Finite Group Databasebeta

← #135×D421 #41003618

Subgroups — #135×D421 #41003618

Order 26944.

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×D421C212yesmaximal, complemented
2#27212389C211yesmaximal, characteristic, complemented
3#27212388C211yesmaximal, characteristic, complemented
4#27212387C212yesmaximal, complemented
5#27212386C212yesmaximal, complemented
6#27212385C211yesmaximal, characteristic, complemented
7Q16×D421C211yesmaximal, characteristic, complemented
8SD16×D421C212yesmaximal, complemented
9D8×D421C211yesmaximal, characteristic, complemented
10C2×C8×D421C211yesmaximal, characteristic, complemented
11#135×C421C211yesmaximal, characteristic, complemented
12#20269772C2212yescomplemented
13#20269771—22no—
14#20269771C2212yescomplemented
15#20269770—22no—
16#20269770C2212yescomplemented
17Q8×D421C2212yes—
18D4×D421—22no—
19D4×D421C2212yes—
20C2×C4×D421—22no—
21C2×C4×D421C2211yescharacteristic
22Dic1684C2211yescharacteristic, complemented
23#20269761C2212yescomplemented
24#20269760C2212yescomplemented
25#20269759C2212yescomplemented
26#20269758C2212yescomplemented
27D3368C2211yescharacteristic, complemented
28#20269756C2212yescomplemented
29C2×#16809890C2211yescharacteristic, complemented
30C8×D421C2211yescharacteristic
31C8×D421C2211yescharacteristic
32C8×D421C2211yescharacteristic, complemented
33C8×D421C2211yescharacteristic, complemented
34#41×C421C2212yescomplemented
35Q16×C421C2211yescharacteristic, complemented
36SD16×C421C2212yescomplemented
37D8×C421C2211yescharacteristic, complemented
38C2×C8×C421C2211yescharacteristic, complemented
39C22×D421—22no—
40C22×D421D411yescharacteristic
41Dic842—22no—
42Dic842C2312yes—
43D1684—22no—
44D1684C2312yes—
45C22⋊D421—22no—
46C22⋊D421—22no—
47C22⋊D421—22no—
48C2×Dic421—22no—
49C2×Dic421D411yescharacteristic
50C4×D421—22no—
51C4×D421—22no—
52C4×D421—22no—
53C4×D421D411yescharacteristic
54C4×D421D411yescharacteristic
55C4×D421C2311yescharacteristic
56C4×D421C2311yescharacteristic
57#16809890C2311yescharacteristic
58#16809890C2311yescharacteristic
59Q8×C421C2312yes—
60D4×C421—22no—
61D4×C421C2312yes—
62C2×C4×C421—22no—
63C2×C4×C421C2311yescharacteristic
64C8×C421C2311yescharacteristic
65C8×C421C2311yescharacteristic
66C22×C421—22no—
67C22×C421C2×D411yescharacteristic
68C2×D421—22no—
69C2×D421—22no—
70C2×D421—44no—
71C2×D421C2×D411yescharacteristic
72C2×D421C2×D411yescharacteristic
73C4×C421—22no—
74C4×C421C2×D411yescharacteristic
75C4×C421C2411yescharacteristic
76Dic421—22no—
77Dic421C2×D411yescharacteristic
78Dic421C2×D411yescharacteristic
79C2×C421—21no—
80C2×C421—42no—
81C2×C421C22×D411yescharacteristic
82D421—21no—
83D421—42no—
84D421#13512yesdirect product factor
85C421C2×#13511yescharacteristic, complemented, Sylow
86C2×#135—4211nomaximal, Sylow
87C2×#41—4212no—
88#135—4212no—
89#135—4211no—
90#135—4212no—
91#135—4211no—
92#135—4211no—
93#135D42111yescharacteristic, direct product factor
94C2×Q16—4211no—
95C2×SD16—4212no—
96C2×D8—4211no—
97C22×C8—4211no—
98#41—8422no—
99#41—8422no—
100#41—4212no—
101#41—4212no—
102#41—4212no—
103#41C2×D42112yescomplemented
104C2×Q8—4212no—
105C2×D4—8422no—
106C2×D4—4212no—
107C22×C4—8422no—
108C22×C4—4211no—
109Q16—4212no—
110Q16—4211no—
111Q16C2×D42111yescharacteristic, complemented
112SD16—4212no—
113SD16—4212no—
114SD16—4212no—
115SD16C2×D42112yescomplemented
116D8—4212no—
117D8—4211no—
118D8C2×D42111yescharacteristic, complemented
119C2×C8—4211no—
120C2×C8—4211no—
121C2×C8—4211no—
122C2×C8—4211no—
123C2×C8—4211no—
124C2×C8C2×D42111yescharacteristic, complemented
125C23—8422no—
126C23—4211no—
127Q8—8422no—
128Q8—4212no—
129Q8C22×D42112yes—
130D4—22no—
131D4—8422no—
132D4—8422no—
133D4—8422no—
134D4—8422no—
135D4—4212no—
136D4C22×D42112yes—
137C2×C4—22no—
138C2×C4—8422no—
139C2×C4—8422no—
140C2×C4—8422no—
141C2×C4—8422no—
142C2×C4—4211no—
143C2×C4—4211no—
144C2×C4—4211no—
145C2×C4—4211no—
146C2×C4—4211no—
147C2×C4C22×D42111yescharacteristic
148C8—4211no—
149C8—4211no—
150C8C22×D42111yescharacteristic
151C8C22×D42111yescharacteristic
152C22—22no—
153C22—8422no—
154C22—8422no—
155C22—16844no—
156C22—4211no—
157C22—4211no—
158C22D4×D42111yescharacteristic
159C4—22no—
160C4—8422no—
161C4—4211no—
162C4—4211no—
163C4D4×D42111yescharacteristic
164C4C23×D42111yescharacteristic
165C2—21no—
166C2—42no—
167C2—8421no—
168C2—16842no—
169C2—4212no—
170C2C2×D4×D42111yescharacteristic
subgroupG/H
centerC4D4×D421
commutator, cosocleC4×C421C24
FrattiniC4C23×D421
Fitting#135×C421C2
socleC2×C421C22×D4
hypercenter#135D421
hypocenterC421C2×#135
LCSG ▷ C4×C421 ▷ C2×C421 ▷ C421
UCSC1 ◁ C4 ◁ C2×C4 ◁ #135

Complements

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

NH ≅ G/Ntype
1. #41×D421C2semidirect
2. #27212389C2semidirect
3. #27212388C2semidirect
4. #27212387C2semidirect
5. #27212386C2semidirect
6. #27212385C2semidirect
7. Q16×D421C2semidirect
8. SD16×D421C2semidirect
9. D8×D421C2semidirect
10. C2×C8×D421C2semidirect
11. #135×C421C2semidirect
12. #20269772C22semidirect
14. #20269771C22semidirect
16. #20269770C22semidirect
22. Dic1684C22semidirect
23. #20269761C22semidirect
24. #20269760C22semidirect
25. #20269759C22semidirect
26. #20269758C22semidirect
27. D3368C22semidirect
28. #20269756C22semidirect
29. C2×#16809890C22semidirect
32. C8×D421C22semidirect
33. C8×D421C22semidirect
34. #41×C421C22semidirect
35. Q16×C421C22semidirect
36. SD16×C421C22semidirect
37. D8×C421C22semidirect
38. C2×C8×C421C22semidirect
84. D421#135direct (N × H)
85. C421C2×#135semidirect
93. #135D421direct (N × H)
103. #41C2×D421semidirect
111. Q16C2×D421semidirect
115. SD16C2×D421semidirect
118. D8C2×D421semidirect
124. C2×C8C2×D421semidirect

Maximal subgroups

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

subgroupindex
#41×D4212
#272123892
#272123882
#272123872
#272123862
#272123852
Q16×D4212
SD16×D4212
D8×D4212
C2×C8×D4212
#135×C4212
C2×#135421