Finite Group Databasebeta

← #36363496

Subgroups — #36363496

Order 22144.

414 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 133 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#26012002C211yesmaximal, characteristic
2C2×#19149638C211yesmaximal, characteristic, complemented
3#26011833C214yesmaximal, complemented
4#26011816C214yesmaximal, complemented
5#26011533C212yesmaximal, complemented
6#26011493C212yesmaximal, complemented
7#407×C173C211yesmaximal, characteristic, complemented
8C2×#14583826C2211yescharacteristic
9#19149662—21no—
10#19149658C2212yes—
11#19149653C2212yes—
12#19149648—21no—
13C4×C22⋊D173—22no—
14#19149638—22no—
15#19149638C2214yes—
16#19149638C2211yescharacteristic
17C2×C4⋊Dic173C2212yes—
18C2×C22⋊Dic173—21no—
19C2×C4×Dic173—21no—
20#19149613C2214yescomplemented
21#19149611C2214yescomplemented
22#19149604C2214yescomplemented
23#19149601C2214yescomplemented
24C23×C4×C173C2211yescharacteristic
25C2×M16×C173C2212yes—
26C22⋊C8×C173C2214yes—
27#14583826—24no—
28C2×Dic346C2311yescharacteristic
29C2×D692C2311yescharacteristic
30C2×C22⋊D173—22no—
31C2×C22⋊D173—21no—
32C22×Dic173—22no—
33C2×C4×D173—21no—
34C4⋊Dic173—22no—
35C4⋊Dic173C2314yes—
36C4⋊Dic173C2312yes—
37#14583805—22no—
38C22⋊Dic173—24no—
39C22⋊Dic173—21no—
40C22⋊Dic173—22no—
41#14583803—22no—
42C4×Dic173—24no—
43C4×Dic173—21no—
44C24×C173D411yescharacteristic
45C22×C4×C173—24no—
46C22×C4×C173D411yescharacteristic
47C22×C4×C173D412yes—
48C22×C4×C173C2312yes—
49C22×C4×C173C2311yescharacteristic
50M16×C173—24no—
51C2×C8×C173C2314yes—
52C22×D173—21no—
53Dic346—22no—
54D692—22no—
55C22⋊D173—44no—
56C22⋊D173—42no—
57C2×Dic173—21no—
58C2×Dic173—24no—
59C2×Dic173—22no—
60C2×Dic173—44no—
61C4×D173—42no—
62C23×C173—22no—
63C23×C173—42no—
64C23×C173C2×D412yes—
65C23×C173C2×D411yescharacteristic
66C2×C4×C173—22no—
67C2×C4×C173—24no—
68C2×C4×C173—24no—
69C2×C4×C173C2×D412yes—
70C2×C4×C173C2×D411yescharacteristic
71C2×C4×C173#4114yes—
72C2×C4×C173C2411yescharacteristic
73C8×C173—24no—
74C22×C173—22no—
75C22×C173—22no—
76C22×C173—22no—
77C22×C173—22no—
78C22×C173—44no—
79C22×C173#12312yes—
80C22×C173C22×D411yescharacteristic
81C2×D173—41no—
82C2×D173—42no—
83C4×C173—22no—
84C4×C173#12312yes—
85C4×C173C2×#4112yes—
86Dic173—41no—
87Dic173—44no—
88Dic173—42no—
89C2×C173—22no—
90C2×C173—42no—
91C2×C173C2×#12311yescharacteristic
92C2×C173#57512yes—
93D173—81no—
94C173#309311yescharacteristic, complemented, Sylow
95#3093—1731nomaximal, Sylow
96#525—1731no—
97C2×#117—1731no—
98#485—1734no—
99#482—1734no—
100#428—1732no—
101#417—1732no—
102#407D17311yescharacteristic, complemented
103C2×#41—1731no—
104C23×C4C2×D17311yescharacteristic
105C2×M16C2×D17312yes—
106#123—3461no—
107C22⋊Q8—1732no—
108#121—1732no—
109#120—3461no—
110C4×D4—3462no—
111#117—3462no—
112#117—1734no—
113#117—1731no—
114C2×C4⋊C4—1732no—
115C2×C22⋊C4—3461no—
116C2×C42—3461no—
117#107—1734no—
118#106—1734no—
119#103—1734no—
120#102—1734no—
121C22⋊C8C2×D17314yes—
122C24D69211yescharacteristic
123#41—3464no—
124C2×Q8—1731no—
125C2×D4—3462no—
126C2×D4—3461no—
127C2×D4—1731no—
128C22×C4—24no—
129C22×C4—3462no—
130C22×C4—3461no—
131C22×C4D69211yescharacteristic
132C22×C4D69212yes—
133C22×C4C22×D17312yes—
134C22×C4C22×D17311yescharacteristic
135M16—24no—
136C2×C8C22×D17314yes—
137C4⋊C4—3462no—
138C4⋊C4—3462no—
139C4⋊C4—1734no—
140C4⋊C4—1732no—
141C22⋊C4—3464no—
142C22⋊C4—3461no—
143C22⋊C4—3462no—
144C22⋊C4—3462no—
145C42—3464no—
146C42—3461no—
147C23—22no—
148C23—42no—
149C23—3461no—
150C23C2×D69212yes—
151C23C2×D69211yescharacteristic
152Q8—3462no—
153D4—3462no—
154D4—6924no—
155D4—6922no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—3464no—
160C2×C4—3462no—
161C2×C4—3461no—
162C2×C4—6924no—
163C2×C4—6922no—
164C2×C4C2×D69212yes—
165C2×C4C2×D69211yescharacteristic
166C2×C4#1458382514yes—
167C2×C4C23×D17311yescharacteristic
168C8—24no—
169C22—22no—
170C22—22no—
171C22—22no—
172C22—22no—
173C22—44no—
174C22—6921no—
175C22—6922no—
176C22#1914966312yes—
177C22C22×D69211yescharacteristic
178C4—22no—
179C4—6921no—
180C4—6924no—
181C4—6922no—
182C4#1914966312yes—
183C4C2×#1458382512yes—
184C2—22no—
185C2—42no—
186C2—13841no—
187C2C2×#1914966311yescharacteristic
188C2#2601229112yes—
subgroupG/H
centerC22C22×D692
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#407×C173C2
socleC22×C173C22×D4
prime radical#26012002C2
hypercenter#407D173
hypocenterC173#3093
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×#19149638C2semidirect
3. #26011833C2semidirect
4. #26011816C2semidirect
5. #26011533C2semidirect
6. #26011493C2semidirect
7. #407×C173C2semidirect
20. #19149613C22semidirect
21. #19149611C22semidirect
22. #19149604C22semidirect
23. #19149601C22semidirect
94. C173#3093semidirect
102. #407D173semidirect

Maximal subgroups

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

subgroupindex
#260120022
C2×#191496382
#260118332
#260118162
#260115332
#260114932
#407×C1732
#3093173