Finite Group Databasebeta

← #36354095

Subgroups — #36354095

Order 22144.

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

normal subgroups only

This group has 362 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#26012058C211yesmaximal, characteristic
2#26011509C212yesmaximal, complemented
3C2×#19149596C211yesmaximal, characteristic, complemented
4#26011247C212yesmaximal, complemented
5#518×C173C211yesmaximal, characteristic, complemented
6C2×#14583826C2211yescharacteristic
7C2×#14583810C2212yes—
8#19149675C412yescomplemented
9#19149666C412yescomplemented
10#19149650—22no—
11#19149648—22no—
12#19149638C2211yescharacteristic
13#19149607—24no—
14#19149596—22no—
15C22×D4×C173C2211yescharacteristic
16C4×D4×C173—24no—
17#117×C173C2212yes—
18C2×C22⋊C4×C173—22no—
19#14583826—22no—
20#14583826—22no—
21C2×Dic346C2×C411yescharacteristic, complemented
22C2×D692C2×C411yescharacteristic, complemented
23C2×C22⋊D173—24no—
24C2×C22⋊D173—21no—
25C2×C4×D173—21no—
26#14583810—22no—
27#14583810—22no—
28C2×#13063436—22no—
29C4⋊Dic173—21no—
30C22⋊Dic173—24no—
31C22⋊Dic173—21no—
32C4×Dic173C2×C412yescomplemented
33C24×C173D412yes—
34C2×D4×C173—22no—
35C2×D4×C173—22no—
36C2×D4×C173C2×C412yes—
37C2×D4×C173D412yes—
38C22×C4×C173—24no—
39C22×C4×C173C2311yescharacteristic
40C4⋊C4×C173—22no—
41C22⋊C4×C173—24no—
42C22⋊C4×C173—22no—
43C42×C173D414yescomplemented
44C22×D173—21no—
45Dic346—22no—
46D692—22no—
47C22⋊D173—41no—
48C22⋊D173—41no—
49C22⋊D173—84no—
50C2×Dic173—21no—
51C2×Dic173—22no—
52C4×D173—42no—
53#13063436—42no—
54C23×C173—22no—
55C23×C173—22no—
56C23×C173—82no—
57C23×C173C22⋊C414yes—
58C23×C173C2×D411yescharacteristic
59D4×C173—24no—
60D4×C173—24no—
61C2×C4×C173—24no—
62C2×C4×C173—84no—
63C2×C4×C173C22×C411yescharacteristic
64C2×C4×C173C2×D412yes—
65C2×C4×C173C2×D411yescharacteristic
66C2×C4×C173#4112yes—
67C22×C173—24no—
68C22×C173—41no—
69C22×C173—44no—
70C22×C173—44no—
71C22×C173—82no—
72C22×C173C2×C22⋊C411yescharacteristic
73C22×C173#12011yescharacteristic
74C22×C173#12311yescharacteristic
75C2×D173—42no—
76C2×D173—41no—
77C4×C173—44no—
78C4×C173C4×D412yes—
79C4×C173#12112yes—
80Dic173—42no—
81Dic173—41no—
82C2×C173—21no—
83C2×C173—21no—
84C2×C173—21no—
85C2×C173—44no—
86C2×C173#38611yescharacteristic
87D173—81no—
88C173#191111yescharacteristic, complemented, Sylow
89#1911—1731nomaximal, Sylow
90#535—1731no—
91#518D17311yescharacteristic, complemented
92#421—1732no—
93C2×#100—1731no—
94H(3,ℤ/4)—1732no—
95C2×#41—1731no—
96C22×D4C2×D17311yescharacteristic
97C2×M16—1732no—
98#127—1732no—
99#124—1732no—
100#121—3462no—
101#120—3462no—
102C4×D4—24no—
103#117—1731no—
104#117C2×D17312yes—
105C2×C22⋊C4—22no—
106#104—3464no—
107#100—3462no—
108C24C22⋊D17312yes—
109#41—3462no—
110#41—3462no—
111C2×Q8—1731no—
112C2×D4—22no—
113C2×D4—22no—
114C2×D4—3464no—
115C2×D4—3461no—
116C2×D4—1731no—
117C2×D4C4×D17312yes—
118C2×D4D69212yes—
119C22×C4—24no—
120C22×C4—3461no—
121C22×C4C22×D17311yescharacteristic
122M16—3462no—
123M16—3462no—
124C2×C8—3462no—
125C4⋊C4—22no—
126C4⋊C4—3461no—
127C22⋊C4—24no—
128C22⋊C4—22no—
129C22⋊C4—3461no—
130C22⋊C4—3464no—
131C42—1732no—
132C42C22⋊D17314yescomplemented
133C23—22no—
134C23—22no—
135C23—82no—
136C23—3461no—
137C23#1458380314yes—
138C23C2×C22⋊D17311yescharacteristic
139Q8—3462no—
140D4—24no—
141D4—24no—
142D4—3462no—
143D4—13844no—
144D4—6921no—
145D4—6921no—
146C2×C4—24no—
147C2×C4—84no—
148C2×C4—3462no—
149C2×C4—3461no—
150C2×C4—6922no—
151C2×C4C2×C4×D17311yescharacteristic
152C2×C4C2×C22⋊D17312yes—
153C2×C4C2×D69211yescharacteristic
154C2×C4#1458382612yes—
155C8—6922no—
156C22—24no—
157C22—41no—
158C22—44no—
159C22—44no—
160C22—82no—
161C22—6922no—
162C22—6921no—
163C22C2×#1458380311yescharacteristic
164C22#1914964811yescharacteristic
165C22#1914966211yescharacteristic
166C4—44no—
167C4—6922no—
168C4—6921no—
169C4C4×C22⋊D17312yes—
170C4#1914965312yes—
171C2—21no—
172C2—21no—
173C2—21no—
174C2—44no—
175C2—13841no—
176C2#2601137011yescharacteristic
subgroupG/H
centerC2#26011370
commutatorC2×C4×C173C22×C4
FrattiniC22×C4C22×D173
Fitting#518×C173C2
socleC2×C173#386
prime radical#26012058C2
cosocleC22×C4×C173C23
hypercenter#518D173
hypocenterC173#1911
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22×C4 ◁ #518

Complements

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

NH ≅ G/Ntype
2. #26011509C2semidirect
3. C2×#19149596C2semidirect
4. #26011247C2semidirect
5. #518×C173C2semidirect
8. #19149675C4semidirect
9. #19149666C4semidirect
21. C2×Dic346C2×C4semidirect
22. C2×D692C2×C4semidirect
32. C4×Dic173C2×C4semidirect
43. C42×C173D4semidirect
88. C173#1911semidirect
91. #518D173semidirect
132. C42C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
#260120582
#260115092
C2×#191495962
#260112472
#518×C1732
#1911173