Finite Group Databasebeta

← #36368968

Subgroups — #36368968

Order 22144.

326 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 60 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#26012463C211yesmaximal, characteristic, complemented
2#26012434C211yesmaximal, characteristic, complemented
3#26012342C214yesmaximal, complemented
4C2×#19149650C211yesmaximal, characteristic
5C2×#14583835—21no—
6C2×C22⋊#10621924—21no—
7#19149736—21no—
8#19149736C2212yescomplemented
9(C2×#10621924)⋊C4—21no—
10(C2×#10621924)⋊C4C2212yescomplemented
11C22×C22⋊D173—21no—
12C22×C4×D173C2211yescharacteristic
13#19149650—21no—
14#19149650—21no—
15#19149650C412yescomplemented
16#19149650C2212yes—
17C2×C4⋊Dic173C411yescharacteristic, complemented
18C2×C22⋊Dic173—21no—
19C22×D4×C173C411yescharacteristic, complemented
20C22×#10621924—22no—
21#14583835—42no—
22C2×#13063443—22no—
23C22⋊#10621924—42no—
24C23×D173D411yescharacteristic
25C2×C22⋊D173—22no—
26C2×C22⋊D173—22no—
27C2×C22⋊D173—44no—
28C22×Dic173—21no—
29C22×Dic173D411yescharacteristic
30C2×C4×D173—42no—
31C2×C4×D173C2×C411yescharacteristic
32C2×C4×D173D411yescharacteristic
33C2×C4×D173D411yescharacteristic
34C2×C4×D173C2311yescharacteristic
35C4⋊Dic173—21no—
36C4⋊Dic173C2×C412yes—
37C22⋊Dic173—22no—
38C22⋊Dic173—22no—
39C24×C173—21no—
40C2×D4×C173—21no—
41C2×D4×C173—42no—
42C2×D4×C173C2×C412yes—
43C22×C4×C173C2×C411yescharacteristic
44C2×#10621924—42no—
45C2×#10621924—42no—
46C2×#10621924—42no—
47#13063443—42no—
48C22×D173—22no—
49C22×D173—22no—
50C22×D173—41no—
51C22×D173C22⋊C411yescharacteristic
52C22×D173C2×D411yescharacteristic
53C22⋊D173—44no—
54C22⋊D173—44no—
55C2×Dic173—22no—
56C2×Dic173—22no—
57C2×Dic173—42no—
58C2×Dic173C22⋊C411yescharacteristic
59C2×Dic173C2×D411yescharacteristic
60C4×D173—41no—
61C4×D173—41no—
62C4×D173—41no—
63C4×D173—41no—
64C23×C173—22no—
65C23×C173—42no—
66C23×C173—42no—
67C23×C173—42no—
68C23×C173C22⋊C411yescharacteristic
69D4×C173—42no—
70D4×C173—42no—
71C2×C4×C173—22no—
72C2×C4×C173C22⋊C411yescharacteristic
73C2×C4×C173C22×C411yescharacteristic
74C22×C173—21no—
75C22×C173—21no—
76C22×C173—42no—
77C22×C173—42no—
78C22×C173—42no—
79C22×C173—84no—
80C22×C173#9912yes—
81C22×C173C2×C22⋊C411yescharacteristic
82C2×D173—21no—
83C2×D173—21no—
84C2×D173—21no—
85C2×D173—21no—
86C2×D173—21no—
87C2×D173—44no—
88C2×D173AGL(1,ℤ/8)12yes—
89#10621924—82no—
90C4×C173—21no—
91C4×C173—21no—
92Dic173—21no—
93Dic173—21no—
94Dic173—42no—
95C2×C173—22no—
96C2×C173—82no—
97C2×C173C2×#9911yescharacteristic
98C2×C173#41811yescharacteristic
99C2×C173#42111yescharacteristic
100D173—22no—
101D173—41no—
102C173#151011yescharacteristic, complemented, Sylow
103#1510—1731nomaximal, Sylow
104C2×#121—1731no—
105#407—1731no—
106#381—1731no—
107#327—1734no—
108C22×D4—3461no—
109C22×D4#1062192411yescharacteristic, complemented
110C23×C4—1731no—
111C2×M16—3461no—
112#121—3461no—
113#121—3461no—
114#121—1732no—
115#121—1732no—
116C2×C4⋊C4—1731no—
117C2×C22⋊C4—3461no—
118C2×C22⋊C4—3461no—
119C22⋊C8—3461no—
120C22⋊C8—1732no—
121#95—3461no—
122#95—1732no—
123C24—21no—
124C24—1731no—
125C2×D4—21no—
126C2×D4—42no—
127C2×D4—3462no—
128C2×D4—3462no—
129C2×D4—6924no—
130C2×D4C2×#1062192412yes—
131C22×C4—3461no—
132C22×C4—3462no—
133C22×C4—1731no—
134C22×C4—1731no—
135C22×C4—1731no—
136C22×C4—1731no—
137C22×C4—1731no—
138C22×C4—6922no—
139C22×C4C2×#1062192411yescharacteristic
140M16—6922no—
141C2×C8—3462no—
142C4⋊C4—3461no—
143C4⋊C4—1732no—
144C22⋊C4—3462no—
145C22⋊C4—3462no—
146C22⋊C4—6922no—
147C23—22no—
148C23—42no—
149C23—42no—
150C23—42no—
151C23—3462no—
152C23—3462no—
153C23—1731no—
154C23—1731no—
155C23—6921no—
156C23C22⋊#1062192411yescharacteristic
157D4—42no—
158D4—42no—
159D4—6924no—
160D4—6924no—
161C2×C4—22no—
162C2×C4—3462no—
163C2×C4—3462no—
164C2×C4—1731no—
165C2×C4—1731no—
166C2×C4—6921no—
167C2×C4—6921no—
168C2×C4—6921no—
169C2×C4—6921no—
170C2×C4—6922no—
171C2×C4—6922no—
172C2×C4—6922no—
173C2×C4—6922no—
174C2×C4C22⋊#1062192411yescharacteristic
175C2×C4C22×#1062192411yescharacteristic
176C8—6922no—
177C22—21no—
178C22—21no—
179C22—42no—
180C22—42no—
181C22—42no—
182C22—84no—
183C22—3461no—
184C22—3461no—
185C22—3461no—
186C22—3461no—
187C22—3461no—
188C22—1732no—
189C22—6924no—
190C22#1914973712yes—
191C22C2×C22⋊#1062192411yescharacteristic
192C4—21no—
193C4—21no—
194C4—3461no—
195C4—3461no—
196C4—13842no—
197C4—6922no—
198C2—22no—
199C2—82no—
200C2—3462no—
201C2—6921no—
202C2C2×#1914973711yescharacteristic
203C2#2601248311yescharacteristic
204C2#2601248911yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
FittingC22×D4×C173C4
socleC22×C173C2×C22⋊C4
prime radicalC2×#19149650C2
cosocleC2×C4×D173C23
hypercenterC22×D4#10621924
hypocenterC173#1510
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. #26012463C2semidirect
2. #26012434C2semidirect
3. #26012342C2semidirect
8. #19149736C22semidirect
10. (C2×#10621924)⋊C4C22semidirect
15. #19149650C4semidirect
17. C2×C4⋊Dic173C4semidirect
19. C22×D4×C173C4semidirect
102. C173#1510semidirect
109. C22×D4#10621924semidirect

Maximal subgroups

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

subgroupindex
#260124632
#260124342
#260123422
C2×#191496502
#1510173