Finite Group Databasebeta

← H(5,𝔽2)×#90673 #28407555

Subgroups — H(5,𝔽2)×#90673 #28407555

Order 14464.

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#20299667C216yesmaximal, complemented
2#20299665C219yesmaximal, complemented
3#41×#90673C216yesmaximal, complemented
4C2×D4×#90673C219yesmaximal, complemented
5H(5,𝔽2)×D113C211yesmaximal, characteristic
6C23×#90673C2216yescomplemented
7Q8×#90673C2212yescomplemented
8Q8×#90673C2216yescomplemented
9D4×#90673C22118yescomplemented
10D4×#90673C22136yescomplemented
11D4×#90673C22118yescomplemented
12#16813701C22118yescomplemented
13C2×C4⋊#90673C2219yescomplemented
14C2×C22⋊#90673C22118yescomplemented
15C2×C4×#90673C2219yescomplemented
16#16813668C416yescomplemented
17#16813667C419yescomplemented
18#41×D113C2216yes—
19C2×D4×D113C2219yes—
20H(5,𝔽2)×C113C411yescharacteristic, complemented
21C22×#90673—412no—
22C22×#90673C2319yes—
23C22×#90673C2319yes—
24C22×#90673C23118yes—
25C4⋊#90673C23118yes—
26C4⋊#90673C2316yes—
27C22⋊#90673C23136yes—
28C4×#90673C2316yes—
29C4×#90673C23118yes—
30C23×D113C2316yes—
31#14268648C2×C416yescomplemented
32#14268647C2×C4118yescomplemented
33#14268646C2×C4118yescomplemented
34Q8×D113C2312yes—
35C2×D452C2×C419yescomplemented
36C2×C22⋊D113C2×C4118yescomplemented
37D4×D113C2×C4136yescomplemented
38D4×D113C23118yes—
39C2×C4×D113C2319yes—
40#41×C113C2×C416yescomplemented
41C2×D4×C113C2×C419yescomplemented
42C23×C113C22×C416yescomplemented
43C22×D113—46no—
44C22×D113—46no—
45C22×D113C22×C4118yescomplemented
46C22×D113C2419yes—
47C2×#90673—218no—
48C2×#90673—418no—
49C2×#90673—49no—
50C2×#90673C2416yes—
51C2×#90673C2419yes—
52C2×#90673C2411yescharacteristic
53Q8×C113C22×C412yescomplemented
54D4×C113C22×C4118yescomplemented
55C2×C4×C113C22×C419yescomplemented
56C22⋊D113C22×C4136yescomplemented
57C2×Dic113C22×C419yescomplemented
58D452C22×C4118yescomplemented
59C4×D113C22×C4118yescomplemented
60C4×D113C2416yes—
61Dic226C22×C416yescomplemented
62C22×C113—46no—
63C22×C113C23×C419yes—
64C2×D113—29no—
65C2×D113—29no—
66C2×D113—418no—
67C2×D113C23×C419yes—
68C2×D113C2511yescharacteristic
69#90673—29no—
70#90673—26no—
71#90673H(5,𝔽2)12yesdirect product factor
72C4×C113C23×C416yes—
73Dic113C23×C416yes—
74C2×C113—29no—
75C2×C113C24×C411yescharacteristic
76D113—29no—
77D113C2×H(5,𝔽2)11yescharacteristic
78D113#58511yescharacteristic
79C4×H(5,𝔽2)—1131nomaximal, Sylow
80C113C4×H(5,𝔽2)11yescharacteristic, complemented, Sylow
81C2×H(5,𝔽2)—1131no—
82#520—1136no—
83#518—1139no—
84C4×#41—1136no—
85C2×C4×D4—1139no—
86H(5,𝔽2)—1136no—
87H(5,𝔽2)—1139no—
88H(5,𝔽2)#9067311yescharacteristic, direct product factor
89C2×#41—1136no—
90C22×D4—1139no—
91C23×C4—1136no—
92C4×Q8—1132no—
93C4×Q8—1136no—
94C4×D4—11318no—
95C4×D4—11336no—
96C4×D4—11318no—
97#117—11318no—
98C2×C4⋊C4—1139no—
99C2×C22⋊C4—11318no—
100C2×C42—1139no—
101C24—1136no—
102#41—1136no—
103#41—11318no—
104#41—11318no—
105#41C2×#9067316yescomplemented
106C2×Q8—1132no—
107C2×D4—11318no—
108C2×D4—11318no—
109C2×D4—11336no—
110C2×D4—1139no—
111C2×D4C2×#9067319yescomplemented
112C22×C4—1139no—
113C22×C4—1139no—
114C22×C4—11318no—
115C22×C4—1139no—
116C22×C4—45212no—
117C4⋊C4—11318no—
118C4⋊C4—1136no—
119C22⋊C4—11336no—
120C42—1136no—
121C42—11318no—
122C23—11318no—
123C23—1139no—
124C23—4526no—
125C23—4526no—
126C23C22×#9067316yescomplemented
127Q8—1136no—
128Q8C22×#9067312yescomplemented
129D4—11336no—
130D4—11318no—
131D4C22×#90673118yescomplemented
132C2×C4—1131no—
133C2×C4—1139no—
134C2×C4—1136no—
135C2×C4—11318no—
136C2×C4—1136no—
137C2×C4—1139no—
138C2×C4—45218no—
139C2×C4—4529no—
140C2×C4—22618no—
141C2×C4C22×#9067319yescomplemented
142C22—46no—
143C22—1131no—
144C22—1139no—
145C22—45218no—
146C22—2269no—
147C22—2269no—
148C22C23×#9067319yes—
149C4—1132no—
150C4—1136no—
151C4—2269no—
152C4—2266no—
153C4C23×#9067316yes—
154C2—29no—
155C2—1131no—
156C2—1131no—
157C2—2269no—
158C2C24×#9067311yescharacteristic
subgroupG/H
center, FrattiniC2C24×#90673
commutator, socleC2×C113C24×C4
FittingH(5,𝔽2)×C113C4
prime radicalH(5,𝔽2)×D113C2
cosocleC2×D113C25
hypercenterH(5,𝔽2)#90673
hypocenterC113C4×H(5,𝔽2)
LCSG ▷ C2×C113 ▷ C113
UCSC1 ◁ C2 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #20299667C2semidirect
2. #20299665C2semidirect
3. #41×#90673C2semidirect
4. C2×D4×#90673C2semidirect
6. C23×#90673C22semidirect
7. Q8×#90673C22semidirect
8. Q8×#90673C22semidirect
9. D4×#90673C22semidirect
10. D4×#90673C22semidirect
11. D4×#90673C22semidirect
12. #16813701C22semidirect
13. C2×C4⋊#90673C22semidirect
14. C2×C22⋊#90673C22semidirect
15. C2×C4×#90673C22semidirect
16. #16813668C4semidirect
17. #16813667C4semidirect
20. H(5,𝔽2)×C113C4semidirect
31. #14268648C2×C4semidirect
32. #14268647C2×C4semidirect
33. #14268646C2×C4semidirect
35. C2×D452C2×C4semidirect
36. C2×C22⋊D113C2×C4semidirect
37. D4×D113C2×C4semidirect
40. #41×C113C2×C4semidirect
41. C2×D4×C113C2×C4semidirect
42. C23×C113C22×C4semidirect
45. C22×D113C22×C4semidirect
53. Q8×C113C22×C4semidirect
54. D4×C113C22×C4semidirect
55. C2×C4×C113C22×C4semidirect
56. C22⋊D113C22×C4semidirect
57. C2×Dic113C22×C4semidirect
58. D452C22×C4semidirect
59. C4×D113C22×C4semidirect
61. Dic226C22×C4semidirect
71. #90673H(5,𝔽2)direct (N × H)
80. C113C4×H(5,𝔽2)semidirect
88. H(5,𝔽2)#90673direct (N × H)
105. #41C2×#90673semidirect
111. C2×D4C2×#90673semidirect
126. C23C22×#90673semidirect
128. Q8C22×#90673semidirect
131. D4C22×#90673semidirect
141. C2×C4C22×#90673semidirect

Maximal subgroups

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

subgroupindex
#202996672
#202996652
#41×#906732
C2×D4×#906732
H(5,𝔽2)×D1132
C4×H(5,𝔽2)113