Finite Group Databasebeta

← C2×#22550713 #30653539

Subgroups — C2×#22550713 #30653539

Order 16256.

5956 conjugacy classes of subgroups; only the 3205 normal subgroups are shown.

normal subgroups only

This group has 5956 conjugacy classes of subgroups — too many to draw. Only its 3205 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#22550713C2132yesmaximal, direct product factor
2C2×#16861257C215yesmaximal, complemented
3C2×#16861255C2110yesmaximal, complemented
4C22×#14523249C215yesmaximal, complemented
5C2×#41×D127C2110yesmaximal, complemented
6C2×2−1+4×C127C211yesmaximal, characteristic, complemented
7#16861257C22180yescomplemented
8#16861255C221160yescomplemented
9C2×#14523249C22110yescomplemented
10C2×#14523249C22120yescomplemented
11C2×#14523249C22180yescomplemented
12C2×#14523248C22130yescomplemented
13C2×#14523247C22110yescomplemented
14#41×D127C221160yescomplemented
15C2×Q8×D127C22110yescomplemented
16C22×D508C22115yescomplemented
17C2×D4×D127C22130yescomplemented
18C22×C4×D127C22115yescomplemented
192−1+4×C127C22116yescomplemented
20C2×#41×C127C22110yescomplemented
21C22×Q8×C127C2215yescomplemented
22C23×D127C23115yes—
23#14523249C23180yescomplemented
24#14523249C231160yescomplemented
25#14523248C231240yescomplemented
26#14523247C23180yescomplemented
27C2×Dic254C23110yes—
28Q8×D127C23180yescomplemented
29C2×D508C23130yes—
30C2×D508C231120yescomplemented
31C2×C22⋊D127C23120yes—
32D4×D127C231240yescomplemented
33C22×Dic127C2315yes—
34C2×C4×D127C23110yes—
35C2×C4×D127C23130yes—
36C2×C4×D127C231120yescomplemented
37#41×C127C23180yescomplemented
38C2×Q8×C127C23110yes—
39C2×Q8×C127C23140yescomplemented
40C2×D4×C127C23110yes—
41C22×C4×C127C23115yes—
42C22×D127—430no—
43C22×D127C24110yes—
44C22×D127C24160yes—
45Dic254C24140yes—
46D508C241120yes—
47C22⋊D127C24180yes—
48C2×Dic127C2411yescharacteristic
49C2×Dic127C2415yes—
50C2×Dic127C24120yes—
51C4×D127C24140yes—
52C4×D127C241120yes—
53C23×C127C2415yes—
54Q8×C127C24140yes—
55D4×C127C24140yes—
56C2×C4×C127C24110yes—
57C2×C4×C127C24160yes—
58C22×C127—210no—
59C22×C127C25110yes—
60C22×C127C2511yescharacteristic
61C2×D127—220no—
62C2×D127—460no—
63C2×D127C25120yes—
64C4×C127C25120yes—
65Dic127C2512yes—
66Dic127C25110yes—
67C2×C127—210no—
68C2×C127#58512yescomplemented
69C2×C127C2611yescharacteristic
70D127—220no—
71C2×#585—1271nomaximal, Sylow
72C127C2×#58511yescharacteristic, complemented, Sylow
73#585—12732no—
74C2×2−1+4—1275no—
75C2×2−1+4D12711yescharacteristic, complemented
76C2×H(5,𝔽2)—12710no—
77C22×#41—12710no—
78C22×#41—1275no—
792−1+4—12780no—
802−1+4C2×D127116yescomplemented
81H(5,𝔽2)—127160no—
82C2×#41—127160no—
83C2×#41—12710no—
84C2×#41—12720no—
85C2×#41—12780no—
86C2×#41—12710no—
87C2×#41—12730no—
88C2×#41C2×D127110yescomplemented
89C22×Q8—12710no—
90C22×Q8C2×D12715yescomplemented
91C22×D4—12730no—
92C22×D4—12715no—
93C23×C4—12715no—
94C24—12715no—
95#41—12780no—
96#41—127160no—
97#41—12780no—
98#41—127240no—
99#41C22×D127180yescomplemented
100C2×Q8—12780no—
101C2×Q8—12710no—
102C2×Q8C22×D127110yes—
103C2×Q8C22×D127140yescomplemented
104C2×D4—12720no—
105C2×D4—127240no—
106C2×D4—12730no—
107C2×D4—127120no—
108C2×D4C22×D127110yes—
109C22×C4—1275no—
110C22×C4—12710no—
111C22×C4—12730no—
112C22×C4—127120no—
113C22×C4C22×D127115yes—
114C23—12760no—
115C23—12710no—
116C23—50830no—
117C23C23×D12715yes—
118Q8—12740no—
119Q8C23×D127140yes—
120D4—12780no—
121D4—127120no—
122D4C23×D127140yes—
123C2×C4—12720no—
124C2×C4—1271no—
125C2×C4—1275no—
126C2×C4—12740no—
127C2×C4—127120no—
128C2×C4C23×D127160yes—
129C2×C4C23×D127110yes—
130C22—210no—
131C22—12720no—
132C22—50860no—
133C22—25420no—
134C22C24×D127110yes—
135C22C24×D12711yescharacteristic
136C4—1272no—
137C4—12710no—
138C4C24×D127120yes—
139C2—210no—
140C2—25420no—
141C2#2255071312yesdirect product factor
142C2C25×D12711yescharacteristic
subgroupG/H
centerC22C24×D127
commutator, cosocleC2×C127C26
FrattiniC2C25×D127
FittingC2×2−1+4×C127C2
socleC22×C127C25
hypercenterC2×2−1+4D127
hypocenterC127C2×#585
LCSG ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. #22550713C2direct (N × H)
2. C2×#16861257C2semidirect
3. C2×#16861255C2semidirect
4. C22×#14523249C2semidirect
5. C2×#41×D127C2semidirect
6. C2×2−1+4×C127C2semidirect
7. #16861257C22semidirect
8. #16861255C22semidirect
9. C2×#14523249C22semidirect
10. C2×#14523249C22semidirect
11. C2×#14523249C22semidirect
12. C2×#14523248C22semidirect
13. C2×#14523247C22semidirect
14. #41×D127C22semidirect
15. C2×Q8×D127C22semidirect
16. C22×D508C22semidirect
17. C2×D4×D127C22semidirect
18. C22×C4×D127C22semidirect
19. 2−1+4×C127C22semidirect
20. C2×#41×C127C22semidirect
21. C22×Q8×C127C22semidirect
23. #14523249C23semidirect
24. #14523249C23semidirect
25. #14523248C23semidirect
26. #14523247C23semidirect
28. Q8×D127C23semidirect
30. C2×D508C23semidirect
32. D4×D127C23semidirect
36. C2×C4×D127C23semidirect
37. #41×C127C23semidirect
39. C2×Q8×C127C23semidirect
68. C2×C127#585semidirect
72. C127C2×#585semidirect
75. C2×2−1+4D127semidirect
80. 2−1+4C2×D127semidirect
88. C2×#41C2×D127semidirect
90. C22×Q8C2×D127semidirect
99. #41C22×D127semidirect
103. C2×Q8C22×D127semidirect
141. C2#22550713direct (N × H)

Maximal subgroups

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

subgroupindex
#225507132
C2×#168612572
C2×#168612552
C22×#145232492
C2×#41×D1272
C2×2−1+4×C1272
C2×#585127