Finite Group Databasebeta

← H(5,𝔽2)×#10621924 #36370544

Subgroups — H(5,𝔽2)×#10621924 #36370544

Order 22144.

1660 conjugacy classes of subgroups; only the 754 normal subgroups are shown.

normal subgroups only

This group has 1660 conjugacy classes of subgroups — too many to draw. Only its 754 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#26012560C216yesmaximal, complemented
2#26012558C219yesmaximal, complemented
3#41×#10621924C216yesmaximal, complemented
4C2×D4×#10621924C219yesmaximal, complemented
5H(5,𝔽2)×D173C211yesmaximal, characteristic
6C23×#10621924C2216yescomplemented
7Q8×#10621924C2212yescomplemented
8Q8×#10621924C2216yescomplemented
9D4×#10621924C22118yescomplemented
10D4×#10621924C22136yescomplemented
11D4×#10621924C22118yescomplemented
12#19149758C22118yescomplemented
13C2×C4⋊#10621924C2219yescomplemented
14C2×C22⋊#10621924C22118yescomplemented
15C2×C4×#10621924C2219yescomplemented
16#19149725C416yescomplemented
17#19149724C419yescomplemented
18#41×D173C2216yes—
19C2×D4×D173C2219yes—
20H(5,𝔽2)×C173C411yescharacteristic, complemented
21C22×#10621924—412no—
22C22×#10621924C2319yes—
23C22×#10621924C2319yes—
24C22×#10621924C23118yes—
25C4⋊#10621924C23118yes—
26C4⋊#10621924C2316yes—
27C22⋊#10621924C23136yes—
28C4×#10621924C2316yes—
29C4×#10621924C23118yes—
30C23×D173C2316yes—
31#14583827C2×C416yescomplemented
32#14583826C2×C4118yescomplemented
33#14583825C2×C4118yescomplemented
34Q8×D173C2312yes—
35C2×D692C2×C419yescomplemented
36C2×C22⋊D173C2×C4118yescomplemented
37D4×D173C2×C4136yescomplemented
38D4×D173C23118yes—
39C2×C4×D173C2319yes—
40#41×C173C2×C416yescomplemented
41C2×D4×C173C2×C419yescomplemented
42C2×#10621924—218no—
43C2×#10621924—418no—
44C2×#10621924—49no—
45C2×#10621924C2416yes—
46C2×#10621924C2419yes—
47C2×#10621924C2411yescharacteristic
48C22×D173—46no—
49C22×D173—46no—
50C22×D173C22×C4118yescomplemented
51C22×D173C2419yes—
52Dic346C22×C416yescomplemented
53D692C22×C4118yescomplemented
54C22⋊D173C22×C4136yescomplemented
55C2×Dic173C22×C419yescomplemented
56C4×D173C22×C4118yescomplemented
57C4×D173C2416yes—
58C23×C173C22×C416yescomplemented
59Q8×C173C22×C412yescomplemented
60D4×C173C22×C4118yescomplemented
61C2×C4×C173C22×C419yescomplemented
62C22×C173—46no—
63C22×C173C23×C419yes—
64C2×D173—29no—
65C2×D173—29no—
66C2×D173—418no—
67C2×D173C23×C419yes—
68C2×D173C2511yescharacteristic
69#10621924—29no—
70#10621924—26no—
71#10621924H(5,𝔽2)12yesdirect product factor
72C4×C173C23×C416yes—
73Dic173C23×C416yes—
74C2×C173—29no—
75C2×C173C24×C411yescharacteristic
76D173—29no—
77D173C2×H(5,𝔽2)11yescharacteristic
78D173#58511yescharacteristic
79C173C4×H(5,𝔽2)11yescharacteristic, complemented, Sylow
80C4×H(5,𝔽2)—1731nomaximal, Sylow
81C2×H(5,𝔽2)—1731no—
82#520—1736no—
83#518—1739no—
84C4×#41—1736no—
85C2×C4×D4—1739no—
86H(5,𝔽2)—1736no—
87H(5,𝔽2)—1739no—
88H(5,𝔽2)#1062192411yescharacteristic, direct product factor
89C2×#41—1736no—
90C22×D4—1739no—
91C23×C4—1736no—
92C4×Q8—1732no—
93C4×Q8—1736no—
94C4×D4—17318no—
95C4×D4—17336no—
96C4×D4—17318no—
97#117—17318no—
98C2×C4⋊C4—1739no—
99C2×C22⋊C4—17318no—
100C2×C42—1739no—
101C24—1736no—
102#41—1736no—
103#41—17318no—
104#41—17318no—
105#41C2×#1062192416yescomplemented
106C2×Q8—1732no—
107C2×D4—17318no—
108C2×D4—17336no—
109C2×D4—17318no—
110C2×D4—1739no—
111C2×D4C2×#1062192419yescomplemented
112C22×C4—1739no—
113C22×C4—1739no—
114C22×C4—1739no—
115C22×C4—17318no—
116C22×C4—69212no—
117C4⋊C4—17318no—
118C4⋊C4—1736no—
119C22⋊C4—17336no—
120C42—1736no—
121C42—17318no—
122C23—1739no—
123C23—17318no—
124C23—6926no—
125C23—6926no—
126C23C22×#1062192416yescomplemented
127Q8—1736no—
128Q8C22×#1062192412yescomplemented
129D4—17336no—
130D4—17318no—
131D4C22×#10621924118yescomplemented
132C2×C4—34618no—
133C2×C4—1736no—
134C2×C4—1739no—
135C2×C4—1731no—
136C2×C4—1739no—
137C2×C4—17318no—
138C2×C4—1736no—
139C2×C4—6929no—
140C2×C4—69218no—
141C2×C4C22×#1062192419yescomplemented
142C22—46no—
143C22—3469no—
144C22—3469no—
145C22—1739no—
146C22—1731no—
147C22—69218no—
148C22C23×#1062192419yes—
149C4—3469no—
150C4—3466no—
151C4—1732no—
152C4—1736no—
153C4C23×#1062192416yes—
154C2—29no—
155C2—3469no—
156C2—1731no—
157C2—1731no—
158C2C24×#1062192411yescharacteristic
subgroupG/H
center, FrattiniC2C24×#10621924
commutator, socleC2×C173C24×C4
FittingH(5,𝔽2)×C173C4
prime radicalH(5,𝔽2)×D173C2
cosocleC2×D173C25
hypercenterH(5,𝔽2)#10621924
hypocenterC173C4×H(5,𝔽2)
LCSG ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #26012560C2semidirect
2. #26012558C2semidirect
3. #41×#10621924C2semidirect
4. C2×D4×#10621924C2semidirect
6. C23×#10621924C22semidirect
7. Q8×#10621924C22semidirect
8. Q8×#10621924C22semidirect
9. D4×#10621924C22semidirect
10. D4×#10621924C22semidirect
11. D4×#10621924C22semidirect
12. #19149758C22semidirect
13. C2×C4⋊#10621924C22semidirect
14. C2×C22⋊#10621924C22semidirect
15. C2×C4×#10621924C22semidirect
16. #19149725C4semidirect
17. #19149724C4semidirect
20. H(5,𝔽2)×C173C4semidirect
31. #14583827C2×C4semidirect
32. #14583826C2×C4semidirect
33. #14583825C2×C4semidirect
35. C2×D692C2×C4semidirect
36. C2×C22⋊D173C2×C4semidirect
37. D4×D173C2×C4semidirect
40. #41×C173C2×C4semidirect
41. C2×D4×C173C2×C4semidirect
50. C22×D173C22×C4semidirect
52. Dic346C22×C4semidirect
53. D692C22×C4semidirect
54. C22⋊D173C22×C4semidirect
55. C2×Dic173C22×C4semidirect
56. C4×D173C22×C4semidirect
58. C23×C173C22×C4semidirect
59. Q8×C173C22×C4semidirect
60. D4×C173C22×C4semidirect
61. C2×C4×C173C22×C4semidirect
71. #10621924H(5,𝔽2)direct (N × H)
79. C173C4×H(5,𝔽2)semidirect
88. H(5,𝔽2)#10621924direct (N × H)
105. #41C2×#10621924semidirect
111. C2×D4C2×#10621924semidirect
126. C23C22×#10621924semidirect
128. Q8C22×#10621924semidirect
131. D4C22×#10621924semidirect
141. C2×C4C22×#10621924semidirect

Maximal subgroups

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

subgroupindex
#260125602
#260125582
#41×#106219242
C2×D4×#106219242
H(5,𝔽2)×D1732
C4×H(5,𝔽2)173