Finite Group Databasebeta

← C2×#24924104 #36338945

Subgroups — C2×#24924104 #36338945

Order 21376.

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#24924104C2132yesmaximal, direct product factor
2C2×#19128879C216yesmaximal, complemented
3C2×#19128877C219yesmaximal, complemented
4C22×#14581949C219yesmaximal, complemented
5C2×#41×D167C216yesmaximal, complemented
6C2×H(5,𝔽2)×C167C211yesmaximal, characteristic, complemented
7#19128879C22196yescomplemented
8#19128877C221144yescomplemented
9C2×#14581951C2212yescomplemented
10C2×#14581950C22118yescomplemented
11C2×#14581949C22118yescomplemented
12C2×#14581949C22136yescomplemented
13C2×#14581949C221144yescomplemented
14#41×D167C22196yescomplemented
15C22×Dic334C2219yescomplemented
16C2×Q8×D167C2216yescomplemented
17C22×C22⋊D167C22118yescomplemented
18C2×D4×D167C22118yescomplemented
19C23×Dic167C2216yescomplemented
20C22×C4×D167C2219yescomplemented
21H(5,𝔽2)×C167C22116yescomplemented
22C2×#41×C167C2216yescomplemented
23C22×D4×C167C2219yescomplemented
24C23×D167C2319yes—
25#14581951C23116yescomplemented
26#14581950C231144yescomplemented
27#14581949C231144yescomplemented
28#14581949C231288yescomplemented
29C2×Dic334C23118yes—
30C2×Dic334C23172yescomplemented
31Q8×D167C23148yescomplemented
32C2×D668C2316yes—
33C2×C22⋊D167C23136yes—
34C2×C22⋊D167C231144yescomplemented
35D4×D167C231144yescomplemented
36C22×Dic167C2319yes—
37C22×Dic167C23118yes—
38C22×Dic167C23148yescomplemented
39C2×C4×D167C2316yes—
40C2×C4×D167C23118yes—
41C2×C4×D167C23172yescomplemented
42C24×C167C2316yes—
43#41×C167C23148yescomplemented
44C2×Q8×C167C2312yes—
45C2×D4×C167C23118yes—
46C2×D4×C167C23172yescomplemented
47C22×C4×C167C2319yes—
48C22×D167—418no—
49C22×D167C2416yes—
50C22×D167C24136yes—
51Dic334C24172yes—
52D668C24124yes—
53C22⋊D167C241144yes—
54C2×Dic167C2411yescharacteristic
55C2×Dic167C2419yes—
56C2×Dic167C24136yes—
57C2×Dic167C24172yes—
58C4×D167C24124yes—
59C4×D167C24172yes—
60C23×C167—412no—
61C23×C167C2419yes—
62C23×C167C24124yes—
63Q8×C167C2418yes—
64D4×C167C24172yes—
65C2×C4×C167C2416yes—
66C2×C4×C167C24136yes—
67C22×C167—218no—
68C22×C167—424no—
69C22×C167C25118yes—
70C22×C167C2511yescharacteristic
71C2×D167—212no—
72C2×D167—436no—
73C2×D167C25112yes—
74C4×C167C25112yes—
75Dic167C2512yes—
76Dic167C25118yes—
77C2×C167—218no—
78C2×C167#58512yescomplemented
79C2×C167C2611yescharacteristic
80D167—212no—
81C167C2×#58511yescharacteristic, complemented, Sylow
82C2×#585—1671nomaximal, Sylow
83#585—16732no—
84C2×2−1+4—1676no—
85C2×H(5,𝔽2)—1679no—
86C2×H(5,𝔽2)D16711yescharacteristic, complemented
87C22×#41—1676no—
88C22×#41—1679no—
892−1+4—16796no—
90H(5,𝔽2)—167144no—
91H(5,𝔽2)C2×D167116yescomplemented
92C2×#41—16796no—
93C2×#41—1672no—
94C2×#41—16718no—
95C2×#41—16736no—
96C2×#41—167144no—
97C2×#41—16718no—
98C2×#41C2×D16716yescomplemented
99C22×Q8—1676no—
100C22×Q8—1679no—
101C22×D4—16718no—
102C22×D4—16718no—
103C22×D4C2×D16719yescomplemented
104C23×C4—1676no—
105C23×C4—1679no—
106C24—1679no—
107C24C22×D16716yes—
108#41—16716no—
109#41—167144no—
110#41—167288no—
111#41—167144no—
112#41C22×D167148yescomplemented
113C2×Q8—16748no—
114C2×Q8—16718no—
115C2×Q8—16772no—
116C2×Q8C22×D16712yes—
117C2×D4—16736no—
118C2×D4—167144no—
119C2×D4—167144no—
120C2×D4—1676no—
121C2×D4C22×D167118yes—
122C2×D4C22×D167172yescomplemented
123C22×C4—1679no—
124C22×C4—16718no—
125C22×C4—16748no—
126C22×C4—1676no—
127C22×C4—16718no—
128C22×C4—16772no—
129C22×C4C22×D16719yes—
130C23—412no—
131C23—66818no—
132C23—1676no—
133C23—16736no—
134C23C23×D16719yes—
135C23C23×D167124yes—
136Q8—16772no—
137Q8C23×D16718yes—
138D4—167144no—
139D4—16724no—
140D4C23×D167172yes—
141C2×C4—16736no—
142C2×C4—16772no—
143C2×C4—1671no—
144C2×C4—1679no—
145C2×C4—16724no—
146C2×C4—16772no—
147C2×C4C23×D167136yes—
148C2×C4C23×D16716yes—
149C22—218no—
150C22—424no—
151C22—33412no—
152C22—66836no—
153C22—16712no—
154C22C24×D167118yes—
155C22C24×D16711yescharacteristic
156C4—1672no—
157C4—16718no—
158C4C24×D167112yes—
159C2—218no—
160C2—33412no—
161C2#2492410412yesdirect product factor
162C2C25×D16711yescharacteristic
subgroupG/H
centerC22C24×D167
commutator, cosocleC2×C167C26
FrattiniC2C25×D167
FittingC2×H(5,𝔽2)×C167C2
socleC22×C167C25
hypercenterC2×H(5,𝔽2)D167
hypocenterC167C2×#585
LCSG ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #24924104C2direct (N × H)
2. C2×#19128879C2semidirect
3. C2×#19128877C2semidirect
4. C22×#14581949C2semidirect
5. C2×#41×D167C2semidirect
6. C2×H(5,𝔽2)×C167C2semidirect
7. #19128879C22semidirect
8. #19128877C22semidirect
9. C2×#14581951C22semidirect
10. C2×#14581950C22semidirect
11. C2×#14581949C22semidirect
12. C2×#14581949C22semidirect
13. C2×#14581949C22semidirect
14. #41×D167C22semidirect
15. C22×Dic334C22semidirect
16. C2×Q8×D167C22semidirect
17. C22×C22⋊D167C22semidirect
18. C2×D4×D167C22semidirect
19. C23×Dic167C22semidirect
20. C22×C4×D167C22semidirect
21. H(5,𝔽2)×C167C22semidirect
22. C2×#41×C167C22semidirect
23. C22×D4×C167C22semidirect
25. #14581951C23semidirect
26. #14581950C23semidirect
27. #14581949C23semidirect
28. #14581949C23semidirect
30. C2×Dic334C23semidirect
31. Q8×D167C23semidirect
34. C2×C22⋊D167C23semidirect
35. D4×D167C23semidirect
38. C22×Dic167C23semidirect
41. C2×C4×D167C23semidirect
43. #41×C167C23semidirect
46. C2×D4×C167C23semidirect
78. C2×C167#585semidirect
81. C167C2×#585semidirect
86. C2×H(5,𝔽2)D167semidirect
91. H(5,𝔽2)C2×D167semidirect
98. C2×#41C2×D167semidirect
103. C22×D4C2×D167semidirect
112. #41C22×D167semidirect
122. C2×D4C22×D167semidirect
161. C2#24924104direct (N × H)

Maximal subgroups

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

subgroupindex
#249241042
C2×#191288792
C2×#191288772
C22×#145819492
C2×#41×D1672
C2×H(5,𝔽2)×C1672
C2×#585167