Finite Group Databasebeta

← C2×#30632524 #44565174

Subgroups — C2×#30632524 #44565174

Order 32128.

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#30632524C2132yesmaximal, direct product factor
2C2×#22548763C215yesmaximal, complemented
3C2×#22548761C2110yesmaximal, complemented
4C22×#16860811C215yesmaximal, complemented
5C2×#41×D251C2110yesmaximal, complemented
6C2×2−1+4×C251C211yesmaximal, characteristic, complemented
7#22548763C22180yescomplemented
8#22548761C221160yescomplemented
9C2×#16860811C22110yescomplemented
10C2×#16860811C22120yescomplemented
11C2×#16860811C22180yescomplemented
12C2×#16860810C22130yescomplemented
13C2×#16860809C22110yescomplemented
14#41×D251C221160yescomplemented
15C2×Q8×D251C22110yescomplemented
16C22×D1004C22115yescomplemented
17C2×D4×D251C22130yescomplemented
18C22×C4×D251C22115yescomplemented
192−1+4×C251C22116yescomplemented
20C2×#41×C251C22110yescomplemented
21C22×Q8×C251C2215yescomplemented
22C23×D251C23115yes—
23#16860811C23180yescomplemented
24#16860811C231160yescomplemented
25#16860810C231240yescomplemented
26#16860809C23180yescomplemented
27C2×Dic502C23110yes—
28Q8×D251C23180yescomplemented
29C2×D1004C23130yes—
30C2×D1004C231120yescomplemented
31C2×C22⋊D251C23120yes—
32D4×D251C231240yescomplemented
33C22×Dic251C2315yes—
34C2×C4×D251C23110yes—
35C2×C4×D251C23130yes—
36C2×C4×D251C231120yescomplemented
37#41×C251C23180yescomplemented
38C2×Q8×C251C23110yes—
39C2×Q8×C251C23140yescomplemented
40C2×D4×C251C23110yes—
41C22×C4×C251C23115yes—
42C22×D251—430no—
43C22×D251C24110yes—
44C22×D251C24160yes—
45Dic502C24140yes—
46D1004C241120yes—
47C22⋊D251C24180yes—
48C2×Dic251C2411yescharacteristic
49C2×Dic251C2415yes—
50C2×Dic251C24120yes—
51C4×D251C24140yes—
52C4×D251C241120yes—
53C23×C251C2415yes—
54Q8×C251C24140yes—
55D4×C251C24140yes—
56C2×C4×C251C24110yes—
57C2×C4×C251C24160yes—
58C22×C251—210no—
59C22×C251C25110yes—
60C22×C251C2511yescharacteristic
61C2×D251—220no—
62C2×D251—460no—
63C2×D251C25120yes—
64C4×C251C25120yes—
65Dic251C2512yes—
66Dic251C25110yes—
67C2×C251—210no—
68C2×C251#58512yescomplemented
69C2×C251C2611yescharacteristic
70D251—220no—
71C251C2×#58511yescharacteristic, complemented, Sylow
72C2×#585—2511nomaximal, Sylow
73#585—25132no—
74C2×2−1+4—2515no—
75C2×2−1+4D25111yescharacteristic, complemented
76C2×H(5,𝔽2)—25110no—
77C22×#41—25110no—
78C22×#41—2515no—
792−1+4—25180no—
802−1+4C2×D251116yescomplemented
81H(5,𝔽2)—251160no—
82C2×#41—251160no—
83C2×#41—25110no—
84C2×#41—25120no—
85C2×#41—25180no—
86C2×#41—25110no—
87C2×#41—25130no—
88C2×#41C2×D251110yescomplemented
89C22×Q8—25110no—
90C22×Q8C2×D25115yescomplemented
91C22×D4—25130no—
92C22×D4—25115no—
93C23×C4—25115no—
94C24—25115no—
95#41—25180no—
96#41—251160no—
97#41—25180no—
98#41—251240no—
99#41C22×D251180yescomplemented
100C2×Q8—25180no—
101C2×Q8—25110no—
102C2×Q8C22×D251110yes—
103C2×Q8C22×D251140yescomplemented
104C2×D4—25120no—
105C2×D4—251240no—
106C2×D4—25130no—
107C2×D4—251120no—
108C2×D4C22×D251110yes—
109C22×C4—2515no—
110C22×C4—25110no—
111C22×C4—25130no—
112C22×C4—251120no—
113C22×C4C22×D251115yes—
114C23—100430no—
115C23—25160no—
116C23—25110no—
117C23C23×D25115yes—
118Q8—25140no—
119Q8C23×D251140yes—
120D4—25180no—
121D4—251120no—
122D4C23×D251140yes—
123C2×C4—25120no—
124C2×C4—2511no—
125C2×C4—2515no—
126C2×C4—25140no—
127C2×C4—251120no—
128C2×C4C23×D251160yes—
129C2×C4C23×D251110yes—
130C22—210no—
131C22—100460no—
132C22—50220no—
133C22—25120no—
134C22C24×D251110yes—
135C22C24×D25111yescharacteristic
136C4—2512no—
137C4—25110no—
138C4C24×D251120yes—
139C2—210no—
140C2—50220no—
141C2#3063252412yesdirect product factor
142C2C25×D25111yescharacteristic
subgroupG/H
centerC22C24×D251
commutator, cosocleC2×C251C26
FrattiniC2C25×D251
FittingC2×2−1+4×C251C2
socleC22×C251C25
hypercenterC2×2−1+4D251
hypocenterC251C2×#585
LCSG ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. #30632524C2direct (N × H)
2. C2×#22548763C2semidirect
3. C2×#22548761C2semidirect
4. C22×#16860811C2semidirect
5. C2×#41×D251C2semidirect
6. C2×2−1+4×C251C2semidirect
7. #22548763C22semidirect
8. #22548761C22semidirect
9. C2×#16860811C22semidirect
10. C2×#16860811C22semidirect
11. C2×#16860811C22semidirect
12. C2×#16860810C22semidirect
13. C2×#16860809C22semidirect
14. #41×D251C22semidirect
15. C2×Q8×D251C22semidirect
16. C22×D1004C22semidirect
17. C2×D4×D251C22semidirect
18. C22×C4×D251C22semidirect
19. 2−1+4×C251C22semidirect
20. C2×#41×C251C22semidirect
21. C22×Q8×C251C22semidirect
23. #16860811C23semidirect
24. #16860811C23semidirect
25. #16860810C23semidirect
26. #16860809C23semidirect
28. Q8×D251C23semidirect
30. C2×D1004C23semidirect
32. D4×D251C23semidirect
36. C2×C4×D251C23semidirect
37. #41×C251C23semidirect
39. C2×Q8×C251C23semidirect
68. C2×C251#585semidirect
71. C251C2×#585semidirect
75. C2×2−1+4D251semidirect
80. 2−1+4C2×D251semidirect
88. C2×#41C2×D251semidirect
90. C22×Q8C2×D251semidirect
99. #41C22×D251semidirect
103. C2×Q8C22×D251semidirect
141. C2#30632524direct (N × H)

Maximal subgroups

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

subgroupindex
#306325242
C2×#225487632
C2×#225487612
C22×#168608112
C2×#41×D2512
C2×2−1+4×C2512
C2×#585251