Finite Group Databasebeta

← #36354085

Subgroups — #36354085

Order 22144.

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

normal subgroups only

This group has 312 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#26012062C211yesmaximal, characteristic
2#26011501C212yesmaximal, complemented
3#26011465C211yesmaximal, characteristic
4#26011248C212yesmaximal, complemented
5#568×C173C211yesmaximal, characteristic
6C22×Dic346C2211yescharacteristic
7#19149677C412yescomplemented
8#19149666C412yescomplemented
9#19149660—22no—
10#19149658—22no—
11#19149638C2212yes—
12#19149638C2211yescharacteristic
13#19149604—24no—
14#19149594—22no—
15C2×#41×C173C2211yescharacteristic
16#131×C173—24no—
17C2×M16×C173—22no—
18C2×M16×C173C2212yes—
19C2×Dic346—21no—
20C2×Dic346—22no—
21C2×Dic346—22no—
22C2×Dic346C2×C412yes—
23C22×Dic173—21no—
24C4⋊Dic173—21no—
25C4⋊Dic173—22no—
26#14583805—24no—
27C22⋊Dic173—24no—
28C22⋊Dic173—21no—
29C22⋊Dic173—22no—
30C4×Dic173—22no—
31C4×Dic173C2×C412yes—
32#41×C173—22no—
33#41×C173—22no—
34C2×Q8×C173C2×C411yescharacteristic
35C2×D4×C173C2×C411yescharacteristic
36C2×D4×C173D412yes—
37C22×C4×C173D412yes—
38C22×C4×C173C2311yescharacteristic
39M16×C173—24no—
40M16×C173—22no—
41M16×C173—22no—
42C2×C8×C173—24no—
43C2×C8×C173D414yes—
44Dic346—24no—
45Dic346—41no—
46Dic346—41no—
47C2×Dic173—22no—
48C2×Dic173—22no—
49C2×Dic173—42no—
50C2×Dic173—42no—
51C23×C173C22⋊C412yes—
52C23×C173C2×D411yescharacteristic
53Q8×C173—22no—
54D4×C173—24no—
55D4×C173—22no—
56C2×C4×C173—22no—
57C2×C4×C173—22no—
58C2×C4×C173C22⋊C412yes—
59C2×C4×C173C22×C411yescharacteristic
60C2×C4×C173C2×D412yes—
61C2×C4×C173C2×D411yescharacteristic
62C2×C4×C173#4112yes—
63C8×C173—24no—
64C22×C173—22no—
65C22×C173—41no—
66C22×C173—42no—
67C22×C173C2×C22⋊C411yescharacteristic
68C22×C173#12112yes—
69C4×C173—22no—
70C4×C173C4×D412yes—
71C4×C173#12011yescharacteristic
72C4×C173#12311yescharacteristic
73Dic173—42no—
74Dic173—42no—
75Dic173—82no—
76C2×C173—21no—
77C2×C173—22no—
78C2×C173—42no—
79C2×C173#38611yescharacteristic
80C173#190911yescharacteristic, complemented, Sylow
81#1909—1731nomaximal, Sylow
82#568D17311yescharacteristic
83#536—1731no—
84#419—1732no—
85#410—1731no—
86H(3,ℤ/4)—1732no—
87C2×#41C2×D17311yescharacteristic
88C22×Q8—1731no—
89#131—24no—
90C2×M16—22no—
91C2×M16C2×D17312yes—
92C4⋊Q8—1732no—
93#124—1732no—
94#123—3462no—
95C22⋊Q8—3462no—
96#117—1732no—
97#117—1731no—
98#103—3464no—
99#99—3462no—
100#41—22no—
101#41—22no—
102C2×Q8—3461no—
103C2×Q8—3462no—
104C2×Q8—3462no—
105C2×Q8—1732no—
106C2×Q8C4×D17311yescharacteristic
107C2×D4C4×D17311yescharacteristic
108C2×D4D69212yes—
109C22×C4—3461no—
110C22×C4C22⋊D17312yes—
111C22×C4C22×D17311yescharacteristic
112M16—24no—
113M16—22no—
114M16—22no—
115C2×C8—24no—
116C2×C8C22⋊D17314yes—
117C4⋊C4—3461no—
118C4⋊C4—3464no—
119C4⋊C4—3462no—
120C22⋊C4—3461no—
121C22⋊C4—3464no—
122C22⋊C4—3462no—
123C42—3462no—
124C42—1732no—
125C23#1458380312yes—
126C23C2×D69211yescharacteristic
127Q8—22no—
128Q8—3464no—
129Q8—6921no—
130Q8—6921no—
131D4—24no—
132D4—22no—
133C2×C4—22no—
134C2×C4—22no—
135C2×C4—3462no—
136C2×C4—3462no—
137C2×C4—6922no—
138C2×C4—6922no—
139C2×C4#1458380312yes—
140C2×C4C2×C4×D17311yescharacteristic
141C2×C4C2×C22⋊D17312yes—
142C2×C4C2×C22⋊D17311yescharacteristic
143C2×C4#1458382612yes—
144C8—24no—
145C22—22no—
146C22—41no—
147C22—42no—
148C22C2×#1458380311yescharacteristic
149C22#1914965312yes—
150C4—22no—
151C4—13842no—
152C4—6922no—
153C4—6922no—
154C4C4×C22⋊D17312yes—
155C4#1914964811yescharacteristic
156C4#1914966211yescharacteristic
157C2—22no—
158C2—21no—
159C2—42no—
160C2#2601137011yescharacteristic
subgroupG/H
centerC2#26011370
commutatorC2×C4×C173C22×C4
FrattiniC22×C4C22×D173
Fitting#568×C173C2
socleC2×C173#386
prime radicalC2×#41×C173C22
cosocleC22×C4×C173C23
hypercenter#568D173
hypocenterC173#1909
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #26011501C2semidirect
4. #26011248C2semidirect
7. #19149677C4semidirect
8. #19149666C4semidirect
80. C173#1909semidirect

Maximal subgroups

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

subgroupindex
#260120622
#260115012
#260114652
#260112482
#568×C1732
#1909173