Finite Group Databasebeta

← C22×#15690875 #27158361

Subgroups — C22×#15690875 #27158361

Order 12416.

856 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 856 conjugacy classes of subgroups — too many to draw. Only its 218 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
1C22×#13095521C211yesmaximal, characteristic, complemented
2C22×C4×#87604C211yesmaximal, characteristic, complemented
3C2×#15690875C2112yesmaximal, direct product factor
4C22×#13095512C211yesmaximal, characteristic
5C23×#87604—21no—
6C2×#13095521—23no—
7C2×#13095521C2216yescomplemented
8C22×#11715897—21no—
9C2×C4×#87604—23no—
10C2×C4×#87604C2216yescomplemented
11#15690875C22116yesdirect product factor
12C2×#13095512—23no—
13C2×#13095512—23no—
14C2×#13095512C416yescomplemented
15C2×#13095512C2216yes—
16C22×D388—21no—
17C22×D388C411yescharacteristic, complemented
18C22×C4×D97—21no—
19C22×C4×D97C2211yescharacteristic
20C22×Q8×C97C411yescharacteristic, complemented
21C22×#87604—28no—
22C22×#87604—26no—
23#13095521—26no—
24#13095521C2314yescomplemented
25C2×#11715897—26no—
26C4×#87604—26no—
27C4×#87604C2314yescomplemented
28C23×D97—21no—
29C23×D97D411yescharacteristic
30#13095512—212no—
31#13095512—26no—
32#13095512—26no—
33#13095512C2×C4112yescomplemented
34#13095512C2314yes—
35C2×D388—26no—
36C2×D388—26no—
37C2×D388—23no—
38C2×D388C2×C416yescomplemented
39C22×Dic97D411yescharacteristic
40C2×C4×D97—26no—
41C2×C4×D97—26no—
42C2×C4×D97C2×C413yes—
43C2×C4×D97D413yes—
44C2×C4×D97D413yes—
45C2×C4×D97C2313yes—
46C2×Q8×C97—23no—
47C2×Q8×C97C2×C416yescomplemented
48C22×C4×C97—21no—
49C22×C4×C97C2×C411yescharacteristic
50C23×C97C22⋊C411yescharacteristic
51C22×D97—24no—
52C22×D97—26no—
53C22×D97—44no—
54C22×D97C22⋊C413yes—
55C22×D97C2×D413yes—
56C2×#87604—224no—
57C2×#87604—24no—
58Q8×C97—26no—
59Q8×C97C22×C414yescomplemented
60C2×C4×C97—26no—
61C2×C4×C97C22⋊C413yes—
62C2×C4×C97C22×C413yes—
63C2×Dic97C22⋊C413yes—
64C2×Dic97C2×D413yes—
65D388—212no—
66D388—24no—
67D388—26no—
68D388C22×C414yescomplemented
69C4×D97—212no—
70C4×D97—24no—
71C4×D97C22⋊C416yes—
72C4×D97C22×C413yes—
73C4×D97C2×D413yes—
74C4×D97C2×D413yes—
75C4×D97C2411yescharacteristic
76#11715897—24no—
77C22×C97#10414yescomplemented
78C22×C97C2×C22⋊C413yes—
79C2×D97—26no—
80C2×D97—26no—
81C2×D97—24no—
82C2×D97—412no—
83C2×D97C2×C22⋊C413yes—
84C2×D97C22×D411yescharacteristic
85#87604—28no—
86C4×C97—24no—
87C4×C97C2×C22⋊C413yes—
88C4×C97C23×C411yescharacteristic
89Dic97C2×C22⋊C413yes—
90Dic97C22×D411yescharacteristic
91C2×C97C2×#10416yescomplemented
92C2×C97C22×C22⋊C411yescharacteristic
93D97—21no—
94D97—23no—
95D97—44no—
96C22×#104—971nomaximal, Sylow
97C97C22×#10411yescharacteristic, complemented, Sylow
98C22×#41—971no—
99C22×M16—971no—
100C22×C42—971no—
101C2×#104—9712no—
102C2×#41—976no—
103C2×#41—976no—
104C2×#41—1943no—
105C2×#41—1943no—
106C22×Q8#8760411yescharacteristic, complemented
107C22×D4—971no—
108C22×D4—1941no—
109C23×C4—971no—
110C23×C4—1941no—
111C23×C4—1941no—
112C2×M16—976no—
113C2×M16—1943no—
114C22×C8—1941no—
115C2×C42—976no—
116C2×C42—1943no—
117#104—9716no—
118C24—971no—
119C24—1941no—
120#41—9712no—
121#41—974no—
122#41—1946no—
123#41—1946no—
124#41—19412no—
125C2×Q8—23no—
126C2×Q8C2×#8760416yescomplemented
127C2×D4—976no—
128C2×D4—1946no—
129C2×D4—1946no—
130C2×D4—1943no—
131C22×C4—21no—
132C22×C4—971no—
133C22×C4—973no—
134C22×C4—973no—
135C22×C4—973no—
136C22×C4—973no—
137C22×C4—1946no—
138C22×C4—1946no—
139C22×C4—1946no—
140C22×C4—1948no—
141C22×C4C2×#8760411yescharacteristic
142M16—974no—
143M16—1946no—
144C2×C8—1946no—
145C42—974no—
146C42—1946no—
147C23—973no—
148C23—973no—
149C23—3884no—
150C23—1946no—
151C23—1944no—
152C23C22⋊#8760411yescharacteristic
153Q8—26no—
154Q8C22×#8760414yescomplemented
155D4—974no—
156D4—1944no—
157D4—19412no—
158D4—1946no—
159C2×C4—26no—
160C2×C4—973no—
161C2×C4—973no—
162C2×C4—973no—
163C2×C4—973no—
164C2×C4—976no—
165C2×C4—971no—
166C2×C4—973no—
167C2×C4—1944no—
168C2×C4—19412no—
169C2×C4—1944no—
170C2×C4—19424no—
171C2×C4C22⋊#8760413yes—
172C2×C4C22×#8760413yes—
173C8—1944no—
174C22—971no—
175C22—973no—
176C22—38812no—
177C22—1944no—
178C22—1946no—
179C22—1946no—
180C22#1569087514yesdirect product factor
181C22C2×C22⋊#8760413yes—
182C4—24no—
183C4—971no—
184C4—973no—
185C4—1948no—
186C4C2×C22⋊#8760413yes—
187C4C23×#8760411yescharacteristic
188C2—3884no—
189C2—1941no—
190C2—1943no—
191C2C2×#1569087516yesdirect product factor
192C2C22×C22⋊#8760411yescharacteristic
subgroupG/H
centerC23C22⋊#87604
commutatorC4×C97C23×C4
FrattiniC4C23×#87604
FittingC22×Q8×C97C4
socleC23×C97C22⋊C4
prime radicalC22×#13095512C2
cosocleC4×D97C24
hypercenterC22×Q8#87604
hypocenterC97C22×#104
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
1. C22×#13095521C2semidirect
2. C22×C4×#87604C2semidirect
3. C2×#15690875C2direct (N × H)
7. C2×#13095521C22semidirect
10. C2×C4×#87604C22semidirect
11. #15690875C22direct (N × H)
14. C2×#13095512C4semidirect
17. C22×D388C4semidirect
20. C22×Q8×C97C4semidirect
24. #13095521C23semidirect
27. C4×#87604C23semidirect
33. #13095512C2×C4semidirect
38. C2×D388C2×C4semidirect
47. C2×Q8×C97C2×C4semidirect
59. Q8×C97C22×C4semidirect
68. D388C22×C4semidirect
77. C22×C97#104semidirect
91. C2×C97C2×#104semidirect
97. C97C22×#104semidirect
106. C22×Q8#87604semidirect
126. C2×Q8C2×#87604semidirect
154. Q8C22×#87604semidirect
180. C22#15690875direct (N × H)
191. C2C2×#15690875direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130955212
C22×C4×#876042
C2×#156908752
C22×#130955122
C22×#10497