Finite Group Databasebeta

← C22×#22576122 #46849534

Subgroups — C22×#22576122 #46849534

Order 34688.

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×D1084C211yesmaximal, characteristic
2C23×C4×D271C212yesmaximal, complemented
3C2×#22576122C2124yesmaximal, direct product factor
4C22×(C2×D271)⋊C4C212yesmaximal, complemented
5C22×C4×Dic271C211yesmaximal, characteristic, complemented
6C22×C4⋊C4×C271C211yesmaximal, characteristic, complemented
7C24×D271C2212yes—
8C22×D1084C4116yescomplemented
9C22×D1084C22112yes—
10C23×Dic271C2211yescharacteristic
11C23×Dic271C2211yescharacteristic
12C22×C4×D271—216no—
13C22×C4×D271C22124yescomplemented
14#22576122C22164yesdirect product factor
15C2×(C2×D271)⋊C4C22124yescomplemented
16C2×C4×Dic271C22112yescomplemented
17C23×C4×C271C2212yes—
18C23×C4×C271C2211yescharacteristic
19C2×C4⋊C4×C271C22112yescomplemented
20C23×D271—28no—
21C23×D271—28no—
22C23×D271C2×C4116yescomplemented
23C23×D271C23112yes—
24C2×D1084C2×C4196yescomplemented
25C2×D1084C23116yes—
26C22×Dic271—24no—
27C22×Dic271D418yescomplemented
28C22×Dic271C2316yes—
29C22×Dic271C2316yes—
30C2×C4×D271—296no—
31C2×C4×D271C23132yescomplemented
32(C2×D271)⋊C4C23132yescomplemented
33C4×Dic271C23116yescomplemented
34C24×C271C2311yescharacteristic
35C22×C4×C271—28no—
36C22×C4×C271C2×C418yescomplemented
37C22×C4×C271C23112yes—
38C22×C4×C271C2316yes—
39C4⋊C4×C271C23116yescomplemented
40C22×D271—264no—
41C22×D271—224no—
42C22×D271—224no—
43C22×D271C22×C4148yescomplemented
44C22×D271C2418yes—
45D1084C22×C4164yescomplemented
46C2×Dic271—212no—
47C2×Dic271C2×D4124yescomplemented
48C2×Dic271C2414yes—
49C2×Dic271C2414yes—
50C4×D271—264no—
51C23×C271C22×C414yes—
52C23×C271C2×D414yes—
53C23×C271#4114yes—
54C23×C271C2413yes—
55C2×C4×C271—224no—
56C2×C4×C271C22×C4124yescomplemented
57C2×C4×C271C2418yes—
58C2×C4×C271C2414yes—
59C22×C271C4×D4116yescomplemented
60C22×C271C23×C416yes—
61C22×C271C22×D416yes—
62C22×C271C2×#4116yes—
63C22×C271C2511yescharacteristic
64C2×D271—296no—
65C2×D271—28no—
66C2×D271—28no—
67C2×D271C23×C4116yescomplemented
68C4×C271—28no—
69C4×C271C23×C418yescomplemented
70Dic271—24no—
71Dic271C22×D418yescomplemented
72C2×C271C2×C4×D4112yescomplemented
73C2×C271C24×C411yescharacteristic
74C2×C271C23×D411yescharacteristic
75C2×C271C22×#4111yescharacteristic
76D271—216no—
77C271C22×C4×D411yescharacteristic, complemented, Sylow
78C22×C4×D4—2711nomaximal, Sylow
79C23×D4—2711no—
80C24×C4—2712no—
81C2×C4×D4—27124no—
82C22×C4⋊C4D27111yescharacteristic, complemented
83C22×C22⋊C4—2712no—
84C22×C42—2711no—
85C25—2712no—
86C22×D4—27116no—
87C22×D4—27112no—
88C23×C4—2711no—
89C23×C4—2711no—
90C23×C4—27124no—
91C23×C4—54216no—
92C23×C4C2×D27112yes—
93C23×C4C2×D27111yescharacteristic
94C4×D4—27164no—
95C2×C4⋊C4C2×D271112yescomplemented
96C2×C22⋊C4—27124no—
97C2×C42—27112no—
98C24—27116no—
99C24—27112no—
100C24—5428no—
101C24—5428no—
102C24C22×D27111yescharacteristic
103C2×D4—27196no—
104C2×D4—27116no—
105C22×C4—28no—
106C22×C4—2718no—
107C22×C4—2716no—
108C22×C4—2716no—
109C22×C4—27132no—
110C22×C4—54296no—
111C22×C4—5424no—
112C22×C4C4×D27118yescomplemented
113C22×C4C22×D271112yes—
114C22×C4C22×D27116yes—
115C4⋊C4C22×D271116yescomplemented
116C22⋊C4—27132no—
117C42—27116no—
118C23—27148no—
119C23—2718no—
120C23—54224no—
121C23—54224no—
122C23—54264no—
123C23C2×C4×D27114yes—
124C23D4×D27114yes—
125C23#1686465514yes—
126C23C23×D27113yes—
127D4—27164no—
128C2×C4—224no—
129C2×C4—27124no—
130C2×C4—2714no—
131C2×C4—2714no—
132C2×C4—54264no—
133C2×C4—54212no—
134C2×C4C2×C4×D271124yescomplemented
135C2×C4C23×D27118yes—
136C2×C4C23×D27114yes—
137C22—27116no—
138C22—5428no—
139C22—54296no—
140C22—5428no—
141C22#22576122116yesdirect product factor
142C22C22×C4×D27116yes—
143C22C2×D4×D27116yes—
144C22C2×#1686465516yes—
145C22C24×D27111yescharacteristic
146C4—28no—
147C4—2718no—
148C4—5424no—
149C4C22×C4×D27118yescomplemented
150C2—54216no—
151C2C2×#22576122112yesdirect product factor
152C2C23×C4×D27111yescharacteristic
153C2C22×D4×D27111yescharacteristic
154C2C22×#1686465511yescharacteristic
subgroupG/H
centerC24C22×D271
commutatorC2×C271C24×C4
FrattiniC22C24×D271
FittingC22×C4⋊C4×C271C2
socleC24×C271C23
prime radicalC23×D1084C2
cosocleC22×C271C25
hypercenterC22×C4⋊C4D271
hypocenterC271C22×C4×D4
LCSG ▷ C2×C271 ▷ C271
UCSC1 ◁ C24 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C23×C4×D271C2semidirect
3. C2×#22576122C2direct (N × H)
4. C22×(C2×D271)⋊C4C2semidirect
5. C22×C4×Dic271C2semidirect
6. C22×C4⋊C4×C271C2semidirect
8. C22×D1084C4semidirect
13. C22×C4×D271C22semidirect
14. #22576122C22direct (N × H)
15. C2×(C2×D271)⋊C4C22semidirect
16. C2×C4×Dic271C22semidirect
19. C2×C4⋊C4×C271C22semidirect
22. C23×D271C2×C4semidirect
24. C2×D1084C2×C4semidirect
27. C22×Dic271D4semidirect
31. C2×C4×D271C23semidirect
32. (C2×D271)⋊C4C23semidirect
33. C4×Dic271C23semidirect
36. C22×C4×C271C2×C4semidirect
39. C4⋊C4×C271C23semidirect
43. C22×D271C22×C4semidirect
45. D1084C22×C4semidirect
47. C2×Dic271C2×D4semidirect
56. C2×C4×C271C22×C4semidirect
59. C22×C271C4×D4semidirect
67. C2×D271C23×C4semidirect
69. C4×C271C23×C4semidirect
71. Dic271C22×D4semidirect
72. C2×C271C2×C4×D4semidirect
77. C271C22×C4×D4semidirect
82. C22×C4⋊C4D271semidirect
95. C2×C4⋊C4C2×D271semidirect
112. C22×C4C4×D271semidirect
115. C4⋊C4C22×D271semidirect
134. C2×C4C2×C4×D271semidirect
141. C22#22576122direct (N × H)
149. C4C22×C4×D271semidirect
151. C2C2×#22576122direct (N × H)

Maximal subgroups

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

subgroupindex
C23×D10842
C23×C4×D2712
C2×#225761222
C22×(C2×D271)⋊C42
C22×C4×Dic2712
C22×C4⋊C4×C2712
C22×C4×D4271