Finite Group Databasebeta

← #135×D401 #38786892

Subgroups — #135×D401 #38786892

Order 25664.

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×D401C212yesmaximal, complemented
2#27165760C211yesmaximal, characteristic, complemented
3#27165759C211yesmaximal, characteristic, complemented
4#27165758C212yesmaximal, complemented
5#27165757C212yesmaximal, complemented
6#27165756C211yesmaximal, characteristic, complemented
7Q16×D401C211yesmaximal, characteristic, complemented
8SD16×D401C212yesmaximal, complemented
9D8×D401C211yesmaximal, characteristic, complemented
10C2×C8×D401C211yesmaximal, characteristic, complemented
11#135×C401C211yesmaximal, characteristic, complemented
12#20264367C2212yescomplemented
13#20264366—22no—
14#20264366C2212yescomplemented
15#20264365—22no—
16#20264365C2212yescomplemented
17Q8×D401C2212yes—
18D4×D401—22no—
19D4×D401C2212yes—
20C2×C4×D401—22no—
21C2×C4×D401C2211yescharacteristic
22Dic1604C2211yescharacteristic, complemented
23#20264356C2212yescomplemented
24#20264355C2212yescomplemented
25#20264354C2212yescomplemented
26#20264353C2212yescomplemented
27D3208C2211yescharacteristic, complemented
28#20264351C2212yescomplemented
29C2×#15691999C2211yescharacteristic, complemented
30C8×D401C2211yescharacteristic
31C8×D401C2211yescharacteristic
32C8×D401C2211yescharacteristic, complemented
33C8×D401C2211yescharacteristic, complemented
34#41×C401C2212yescomplemented
35Q16×C401C2211yescharacteristic, complemented
36SD16×C401C2212yescomplemented
37D8×C401C2211yescharacteristic, complemented
38C2×C8×C401C2211yescharacteristic, complemented
39C22×D401—22no—
40C22×D401D411yescharacteristic
41Dic802—22no—
42Dic802C2312yes—
43D1604—22no—
44D1604C2312yes—
45C22⋊D401—22no—
46C22⋊D401—22no—
47C22⋊D401—22no—
48C2×Dic401—22no—
49C2×Dic401D411yescharacteristic
50C4×D401—22no—
51C4×D401—22no—
52C4×D401—22no—
53C4×D401D411yescharacteristic
54C4×D401D411yescharacteristic
55C4×D401C2311yescharacteristic
56C4×D401C2311yescharacteristic
57#15691999C2311yescharacteristic
58#15691999C2311yescharacteristic
59Q8×C401C2312yes—
60D4×C401—22no—
61D4×C401C2312yes—
62C2×C4×C401—22no—
63C2×C4×C401C2311yescharacteristic
64C8×C401C2311yescharacteristic
65C8×C401C2311yescharacteristic
66C22×C401—22no—
67C22×C401C2×D411yescharacteristic
68C2×D401—22no—
69C2×D401—22no—
70C2×D401—44no—
71C2×D401C2×D411yescharacteristic
72C2×D401C2×D411yescharacteristic
73C4×C401—22no—
74C4×C401C2×D411yescharacteristic
75C4×C401C2411yescharacteristic
76Dic401—22no—
77Dic401C2×D411yescharacteristic
78Dic401C2×D411yescharacteristic
79C2×C401—21no—
80C2×C401—42no—
81C2×C401C22×D411yescharacteristic
82D401—21no—
83D401—42no—
84D401#13512yesdirect product factor
85C401C2×#13511yescharacteristic, complemented, Sylow
86C2×#135—4011nomaximal, Sylow
87C2×#41—4012no—
88#135—4012no—
89#135—4011no—
90#135—4012no—
91#135—4011no—
92#135—4011no—
93#135D40111yescharacteristic, direct product factor
94C2×Q16—4011no—
95C2×SD16—4012no—
96C2×D8—4011no—
97C22×C8—4011no—
98#41—8022no—
99#41—8022no—
100#41—4012no—
101#41—4012no—
102#41—4012no—
103#41C2×D40112yescomplemented
104C2×Q8—4012no—
105C2×D4—8022no—
106C2×D4—4012no—
107C22×C4—8022no—
108C22×C4—4011no—
109Q16—4012no—
110Q16—4011no—
111Q16C2×D40111yescharacteristic, complemented
112SD16—4012no—
113SD16—4012no—
114SD16—4012no—
115SD16C2×D40112yescomplemented
116D8—4012no—
117D8—4011no—
118D8C2×D40111yescharacteristic, complemented
119C2×C8—4011no—
120C2×C8—4011no—
121C2×C8—4011no—
122C2×C8—4011no—
123C2×C8—4011no—
124C2×C8C2×D40111yescharacteristic, complemented
125C23—8022no—
126C23—4011no—
127Q8—8022no—
128Q8—4012no—
129Q8C22×D40112yes—
130D4—22no—
131D4—8022no—
132D4—8022no—
133D4—8022no—
134D4—8022no—
135D4—4012no—
136D4C22×D40112yes—
137C2×C4—22no—
138C2×C4—8022no—
139C2×C4—8022no—
140C2×C4—8022no—
141C2×C4—8022no—
142C2×C4—4011no—
143C2×C4—4011no—
144C2×C4—4011no—
145C2×C4—4011no—
146C2×C4—4011no—
147C2×C4C22×D40111yescharacteristic
148C8—4011no—
149C8—4011no—
150C8C22×D40111yescharacteristic
151C8C22×D40111yescharacteristic
152C22—22no—
153C22—8022no—
154C22—8022no—
155C22—16044no—
156C22—4011no—
157C22—4011no—
158C22D4×D40111yescharacteristic
159C4—22no—
160C4—8022no—
161C4—4011no—
162C4—4011no—
163C4D4×D40111yescharacteristic
164C4C23×D40111yescharacteristic
165C2—21no—
166C2—42no—
167C2—8021no—
168C2—16042no—
169C2—4012no—
170C2C2×D4×D40111yescharacteristic
subgroupG/H
centerC4D4×D401
commutator, cosocleC4×C401C24
FrattiniC4C23×D401
Fitting#135×C401C2
socleC2×C401C22×D4
hypercenter#135D401
hypocenterC401C2×#135
LCSG ▷ C4×C401 ▷ C2×C401 ▷ C401
UCSC1 ◁ C4 ◁ C2×C4 ◁ #135

Complements

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

NH ≅ G/Ntype
1. #41×D401C2semidirect
2. #27165760C2semidirect
3. #27165759C2semidirect
4. #27165758C2semidirect
5. #27165757C2semidirect
6. #27165756C2semidirect
7. Q16×D401C2semidirect
8. SD16×D401C2semidirect
9. D8×D401C2semidirect
10. C2×C8×D401C2semidirect
11. #135×C401C2semidirect
12. #20264367C22semidirect
14. #20264366C22semidirect
16. #20264365C22semidirect
22. Dic1604C22semidirect
23. #20264356C22semidirect
24. #20264355C22semidirect
25. #20264354C22semidirect
26. #20264353C22semidirect
27. D3208C22semidirect
28. #20264351C22semidirect
29. C2×#15691999C22semidirect
32. C8×D401C22semidirect
33. C8×D401C22semidirect
34. #41×C401C22semidirect
35. Q16×C401C22semidirect
36. SD16×C401C22semidirect
37. D8×C401C22semidirect
38. C2×C8×C401C22semidirect
84. D401#135direct (N × H)
85. C401C2×#135semidirect
93. #135D401direct (N × H)
103. #41C2×D401semidirect
111. Q16C2×D401semidirect
115. SD16C2×D401semidirect
118. D8C2×D401semidirect
124. C2×C8C2×D401semidirect

Maximal subgroups

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

subgroupindex
#41×D4012
#271657602
#271657592
#271657582
#271657572
#271657562
Q16×D4012
SD16×D4012
D8×D4012
C2×C8×D4012
#135×C4012
C2×#135401