Finite Group Databasebeta

← C2×#20295861 #28377839

Subgroups — C2×#20295861 #28377839

Order 13952.

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#20295861C2132yesmaximal, direct product factor
2C2×#16812746C216yesmaximal, complemented
3C2×#16812744C219yesmaximal, complemented
4C22×#13182028C219yesmaximal, complemented
5C2×#41×D109C216yesmaximal, complemented
6C2×H(5,𝔽2)×C109C211yesmaximal, characteristic, complemented
7#16812746C22196yescomplemented
8#16812744C221144yescomplemented
9C2×#13182030C2212yescomplemented
10C2×#13182029C22118yescomplemented
11C2×#13182028C22118yescomplemented
12C2×#13182028C22136yescomplemented
13C2×#13182028C221144yescomplemented
14#41×D109C22196yescomplemented
15C22×Dic218C2219yescomplemented
16C2×Q8×D109C2216yescomplemented
17C22×C22⋊D109C22118yescomplemented
18C2×D4×D109C22118yescomplemented
19C23×Dic109C2216yescomplemented
20C22×C4×D109C2219yescomplemented
21H(5,𝔽2)×C109C22116yescomplemented
22C2×#41×C109C2216yescomplemented
23C22×D4×C109C2219yescomplemented
24C23×D109C2319yes—
25#13182030C23116yescomplemented
26#13182029C231144yescomplemented
27#13182028C231144yescomplemented
28#13182028C231288yescomplemented
29C2×Dic218C23118yes—
30C2×Dic218C23172yescomplemented
31Q8×D109C23148yescomplemented
32C2×D436C2316yes—
33C2×C22⋊D109C23136yes—
34C2×C22⋊D109C231144yescomplemented
35D4×D109C231144yescomplemented
36C22×Dic109C2319yes—
37C22×Dic109C23118yes—
38C22×Dic109C23148yescomplemented
39C2×C4×D109C2316yes—
40C2×C4×D109C23118yes—
41C2×C4×D109C23172yescomplemented
42C24×C109C2316yes—
43#41×C109C23148yescomplemented
44C2×Q8×C109C2312yes—
45C2×D4×C109C23118yes—
46C2×D4×C109C23172yescomplemented
47C22×C4×C109C2319yes—
48C23×C109—412no—
49C23×C109C2419yes—
50C23×C109C24124yes—
51C22×D109—418no—
52C22×D109C2416yes—
53C22×D109C24136yes—
54Q8×C109C2418yes—
55D4×C109C24172yes—
56C2×C4×C109C2416yes—
57C2×C4×C109C24136yes—
58C22⋊D109C241144yes—
59C2×Dic109C2411yescharacteristic
60C2×Dic109C2419yes—
61C2×Dic109C24136yes—
62C2×Dic109C24172yes—
63D436C24124yes—
64C4×D109C24124yes—
65C4×D109C24172yes—
66Dic218C24172yes—
67C22×C109—218no—
68C22×C109—424no—
69C22×C109C25118yes—
70C22×C109C2511yescharacteristic
71C2×D109—212no—
72C2×D109—436no—
73C2×D109C25112yes—
74C4×C109C25112yes—
75Dic109C25118yes—
76Dic109C2512yes—
77C2×C109—218no—
78C2×C109#58512yescomplemented
79C2×C109C2611yescharacteristic
80D109—212no—
81C2×#585—1091nomaximal, Sylow
82C109C2×#58511yescharacteristic, complemented, Sylow
83#585—10932no—
84C2×2−1+4—1096no—
85C2×H(5,𝔽2)—1099no—
86C2×H(5,𝔽2)D10911yescharacteristic, complemented
87C22×#41—1096no—
88C22×#41—1099no—
892−1+4—10996no—
90H(5,𝔽2)—109144no—
91H(5,𝔽2)C2×D109116yescomplemented
92C2×#41—10996no—
93C2×#41—1092no—
94C2×#41—10918no—
95C2×#41—10936no—
96C2×#41—109144no—
97C2×#41—10918no—
98C2×#41C2×D10916yescomplemented
99C22×Q8—1096no—
100C22×Q8—1099no—
101C22×D4—10918no—
102C22×D4—10918no—
103C22×D4C2×D10919yescomplemented
104C23×C4—1096no—
105C23×C4—1099no—
106C24—1099no—
107C24C22×D10916yes—
108#41—10916no—
109#41—109144no—
110#41—109288no—
111#41—109144no—
112#41C22×D109148yescomplemented
113C2×Q8—10948no—
114C2×Q8—10918no—
115C2×Q8—10972no—
116C2×Q8C22×D10912yes—
117C2×D4—10936no—
118C2×D4—109144no—
119C2×D4—109144no—
120C2×D4—1096no—
121C2×D4C22×D109118yes—
122C2×D4C22×D109172yescomplemented
123C22×C4—1099no—
124C22×C4—10918no—
125C22×C4—10948no—
126C22×C4—1096no—
127C22×C4—10918no—
128C22×C4—10972no—
129C22×C4C22×D10919yes—
130C23—412no—
131C23—10936no—
132C23—1096no—
133C23—43618no—
134C23C23×D10919yes—
135C23C23×D109124yes—
136Q8—10972no—
137Q8C23×D10918yes—
138D4—109144no—
139D4—10924no—
140D4C23×D109172yes—
141C2×C4—10936no—
142C2×C4—10972no—
143C2×C4—1091no—
144C2×C4—1099no—
145C2×C4—10924no—
146C2×C4—10972no—
147C2×C4C23×D109136yes—
148C2×C4C23×D10916yes—
149C22—218no—
150C22—424no—
151C22—10912no—
152C22—43636no—
153C22—21812no—
154C22C24×D109118yes—
155C22C24×D10911yescharacteristic
156C4—1092no—
157C4—10918no—
158C4C24×D109112yes—
159C2—218no—
160C2—21812no—
161C2#2029586112yesdirect product factor
162C2C25×D10911yescharacteristic
subgroupG/H
centerC22C24×D109
commutator, cosocleC2×C109C26
FrattiniC2C25×D109
FittingC2×H(5,𝔽2)×C109C2
socleC22×C109C25
hypercenterC2×H(5,𝔽2)D109
hypocenterC109C2×#585
LCSG ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #20295861C2direct (N × H)
2. C2×#16812746C2semidirect
3. C2×#16812744C2semidirect
4. C22×#13182028C2semidirect
5. C2×#41×D109C2semidirect
6. C2×H(5,𝔽2)×C109C2semidirect
7. #16812746C22semidirect
8. #16812744C22semidirect
9. C2×#13182030C22semidirect
10. C2×#13182029C22semidirect
11. C2×#13182028C22semidirect
12. C2×#13182028C22semidirect
13. C2×#13182028C22semidirect
14. #41×D109C22semidirect
15. C22×Dic218C22semidirect
16. C2×Q8×D109C22semidirect
17. C22×C22⋊D109C22semidirect
18. C2×D4×D109C22semidirect
19. C23×Dic109C22semidirect
20. C22×C4×D109C22semidirect
21. H(5,𝔽2)×C109C22semidirect
22. C2×#41×C109C22semidirect
23. C22×D4×C109C22semidirect
25. #13182030C23semidirect
26. #13182029C23semidirect
27. #13182028C23semidirect
28. #13182028C23semidirect
30. C2×Dic218C23semidirect
31. Q8×D109C23semidirect
34. C2×C22⋊D109C23semidirect
35. D4×D109C23semidirect
38. C22×Dic109C23semidirect
41. C2×C4×D109C23semidirect
43. #41×C109C23semidirect
46. C2×D4×C109C23semidirect
78. C2×C109#585semidirect
82. C109C2×#585semidirect
86. C2×H(5,𝔽2)D109semidirect
91. H(5,𝔽2)C2×D109semidirect
98. C2×#41C2×D109semidirect
103. C22×D4C2×D109semidirect
112. #41C22×D109semidirect
122. C2×D4C22×D109semidirect
161. C2#20295861direct (N × H)

Maximal subgroups

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

subgroupindex
#202958612
C2×#168127462
C2×#168127442
C22×#131820282
C2×#41×D1092
C2×H(5,𝔽2)×C1092
C2×#585109