Finite Group Databasebeta

← #36354110

Subgroups — #36354110

Order 22144.

340 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 89 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#26012055C211yesmaximal, characteristic
2#26011475C211yesmaximal, characteristic, complemented
3C2×#19149595C212yesmaximal, complemented
4#26011263C212yesmaximal, complemented
5#568×C173C211yesmaximal, characteristic, complemented
6C22×D692C2211yescharacteristic
7C2×#14583810C2212yes—
8#19149675—21no—
9#19149668—21no—
10#19149653—22no—
11#19149647C414yes—
12#19149638C2211yescharacteristic
13#19149598—21no—
14#19149596—21no—
15#19149595—24no—
16C2×#41×C173C2211yescharacteristic
17#131×C173—24no—
18C2×M16×C173—22no—
19C2×M16×C173C2212yes—
20C23×D173C2×C412yes—
21C2×D692—22no—
22C2×D692—21no—
23C2×D692—21no—
24C2×D692—22no—
25C2×C22⋊D173—24no—
26#14583810—22no—
27#14583810—22no—
28C2×#13063436—22no—
29C4⋊Dic173—21no—
30C22⋊Dic173C2×C412yes—
31#14583803—24no—
32C4×Dic173—21no—
33#41×C173—22no—
34#41×C173—22no—
35C2×Q8×C173D411yescharacteristic
36C2×D4×C173C2×C412yes—
37C2×D4×C173D411yescharacteristic
38C22×C4×C173D412yes—
39C22×C4×C173C2311yescharacteristic
40M16×C173—24no—
41M16×C173—22no—
42M16×C173D414yescomplemented
43C2×C8×C173—24no—
44C2×C8×C173—22no—
45C22×D173—22no—
46C22×D173—24no—
47C22×D173—82no—
48D692—41no—
49D692—42no—
50D692—41no—
51C22⋊D173—84no—
52C2×Dic173—22no—
53#13063436—42no—
54C23×C173C22⋊C412yes—
55C23×C173C22×C411yescharacteristic
56Q8×C173—22no—
57D4×C173—24no—
58D4×C173—22no—
59C2×C4×C173—22no—
60C2×C4×C173—22no—
61C2×C4×C173C22⋊C412yes—
62C2×C4×C173C2×D412yes—
63C2×C4×C173C2×D411yescharacteristic
64C2×C4×C173C2×D411yescharacteristic
65C2×C4×C173#4112yes—
66C8×C173—24no—
67C22×C173—22no—
68C22×C173—41no—
69C22×C173—42no—
70C22×C173C2×C22⋊C411yescharacteristic
71C22×C173C4×D412yes—
72C2×D173—44no—
73C2×D173—42no—
74C2×D173—84no—
75C4×C173—22no—
76C4×C173#12011yescharacteristic
77C4×C173#12112yes—
78C4×C173#12311yescharacteristic
79Dic173—42no—
80C2×C173—21no—
81C2×C173—22no—
82C2×C173—42no—
83C2×C173#38611yescharacteristic
84D173—82no—
85C173#191411yescharacteristic, complemented, Sylow
86#1914—1731nomaximal, Sylow
87#568D17311yescharacteristic, complemented
88#535—1731no—
89#413—1731no—
90C2×#100—1732no—
91#344—1732no—
92C2×#41C2×D17311yescharacteristic
93C22×D4—1731no—
94#131—24no—
95C2×M16—22no—
96C2×M16—1732no—
97C2×M16C2×D17312yes—
98#127—3461no—
99#124—3461no—
100#121—3462no—
101#120—1734no—
102#117—1731no—
103#101—3461no—
104#100—3461no—
105#100—3464no—
106C24—1732no—
107#41—22no—
108#41—22no—
109C2×Q8D69211yescharacteristic
110C2×D4—3464no—
111C2×D4—3462no—
112C2×D4—3461no—
113C2×D4—3461no—
114C2×D4—3462no—
115C2×D4C4×D17312yes—
116C2×D4D69211yescharacteristic
117C22×C4C22⋊D17312yes—
118C22×C4C22×D17311yescharacteristic
119M16—24no—
120M16—22no—
121M16—3462no—
122M16—3462no—
123M16C22⋊D17314yescomplemented
124C2×C8—24no—
125C2×C8—22no—
126C2×C8—3462no—
127C4⋊C4—3461no—
128C22⋊C4—3464no—
129C22⋊C4—1732no—
130C42—3461no—
131C23—3464no—
132C23—3462no—
133C23—13842no—
134C23#1458380312yes—
135C23C2×C4×D17311yescharacteristic
136Q8—22no—
137D4—24no—
138D4—22no—
139D4—13844no—
140D4—6921no—
141D4—6922no—
142D4—6921no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—3462no—
146C2×C4#1458380312yes—
147C2×C4C2×C22⋊D17312yes—
148C2×C4C2×C22⋊D17311yescharacteristic
149C2×C4C2×D69211yescharacteristic
150C2×C4#1458382612yes—
151C8—24no—
152C8—6922no—
153C22—22no—
154C22—41no—
155C22—42no—
156C22—13844no—
157C22—6924no—
158C22—6922no—
159C22C2×#1458380311yescharacteristic
160C22C4×C22⋊D17312yes—
161C4—22no—
162C4—6922no—
163C4#1914964811yescharacteristic
164C4#1914965312yes—
165C4#1914966211yescharacteristic
166C2—21no—
167C2—22no—
168C2—42no—
169C2—13842no—
170C2#2601137011yescharacteristic
subgroupG/H
centerC2#26011370
commutatorC23×C173C22×C4
FrattiniC22×C4C22×D173
Fitting#568×C173C2
socleC2×C173#386
prime radical#26012055C2
cosocleC22×C4×C173C23
hypercenter#568D173
hypocenterC173#1914
LCSG ▷ C23×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #26011475C2semidirect
3. C2×#19149595C2semidirect
4. #26011263C2semidirect
5. #568×C173C2semidirect
42. M16×C173D4semidirect
85. C173#1914semidirect
87. #568D173semidirect
123. M16C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
#260120552
#260114752
C2×#191495952
#260112632
#568×C1732
#1914173