Finite Group Databasebeta

← H(5,𝔽2)×#10597010 #34061672

Subgroups — H(5,𝔽2)×#10597010 #34061672

Order 19072.

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#23750694C216yesmaximal, complemented
2#23750692C219yesmaximal, complemented
3#41×#10597010C216yesmaximal, complemented
4C2×D4×#10597010C219yesmaximal, complemented
5H(5,𝔽2)×D149C211yesmaximal, characteristic
6C23×#10597010C2216yescomplemented
7Q8×#10597010C2212yescomplemented
8Q8×#10597010C2216yescomplemented
9D4×#10597010C22118yescomplemented
10D4×#10597010C22136yescomplemented
11D4×#10597010C22118yescomplemented
12#18011990C22118yescomplemented
13C2×C4⋊#10597010C2219yescomplemented
14C2×C22⋊#10597010C22118yescomplemented
15C2×C4×#10597010C2219yescomplemented
16#18011957C416yescomplemented
17#18011956C419yescomplemented
18#41×D149C2216yes—
19C2×D4×D149C2219yes—
20H(5,𝔽2)×C149C411yescharacteristic, complemented
21C22×#10597010—412no—
22C22×#10597010C2319yes—
23C22×#10597010C2319yes—
24C22×#10597010C23118yes—
25C4⋊#10597010C23118yes—
26C4⋊#10597010C2316yes—
27C22⋊#10597010C23136yes—
28C4×#10597010C2316yes—
29C4×#10597010C23118yes—
30C23×D149C2316yes—
31#14558737C2×C416yescomplemented
32#14558736C2×C4118yescomplemented
33#14558735C2×C4118yescomplemented
34Q8×D149C2312yes—
35C2×D596C2×C419yescomplemented
36C2×C22⋊D149C2×C4118yescomplemented
37D4×D149C2×C4136yescomplemented
38D4×D149C23118yes—
39C2×C4×D149C2319yes—
40#41×C149C2×C416yescomplemented
41C2×D4×C149C2×C419yescomplemented
42C2×#10597010—218no—
43C2×#10597010—418no—
44C2×#10597010—49no—
45C2×#10597010C2416yes—
46C2×#10597010C2419yes—
47C2×#10597010C2411yescharacteristic
48C22×D149—46no—
49C22×D149—46no—
50C22×D149C22×C4118yescomplemented
51C22×D149C2419yes—
52Dic298C22×C416yescomplemented
53D596C22×C4118yescomplemented
54C22⋊D149C22×C4136yescomplemented
55C2×Dic149C22×C419yescomplemented
56C4×D149C22×C4118yescomplemented
57C4×D149C2416yes—
58C23×C149C22×C416yescomplemented
59Q8×C149C22×C412yescomplemented
60D4×C149C22×C4118yescomplemented
61C2×C4×C149C22×C419yescomplemented
62C22×C149—46no—
63C22×C149C23×C419yes—
64C2×D149—29no—
65C2×D149—29no—
66C2×D149—418no—
67C2×D149C23×C419yes—
68C2×D149C2511yescharacteristic
69#10597010—29no—
70#10597010—26no—
71#10597010H(5,𝔽2)12yesdirect product factor
72C4×C149C23×C416yes—
73Dic149C23×C416yes—
74C2×C149—29no—
75C2×C149C24×C411yescharacteristic
76D149—29no—
77D149C2×H(5,𝔽2)11yescharacteristic
78D149#58511yescharacteristic
79C149C4×H(5,𝔽2)11yescharacteristic, complemented, Sylow
80C4×H(5,𝔽2)—1491nomaximal, Sylow
81C2×H(5,𝔽2)—1491no—
82#520—1496no—
83#518—1499no—
84C4×#41—1496no—
85C2×C4×D4—1499no—
86H(5,𝔽2)—1496no—
87H(5,𝔽2)—1499no—
88H(5,𝔽2)#1059701011yescharacteristic, direct product factor
89C2×#41—1496no—
90C22×D4—1499no—
91C23×C4—1496no—
92C4×Q8—1492no—
93C4×Q8—1496no—
94C4×D4—14918no—
95C4×D4—14936no—
96C4×D4—14918no—
97#117—14918no—
98C2×C4⋊C4—1499no—
99C2×C22⋊C4—14918no—
100C2×C42—1499no—
101C24—1496no—
102#41—1496no—
103#41—14918no—
104#41—14918no—
105#41C2×#1059701016yescomplemented
106C2×Q8—1492no—
107C2×D4—14918no—
108C2×D4—14936no—
109C2×D4—14918no—
110C2×D4—1499no—
111C2×D4C2×#1059701019yescomplemented
112C22×C4—59612no—
113C22×C4—1499no—
114C22×C4—1499no—
115C22×C4—14918no—
116C22×C4—1499no—
117C4⋊C4—14918no—
118C4⋊C4—1496no—
119C22⋊C4—14936no—
120C42—1496no—
121C42—14918no—
122C23—5966no—
123C23—5966no—
124C23—1499no—
125C23—14918no—
126C23C22×#1059701016yescomplemented
127Q8—1496no—
128Q8C22×#1059701012yescomplemented
129D4—14936no—
130D4—14918no—
131D4C22×#10597010118yescomplemented
132C2×C4—29818no—
133C2×C4—59618no—
134C2×C4—5969no—
135C2×C4—1496no—
136C2×C4—1499no—
137C2×C4—1491no—
138C2×C4—1499no—
139C2×C4—14918no—
140C2×C4—1496no—
141C2×C4C22×#1059701019yescomplemented
142C22—46no—
143C22—2989no—
144C22—2989no—
145C22—59618no—
146C22—1499no—
147C22—1491no—
148C22C23×#1059701019yes—
149C4—2989no—
150C4—2986no—
151C4—1492no—
152C4—1496no—
153C4C23×#1059701016yes—
154C2—29no—
155C2—2989no—
156C2—1491no—
157C2—1491no—
158C2C24×#1059701011yescharacteristic
subgroupG/H
center, FrattiniC2C24×#10597010
commutator, socleC2×C149C24×C4
FittingH(5,𝔽2)×C149C4
prime radicalH(5,𝔽2)×D149C2
cosocleC2×D149C25
hypercenterH(5,𝔽2)#10597010
hypocenterC149C4×H(5,𝔽2)
LCSG ▷ C2×C149 ▷ C149
UCSC1 ◁ C2 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #23750694C2semidirect
2. #23750692C2semidirect
3. #41×#10597010C2semidirect
4. C2×D4×#10597010C2semidirect
6. C23×#10597010C22semidirect
7. Q8×#10597010C22semidirect
8. Q8×#10597010C22semidirect
9. D4×#10597010C22semidirect
10. D4×#10597010C22semidirect
11. D4×#10597010C22semidirect
12. #18011990C22semidirect
13. C2×C4⋊#10597010C22semidirect
14. C2×C22⋊#10597010C22semidirect
15. C2×C4×#10597010C22semidirect
16. #18011957C4semidirect
17. #18011956C4semidirect
20. H(5,𝔽2)×C149C4semidirect
31. #14558737C2×C4semidirect
32. #14558736C2×C4semidirect
33. #14558735C2×C4semidirect
35. C2×D596C2×C4semidirect
36. C2×C22⋊D149C2×C4semidirect
37. D4×D149C2×C4semidirect
40. #41×C149C2×C4semidirect
41. C2×D4×C149C2×C4semidirect
50. C22×D149C22×C4semidirect
52. Dic298C22×C4semidirect
53. D596C22×C4semidirect
54. C22⋊D149C22×C4semidirect
55. C2×Dic149C22×C4semidirect
56. C4×D149C22×C4semidirect
58. C23×C149C22×C4semidirect
59. Q8×C149C22×C4semidirect
60. D4×C149C22×C4semidirect
61. C2×C4×C149C22×C4semidirect
71. #10597010H(5,𝔽2)direct (N × H)
79. C149C4×H(5,𝔽2)semidirect
88. H(5,𝔽2)#10597010direct (N × H)
105. #41C2×#10597010semidirect
111. C2×D4C2×#10597010semidirect
126. C23C22×#10597010semidirect
128. Q8C22×#10597010semidirect
131. D4C22×#10597010semidirect
141. C2×C4C22×#10597010semidirect

Maximal subgroups

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

subgroupindex
#237506942
#237506922
#41×#105970102
C2×D4×#105970102
H(5,𝔽2)×D1492
C4×H(5,𝔽2)149