Finite Group Databasebeta

← C22×#15692217 #27184302

Subgroups — C22×#15692217 #27184302

Order 12928.

2528 conjugacy classes of subgroups; only the 941 normal subgroups are shown.

normal subgroups only

This group has 2528 conjugacy classes of subgroups — too many to draw. Only its 941 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
1C23×D404C211yesmaximal, characteristic
2C23×C4×D101C212yesmaximal, complemented
3C2×#15692217C2124yesmaximal, direct product factor
4C22×#13108051C212yesmaximal, complemented
5C22×C4×Dic101C211yesmaximal, characteristic, complemented
6C22×C4⋊C4×C101C211yesmaximal, characteristic, complemented
7C24×D101C2212yes—
8C22×D404C4116yescomplemented
9C22×D404C22112yes—
10C23×Dic101C2211yescharacteristic
11C23×Dic101C2211yescharacteristic
12C22×C4×D101—216no—
13C22×C4×D101C22124yescomplemented
14#15692217C22164yesdirect product factor
15C2×#13108051C22124yescomplemented
16C2×C4×Dic101C22112yescomplemented
17C23×C4×C101C2212yes—
18C23×C4×C101C2211yescharacteristic
19C2×C4⋊C4×C101C22112yescomplemented
20C23×D101—28no—
21C23×D101—28no—
22C23×D101C2×C4116yescomplemented
23C23×D101C23112yes—
24C2×D404C2×C4196yescomplemented
25C2×D404C23116yes—
26C22×Dic101—24no—
27C22×Dic101D418yescomplemented
28C22×Dic101C2316yes—
29C22×Dic101C2316yes—
30C2×C4×D101—296no—
31C2×C4×D101C23132yescomplemented
32#13108051C23132yescomplemented
33C4×Dic101C23116yescomplemented
34C24×C101C2311yescharacteristic
35C22×C4×C101—28no—
36C22×C4×C101C2×C418yescomplemented
37C22×C4×C101C23112yes—
38C22×C4×C101C2316yes—
39C4⋊C4×C101C23116yescomplemented
40C23×C101C22×C414yes—
41C23×C101C2×D414yes—
42C23×C101#4114yes—
43C23×C101C2413yes—
44C22×D101—264no—
45C22×D101—224no—
46C22×D101—224no—
47C22×D101C22×C4148yescomplemented
48C22×D101C2418yes—
49C2×C4×C101—224no—
50C2×C4×C101C22×C4124yescomplemented
51C2×C4×C101C2418yes—
52C2×C4×C101C2414yes—
53C2×Dic101—212no—
54C2×Dic101C2×D4124yescomplemented
55C2×Dic101C2414yes—
56C2×Dic101C2414yes—
57D404C22×C4164yescomplemented
58C4×D101—264no—
59C22×C101C4×D4116yescomplemented
60C22×C101C23×C416yes—
61C22×C101C22×D416yes—
62C22×C101C2×#4116yes—
63C22×C101C2511yescharacteristic
64C2×D101—296no—
65C2×D101—28no—
66C2×D101—28no—
67C2×D101C23×C4116yescomplemented
68C4×C101—28no—
69C4×C101C23×C418yescomplemented
70Dic101—24no—
71Dic101C22×D418yescomplemented
72C2×C101C2×C4×D4112yescomplemented
73C2×C101C24×C411yescharacteristic
74C2×C101C23×D411yescharacteristic
75C2×C101C22×#4111yescharacteristic
76D101—216no—
77C22×C4×D4—1011nomaximal, Sylow
78C101C22×C4×D411yescharacteristic, complemented, Sylow
79C23×D4—1011no—
80C24×C4—1012no—
81C2×C4×D4—10124no—
82C22×C4⋊C4D10111yescharacteristic, complemented
83C22×C22⋊C4—1012no—
84C22×C42—1011no—
85C25—1012no—
86C22×D4—10116no—
87C22×D4—10112no—
88C23×C4—1011no—
89C23×C4—1011no—
90C23×C4—10124no—
91C23×C4—20216no—
92C23×C4C2×D10112yes—
93C23×C4C2×D10111yescharacteristic
94C4×D4—10164no—
95C2×C4⋊C4C2×D101112yescomplemented
96C2×C22⋊C4—10124no—
97C2×C42—10112no—
98C24—10116no—
99C24—10112no—
100C24—2028no—
101C24—2028no—
102C24C22×D10111yescharacteristic
103C2×D4—10196no—
104C2×D4—10116no—
105C22×C4—28no—
106C22×C4—1018no—
107C22×C4—1016no—
108C22×C4—1016no—
109C22×C4—10132no—
110C22×C4—20296no—
111C22×C4—2024no—
112C22×C4C4×D10118yescomplemented
113C22×C4C22×D101112yes—
114C22×C4C22×D10116yes—
115C4⋊C4C22×D101116yescomplemented
116C22⋊C4—10132no—
117C42—10116no—
118C23—10148no—
119C23—1018no—
120C23—20224no—
121C23—20224no—
122C23—20264no—
123C23C2×C4×D10114yes—
124C23D4×D10114yes—
125C23#1310807514yes—
126C23C23×D10113yes—
127D4—10164no—
128C2×C4—224no—
129C2×C4—10124no—
130C2×C4—1014no—
131C2×C4—1014no—
132C2×C4—20264no—
133C2×C4—20212no—
134C2×C4C2×C4×D101124yescomplemented
135C2×C4C23×D10118yes—
136C2×C4C23×D10114yes—
137C22—10116no—
138C22—2028no—
139C22—2028no—
140C22—20296no—
141C22#15692217116yesdirect product factor
142C22C22×C4×D10116yes—
143C22C2×D4×D10116yes—
144C22C2×#1310807516yes—
145C22C24×D10111yescharacteristic
146C4—28no—
147C4—1018no—
148C4—2024no—
149C4C22×C4×D10118yescomplemented
150C2—20216no—
151C2C2×#15692217112yesdirect product factor
152C2C23×C4×D10111yescharacteristic
153C2C22×D4×D10111yescharacteristic
154C2C22×#1310807511yescharacteristic
subgroupG/H
centerC24C22×D101
commutatorC2×C101C24×C4
FrattiniC22C24×D101
FittingC22×C4⋊C4×C101C2
socleC24×C101C23
prime radicalC23×D404C2
cosocleC22×C101C25
hypercenterC22×C4⋊C4D101
hypocenterC101C22×C4×D4
LCSG ▷ C2×C101 ▷ C101
UCSC1 ◁ C24 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C23×C4×D101C2semidirect
3. C2×#15692217C2direct (N × H)
4. C22×#13108051C2semidirect
5. C22×C4×Dic101C2semidirect
6. C22×C4⋊C4×C101C2semidirect
8. C22×D404C4semidirect
13. C22×C4×D101C22semidirect
14. #15692217C22direct (N × H)
15. C2×#13108051C22semidirect
16. C2×C4×Dic101C22semidirect
19. C2×C4⋊C4×C101C22semidirect
22. C23×D101C2×C4semidirect
24. C2×D404C2×C4semidirect
27. C22×Dic101D4semidirect
31. C2×C4×D101C23semidirect
32. #13108051C23semidirect
33. C4×Dic101C23semidirect
36. C22×C4×C101C2×C4semidirect
39. C4⋊C4×C101C23semidirect
47. C22×D101C22×C4semidirect
50. C2×C4×C101C22×C4semidirect
54. C2×Dic101C2×D4semidirect
57. D404C22×C4semidirect
59. C22×C101C4×D4semidirect
67. C2×D101C23×C4semidirect
69. C4×C101C23×C4semidirect
71. Dic101C22×D4semidirect
72. C2×C101C2×C4×D4semidirect
78. C101C22×C4×D4semidirect
82. C22×C4⋊C4D101semidirect
95. C2×C4⋊C4C2×D101semidirect
112. C22×C4C4×D101semidirect
115. C4⋊C4C22×D101semidirect
134. C2×C4C2×C4×D101semidirect
141. C22#15692217direct (N × H)
149. C4C22×C4×D101semidirect
151. C2C2×#15692217direct (N × H)

Maximal subgroups

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

subgroupindex
C23×D4042
C23×C4×D1012
C2×#156922172
C22×#131080512
C22×C4×Dic1012
C22×C4⋊C4×C1012
C22×C4×D4101