Finite Group Databasebeta

← C2×#14521884 #16860518

Subgroups — C2×#14521884 #16860518

Order 3968.

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#14521884C2132yesmaximal, direct product factor
2C2×#11758567C216yesmaximal, complemented
3C2×#11758561C219yesmaximal, complemented
4C22×#92497C219yesmaximal, complemented
5C2×#41×D31C216yesmaximal, complemented
6C2×C31×H(5,𝔽2)C211yesmaximal, characteristic, complemented
7C31×H(5,𝔽2)C22116yescomplemented
8C2×#41×C31C2216yescomplemented
9C22×D4×C31C2219yescomplemented
10C22×C22⋊D31C22118yescomplemented
11C23×Dic31C2216yescomplemented
12#11758567C22196yescomplemented
13#41×D31C22196yescomplemented
14C2×#92499C2212yescomplemented
15C2×Q8×D31C2216yescomplemented
16#11758561C221144yescomplemented
17C2×#92497C22118yescomplemented
18C2×#92497C22136yescomplemented
19C2×#92497C221144yescomplemented
20C2×D4×D31C22118yescomplemented
21C2×#92495C22118yescomplemented
22C22×C4×D31C2219yescomplemented
23C22×Dic62C2219yescomplemented
24C24×C31C2316yes—
25C23×D31C2319yes—
26#41×C31C23148yescomplemented
27C2×Q8×C31C2312yes—
28C2×D4×C31C23118yes—
29C2×D4×C31C23172yescomplemented
30C22×C4×C31C2319yes—
31C2×C22⋊D31C23136yes—
32C2×C22⋊D31C231144yescomplemented
33C22×Dic31C23118yes—
34C22×Dic31C2319yes—
35C22×Dic31C23148yescomplemented
36#92499C23116yescomplemented
37Q8×D31C23148yescomplemented
38#92497C231144yescomplemented
39#92497C231288yescomplemented
40D4×D31C231144yescomplemented
41#92495C231144yescomplemented
42C2×D124C2316yes—
43C2×C4×D31C2316yes—
44C2×C4×D31C23118yes—
45C2×C4×D31C23172yescomplemented
46C2×Dic62C23118yes—
47C2×Dic62C23172yescomplemented
48C23×C31—412no—
49C23×C31C2419yes—
50C23×C31C24124yes—
51C22×D31—418no—
52C22×D31C2416yes—
53C22×D31C24136yes—
54Q8×C31C2418yes—
55D4×C31C24172yes—
56C2×C4×C31C2416yes—
57C2×C4×C31C24136yes—
58C22⋊D31C241144yes—
59C2×Dic31C2419yes—
60C2×Dic31C24172yes—
61C2×Dic31C2411yescharacteristic
62C2×Dic31C24136yes—
63D124C24124yes—
64C4×D31C24124yes—
65C4×D31C24172yes—
66Dic62C24172yes—
67C2×#585—311nomaximal, Sylow
68C22×C31—218no—
69C22×C31—424no—
70C22×C31C25118yes—
71C22×C31C2511yescharacteristic
72C2×D31—212no—
73C2×D31—436no—
74C2×D31C25112yes—
75C4×C31C25112yes—
76Dic31C25118yes—
77Dic31C2512yes—
78#585—3132no—
79C2×2−1+4—316no—
80C2×H(5,𝔽2)—319no—
81C2×H(5,𝔽2)D3111yescharacteristic, complemented
82C22×#41—316no—
83C22×#41—319no—
84C2×C31—218no—
85C2×C31#58512yescomplemented
86C2×C31C2611yescharacteristic
87D31—212no—
882−1+4—3196no—
89H(5,𝔽2)—31144no—
90H(5,𝔽2)C2×D31116yescomplemented
91C2×#41—3196no—
92C2×#41—312no—
93C2×#41—3118no—
94C2×#41—3136no—
95C2×#41—31144no—
96C2×#41—3118no—
97C2×#41C2×D3116yescomplemented
98C22×Q8—316no—
99C22×Q8—319no—
100C22×D4—3118no—
101C22×D4—3118no—
102C22×D4C2×D3119yescomplemented
103C23×C4—316no—
104C23×C4—319no—
105C31C2×#58511yescharacteristic, complemented, Sylow
106C24—319no—
107C24C22×D3116yes—
108#41—3116no—
109#41—31144no—
110#41—31288no—
111#41—31144no—
112#41C22×D31148yescomplemented
113C2×Q8—3148no—
114C2×Q8—3118no—
115C2×Q8—3172no—
116C2×Q8C22×D3112yes—
117C2×D4—3136no—
118C2×D4—31144no—
119C2×D4—31144no—
120C2×D4—316no—
121C2×D4C22×D31118yes—
122C2×D4C22×D31172yescomplemented
123C22×C4—319no—
124C22×C4—3118no—
125C22×C4—3148no—
126C22×C4—316no—
127C22×C4—3118no—
128C22×C4—3172no—
129C22×C4C22×D3119yes—
130C23—412no—
131C23—316no—
132C23—3136no—
133C23—12418no—
134C23C23×D3119yes—
135C23C23×D31124yes—
136Q8—3172no—
137Q8C23×D3118yes—
138D4—31144no—
139D4—3124no—
140D4C23×D31172yes—
141C2×C4—3136no—
142C2×C4—3172no—
143C2×C4—311no—
144C2×C4—319no—
145C2×C4—3124no—
146C2×C4—3172no—
147C2×C4C23×D31136yes—
148C2×C4C23×D3116yes—
149C22—218no—
150C22—424no—
151C22—3112no—
152C22—6212no—
153C22—12436no—
154C22C24×D31118yes—
155C22C24×D3111yescharacteristic
156C4—312no—
157C4—3118no—
158C4C24×D31112yes—
159C2—218no—
160C2—6212no—
161C2#1452188412yesdirect product factor
162C2C25×D3111yescharacteristic
subgroupG/H
centerC22C24×D31
commutator, cosocleC2×C31C26
FrattiniC2C25×D31
FittingC2×C31×H(5,𝔽2)C2
socleC22×C31C25
hypercenterC2×H(5,𝔽2)D31
hypocenterC31C2×#585
LCSG ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #14521884C2direct (N × H)
2. C2×#11758567C2semidirect
3. C2×#11758561C2semidirect
4. C22×#92497C2semidirect
5. C2×#41×D31C2semidirect
6. C2×C31×H(5,𝔽2)C2semidirect
7. C31×H(5,𝔽2)C22semidirect
8. C2×#41×C31C22semidirect
9. C22×D4×C31C22semidirect
10. C22×C22⋊D31C22semidirect
11. C23×Dic31C22semidirect
12. #11758567C22semidirect
13. #41×D31C22semidirect
14. C2×#92499C22semidirect
15. C2×Q8×D31C22semidirect
16. #11758561C22semidirect
17. C2×#92497C22semidirect
18. C2×#92497C22semidirect
19. C2×#92497C22semidirect
20. C2×D4×D31C22semidirect
21. C2×#92495C22semidirect
22. C22×C4×D31C22semidirect
23. C22×Dic62C22semidirect
26. #41×C31C23semidirect
29. C2×D4×C31C23semidirect
32. C2×C22⋊D31C23semidirect
35. C22×Dic31C23semidirect
36. #92499C23semidirect
37. Q8×D31C23semidirect
38. #92497C23semidirect
39. #92497C23semidirect
40. D4×D31C23semidirect
41. #92495C23semidirect
45. C2×C4×D31C23semidirect
47. C2×Dic62C23semidirect
81. C2×H(5,𝔽2)D31semidirect
85. C2×C31#585semidirect
90. H(5,𝔽2)C2×D31semidirect
97. C2×#41C2×D31semidirect
102. C22×D4C2×D31semidirect
105. C31C2×#585semidirect
112. #41C22×D31semidirect
122. C2×D4C22×D31semidirect
161. C2#14521884direct (N × H)

Maximal subgroups

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

subgroupindex
#145218842
C2×#117585672
C2×#117585612
C22×#924972
C2×#41×D312
C2×C31×H(5,𝔽2)2
C2×#58531