Finite Group Databasebeta

← #36368647

Subgroups — #36368647

Order 22144.

234 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 44 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
1C2×#19149739C211yesmaximal, characteristic
2C2×#19149737C211yesmaximal, characteristic
3#26011370C211yesmaximal, characteristic
4C2×#14583835—21no—
5C2×C22⋊#10621924—21no—
6#19149739—21no—
7#19149739C412yescomplemented
8#19149737—21no—
9#19149737C412yescomplemented
10C22×C22⋊D173C2211yescharacteristic
11C2×C22⋊Dic173C411yescharacteristic, complemented
12C2×#14583803—21no—
13#19149586—21no—
14C23×C4×C173C411yescharacteristic, complemented
15C22×#10621924—41no—
16#14583835—22no—
17#14583835—41no—
18C2×#13063443—41no—
19C22⋊#10621924—22no—
20C22⋊#10621924—41no—
21C23×D173D411yescharacteristic
22C2×C22⋊D173—41no—
23C2×C22⋊D173—41no—
24C2×C22⋊D173C2×C411yescharacteristic
25C2×C22⋊D173C2×C411yescharacteristic
26C2×C22⋊D173D411yescharacteristic
27C2×C22⋊D173Q811yescharacteristic
28C22×Dic173—21no—
29C22×Dic173D411yescharacteristic
30C22⋊Dic173—41no—
31#14583803—42no—
32C24×C173C2×C411yescharacteristic
33C22×C4×C173—21no—
34C22×C4×C173—42no—
35C22×C4×C173—41no—
36C2×#10621924—42no—
37C2×#10621924—81no—
38C2×#10621924—81no—
39#13063443—42no—
40C22×D173—21no—
41C22×D173—21no—
42C22×D173—41no—
43C22×D173—41no—
44C22×D173C22⋊C411yescharacteristic
45C22×D173C4⋊C411yescharacteristic
46C22⋊D173—41no—
47C22⋊D173—41no—
48C22⋊D173—41no—
49C22⋊D173—41no—
50C2×Dic173—21no—
51C2×Dic173—21no—
52C2×Dic173—41no—
53C2×Dic173—41no—
54C2×Dic173—41no—
55C2×Dic173C22⋊C411yescharacteristic
56C2×Dic173C4⋊C411yescharacteristic
57C23×C173—21no—
58C23×C173—21no—
59C23×C173—41no—
60C23×C173—41no—
61C23×C173C4211yescharacteristic
62C23×C173C22⋊C411yescharacteristic
63C23×C173C4⋊C411yescharacteristic
64C2×C4×C173—22no—
65C2×C4×C173—41no—
66C2×C4×C173—42no—
67C2×C4×C173—41no—
68C2×C4×C173—42no—
69C22×C173—21no—
70C22×C173—21no—
71C22×C173—21no—
72C22×C173—21no—
73C22×C173—42no—
74C22×C173—41no—
75C22×C173—41no—
76C22×C173—41no—
77C22×C173—41no—
78C22×C173#9511yescharacteristic
79C22×C173#9911yescharacteristic
80C22×C173#9911yescharacteristic
81C2×D173—21no—
82C2×D173—21no—
83C2×D173—41no—
84C2×D173—41no—
85C2×D173—41no—
86C2×D173—41no—
87C2×D173—81no—
88#10621924—82no—
89C4×C173—42no—
90Dic173—21no—
91Dic173—21no—
92Dic173—81no—
93C2×C173—21no—
94C2×C173—21no—
95C2×C173—41no—
96C2×C173—41no—
97C2×C173#34211yescharacteristic
98C2×C173AΣL(1,𝔽16)11yescharacteristic
99C2×C173#35211yescharacteristic
100D173—41no—
101D173—41no—
102C173#139011yescharacteristic, complemented, Sylow
103#1390—1731nomaximal, Sylow
104C2×#100—1731no—
105C2×#99—1731no—
106#386—1731no—
107C22×D4—1731no—
108C23×C4#1062192411yescharacteristic, complemented
109C2×M16—3461no—
110C2×C22⋊C4—3461no—
111C2×C22⋊C4—3461no—
112C2×C22⋊C4—1731no—
113#100—3461no—
114#100—1732no—
115#99—3461no—
116#99—1732no—
117#95—3461no—
118C24—1731no—
119C24C2×#1062192411yescharacteristic
120C2×D4—1731no—
121C2×D4—1731no—
122C2×D4—1731no—
123C2×D4—1731no—
124C2×D4—6921no—
125C2×D4—6921no—
126C22×C4—21no—
127C22×C4—42no—
128C22×C4—41no—
129C22×C4—3461no—
130C22×C4—1731no—
131C22×C4—6921no—
132M16—3462no—
133M16—6921no—
134C2×C8—6921no—
135C22⋊C4—3462no—
136C22⋊C4—6922no—
137C22⋊C4—6921no—
138C22⋊C4—6921no—
139C23—21no—
140C23—21no—
141C23—41no—
142C23—41no—
143C23—3461no—
144C23—3461no—
145C23—1731no—
146C23—1731no—
147C23—6921no—
148C23—6921no—
149C23C4×#1062192411yescharacteristic
150C23C22⋊#1062192411yescharacteristic
151C23C4⋊#1062192411yescharacteristic
152D4—6921no—
153D4—6921no—
154D4—6921no—
155D4—6921no—
156C2×C4—22no—
157C2×C4—41no—
158C2×C4—42no—
159C2×C4—41no—
160C2×C4—42no—
161C2×C4—3461no—
162C2×C4—3461no—
163C2×C4—13841no—
164C2×C4—13841no—
165C2×C4—1731no—
166C2×C4—1731no—
167C2×C4—6921no—
168C2×C4—6921no—
169C2×C4—6922no—
170C2×C4—6921no—
171C8—6922no—
172C22—21no—
173C22—21no—
174C22—21no—
175C22—21no—
176C22—41no—
177C22—41no—
178C22—42no—
179C22—41no—
180C22—41no—
181C22—3461no—
182C22—3461no—
183C22—13841no—
184C22—6921no—
185C22—6921no—
186C22—6921no—
187C22—6921no—
188C22(C2×#10621924)⋊C411yescharacteristic
189C22#1914973711yescharacteristic
190C22#1914973811yescharacteristic
191C4—42no—
192C4—3461no—
193C4—3461no—
194C4—13841no—
195C4—13842no—
196C2—21no—
197C2—21no—
198C2—41no—
199C2—41no—
200C2—6921no—
201C2—6921no—
202C2#2601236911yescharacteristic
203C2#2601239111yescharacteristic
204C2#2601239311yescharacteristic
subgroupG/H
centerC22#19149737
commutatorC23×C173C42
FrattiniC24C2×#10621924
FittingC23×C4×C173C4
socleC22×C173#99
prime radical, cosocleC22×C22⋊D173C22
hypercenterC23×C4#10621924
hypocenterC173#1390
LCSG ▷ C23×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C23 ◁ C24 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
7. #19149739C4semidirect
9. #19149737C4semidirect
11. C2×C22⋊Dic173C4semidirect
14. C23×C4×C173C4semidirect
102. C173#1390semidirect
108. C23×C4#10621924semidirect

Maximal subgroups

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

subgroupindex
C2×#191497392
C2×#191497372
#260113702
#1390173