Finite Group Databasebeta

← H(5,𝔽2)×#3866 #18011577

Subgroups — H(5,𝔽2)×#3866 #18011577

Order 4736.

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#14558602C216yesmaximal, complemented
2#14558600C219yesmaximal, complemented
3#41×#3866C216yesmaximal, complemented
4C2×D4×#3866C219yesmaximal, complemented
5H(5,𝔽2)×D37C211yesmaximal, characteristic
6C23×#3866C2216yescomplemented
7Q8×#3866C2212yescomplemented
8Q8×#3866C2216yescomplemented
9D4×#3866C22118yescomplemented
10D4×#3866C22136yescomplemented
11D4×#3866C22118yescomplemented
12#11924615C22118yescomplemented
13C2×C4⋊#3866C2219yescomplemented
14C2×C22⋊#3866C22118yescomplemented
15C2×C4×#3866C2219yescomplemented
16#11924582C416yescomplemented
17#11924581C419yescomplemented
18#41×D37C2216yes—
19C2×D4×D37C2219yes—
20H(5,𝔽2)×C37C411yescharacteristic, complemented
21C23×D37C2316yes—
22C22×#3866—412no—
23C22×#3866C2319yes—
24C22×#3866C2319yes—
25C22×#3866C23118yes—
26#41×C37C2×C416yescomplemented
27C2×D4×C37C2×C419yescomplemented
28C2×C22⋊D37C2×C4118yescomplemented
29#10596966C2×C416yescomplemented
30Q8×D37C2312yes—
31#10596964C2×C4118yescomplemented
32D4×D37C2×C4136yescomplemented
33D4×D37C23118yes—
34#10596962C2×C4118yescomplemented
35C2×D148C2×C419yescomplemented
36C2×C4×D37C2319yes—
37C22⋊#3866C23136yes—
38C4⋊#3866C23118yes—
39C4⋊#3866C2316yes—
40C4×#3866C2316yes—
41C4×#3866C23118yes—
42C23×C37C22×C416yescomplemented
43C22×D37—46no—
44C22×D37—46no—
45C22×D37C22×C4118yescomplemented
46C22×D37C2419yes—
47C2×#3866—218no—
48C2×#3866—418no—
49C2×#3866—49no—
50C2×#3866C2416yes—
51C2×#3866C2419yes—
52C2×#3866C2411yescharacteristic
53Q8×C37C22×C412yescomplemented
54D4×C37C22×C4118yescomplemented
55C2×C4×C37C22×C419yescomplemented
56C22⋊D37C22×C4136yescomplemented
57C2×Dic37C22×C419yescomplemented
58D148C22×C4118yescomplemented
59C4×D37C22×C4118yescomplemented
60C4×D37C2416yes—
61Dic74C22×C416yescomplemented
62C22×C37—46no—
63C22×C37C23×C419yes—
64C2×D37—29no—
65C2×D37—29no—
66C2×D37—418no—
67C2×D37C23×C419yes—
68C2×D37C2511yescharacteristic
69#3866—29no—
70#3866—26no—
71#3866H(5,𝔽2)12yesdirect product factor
72C4×C37C23×C416yes—
73Dic37C23×C416yes—
74C4×H(5,𝔽2)—371nomaximal, Sylow
75C2×C37—29no—
76C2×C37C24×C411yescharacteristic
77D37—29no—
78D37C2×H(5,𝔽2)11yescharacteristic
79D37#58511yescharacteristic
80C2×H(5,𝔽2)—371no—
81#520—376no—
82#518—379no—
83C4×#41—376no—
84C2×C4×D4—379no—
85C37C4×H(5,𝔽2)11yescharacteristic, complemented, Sylow
86H(5,𝔽2)—376no—
87H(5,𝔽2)—379no—
88H(5,𝔽2)#386611yescharacteristic, direct product factor
89C2×#41—376no—
90C22×D4—379no—
91C23×C4—376no—
92C4×Q8—372no—
93C4×Q8—376no—
94C4×D4—3718no—
95C4×D4—3736no—
96C4×D4—3718no—
97#117—3718no—
98C2×C4⋊C4—379no—
99C2×C22⋊C4—3718no—
100C2×C42—379no—
101C24—376no—
102#41—376no—
103#41—3718no—
104#41—3718no—
105#41C2×#386616yescomplemented
106C2×Q8—372no—
107C2×D4—3718no—
108C2×D4—3736no—
109C2×D4—3718no—
110C2×D4—379no—
111C2×D4C2×#386619yescomplemented
112C22×C4—379no—
113C22×C4—379no—
114C22×C4—379no—
115C22×C4—3718no—
116C22×C4—14812no—
117C4⋊C4—3718no—
118C4⋊C4—376no—
119C22⋊C4—3736no—
120C42—376no—
121C42—3718no—
122C23—379no—
123C23—3718no—
124C23—1486no—
125C23—1486no—
126C23C22×#386616yescomplemented
127Q8—376no—
128Q8C22×#386612yescomplemented
129D4—3736no—
130D4—3718no—
131D4C22×#3866118yescomplemented
132C2×C4—371no—
133C2×C4—379no—
134C2×C4—376no—
135C2×C4—3718no—
136C2×C4—376no—
137C2×C4—379no—
138C2×C4—7418no—
139C2×C4—14818no—
140C2×C4—1489no—
141C2×C4C22×#386619yescomplemented
142C22—46no—
143C22—371no—
144C22—379no—
145C22—749no—
146C22—749no—
147C22—14818no—
148C22C23×#386619yes—
149C4—372no—
150C4—376no—
151C4—749no—
152C4—746no—
153C4C23×#386616yes—
154C2—29no—
155C2—371no—
156C2—371no—
157C2—749no—
158C2C24×#386611yescharacteristic
subgroupG/H
center, FrattiniC2C24×#3866
commutator, socleC2×C37C24×C4
FittingH(5,𝔽2)×C37C4
prime radicalH(5,𝔽2)×D37C2
cosocleC2×D37C25
hypercenterH(5,𝔽2)#3866
hypocenterC37C4×H(5,𝔽2)
LCSG ▷ C2×C37 ▷ C37
UCSC1 ◁ C2 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #14558602C2semidirect
2. #14558600C2semidirect
3. #41×#3866C2semidirect
4. C2×D4×#3866C2semidirect
6. C23×#3866C22semidirect
7. Q8×#3866C22semidirect
8. Q8×#3866C22semidirect
9. D4×#3866C22semidirect
10. D4×#3866C22semidirect
11. D4×#3866C22semidirect
12. #11924615C22semidirect
13. C2×C4⋊#3866C22semidirect
14. C2×C22⋊#3866C22semidirect
15. C2×C4×#3866C22semidirect
16. #11924582C4semidirect
17. #11924581C4semidirect
20. H(5,𝔽2)×C37C4semidirect
26. #41×C37C2×C4semidirect
27. C2×D4×C37C2×C4semidirect
28. C2×C22⋊D37C2×C4semidirect
29. #10596966C2×C4semidirect
31. #10596964C2×C4semidirect
32. D4×D37C2×C4semidirect
34. #10596962C2×C4semidirect
35. C2×D148C2×C4semidirect
42. C23×C37C22×C4semidirect
45. C22×D37C22×C4semidirect
53. Q8×C37C22×C4semidirect
54. D4×C37C22×C4semidirect
55. C2×C4×C37C22×C4semidirect
56. C22⋊D37C22×C4semidirect
57. C2×Dic37C22×C4semidirect
58. D148C22×C4semidirect
59. C4×D37C22×C4semidirect
61. Dic74C22×C4semidirect
71. #3866H(5,𝔽2)direct (N × H)
85. C37C4×H(5,𝔽2)semidirect
88. H(5,𝔽2)#3866direct (N × H)
105. #41C2×#3866semidirect
111. C2×D4C2×#3866semidirect
126. C23C22×#3866semidirect
128. Q8C22×#3866semidirect
131. D4C22×#3866semidirect
141. C2×C4C22×#3866semidirect

Maximal subgroups

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

subgroupindex
#145586022
#145586002
#41×#38662
C2×D4×#38662
H(5,𝔽2)×D372
C4×H(5,𝔽2)37