Finite Group Databasebeta

← H(5,𝔽2)×#87604 #27158717

Subgroups — H(5,𝔽2)×#87604 #27158717

Order 12416.

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#20261804C216yesmaximal, complemented
2#20261802C219yesmaximal, complemented
3#41×#87604C216yesmaximal, complemented
4C2×D4×#87604C219yesmaximal, complemented
5H(5,𝔽2)×D97C211yesmaximal, characteristic
6C23×#87604C2216yescomplemented
7Q8×#87604C2212yescomplemented
8Q8×#87604C2216yescomplemented
9D4×#87604C22118yescomplemented
10D4×#87604C22136yescomplemented
11D4×#87604C22118yescomplemented
12#15690889C22118yescomplemented
13C2×C4⋊#87604C2219yescomplemented
14C2×C22⋊#87604C22118yescomplemented
15C2×C4×#87604C2219yescomplemented
16#15690856C416yescomplemented
17#15690855C419yescomplemented
18#41×D97C2216yes—
19C2×D4×D97C2219yes—
20H(5,𝔽2)×C97C411yescharacteristic, complemented
21C22×#87604—412no—
22C22×#87604C2319yes—
23C22×#87604C2319yes—
24C22×#87604C23118yes—
25C4⋊#87604C23118yes—
26C4⋊#87604C2316yes—
27C22⋊#87604C23136yes—
28C4×#87604C2316yes—
29C4×#87604C23118yes—
30C23×D97C2316yes—
31#13095512C2×C416yescomplemented
32#13095511C2×C4118yescomplemented
33#13095510C2×C4118yescomplemented
34Q8×D97C2312yes—
35C2×D388C2×C419yescomplemented
36C2×C22⋊D97C2×C4118yescomplemented
37D4×D97C2×C4136yescomplemented
38D4×D97C23118yes—
39C2×C4×D97C2319yes—
40#41×C97C2×C416yescomplemented
41C2×D4×C97C2×C419yescomplemented
42C23×C97C22×C416yescomplemented
43C22×D97—46no—
44C22×D97—46no—
45C22×D97C22×C4118yescomplemented
46C22×D97C2419yes—
47C2×#87604—218no—
48C2×#87604—418no—
49C2×#87604—49no—
50C2×#87604C2416yes—
51C2×#87604C2419yes—
52C2×#87604C2411yescharacteristic
53Q8×C97C22×C412yescomplemented
54D4×C97C22×C4118yescomplemented
55C2×C4×C97C22×C419yescomplemented
56C22⋊D97C22×C4136yescomplemented
57C2×Dic97C22×C419yescomplemented
58D388C22×C4118yescomplemented
59C4×D97C22×C4118yescomplemented
60C4×D97C2416yes—
61Dic194C22×C416yescomplemented
62C22×C97—46no—
63C22×C97C23×C419yes—
64C2×D97—29no—
65C2×D97—29no—
66C2×D97—418no—
67C2×D97C23×C419yes—
68C2×D97C2511yescharacteristic
69#87604—29no—
70#87604—26no—
71#87604H(5,𝔽2)12yesdirect product factor
72C4×C97C23×C416yes—
73Dic97C23×C416yes—
74C2×C97—29no—
75C2×C97C24×C411yescharacteristic
76D97—29no—
77D97C2×H(5,𝔽2)11yescharacteristic
78D97#58511yescharacteristic
79C4×H(5,𝔽2)—971nomaximal, Sylow
80C97C4×H(5,𝔽2)11yescharacteristic, complemented, Sylow
81C2×H(5,𝔽2)—971no—
82#520—976no—
83#518—979no—
84C4×#41—976no—
85C2×C4×D4—979no—
86H(5,𝔽2)—976no—
87H(5,𝔽2)—979no—
88H(5,𝔽2)#8760411yescharacteristic, direct product factor
89C2×#41—976no—
90C22×D4—979no—
91C23×C4—976no—
92C4×Q8—972no—
93C4×Q8—976no—
94C4×D4—9718no—
95C4×D4—9736no—
96C4×D4—9718no—
97#117—9718no—
98C2×C4⋊C4—979no—
99C2×C22⋊C4—9718no—
100C2×C42—979no—
101C24—976no—
102#41—976no—
103#41—9718no—
104#41—9718no—
105#41C2×#8760416yescomplemented
106C2×Q8—972no—
107C2×D4—9718no—
108C2×D4—9736no—
109C2×D4—9718no—
110C2×D4—979no—
111C2×D4C2×#8760419yescomplemented
112C22×C4—979no—
113C22×C4—979no—
114C22×C4—9718no—
115C22×C4—979no—
116C22×C4—38812no—
117C4⋊C4—9718no—
118C4⋊C4—976no—
119C22⋊C4—9736no—
120C42—976no—
121C42—9718no—
122C23—9718no—
123C23—979no—
124C23—3886no—
125C23—3886no—
126C23C22×#8760416yescomplemented
127Q8—976no—
128Q8C22×#8760412yescomplemented
129D4—9736no—
130D4—9718no—
131D4C22×#87604118yescomplemented
132C2×C4—971no—
133C2×C4—979no—
134C2×C4—976no—
135C2×C4—9718no—
136C2×C4—976no—
137C2×C4—979no—
138C2×C4—38818no—
139C2×C4—3889no—
140C2×C4—19418no—
141C2×C4C22×#8760419yescomplemented
142C22—46no—
143C22—971no—
144C22—979no—
145C22—38818no—
146C22—1949no—
147C22—1949no—
148C22C23×#8760419yes—
149C4—972no—
150C4—976no—
151C4—1949no—
152C4—1946no—
153C4C23×#8760416yes—
154C2—29no—
155C2—971no—
156C2—971no—
157C2—1949no—
158C2C24×#8760411yescharacteristic
subgroupG/H
center, FrattiniC2C24×#87604
commutator, socleC2×C97C24×C4
FittingH(5,𝔽2)×C97C4
prime radicalH(5,𝔽2)×D97C2
cosocleC2×D97C25
hypercenterH(5,𝔽2)#87604
hypocenterC97C4×H(5,𝔽2)
LCSG ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #20261804C2semidirect
2. #20261802C2semidirect
3. #41×#87604C2semidirect
4. C2×D4×#87604C2semidirect
6. C23×#87604C22semidirect
7. Q8×#87604C22semidirect
8. Q8×#87604C22semidirect
9. D4×#87604C22semidirect
10. D4×#87604C22semidirect
11. D4×#87604C22semidirect
12. #15690889C22semidirect
13. C2×C4⋊#87604C22semidirect
14. C2×C22⋊#87604C22semidirect
15. C2×C4×#87604C22semidirect
16. #15690856C4semidirect
17. #15690855C4semidirect
20. H(5,𝔽2)×C97C4semidirect
31. #13095512C2×C4semidirect
32. #13095511C2×C4semidirect
33. #13095510C2×C4semidirect
35. C2×D388C2×C4semidirect
36. C2×C22⋊D97C2×C4semidirect
37. D4×D97C2×C4semidirect
40. #41×C97C2×C4semidirect
41. C2×D4×C97C2×C4semidirect
42. C23×C97C22×C4semidirect
45. C22×D97C22×C4semidirect
53. Q8×C97C22×C4semidirect
54. D4×C97C22×C4semidirect
55. C2×C4×C97C22×C4semidirect
56. C22⋊D97C22×C4semidirect
57. C2×Dic97C22×C4semidirect
58. D388C22×C4semidirect
59. C4×D97C22×C4semidirect
61. Dic194C22×C4semidirect
71. #87604H(5,𝔽2)direct (N × H)
80. C97C4×H(5,𝔽2)semidirect
88. H(5,𝔽2)#87604direct (N × H)
105. #41C2×#87604semidirect
111. C2×D4C2×#87604semidirect
126. C23C22×#87604semidirect
128. Q8C22×#87604semidirect
131. D4C22×#87604semidirect
141. C2×C4C22×#87604semidirect

Maximal subgroups

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

subgroupindex
#202618042
#202618022
#41×#876042
C2×D4×#876042
H(5,𝔽2)×D972
C4×H(5,𝔽2)97