Finite Group Databasebeta

← C22×D8×D173 #36368277

Subgroups — C22×D8×D173 #36368277

Order 22144.

1976 conjugacy classes of subgroups; only the 565 normal subgroups are shown.

normal subgroups only

This group has 1976 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×D4×D173C212yesmaximal, complemented
2C22×D1384C211yesmaximal, characteristic, complemented
3C22×#14583811C212yesmaximal, complemented
4C2×D8×D173C2124yesmaximal, direct product factor
5C22×C8×D173C211yesmaximal, characteristic, complemented
6C22×D8×C173C211yesmaximal, characteristic, complemented
7C24×D173—22no—
8C22×D692C2212yescomplemented
9C22×C22⋊D173—22no—
10C2×D4×D173—212no—
11C2×D4×D173C22124yescomplemented
12C22×C4×D173C2211yescharacteristic
13C2×D1384C22112yescomplemented
14C2×#14583811C22124yescomplemented
15D8×D173C22164yesdirect product factor
16C22×#13063436C2211yescharacteristic, complemented
17C2×C8×D173C22112yescomplemented
18C22×D4×C173C2212yescomplemented
19C2×D8×C173C22112yescomplemented
20C22×C8×C173C2211yescharacteristic, complemented
21C23×D173—224no—
22C23×D173—22no—
23C23×D173—416no—
24C23×D173D411yescharacteristic
25C2×D692—26no—
26C2×D692C23112yescomplemented
27C2×C22⋊D173—224no—
28D4×D173—248no—
29D4×D173C23132yescomplemented
30C22×Dic173D411yescharacteristic
31C2×C4×D173D416yes—
32C2×C4×D173C2316yes—
33D1384C23116yescomplemented
34#14583811C23132yescomplemented
35C2×#13063436C2316yescomplemented
36C8×D173C23116yescomplemented
37C24×C173—22no—
38C2×D4×C173—26no—
39C2×D4×C173C23112yescomplemented
40C22×C4×C173C2311yescharacteristic
41D8×C173C23116yescomplemented
42C2×C8×C173C2316yescomplemented
43C22×D173—212no—
44C22×D173—232no—
45C22×D173—48no—
46C22×D173—496no—
47C22×D173D818yesdirect product factor
48C22×D173C2×D416yes—
49D692—212no—
50D692C2418yescomplemented
51C22⋊D173—232no—
52C2×Dic173C2×D416yes—
53C4×D173C2×D4112yes—
54C4×D173C2414yes—
55#13063436C2414yescomplemented
56C23×C173—212no—
57C23×C173—48no—
58C23×C173C2×D411yescharacteristic
59D4×C173—212no—
60D4×C173C2418yescomplemented
61C2×C4×C173C2×D413yes—
62C2×C4×C173C2413yes—
63C8×C173C2414yescomplemented
64C22×C173—28no—
65C22×C173—424no—
66C22×C173C2×D814yescomplemented
67C22×C173C22×D413yes—
68C2×D173—28no—
69C2×D173—424no—
70C2×D173—464no—
71C2×D173C2×D8124yesdirect product factor
72C2×D173C22×D414yes—
73C4×C173C22×D413yes—
74C4×C173C2511yescharacteristic
75Dic173C22×D414yes—
76C2×C173—48no—
77C2×C173C22×D816yescomplemented
78C2×C173C23×D411yescharacteristic
79D173—48no—
80D173C22×D818yesdirect product factor
81C173C23×D811yescharacteristic, complemented, Sylow
82C23×D8—1731nomaximal, Sylow
83C23×D4—1732no—
84C22×D8—1732no—
85C22×D8—17324no—
86C22×D8—1731no—
87C22×D8D17311yescharacteristic, direct product factor
88C23×C8—1731no—
89C25—3462no—
90C22×D4—34612no—
91C22×D4—3462no—
92C22×D4—17324no—
93C22×D4—1732no—
94C22×D4C2×D17312yescomplemented
95C23×C4—1731no—
96C2×D8—17324no—
97C2×D8—17364no—
98C2×D8—17312no—
99C2×D8C2×D173112yesdirect product factor
100C22×C8—1731no—
101C22×C8—17312no—
102C22×C8C2×D17311yescharacteristic, complemented
103C24—22no—
104C24—34624no—
105C24—3462no—
106C24—1731no—
107C24—69216no—
108C2×D4—26no—
109C2×D4—34648no—
110C2×D4—34624no—
111C2×D4—3466no—
112C2×D4—17332no—
113C2×D4—17312no—
114C2×D4C22×D173112yescomplemented
115C22×C4—1731no—
116C22×C4—1736no—
117C22×C4—1736no—
118C22×C4C22×D17311yescharacteristic
119D8—17332no—
120D8—17316no—
121D8C22×D173116yesdirect product factor
122C2×C8—1736no—
123C2×C8—17316no—
124C2×C8C22×D17316yescomplemented
125C23—212no—
126C23—48no—
127C23—34632no—
128C23—34612no—
129C23—1738no—
130C23—1736no—
131C23—69296no—
132C23—6928no—
133C23D4×D17311yescharacteristic
134D4—212no—
135D4—34632no—
136D4—34612no—
137D4—1738no—
138D4C23×D17318yescomplemented
139C2×C4—1736no—
140C2×C4—17312no—
141C2×C4—1734no—
142C2×C4D4×D17313yes—
143C2×C4C23×D17313yes—
144C8—1734no—
145C8C23×D17314yescomplemented
146C22—28no—
147C22—424no—
148C22—3468no—
149C22—17324no—
150C22—1734no—
151C22—69264no—
152C22—69224no—
153C22D8×D17314yesdirect product factor
154C22C2×D4×D17313yes—
155C4—1734no—
156C4C2×D4×D17313yes—
157C4C24×D17311yescharacteristic
158C2—48no—
159C2—1738no—
160C2—6928no—
161C2C2×D8×D17316yesdirect product factor
162C2C22×D4×D17311yescharacteristic
subgroupG/H
centerC23D4×D173
commutator, cosocleC4×C173C25
FrattiniC4C24×D173
FittingC22×D8×C173C2
socleC23×C173C2×D4
hypercenterC22×D8D173
hypocenterC173C23×D8
LCSG ▷ C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C22×D4×D173C2semidirect
2. C22×D1384C2semidirect
3. C22×#14583811C2semidirect
4. C2×D8×D173C2direct (N × H)
5. C22×C8×D173C2semidirect
6. C22×D8×C173C2semidirect
8. C22×D692C22semidirect
11. C2×D4×D173C22semidirect
13. C2×D1384C22semidirect
14. C2×#14583811C22semidirect
15. D8×D173C22direct (N × H)
16. C22×#13063436C22semidirect
17. C2×C8×D173C22semidirect
18. C22×D4×C173C22semidirect
19. C2×D8×C173C22semidirect
20. C22×C8×C173C22semidirect
26. C2×D692C23semidirect
29. D4×D173C23semidirect
33. D1384C23semidirect
34. #14583811C23semidirect
35. C2×#13063436C23semidirect
36. C8×D173C23semidirect
39. C2×D4×C173C23semidirect
41. D8×C173C23semidirect
42. C2×C8×C173C23semidirect
47. C22×D173D8direct (N × H)
50. D692C24semidirect
55. #13063436C24semidirect
60. D4×C173C24semidirect
63. C8×C173C24semidirect
66. C22×C173C2×D8semidirect
71. C2×D173C2×D8direct (N × H)
77. C2×C173C22×D8semidirect
80. D173C22×D8direct (N × H)
81. C173C23×D8semidirect
87. C22×D8D173direct (N × H)
94. C22×D4C2×D173semidirect
99. C2×D8C2×D173direct (N × H)
102. C22×C8C2×D173semidirect
114. C2×D4C22×D173semidirect
121. D8C22×D173direct (N × H)
124. C2×C8C22×D173semidirect
138. D4C23×D173semidirect
145. C8C23×D173semidirect
153. C22D8×D173direct (N × H)
161. C2C2×D8×D173direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D4×D1732
C22×D13842
C22×#145838112
C2×D8×D1732
C22×C8×D1732
C22×D8×C1732
C23×D8173