Finite Group Databasebeta

← C22×#19128878 #36338940

Subgroups — C22×#19128878 #36338940

Order 21376.

6560 conjugacy classes of subgroups; only the 3221 normal subgroups are shown.

normal subgroups only

This group has 6560 conjugacy classes of subgroups — too many to draw. Only its 3221 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
1C2×#19128878C2148yesmaximal, direct product factor
2C22×#14581951C212yesmaximal, complemented
3C22×#14581950C213yesmaximal, complemented
4C23×D668C213yesmaximal, complemented
5C22×D4×D167C216yesmaximal, complemented
6C22×#41×C167C211yesmaximal, characteristic, complemented
7C24×D167C2216yescomplemented
8#19128878C221256yesdirect product factor
9C2×#14581951C22148yescomplemented
10C2×#14581950C22172yescomplemented
11C22×Dic334C2211yescharacteristic, complemented
12C22×D668C2216yescomplemented
13C22×D668C2213yescomplemented
14C22×D668C22172yescomplemented
15C22×C22⋊D167C2216yescomplemented
16C2×D4×D167C221144yescomplemented
17C22×C4×D167C2216yescomplemented
18C2×#41×C167C22124yescomplemented
19C22×Q8×C167C2211yescharacteristic, complemented
20C22×D4×C167C2213yescomplemented
21C23×C4×C167C2213yescomplemented
22C23×D167—424no—
23C23×D167C2316yes—
24C23×D167C23172yescomplemented
25#14581951C231128yescomplemented
26#14581950C231192yescomplemented
27C2×Dic334C23112yescomplemented
28C2×D668C23172yescomplemented
29C2×D668C23136yescomplemented
30C2×D668C231192yescomplemented
31C2×C22⋊D167C23172yescomplemented
32D4×D167C231384yescomplemented
33C22×Dic167C2312yes—
34C2×C4×D167C23172yescomplemented
35C24×C167C2313yes—
36#41×C167C23164yescomplemented
37C2×Q8×C167C23112yescomplemented
38C2×D4×C167C23136yescomplemented
39C22×C4×C167C2313yes—
40C22×C4×C167C2311yescharacteristic
41C22×C4×C167C23136yescomplemented
42C22×D167—224no—
43C22×D167—4144no—
44C22×D167C24136yes—
45C22×D167C24196yescomplemented
46Dic334C24116yescomplemented
47D668C24196yescomplemented
48D668C24148yescomplemented
49C22⋊D167C24196yescomplemented
50C2×Dic167C24112yes—
51C4×D167C24196yescomplemented
52C23×C167—212no—
53C23×C167C24118yes—
54C23×C167C2411yescharacteristic
55Q8×C167C24116yescomplemented
56D4×C167C24148yescomplemented
57C2×C4×C167C2416yes—
58C2×C4×C167C24118yes—
59C2×C4×C167C24148yescomplemented
60C22×C167—236no—
61C22×C167H(5,𝔽2)14yescomplemented
62C22×C167C25112yes—
63C22×C167C2513yes—
64C2×D167—272no—
65C2×D167—496no—
66C2×D167C25124yes—
67C4×C167C2514yes—
68C4×C167C25112yes—
69Dic167C2518yes—
70C2×C167—212no—
71C2×C167C2×H(5,𝔽2)16yescomplemented
72C2×C167C2611yescharacteristic
73D167—224no—
74C167C22×H(5,𝔽2)11yescharacteristic, complemented, Sylow
75C22×H(5,𝔽2)—1671nomaximal, Sylow
76C2×H(5,𝔽2)—16748no—
77C22×#41—1672no—
78C22×#41—1673no—
79C22×#41D16711yescharacteristic, complemented
80C23×D4—1676no—
81C23×D4—1673no—
82C25—1676no—
83H(5,𝔽2)—167256no—
84C2×#41—16748no—
85C2×#41—16772no—
86C2×#41C2×D167124yescomplemented
87C22×Q8—1671no—
88C22×Q8C2×D16711yescharacteristic, complemented
89C22×D4—1676no—
90C22×D4—167144no—
91C22×D4—1676no—
92C22×D4—1673no—
93C22×D4—16772no—
94C22×D4C2×D16713yescomplemented
95C23×C4—1676no—
96C23×C4C2×D16713yescomplemented
97C24—66824no—
98C24—1676no—
99C24—16772no—
100C24C22×D16713yes—
101#41—167128no—
102#41—167192no—
103#41C22×D167164yescomplemented
104C2×Q8—16712no—
105C2×Q8C22×D167112yescomplemented
106C2×D4—16772no—
107C2×D4—167384no—
108C2×D4—16772no—
109C2×D4—16736no—
110C2×D4—167192no—
111C2×D4C22×D167136yescomplemented
112C22×C4—1672no—
113C22×C4—16772no—
114C22×C4C22×D16713yes—
115C22×C4C22×D16711yescharacteristic
116C22×C4C22×D167136yescomplemented
117C23—212no—
118C23—33424no—
119C23—668144no—
120C23—16736no—
121C23—16796no—
122C23C23×D167118yes—
123C23C23×D16711yescharacteristic
124Q8—16716no—
125Q8C23×D167116yescomplemented
126D4—16796no—
127D4—16796no—
128D4—16748no—
129D4C23×D167148yescomplemented
130C2×C4—16712no—
131C2×C4—16796no—
132C2×C4C23×D167118yes—
133C2×C4C23×D16716yes—
134C2×C4C23×D167148yescomplemented
135C22—236no—
136C22—33472no—
137C22—66896no—
138C22—16724no—
139C22#1912887814yesdirect product factor
140C22C24×D167112yes—
141C22C24×D16713yes—
142C4—1678no—
143C4C24×D167112yes—
144C4C24×D16714yes—
145C2—212no—
146C2—33424no—
147C2C2×#1912887816yesdirect product factor
148C2C25×D16711yescharacteristic
subgroupG/H
centerC23C23×D167
commutator, cosocleC2×C167C26
FrattiniC2C25×D167
FittingC22×#41×C167C2
socleC23×C167C24
hypercenterC22×#41D167
hypocenterC167C22×H(5,𝔽2)
LCSG ▷ C2×C167 ▷ C167
UCSC1 ◁ C23 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#19128878C2direct (N × H)
2. C22×#14581951C2semidirect
3. C22×#14581950C2semidirect
4. C23×D668C2semidirect
5. C22×D4×D167C2semidirect
6. C22×#41×C167C2semidirect
7. C24×D167C22semidirect
8. #19128878C22direct (N × H)
9. C2×#14581951C22semidirect
10. C2×#14581950C22semidirect
11. C22×Dic334C22semidirect
12. C22×D668C22semidirect
13. C22×D668C22semidirect
14. C22×D668C22semidirect
15. C22×C22⋊D167C22semidirect
16. C2×D4×D167C22semidirect
17. C22×C4×D167C22semidirect
18. C2×#41×C167C22semidirect
19. C22×Q8×C167C22semidirect
20. C22×D4×C167C22semidirect
21. C23×C4×C167C22semidirect
24. C23×D167C23semidirect
25. #14581951C23semidirect
26. #14581950C23semidirect
27. C2×Dic334C23semidirect
28. C2×D668C23semidirect
29. C2×D668C23semidirect
30. C2×D668C23semidirect
31. C2×C22⋊D167C23semidirect
32. D4×D167C23semidirect
34. C2×C4×D167C23semidirect
36. #41×C167C23semidirect
37. C2×Q8×C167C23semidirect
38. C2×D4×C167C23semidirect
41. C22×C4×C167C23semidirect
45. C22×D167C24semidirect
46. Dic334C24semidirect
47. D668C24semidirect
48. D668C24semidirect
49. C22⋊D167C24semidirect
51. C4×D167C24semidirect
55. Q8×C167C24semidirect
56. D4×C167C24semidirect
59. C2×C4×C167C24semidirect
61. C22×C167H(5,𝔽2)semidirect
71. C2×C167C2×H(5,𝔽2)semidirect
74. C167C22×H(5,𝔽2)semidirect
79. C22×#41D167semidirect
86. C2×#41C2×D167semidirect
88. C22×Q8C2×D167semidirect
94. C22×D4C2×D167semidirect
96. C23×C4C2×D167semidirect
103. #41C22×D167semidirect
105. C2×Q8C22×D167semidirect
111. C2×D4C22×D167semidirect
116. C22×C4C22×D167semidirect
125. Q8C23×D167semidirect
129. D4C23×D167semidirect
134. C2×C4C23×D167semidirect
139. C22#19128878direct (N × H)
147. C2C2×#19128878direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#191288782
C22×#145819512
C22×#145819502
C23×D6682
C22×D4×D1672
C22×#41×C1672
C22×H(5,𝔽2)167