Finite Group Databasebeta

← #135×D223 #28384358

Subgroups — #135×D223 #28384358

Order 14272.

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×D223C212yesmaximal, complemented
2#20297327C211yesmaximal, characteristic, complemented
3#20297326C211yesmaximal, characteristic, complemented
4#20297325C212yesmaximal, complemented
5#20297324C212yesmaximal, complemented
6#20297323C211yesmaximal, characteristic, complemented
7Q16×D223C211yesmaximal, characteristic, complemented
8SD16×D223C212yesmaximal, complemented
9D8×D223C211yesmaximal, characteristic, complemented
10C2×C8×D223C211yesmaximal, characteristic, complemented
11#135×C223C211yesmaximal, characteristic, complemented
12#16813171C2212yescomplemented
13#16813170—22no—
14#16813170C2212yescomplemented
15#16813169—22no—
16#16813169C2212yescomplemented
17Q8×D223C2212yes—
18D4×D223—22no—
19D4×D223C2212yes—
20C2×C4×D223—22no—
21C2×C4×D223C2211yescharacteristic
22Dic892C2211yescharacteristic, complemented
23#16813160C2212yescomplemented
24#16813159C2212yescomplemented
25#16813158C2212yescomplemented
26#16813157C2212yescomplemented
27D1784C2211yescharacteristic, complemented
28#16813155C2212yescomplemented
29C2×C223⋊C8C2211yescharacteristic, complemented
30C8×D223C2211yescharacteristic
31C8×D223C2211yescharacteristic
32C8×D223C2211yescharacteristic, complemented
33C8×D223C2211yescharacteristic, complemented
34#41×C223C2212yescomplemented
35Q16×C223C2211yescharacteristic, complemented
36SD16×C223C2212yescomplemented
37D8×C223C2211yescharacteristic, complemented
38C2×C8×C223C2211yescharacteristic, complemented
39C22×D223—22no—
40C22×D223D411yescharacteristic
41Dic446—22no—
42Dic446C2312yes—
43D892—22no—
44D892C2312yes—
45C22⋊D223—22no—
46C22⋊D223—22no—
47C22⋊D223—22no—
48C2×Dic223—22no—
49C2×Dic223D411yescharacteristic
50C4×D223—22no—
51C4×D223—22no—
52C4×D223—22no—
53C4×D223D411yescharacteristic
54C4×D223D411yescharacteristic
55C4×D223C2311yescharacteristic
56C4×D223C2311yescharacteristic
57C223⋊C8C2311yescharacteristic
58C223⋊C8C2311yescharacteristic
59Q8×C223C2312yes—
60D4×C223—22no—
61D4×C223C2312yes—
62C2×C4×C223—22no—
63C2×C4×C223C2311yescharacteristic
64C8×C223C2311yescharacteristic
65C8×C223C2311yescharacteristic
66C22×C223—22no—
67C22×C223C2×D411yescharacteristic
68C2×D223—22no—
69C2×D223—22no—
70C2×D223—44no—
71C2×D223C2×D411yescharacteristic
72C2×D223C2×D411yescharacteristic
73C4×C223—22no—
74C4×C223C2×D411yescharacteristic
75C4×C223C2411yescharacteristic
76Dic223—22no—
77Dic223C2×D411yescharacteristic
78Dic223C2×D411yescharacteristic
79C2×C223—21no—
80C2×C223—42no—
81C2×C223C22×D411yescharacteristic
82D223—21no—
83D223—42no—
84D223#13512yesdirect product factor
85C223C2×#13511yescharacteristic, complemented, Sylow
86C2×#135—2231nomaximal, Sylow
87C2×#41—2232no—
88#135—2232no—
89#135—2231no—
90#135—2232no—
91#135—2231no—
92#135—2231no—
93#135D22311yescharacteristic, direct product factor
94C2×Q16—2231no—
95C2×SD16—2232no—
96C2×D8—2231no—
97C22×C8—2231no—
98#41—4462no—
99#41—4462no—
100#41—2232no—
101#41—2232no—
102#41—2232no—
103#41C2×D22312yescomplemented
104C2×Q8—2232no—
105C2×D4—4462no—
106C2×D4—2232no—
107C22×C4—4462no—
108C22×C4—2231no—
109Q16—2232no—
110Q16—2231no—
111Q16C2×D22311yescharacteristic, complemented
112SD16—2232no—
113SD16—2232no—
114SD16—2232no—
115SD16C2×D22312yescomplemented
116D8—2232no—
117D8—2231no—
118D8C2×D22311yescharacteristic, complemented
119C2×C8—2231no—
120C2×C8—2231no—
121C2×C8—2231no—
122C2×C8—2231no—
123C2×C8—2231no—
124C2×C8C2×D22311yescharacteristic, complemented
125C23—4462no—
126C23—2231no—
127Q8—4462no—
128Q8—2232no—
129Q8C22×D22312yes—
130D4—22no—
131D4—4462no—
132D4—4462no—
133D4—4462no—
134D4—4462no—
135D4—2232no—
136D4C22×D22312yes—
137C2×C4—22no—
138C2×C4—4462no—
139C2×C4—4462no—
140C2×C4—4462no—
141C2×C4—4462no—
142C2×C4—2231no—
143C2×C4—2231no—
144C2×C4—2231no—
145C2×C4—2231no—
146C2×C4—2231no—
147C2×C4C22×D22311yescharacteristic
148C8—2231no—
149C8—2231no—
150C8C22×D22311yescharacteristic
151C8C22×D22311yescharacteristic
152C22—22no—
153C22—8924no—
154C22—4462no—
155C22—4462no—
156C22—2231no—
157C22—2231no—
158C22D4×D22311yescharacteristic
159C4—22no—
160C4—4462no—
161C4—2231no—
162C4—2231no—
163C4D4×D22311yescharacteristic
164C4C23×D22311yescharacteristic
165C2—21no—
166C2—42no—
167C2—8922no—
168C2—4461no—
169C2—2232no—
170C2C2×D4×D22311yescharacteristic
subgroupG/H
centerC4D4×D223
commutator, cosocleC4×C223C24
FrattiniC4C23×D223
Fitting#135×C223C2
socleC2×C223C22×D4
hypercenter#135D223
hypocenterC223C2×#135
LCSG ▷ C4×C223 ▷ C2×C223 ▷ C223
UCSC1 ◁ C4 ◁ C2×C4 ◁ #135

Complements

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

NH ≅ G/Ntype
1. #41×D223C2semidirect
2. #20297327C2semidirect
3. #20297326C2semidirect
4. #20297325C2semidirect
5. #20297324C2semidirect
6. #20297323C2semidirect
7. Q16×D223C2semidirect
8. SD16×D223C2semidirect
9. D8×D223C2semidirect
10. C2×C8×D223C2semidirect
11. #135×C223C2semidirect
12. #16813171C22semidirect
14. #16813170C22semidirect
16. #16813169C22semidirect
22. Dic892C22semidirect
23. #16813160C22semidirect
24. #16813159C22semidirect
25. #16813158C22semidirect
26. #16813157C22semidirect
27. D1784C22semidirect
28. #16813155C22semidirect
29. C2×C223⋊C8C22semidirect
32. C8×D223C22semidirect
33. C8×D223C22semidirect
34. #41×C223C22semidirect
35. Q16×C223C22semidirect
36. SD16×C223C22semidirect
37. D8×C223C22semidirect
38. C2×C8×C223C22semidirect
84. D223#135direct (N × H)
85. C223C2×#135semidirect
93. #135D223direct (N × H)
103. #41C2×D223semidirect
111. Q16C2×D223semidirect
115. SD16C2×D223semidirect
118. D8C2×D223semidirect
124. C2×C8C2×D223semidirect

Maximal subgroups

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

subgroupindex
#41×D2232
#202973272
#202973262
#202973252
#202973242
#202973232
Q16×D2232
SD16×D2232
D8×D2232
C2×C8×D2232
#135×C2232
C2×#135223