Finite Group Databasebeta

← C2×#20261570 #27156543

Subgroups — C2×#20261570 #27156543

Order 12416.

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#20261570C2132yesmaximal, direct product factor
2C2×#15690857C216yesmaximal, complemented
3C2×#15690855C219yesmaximal, complemented
4C22×#13095510C219yesmaximal, complemented
5C2×#41×D97C216yesmaximal, complemented
6C2×H(5,𝔽2)×C97C211yesmaximal, characteristic, complemented
7#15690857C22196yescomplemented
8#15690855C221144yescomplemented
9C2×#13095512C2212yescomplemented
10C2×#13095511C22118yescomplemented
11C2×#13095510C22118yescomplemented
12C2×#13095510C22136yescomplemented
13C2×#13095510C221144yescomplemented
14#41×D97C22196yescomplemented
15C22×Dic194C2219yescomplemented
16C2×Q8×D97C2216yescomplemented
17C22×C22⋊D97C22118yescomplemented
18C2×D4×D97C22118yescomplemented
19C23×Dic97C2216yescomplemented
20C22×C4×D97C2219yescomplemented
21H(5,𝔽2)×C97C22116yescomplemented
22C2×#41×C97C2216yescomplemented
23C22×D4×C97C2219yescomplemented
24C23×D97C2319yes—
25#13095512C23116yescomplemented
26#13095511C231144yescomplemented
27#13095510C231144yescomplemented
28#13095510C231288yescomplemented
29C2×Dic194C23118yes—
30C2×Dic194C23172yescomplemented
31Q8×D97C23148yescomplemented
32C2×D388C2316yes—
33C2×C22⋊D97C23136yes—
34C2×C22⋊D97C231144yescomplemented
35D4×D97C231144yescomplemented
36C22×Dic97C2319yes—
37C22×Dic97C23118yes—
38C22×Dic97C23148yescomplemented
39C2×C4×D97C2316yes—
40C2×C4×D97C23118yes—
41C2×C4×D97C23172yescomplemented
42C24×C97C2316yes—
43#41×C97C23148yescomplemented
44C2×Q8×C97C2312yes—
45C2×D4×C97C23118yes—
46C2×D4×C97C23172yescomplemented
47C22×C4×C97C2319yes—
48C23×C97—412no—
49C23×C97C2419yes—
50C23×C97C24124yes—
51C22×D97—418no—
52C22×D97C2416yes—
53C22×D97C24136yes—
54Q8×C97C2418yes—
55D4×C97C24172yes—
56C2×C4×C97C2416yes—
57C2×C4×C97C24136yes—
58C22⋊D97C241144yes—
59C2×Dic97C2411yescharacteristic
60C2×Dic97C2419yes—
61C2×Dic97C24136yes—
62C2×Dic97C24172yes—
63D388C24124yes—
64C4×D97C24124yes—
65C4×D97C24172yes—
66Dic194C24172yes—
67C22×C97—218no—
68C22×C97—424no—
69C22×C97C25118yes—
70C22×C97C2511yescharacteristic
71C2×D97—212no—
72C2×D97—436no—
73C2×D97C25112yes—
74C4×C97C25112yes—
75Dic97C25118yes—
76Dic97C2512yes—
77C2×C97—218no—
78C2×C97#58512yescomplemented
79C2×C97C2611yescharacteristic
80D97—212no—
81C2×#585—971nomaximal, Sylow
82C97C2×#58511yescharacteristic, complemented, Sylow
83#585—9732no—
84C2×2−1+4—976no—
85C2×H(5,𝔽2)—979no—
86C2×H(5,𝔽2)D9711yescharacteristic, complemented
87C22×#41—976no—
88C22×#41—979no—
892−1+4—9796no—
90H(5,𝔽2)—97144no—
91H(5,𝔽2)C2×D97116yescomplemented
92C2×#41—9796no—
93C2×#41—972no—
94C2×#41—9718no—
95C2×#41—9736no—
96C2×#41—97144no—
97C2×#41—9718no—
98C2×#41C2×D9716yescomplemented
99C22×Q8—976no—
100C22×Q8—979no—
101C22×D4—9718no—
102C22×D4—9718no—
103C22×D4C2×D9719yescomplemented
104C23×C4—976no—
105C23×C4—979no—
106C24—979no—
107C24C22×D9716yes—
108#41—9716no—
109#41—97144no—
110#41—97288no—
111#41—97144no—
112#41C22×D97148yescomplemented
113C2×Q8—9748no—
114C2×Q8—9718no—
115C2×Q8—9772no—
116C2×Q8C22×D9712yes—
117C2×D4—9736no—
118C2×D4—97144no—
119C2×D4—97144no—
120C2×D4—976no—
121C2×D4C22×D97118yes—
122C2×D4C22×D97172yescomplemented
123C22×C4—979no—
124C22×C4—9718no—
125C22×C4—9748no—
126C22×C4—976no—
127C22×C4—9718no—
128C22×C4—9772no—
129C22×C4C22×D9719yes—
130C23—412no—
131C23—9736no—
132C23—976no—
133C23—38818no—
134C23C23×D9719yes—
135C23C23×D97124yes—
136Q8—9772no—
137Q8C23×D9718yes—
138D4—97144no—
139D4—9724no—
140D4C23×D97172yes—
141C2×C4—9736no—
142C2×C4—9772no—
143C2×C4—971no—
144C2×C4—979no—
145C2×C4—9724no—
146C2×C4—9772no—
147C2×C4C23×D97136yes—
148C2×C4C23×D9716yes—
149C22—218no—
150C22—424no—
151C22—9712no—
152C22—38836no—
153C22—19412no—
154C22C24×D97118yes—
155C22C24×D9711yescharacteristic
156C4—972no—
157C4—9718no—
158C4C24×D97112yes—
159C2—218no—
160C2—19412no—
161C2#2026157012yesdirect product factor
162C2C25×D9711yescharacteristic
subgroupG/H
centerC22C24×D97
commutator, cosocleC2×C97C26
FrattiniC2C25×D97
FittingC2×H(5,𝔽2)×C97C2
socleC22×C97C25
hypercenterC2×H(5,𝔽2)D97
hypocenterC97C2×#585
LCSG ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #20261570C2direct (N × H)
2. C2×#15690857C2semidirect
3. C2×#15690855C2semidirect
4. C22×#13095510C2semidirect
5. C2×#41×D97C2semidirect
6. C2×H(5,𝔽2)×C97C2semidirect
7. #15690857C22semidirect
8. #15690855C22semidirect
9. C2×#13095512C22semidirect
10. C2×#13095511C22semidirect
11. C2×#13095510C22semidirect
12. C2×#13095510C22semidirect
13. C2×#13095510C22semidirect
14. #41×D97C22semidirect
15. C22×Dic194C22semidirect
16. C2×Q8×D97C22semidirect
17. C22×C22⋊D97C22semidirect
18. C2×D4×D97C22semidirect
19. C23×Dic97C22semidirect
20. C22×C4×D97C22semidirect
21. H(5,𝔽2)×C97C22semidirect
22. C2×#41×C97C22semidirect
23. C22×D4×C97C22semidirect
25. #13095512C23semidirect
26. #13095511C23semidirect
27. #13095510C23semidirect
28. #13095510C23semidirect
30. C2×Dic194C23semidirect
31. Q8×D97C23semidirect
34. C2×C22⋊D97C23semidirect
35. D4×D97C23semidirect
38. C22×Dic97C23semidirect
41. C2×C4×D97C23semidirect
43. #41×C97C23semidirect
46. C2×D4×C97C23semidirect
78. C2×C97#585semidirect
82. C97C2×#585semidirect
86. C2×H(5,𝔽2)D97semidirect
91. H(5,𝔽2)C2×D97semidirect
98. C2×#41C2×D97semidirect
103. C22×D4C2×D97semidirect
112. #41C22×D97semidirect
122. C2×D4C22×D97semidirect
161. C2#20261570direct (N × H)

Maximal subgroups

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

subgroupindex
#202615702
C2×#156908572
C2×#156908552
C22×#130955102
C2×#41×D972
C2×H(5,𝔽2)×C972
C2×#58597