Finite Group Databasebeta

← #135×D419 #41001139

Subgroups — #135×D419 #41001139

Order 26816.

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×D419C212yesmaximal, complemented
2#27211741C211yesmaximal, characteristic, complemented
3#27211740C211yesmaximal, characteristic, complemented
4#27211739C212yesmaximal, complemented
5#27211738C212yesmaximal, complemented
6#27211737C211yesmaximal, characteristic, complemented
7Q16×D419C211yesmaximal, characteristic, complemented
8SD16×D419C212yesmaximal, complemented
9D8×D419C211yesmaximal, characteristic, complemented
10C2×C8×D419C211yesmaximal, characteristic, complemented
11#135×C419C211yesmaximal, characteristic, complemented
12#20269412C2212yescomplemented
13#20269411—22no—
14#20269411C2212yescomplemented
15#20269410—22no—
16#20269410C2212yescomplemented
17Q8×D419C2212yes—
18D4×D419—22no—
19D4×D419C2212yes—
20C2×C4×D419—22no—
21C2×C4×D419C2211yescharacteristic
22Dic1676C2211yescharacteristic, complemented
23#20269401C2212yescomplemented
24#20269400C2212yescomplemented
25#20269399C2212yescomplemented
26#20269398C2212yescomplemented
27D3352C2211yescharacteristic, complemented
28#20269396C2212yescomplemented
29C2×C419⋊C8C2211yescharacteristic, complemented
30C8×D419C2211yescharacteristic
31C8×D419C2211yescharacteristic
32C8×D419C2211yescharacteristic, complemented
33C8×D419C2211yescharacteristic, complemented
34#41×C419C2212yescomplemented
35Q16×C419C2211yescharacteristic, complemented
36SD16×C419C2212yescomplemented
37D8×C419C2211yescharacteristic, complemented
38C2×C8×C419C2211yescharacteristic, complemented
39C22×D419—22no—
40C22×D419D411yescharacteristic
41Dic838—22no—
42Dic838C2312yes—
43D1676—22no—
44D1676C2312yes—
45C22⋊D419—22no—
46C22⋊D419—22no—
47C22⋊D419—22no—
48C2×Dic419—22no—
49C2×Dic419D411yescharacteristic
50C4×D419—22no—
51C4×D419—22no—
52C4×D419—22no—
53C4×D419D411yescharacteristic
54C4×D419D411yescharacteristic
55C4×D419C2311yescharacteristic
56C4×D419C2311yescharacteristic
57C419⋊C8C2311yescharacteristic
58C419⋊C8C2311yescharacteristic
59Q8×C419C2312yes—
60D4×C419—22no—
61D4×C419C2312yes—
62C2×C4×C419—22no—
63C2×C4×C419C2311yescharacteristic
64C8×C419C2311yescharacteristic
65C8×C419C2311yescharacteristic
66C22×C419—22no—
67C22×C419C2×D411yescharacteristic
68C2×D419—22no—
69C2×D419—22no—
70C2×D419—44no—
71C2×D419C2×D411yescharacteristic
72C2×D419C2×D411yescharacteristic
73C4×C419—22no—
74C4×C419C2×D411yescharacteristic
75C4×C419C2411yescharacteristic
76Dic419—22no—
77Dic419C2×D411yescharacteristic
78Dic419C2×D411yescharacteristic
79C2×C419—21no—
80C2×C419—42no—
81C2×C419C22×D411yescharacteristic
82D419—21no—
83D419—42no—
84D419#13512yesdirect product factor
85C419C2×#13511yescharacteristic, complemented, Sylow
86C2×#135—4191nomaximal, Sylow
87C2×#41—4192no—
88#135—4192no—
89#135—4191no—
90#135—4192no—
91#135—4191no—
92#135—4191no—
93#135D41911yescharacteristic, direct product factor
94C2×Q16—4191no—
95C2×SD16—4192no—
96C2×D8—4191no—
97C22×C8—4191no—
98#41—8382no—
99#41—8382no—
100#41—4192no—
101#41—4192no—
102#41—4192no—
103#41C2×D41912yescomplemented
104C2×Q8—4192no—
105C2×D4—8382no—
106C2×D4—4192no—
107C22×C4—8382no—
108C22×C4—4191no—
109Q16—4192no—
110Q16—4191no—
111Q16C2×D41911yescharacteristic, complemented
112SD16—4192no—
113SD16—4192no—
114SD16—4192no—
115SD16C2×D41912yescomplemented
116D8—4192no—
117D8—4191no—
118D8C2×D41911yescharacteristic, complemented
119C2×C8—4191no—
120C2×C8—4191no—
121C2×C8—4191no—
122C2×C8—4191no—
123C2×C8—4191no—
124C2×C8C2×D41911yescharacteristic, complemented
125C23—8382no—
126C23—4191no—
127Q8—8382no—
128Q8—4192no—
129Q8C22×D41912yes—
130D4—22no—
131D4—8382no—
132D4—8382no—
133D4—8382no—
134D4—8382no—
135D4—4192no—
136D4C22×D41912yes—
137C2×C4—22no—
138C2×C4—8382no—
139C2×C4—8382no—
140C2×C4—8382no—
141C2×C4—8382no—
142C2×C4—4191no—
143C2×C4—4191no—
144C2×C4—4191no—
145C2×C4—4191no—
146C2×C4—4191no—
147C2×C4C22×D41911yescharacteristic
148C8—4191no—
149C8—4191no—
150C8C22×D41911yescharacteristic
151C8C22×D41911yescharacteristic
152C22—22no—
153C22—8382no—
154C22—8382no—
155C22—16764no—
156C22—4191no—
157C22—4191no—
158C22D4×D41911yescharacteristic
159C4—22no—
160C4—8382no—
161C4—4191no—
162C4—4191no—
163C4D4×D41911yescharacteristic
164C4C23×D41911yescharacteristic
165C2—21no—
166C2—42no—
167C2—8381no—
168C2—16762no—
169C2—4192no—
170C2C2×D4×D41911yescharacteristic
subgroupG/H
centerC4D4×D419
commutator, cosocleC4×C419C24
FrattiniC4C23×D419
Fitting#135×C419C2
socleC2×C419C22×D4
hypercenter#135D419
hypocenterC419C2×#135
LCSG ▷ C4×C419 ▷ C2×C419 ▷ C419
UCSC1 ◁ C4 ◁ C2×C4 ◁ #135

Complements

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

NH ≅ G/Ntype
1. #41×D419C2semidirect
2. #27211741C2semidirect
3. #27211740C2semidirect
4. #27211739C2semidirect
5. #27211738C2semidirect
6. #27211737C2semidirect
7. Q16×D419C2semidirect
8. SD16×D419C2semidirect
9. D8×D419C2semidirect
10. C2×C8×D419C2semidirect
11. #135×C419C2semidirect
12. #20269412C22semidirect
14. #20269411C22semidirect
16. #20269410C22semidirect
22. Dic1676C22semidirect
23. #20269401C22semidirect
24. #20269400C22semidirect
25. #20269399C22semidirect
26. #20269398C22semidirect
27. D3352C22semidirect
28. #20269396C22semidirect
29. C2×C419⋊C8C22semidirect
32. C8×D419C22semidirect
33. C8×D419C22semidirect
34. #41×C419C22semidirect
35. Q16×C419C22semidirect
36. SD16×C419C22semidirect
37. D8×C419C22semidirect
38. C2×C8×C419C22semidirect
84. D419#135direct (N × H)
85. C419C2×#135semidirect
93. #135D419direct (N × H)
103. #41C2×D419semidirect
111. Q16C2×D419semidirect
115. SD16C2×D419semidirect
118. D8C2×D419semidirect
124. C2×C8C2×D419semidirect

Maximal subgroups

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

subgroupindex
#41×D4192
#272117412
#272117402
#272117392
#272117382
#272117372
Q16×D4192
SD16×D4192
D8×D4192
C2×C8×D4192
#135×C4192
C2×#135419