Finite Group Databasebeta

← #36354111

Subgroups — #36354111

Order 22144.

340 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 93 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#26012258C211yesmaximal, characteristic, complemented
2#26011475C211yesmaximal, characteristic, complemented
3C2×#19149596C212yesmaximal, complemented
4#26011261C212yesmaximal, complemented
5#535×C173C211yesmaximal, characteristic
6#19149686—24no—
7C2×#14583810—22no—
8C2×#14583810C2212yes—
9C2×#14583810C2212yes—
10#19149598—21no—
11#19149596—21no—
12#19149596—24no—
13C2×#41×C173C2211yescharacteristic
14C22×D4×C173C2211yescharacteristic
15#127×C173—21no—
16#124×C173—21no—
17#121×C173—22no—
18#120×C173C414yes—
19#117×C173C2211yescharacteristic
20#14583810—24no—
21#14583810—22no—
22#14583810—22no—
23#14583810—22no—
24#14583810D414yescomplemented
25C2×#13063436—24no—
26C2×#13063436—22no—
27C2×#13063436—22no—
28C24×C173C2×C412yes—
29#41×C173—22no—
30#41×C173—22no—
31C2×Q8×C173D411yescharacteristic
32C2×D4×C173—24no—
33C2×D4×C173—22no—
34C2×D4×C173—21no—
35C2×D4×C173—21no—
36C2×D4×C173—22no—
37C2×D4×C173C2×C412yes—
38C2×D4×C173D411yescharacteristic
39C22×C4×C173D412yes—
40C22×C4×C173C2311yescharacteristic
41C4⋊C4×C173—21no—
42C22⋊C4×C173—24no—
43C22⋊C4×C173C2×C412yes—
44C42×C173—21no—
45#13063436—24no—
46#13063436—42no—
47C23×C173—22no—
48C23×C173—24no—
49C23×C173—82no—
50C23×C173C22⋊C412yes—
51C23×C173C22×C411yescharacteristic
52Q8×C173—22no—
53D4×C173—24no—
54D4×C173—22no—
55D4×C173—41no—
56D4×C173—42no—
57D4×C173—41no—
58D4×C173—84no—
59C2×C4×C173—22no—
60C2×C4×C173—22no—
61C2×C4×C173—22no—
62C2×C4×C173C22⋊C412yes—
63C2×C4×C173C2×D412yes—
64C2×C4×C173C2×D411yescharacteristic
65C2×C4×C173C2×D411yescharacteristic
66C2×C4×C173#4112yes—
67C22×C173—22no—
68C22×C173—41no—
69C22×C173—42no—
70C22×C173—44no—
71C22×C173—42no—
72C22×C173—84no—
73C22×C173C2×C22⋊C411yescharacteristic
74C22×C173C4×D412yes—
75C4×C173—22no—
76C4×C173—42no—
77C4×C173#12011yescharacteristic
78C4×C173#12112yes—
79C4×C173#12311yescharacteristic
80C2×C173—21no—
81C2×C173—22no—
82C2×C173—42no—
83C2×C173—82no—
84C2×C173#38611yescharacteristic
85C173#191411yescharacteristic, complemented, Sylow
86#1914—1731nomaximal, Sylow
87#568—1731no—
88#535D17311yescharacteristic
89#413—1731no—
90C2×#100—1732no—
91#344—1732no—
92C2×#41C2×D17311yescharacteristic
93C22×D4C2×D17311yescharacteristic
94#131—3464no—
95C2×M16—3462no—
96C2×M16—1732no—
97C2×M16—1732no—
98#127—21no—
99#124—21no—
100#121—22no—
101#120Dic17314yes—
102#117C2×D17311yescharacteristic
103#101—3461no—
104#100—3461no—
105#100—3464no—
106C24C2×Dic17312yes—
107#41—22no—
108#41—22no—
109C2×Q8C22⋊D17311yescharacteristic
110C2×D4—24no—
111C2×D4—22no—
112C2×D4—21no—
113C2×D4—21no—
114C2×D4—22no—
115C2×D4C2×Dic17312yes—
116C2×D4C22⋊D17311yescharacteristic
117C22×C4C22⋊D17312yes—
118C22×C4C22×D17311yescharacteristic
119M16—3464no—
120M16—3462no—
121M16—3462no—
122M16—3462no—
123M16—1734no—
124C2×C8—3464no—
125C2×C8—3462no—
126C2×C8—3462no—
127C4⋊C4—21no—
128C22⋊C4—24no—
129C22⋊C4C2×Dic17312yes—
130C42—21no—
131C23—22no—
132C23—24no—
133C23—82no—
134C23C22⋊Dic17312yes—
135C23C22×Dic17311yescharacteristic
136Q8—22no—
137D4—24no—
138D4—22no—
139D4—41no—
140D4—42no—
141D4—41no—
142D4—84no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4C22⋊Dic17312yes—
147C2×C4D4×D17312yes—
148C2×C4C2×C22⋊D17311yescharacteristic
149C2×C4C2×C22⋊D17311yescharacteristic
150C2×C4#1458382512yes—
151C8—3464no—
152C8—6922no—
153C22—22no—
154C22—41no—
155C22—42no—
156C22—44no—
157C22—42no—
158C22—84no—
159C22C2×C22⋊Dic17311yescharacteristic
160C22D4×Dic17312yes—
161C4—22no—
162C4—42no—
163C4#1914964711yescharacteristic
164C4#1914965112yes—
165C4#1914966011yescharacteristic
166C2—21no—
167C2—22no—
168C2—42no—
169C2—82no—
170C2#2601137111yescharacteristic
subgroupG/H
centerC2#26011371
commutatorC23×C173C22×C4
FrattiniC22×C4C22×D173
Fitting, prime radical#535×C173C2
socleC2×C173#386
cosocleC22×C4×C173C23
hypercenter#535D173
hypocenterC173#1914
LCSG ▷ C23×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22×C4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #26012258C2semidirect
2. #26011475C2semidirect
3. C2×#19149596C2semidirect
4. #26011261C2semidirect
24. #14583810D4semidirect
85. C173#1914semidirect

Maximal subgroups

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

subgroupindex
#260122582
#260114752
C2×#191495962
#260112612
#535×C1732
#1914173