Finite Group Databasebeta

← C2×#31827300 #49136068

Subgroups — C2×#31827300 #49136068

Order 36224.

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#31827300C2132yesmaximal, direct product factor
2C2×#22583657C216yesmaximal, complemented
3C2×#22583655C219yesmaximal, complemented
4C22×#17985391C219yesmaximal, complemented
5C2×#41×D283C216yesmaximal, complemented
6C2×H(5,𝔽2)×C283C211yesmaximal, characteristic, complemented
7#22583657C22196yescomplemented
8#22583655C221144yescomplemented
9C2×#17985393C2212yescomplemented
10C2×#17985392C22118yescomplemented
11C2×#17985391C22118yescomplemented
12C2×#17985391C22136yescomplemented
13C2×#17985391C221144yescomplemented
14#41×D283C22196yescomplemented
15C22×Dic566C2219yescomplemented
16C2×Q8×D283C2216yescomplemented
17C22×C22⋊D283C22118yescomplemented
18C2×D4×D283C22118yescomplemented
19C23×Dic283C2216yescomplemented
20C22×C4×D283C2219yescomplemented
21H(5,𝔽2)×C283C22116yescomplemented
22C2×#41×C283C2216yescomplemented
23C22×D4×C283C2219yescomplemented
24C23×D283C2319yes—
25#17985393C23116yescomplemented
26#17985392C231144yescomplemented
27#17985391C231144yescomplemented
28#17985391C231288yescomplemented
29C2×Dic566C23118yes—
30C2×Dic566C23172yescomplemented
31Q8×D283C23148yescomplemented
32C2×D1132C2316yes—
33C2×C22⋊D283C23136yes—
34C2×C22⋊D283C231144yescomplemented
35D4×D283C231144yescomplemented
36C22×Dic283C2319yes—
37C22×Dic283C23118yes—
38C22×Dic283C23148yescomplemented
39C2×C4×D283C2316yes—
40C2×C4×D283C23118yes—
41C2×C4×D283C23172yescomplemented
42C24×C283C2316yes—
43#41×C283C23148yescomplemented
44C2×Q8×C283C2312yes—
45C2×D4×C283C23118yes—
46C2×D4×C283C23172yescomplemented
47C22×C4×C283C2319yes—
48C22×D283—418no—
49C22×D283C2416yes—
50C22×D283C24136yes—
51Dic566C24172yes—
52D1132C24124yes—
53C22⋊D283C241144yes—
54C2×Dic283C2411yescharacteristic
55C2×Dic283C2419yes—
56C2×Dic283C24136yes—
57C2×Dic283C24172yes—
58C4×D283C24124yes—
59C4×D283C24172yes—
60C23×C283—412no—
61C23×C283C2419yes—
62C23×C283C24124yes—
63Q8×C283C2418yes—
64D4×C283C24172yes—
65C2×C4×C283C2416yes—
66C2×C4×C283C24136yes—
67C22×C283—218no—
68C22×C283—424no—
69C22×C283C25118yes—
70C22×C283C2511yescharacteristic
71C2×D283—212no—
72C2×D283—436no—
73C2×D283C25112yes—
74C4×C283C25112yes—
75Dic283C2512yes—
76Dic283C25118yes—
77C2×C283—218no—
78C2×C283#58512yescomplemented
79C2×C283C2611yescharacteristic
80D283—212no—
81C283C2×#58511yescharacteristic, complemented, Sylow
82C2×#585—2831nomaximal, Sylow
83#585—28332no—
84C2×2−1+4—2836no—
85C2×H(5,𝔽2)—2839no—
86C2×H(5,𝔽2)D28311yescharacteristic, complemented
87C22×#41—2836no—
88C22×#41—2839no—
892−1+4—28396no—
90H(5,𝔽2)—283144no—
91H(5,𝔽2)C2×D283116yescomplemented
92C2×#41—28396no—
93C2×#41—2832no—
94C2×#41—28318no—
95C2×#41—28336no—
96C2×#41—283144no—
97C2×#41—28318no—
98C2×#41C2×D28316yescomplemented
99C22×Q8—2836no—
100C22×Q8—2839no—
101C22×D4—28318no—
102C22×D4—28318no—
103C22×D4C2×D28319yescomplemented
104C23×C4—2836no—
105C23×C4—2839no—
106C24—2839no—
107C24C22×D28316yes—
108#41—28316no—
109#41—283144no—
110#41—283288no—
111#41—283144no—
112#41C22×D283148yescomplemented
113C2×Q8—28348no—
114C2×Q8—28318no—
115C2×Q8—28372no—
116C2×Q8C22×D28312yes—
117C2×D4—28336no—
118C2×D4—283144no—
119C2×D4—283144no—
120C2×D4—2836no—
121C2×D4C22×D283118yes—
122C2×D4C22×D283172yescomplemented
123C22×C4—2839no—
124C22×C4—28318no—
125C22×C4—28348no—
126C22×C4—2836no—
127C22×C4—28318no—
128C22×C4—28372no—
129C22×C4C22×D28319yes—
130C23—412no—
131C23—2836no—
132C23—28336no—
133C23—113218no—
134C23C23×D28319yes—
135C23C23×D283124yes—
136Q8—28372no—
137Q8C23×D28318yes—
138D4—283144no—
139D4—28324no—
140D4C23×D283172yes—
141C2×C4—28336no—
142C2×C4—28372no—
143C2×C4—2831no—
144C2×C4—2839no—
145C2×C4—28324no—
146C2×C4—28372no—
147C2×C4C23×D283136yes—
148C2×C4C23×D28316yes—
149C22—218no—
150C22—424no—
151C22—28312no—
152C22—56612no—
153C22—113236no—
154C22C24×D283118yes—
155C22C24×D28311yescharacteristic
156C4—2832no—
157C4—28318no—
158C4C24×D283112yes—
159C2—218no—
160C2—56612no—
161C2#3182730012yesdirect product factor
162C2C25×D28311yescharacteristic
subgroupG/H
centerC22C24×D283
commutator, cosocleC2×C283C26
FrattiniC2C25×D283
FittingC2×H(5,𝔽2)×C283C2
socleC22×C283C25
hypercenterC2×H(5,𝔽2)D283
hypocenterC283C2×#585
LCSG ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #31827300C2direct (N × H)
2. C2×#22583657C2semidirect
3. C2×#22583655C2semidirect
4. C22×#17985391C2semidirect
5. C2×#41×D283C2semidirect
6. C2×H(5,𝔽2)×C283C2semidirect
7. #22583657C22semidirect
8. #22583655C22semidirect
9. C2×#17985393C22semidirect
10. C2×#17985392C22semidirect
11. C2×#17985391C22semidirect
12. C2×#17985391C22semidirect
13. C2×#17985391C22semidirect
14. #41×D283C22semidirect
15. C22×Dic566C22semidirect
16. C2×Q8×D283C22semidirect
17. C22×C22⋊D283C22semidirect
18. C2×D4×D283C22semidirect
19. C23×Dic283C22semidirect
20. C22×C4×D283C22semidirect
21. H(5,𝔽2)×C283C22semidirect
22. C2×#41×C283C22semidirect
23. C22×D4×C283C22semidirect
25. #17985393C23semidirect
26. #17985392C23semidirect
27. #17985391C23semidirect
28. #17985391C23semidirect
30. C2×Dic566C23semidirect
31. Q8×D283C23semidirect
34. C2×C22⋊D283C23semidirect
35. D4×D283C23semidirect
38. C22×Dic283C23semidirect
41. C2×C4×D283C23semidirect
43. #41×C283C23semidirect
46. C2×D4×C283C23semidirect
78. C2×C283#585semidirect
81. C283C2×#585semidirect
86. C2×H(5,𝔽2)D283semidirect
91. H(5,𝔽2)C2×D283semidirect
98. C2×#41C2×D283semidirect
103. C22×D4C2×D283semidirect
112. #41C22×D283semidirect
122. C2×D4C22×D283semidirect
161. C2#31827300direct (N × H)

Maximal subgroups

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

subgroupindex
#318273002
C2×#225836572
C2×#225836552
C22×#179853912
C2×#41×D2832
C2×H(5,𝔽2)×C2832
C2×#585283