Finite Group Databasebeta

← C2×#22554147 #30680768

Subgroups — C2×#22554147 #30680768

Order 16768.

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#22554147C2132yesmaximal, direct product factor
2C2×#16862293C216yesmaximal, complemented
3C2×#16862291C219yesmaximal, complemented
4C22×#14523512C219yesmaximal, complemented
5C2×#41×D131C216yesmaximal, complemented
6C2×H(5,𝔽2)×C131C211yesmaximal, characteristic, complemented
7#16862293C22196yescomplemented
8#16862291C221144yescomplemented
9C2×#14523514C2212yescomplemented
10C2×#14523513C22118yescomplemented
11C2×#14523512C22118yescomplemented
12C2×#14523512C22136yescomplemented
13C2×#14523512C221144yescomplemented
14#41×D131C22196yescomplemented
15C22×Dic262C2219yescomplemented
16C2×Q8×D131C2216yescomplemented
17C22×C22⋊D131C22118yescomplemented
18C2×D4×D131C22118yescomplemented
19C23×Dic131C2216yescomplemented
20C22×C4×D131C2219yescomplemented
21H(5,𝔽2)×C131C22116yescomplemented
22C2×#41×C131C2216yescomplemented
23C22×D4×C131C2219yescomplemented
24C23×D131C2319yes—
25#14523514C23116yescomplemented
26#14523513C231144yescomplemented
27#14523512C231144yescomplemented
28#14523512C231288yescomplemented
29C2×Dic262C23118yes—
30C2×Dic262C23172yescomplemented
31Q8×D131C23148yescomplemented
32C2×D524C2316yes—
33C2×C22⋊D131C23136yes—
34C2×C22⋊D131C231144yescomplemented
35D4×D131C231144yescomplemented
36C22×Dic131C2319yes—
37C22×Dic131C23118yes—
38C22×Dic131C23148yescomplemented
39C2×C4×D131C2316yes—
40C2×C4×D131C23118yes—
41C2×C4×D131C23172yescomplemented
42C24×C131C2316yes—
43#41×C131C23148yescomplemented
44C2×Q8×C131C2312yes—
45C2×D4×C131C23118yes—
46C2×D4×C131C23172yescomplemented
47C22×C4×C131C2319yes—
48C22×D131—418no—
49C22×D131C2416yes—
50C22×D131C24136yes—
51Dic262C24172yes—
52D524C24124yes—
53C22⋊D131C241144yes—
54C2×Dic131C2411yescharacteristic
55C2×Dic131C2419yes—
56C2×Dic131C24136yes—
57C2×Dic131C24172yes—
58C4×D131C24124yes—
59C4×D131C24172yes—
60C23×C131—412no—
61C23×C131C2419yes—
62C23×C131C24124yes—
63Q8×C131C2418yes—
64D4×C131C24172yes—
65C2×C4×C131C2416yes—
66C2×C4×C131C24136yes—
67C22×C131—218no—
68C22×C131—424no—
69C22×C131C25118yes—
70C22×C131C2511yescharacteristic
71C2×D131—212no—
72C2×D131—436no—
73C2×D131C25112yes—
74C4×C131C25112yes—
75Dic131C2512yes—
76Dic131C25118yes—
77C2×C131—218no—
78C2×C131#58512yescomplemented
79C2×C131C2611yescharacteristic
80D131—212no—
81C131C2×#58511yescharacteristic, complemented, Sylow
82C2×#585—1311nomaximal, Sylow
83#585—13132no—
84C2×2−1+4—1316no—
85C2×H(5,𝔽2)—1319no—
86C2×H(5,𝔽2)D13111yescharacteristic, complemented
87C22×#41—1316no—
88C22×#41—1319no—
892−1+4—13196no—
90H(5,𝔽2)—131144no—
91H(5,𝔽2)C2×D131116yescomplemented
92C2×#41—13196no—
93C2×#41—1312no—
94C2×#41—13118no—
95C2×#41—13136no—
96C2×#41—131144no—
97C2×#41—13118no—
98C2×#41C2×D13116yescomplemented
99C22×Q8—1316no—
100C22×Q8—1319no—
101C22×D4—13118no—
102C22×D4—13118no—
103C22×D4C2×D13119yescomplemented
104C23×C4—1316no—
105C23×C4—1319no—
106C24—1319no—
107C24C22×D13116yes—
108#41—13116no—
109#41—131144no—
110#41—131288no—
111#41—131144no—
112#41C22×D131148yescomplemented
113C2×Q8—13148no—
114C2×Q8—13118no—
115C2×Q8—13172no—
116C2×Q8C22×D13112yes—
117C2×D4—13136no—
118C2×D4—131144no—
119C2×D4—131144no—
120C2×D4—1316no—
121C2×D4C22×D131118yes—
122C2×D4C22×D131172yescomplemented
123C22×C4—1319no—
124C22×C4—13118no—
125C22×C4—13148no—
126C22×C4—1316no—
127C22×C4—13118no—
128C22×C4—13172no—
129C22×C4C22×D13119yes—
130C23—412no—
131C23—52418no—
132C23—1316no—
133C23—13136no—
134C23C23×D13119yes—
135C23C23×D131124yes—
136Q8—13172no—
137Q8C23×D13118yes—
138D4—131144no—
139D4—13124no—
140D4C23×D131172yes—
141C2×C4—13136no—
142C2×C4—13172no—
143C2×C4—1311no—
144C2×C4—1319no—
145C2×C4—13124no—
146C2×C4—13172no—
147C2×C4C23×D131136yes—
148C2×C4C23×D13116yes—
149C22—218no—
150C22—424no—
151C22—26212no—
152C22—52436no—
153C22—13112no—
154C22C24×D131118yes—
155C22C24×D13111yescharacteristic
156C4—1312no—
157C4—13118no—
158C4C24×D131112yes—
159C2—218no—
160C2—26212no—
161C2#2255414712yesdirect product factor
162C2C25×D13111yescharacteristic
subgroupG/H
centerC22C24×D131
commutator, cosocleC2×C131C26
FrattiniC2C25×D131
FittingC2×H(5,𝔽2)×C131C2
socleC22×C131C25
hypercenterC2×H(5,𝔽2)D131
hypocenterC131C2×#585
LCSG ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #22554147C2direct (N × H)
2. C2×#16862293C2semidirect
3. C2×#16862291C2semidirect
4. C22×#14523512C2semidirect
5. C2×#41×D131C2semidirect
6. C2×H(5,𝔽2)×C131C2semidirect
7. #16862293C22semidirect
8. #16862291C22semidirect
9. C2×#14523514C22semidirect
10. C2×#14523513C22semidirect
11. C2×#14523512C22semidirect
12. C2×#14523512C22semidirect
13. C2×#14523512C22semidirect
14. #41×D131C22semidirect
15. C22×Dic262C22semidirect
16. C2×Q8×D131C22semidirect
17. C22×C22⋊D131C22semidirect
18. C2×D4×D131C22semidirect
19. C23×Dic131C22semidirect
20. C22×C4×D131C22semidirect
21. H(5,𝔽2)×C131C22semidirect
22. C2×#41×C131C22semidirect
23. C22×D4×C131C22semidirect
25. #14523514C23semidirect
26. #14523513C23semidirect
27. #14523512C23semidirect
28. #14523512C23semidirect
30. C2×Dic262C23semidirect
31. Q8×D131C23semidirect
34. C2×C22⋊D131C23semidirect
35. D4×D131C23semidirect
38. C22×Dic131C23semidirect
41. C2×C4×D131C23semidirect
43. #41×C131C23semidirect
46. C2×D4×C131C23semidirect
78. C2×C131#585semidirect
81. C131C2×#585semidirect
86. C2×H(5,𝔽2)D131semidirect
91. H(5,𝔽2)C2×D131semidirect
98. C2×#41C2×D131semidirect
103. C22×D4C2×D131semidirect
112. #41C22×D131semidirect
122. C2×D4C22×D131semidirect
161. C2#22554147direct (N × H)

Maximal subgroups

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

subgroupindex
#225541472
C2×#168622932
C2×#168622912
C22×#145235122
C2×#41×D1312
C2×H(5,𝔽2)×C1312
C2×#585131