Finite Group Databasebeta

← C22×C4⋊C4×D173 #36366722

Subgroups — C22×C4⋊C4×D173 #36366722

Order 22144.

2488 conjugacy classes of subgroups; only the 1157 normal subgroups are shown.

normal subgroups only

This group has 2488 conjugacy classes of subgroups — too many to draw. Only its 1157 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×C4×D173C212yesmaximal
2C23×C4×D173C211yesmaximal, characteristic
3C22×C4⋊Dic173C211yesmaximal, characteristic, complemented
4C22×#14583805C212yesmaximal, complemented
5C2×C4⋊C4×D173C2124yesmaximal, direct product factor
6C22×C4⋊C4×C173C211yesmaximal, characteristic, complemented
7C24×D173C2211yescharacteristic
8C23×Dic173C2212yes—
9C23×Dic173C2211yescharacteristic
10C22×C4×D173—216no—
11C22×C4×D173C4116yescomplemented
12C22×C4×D173C22124yes—
13C22×C4×D173C22112yes—
14C2×C4⋊Dic173C22112yescomplemented
15C2×#14583805C22124yescomplemented
16C4⋊C4×D173C22164yesdirect product factor
17C23×C4×C173C2212yes—
18C23×C4×C173C2211yescharacteristic
19C2×C4⋊C4×C173C22112yescomplemented
20C23×D173C2×C418yes—
21C23×D173D418yes—
22C23×D173Q818yes—
23C23×D173C2316yes—
24C22×Dic173—28no—
25C22×Dic173C2×C418yescomplemented
26C22×Dic173C23112yes—
27C22×Dic173C2316yes—
28C2×C4×D173—296no—
29C2×C4×D173C2×C4196yescomplemented
30C2×C4×D173C23132yes—
31C2×C4×D173C23116yes—
32C4⋊Dic173C23116yescomplemented
33#14583805C23132yescomplemented
34C24×C173C2311yescharacteristic
35C22×C4×C173—28no—
36C22×C4×C173C2×C418yescomplemented
37C22×C4×C173C23112yes—
38C22×C4×C173C2316yes—
39C4⋊C4×C173C23116yescomplemented
40C22×D173C4⋊C4164yesdirect product factor
41C22×D173C22×C4124yes—
42C22×D173C2×D4124yes—
43C22×D173C2×Q8124yes—
44C22×D173C2414yes—
45C2×Dic173—224no—
46C2×Dic173C22×C4124yescomplemented
47C2×Dic173C2418yes—
48C2×Dic173C2414yes—
49C4×D173—264no—
50C4×D173C22×C4164yescomplemented
51C23×C173C22×C414yes—
52C23×C173C2×D414yes—
53C23×C173C2×Q814yes—
54C23×C173C2413yes—
55C2×C4×C173—224no—
56C2×C4×C173C22×C4124yescomplemented
57C2×C4×C173C2418yes—
58C2×C4×C173C2414yes—
59C22×C173C2×C4⋊C4116yescomplemented
60C22×C173C23×C416yes—
61C22×C173C22×D416yes—
62C22×C173C22×Q816yes—
63C22×C173C2511yescharacteristic
64C2×D173C2×C4⋊C4196yesdirect product factor
65C2×D173C23×C418yes—
66C2×D173C22×D418yes—
67C2×D173C22×Q818yes—
68C4×C173—28no—
69C4×C173C23×C418yescomplemented
70Dic173—28no—
71Dic173C23×C418yescomplemented
72C2×C173C22×C4⋊C4112yescomplemented
73C2×C173C24×C411yescharacteristic
74C2×C173C23×D411yescharacteristic
75C2×C173C23×Q811yescharacteristic
76D173C22×C4⋊C4116yesdirect product factor
77C173C23×C4⋊C411yescharacteristic, complemented, Sylow
78C23×C4⋊C4—1731nomaximal, Sylow
79C24×C4—1732no—
80C24×C4—1731no—
81C22×C4⋊C4—1732no—
82C22×C4⋊C4—1731no—
83C22×C4⋊C4—17324no—
84C22×C4⋊C4D17311yescharacteristic, direct product factor
85C25—1731no—
86C23×C4—34616no—
87C23×C4—1732no—
88C23×C4—17324no—
89C23×C4—1731no—
90C23×C4—17316no—
91C23×C4—17312no—
92C23×C4C2×D17312yes—
93C23×C4C2×D17311yescharacteristic
94C2×C4⋊C4—17324no—
95C2×C4⋊C4—17312no—
96C2×C4⋊C4—17364no—
97C2×C4⋊C4C2×D173112yesdirect product factor
98C24—1738no—
99C24—1738no—
100C24—1738no—
101C24—1736no—
102C24C22×D17311yescharacteristic
103C22×C4—28no—
104C22×C4—34696no—
105C22×C4—3468no—
106C22×C4—17312no—
107C22×C4—17332no—
108C22×C4—1738no—
109C22×C4—1736no—
110C22×C4—17396no—
111C22×C4—17316no—
112C22×C4C4×D17318yescomplemented
113C22×C4C22×D173112yes—
114C22×C4C22×D17316yes—
115C4⋊C4—17332no—
116C4⋊C4—17316no—
117C4⋊C4C22×D173116yesdirect product factor
118C23—17364no—
119C23—17324no—
120C23—17324no—
121C23—17324no—
122C23—1734no—
123C23C2×C4×D17314yes—
124C23D4×D17314yes—
125C23Q8×D17314yes—
126C23C23×D17313yes—
127C2×C4—224no—
128C2×C4—34664no—
129C2×C4—34624no—
130C2×C4—1738no—
131C2×C4—17324no—
132C2×C4—1734no—
133C2×C4—17364no—
134C2×C4C2×C4×D173124yescomplemented
135C2×C4C23×D17318yes—
136C2×C4C23×D17314yes—
137C22—17396no—
138C22—1738no—
139C22—1738no—
140C22—1738no—
141C22C4⋊C4×D173116yesdirect product factor
142C22C22×C4×D17316yes—
143C22C2×D4×D17316yes—
144C22C2×Q8×D17316yes—
145C22C24×D17311yescharacteristic
146C4—28no—
147C4—3468no—
148C4—1738no—
149C4C22×C4×D17318yescomplemented
150C2—17316no—
151C2C2×C4⋊C4×D173112yesdirect product factor
152C2C23×C4×D17311yescharacteristic
153C2C22×D4×D17311yescharacteristic
154C2C22×Q8×D17311yescharacteristic
subgroupG/H
centerC24C22×D173
commutatorC2×C173C24×C4
FrattiniC22C24×D173
FittingC22×C4⋊C4×C173C2
socleC24×C173C23
prime radicalC24×D173C22
cosocleC22×C173C25
hypercenterC22×C4⋊C4D173
hypocenterC173C23×C4⋊C4
LCSG ▷ C2×C173 ▷ C173
UCSC1 ◁ C24 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C22×C4⋊Dic173C2semidirect
4. C22×#14583805C2semidirect
5. C2×C4⋊C4×D173C2direct (N × H)
6. C22×C4⋊C4×C173C2semidirect
11. C22×C4×D173C4semidirect
14. C2×C4⋊Dic173C22semidirect
15. C2×#14583805C22semidirect
16. C4⋊C4×D173C22direct (N × H)
19. C2×C4⋊C4×C173C22semidirect
25. C22×Dic173C2×C4semidirect
29. C2×C4×D173C2×C4semidirect
32. C4⋊Dic173C23semidirect
33. #14583805C23semidirect
36. C22×C4×C173C2×C4semidirect
39. C4⋊C4×C173C23semidirect
40. C22×D173C4⋊C4direct (N × H)
46. C2×Dic173C22×C4semidirect
50. C4×D173C22×C4semidirect
56. C2×C4×C173C22×C4semidirect
59. C22×C173C2×C4⋊C4semidirect
64. C2×D173C2×C4⋊C4direct (N × H)
69. C4×C173C23×C4semidirect
71. Dic173C23×C4semidirect
72. C2×C173C22×C4⋊C4semidirect
76. D173C22×C4⋊C4direct (N × H)
77. C173C23×C4⋊C4semidirect
84. C22×C4⋊C4D173direct (N × H)
97. C2×C4⋊C4C2×D173direct (N × H)
112. C22×C4C4×D173semidirect
117. C4⋊C4C22×D173direct (N × H)
134. C2×C4C2×C4×D173semidirect
141. C22C4⋊C4×D173direct (N × H)
149. C4C22×C4×D173semidirect
151. C2C2×C4⋊C4×D173direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C4×D1732
C22×C4⋊Dic1732
C22×#145838052
C2×C4⋊C4×D1732
C22×C4⋊C4×C1732
C23×C4⋊C4173