Finite Group Databasebeta

← #135×D233 #28417213

Subgroups — #135×D233 #28417213

Order 14912.

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×D233C212yesmaximal, complemented
2#21418558C211yesmaximal, characteristic, complemented
3#21418557C211yesmaximal, characteristic, complemented
4#21418556C212yesmaximal, complemented
5#21418555C212yesmaximal, complemented
6#21418554C211yesmaximal, characteristic, complemented
7Q16×D233C211yesmaximal, characteristic, complemented
8SD16×D233C212yesmaximal, complemented
9D8×D233C211yesmaximal, characteristic, complemented
10C2×C8×D233C211yesmaximal, characteristic, complemented
11#135×C233C211yesmaximal, characteristic, complemented
12#16836595C2212yescomplemented
13#16836594—22no—
14#16836594C2212yescomplemented
15#16836593—22no—
16#16836593C2212yescomplemented
17Q8×D233C2212yes—
18D4×D233—22no—
19D4×D233C2212yes—
20C2×C4×D233—22no—
21C2×C4×D233C2211yescharacteristic
22Dic932C2211yescharacteristic, complemented
23#16836584C2212yescomplemented
24#16836583C2212yescomplemented
25#16836582C2212yescomplemented
26#16836581C2212yescomplemented
27D1864C2211yescharacteristic, complemented
28#16836579C2212yescomplemented
29C2×#14272338C2211yescharacteristic, complemented
30C8×D233C2211yescharacteristic
31C8×D233C2211yescharacteristic
32C8×D233C2211yescharacteristic, complemented
33C8×D233C2211yescharacteristic, complemented
34#41×C233C2212yescomplemented
35Q16×C233C2211yescharacteristic, complemented
36SD16×C233C2212yescomplemented
37D8×C233C2211yescharacteristic, complemented
38C2×C8×C233C2211yescharacteristic, complemented
39C22×D233—22no—
40C22×D233D411yescharacteristic
41Dic466—22no—
42Dic466C2312yes—
43D932—22no—
44D932C2312yes—
45C22⋊D233—22no—
46C22⋊D233—22no—
47C22⋊D233—22no—
48C2×Dic233—22no—
49C2×Dic233D411yescharacteristic
50C4×D233—22no—
51C4×D233—22no—
52C4×D233—22no—
53C4×D233D411yescharacteristic
54C4×D233D411yescharacteristic
55C4×D233C2311yescharacteristic
56C4×D233C2311yescharacteristic
57#14272338C2311yescharacteristic
58#14272338C2311yescharacteristic
59Q8×C233C2312yes—
60D4×C233—22no—
61D4×C233C2312yes—
62C2×C4×C233—22no—
63C2×C4×C233C2311yescharacteristic
64C8×C233C2311yescharacteristic
65C8×C233C2311yescharacteristic
66C22×C233—22no—
67C22×C233C2×D411yescharacteristic
68C2×D233—22no—
69C2×D233—22no—
70C2×D233—44no—
71C2×D233C2×D411yescharacteristic
72C2×D233C2×D411yescharacteristic
73C4×C233—22no—
74C4×C233C2×D411yescharacteristic
75C4×C233C2411yescharacteristic
76Dic233—22no—
77Dic233C2×D411yescharacteristic
78Dic233C2×D411yescharacteristic
79C2×C233—21no—
80C2×C233—42no—
81C2×C233C22×D411yescharacteristic
82D233—21no—
83D233—42no—
84D233#13512yesdirect product factor
85C233C2×#13511yescharacteristic, complemented, Sylow
86C2×#135—2331nomaximal, Sylow
87C2×#41—2332no—
88#135—2332no—
89#135—2331no—
90#135—2332no—
91#135—2331no—
92#135—2331no—
93#135D23311yescharacteristic, direct product factor
94C2×Q16—2331no—
95C2×SD16—2332no—
96C2×D8—2331no—
97C22×C8—2331no—
98#41—4662no—
99#41—4662no—
100#41—2332no—
101#41—2332no—
102#41—2332no—
103#41C2×D23312yescomplemented
104C2×Q8—2332no—
105C2×D4—4662no—
106C2×D4—2332no—
107C22×C4—4662no—
108C22×C4—2331no—
109Q16—2332no—
110Q16—2331no—
111Q16C2×D23311yescharacteristic, complemented
112SD16—2332no—
113SD16—2332no—
114SD16—2332no—
115SD16C2×D23312yescomplemented
116D8—2332no—
117D8—2331no—
118D8C2×D23311yescharacteristic, complemented
119C2×C8—2331no—
120C2×C8—2331no—
121C2×C8—2331no—
122C2×C8—2331no—
123C2×C8—2331no—
124C2×C8C2×D23311yescharacteristic, complemented
125C23—4662no—
126C23—2331no—
127Q8—4662no—
128Q8—2332no—
129Q8C22×D23312yes—
130D4—22no—
131D4—4662no—
132D4—4662no—
133D4—4662no—
134D4—4662no—
135D4—2332no—
136D4C22×D23312yes—
137C2×C4—22no—
138C2×C4—4662no—
139C2×C4—4662no—
140C2×C4—4662no—
141C2×C4—4662no—
142C2×C4—2331no—
143C2×C4—2331no—
144C2×C4—2331no—
145C2×C4—2331no—
146C2×C4—2331no—
147C2×C4C22×D23311yescharacteristic
148C8—2331no—
149C8—2331no—
150C8C22×D23311yescharacteristic
151C8C22×D23311yescharacteristic
152C22—22no—
153C22—9324no—
154C22—4662no—
155C22—4662no—
156C22—2331no—
157C22—2331no—
158C22D4×D23311yescharacteristic
159C4—22no—
160C4—4662no—
161C4—2331no—
162C4—2331no—
163C4D4×D23311yescharacteristic
164C4C23×D23311yescharacteristic
165C2—21no—
166C2—42no—
167C2—9322no—
168C2—4661no—
169C2—2332no—
170C2C2×D4×D23311yescharacteristic
subgroupG/H
centerC4D4×D233
commutator, cosocleC4×C233C24
FrattiniC4C23×D233
Fitting#135×C233C2
socleC2×C233C22×D4
hypercenter#135D233
hypocenterC233C2×#135
LCSG ▷ C4×C233 ▷ C2×C233 ▷ C233
UCSC1 ◁ C4 ◁ C2×C4 ◁ #135

Complements

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

NH ≅ G/Ntype
1. #41×D233C2semidirect
2. #21418558C2semidirect
3. #21418557C2semidirect
4. #21418556C2semidirect
5. #21418555C2semidirect
6. #21418554C2semidirect
7. Q16×D233C2semidirect
8. SD16×D233C2semidirect
9. D8×D233C2semidirect
10. C2×C8×D233C2semidirect
11. #135×C233C2semidirect
12. #16836595C22semidirect
14. #16836594C22semidirect
16. #16836593C22semidirect
22. Dic932C22semidirect
23. #16836584C22semidirect
24. #16836583C22semidirect
25. #16836582C22semidirect
26. #16836581C22semidirect
27. D1864C22semidirect
28. #16836579C22semidirect
29. C2×#14272338C22semidirect
32. C8×D233C22semidirect
33. C8×D233C22semidirect
34. #41×C233C22semidirect
35. Q16×C233C22semidirect
36. SD16×C233C22semidirect
37. D8×C233C22semidirect
38. C2×C8×C233C22semidirect
84. D233#135direct (N × H)
85. C233C2×#135semidirect
93. #135D233direct (N × H)
103. #41C2×D233semidirect
111. Q16C2×D233semidirect
115. SD16C2×D233semidirect
118. D8C2×D233semidirect
124. C2×C8C2×D233semidirect

Maximal subgroups

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

subgroupindex
#41×D2332
#214185582
#214185572
#214185562
#214185552
#214185542
Q16×D2332
SD16×D2332
D8×D2332
C2×C8×D2332
#135×C2332
C2×#135233