Finite Group Databasebeta

← #135×D521 #45688210

Subgroups — #135×D521 #45688210

Order 33344.

266 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 101 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#41×D521C212yesmaximal, complemented
2#30659731C211yesmaximal, characteristic, complemented
3#30659730C211yesmaximal, characteristic, complemented
4#30659729C212yesmaximal, complemented
5#30659728C212yesmaximal, complemented
6#30659727C211yesmaximal, characteristic, complemented
7Q16×D521C211yesmaximal, characteristic, complemented
8SD16×D521C212yesmaximal, complemented
9D8×D521C211yesmaximal, characteristic, complemented
10C2×C8×D521C211yesmaximal, characteristic, complemented
11#135×C521C211yesmaximal, characteristic, complemented
12#22552365C2212yescomplemented
13#22552364—22no—
14#22552364C2212yescomplemented
15#22552363—22no—
16#22552363C2212yescomplemented
17Q8×D521C2212yes—
18D4×D521—22no—
19D4×D521C2212yes—
20C2×C4×D521—22no—
21C2×C4×D521C2211yescharacteristic
22Dic2084C2211yescharacteristic, complemented
23#22552354C2212yescomplemented
24#22552353C2212yescomplemented
25#22552352C2212yescomplemented
26#22552351C2212yescomplemented
27D4168C2211yescharacteristic, complemented
28#22552349C2212yescomplemented
29C2×#16861960C2211yescharacteristic, complemented
30C8×D521C2211yescharacteristic
31C8×D521C2211yescharacteristic
32C8×D521C2211yescharacteristic, complemented
33C8×D521C2211yescharacteristic, complemented
34#41×C521C2212yescomplemented
35Q16×C521C2211yescharacteristic, complemented
36SD16×C521C2212yescomplemented
37D8×C521C2211yescharacteristic, complemented
38C2×C8×C521C2211yescharacteristic, complemented
39C22×D521—22no—
40C22×D521D411yescharacteristic
41Dic1042—22no—
42Dic1042C2312yes—
43D2084—22no—
44D2084C2312yes—
45C22⋊D521—22no—
46C22⋊D521—22no—
47C22⋊D521—22no—
48C2×Dic521—22no—
49C2×Dic521D411yescharacteristic
50C4×D521—22no—
51C4×D521—22no—
52C4×D521—22no—
53C4×D521D411yescharacteristic
54C4×D521D411yescharacteristic
55C4×D521C2311yescharacteristic
56C4×D521C2311yescharacteristic
57#16861960C2311yescharacteristic
58#16861960C2311yescharacteristic
59Q8×C521C2312yes—
60D4×C521—22no—
61D4×C521C2312yes—
62C2×C4×C521—22no—
63C2×C4×C521C2311yescharacteristic
64C8×C521C2311yescharacteristic
65C8×C521C2311yescharacteristic
66C22×C521—22no—
67C22×C521C2×D411yescharacteristic
68C2×D521—22no—
69C2×D521—22no—
70C2×D521—44no—
71C2×D521C2×D411yescharacteristic
72C2×D521C2×D411yescharacteristic
73C4×C521—22no—
74C4×C521C2×D411yescharacteristic
75C4×C521C2411yescharacteristic
76Dic521—22no—
77Dic521C2×D411yescharacteristic
78Dic521C2×D411yescharacteristic
79C2×C521—21no—
80C2×C521—42no—
81C2×C521C22×D411yescharacteristic
82D521—21no—
83D521—42no—
84D521#13512yesdirect product factor
85C521C2×#13511yescharacteristic, complemented, Sylow
86C2×#135—5211nomaximal, Sylow
87C2×#41—5212no—
88#135—5212no—
89#135—5211no—
90#135—5212no—
91#135—5211no—
92#135—5211no—
93#135D52111yescharacteristic, direct product factor
94C2×Q16—5211no—
95C2×SD16—5212no—
96C2×D8—5211no—
97C22×C8—5211no—
98#41—5212no—
99#41—5212no—
100#41—5212no—
101#41—10422no—
102#41—10422no—
103#41C2×D52112yescomplemented
104C2×Q8—5212no—
105C2×D4—5212no—
106C2×D4—10422no—
107C22×C4—5211no—
108C22×C4—10422no—
109Q16—5212no—
110Q16—5211no—
111Q16C2×D52111yescharacteristic, complemented
112SD16—5212no—
113SD16—5212no—
114SD16—5212no—
115SD16C2×D52112yescomplemented
116D8—5212no—
117D8—5211no—
118D8C2×D52111yescharacteristic, complemented
119C2×C8—5211no—
120C2×C8—5211no—
121C2×C8—5211no—
122C2×C8—5211no—
123C2×C8—5211no—
124C2×C8C2×D52111yescharacteristic, complemented
125C23—5211no—
126C23—10422no—
127Q8—5212no—
128Q8—10422no—
129Q8C22×D52112yes—
130D4—22no—
131D4—5212no—
132D4—10422no—
133D4—10422no—
134D4—10422no—
135D4—10422no—
136D4C22×D52112yes—
137C2×C4—22no—
138C2×C4—5211no—
139C2×C4—5211no—
140C2×C4—5211no—
141C2×C4—5211no—
142C2×C4—5211no—
143C2×C4—10422no—
144C2×C4—10422no—
145C2×C4—10422no—
146C2×C4—10422no—
147C2×C4C22×D52111yescharacteristic
148C8—5211no—
149C8—5211no—
150C8C22×D52111yescharacteristic
151C8C22×D52111yescharacteristic
152C22—22no—
153C22—5211no—
154C22—5211no—
155C22—10422no—
156C22—10422no—
157C22—20844no—
158C22D4×D52111yescharacteristic
159C4—22no—
160C4—5211no—
161C4—5211no—
162C4—10422no—
163C4D4×D52111yescharacteristic
164C4C23×D52111yescharacteristic
165C2—21no—
166C2—42no—
167C2—5212no—
168C2—10421no—
169C2—20842no—
170C2C2×D4×D52111yescharacteristic
subgroupG/H
centerC4D4×D521
commutator, cosocleC4×C521C24
FrattiniC4C23×D521
Fitting#135×C521C2
socleC2×C521C22×D4
hypercenter#135D521
hypocenterC521C2×#135
LCSG ▷ C4×C521 ▷ C2×C521 ▷ C521
UCSC1 ◁ C4 ◁ C2×C4 ◁ #135

Complements

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

NH ≅ G/Ntype
1. #41×D521C2semidirect
2. #30659731C2semidirect
3. #30659730C2semidirect
4. #30659729C2semidirect
5. #30659728C2semidirect
6. #30659727C2semidirect
7. Q16×D521C2semidirect
8. SD16×D521C2semidirect
9. D8×D521C2semidirect
10. C2×C8×D521C2semidirect
11. #135×C521C2semidirect
12. #22552365C22semidirect
14. #22552364C22semidirect
16. #22552363C22semidirect
22. Dic2084C22semidirect
23. #22552354C22semidirect
24. #22552353C22semidirect
25. #22552352C22semidirect
26. #22552351C22semidirect
27. D4168C22semidirect
28. #22552349C22semidirect
29. C2×#16861960C22semidirect
32. C8×D521C22semidirect
33. C8×D521C22semidirect
34. #41×C521C22semidirect
35. Q16×C521C22semidirect
36. SD16×C521C22semidirect
37. D8×C521C22semidirect
38. C2×C8×C521C22semidirect
84. D521#135direct (N × H)
85. C521C2×#135semidirect
93. #135D521direct (N × H)
103. #41C2×D521semidirect
111. Q16C2×D521semidirect
115. SD16C2×D521semidirect
118. D8C2×D521semidirect
124. C2×C8C2×D521semidirect

Maximal subgroups

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

subgroupindex
#41×D5212
#306597312
#306597302
#306597292
#306597282
#306597272
Q16×D5212
SD16×D5212
D8×D5212
C2×C8×D5212
#135×C5212
C2×#135521