Finite Group Databasebeta

← C2×#26012053 #36366866

Subgroups — C2×#26012053 #36366866

Order 22144.

1984 conjugacy classes of subgroups; only the 565 normal subgroups are shown.

normal subgroups only

This group has 1984 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
1C23×C22⋊D173C212yesmaximal, complemented
2#26012053C2116yesmaximal, direct product factor
3C2×#19149663C214yesmaximal, complemented
4C2×#19149653C214yesmaximal, complemented
5C2×#19149649C214yesmaximal, complemented
6C22×C22⋊C4×C173C211yesmaximal, characteristic, complemented
7C24×D173C2212yescomplemented
8C22×D692C2214yescomplemented
9C22×C22⋊D173—216no—
10C22×C22⋊D173C2214yes—
11C22×C22⋊D173C2214yescomplemented
12C22×C22⋊D173C22116yescomplemented
13C23×Dic173C2212yescomplemented
14#19149663C22132yescomplemented
15#19149653C22132yescomplemented
16#19149649C22132yescomplemented
17C2×C4⋊Dic173C2214yescomplemented
18C2×#14583803C2218yescomplemented
19C25×C173C2211yescharacteristic
20C23×C4×C173C2212yescomplemented
21C2×C22⋊C4×C173C2214yescomplemented
22C2×C22⋊C4×C173C2218yescomplemented
23C23×D173—28no—
24C23×D173—48no—
25C23×D173C2314yes—
26C23×D173C2318yescomplemented
27C2×D692—48no—
28C2×D692C23116yescomplemented
29C2×C22⋊D173—216no—
30C2×C22⋊D173—216no—
31C2×C22⋊D173—264no—
32C2×C22⋊D173C23116yes—
33C2×C22⋊D173C23116yescomplemented
34C22×Dic173—28no—
35C22×Dic173C2314yes—
36C22×Dic173C2318yescomplemented
37C4⋊Dic173C23116yescomplemented
38#14583803C23132yescomplemented
39C24×C173—44no—
40C24×C173D418yescomplemented
41C24×C173C2311yescharacteristic
42C24×C173C2312yes—
43C24×C173C2314yes—
44C22×C4×C173—44no—
45C22×C4×C173C2314yes—
46C22×C4×C173C2318yescomplemented
47C22⋊C4×C173C23116yescomplemented
48C22×D173—28no—
49C22×D173—216no—
50C22×D173—216no—
51C22×D173—416no—
52C22×D173—432no—
53C22×D173C2418yes—
54D692—416no—
55C22⋊D173—232no—
56C22⋊D173—232no—
57C2×Dic173—28no—
58C2×Dic173—216no—
59C2×Dic173C2418yes—
60C23×C173—28no—
61C23×C173—24no—
62C23×C173—416no—
63C23×C173—48no—
64C23×C173C2×D418yes—
65C23×C173C2×D414yes—
66C23×C173C2×D4116yescomplemented
67C23×C173C2414yes—
68C23×C173C2412yes—
69C23×C173C2411yescharacteristic
70C2×C4×C173—28no—
71C2×C4×C173—48no—
72C2×C4×C173C2418yes—
73C22×C173—28no—
74C22×C173—216no—
75C22×C173—24no—
76C22×C173—28no—
77C22×C173—416no—
78C22×C173C22×D418yes—
79C22×C173C22×D414yes—
80C22×C173C22×D412yes—
81C22×C173H(5,𝔽2)14yes—
82C22×C173C2511yescharacteristic
83C2×D173—28no—
84C2×D173—216no—
85C2×D173—432no—
86C2×D173—416no—
87C4×C173—28no—
88Dic173—28no—
89C2×C173—28no—
90C2×C173—24no—
91C2×C173#53414yescomplemented
92C2×C173C23×D411yescharacteristic
93C2×C173C2×H(5,𝔽2)12yes—
94D173—48no—
95C173C2×#53411yescharacteristic, complemented, Sylow
96C2×#534—1731nomaximal, Sylow
97C23×D4—1732no—
98#534—17316no—
99C2×#123—1734no—
100C2×#121—1734no—
101C2×#120—1734no—
102C22×C22⋊C4D17311yescharacteristic, complemented
103C25—1732no—
104C25C2×D17311yescharacteristic
105C22×D4—34616no—
106C22×D4—1734no—
107C22×D4—1734no—
108C22×D4—17316no—
109C22×D4—1734no—
110C23×C4—1732no—
111C23×C4C2×D17312yescomplemented
112#123—17332no—
113#121—17332no—
114#120—17332no—
115C2×C4⋊C4—1734no—
116C2×C22⋊C4—1738no—
117C2×C22⋊C4C2×D17314yescomplemented
118C2×C22⋊C4C2×D17318yescomplemented
119C24—44no—
120C24—3468no—
121C24—1734no—
122C24—1738no—
123C24—6928no—
124C24D69218yescomplemented
125C24C22×D17312yes—
126C24C22×D17311yescharacteristic
127C24C22×D17314yes—
128C2×D4—34616no—
129C2×D4—34616no—
130C2×D4—34664no—
131C2×D4—17316no—
132C2×D4—17316no—
133C2×D4—17316no—
134C2×D4—6928no—
135C22×C4—44no—
136C22×C4—3468no—
137C22×C4—1738no—
138C22×C4—1734no—
139C22×C4C22×D17314yes—
140C22×C4C22×D17318yescomplemented
141C4⋊C4—17316no—
142C22⋊C4—17332no—
143C22⋊C4C22×D173116yescomplemented
144C23—24no—
145C23—28no—
146C23—416no—
147C23—48no—
148C23—34616no—
149C23—3468no—
150C23—34616no—
151C23—1738no—
152C23—69216no—
153C23—69232no—
154C23C2×D69218yes—
155C23C2×D69214yes—
156C23C2×D692116yescomplemented
157C23C23×D17314yes—
158C23C23×D17312yes—
159C23C23×D17311yescharacteristic
160D4—34632no—
161D4—34632no—
162D4—69216no—
163C2×C4—28no—
164C2×C4—48no—
165C2×C4—34616no—
166C2×C4—3468no—
167C2×C4—1738no—
168C2×C4C23×D17318yes—
169C22—216no—
170C22—28no—
171C22—24no—
172C22—28no—
173C22—416no—
174C22—3468no—
175C22—34616no—
176C22—69232no—
177C22—69216no—
178C22C22×D69218yes—
179C22C22×D69214yes—
180C22C22×D69212yes—
181C22#1914972414yes—
182C22C24×D17311yescharacteristic
183C4—28no—
184C4—3468no—
185C2—28no—
186C2—24no—
187C2—6928no—
188C2#2601205314yesdirect product factor
189C2C23×D69211yescharacteristic
190C2C2×#1914972412yes—
subgroupG/H
centerC23C23×D173
commutator, cosocleC22×C173C25
FrattiniC22C24×D173
FittingC22×C22⋊C4×C173C2
socleC23×C173C24
hypercenterC22×C22⋊C4D173
hypocenterC173C2×#534
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C23×C22⋊D173C2semidirect
2. #26012053C2direct (N × H)
3. C2×#19149663C2semidirect
4. C2×#19149653C2semidirect
5. C2×#19149649C2semidirect
6. C22×C22⋊C4×C173C2semidirect
7. C24×D173C22semidirect
8. C22×D692C22semidirect
11. C22×C22⋊D173C22semidirect
12. C22×C22⋊D173C22semidirect
13. C23×Dic173C22semidirect
14. #19149663C22semidirect
15. #19149653C22semidirect
16. #19149649C22semidirect
17. C2×C4⋊Dic173C22semidirect
18. C2×#14583803C22semidirect
20. C23×C4×C173C22semidirect
21. C2×C22⋊C4×C173C22semidirect
22. C2×C22⋊C4×C173C22semidirect
26. C23×D173C23semidirect
28. C2×D692C23semidirect
33. C2×C22⋊D173C23semidirect
36. C22×Dic173C23semidirect
37. C4⋊Dic173C23semidirect
38. #14583803C23semidirect
40. C24×C173D4semidirect
46. C22×C4×C173C23semidirect
47. C22⋊C4×C173C23semidirect
66. C23×C173C2×D4semidirect
91. C2×C173#534semidirect
95. C173C2×#534semidirect
102. C22×C22⋊C4D173semidirect
111. C23×C4C2×D173semidirect
117. C2×C22⋊C4C2×D173semidirect
118. C2×C22⋊C4C2×D173semidirect
124. C24D692semidirect
140. C22×C4C22×D173semidirect
143. C22⋊C4C22×D173semidirect
156. C23C2×D692semidirect
188. C2#26012053direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C22⋊D1732
#260120532
C2×#191496632
C2×#191496532
C2×#191496492
C22×C22⋊C4×C1732
C2×#534173