Finite Group Databasebeta

← #36354001

Subgroups — #36354001

Order 22144.

412 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 109 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
1C2×#19149653C211yesmaximal, characteristic
2C2×#19149601C212yesmaximal, complemented
3#26011351C211yesmaximal, characteristic, complemented
4#26011254C212yesmaximal, complemented
5C22×M16×C173C211yesmaximal, characteristic, complemented
6C22×D692C2211yescharacteristic
7C22×C22⋊D173—21no—
8#19149653—22no—
9#19149653C414yescomplemented
10C2×C4⋊Dic173C2212yes—
11C2×C4⋊Dic173C2211yescharacteristic
12C2×C22⋊Dic173—21no—
13C2×#14583803—21no—
14#19149601—24no—
15#19149586—21no—
16C23×C4×C173C2211yescharacteristic
17C2×M16×C173—22no—
18C2×M16×C173—24no—
19C22×C8×C173C2212yes—
20C23×D173—21no—
21C2×D692—22no—
22C2×D692—21no—
23C2×D692—22no—
24C2×D692C2×C412yes—
25C2×C22⋊D173—24no—
26C2×C22⋊D173—44no—
27C22×Dic173—21no—
28C22×Dic173—22no—
29C4⋊Dic173—21no—
30C4⋊Dic173—22no—
31C4⋊Dic173—22no—
32C4⋊Dic173C2×C412yes—
33C22⋊Dic173—42no—
34#14583803—24no—
35C24×C173D411yescharacteristic
36C22×C4×C173—22no—
37C22×C4×C173—22no—
38C22×C4×C173C2×C412yes—
39C22×C4×C173D411yescharacteristic
40C22×C4×C173D412yes—
41C22×C4×C173C2311yescharacteristic
42M16×C173—28no—
43C2×C8×C173—22no—
44C2×C8×C173—22no—
45C2×C8×C173D414yescomplemented
46C22×D173—22no—
47C22×D173—42no—
48C22×D173—42no—
49C22×D173—41no—
50C22×D173—41no—
51D692—24no—
52D692—41no—
53D692—41no—
54C22⋊D173—48no—
55C2×Dic173—22no—
56C2×Dic173—42no—
57C2×Dic173—42no—
58C2×Dic173—42no—
59C2×Dic173—42no—
60C23×C173—22no—
61C23×C173—22no—
62C23×C173—21no—
63C23×C173—21no—
64C23×C173C22⋊C412yes—
65C23×C173C2×D411yescharacteristic
66C2×C4×C173—22no—
67C2×C4×C173—24no—
68C2×C4×C173—22no—
69C2×C4×C173—22no—
70C2×C4×C173C22⋊C412yes—
71C2×C4×C173C22×C411yescharacteristic
72C2×C4×C173C2×D412yes—
73C2×C4×C173C2×D411yescharacteristic
74C2×C4×C173#4112yes—
75C8×C173—24no—
76C22×C173—22no—
77C22×C173—22no—
78C22×C173—22no—
79C22×C173—42no—
80C22×C173—42no—
81C22×C173C2×C22⋊C411yescharacteristic
82C22×C173#12112yes—
83C22×C173AGL(1,ℤ/8)11yescharacteristic
84C22×C173AGL(1,ℤ/8)12yes—
85C22×C173#13711yescharacteristic
86C2×D173—42no—
87C2×D173—42no—
88C2×D173—42no—
89C2×D173—82no—
90C2×D173—82no—
91C4×C173—22no—
92C4×C173C4×D412yes—
93C4×C173#12011yescharacteristic
94C4×C173#12311yescharacteristic
95Dic173—42no—
96Dic173—82no—
97C2×C173—42no—
98C2×C173#38611yescharacteristic
99C2×C173#41711yescharacteristic
100C2×C173#41811yescharacteristic
101C2×C173#46812yes—
102C2×C173#46912yes—
103D173—82no—
104C173#189511yescharacteristic, complemented, Sylow
105#1895—1731nomaximal, Sylow
106C22×M16D17311yescharacteristic, complemented
107C2×#121—1731no—
108C2×#102—1732no—
109#380—1731no—
110#340—1732no—
111C22×D4—3461no—
112C22×D4—1731no—
113C23×C4C2×D17311yescharacteristic
114C2×M16—22no—
115C2×M16—24no—
116C22×C8C2×D17312yes—
117#121—3462no—
118#121—1734no—
119C2×C4⋊C4—1732no—
120C2×C4⋊C4—1731no—
121C2×C22⋊C4—3461no—
122C2×C22⋊C4—3461no—
123#102—3464no—
124#95—3461no—
125C24—3461no—
126C24D69211yescharacteristic
127C2×D4—3464no—
128C2×D4—3462no—
129C2×D4—3461no—
130C2×D4—3462no—
131C2×D4—1732no—
132C2×D4—6924no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—3461no—
136C22×C4—3462no—
137C22×C4C4×D17312yes—
138C22×C4C22⋊D17312yes—
139C22×C4D69211yescharacteristic
140C22×C4C22×D17311yescharacteristic
141M16—28no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8C22⋊D17314yescomplemented
145C4⋊C4—3461no—
146C4⋊C4—3462no—
147C4⋊C4—3462no—
148C4⋊C4—1732no—
149C22⋊C4—3464no—
150C22⋊C4—6922no—
151C23—22no—
152C23—21no—
153C23—22no—
154C23—21no—
155C23—3462no—
156C23—6922no—
157C23—6922no—
158C23—6921no—
159C23—6921no—
160C23#1458380312yes—
161C23C2×D69211yescharacteristic
162D4—3464no—
163D4—6928no—
164D4—6921no—
165D4—6921no—
166C2×C4—22no—
167C2×C4—24no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—3462no—
171C2×C4—6922no—
172C2×C4—6922no—
173C2×C4—6922no—
174C2×C4—6922no—
175C2×C4#1458380312yes—
176C2×C4C2×C4×D17311yescharacteristic
177C2×C4C2×C22⋊D17312yes—
178C2×C4C2×C22⋊D17311yescharacteristic
179C2×C4#1458382612yes—
180C8—24no—
181C22—22no—
182C22—22no—
183C22—22no—
184C22—42no—
185C22—42no—
186C22—13842no—
187C22—13842no—
188C22—6922no—
189C22—6922no—
190C22—6922no—
191C22C2×#1458380311yescharacteristic
192C22#1914965312yes—
193C22#1914970712yes—
194C22#1914970711yescharacteristic
195C22#1914971211yescharacteristic
196C4—22no—
197C4—13842no—
198C4—6922no—
199C4C4×C22⋊D17312yes—
200C4#1914964811yescharacteristic
201C4#1914966211yescharacteristic
202C2—42no—
203C2—13842no—
204C2#2601137011yescharacteristic
205C2#2601149311yescharacteristic
206C2#2601149711yescharacteristic
207C2#2601173712yes—
208C2#2601174412yes—
subgroupG/H
centerC23C2×D692
commutatorC2×C4×C173C22×C4
FrattiniC22×C4C22×D173
FittingC22×M16×C173C2
socleC23×C173C2×D4
prime radicalC2×#19149653C2
cosocleC22×C4×C173C23
hypercenterC22×M16D173
hypocenterC173#1895
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
2. C2×#19149601C2semidirect
3. #26011351C2semidirect
4. #26011254C2semidirect
5. C22×M16×C173C2semidirect
9. #19149653C4semidirect
45. C2×C8×C173D4semidirect
104. C173#1895semidirect
106. C22×M16D173semidirect
144. C2×C8C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191496532
C2×#191496012
#260113512
#260112542
C22×M16×C1732
#1895173