Finite Group Databasebeta

← C2×#26012234 #36367684

Subgroups — C2×#26012234 #36367684

Order 22144.

1408 conjugacy classes of subgroups; only the 469 normal subgroups are shown.

normal subgroups only

This group has 1408 conjugacy classes of subgroups — too many to draw. Only its 469 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#26012234C2116yesmaximal, direct product factor
2C2×#19149666C213yesmaximal, complemented
3C2×#19149665C216yesmaximal, complemented
4C2×#19149648C212yesmaximal, complemented
5C2×#19149647C213yesmaximal, complemented
6C2×#120×C173C211yesmaximal, characteristic, complemented
7C24×D173C2211yescharacteristic, complemented
8C22×Dic346C2213yescomplemented
9C22×C22⋊D173C2216yes—
10#19149666C22124yescomplemented
11#19149665C22148yescomplemented
12#19149648C22116yescomplemented
13#19149647C22124yescomplemented
14C2×C22⋊Dic173C2213yescomplemented
15C2×C22⋊Dic173C2216yescomplemented
16C2×#14583803C2216yescomplemented
17C2×C4×Dic173C2213yescomplemented
18C25×C173C2211yescharacteristic, complemented
19C22×D4×C173C2213yes—
20#120×C173C2218yescomplemented
21C2×C22⋊C4×C173C2213yescomplemented
22C23×D173—46no—
23C23×D173C2312yes—
24C23×D173C2314yescomplemented
25C2×Dic346—46no—
26C2×Dic346C23112yescomplemented
27C2×C22⋊D173—412no—
28C2×C22⋊D173C23124yes—
29C22×Dic173C2316yes—
30C22⋊Dic173C23112yescomplemented
31C22⋊Dic173C23124yescomplemented
32#14583803C23124yescomplemented
33C4×Dic173C23112yescomplemented
34C24×C173—46no—
35C24×C173C2311yescharacteristic
36C24×C173C2313yes—
37C24×C173C2314yescomplemented
38C2×D4×C173—46no—
39C2×D4×C173C23112yes—
40C22×C4×C173C2313yes—
41C22⋊C4×C173C23112yescomplemented
42C22×D173—212no—
43C22×D173—412no—
44C22×D173—416no—
45C22×D173C2414yes—
46Dic346—412no—
47C22⋊D173—424no—
48C2×Dic173—212no—
49C2×Dic173C24112yes—
50C23×C173—212no—
51C23×C173—26no—
52C23×C173—26no—
53C23×C173—412no—
54C23×C173—416no—
55C23×C173C2416yes—
56C23×C173C2412yes—
57C23×C173C2411yescharacteristic
58D4×C173—412no—
59C2×C4×C173—26no—
60C2×C4×C173C2416yes—
61C22×C173—26no—
62C22×C173—26no—
63C22×C173—212no—
64C22×C173—416no—
65C22×C173—412no—
66C22×C173—44no—
67C22×C173H(5,𝔽2)16yes—
68C22×C173C2511yescharacteristic
69C2×D173—212no—
70C2×D173—416no—
71C2×D173—48no—
72C4×C173—26no—
73Dic173—212no—
74C2×C173—42no—
75C2×C173—46no—
76C2×C173H(3,𝔽4)14yescomplemented
77C2×C173C2×H(5,𝔽2)13yes—
78D173—44no—
79C173C2×H(3,𝔽4)11yescharacteristic, complemented, Sylow
80C2×H(3,𝔽4)—1731nomaximal, Sylow
81H(3,𝔽4)—17316no—
82C2×#124—1733no—
83C2×#124—1736no—
84C2×#120—1732no—
85C2×#120—1733no—
86C2×#120D17311yescharacteristic, complemented
87C25—1731no—
88C25C2×D17311yescharacteristic, complemented
89C22×Q8—1733no—
90C22×D4—1736no—
91C22×D4C2×D17313yes—
92#124—17324no—
93#124—17348no—
94#120—17316no—
95#120—17324no—
96#120C2×D17318yescomplemented
97C2×C22⋊C4—1733no—
98C2×C22⋊C4—1736no—
99C2×C22⋊C4—1736no—
100C2×C22⋊C4C2×D17313yescomplemented
101C2×C42—1733no—
102C24—46no—
103C24—1732no—
104C24—1734no—
105C24—6926no—
106C24C22×D17311yescharacteristic
107C24C22×D17313yes—
108C24C22×D17314yescomplemented
109C2×Q8—17312no—
110C2×Q8—6926no—
111C2×D4—46no—
112C2×D4—17324no—
113C2×D4—69212no—
114C2×D4C22×D173112yes—
115C22×C4—1736no—
116C22×C4C22×D17313yes—
117C22⋊C4—17312no—
118C22⋊C4—17324no—
119C22⋊C4—17324no—
120C22⋊C4C22×D173112yescomplemented
121C42—17312no—
122C23—26no—
123C23—26no—
124C23—212no—
125C23—412no—
126C23—416no—
127C23—34612no—
128C23—1734no—
129C23—69212no—
130C23—69216no—
131C23C23×D17312yes—
132C23C23×D17316yes—
133C23C23×D17311yescharacteristic
134Q8—69212no—
135D4—412no—
136D4—69224no—
137C2×C4—26no—
138C2×C4—34612no—
139C2×C4—17312no—
140C2×C4C23×D17316yes—
141C22—26no—
142C22—26no—
143C22—212no—
144C22—416no—
145C22—412no—
146C22—44no—
147C22—34612no—
148C22—69216no—
149C22—6928no—
150C22#1914972416yes—
151C22C24×D17311yescharacteristic
152C4—26no—
153C4—34612no—
154C2—46no—
155C2—42no—
156C2—6924no—
157C2#2601223414yesdirect product factor
158C2C2×#1914972413yes—
subgroupG/H
centerC23C23×D173
commutator, cosocleC22×C173C25
FrattiniC22C24×D173
FittingC2×#120×C173C2
socleC23×C173C24
hypercenterC2×#120D173
hypocenterC173C2×H(3,𝔽4)
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. #26012234C2direct (N × H)
2. C2×#19149666C2semidirect
3. C2×#19149665C2semidirect
4. C2×#19149648C2semidirect
5. C2×#19149647C2semidirect
6. C2×#120×C173C2semidirect
7. C24×D173C22semidirect
8. C22×Dic346C22semidirect
10. #19149666C22semidirect
11. #19149665C22semidirect
12. #19149648C22semidirect
13. #19149647C22semidirect
14. C2×C22⋊Dic173C22semidirect
15. C2×C22⋊Dic173C22semidirect
16. C2×#14583803C22semidirect
17. C2×C4×Dic173C22semidirect
18. C25×C173C22semidirect
20. #120×C173C22semidirect
21. C2×C22⋊C4×C173C22semidirect
24. C23×D173C23semidirect
26. C2×Dic346C23semidirect
30. C22⋊Dic173C23semidirect
31. C22⋊Dic173C23semidirect
32. #14583803C23semidirect
33. C4×Dic173C23semidirect
37. C24×C173C23semidirect
41. C22⋊C4×C173C23semidirect
76. C2×C173H(3,𝔽4)semidirect
79. C173C2×H(3,𝔽4)semidirect
86. C2×#120D173semidirect
88. C25C2×D173semidirect
96. #120C2×D173semidirect
100. C2×C22⋊C4C2×D173semidirect
108. C24C22×D173semidirect
120. C22⋊C4C22×D173semidirect
157. C2#26012234direct (N × H)

Maximal subgroups

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

subgroupindex
#260122342
C2×#191496662
C2×#191496652
C2×#191496482
C2×#191496472
C2×#120×C1732
C2×H(3,𝔽4)173