic83C2216yescomplemented
20C22×C4×D83C2219yescomplemented
21H(5,𝔽2)×C83C22116yescomplemented
22C2×#41×C83C2216yescomplemented
23C22×D4×C83C2219yescomplemented
24C23×D83C2319yes—
25#13050591C23116yescomplemented
26#13050590C231144yescomplemented
27#13050589C231144yescomplemented
28#13050589C231288yescomplemented
29C2×Dic166C23118yes—
30C2×Dic166C23172yescomplemented
31Q8×D83C23148yescomplemented
32C2×D332C2316yes—
33C2×C22⋊D83C23136yes—
34C2×C22⋊D83C231144yescomplemented
35D4×D83C231144yescomplemented
36C22×Dic83C2319yes—
37C22×Dic83C23118yes—
38C22×Dic83C23148yescomplemented
39C2×C4×D83C2316yes—
40C2×C4×D83C23118yes—
41C2×C4×D83C23172yescomplemented
42C24×C83C2316yes—
43#41×C83C23148yescomplemented
44C2×Q8×C83C2312yes—
45C2×D4×C83C23118yes—
46C2×D4×C83C23172yescomplemented
47C22×C4×C83C2319yes—
48C23×C83—412no—
49C23×C83C2419yes—
50C23×C83C24124yes—
51C22×D83—418no—
52C22×D83C2416yes—
53C22×D83C24136yes—
54Q8×C83C2418yes—
55D4×C83C24172yes—
56C2×C4×C83C2416yes—
57C2×C4×C83C24136yes—
58C22⋊D83C241144yes—
59C2×Dic83C2411yescharacteristic
60C2×Dic83C2419yes—
61C2×Dic83C24136yes—
62C2×Dic83C24172yes—
63D332C24124yes—
64C4×D83C24124yes—
65C4×D83C24172yes—
66Dic166C24172yes—
67C22×C83—218no—
68C22×C83—424no—
69C22×C83C25118yes—
70C22×C83C2511yescharacteristic
71C2×D83—212no—
72C2×D83—436no—
73C2×D83C25112yes—
74C4×C83C25112yes—
75Dic83C25118yes—
76Dic83C2512yes—
77C2×C83—218no—
78C2×C83#58512yescomplemented
79C2×C83C2611yescharacteristic
80D83—212no—
81C2×#585—831nomaximal, Sylow
82C83C2×#58511yescharacteristic, complemented, Sylow
83#585—8332no—
84C2×2−1+4—836no—
85C2×H(5,𝔽2)—839no—
86C2×H(5,𝔽2)D8311yescharacteristic, complemented
87C22×#41—836no—
88C22×#41—839no—
892−1+4—8396no—
90H(5,𝔽2)—83144no—
91H(5,𝔽2)C2×D83116yescomplemented
92C2×#41—8396no—
93C2×#41—832no—
94C2×#41—8318no—
95C2×#41—8336no—
96C2×#41—83144no—
97C2×#41—8318no—
98C2×#41C2×D8316yescomplemented
99C22×Q8—836no—
100C22×Q8—839no—
101C22×D4—8318no—
102C22×D4—8318no—
103C22×D4C2×D8319yescomplemented
104C23×C4—836no—
105C23×C4—839no—
106C24—839no—
107C24C22×D8316yes—
108#41—8316no—
109#41—83144no—
110#41—83288no—
111#41—83144no—
112#41C22×D83148yescomplemented
113C2×Q8—8348no—
114C2×Q8—8318no—
115C2×Q8—8372no—
116C2×Q8C22×D8312yes—
117C2×D4—8336no—
118C2×D4—83144no—
119C2×D4—83144no—
120C2×D4—836no—
121C2×D4C22×D83118yes—
122C2×D4C22×D83172yescomplemented
123C22×C4—839no—
124C22×C4—8318no—
125C22×C4—8348no—
126C22×C4—836no—
127C22×C4—8318no—
128C22×C4—8372no—
129C22×C4C22×D8319yes—
130C23—412no—
131C23—33218no—
132C23—8336no—
133C23—836no—
134C23C23×D8319yes—
135C23C23×D83124yes—
136Q8—8372no—
137Q8C23×D8318yes—
138D4—83144no—
139D4—8324no—
140D4C23×D83172yes—
141C2×C4—8336no—
142C2×C4—8372no—
143C2×C4—831no—
144C2×C4—839no—
145C2×C4—8324no—
146C2×C4—8372no—
147C2×C4C23×D83136yes—
148C2×C4C23×D8316yes—
149C22—218no—
150C22—424no—
151C22—33236no—
152C22—8312no—
153C22—16612no—
154C22C24×D83118yes—
155C22C24×D8311yescharacteristic
156C4—832no—
157C4—8318no—
158C4C24×D83112yes—
159C2—218no—
160C2—16612no—
161C2#1912853612yesdirect product factor
162C2C25×D8311yescharacteristic
subgroupG/H
centerC22C24×D83
commutator, cosocleC2×C83C26
FrattiniC2C25×D83
FittingC2×H(5,𝔽2)×C83C2
socleC22×C83C25
hypercenterC2×H(5,𝔽2)D83
hypocenterC83C2×#585
LCSG ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #19128536C2direct (N × H)
2. C2×#14581839C2semidirect
3. C2×#14581837C2semidirect
4. C22×#13050589C2semidirect
5. C2×#41×D83C2semidirect
6. C2×H(5,𝔽2)×C83C2semidirect
7. #14581839C22semidirect
8. #14581837C22semidirect
9. C2×#13050591C22semidirect
10. C2×#13050590C22semidirect
11. C2×#13050589C22semidirect
12. C2×#13050589C22semidirect
13. C2×#13050589C22semidirect
14. #41×D83C22semidirect
15. C22×Dic166C22semidirect
16. C2×Q8×D83C22semidirect
17. C22×C22⋊D83C22semidirect
18. C2×D4×D83C22semidirect
19. C23×Dic83C22semidirect
20. C22×C4×D83C22semidirect
21. H(5,𝔽2)×C83C22semidirect
22. C2×#41×C83C22semidirect
23. C22×D4×C83C22semidirect
25. #13050591C23semidirect
26. #13050590C23semidirect
27. #13050589C23semidirect
28. #13050589C23semidirect
30. C2×Dic166C23semidirect
31. Q8×D83C23semidirect
34. C2×C22⋊D83C23semidirect
35. D4×D83C23semidirect
38. C22×Dic83C23semidirect
41. C2×C4×D83C23semidirect
43. #41×C83C23semidirect
46. C2×D4×C83C23semidirect
78. C2×C83#585semidirect
82. C83C2×#585semidirect
86. C2×H(5,𝔽2)D83semidirect
91. H(5,𝔽2)C2×D83semidirect
98. C2×#41C2×D83semidirect
103. C22×D4C2×D83semidirect
112. #41C22×D83semidirect
122. C2×D4C22×D83semidirect
161. C2#19128536direct (N × H)

Maximal subgroups

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

subgroupindex
#191285362
C2×#145818392
C2×#145818372
C22×#130505892
C2×#41×D832
C2×H(5,𝔽2)×C832
C2×#58583