Finite Group Databasebeta

← C2×#26012326 #36368370

Subgroups — C2×#26012326 #36368370

Order 22144.

5956 conjugacy classes of subgroups; only the 3205 normal subgroups are shown.

normal subgroups only

This group has 5956 conjugacy classes of subgroups — too many to draw. Only its 3205 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#26012326C2132yesmaximal, direct product factor
2C2×#19149726C216yesmaximal, complemented
3C2×#19149724C219yesmaximal, complemented
4C22×#14583825C219yesmaximal, complemented
5C2×#41×D173C216yesmaximal, complemented
6C2×H(5,𝔽2)×C173C211yesmaximal, characteristic, complemented
7#19149726C22196yescomplemented
8#19149724C221144yescomplemented
9C2×#14583827C2212yescomplemented
10C2×#14583826C22118yescomplemented
11C2×#14583825C22118yescomplemented
12C2×#14583825C22136yescomplemented
13C2×#14583825C221144yescomplemented
14#41×D173C22196yescomplemented
15C22×Dic346C2219yescomplemented
16C2×Q8×D173C2216yescomplemented
17C22×C22⋊D173C22118yescomplemented
18C2×D4×D173C22118yescomplemented
19C23×Dic173C2216yescomplemented
20C22×C4×D173C2219yescomplemented
21H(5,𝔽2)×C173C22116yescomplemented
22C2×#41×C173C2216yescomplemented
23C22×D4×C173C2219yescomplemented
24C23×D173C2319yes—
25#14583827C23116yescomplemented
26#14583826C231144yescomplemented
27#14583825C231144yescomplemented
28#14583825C231288yescomplemented
29C2×Dic346C23118yes—
30C2×Dic346C23172yescomplemented
31Q8×D173C23148yescomplemented
32C2×D692C2316yes—
33C2×C22⋊D173C23136yes—
34C2×C22⋊D173C231144yescomplemented
35D4×D173C231144yescomplemented
36C22×Dic173C2319yes—
37C22×Dic173C23118yes—
38C22×Dic173C23148yescomplemented
39C2×C4×D173C2316yes—
40C2×C4×D173C23118yes—
41C2×C4×D173C23172yescomplemented
42C24×C173C2316yes—
43#41×C173C23148yescomplemented
44C2×Q8×C173C2312yes—
45C2×D4×C173C23118yes—
46C2×D4×C173C23172yescomplemented
47C22×C4×C173C2319yes—
48C22×D173—418no—
49C22×D173C2416yes—
50C22×D173C24136yes—
51Dic346C24172yes—
52D692C24124yes—
53C22⋊D173C241144yes—
54C2×Dic173C2411yescharacteristic
55C2×Dic173C2419yes—
56C2×Dic173C24136yes—
57C2×Dic173C24172yes—
58C4×D173C24124yes—
59C4×D173C24172yes—
60C23×C173—412no—
61C23×C173C2419yes—
62C23×C173C24124yes—
63Q8×C173C2418yes—
64D4×C173C24172yes—
65C2×C4×C173C2416yes—
66C2×C4×C173C24136yes—
67C22×C173—218no—
68C22×C173—424no—
69C22×C173C25118yes—
70C22×C173C2511yescharacteristic
71C2×D173—212no—
72C2×D173—436no—
73C2×D173C25112yes—
74C4×C173C25112yes—
75Dic173C2512yes—
76Dic173C25118yes—
77C2×C173—218no—
78C2×C173#58512yescomplemented
79C2×C173C2611yescharacteristic
80D173—212no—
81C173C2×#58511yescharacteristic, complemented, Sylow
82C2×#585—1731nomaximal, Sylow
83#585—17332no—
84C2×2−1+4—1736no—
85C2×H(5,𝔽2)—1739no—
86C2×H(5,𝔽2)D17311yescharacteristic, complemented
87C22×#41—1736no—
88C22×#41—1739no—
892−1+4—17396no—
90H(5,𝔽2)—173144no—
91H(5,𝔽2)C2×D173116yescomplemented
92C2×#41—17396no—
93C2×#41—1732no—
94C2×#41—17318no—
95C2×#41—17336no—
96C2×#41—173144no—
97C2×#41—17318no—
98C2×#41C2×D17316yescomplemented
99C22×Q8—1736no—
100C22×Q8—1739no—
101C22×D4—17318no—
102C22×D4—17318no—
103C22×D4C2×D17319yescomplemented
104C23×C4—1736no—
105C23×C4—1739no—
106C24—1739no—
107C24C22×D17316yes—
108#41—17316no—
109#41—173144no—
110#41—173288no—
111#41—173144no—
112#41C22×D173148yescomplemented
113C2×Q8—17348no—
114C2×Q8—17318no—
115C2×Q8—17372no—
116C2×Q8C22×D17312yes—
117C2×D4—17336no—
118C2×D4—173144no—
119C2×D4—173144no—
120C2×D4—1736no—
121C2×D4C22×D173118yes—
122C2×D4C22×D173172yescomplemented
123C22×C4—1739no—
124C22×C4—17318no—
125C22×C4—17348no—
126C22×C4—1736no—
127C22×C4—17318no—
128C22×C4—17372no—
129C22×C4C22×D17319yes—
130C23—412no—
131C23—1736no—
132C23—17336no—
133C23—69218no—
134C23C23×D17319yes—
135C23C23×D173124yes—
136Q8—17372no—
137Q8C23×D17318yes—
138D4—173144no—
139D4—17324no—
140D4C23×D173172yes—
141C2×C4—17336no—
142C2×C4—17372no—
143C2×C4—1731no—
144C2×C4—1739no—
145C2×C4—17324no—
146C2×C4—17372no—
147C2×C4C23×D173136yes—
148C2×C4C23×D17316yes—
149C22—218no—
150C22—424no—
151C22—34612no—
152C22—17312no—
153C22—69236no—
154C22C24×D173118yes—
155C22C24×D17311yescharacteristic
156C4—1732no—
157C4—17318no—
158C4C24×D173112yes—
159C2—218no—
160C2—34612no—
161C2#2601232612yesdirect product factor
162C2C25×D17311yescharacteristic
subgroupG/H
centerC22C24×D173
commutator, cosocleC2×C173C26
FrattiniC2C25×D173
FittingC2×H(5,𝔽2)×C173C2
socleC22×C173C25
hypercenterC2×H(5,𝔽2)D173
hypocenterC173C2×#585
LCSG ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #26012326C2direct (N × H)
2. C2×#19149726C2semidirect
3. C2×#19149724C2semidirect
4. C22×#14583825C2semidirect
5. C2×#41×D173C2semidirect
6. C2×H(5,𝔽2)×C173C2semidirect
7. #19149726C22semidirect
8. #19149724C22semidirect
9. C2×#14583827C22semidirect
10. C2×#14583826C22semidirect
11. C2×#14583825C22semidirect
12. C2×#14583825C22semidirect
13. C2×#14583825C22semidirect
14. #41×D173C22semidirect
15. C22×Dic346C22semidirect
16. C2×Q8×D173C22semidirect
17. C22×C22⋊D173C22semidirect
18. C2×D4×D173C22semidirect
19. C23×Dic173C22semidirect
20. C22×C4×D173C22semidirect
21. H(5,𝔽2)×C173C22semidirect
22. C2×#41×C173C22semidirect
23. C22×D4×C173C22semidirect
25. #14583827C23semidirect
26. #14583826C23semidirect
27. #14583825C23semidirect
28. #14583825C23semidirect
30. C2×Dic346C23semidirect
31. Q8×D173C23semidirect
34. C2×C22⋊D173C23semidirect
35. D4×D173C23semidirect
38. C22×Dic173C23semidirect
41. C2×C4×D173C23semidirect
43. #41×C173C23semidirect
46. C2×D4×C173C23semidirect
78. C2×C173#585semidirect
81. C173C2×#585semidirect
86. C2×H(5,𝔽2)D173semidirect
91. H(5,𝔽2)C2×D173semidirect
98. C2×#41C2×D173semidirect
103. C22×D4C2×D173semidirect
112. #41C22×D173semidirect
122. C2×D4C22×D173semidirect
161. C2#26012326direct (N × H)

Maximal subgroups

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

subgroupindex
#260123262
C2×#191497262
C2×#191497242
C22×#145838252
C2×#41×D1732
C2×H(5,𝔽2)×C1732
C2×#585173