Finite Group Databasebeta

← C2×#26042114 #37548528

Subgroups — C2×#26042114 #37548528

Order 23168.

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#26042114C2132yesmaximal, direct product factor
2C2×#19153863C216yesmaximal, complemented
3C2×#19153861C219yesmaximal, complemented
4C22×#15668448C219yesmaximal, complemented
5C2×#41×D181C216yesmaximal, complemented
6C2×H(5,𝔽2)×C181C211yesmaximal, characteristic, complemented
7#19153863C22196yescomplemented
8#19153861C221144yescomplemented
9C2×#15668450C2212yescomplemented
10C2×#15668449C22118yescomplemented
11C2×#15668448C22136yescomplemented
12C2×#15668448C22118yescomplemented
13C2×#15668448C221144yescomplemented
14#41×D181C22196yescomplemented
15C22×Dic362C2219yescomplemented
16C2×Q8×D181C2216yescomplemented
17C22×C22⋊D181C22118yescomplemented
18C2×D4×D181C22118yescomplemented
19C23×Dic181C2216yescomplemented
20C22×C4×D181C2219yescomplemented
21H(5,𝔽2)×C181C22116yescomplemented
22C2×#41×C181C2216yescomplemented
23C22×D4×C181C2219yescomplemented
24C23×D181C2319yes—
25#15668450C23116yescomplemented
26#15668449C231144yescomplemented
27#15668448C231144yescomplemented
28#15668448C231288yescomplemented
29C2×Dic362C23118yes—
30C2×Dic362C23172yescomplemented
31Q8×D181C23148yescomplemented
32C2×D724C2316yes—
33C2×C22⋊D181C23136yes—
34C2×C22⋊D181C231144yescomplemented
35D4×D181C231144yescomplemented
36C22×Dic181C2319yes—
37C22×Dic181C23118yes—
38C22×Dic181C23148yescomplemented
39C2×C4×D181C2316yes—
40C2×C4×D181C23118yes—
41C2×C4×D181C23172yescomplemented
42C24×C181C2316yes—
43#41×C181C23148yescomplemented
44C2×Q8×C181C2312yes—
45C2×D4×C181C23118yes—
46C2×D4×C181C23172yescomplemented
47C22×C4×C181C2319yes—
48C22×D181—418no—
49C22×D181C2416yes—
50C22×D181C24136yes—
51Dic362C24172yes—
52D724C24124yes—
53C22⋊D181C241144yes—
54C2×Dic181C2411yescharacteristic
55C2×Dic181C2419yes—
56C2×Dic181C24136yes—
57C2×Dic181C24172yes—
58C4×D181C24124yes—
59C4×D181C24172yes—
60C23×C181—412no—
61C23×C181C2419yes—
62C23×C181C24124yes—
63Q8×C181C2418yes—
64D4×C181C24172yes—
65C2×C4×C181C2416yes—
66C2×C4×C181C24136yes—
67C22×C181—218no—
68C22×C181—424no—
69C22×C181C25118yes—
70C22×C181C2511yescharacteristic
71C2×D181—212no—
72C2×D181—436no—
73C2×D181C25112yes—
74C4×C181C25112yes—
75Dic181C2512yes—
76Dic181C25118yes—
77C2×C181—218no—
78C2×C181#58512yescomplemented
79C2×C181C2611yescharacteristic
80D181—212no—
81C181C2×#58511yescharacteristic, complemented, Sylow
82C2×#585—1811nomaximal, Sylow
83#585—18132no—
84C2×2−1+4—1816no—
85C2×H(5,𝔽2)—1819no—
86C2×H(5,𝔽2)D18111yescharacteristic, complemented
87C22×#41—1816no—
88C22×#41—1819no—
892−1+4—18196no—
90H(5,𝔽2)—181144no—
91H(5,𝔽2)C2×D181116yescomplemented
92C2×#41—18196no—
93C2×#41—1812no—
94C2×#41—18118no—
95C2×#41—18136no—
96C2×#41—181144no—
97C2×#41—18118no—
98C2×#41C2×D18116yescomplemented
99C22×Q8—1816no—
100C22×Q8—1819no—
101C22×D4—18118no—
102C22×D4—18118no—
103C22×D4C2×D18119yescomplemented
104C23×C4—1816no—
105C23×C4—1819no—
106C24—1819no—
107C24C22×D18116yes—
108#41—18116no—
109#41—181144no—
110#41—181288no—
111#41—181144no—
112#41C22×D181148yescomplemented
113C2×Q8—18148no—
114C2×Q8—18118no—
115C2×Q8—18172no—
116C2×Q8C22×D18112yes—
117C2×D4—18136no—
118C2×D4—181144no—
119C2×D4—181144no—
120C2×D4—1816no—
121C2×D4C22×D181118yes—
122C2×D4C22×D181172yescomplemented
123C22×C4—1819no—
124C22×C4—18118no—
125C22×C4—18148no—
126C22×C4—1816no—
127C22×C4—18118no—
128C22×C4—18172no—
129C22×C4C22×D18119yes—
130C23—412no—
131C23—1816no—
132C23—18136no—
133C23—72418no—
134C23C23×D18119yes—
135C23C23×D181124yes—
136Q8—18172no—
137Q8C23×D18118yes—
138D4—181144no—
139D4—18124no—
140D4C23×D181172yes—
141C2×C4—18136no—
142C2×C4—18172no—
143C2×C4—1811no—
144C2×C4—1819no—
145C2×C4—18124no—
146C2×C4—18172no—
147C2×C4C23×D181136yes—
148C2×C4C23×D18116yes—
149C22—218no—
150C22—424no—
151C22—36212no—
152C22—18112no—
153C22—72436no—
154C22C24×D181118yes—
155C22C24×D18111yescharacteristic
156C4—1812no—
157C4—18118no—
158C4C24×D181112yes—
159C2—218no—
160C2—36212no—
161C2#2604211412yesdirect product factor
162C2C25×D18111yescharacteristic
subgroupG/H
centerC22C24×D181
commutator, cosocleC2×C181C26
FrattiniC2C25×D181
FittingC2×H(5,𝔽2)×C181C2
socleC22×C181C25
hypercenterC2×H(5,𝔽2)D181
hypocenterC181C2×#585
LCSG ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #26042114C2direct (N × H)
2. C2×#19153863C2semidirect
3. C2×#19153861C2semidirect
4. C22×#15668448C2semidirect
5. C2×#41×D181C2semidirect
6. C2×H(5,𝔽2)×C181C2semidirect
7. #19153863C22semidirect
8. #19153861C22semidirect
9. C2×#15668450C22semidirect
10. C2×#15668449C22semidirect
11. C2×#15668448C22semidirect
12. C2×#15668448C22semidirect
13. C2×#15668448C22semidirect
14. #41×D181C22semidirect
15. C22×Dic362C22semidirect
16. C2×Q8×D181C22semidirect
17. C22×C22⋊D181C22semidirect
18. C2×D4×D181C22semidirect
19. C23×Dic181C22semidirect
20. C22×C4×D181C22semidirect
21. H(5,𝔽2)×C181C22semidirect
22. C2×#41×C181C22semidirect
23. C22×D4×C181C22semidirect
25. #15668450C23semidirect
26. #15668449C23semidirect
27. #15668448C23semidirect
28. #15668448C23semidirect
30. C2×Dic362C23semidirect
31. Q8×D181C23semidirect
34. C2×C22⋊D181C23semidirect
35. D4×D181C23semidirect
38. C22×Dic181C23semidirect
41. C2×C4×D181C23semidirect
43. #41×C181C23semidirect
46. C2×D4×C181C23semidirect
78. C2×C181#585semidirect
81. C181C2×#585semidirect
86. C2×H(5,𝔽2)D181semidirect
91. H(5,𝔽2)C2×D181semidirect
98. C2×#41C2×D181semidirect
103. C22×D4C2×D181semidirect
112. #41C22×D181semidirect
122. C2×D4C22×D181semidirect
161. C2#26042114direct (N × H)

Maximal subgroups

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

subgroupindex
#260421142
C2×#191538632
C2×#191538612
C22×#156684482
C2×#41×D1812
C2×H(5,𝔽2)×C1812
C2×#585181