Finite Group Databasebeta

← #36354006

Subgroups — #36354006

Order 22144.

442 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 101 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#26012002C211yesmaximal, characteristic
2C2×#19149607C212yesmaximal, complemented
3#26011455C211yesmaximal, characteristic, complemented
4#26011251C212yesmaximal, complemented
5C22×C42×C173C211yesmaximal, characteristic, complemented
6C2×#14583826C2211yescharacteristic
7C2×#14583810C2212yes—
8#19149662—21no—
9#19149658C412yescomplemented
10#19149653C412yescomplemented
11#19149648—21no—
12C4×C22⋊D173—22no—
13#19149638C2211yescharacteristic
14#19149607—24no—
15C22⋊#13063436—22no—
16C23×C4×C173—21no—
17C23×C4×C173C2211yescharacteristic
18C2×C42×C173—21no—
19C2×C42×C173—24no—
20C2×C42×C173C2212yes—
21#14583826—22no—
22#14583826—22no—
23C2×Dic346C2×C411yescharacteristic, complemented
24C2×D692C2×C411yescharacteristic, complemented
25C2×C22⋊D173—22no—
26C2×C22⋊D173—21no—
27C2×C4×D173—21no—
28#14583810—22no—
29#14583810—22no—
30C2×#13063436—22no—
31C4⋊Dic173C2×C412yescomplemented
32#14583805—22no—
33C22⋊Dic173—22no—
34C22⋊Dic173—21no—
35#14583803—22no—
36C4×Dic173—21no—
37C24×C173D411yescharacteristic
38C22×C4×C173—28no—
39C22×C4×C173—24no—
40C22×C4×C173—22no—
41C22×C4×C173—22no—
42C22×C4×C173—22no—
43C22×C4×C173C2×C412yes—
44C22×C4×C173D411yescharacteristic
45C22×C4×C173D412yes—
46C22×C4×C173C2311yescharacteristic
47C42×C173—24no—
48C42×C173—22no—
49C42×C173D414yescomplemented
50C22×D173—21no—
51Dic346—22no—
52D692—22no—
53C22⋊D173—44no—
54C22⋊D173—41no—
55C22⋊D173—41no—
56C2×Dic173—21no—
57C2×Dic173—22no—
58C4×D173—42no—
59#13063436—42no—
60C23×C173—24no—
61C23×C173—21no—
62C23×C173—21no—
63C23×C173C22⋊C412yes—
64C23×C173C2×D411yescharacteristic
65C2×C4×C173—216no—
66C2×C4×C173—28no—
67C2×C4×C173—21no—
68C2×C4×C173—24no—
69C2×C4×C173—21no—
70C2×C4×C173—24no—
71C2×C4×C173—24no—
72C2×C4×C173C22⋊C412yes—
73C2×C4×C173C22×C411yescharacteristic
74C2×C4×C173C2×D412yes—
75C2×C4×C173C2×D411yescharacteristic
76C2×C4×C173#4112yes—
77C22×C173—24no—
78C22×C173—24no—
79C22×C173—22no—
80C22×C173—22no—
81C22×C173—22no—
82C22×C173#10414yescomplemented
83C22×C173C2×C22⋊C411yescharacteristic
84C22×C173#12011yescharacteristic
85C22×C173#12311yescharacteristic
86C2×D173—41no—
87C2×D173—42no—
88C4×C173—28no—
89C4×C173—22no—
90C4×C173C4×D412yes—
91C4×C173#12112yes—
92Dic173—41no—
93Dic173—42no—
94C2×C173—21no—
95C2×C173—24no—
96C2×C173—21no—
97C2×C173#38611yescharacteristic
98C2×C173C2×#10412yes—
99D173—81no—
100C173#189611yescharacteristic, complemented, Sylow
101#1896—1731nomaximal, Sylow
102#525—1731no—
103C22×C42D17311yescharacteristic, complemented
104C2×#104—1732no—
105#407—1731no—
106#339—1732no—
107C2×#41—1731no—
108C23×C4—21no—
109C23×C4C2×D17311yescharacteristic
110C2×M16—1732no—
111#123—3461no—
112C22⋊Q8—1732no—
113#121—1732no—
114#120—3461no—
115C4×D4—3462no—
116#117—1731no—
117C2×C42—21no—
118C2×C42—24no—
119C2×C42C2×D17312yes—
120#104—3464no—
121C22⋊C8—3462no—
122C24C22⋊D17311yescharacteristic
123#41—3462no—
124#41—3462no—
125C2×Q8—1731no—
126C2×D4—3461no—
127C2×D4—3462no—
128C2×D4—1731no—
129C22×C4—24no—
130C22×C4—28no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—3461no—
135C22×C4C4×D17312yes—
136C22×C4C22⋊D17311yescharacteristic
137C22×C4D69212yes—
138C22×C4C22×D17311yescharacteristic
139M16—3462no—
140M16—3462no—
141C2×C8—3462no—
142C4⋊C4—3462no—
143C4⋊C4—1732no—
144C22⋊C4—3461no—
145C22⋊C4—3462no—
146C22⋊C4—3462no—
147C42—24no—
148C42—22no—
149C42—3461no—
150C42C22⋊D17314yescomplemented
151C23—24no—
152C23—21no—
153C23—21no—
154C23—3461no—
155C23#1458380312yes—
156C23C2×C22⋊D17311yescharacteristic
157Q8—3462no—
158D4—3462no—
159D4—6924no—
160D4—6921no—
161D4—6921no—
162C2×C4—216no—
163C2×C4—28no—
164C2×C4—21no—
165C2×C4—24no—
166C2×C4—21no—
167C2×C4—24no—
168C2×C4—24no—
169C2×C4—3462no—
170C2×C4—3461no—
171C2×C4—6922no—
172C2×C4#1458380312yes—
173C2×C4C2×C4×D17311yescharacteristic
174C2×C4C2×C22⋊D17312yes—
175C2×C4C2×D69211yescharacteristic
176C2×C4#1458382612yes—
177C8—6922no—
178C22—24no—
179C22—24no—
180C22—22no—
181C22—22no—
182C22—22no—
183C22—6922no—
184C22—6921no—
185C22#1914960714yescomplemented
186C22C2×#1458380311yescharacteristic
187C22#1914964811yescharacteristic
188C22#1914966211yescharacteristic
189C4—28no—
190C4—22no—
191C4—6922no—
192C4—6921no—
193C4C4×C22⋊D17312yes—
194C4#1914965312yes—
195C2—21no—
196C2—24no—
197C2—21no—
198C2—13841no—
199C2#2601137011yescharacteristic
200C2C2×#1914960712yes—
subgroupG/H
centerC2×C4C2×D692
commutatorC2×C4×C173C22×C4
FrattiniC22×C4C22×D173
FittingC22×C42×C173C2
socleC22×C173C2×C22⋊C4
prime radical#26012002C2
cosocleC22×C4×C173C23
hypercenterC22×C42D173
hypocenterC173#1896
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
2. C2×#19149607C2semidirect
3. #26011455C2semidirect
4. #26011251C2semidirect
5. C22×C42×C173C2semidirect
9. #19149658C4semidirect
10. #19149653C4semidirect
23. C2×Dic346C2×C4semidirect
24. C2×D692C2×C4semidirect
31. C4⋊Dic173C2×C4semidirect
49. C42×C173D4semidirect
82. C22×C173#104semidirect
100. C173#1896semidirect
103. C22×C42D173semidirect
150. C42C22⋊D173semidirect
185. C22#19149607semidirect

Maximal subgroups

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

subgroupindex
#260120022
C2×#191496072
#260114552
#260112512
C22×C42×C1732
#1896173