Finite Group Databasebeta

← #36354086

Subgroups — #36354086

Order 22144.

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

normal subgroups only

This group has 312 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
2#26011500C212yesmaximal, complemented
3#26011465C211yesmaximal, characteristic
4#26011247C212yesmaximal, complemented
5#536×C173C211yesmaximal, characteristic
6#19149686—24no—
7C2×#14583810—22no—
8C2×#14583810C2212yes—
9#19149638C2212yes—
10#19149603—24no—
11#19149594—22no—
12C2×#41×C173C2211yescharacteristic
13C22×Q8×C173C2211yescharacteristic
14C4⋊Q8×C173C412yescomplemented
15#124×C173C412yescomplemented
16#123×C173—22no—
17C22⋊Q8×C173—22no—
18#117×C173C2211yescharacteristic
19#14583810—24no—
20#14583810—22no—
21#14583810—22no—
22C2×#13063436—24no—
23C2×#13063436D414yes—
24C4⋊Dic173—22no—
25C22⋊Dic173—22no—
26C4×Dic173—22no—
27#41×C173—22no—
28#41×C173—22no—
29C2×Q8×C173—21no—
30C2×Q8×C173—22no—
31C2×Q8×C173—22no—
32C2×Q8×C173C2×C412yes—
33C2×Q8×C173C2×C411yescharacteristic
34C2×D4×C173C2×C411yescharacteristic
35C2×D4×C173D412yes—
36C22×C4×C173—21no—
37C22×C4×C173D412yes—
38C22×C4×C173C2311yescharacteristic
39C4⋊C4×C173—24no—
40C4⋊C4×C173—21no—
41C22⋊C4×C173—24no—
42C22⋊C4×C173—21no—
43C42×C173C2×C412yes—
44C2×Dic173—42no—
45#13063436—24no—
46C23×C173C22⋊C412yes—
47C23×C173C2×D411yescharacteristic
48Q8×C173—22no—
49Q8×C173—24no—
50Q8×C173—41no—
51Q8×C173—41no—
52D4×C173—24no—
53D4×C173—22no—
54C2×C4×C173—22no—
55C2×C4×C173—22no—
56C2×C4×C173—22no—
57C2×C4×C173—22no—
58C2×C4×C173—42no—
59C2×C4×C173C22⋊C412yes—
60C2×C4×C173C22×C411yescharacteristic
61C2×C4×C173C2×D412yes—
62C2×C4×C173C2×D411yescharacteristic
63C2×C4×C173#4112yes—
64C22×C173—22no—
65C22×C173—41no—
66C22×C173—42no—
67C22×C173C2×C22⋊C411yescharacteristic
68C22×C173#12112yes—
69C4×C173—22no—
70C4×C173—42no—
71C4×C173—42no—
72C4×C173C4×D412yes—
73C4×C173#12011yescharacteristic
74C4×C173#12311yescharacteristic
75Dic173—82no—
76C2×C173—22no—
77C2×C173—21no—
78C2×C173—42no—
79C2×C173#38611yescharacteristic
80C173#190911yescharacteristic, complemented, Sylow
81#1909—1731nomaximal, Sylow
82#568—1731no—
83#536D17311yescharacteristic
84#419—1732no—
85#410—1731no—
86H(3,ℤ/4)—1732no—
87C2×#41C2×D17311yescharacteristic
88C22×Q8C2×D17311yescharacteristic
89#131—3464no—
90C2×M16—3462no—
91C2×M16—1732no—
92C4⋊Q8Dic17312yescomplemented
93#124Dic17312yescomplemented
94#123—22no—
95C22⋊Q8—22no—
96#117—1732no—
97#117C2×D17311yescharacteristic
98#103—3464no—
99#99—3462no—
100#41—22no—
101#41—22no—
102C2×Q8—21no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8C2×Dic17312yes—
106C2×Q8C2×Dic17311yescharacteristic
107C2×D4C2×Dic17311yescharacteristic
108C2×D4C22⋊D17312yes—
109C22×C4—21no—
110C22×C4C22⋊D17312yes—
111C22×C4C22×D17311yescharacteristic
112M16—3464no—
113M16—3462no—
114M16—3462no—
115C2×C8—3464no—
116C2×C8—1734no—
117C4⋊C4—21no—
118C4⋊C4—24no—
119C4⋊C4—3462no—
120C22⋊C4—21no—
121C22⋊C4—24no—
122C22⋊C4—3462no—
123C42—3462no—
124C42C2×Dic17312yes—
125C23C22⋊Dic17312yes—
126C23C2×C22⋊D17311yescharacteristic
127Q8—22no—
128Q8—24no—
129Q8—41no—
130Q8—41no—
131D4—24no—
132D4—22no—
133C2×C4—22no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—42no—
138C2×C4—6922no—
139C2×C4C22⋊Dic17312yes—
140C2×C4C22×Dic17311yescharacteristic
141C2×C4D4×D17312yes—
142C2×C4C2×C22⋊D17311yescharacteristic
143C2×C4#1458382512yes—
144C8—3464no—
145C22—22no—
146C22—41no—
147C22—42no—
148C22C2×C22⋊Dic17311yescharacteristic
149C22#1914965112yes—
150C4—22no—
151C4—42no—
152C4—42no—
153C4—13842no—
154C4D4×Dic17312yes—
155C4#1914964711yescharacteristic
156C4#1914966011yescharacteristic
157C2—21no—
158C2—22no—
159C2—42no—
160C2#2601137111yescharacteristic
subgroupG/H
centerC2#26011371
commutatorC2×C4×C173C22×C4
FrattiniC22×C4C22×D173
Fitting#536×C173C2
socleC2×C173#386
prime radicalC2×#41×C173C22
cosocleC22×C4×C173C23
hypercenter#536D173
hypocenterC173#1909
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
2. #26011500C2semidirect
4. #26011247C2semidirect
14. C4⋊Q8×C173C4semidirect
15. #124×C173C4semidirect
80. C173#1909semidirect
92. C4⋊Q8Dic173semidirect
93. #124Dic173semidirect

Maximal subgroups

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

subgroupindex
#260122582
#260115002
#260114652
#260112472
#536×C1732
#1909173