Finite Group Databasebeta

← C2×#26011733 #36363138

Subgroups — C2×#26011733 #36363138

Order 22144.

512 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 165 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×Dic692C211yesmaximal, characteristic, complemented
2C22×C8×D173C211yesmaximal, characteristic
3C2×#19149659C212yesmaximal
4#26011733C218yesmaximal, direct product factor
5C2×#19149602C212yesmaximal, complemented
6C2×#107×C173C211yesmaximal, characteristic, complemented
7C22×Dic346C2212yes—
8C22×C4×D173C2211yescharacteristic
9C2×Dic692—24no—
10C2×Dic692C2214yescomplemented
11C22×#13063436C2211yescharacteristic
12C2×C8×D173—24no—
13C2×C8×D173C2214yes—
14#19149659—24no—
15#19149659C2218yes—
16C2×C4⋊Dic173—22no—
17C2×#14583803—22no—
18#19149602C2218yescomplemented
19C22×C8×C173C2211yescharacteristic
20C2×C4⋊C4×C173C2212yes—
21#107×C173C2214yescomplemented
22C23×D173D411yescharacteristic
23C2×Dic346—22no—
24C2×Dic346—24no—
25C2×Dic346—24no—
26C2×Dic346C2314yes—
27C22×Dic173—22no—
28C22×Dic173D411yescharacteristic
29C2×C4×D173—22no—
30C2×C4×D173—22no—
31C2×C4×D173D412yes—
32C2×C4×D173C2312yes—
33Dic692—28no—
34C2×#13063436—22no—
35C2×#13063436C2312yes—
36C8×D173—28no—
37C4⋊Dic173—28no—
38#14583803—28no—
39C22×C4×C173—22no—
40C22×C4×C173C2311yescharacteristic
41C2×C8×C173D414yes—
42C2×C8×C173C2312yes—
43C4⋊C4×C173—22no—
44C4⋊C4×C173C2314yes—
45C22×D173—22no—
46C22×D173—22no—
47C22×D173Q1614yescomplemented
48C22×D173C2×D412yes—
49Dic346—24no—
50Dic346—24no—
51Dic346—44no—
52C2×Dic173—22no—
53C2×Dic173—24no—
54C2×Dic173—44no—
55C2×Dic173C2×D412yes—
56C4×D173—24no—
57C4×D173—22no—
58C4×D173—22no—
59#13063436—22no—
60C23×C173C2×D411yescharacteristic
61C2×C4×C173—24no—
62C2×C4×C173—44no—
63C2×C4×C173C2×D411yescharacteristic
64C2×C4×C173C2×D412yes—
65C2×C4×C173#4112yes—
66C2×C4×C173C2411yescharacteristic
67C8×C173C2×D414yes—
68C22×C173#12112yes—
69C22×C173C2×Q1612yes—
70C22×C173#13512yes—
71C22×C173C22×D411yescharacteristic
72C2×D173—22no—
73C2×D173—22no—
74C2×D173—28no—
75C2×D173C2×Q1614yescomplemented
76C4×C173—44no—
77C4×C173#12112yes—
78C4×C173C22×D411yescharacteristic
79C4×C173C2×#4111yescharacteristic
80Dic173—22no—
81Dic173—44no—
82C2×C173#46714yescomplemented
83C2×C173C2×#12111yescharacteristic
84C2×C173C22×Q1611yescharacteristic
85C2×C173C2×#13511yescharacteristic
86D173—24no—
87C173C2×#46711yescharacteristic, complemented, Sylow
88C2×#467—1731nomaximal, Sylow
89C22×Q16—1731no—
90C23×C8—1731no—
91C2×C22⋊Q8—1732no—
92#467—1738no—
93C2×#107D17311yescharacteristic, complemented
94C2×#103—1732no—
95C22×Q8—1732no—
96C23×C4—1731no—
97C2×Q16—3464no—
98C2×Q16—1734no—
99C22×C8—3464no—
100C22×C8—1731no—
101C22×C8—1734no—
102C22×C8C2×D17311yescharacteristic
103C22⋊Q8—3464no—
104C22⋊Q8—1738no—
105C2×C4⋊C4—3462no—
106C2×C4⋊C4C2×D17312yes—
107C2×C22⋊C4—3462no—
108#107C2×D17314yescomplemented
109#103—1738no—
110C24—1731no—
111C2×Q8—3462no—
112C2×Q8—3464no—
113C2×Q8—3464no—
114C2×Q8—1734no—
115C22×C4—22no—
116C22×C4—3462no—
117C22×C4—3462no—
118C22×C4—3462no—
119C22×C4—1731no—
120C22×C4—1732no—
121C22×C4—1732no—
122C22×C4C22×D17311yescharacteristic
123Q16—3468no—
124C2×C8—3462no—
125C2×C8—3468no—
126C2×C8—1732no—
127C2×C8D69214yes—
128C2×C8C22×D17312yes—
129C4⋊C4—22no—
130C4⋊C4—3468no—
131C4⋊C4C22×D17314yes—
132C22⋊C4—3468no—
133C23—3462no—
134C23—3462no—
135C23—1734no—
136C23—1732no—
137C23D4×D17311yescharacteristic
138Q8—3464no—
139Q8—3464no—
140Q8—6924no—
141C2×C4—24no—
142C2×C4—44no—
143C2×C4—3462no—
144C2×C4—3464no—
145C2×C4—3462no—
146C2×C4—3462no—
147C2×C4—3464no—
148C2×C4—1732no—
149C2×C4—6924no—
150C2×C4D4×D17311yescharacteristic
151C2×C4C2×D69212yes—
152C2×C4#1458382712yes—
153C2×C4C23×D17311yescharacteristic
154C8—3462no—
155C8C2×D69214yes—
156C22—3462no—
157C22—3468no—
158C22—3462no—
159C22—1734no—
160C22#1914965412yes—
161C22Q16×D17312yes—
162C22#1914969812yes—
163C22C2×D4×D17311yescharacteristic
164C4—44no—
165C4—3462no—
166C4—6924no—
167C4#1914965412yes—
168C4C22×D69211yescharacteristic
169C4C2×#1458382711yescharacteristic
170C2—3464no—
171C2#2601173314yesdirect product factor
172C2C2×#1914965411yescharacteristic
173C2C2×Q16×D17311yescharacteristic
174C2C2×#1914969811yescharacteristic
subgroupG/H
centerC23D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC2×#107×C173C2
socleC23×C173C2×D4
prime radicalC23×D173D4
hypercenterC2×#107D173
hypocenterC173C2×#467
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#107

Complements

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

NH ≅ G/Ntype
1. C22×Dic692C2semidirect
4. #26011733C2direct (N × H)
5. C2×#19149602C2semidirect
6. C2×#107×C173C2semidirect
10. C2×Dic692C22semidirect
18. #19149602C22semidirect
21. #107×C173C22semidirect
47. C22×D173Q16semidirect
75. C2×D173C2×Q16semidirect
82. C2×C173#467semidirect
87. C173C2×#467semidirect
93. C2×#107D173semidirect
108. #107C2×D173semidirect
171. C2#26011733direct (N × H)

Maximal subgroups

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

subgroupindex
C22×Dic6922
C22×C8×D1732
C2×#191496592
#260117332
C2×#191496022
C2×#107×C1732
C2×#467173