Finite Group Databasebeta

← C22×#19153842 #37548454

Subgroups — C22×#19153842 #37548454

Order 23168.

1720 conjugacy classes of subgroups; only the 565 normal subgroups are shown.

normal subgroups only

This group has 1720 conjugacy classes of subgroups — too many to draw. Only its 565 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×#15668449C211yesmaximal, characteristic, complemented
2C2×#19153842C2124yesmaximal, direct product factor
3C22×#15668437C212yesmaximal, complemented
4C22×#15668434C212yesmaximal, complemented
5C22×#15668433C211yesmaximal, characteristic, complemented
6C23×D4×C181C211yesmaximal, characteristic, complemented
7C2×#15668449—26no—
8C2×#15668449C22112yescomplemented
9C22×Dic362C2211yescharacteristic, complemented
10C22×D724C2211yescharacteristic, complemented
11C22×C22⋊D181—21no—
12C22×C4×D181—21no—
13#19153842C22164yesdirect product factor
14C2×#15668437C22124yescomplemented
15C2×#15668434C22124yescomplemented
16C2×#15668433C22112yescomplemented
17C22×#13089678C2212yescomplemented
18C25×C181—21no—
19C22×D4×C181—21no—
20C22×D4×C181—26no—
21C22×D4×C181C2212yes—
22C22×D4×C181C22112yescomplemented
23C23×C4×C181C2211yescharacteristic
24C23×D181—21no—
25#15668449—224no—
26#15668449C23116yescomplemented
27C2×Dic362—23no—
28C2×Dic362C2316yescomplemented
29C2×D724—23no—
30C2×D724C2316yescomplemented
31C2×C22⋊D181—212no—
32C22×Dic181—21no—
33C2×C4×D181—212no—
34#15668437C23132yescomplemented
35#15668434C23132yescomplemented
36#15668433C23116yescomplemented
37C2×#13089678C23112yescomplemented
38C24×C181—22no—
39C24×C181—212no—
40C24×C181—48no—
41C24×C181D411yescharacteristic
42C2×D4×C181—26no—
43C2×D4×C181—212no—
44C2×D4×C181—224no—
45C2×D4×C181C23112yes—
46C2×D4×C181C23116yescomplemented
47C22×C4×C181D411yescharacteristic
48C22×C4×C181D416yes—
49C22×C4×C181C2311yescharacteristic
50C22×C4×C181C2316yes—
51C22×D181—26no—
52C22×D181—44no—
53Dic362—26no—
54Dic362C2414yescomplemented
55D724—26no—
56D724C2414yescomplemented
57C22⋊D181—216no—
58C2×Dic181—26no—
59C4×D181—216no—
60#13089678C2418yescomplemented
61C23×C181—24no—
62C23×C181—212no—
63C23×C181—216no—
64C23×C181—448no—
65C23×C181—48no—
66C23×C181C2×D416yes—
67C23×C181C2×D411yescharacteristic
68D4×C181—212no—
69D4×C181—216no—
70D4×C181C2418yes—
71C2×C4×C181C2×D416yes—
72C2×C4×C181C2×D413yes—
73C2×C4×C181C2×D4112yes—
74C2×C4×C181C2413yes—
75C2×C4×C181C2414yes—
76C22×C181—212no—
77C22×C181—28no—
78C22×C181—432no—
79C22×C181—424no—
80C22×C181AGL(1,ℤ/8)14yescomplemented
81C22×C181C22×D414yes—
82C22×C181C22×D413yes—
83C2×D181—24no—
84C2×D181—412no—
85C4×C181C22×D414yes—
86C4×C181C22×D413yes—
87C4×C181C2511yescharacteristic
88Dic181—24no—
89C2×C181—24no—
90C2×C181—48no—
91C2×C181C2×AGL(1,ℤ/8)16yescomplemented
92C2×C181C23×D411yescharacteristic
93D181—44no—
94C181C22×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
95C22×AGL(1,ℤ/8)—1811nomaximal, Sylow
96C22×#41—1811no—
97C23×D4D18111yescharacteristic, complemented
98C2×AGL(1,ℤ/8)—18124no—
99C22×SD16—1812no—
100C22×D8—1812no—
101C22×M16—1811no—
102C25—21no—
103C2×#41—3626no—
104C2×#41—18112no—
105C22×Q8—1811no—
106C22×D4—26no—
107C22×D4—21no—
108C22×D4—3621no—
109C22×D4—1811no—
110C22×D4C2×D18112yes—
111C22×D4C2×D181112yescomplemented
112C23×C4—3621no—
113C23×C4C2×D18111yescharacteristic
114AGL(1,ℤ/8)—18164no—
115C2×SD16—18124no—
116C2×D8—18124no—
117C2×M16—18112no—
118C22×C8—1812no—
119C24—212no—
120C24—22no—
121C24—48no—
122C24—3621no—
123C24C22⋊D18111yescharacteristic
124#41—36224no—
125#41—18116no—
126C2×Q8—3623no—
127C2×Q8—1816no—
128C2×D4—26no—
129C2×D4—212no—
130C2×D4—224no—
131C2×D4—36212no—
132C2×D4—3623no—
133C2×D4—1816no—
134C2×D4C22×D181112yes—
135C2×D4C22×D181116yescomplemented
136C22×C4—3621no—
137C22×C4—36212no—
138C22×C4C22⋊D18116yes—
139C22×C4C22⋊D18111yescharacteristic
140C22×C4C22×D18116yes—
141C22×C4C22×D18111yescharacteristic
142SD16—18132no—
143D8—18132no—
144M16—18116no—
145C2×C8—18112no—
146C23—24no—
147C23—212no—
148C23—216no—
149C23—448no—
150C23—48no—
151C23—3626no—
152C23—7244no—
153C23C2×C22⋊D18116yes—
154C23C2×C22⋊D18111yescharacteristic
155Q8—3626no—
156Q8—1814no—
157D4—212no—
158D4—216no—
159D4—36216no—
160D4—3626no—
161D4—1814no—
162D4C23×D18118yes—
163C2×C4—3626no—
164C2×C4—36216no—
165C2×C4C2×C22⋊D18116yes—
166C2×C4C2×C22⋊D18113yes—
167C2×C4C2×C22⋊D181112yes—
168C2×C4C23×D18113yes—
169C2×C4C23×D18114yes—
170C8—1818no—
171C22—212no—
172C22—28no—
173C22—432no—
174C22—424no—
175C22—3624no—
176C22—72412no—
177C22#1915384214yesdirect product factor
178C22C22×C22⋊D18114yes—
179C22C22×C22⋊D18113yes—
180C4—3624no—
181C4C22×C22⋊D18114yes—
182C4C22×C22⋊D18113yes—
183C4C24×D18111yescharacteristic
184C2—24no—
185C2—48no—
186C2—7244no—
187C2C2×#1915384216yesdirect product factor
188C2C23×C22⋊D18111yescharacteristic
subgroupG/H
centerC23C2×C22⋊D181
commutator, cosocleC4×C181C25
FrattiniC4C24×D181
FittingC23×D4×C181C2
socleC23×C181C2×D4
hypercenterC23×D4D181
hypocenterC181C22×AGL(1,ℤ/8)
LCSG ▷ C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C22×#15668449C2semidirect
2. C2×#19153842C2direct (N × H)
3. C22×#15668437C2semidirect
4. C22×#15668434C2semidirect
5. C22×#15668433C2semidirect
6. C23×D4×C181C2semidirect
8. C2×#15668449C22semidirect
9. C22×Dic362C22semidirect
10. C22×D724C22semidirect
13. #19153842C22direct (N × H)
14. C2×#15668437C22semidirect
15. C2×#15668434C22semidirect
16. C2×#15668433C22semidirect
17. C22×#13089678C22semidirect
22. C22×D4×C181C22semidirect
26. #15668449C23semidirect
28. C2×Dic362C23semidirect
30. C2×D724C23semidirect
34. #15668437C23semidirect
35. #15668434C23semidirect
36. #15668433C23semidirect
37. C2×#13089678C23semidirect
46. C2×D4×C181C23semidirect
54. Dic362C24semidirect
56. D724C24semidirect
60. #13089678C24semidirect
80. C22×C181AGL(1,ℤ/8)semidirect
91. C2×C181C2×AGL(1,ℤ/8)semidirect
94. C181C22×AGL(1,ℤ/8)semidirect
97. C23×D4D181semidirect
111. C22×D4C2×D181semidirect
135. C2×D4C22×D181semidirect
177. C22#19153842direct (N × H)
187. C2C2×#19153842direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#156684492
C2×#191538422
C22×#156684372
C22×#156684342
C22×#156684332
C23×D4×C1812
C22×AGL(1,ℤ/8)181