Finite Group Databasebeta

← H(5,𝔽2)×#261 #13131817

Subgroups — H(5,𝔽2)×#261 #13131817

Order 1664.

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
1D13×H(5,𝔽2)C211yesmaximal, characteristic
2#11719631C216yesmaximal, complemented
3#41×#261C216yesmaximal, complemented
4#11719623C219yesmaximal, complemented
5C2×D4×#261C219yesmaximal, complemented
6C23×#261C2216yescomplemented
7C13×H(5,𝔽2)C411yescharacteristic, complemented
8#88256C416yescomplemented
9#41×D13C2216yes—
10#88251C419yescomplemented
11C2×D4×D13C2219yes—
12C2×C22⋊#261C22118yescomplemented
13Q8×#261C2212yescomplemented
14Q8×#261C2216yescomplemented
15D4×#261C22118yescomplemented
16D4×#261C22136yescomplemented
17D4×#261C22118yescomplemented
18#88237C22118yescomplemented
19C2×C4⋊#261C2219yescomplemented
20C2×C4×#261C2219yescomplemented
21C23×D13C2316yes—
22C22×#261—412no—
23C22×#261C2319yes—
24C22×#261C2319yes—
25C22×#261C23118yes—
26C13×#41C2×C416yescomplemented
27C2×D4×C13C2×C419yescomplemented
28C2×C22⋊D13C2×C4118yescomplemented
29#6131C2×C416yescomplemented
30Q8×D13C2312yes—
31#6129C2×C4118yescomplemented
32D4×D13C2×C4136yescomplemented
33D4×D13C23118yes—
34#6127C2×C4118yescomplemented
35C2×D52C2×C419yescomplemented
36C2×C4×D13C2319yes—
37C22⋊#261C23136yes—
38C4⋊#261C23118yes—
39C4⋊#261C2316yes—
40C4×#261C2316yes—
41C4×#261C23118yes—
42C4×H(5,𝔽2)—131nomaximal, Sylow
43C23×C13C22×C416yescomplemented
44C22×D13—46no—
45C22×D13—46no—
46C22×D13C22×C4118yescomplemented
47C22×D13C2419yes—
48C2×#261—218no—
49C2×#261—418no—
50C2×#261—49no—
51C2×#261C2416yes—
52C2×#261C2419yes—
53C2×#261C2411yescharacteristic
54Q8×C13C22×C412yescomplemented
55D4×C13C22×C4118yescomplemented
56C2×C4×C13C22×C419yescomplemented
57C22⋊D13C22×C4136yescomplemented
58C2×Dic13C22×C419yescomplemented
59D52C22×C4118yescomplemented
60C4×D13C22×C4118yescomplemented
61C4×D13C2416yes—
62Dic26C22×C416yescomplemented
63C2×H(5,𝔽2)—131no—
64#520—136no—
65#518—139no—
66C4×#41—136no—
67C2×C4×D4—139no—
68C22×C13—46no—
69C22×C13C23×C419yes—
70C2×D13—29no—
71C2×D13—29no—
72C2×D13—418no—
73C2×D13C23×C419yes—
74C2×D13C2511yescharacteristic
75#261—29no—
76#261—26no—
77#261H(5,𝔽2)12yesdirect product factor
78C4×C13C23×C416yes—
79Dic13C23×C416yes—
80H(5,𝔽2)—136no—
81H(5,𝔽2)—139no—
82H(5,𝔽2)#26111yescharacteristic, direct product factor
83C2×#41—136no—
84C22×D4—139no—
85C23×C4—136no—
86C4×Q8—132no—
87C4×Q8—136no—
88C4×D4—1318no—
89C4×D4—1336no—
90C4×D4—1318no—
91#117—1318no—
92C2×C4⋊C4—139no—
93C2×C22⋊C4—1318no—
94C2×C42—139no—
95C2×C13—29no—
96C2×C13C24×C411yescharacteristic
97D13—29no—
98D13C2×H(5,𝔽2)11yescharacteristic
99D13#58511yescharacteristic
100C24—136no—
101#41—136no—
102#41—1318no—
103#41—1318no—
104#41C2×#26116yescomplemented
105C2×Q8—132no—
106C2×D4—1318no—
107C2×D4—1318no—
108C2×D4—1336no—
109C2×D4—139no—
110C2×D4C2×#26119yescomplemented
111C22×C4—139no—
112C22×C4—139no—
113C22×C4—139no—
114C22×C4—1318no—
115C22×C4—5212no—
116C4⋊C4—1318no—
117C4⋊C4—136no—
118C22⋊C4—1336no—
119C42—136no—
120C42—1318no—
121C13C4×H(5,𝔽2)11yescharacteristic, complemented, Sylow
122C23—139no—
123C23—1318no—
124C23—526no—
125C23—526no—
126C23C22×#26116yescomplemented
127Q8—136no—
128Q8C22×#26112yescomplemented
129D4—1336no—
130D4—1318no—
131D4C22×#261118yescomplemented
132C2×C4—131no—
133C2×C4—139no—
134C2×C4—136no—
135C2×C4—1318no—
136C2×C4—136no—
137C2×C4—139no—
138C2×C4—2618no—
139C2×C4—5218no—
140C2×C4—529no—
141C2×C4C22×#26119yescomplemented
142C22—46no—
143C22—131no—
144C22—139no—
145C22—269no—
146C22—269no—
147C22—5218no—
148C22C23×#26119yes—
149C4—132no—
150C4—136no—
151C4—269no—
152C4—266no—
153C4C23×#26116yes—
154C2—29no—
155C2—131no—
156C2—131no—
157C2—269no—
158C2C24×#26111yescharacteristic
subgroupG/H
center, FrattiniC2C24×#261
commutator, socleC2×C13C24×C4
FittingC13×H(5,𝔽2)C4
prime radicalD13×H(5,𝔽2)C2
cosocleC2×D13C25
hypercenterH(5,𝔽2)#261
hypocenterC13C4×H(5,𝔽2)
LCSG ▷ C2×C13 ▷ C13
UCSC1 ◁ C2 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
2. #11719631C2semidirect
3. #41×#261C2semidirect
4. #11719623C2semidirect
5. C2×D4×#261C2semidirect
6. C23×#261C22semidirect
7. C13×H(5,𝔽2)C4semidirect
8. #88256C4semidirect
10. #88251C4semidirect
12. C2×C22⋊#261C22semidirect
13. Q8×#261C22semidirect
14. Q8×#261C22semidirect
15. D4×#261C22semidirect
16. D4×#261C22semidirect
17. D4×#261C22semidirect
18. #88237C22semidirect
19. C2×C4⋊#261C22semidirect
20. C2×C4×#261C22semidirect
26. C13×#41C2×C4semidirect
27. C2×D4×C13C2×C4semidirect
28. C2×C22⋊D13C2×C4semidirect
29. #6131C2×C4semidirect
31. #6129C2×C4semidirect
32. D4×D13C2×C4semidirect
34. #6127C2×C4semidirect
35. C2×D52C2×C4semidirect
43. C23×C13C22×C4semidirect
46. C22×D13C22×C4semidirect
54. Q8×C13C22×C4semidirect
55. D4×C13C22×C4semidirect
56. C2×C4×C13C22×C4semidirect
57. C22⋊D13C22×C4semidirect
58. C2×Dic13C22×C4semidirect
59. D52C22×C4semidirect
60. C4×D13C22×C4semidirect
62. Dic26C22×C4semidirect
77. #261H(5,𝔽2)direct (N × H)
82. H(5,𝔽2)#261direct (N × H)
104. #41C2×#261semidirect
110. C2×D4C2×#261semidirect
121. C13C4×H(5,𝔽2)semidirect
126. C23C22×#261semidirect
128. Q8C22×#261semidirect
131. D4C22×#261semidirect
141. C2×C4C22×#261semidirect

Maximal subgroups

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

subgroupindex
D13×H(5,𝔽2)2
#117196312
#41×#2612
#117196232
C2×D4×#2612
C4×H(5,𝔽2)13