Finite Group Databasebeta

← #585×D173 #36368369

Subgroups — #585×D173 #36368369

Order 22144.

5956 conjugacy classes of subgroups; only the 3203 normal subgroups are shown.

normal subgroups only

This group has 5956 conjugacy classes of subgroups — too many to draw. Only its 3203 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#26012328C216yesmaximal, complemented
2#26012327C2115yesmaximal, complemented
3#26012326C2110yesmaximal, complemented
42−1+4×D173C216yesmaximal, complemented
5H(5,𝔽2)×D173C2110yesmaximal, complemented
6C2×#41×D173C2115yesmaximal, complemented
7#585×C173C211yesmaximal, characteristic, complemented
8#19149727C22130yescomplemented
9#19149726C22160yescomplemented
10#19149725C22160yescomplemented
11#19149724C22190yescomplemented
12C2×#14583827C22115yescomplemented
13C2×#14583826C22145yescomplemented
14C2×#14583825C22145yescomplemented
15#41×D173C22120yescomplemented
16#41×D173C22160yescomplemented
17#41×D173C22160yescomplemented
18#41×D173C22160yescomplemented
19C2×Q8×D173C22115yescomplemented
20C2×D4×D173C22145yescomplemented
21C22×C4×D173C22115yescomplemented
222−1+4×C173C2216yescomplemented
23H(5,𝔽2)×C173C22110yescomplemented
24C2×#41×C173C22115yescomplemented
25C23×D173C23115yes—
26#14583827C23120yescomplemented
27#14583827C23160yescomplemented
28#14583826C23160yescomplemented
29#14583826C231180yescomplemented
30#14583825C23160yescomplemented
31#14583825C231180yescomplemented
32C2×Dic346C23145yescomplemented
33Q8×D173C23120yes—
34Q8×D173C23160yescomplemented
35C2×D692C23145yescomplemented
36C2×C22⋊D173C23190yescomplemented
37D4×D173C23160yes—
38D4×D173C231180yescomplemented
39C22×Dic173C23115yescomplemented
40C2×C4×D173C23115yes—
41C2×C4×D173C23145yes—
42C2×C4×D173C23145yescomplemented
43C2×C4×D173C23145yescomplemented
44#41×C173C23120yescomplemented
45#41×C173C23160yescomplemented
46C2×Q8×C173C23115yescomplemented
47C2×D4×C173C23145yescomplemented
48C22×C4×C173C23115yescomplemented
49C22×D173—430no—
50C22×D173C24115yes—
51C22×D173C24145yes—
52Dic346C24160yes—
53D692C24160yes—
54C22⋊D173C241120yes—
55C2×Dic173C24115yes—
56C2×Dic173C24145yes—
57C4×D173C2411yescharacteristic
58C4×D173C24115yes—
59C4×D173C24115yes—
60C4×D173C24115yes—
61C4×D173C24190yes—
62C23×C173C24115yes—
63Q8×C173C24120yes—
64D4×C173C24160yes—
65C2×C4×C173C24115yes—
66C2×C4×C173C24145yes—
67C22×C173—415no—
68C22×C173C25115yes—
69C2×D173—230no—
70C2×D173—445no—
71C2×D173C25115yes—
72C2×D173C2511yescharacteristic
73C4×C173C25115yes—
74C4×C173C2511yescharacteristic
75Dic173C2511yescharacteristic
76Dic173C25115yes—
77C2×C173—215no—
78C2×C173C2611yescharacteristic
79D173—215no—
80D173#58512yesdirect product factor
81C173C2×#58511yescharacteristic, complemented, Sylow
82C2×#585—1731nomaximal, Sylow
83#585—1736no—
84#585—17310no—
85#585—17315no—
86#585D17311yescharacteristic, direct product factor
87C2×2−1+4—1736no—
88C2×H(5,𝔽2)—17310no—
89C22×#41—17315no—
902−1+4—17360no—
912−1+4—17330no—
922−1+4C2×D17316yescomplemented
93H(5,𝔽2)—17360no—
94H(5,𝔽2)—17390no—
95H(5,𝔽2)C2×D173110yescomplemented
96C2×#41—17320no—
97C2×#41—17360no—
98C2×#41—17360no—
99C2×#41—17360no—
100C2×#41—17315no—
101C2×#41—17345no—
102C2×#41—17345no—
103C2×#41C2×D173115yescomplemented
104C22×Q8—17315no—
105C22×D4—17345no—
106C23×C4—17315no—
107C24—17315no—
108#41—17320no—
109#41—17360no—
110#41—17360no—
111#41—173180no—
112#41—17360no—
113#41—173180no—
114#41C22×D173120yescomplemented
115#41C22×D173160yescomplemented
116C2×Q8—17320no—
117C2×Q8—17360no—
118C2×Q8—17345no—
119C2×Q8C22×D173115yescomplemented
120C2×D4—17390no—
121C2×D4—17360no—
122C2×D4—173180no—
123C2×D4—17345no—
124C2×D4C22×D173145yescomplemented
125C22×C4—17315no—
126C22×C4—17315no—
127C22×C4—17345no—
128C22×C4—17345no—
129C22×C4—17345no—
130C22×C4C22×D173115yescomplemented
131C23—17315no—
132C23—17345no—
133C23—69230no—
134C23C23×D173115yes—
135Q8—17360no—
136Q8C23×D173120yes—
137D4—173120no—
138D4—17360no—
139D4C23×D173160yes—
140C2×C4—17315no—
141C2×C4—17345no—
142C2×C4—17315no—
143C2×C4—17315no—
144C2×C4—17390no—
145C2×C4—1731no—
146C2×C4—17315no—
147C2×C4C23×D173145yes—
148C2×C4C23×D173115yes—
149C22—415no—
150C22—34630no—
151C22—17315no—
152C22—1731no—
153C22—69245no—
154C22C24×D173115yes—
155C4—1731no—
156C4—17315no—
157C4C24×D173115yes—
158C4C24×D17311yescharacteristic
159C2—215no—
160C2—34615no—
161C2—1732no—
162C2C25×D17311yescharacteristic
subgroupG/H
centerC4C24×D173
commutator, socle, cosocleC2×C173C26
FrattiniC2C25×D173
Fitting#585×C173C2
hypercenter#585D173
hypocenterC173C2×#585
LCSG ▷ C2×C173 ▷ C173
UCSC1 ◁ C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. #26012328C2semidirect
2. #26012327C2semidirect
3. #26012326C2semidirect
4. 2−1+4×D173C2semidirect
5. H(5,𝔽2)×D173C2semidirect
6. C2×#41×D173C2semidirect
7. #585×C173C2semidirect
8. #19149727C22semidirect
9. #19149726C22semidirect
10. #19149725C22semidirect
11. #19149724C22semidirect
12. C2×#14583827C22semidirect
13. C2×#14583826C22semidirect
14. C2×#14583825C22semidirect
15. #41×D173C22semidirect
16. #41×D173C22semidirect
17. #41×D173C22semidirect
18. #41×D173C22semidirect
19. C2×Q8×D173C22semidirect
20. C2×D4×D173C22semidirect
21. C22×C4×D173C22semidirect
22. 2−1+4×C173C22semidirect
23. H(5,𝔽2)×C173C22semidirect
24. C2×#41×C173C22semidirect
26. #14583827C23semidirect
27. #14583827C23semidirect
28. #14583826C23semidirect
29. #14583826C23semidirect
30. #14583825C23semidirect
31. #14583825C23semidirect
32. C2×Dic346C23semidirect
34. Q8×D173C23semidirect
35. C2×D692C23semidirect
36. C2×C22⋊D173C23semidirect
38. D4×D173C23semidirect
39. C22×Dic173C23semidirect
42. C2×C4×D173C23semidirect
43. C2×C4×D173C23semidirect
44. #41×C173C23semidirect
45. #41×C173C23semidirect
46. C2×Q8×C173C23semidirect
47. C2×D4×C173C23semidirect
48. C22×C4×C173C23semidirect
80. D173#585direct (N × H)
81. C173C2×#585semidirect
86. #585D173direct (N × H)
92. 2−1+4C2×D173semidirect
95. H(5,𝔽2)C2×D173semidirect
103. C2×#41C2×D173semidirect
114. #41C22×D173semidirect
115. #41C22×D173semidirect
119. C2×Q8C22×D173semidirect
124. C2×D4C22×D173semidirect
130. C22×C4C22×D173semidirect

Maximal subgroups

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

subgroupindex
#260123282
#260123272
#260123262
2−1+4×D1732
H(5,𝔽2)×D1732
C2×#41×D1732
#585×C1732
C2×#585173