Finite Group Databasebeta

← H(5,𝔽2)×#89137 #28380013

Subgroups — H(5,𝔽2)×#89137 #28380013

Order 13952.

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#20296095C216yesmaximal, complemented
2#20296093C219yesmaximal, complemented
3#41×#89137C216yesmaximal, complemented
4C2×D4×#89137C219yesmaximal, complemented
5H(5,𝔽2)×D109C211yesmaximal, characteristic
6C23×#89137C2216yescomplemented
7Q8×#89137C2212yescomplemented
8Q8×#89137C2216yescomplemented
9D4×#89137C22118yescomplemented
10D4×#89137C22136yescomplemented
11D4×#89137C22118yescomplemented
12#16812778C22118yescomplemented
13C2×C4⋊#89137C2219yescomplemented
14C2×C22⋊#89137C22118yescomplemented
15C2×C4×#89137C2219yescomplemented
16#16812745C416yescomplemented
17#16812744C419yescomplemented
18#41×D109C2216yes—
19C2×D4×D109C2219yes—
20H(5,𝔽2)×C109C411yescharacteristic, complemented
21C22×#89137—412no—
22C22×#89137C2319yes—
23C22×#89137C2319yes—
24C22×#89137C23118yes—
25C4⋊#89137C23118yes—
26C4⋊#89137C2316yes—
27C22⋊#89137C23136yes—
28C4×#89137C2316yes—
29C4×#89137C23118yes—
30C23×D109C2316yes—
31#13182030C2×C416yescomplemented
32#13182029C2×C4118yescomplemented
33#13182028C2×C4118yescomplemented
34Q8×D109C2312yes—
35C2×D436C2×C419yescomplemented
36C2×C22⋊D109C2×C4118yescomplemented
37D4×D109C2×C4136yescomplemented
38D4×D109C23118yes—
39C2×C4×D109C2319yes—
40#41×C109C2×C416yescomplemented
41C2×D4×C109C2×C419yescomplemented
42C23×C109C22×C416yescomplemented
43C22×D109—46no—
44C22×D109—46no—
45C22×D109C22×C4118yescomplemented
46C22×D109C2419yes—
47C2×#89137—218no—
48C2×#89137—418no—
49C2×#89137—49no—
50C2×#89137C2416yes—
51C2×#89137C2419yes—
52C2×#89137C2411yescharacteristic
53Q8×C109C22×C412yescomplemented
54D4×C109C22×C4118yescomplemented
55C2×C4×C109C22×C419yescomplemented
56C22⋊D109C22×C4136yescomplemented
57C2×Dic109C22×C419yescomplemented
58D436C22×C4118yescomplemented
59C4×D109C22×C4118yescomplemented
60C4×D109C2416yes—
61Dic218C22×C416yescomplemented
62C22×C109—46no—
63C22×C109C23×C419yes—
64C2×D109—29no—
65C2×D109—29no—
66C2×D109—418no—
67C2×D109C23×C419yes—
68C2×D109C2511yescharacteristic
69#89137—29no—
70#89137—26no—
71#89137H(5,𝔽2)12yesdirect product factor
72C4×C109C23×C416yes—
73Dic109C23×C416yes—
74C2×C109—29no—
75C2×C109C24×C411yescharacteristic
76D109—29no—
77D109C2×H(5,𝔽2)11yescharacteristic
78D109#58511yescharacteristic
79C4×H(5,𝔽2)—1091nomaximal, Sylow
80C109C4×H(5,𝔽2)11yescharacteristic, complemented, Sylow
81C2×H(5,𝔽2)—1091no—
82#520—1096no—
83#518—1099no—
84C4×#41—1096no—
85C2×C4×D4—1099no—
86H(5,𝔽2)—1096no—
87H(5,𝔽2)—1099no—
88H(5,𝔽2)#8913711yescharacteristic, direct product factor
89C2×#41—1096no—
90C22×D4—1099no—
91C23×C4—1096no—
92C4×Q8—1092no—
93C4×Q8—1096no—
94C4×D4—10918no—
95C4×D4—10936no—
96C4×D4—10918no—
97#117—10918no—
98C2×C4⋊C4—1099no—
99C2×C22⋊C4—10918no—
100C2×C42—1099no—
101C24—1096no—
102#41—1096no—
103#41—10918no—
104#41—10918no—
105#41C2×#8913716yescomplemented
106C2×Q8—1092no—
107C2×D4—10918no—
108C2×D4—10936no—
109C2×D4—10918no—
110C2×D4—1099no—
111C2×D4C2×#8913719yescomplemented
112C22×C4—1099no—
113C22×C4—1099no—
114C22×C4—10918no—
115C22×C4—1099no—
116C22×C4—43612no—
117C4⋊C4—10918no—
118C4⋊C4—1096no—
119C22⋊C4—10936no—
120C42—1096no—
121C42—10918no—
122C23—10918no—
123C23—1099no—
124C23—4366no—
125C23—4366no—
126C23C22×#8913716yescomplemented
127Q8—1096no—
128Q8C22×#8913712yescomplemented
129D4—10936no—
130D4—10918no—
131D4C22×#89137118yescomplemented
132C2×C4—1091no—
133C2×C4—1099no—
134C2×C4—1096no—
135C2×C4—10918no—
136C2×C4—1096no—
137C2×C4—1099no—
138C2×C4—43618no—
139C2×C4—4369no—
140C2×C4—21818no—
141C2×C4C22×#8913719yescomplemented
142C22—46no—
143C22—1091no—
144C22—1099no—
145C22—43618no—
146C22—2189no—
147C22—2189no—
148C22C23×#8913719yes—
149C4—1092no—
150C4—1096no—
151C4—2189no—
152C4—2186no—
153C4C23×#8913716yes—
154C2—29no—
155C2—1091no—
156C2—1091no—
157C2—2189no—
158C2C24×#8913711yescharacteristic
subgroupG/H
center, FrattiniC2C24×#89137
commutator, socleC2×C109C24×C4
FittingH(5,𝔽2)×C109C4
prime radicalH(5,𝔽2)×D109C2
cosocleC2×D109C25
hypercenterH(5,𝔽2)#89137
hypocenterC109C4×H(5,𝔽2)
LCSG ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #20296095C2semidirect
2. #20296093C2semidirect
3. #41×#89137C2semidirect
4. C2×D4×#89137C2semidirect
6. C23×#89137C22semidirect
7. Q8×#89137C22semidirect
8. Q8×#89137C22semidirect
9. D4×#89137C22semidirect
10. D4×#89137C22semidirect
11. D4×#89137C22semidirect
12. #16812778C22semidirect
13. C2×C4⋊#89137C22semidirect
14. C2×C22⋊#89137C22semidirect
15. C2×C4×#89137C22semidirect
16. #16812745C4semidirect
17. #16812744C4semidirect
20. H(5,𝔽2)×C109C4semidirect
31. #13182030C2×C4semidirect
32. #13182029C2×C4semidirect
33. #13182028C2×C4semidirect
35. C2×D436C2×C4semidirect
36. C2×C22⋊D109C2×C4semidirect
37. D4×D109C2×C4semidirect
40. #41×C109C2×C4semidirect
41. C2×D4×C109C2×C4semidirect
42. C23×C109C22×C4semidirect
45. C22×D109C22×C4semidirect
53. Q8×C109C22×C4semidirect
54. D4×C109C22×C4semidirect
55. C2×C4×C109C22×C4semidirect
56. C22⋊D109C22×C4semidirect
57. C2×Dic109C22×C4semidirect
58. D436C22×C4semidirect
59. C4×D109C22×C4semidirect
61. Dic218C22×C4semidirect
71. #89137H(5,𝔽2)direct (N × H)
80. C109C4×H(5,𝔽2)semidirect
88. H(5,𝔽2)#89137direct (N × H)
105. #41C2×#89137semidirect
111. C2×D4C2×#89137semidirect
126. C23C22×#89137semidirect
128. Q8C22×#89137semidirect
131. D4C22×#89137semidirect
141. C2×C4C22×#89137semidirect

Maximal subgroups

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

subgroupindex
#202960952
#202960932
#41×#891372
C2×D4×#891372
H(5,𝔽2)×D1092
C4×H(5,𝔽2)109