Finite Group Databasebeta

← C2×2−1+4×D101 #27185935

Subgroups — C2×2−1+4×D101 #27185935

Order 12928.

5872 conjugacy classes of subgroups; only the 3213 normal subgroups are shown.

normal subgroups only

This group has 5872 conjugacy classes of subgroups — too many to draw. Only its 3213 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
1C2×#15692303C215yesmaximal, complemented
2C2×#15692302C2110yesmaximal, complemented
32−1+4×D101C2132yesmaximal, direct product factor
4C2×#41×D101C2110yesmaximal, complemented
5C22×Q8×D101C215yesmaximal, complemented
6C2×2−1+4×C101C211yesmaximal, characteristic, complemented
7#15692303C22180yescomplemented
8#15692302C221160yescomplemented
9C2×#13108075C22110yescomplemented
10C2×#13108074C22130yescomplemented
11C2×#13108073C22130yescomplemented
12#41×D101C221160yescomplemented
13C22×Dic202C22115yescomplemented
14C2×Q8×D101C22110yes—
15C2×Q8×D101C22120yescomplemented
16C2×Q8×D101C22180yescomplemented
17C2×D4×D101C22110yes—
18C22×C4×D101C22115yes—
192−1+4×C101C22116yescomplemented
20C2×#41×C101C22110yescomplemented
21C22×Q8×C101C2215yescomplemented
22C23×D101C2315yes—
23#13108075C23180yescomplemented
24#13108074C231240yescomplemented
25#13108073C231240yescomplemented
26C2×Dic202C23130yes—
27C2×Dic202C231120yescomplemented
28Q8×D101C23180yes—
29Q8×D101C231160yescomplemented
30C2×D404C23110yes—
31C2×C22⋊D101C23120yes—
32D4×D101C23180yes—
33C22×Dic101C23115yes—
34C2×C4×D101C23110yes—
35C2×C4×D101C231120yes—
36C2×C4×D101C23130yes—
37#41×C101C23180yescomplemented
38C2×Q8×C101C23110yes—
39C2×Q8×C101C23140yescomplemented
40C2×D4×C101C23110yes—
41C22×C4×C101C23115yes—
42C23×C101C2415yes—
43C22×D101—220no—
44C22×D101C24120yes—
45C22×D101C2415yes—
46C22×D101C2411yescharacteristic
47Q8×C101C24140yes—
48D4×C101C24140yes—
49C2×C4×C101C24110yes—
50C2×C4×C101C24160yes—
51C22⋊D101C24180yes—
52C2×Dic101C24110yes—
53C2×Dic101C24160yes—
54D404C24140yes—
55C4×D101C24140yes—
56C4×D101C241120yes—
57Dic202C241120yes—
58C22×C101—210no—
59C22×C101C25110yes—
60C22×C101C2511yescharacteristic
61C2×D101—210no—
62C2×D101—240no—
63C2×D1012−1+414yesdirect product factor
64C2×D101C25110yes—
65C2×D101C2512yes—
66C4×C101C25120yes—
67Dic101C25120yes—
68C2×C101—210no—
69C2×C101C2×2−1+412yescomplemented
70C2×C101C2611yescharacteristic
71D101—210no—
72D101C2×2−1+414yesdirect product factor
73C22×2−1+4—1011nomaximal, Sylow
74C101C22×2−1+411yescharacteristic, complemented, Sylow
75C2×2−1+4—10132no—
76C2×2−1+4—10110no—
77C2×2−1+4—1015no—
78C2×2−1+4D10111yescharacteristic, direct product factor
79C22×#41—10110no—
80C23×Q8—1015no—
812−1+4—101160no—
822−1+4—10180no—
832−1+4C2×D101116yesdirect product factor
84C2×#41—101160no—
85C2×#41—10110no—
86C2×#41—10130no—
87C2×#41—10130no—
88C2×#41C2×D101110yescomplemented
89C22×Q8—10110no—
90C22×Q8—10120no—
91C22×Q8—10180no—
92C22×Q8—10115no—
93C22×Q8C2×D10115yescomplemented
94C22×D4—10110no—
95C23×C4—10115no—
96C24—1015no—
97#41—10180no—
98#41—101240no—
99#41—101240no—
100#41C22×D101180yescomplemented
101C2×Q8—10180no—
102C2×Q8—101160no—
103C2×Q8—10130no—
104C2×Q8—101120no—
105C2×Q8C22×D101110yes—
106C2×Q8C22×D101140yescomplemented
107C2×D4—10120no—
108C2×D4—10180no—
109C2×D4—10110no—
110C2×D4C22×D101110yes—
111C22×C4—10115no—
112C22×C4—10110no—
113C22×C4—101120no—
114C22×C4—10130no—
115C22×C4C22×D101115yes—
116C23—1011no—
117C23—10120no—
118C23—1015no—
119C23—20220no—
120C23C23×D10115yes—
121Q8—101120no—
122Q8C23×D101140yes—
123D4—10180no—
124D4—10140no—
125D4C23×D101140yes—
126C2×C4—10160no—
127C2×C4—10110no—
128C2×C4—10140no—
129C2×C4—101120no—
130C2×C4C23×D101160yes—
131C2×C4C23×D101110yes—
132C22—210no—
133C22—1014no—
134C22—1012no—
135C22—10110no—
136C22—20240no—
137C22—20210no—
138C22C24×D101110yes—
139C22C24×D10111yescharacteristic
140C4—10120no—
141C4C24×D101120yes—
142C2—210no—
143C2—1014no—
144C2—20210no—
145C22−1+4×D10112yesdirect product factor
146C2C25×D10111yescharacteristic
subgroupG/H
centerC22C24×D101
commutator, cosocleC2×C101C26
FrattiniC2C25×D101
FittingC2×2−1+4×C101C2
socleC22×C101C25
hypercenterC2×2−1+4D101
hypocenterC101C22×2−1+4
LCSG ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. C2×#15692303C2semidirect
2. C2×#15692302C2semidirect
3. 2−1+4×D101C2direct (N × H)
4. C2×#41×D101C2semidirect
5. C22×Q8×D101C2semidirect
6. C2×2−1+4×C101C2semidirect
7. #15692303C22semidirect
8. #15692302C22semidirect
9. C2×#13108075C22semidirect
10. C2×#13108074C22semidirect
11. C2×#13108073C22semidirect
12. #41×D101C22semidirect
13. C22×Dic202C22semidirect
15. C2×Q8×D101C22semidirect
16. C2×Q8×D101C22semidirect
19. 2−1+4×C101C22semidirect
20. C2×#41×C101C22semidirect
21. C22×Q8×C101C22semidirect
23. #13108075C23semidirect
24. #13108074C23semidirect
25. #13108073C23semidirect
27. C2×Dic202C23semidirect
29. Q8×D101C23semidirect
37. #41×C101C23semidirect
39. C2×Q8×C101C23semidirect
63. C2×D1012−1+4direct (N × H)
69. C2×C101C2×2−1+4semidirect
72. D101C2×2−1+4direct (N × H)
74. C101C22×2−1+4semidirect
78. C2×2−1+4D101direct (N × H)
83. 2−1+4C2×D101direct (N × H)
88. C2×#41C2×D101semidirect
93. C22×Q8C2×D101semidirect
100. #41C22×D101semidirect
106. C2×Q8C22×D101semidirect
145. C22−1+4×D101direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156923032
C2×#156923022
2−1+4×D1012
C2×#41×D1012
C22×Q8×D1012
C2×2−1+4×C1012
C22×2−1+4101