Finite Group Databasebeta

← #135×D353 #36378565

Subgroups — #135×D353 #36378565

Order 22592.

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×D353C212yesmaximal, complemented
2#26015015C211yesmaximal, characteristic, complemented
3#26015014C211yesmaximal, characteristic, complemented
4#26015013C212yesmaximal, complemented
5#26015012C212yesmaximal, complemented
6#26015011C211yesmaximal, characteristic, complemented
7Q16×D353C211yesmaximal, characteristic, complemented
8SD16×D353C212yesmaximal, complemented
9D8×D353C211yesmaximal, characteristic, complemented
10C2×C8×D353C211yesmaximal, characteristic, complemented
11#135×C353C211yesmaximal, characteristic, complemented
12#19150738C2212yescomplemented
13#19150737—22no—
14#19150737C2212yescomplemented
15#19150736—22no—
16#19150736C2212yescomplemented
17Q8×D353C2212yes—
18D4×D353—22no—
19D4×D353C2212yes—
20C2×C4×D353—22no—
21C2×C4×D353C2211yescharacteristic
22Dic1412C2211yescharacteristic, complemented
23#19150727C2212yescomplemented
24#19150726C2212yescomplemented
25#19150725C2212yescomplemented
26#19150724C2212yescomplemented
27D2824C2211yescharacteristic, complemented
28#19150722C2212yescomplemented
29C2×#15667725C2211yescharacteristic, complemented
30C8×D353C2211yescharacteristic
31C8×D353C2211yescharacteristic
32C8×D353C2211yescharacteristic, complemented
33C8×D353C2211yescharacteristic, complemented
34#41×C353C2212yescomplemented
35Q16×C353C2211yescharacteristic, complemented
36SD16×C353C2212yescomplemented
37D8×C353C2211yescharacteristic, complemented
38C2×C8×C353C2211yescharacteristic, complemented
39C22×D353—22no—
40C22×D353D411yescharacteristic
41Dic706—22no—
42Dic706C2312yes—
43D1412—22no—
44D1412C2312yes—
45C22⋊D353—22no—
46C22⋊D353—22no—
47C22⋊D353—22no—
48C2×Dic353—22no—
49C2×Dic353D411yescharacteristic
50C4×D353—22no—
51C4×D353—22no—
52C4×D353—22no—
53C4×D353D411yescharacteristic
54C4×D353D411yescharacteristic
55C4×D353C2311yescharacteristic
56C4×D353C2311yescharacteristic
57#15667725C2311yescharacteristic
58#15667725C2311yescharacteristic
59Q8×C353C2312yes—
60D4×C353—22no—
61D4×C353C2312yes—
62C2×C4×C353—22no—
63C2×C4×C353C2311yescharacteristic
64C8×C353C2311yescharacteristic
65C8×C353C2311yescharacteristic
66C22×C353—22no—
67C22×C353C2×D411yescharacteristic
68C2×D353—22no—
69C2×D353—22no—
70C2×D353—44no—
71C2×D353C2×D411yescharacteristic
72C2×D353C2×D411yescharacteristic
73C4×C353—22no—
74C4×C353C2×D411yescharacteristic
75C4×C353C2411yescharacteristic
76Dic353—22no—
77Dic353C2×D411yescharacteristic
78Dic353C2×D411yescharacteristic
79C2×C353—21no—
80C2×C353—42no—
81C2×C353C22×D411yescharacteristic
82D353—21no—
83D353—42no—
84D353#13512yesdirect product factor
85C353C2×#13511yescharacteristic, complemented, Sylow
86C2×#135—3531nomaximal, Sylow
87C2×#41—3532no—
88#135—3532no—
89#135—3531no—
90#135—3532no—
91#135—3531no—
92#135—3531no—
93#135D35311yescharacteristic, direct product factor
94C2×Q16—3531no—
95C2×SD16—3532no—
96C2×D8—3531no—
97C22×C8—3531no—
98#41—3532no—
99#41—3532no—
100#41—3532no—
101#41—7062no—
102#41—7062no—
103#41C2×D35312yescomplemented
104C2×Q8—3532no—
105C2×D4—3532no—
106C2×D4—7062no—
107C22×C4—3531no—
108C22×C4—7062no—
109Q16—3532no—
110Q16—3531no—
111Q16C2×D35311yescharacteristic, complemented
112SD16—3532no—
113SD16—3532no—
114SD16—3532no—
115SD16C2×D35312yescomplemented
116D8—3532no—
117D8—3531no—
118D8C2×D35311yescharacteristic, complemented
119C2×C8—3531no—
120C2×C8—3531no—
121C2×C8—3531no—
122C2×C8—3531no—
123C2×C8—3531no—
124C2×C8C2×D35311yescharacteristic, complemented
125C23—3531no—
126C23—7062no—
127Q8—3532no—
128Q8—7062no—
129Q8C22×D35312yes—
130D4—22no—
131D4—3532no—
132D4—7062no—
133D4—7062no—
134D4—7062no—
135D4—7062no—
136D4C22×D35312yes—
137C2×C4—22no—
138C2×C4—3531no—
139C2×C4—3531no—
140C2×C4—3531no—
141C2×C4—3531no—
142C2×C4—3531no—
143C2×C4—7062no—
144C2×C4—7062no—
145C2×C4—7062no—
146C2×C4—7062no—
147C2×C4C22×D35311yescharacteristic
148C8—3531no—
149C8—3531no—
150C8C22×D35311yescharacteristic
151C8C22×D35311yescharacteristic
152C22—22no—
153C22—3531no—
154C22—3531no—
155C22—14124no—
156C22—7062no—
157C22—7062no—
158C22D4×D35311yescharacteristic
159C4—22no—
160C4—3531no—
161C4—3531no—
162C4—7062no—
163C4D4×D35311yescharacteristic
164C4C23×D35311yescharacteristic
165C2—21no—
166C2—42no—
167C2—3532no—
168C2—14122no—
169C2—7061no—
170C2C2×D4×D35311yescharacteristic
subgroupG/H
centerC4D4×D353
commutator, cosocleC4×C353C24
FrattiniC4C23×D353
Fitting#135×C353C2
socleC2×C353C22×D4
hypercenter#135D353
hypocenterC353C2×#135
LCSG ▷ C4×C353 ▷ C2×C353 ▷ C353
UCSC1 ◁ C4 ◁ C2×C4 ◁ #135

Complements

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

NH ≅ G/Ntype
1. #41×D353C2semidirect
2. #26015015C2semidirect
3. #26015014C2semidirect
4. #26015013C2semidirect
5. #26015012C2semidirect
6. #26015011C2semidirect
7. Q16×D353C2semidirect
8. SD16×D353C2semidirect
9. D8×D353C2semidirect
10. C2×C8×D353C2semidirect
11. #135×C353C2semidirect
12. #19150738C22semidirect
14. #19150737C22semidirect
16. #19150736C22semidirect
22. Dic1412C22semidirect
23. #19150727C22semidirect
24. #19150726C22semidirect
25. #19150725C22semidirect
26. #19150724C22semidirect
27. D2824C22semidirect
28. #19150722C22semidirect
29. C2×#15667725C22semidirect
32. C8×D353C22semidirect
33. C8×D353C22semidirect
34. #41×C353C22semidirect
35. Q16×C353C22semidirect
36. SD16×C353C22semidirect
37. D8×C353C22semidirect
38. C2×C8×C353C22semidirect
84. D353#135direct (N × H)
85. C353C2×#135semidirect
93. #135D353direct (N × H)
103. #41C2×D353semidirect
111. Q16C2×D353semidirect
115. SD16C2×D353semidirect
118. D8C2×D353semidirect
124. C2×C8C2×D353semidirect

Maximal subgroups

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

subgroupindex
#41×D3532
#260150152
#260150142
#260150132
#260150122
#260150112
Q16×D3532
SD16×D3532
D8×D3532
C2×C8×D3532
#135×C3532
C2×#135353