Finite Group Databasebeta

← #36369753

Subgroups — #36369753

Order 22144.

324 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 74 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#26012433C211yesmaximal, characteristic
2C4×C22⋊#10621924C211yesmaximal, characteristic
3#26012369C214yesmaximal
4C2×#19149659C211yesmaximal, characteristic
5C2×C4⋊#10621924—21no—
6C2×C22⋊#10621924C2212yes—
7C2×C22⋊#10621924C2212yes—
8C2×C4×#10621924—21no—
9(C2×#10621924)⋊C4—21no—
10(C2×#10621924)⋊C4—21no—
11C22×Dic346C411yescharacteristic, complemented
12C22×C4×D173C2211yescharacteristic
13#19149659—21no—
14#19149659—21no—
15#19149659—21no—
16#19149659—21no—
17C2×C4⋊Dic173C412yescomplemented
18C2×#14583803C2212yes—
19C2×C4⋊C4×C173C411yescharacteristic, complemented
20C22×#10621924—22no—
21C22×#10621924—22no—
22C4⋊#10621924—42no—
23C22⋊#10621924—22no—
24C22⋊#10621924—24no—
25C22⋊#10621924Q814yescomplemented
26C4×#10621924—42no—
27C23×D173C2311yescharacteristic
28C2×Dic346—21no—
29C2×Dic346—21no—
30C2×Dic346—42no—
31C22×Dic173C2×C412yes—
32C22×Dic173C2×C411yescharacteristic
33C2×C4×D173—42no—
34C2×C4×D173D411yescharacteristic
35C2×C4×D173D411yescharacteristic
36C2×C4×D173D411yescharacteristic
37C2×C4×D173D411yescharacteristic
38C4⋊Dic173—22no—
39C4⋊Dic173—22no—
40#14583803—22no—
41#14583803—22no—
42C22×C4×C173C2×C411yescharacteristic
43C22×C4×C173C2×C412yes—
44C4⋊C4×C173—21no—
45C4⋊C4×C173—21no—
46C2×#10621924—24no—
47C2×#10621924—42no—
48C2×#10621924—42no—
49C2×#10621924—42no—
50C2×#10621924—42no—
51C2×#10621924—42no—
52C22×D173—41no—
53C22×D173—41no—
54C22×D173C2×D411yescharacteristic
55C22×D173C2×D411yescharacteristic
56C22×D173C2×Q812yes—
57C22×D173#4112yes—
58Dic346—42no—
59Dic346—42no—
60C2×Dic173—22no—
61C2×Dic173—41no—
62C2×Dic173—42no—
63C2×Dic173C22⋊C411yescharacteristic
64C2×Dic173C22⋊C411yescharacteristic
65C4×D173—22no—
66C4×D173—22no—
67C4×D173—41no—
68C4×D173—41no—
69C23×C173C22×C411yescharacteristic
70C2×C4×C173—22no—
71C2×C4×C173—22no—
72C2×C4×C173—42no—
73C2×C4×C173C22⋊C411yescharacteristic
74C2×C4×C173C22⋊C411yescharacteristic
75C22×C173—21no—
76C22×C173—21no—
77C22×C173C2×C22⋊C411yescharacteristic
78C22×C173C4×Q812yes—
79C2×D173—21no—
80C2×D173—21no—
81C2×D173—21no—
82C2×D173—21no—
83C2×D173—42no—
84C2×D173—42no—
85C2×D173C22⋊Q812yes—
86C2×D173#12411yescharacteristic
87C2×D173C4⋊Q811yescharacteristic
88#10621924—44no—
89#10621924—82no—
90C4×C173—21no—
91C4×C173—42no—
92C4×C173#9912yes—
93Dic173—21no—
94Dic173—21no—
95Dic173—42no—
96C2×C173—22no—
97C2×C173#39111yescharacteristic
98C2×C173C2×#9911yescharacteristic
99C2×C173#41011yescharacteristic
100D173—22no—
101D173—21no—
102D173—21no—
103C173#199611yescharacteristic, complemented, Sylow
104#1996—1731nomaximal, Sylow
105C2×C22⋊Q8—1731no—
106#380—1731no—
107C4×C22⋊C4—1731no—
108#342—1734no—
109C22×Q8—1731no—
110C23×C4—1731no—
111C22⋊Q8—3461no—
112C22⋊Q8—3461no—
113C22⋊Q8—3461no—
114C22⋊Q8—3461no—
115C2×C4⋊C4—3461no—
116C2×C4⋊C4—1732no—
117C2×C4⋊C4#1062192411yescharacteristic, complemented
118C2×C22⋊C4—1732no—
119C2×C22⋊C4—1732no—
120C2×C22⋊C4—1732no—
121C2×C42—3461no—
122#95—3461no—
123#95—3461no—
124C24—1731no—
125C2×Q8—3461no—
126C2×Q8—3461no—
127C2×Q8—6922no—
128C22×C4—3462no—
129C22×C4—3462no—
130C22×C4—1731no—
131C22×C4—1731no—
132C22×C4—1732no—
133C22×C4—1731no—
134C22×C4—1731no—
135C22×C4—1731no—
136C22×C4—6922no—
137C22×C4C2×#1062192412yes—
138C22×C4C2×#1062192411yescharacteristic
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4—3462no—
142C4⋊C4—3462no—
143C4⋊C4—6922no—
144C22⋊C4—3462no—
145C22⋊C4—3462no—
146C22⋊C4—3462no—
147C22⋊C4—3464no—
148C22⋊C4—1734no—
149C42—6922no—
150C23—1731no—
151C23—1731no—
152C23—1732no—
153C23—1732no—
154C23—6921no—
155C23—6921no—
156C23C22×#1062192411yescharacteristic
157Q8—6922no—
158Q8—6922no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—42no—
162C2×C4—3462no—
163C2×C4—3462no—
164C2×C4—3464no—
165C2×C4—3462no—
166C2×C4—1731no—
167C2×C4—1731no—
168C2×C4—6921no—
169C2×C4—6921no—
170C2×C4—6921no—
171C2×C4—6922no—
172C2×C4—6922no—
173C2×C4—6922no—
174C2×C4—6922no—
175C2×C4—6922no—
176C2×C4—6922no—
177C2×C4C22⋊#1062192411yescharacteristic
178C2×C4C22⋊#1062192411yescharacteristic
179C22—21no—
180C22—21no—
181C22—3461no—
182C22—3461no—
183C22—3461no—
184C22—3461no—
185C22—1732no—
186C22—1731no—
187C22—1731no—
188C22—6922no—
189C22—6922no—
190C22C2×C22⋊#1062192411yescharacteristic
191C22Q8×#1062192412yes—
192C4—21no—
193C4—42no—
194C4—3461no—
195C4—3461no—
196C4—13842no—
197C4—6924no—
198C4—6922no—
199C4#1914973812yes—
200C2—22no—
201C2—3461no—
202C2—3462no—
203C2—3461no—
204C2(C2×#10621924)⋊Q811yescharacteristic
205C2C2×#1914973811yescharacteristic
206C2#2601247111yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC23×C173C22×C4
FrattiniC23C22×#10621924
FittingC2×C4⋊C4×C173C4
socleC22×C173C2×C22⋊C4
prime radical, cosocleC23×D173C23
hypercenterC2×C4⋊C4#10621924
hypocenterC173#1996
LCSG ▷ C23×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
11. C22×Dic346C4semidirect
17. C2×C4⋊Dic173C4semidirect
19. C2×C4⋊C4×C173C4semidirect
25. C22⋊#10621924Q8semidirect
103. C173#1996semidirect
117. C2×C4⋊C4#10621924semidirect

Maximal subgroups

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

subgroupindex
#260124332
C4×C22⋊#106219242
#260123692
C2×#191496592
#1996173