Finite Group Databasebeta

← C2×#22578238 #31795596

Subgroups — C2×#22578238 #31795596

Order 17536.

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#22578238C2132yesmaximal, direct product factor
2C2×#17984135C216yesmaximal, complemented
3C2×#17984133C219yesmaximal, complemented
4C22×#14555509C219yesmaximal, complemented
5C2×#41×D137C216yesmaximal, complemented
6C2×H(5,𝔽2)×C137C211yesmaximal, characteristic, complemented
7#17984135C22196yescomplemented
8#17984133C221144yescomplemented
9C2×#14555511C2212yescomplemented
10C2×#14555510C22118yescomplemented
11C2×#14555509C22118yescomplemented
12C2×#14555509C22136yescomplemented
13C2×#14555509C221144yescomplemented
14#41×D137C22196yescomplemented
15C22×Dic274C2219yescomplemented
16C2×Q8×D137C2216yescomplemented
17C22×C22⋊D137C22118yescomplemented
18C2×D4×D137C22118yescomplemented
19C23×Dic137C2216yescomplemented
20C22×C4×D137C2219yescomplemented
21H(5,𝔽2)×C137C22116yescomplemented
22C2×#41×C137C2216yescomplemented
23C22×D4×C137C2219yescomplemented
24C23×D137C2319yes—
25#14555511C23116yescomplemented
26#14555510C231144yescomplemented
27#14555509C231144yescomplemented
28#14555509C231288yescomplemented
29C2×Dic274C23118yes—
30C2×Dic274C23172yescomplemented
31Q8×D137C23148yescomplemented
32C2×D548C2316yes—
33C2×C22⋊D137C23136yes—
34C2×C22⋊D137C231144yescomplemented
35D4×D137C231144yescomplemented
36C22×Dic137C2319yes—
37C22×Dic137C23118yes—
38C22×Dic137C23148yescomplemented
39C2×C4×D137C2316yes—
40C2×C4×D137C23118yes—
41C2×C4×D137C23172yescomplemented
42C24×C137C2316yes—
43#41×C137C23148yescomplemented
44C2×Q8×C137C2312yes—
45C2×D4×C137C23118yes—
46C2×D4×C137C23172yescomplemented
47C22×C4×C137C2319yes—
48C22×D137—418no—
49C22×D137C2416yes—
50C22×D137C24136yes—
51Dic274C24172yes—
52D548C24124yes—
53C22⋊D137C241144yes—
54C2×Dic137C2411yescharacteristic
55C2×Dic137C2419yes—
56C2×Dic137C24136yes—
57C2×Dic137C24172yes—
58C4×D137C24124yes—
59C4×D137C24172yes—
60C23×C137—412no—
61C23×C137C2419yes—
62C23×C137C24124yes—
63Q8×C137C2418yes—
64D4×C137C24172yes—
65C2×C4×C137C2416yes—
66C2×C4×C137C24136yes—
67C22×C137—218no—
68C22×C137—424no—
69C22×C137C25118yes—
70C22×C137C2511yescharacteristic
71C2×D137—212no—
72C2×D137—436no—
73C2×D137C25112yes—
74C4×C137C25112yes—
75Dic137C2512yes—
76Dic137C25118yes—
77C2×C137—218no—
78C2×C137#58512yescomplemented
79C2×C137C2611yescharacteristic
80D137—212no—
81C137C2×#58511yescharacteristic, complemented, Sylow
82C2×#585—1371nomaximal, Sylow
83#585—13732no—
84C2×2−1+4—1376no—
85C2×H(5,𝔽2)—1379no—
86C2×H(5,𝔽2)D13711yescharacteristic, complemented
87C22×#41—1376no—
88C22×#41—1379no—
892−1+4—13796no—
90H(5,𝔽2)—137144no—
91H(5,𝔽2)C2×D137116yescomplemented
92C2×#41—13796no—
93C2×#41—1372no—
94C2×#41—13718no—
95C2×#41—13736no—
96C2×#41—137144no—
97C2×#41—13718no—
98C2×#41C2×D13716yescomplemented
99C22×Q8—1376no—
100C22×Q8—1379no—
101C22×D4—13718no—
102C22×D4—13718no—
103C22×D4C2×D13719yescomplemented
104C23×C4—1376no—
105C23×C4—1379no—
106C24—1379no—
107C24C22×D13716yes—
108#41—13716no—
109#41—137144no—
110#41—137288no—
111#41—137144no—
112#41C22×D137148yescomplemented
113C2×Q8—13748no—
114C2×Q8—13718no—
115C2×Q8—13772no—
116C2×Q8C22×D13712yes—
117C2×D4—13736no—
118C2×D4—137144no—
119C2×D4—137144no—
120C2×D4—1376no—
121C2×D4C22×D137118yes—
122C2×D4C22×D137172yescomplemented
123C22×C4—1379no—
124C22×C4—13718no—
125C22×C4—13748no—
126C22×C4—1376no—
127C22×C4—13718no—
128C22×C4—13772no—
129C22×C4C22×D13719yes—
130C23—412no—
131C23—54818no—
132C23—1376no—
133C23—13736no—
134C23C23×D13719yes—
135C23C23×D137124yes—
136Q8—13772no—
137Q8C23×D13718yes—
138D4—137144no—
139D4—13724no—
140D4C23×D137172yes—
141C2×C4—13736no—
142C2×C4—13772no—
143C2×C4—1371no—
144C2×C4—1379no—
145C2×C4—13724no—
146C2×C4—13772no—
147C2×C4C23×D137136yes—
148C2×C4C23×D13716yes—
149C22—218no—
150C22—424no—
151C22—27412no—
152C22—54836no—
153C22—13712no—
154C22C24×D137118yes—
155C22C24×D13711yescharacteristic
156C4—1372no—
157C4—13718no—
158C4C24×D137112yes—
159C2—218no—
160C2—27412no—
161C2#2257823812yesdirect product factor
162C2C25×D13711yescharacteristic
subgroupG/H
centerC22C24×D137
commutator, cosocleC2×C137C26
FrattiniC2C25×D137
FittingC2×H(5,𝔽2)×C137C2
socleC22×C137C25
hypercenterC2×H(5,𝔽2)D137
hypocenterC137C2×#585
LCSG ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #22578238C2direct (N × H)
2. C2×#17984135C2semidirect
3. C2×#17984133C2semidirect
4. C22×#14555509C2semidirect
5. C2×#41×D137C2semidirect
6. C2×H(5,𝔽2)×C137C2semidirect
7. #17984135C22semidirect
8. #17984133C22semidirect
9. C2×#14555511C22semidirect
10. C2×#14555510C22semidirect
11. C2×#14555509C22semidirect
12. C2×#14555509C22semidirect
13. C2×#14555509C22semidirect
14. #41×D137C22semidirect
15. C22×Dic274C22semidirect
16. C2×Q8×D137C22semidirect
17. C22×C22⋊D137C22semidirect
18. C2×D4×D137C22semidirect
19. C23×Dic137C22semidirect
20. C22×C4×D137C22semidirect
21. H(5,𝔽2)×C137C22semidirect
22. C2×#41×C137C22semidirect
23. C22×D4×C137C22semidirect
25. #14555511C23semidirect
26. #14555510C23semidirect
27. #14555509C23semidirect
28. #14555509C23semidirect
30. C2×Dic274C23semidirect
31. Q8×D137C23semidirect
34. C2×C22⋊D137C23semidirect
35. D4×D137C23semidirect
38. C22×Dic137C23semidirect
41. C2×C4×D137C23semidirect
43. #41×C137C23semidirect
46. C2×D4×C137C23semidirect
78. C2×C137#585semidirect
81. C137C2×#585semidirect
86. C2×H(5,𝔽2)D137semidirect
91. H(5,𝔽2)C2×D137semidirect
98. C2×#41C2×D137semidirect
103. C22×D4C2×D137semidirect
112. #41C22×D137semidirect
122. C2×D4C22×D137semidirect
161. C2#22578238direct (N × H)

Maximal subgroups

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

subgroupindex
#225782382
C2×#179841352
C2×#179841332
C22×#145555092
C2×#41×D1372
C2×H(5,𝔽2)×C1372
C2×#585137