Finite Group Databasebeta

← C2×#26012232 #36367681

Subgroups — C2×#26012232 #36367681

Order 22144.

1344 conjugacy classes of subgroups; only the 469 normal subgroups are shown.

normal subgroups only

This group has 1344 conjugacy classes of subgroups — too many to draw. Only its 469 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
1#26012232C2116yesmaximal, direct product factor
2C2×#19149676C211yesmaximal, characteristic, complemented
3C2×#19149673C211yesmaximal, characteristic, complemented
4C2×#19149664C213yesmaximal, complemented
5C2×#19149654C213yesmaximal, complemented
6C2×#19149652C213yesmaximal, complemented
7C2×#19149649C213yesmaximal, complemented
8C2×#126×C173C211yesmaximal, characteristic, complemented
9C24×D173C2211yescharacteristic
10C22×D692C2213yes—
11C22×D692C2213yes—
12C22×D692C2213yescomplemented
13C22×C22⋊D173C2213yes—
14C22×C4×D173C2213yescomplemented
15#19149676C2218yescomplemented
16#19149673C2218yescomplemented
17#19149664C22124yescomplemented
18#19149654C22124yescomplemented
19#19149652C22124yescomplemented
20#19149649C22124yescomplemented
21C2×#14583805C2213yescomplemented
22C2×#14583803C2213yes—
23C2×#14583803C2213yes—
24C2×#14583803C2213yescomplemented
25#126×C173C2218yescomplemented
26C2×C4⋊C4×C173C2213yescomplemented
27C2×C22⋊C4×C173C2213yes—
28C2×C42×C173C2211yescharacteristic, complemented
29C23×D173—46no—
30C23×D173C2311yescharacteristic
31C23×D173C2311yescharacteristic
32C23×D173C2313yes—
33C23×D173C2314yes—
34C2×D692—46no—
35C2×D692—46no—
36C2×D692—46no—
37C2×D692C23112yes—
38C2×D692C23112yes—
39C2×D692C23112yescomplemented
40C2×C22⋊D173—46no—
41C2×C22⋊D173C23112yes—
42C22×Dic173C2313yes—
43C2×C4×D173—46no—
44C2×C4×D173C23112yescomplemented
45#14583805C23112yescomplemented
46#14583803C23112yes—
47#14583803C23112yes—
48#14583803C23112yescomplemented
49C24×C173C2311yescharacteristic
50C22×C4×C173C2313yes—
51C22×C4×C173C2313yes—
52C4⋊C4×C173C23112yescomplemented
53C22⋊C4×C173C23112yes—
54C42×C173C2314yescomplemented
55C22×D173—26no—
56C22×D173—26no—
57C22×D173—26no—
58C22×D173—26no—
59C22×D173—26no—
60C22×D173—412no—
61C22×D173—416no—
62C22×D173C2416yes—
63C22×D173C2412yes—
64C22×D173C2412yes—
65D692—412no—
66D692—412no—
67D692—412no—
68C22⋊D173—412no—
69C2×Dic173—26no—
70C2×Dic173C2416yes—
71C4×D173—412no—
72C23×C173—26no—
73C23×C173C2412yes—
74C23×C173C2411yescharacteristic
75C2×C4×C173—26no—
76C2×C4×C173—26no—
77C2×C4×C173C2416yes—
78C2×C4×C173C2416yes—
79C22×C173—26no—
80C22×C173—44no—
81C22×C173H(5,𝔽2)16yes—
82C22×C173C2511yescharacteristic
83C2×D173—26no—
84C2×D173—26no—
85C2×D173—26no—
86C2×D173—26no—
87C2×D173—26no—
88C2×D173—416no—
89C2×D173—412no—
90C2×D173—44no—
91C2×D173—44no—
92C4×C173—26no—
93C4×C173—26no—
94Dic173—26no—
95C2×C173—42no—
96C2×C173#56014yescomplemented
97C2×C173C2×H(5,𝔽2)13yes—
98D173—46no—
99D173—42no—
100D173—42no—
101C173C2×#56011yescharacteristic, complemented, Sylow
102C2×#560—1731nomaximal, Sylow
103#560—17316no—
104C2×#127—1731no—
105C2×#126—1731no—
106C2×#126D17311yescharacteristic, complemented
107C2×#123—1733no—
108C2×#121—1733no—
109C2×#121—1733no—
110C2×#120—1733no—
111C25—1731no—
112C22×D4—1733no—
113C22×D4—1733no—
114C22×D4—1733no—
115C22×D4—1733no—
116C23×C4—1733no—
117#127—1738no—
118#126—1738no—
119#126C2×D17318yescomplemented
120#123—17324no—
121#121—17324no—
122#121—17324no—
123#120—17324no—
124C2×C4⋊C4—1733no—
125C2×C4⋊C4C2×D17313yescomplemented
126C2×C22⋊C4—1733no—
127C2×C22⋊C4—1733no—
128C2×C22⋊C4—1733no—
129C2×C22⋊C4C2×D17313yes—
130C2×C42C2×D17311yescharacteristic, complemented
131C24—1731no—
132C24—1731no—
133C24—1733no—
134C24—1734no—
135C24—6926no—
136C24C22×D17311yescharacteristic
137C2×D4—17312no—
138C2×D4—17312no—
139C2×D4—17312no—
140C2×D4—17312no—
141C2×D4—6926no—
142C2×D4—6926no—
143C2×D4—6926no—
144C2×D4—6926no—
145C22×C4—1733no—
146C22×C4—17312no—
147C22×C4—6926no—
148C22×C4C22×D17313yes—
149C22×C4C22×D17313yes—
150C4⋊C4—17312no—
151C4⋊C4C22×D173112yescomplemented
152C22⋊C4—17312no—
153C22⋊C4—17312no—
154C22⋊C4—17312no—
155C22⋊C4C22×D173112yes—
156C42C22×D17314yescomplemented
157C23—26no—
158C23—3466no—
159C23—3466no—
160C23—3466no—
161C23—3466no—
162C23—3466no—
163C23—1732no—
164C23—1732no—
165C23—1736no—
166C23—69212no—
167C23—69216no—
168C23C23×D17312yes—
169C23C23×D17311yescharacteristic
170D4—69212no—
171D4—69212no—
172D4—69212no—
173D4—69212no—
174C2×C4—26no—
175C2×C4—26no—
176C2×C4—3466no—
177C2×C4—1736no—
178C2×C4—69212no—
179C2×C4C23×D17316yes—
180C2×C4C23×D17316yes—
181C22—26no—
182C22—44no—
183C22—3466no—
184C22—3466no—
185C22—3466no—
186C22—3466no—
187C22—3466no—
188C22—69216no—
189C22—6924no—
190C22—6924no—
191C22—69212no—
192C22#1914972516yes—
193C22C24×D17311yescharacteristic
194C4—26no—
195C4—26no—
196C4—3466no—
197C2—42no—
198C2—6926no—
199C2—6922no—
200C2—6922no—
201C2#2601223214yesdirect product factor
202C2C2×#1914972513yes—
subgroupG/H
centerC23C23×D173
commutator, cosocleC22×C173C25
FrattiniC22C24×D173
FittingC2×#126×C173C2
socleC23×C173C24
hypercenterC2×#126D173
hypocenterC173C2×#560
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. #26012232C2direct (N × H)
2. C2×#19149676C2semidirect
3. C2×#19149673C2semidirect
4. C2×#19149664C2semidirect
5. C2×#19149654C2semidirect
6. C2×#19149652C2semidirect
7. C2×#19149649C2semidirect
8. C2×#126×C173C2semidirect
12. C22×D692C22semidirect
14. C22×C4×D173C22semidirect
15. #19149676C22semidirect
16. #19149673C22semidirect
17. #19149664C22semidirect
18. #19149654C22semidirect
19. #19149652C22semidirect
20. #19149649C22semidirect
21. C2×#14583805C22semidirect
24. C2×#14583803C22semidirect
25. #126×C173C22semidirect
26. C2×C4⋊C4×C173C22semidirect
28. C2×C42×C173C22semidirect
39. C2×D692C23semidirect
44. C2×C4×D173C23semidirect
45. #14583805C23semidirect
48. #14583803C23semidirect
52. C4⋊C4×C173C23semidirect
54. C42×C173C23semidirect
96. C2×C173#560semidirect
101. C173C2×#560semidirect
106. C2×#126D173semidirect
119. #126C2×D173semidirect
125. C2×C4⋊C4C2×D173semidirect
130. C2×C42C2×D173semidirect
151. C4⋊C4C22×D173semidirect
156. C42C22×D173semidirect
201. C2#26012232direct (N × H)

Maximal subgroups

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

subgroupindex
#260122322
C2×#191496762
C2×#191496732
C2×#191496642
C2×#191496542
C2×#191496522
C2×#191496492
C2×#126×C1732
C2×#560173