Finite Group Databasebeta

← H(5,𝔽2)×#6923 #21462800

Subgroups — H(5,𝔽2)×#6923 #21462800

Order 7808.

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#16840831C216yesmaximal, complemented
2#16840829C219yesmaximal, complemented
3#41×#6923C216yesmaximal, complemented
4C2×D4×#6923C219yesmaximal, complemented
5H(5,𝔽2)×D61C211yesmaximal, characteristic
6C23×#6923C2216yescomplemented
7Q8×#6923C2212yescomplemented
8Q8×#6923C2216yescomplemented
9D4×#6923C22118yescomplemented
10D4×#6923C22136yescomplemented
11D4×#6923C22118yescomplemented
12#14519856C22118yescomplemented
13C2×C4⋊#6923C2219yescomplemented
14C2×C22⋊#6923C22118yescomplemented
15C2×C4×#6923C2219yescomplemented
16#14519823C416yescomplemented
17#14519822C419yescomplemented
18#41×D61C2216yes—
19C2×D4×D61C2219yes—
20H(5,𝔽2)×C61C411yescharacteristic, complemented
21C23×D61C2316yes—
22C22×#6923—412no—
23C22×#6923C2319yes—
24C22×#6923C2319yes—
25C22×#6923C23118yes—
26#41×C61C2×C416yescomplemented
27C2×D4×C61C2×C419yescomplemented
28C2×C22⋊D61C2×C4118yescomplemented
29#11758226C2×C416yescomplemented
30Q8×D61C2312yes—
31#11758224C2×C4118yescomplemented
32D4×D61C2×C4136yescomplemented
33D4×D61C23118yes—
34#11758222C2×C4118yescomplemented
35C2×D244C2×C419yescomplemented
36C2×C4×D61C2319yes—
37C22⋊#6923C23136yes—
38C4⋊#6923C23118yes—
39C4⋊#6923C2316yes—
40C4×#6923C2316yes—
41C4×#6923C23118yes—
42C23×C61C22×C416yescomplemented
43C22×D61—46no—
44C22×D61—46no—
45C22×D61C22×C4118yescomplemented
46C22×D61C2419yes—
47C2×#6923—218no—
48C2×#6923—418no—
49C2×#6923—49no—
50C2×#6923C2416yes—
51C2×#6923C2419yes—
52C2×#6923C2411yescharacteristic
53Q8×C61C22×C412yescomplemented
54D4×C61C22×C4118yescomplemented
55C2×C4×C61C22×C419yescomplemented
56C22⋊D61C22×C4136yescomplemented
57C2×Dic61C22×C419yescomplemented
58D244C22×C4118yescomplemented
59C4×D61C22×C4118yescomplemented
60C4×D61C2416yes—
61Dic122C22×C416yescomplemented
62C22×C61—46no—
63C22×C61C23×C419yes—
64C2×D61—29no—
65C2×D61—29no—
66C2×D61—418no—
67C2×D61C23×C419yes—
68C2×D61C2511yescharacteristic
69#6923—29no—
70#6923—26no—
71#6923H(5,𝔽2)12yesdirect product factor
72C4×C61C23×C416yes—
73Dic61C23×C416yes—
74C4×H(5,𝔽2)—611nomaximal, Sylow
75C2×C61—29no—
76C2×C61C24×C411yescharacteristic
77D61—29no—
78D61C2×H(5,𝔽2)11yescharacteristic
79D61#58511yescharacteristic
80C2×H(5,𝔽2)—611no—
81#520—616no—
82#518—619no—
83C4×#41—616no—
84C2×C4×D4—619no—
85C61C4×H(5,𝔽2)11yescharacteristic, complemented, Sylow
86H(5,𝔽2)—616no—
87H(5,𝔽2)—619no—
88H(5,𝔽2)#692311yescharacteristic, direct product factor
89C2×#41—616no—
90C22×D4—619no—
91C23×C4—616no—
92C4×Q8—612no—
93C4×Q8—616no—
94C4×D4—6118no—
95C4×D4—6136no—
96C4×D4—6118no—
97#117—6118no—
98C2×C4⋊C4—619no—
99C2×C22⋊C4—6118no—
100C2×C42—619no—
101C24—616no—
102#41—616no—
103#41—6118no—
104#41—6118no—
105#41C2×#692316yescomplemented
106C2×Q8—612no—
107C2×D4—6118no—
108C2×D4—6118no—
109C2×D4—6136no—
110C2×D4—619no—
111C2×D4C2×#692319yescomplemented
112C22×C4—619no—
113C22×C4—619no—
114C22×C4—619no—
115C22×C4—6118no—
116C22×C4—24412no—
117C4⋊C4—6118no—
118C4⋊C4—616no—
119C22⋊C4—6136no—
120C42—616no—
121C42—6118no—
122C23—619no—
123C23—6118no—
124C23—2446no—
125C23—2446no—
126C23C22×#692316yescomplemented
127Q8—616no—
128Q8C22×#692312yescomplemented
129D4—6136no—
130D4—6118no—
131D4C22×#6923118yescomplemented
132C2×C4—611no—
133C2×C4—619no—
134C2×C4—6118no—
135C2×C4—616no—
136C2×C4—616no—
137C2×C4—619no—
138C2×C4—12218no—
139C2×C4—2449no—
140C2×C4—24418no—
141C2×C4C22×#692319yescomplemented
142C22—46no—
143C22—611no—
144C22—619no—
145C22—1229no—
146C22—1229no—
147C22—24418no—
148C22C23×#692319yes—
149C4—612no—
150C4—616no—
151C4—1229no—
152C4—1226no—
153C4C23×#692316yes—
154C2—29no—
155C2—611no—
156C2—611no—
157C2—1229no—
158C2C24×#692311yescharacteristic
subgroupG/H
center, FrattiniC2C24×#6923
commutator, socleC2×C61C24×C4
FittingH(5,𝔽2)×C61C4
hypercenterH(5,𝔽2)#6923
hypocenterC61C4×H(5,𝔽2)
LCSG ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #16840831C2semidirect
2. #16840829C2semidirect
3. #41×#6923C2semidirect
4. C2×D4×#6923C2semidirect
6. C23×#6923C22semidirect
7. Q8×#6923C22semidirect
8. Q8×#6923C22semidirect
9. D4×#6923C22semidirect
10. D4×#6923C22semidirect
11. D4×#6923C22semidirect
12. #14519856C22semidirect
13. C2×C4⋊#6923C22semidirect
14. C2×C22⋊#6923C22semidirect
15. C2×C4×#6923C22semidirect
16. #14519823C4semidirect
17. #14519822C4semidirect
20. H(5,𝔽2)×C61C4semidirect
26. #41×C61C2×C4semidirect
27. C2×D4×C61C2×C4semidirect
28. C2×C22⋊D61C2×C4semidirect
29. #11758226C2×C4semidirect
31. #11758224C2×C4semidirect
32. D4×D61C2×C4semidirect
34. #11758222C2×C4semidirect
35. C2×D244C2×C4semidirect
42. C23×C61C22×C4semidirect
45. C22×D61C22×C4semidirect
53. Q8×C61C22×C4semidirect
54. D4×C61C22×C4semidirect
55. C2×C4×C61C22×C4semidirect
56. C22⋊D61C22×C4semidirect
57. C2×Dic61C22×C4semidirect
58. D244C22×C4semidirect
59. C4×D61C22×C4semidirect
61. Dic122C22×C4semidirect
71. #6923H(5,𝔽2)direct (N × H)
85. C61C4×H(5,𝔽2)semidirect
88. H(5,𝔽2)#6923direct (N × H)
105. #41C2×#6923semidirect
111. C2×D4C2×#6923semidirect
126. C23C22×#6923semidirect
128. Q8C22×#6923semidirect
131. D4C22×#6923semidirect
141. C2×C4C22×#6923semidirect