Finite Group Databasebeta

← #36368842

Subgroups — #36368842

Order 22144.

270 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 64 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#26012461C211yesmaximal, characteristic
2C2×C22⋊#13063443C211yesmaximal, characteristic
3#26012337C212yesmaximal, complemented
4#26012336C212yesmaximal, complemented
5C2×#19149656C211yesmaximal, characteristic
6C2×#14583835—21no—
7C22×#13063443—21no—
8C22⋊#13063443—21no—
9C22⋊#13063443—21no—
10C22⋊#13063443C2212yes—
11C22⋊#13063443C2212yes—
12C22×Dic346C411yescharacteristic
13C23×Dic173C2211yescharacteristic
14#19149656—21no—
15#19149656—21no—
16#19149656—21no—
17#19149656—21no—
18C2×C4⋊Dic173C2211yescharacteristic
19C2×#14583805C411yescharacteristic
20C2×#14583805C2211yescharacteristic
21C2×C22⋊Dic173C411yescharacteristic
22C2×C22⋊C4×C173C411yescharacteristic
23#14583835—42no—
24C2×#13063443—22no—
25C2×#13063443—22no—
26C2×#13063443—42no—
27C2×Dic346—21no—
28C2×Dic346—21no—
29C2×Dic346—42no—
30C22×Dic173—42no—
31C22×Dic173C2×C411yescharacteristic
32C22×Dic173C2×C411yescharacteristic
33C22×Dic173C2×C411yescharacteristic
34C22×Dic173D411yescharacteristic
35C22×Dic173D411yescharacteristic
36C22×Dic173D411yescharacteristic
37C22×Dic173D411yescharacteristic
38C22×Dic173C2311yescharacteristic
39C4⋊Dic173—21no—
40C4⋊Dic173—21no—
41#14583805—21no—
42#14583805—21no—
43#14583805—21no—
44#14583805—21no—
45C22⋊Dic173—21no—
46C22⋊Dic173—21no—
47C24×C173C2×C411yescharacteristic
48C22×C4×C173C2×C411yescharacteristic
49C22×C4×C173C2×C411yescharacteristic
50C22⋊C4×C173—21no—
51C22⋊C4×C173—21no—
52#13063443—42no—
53#13063443—42no—
54Dic346—42no—
55Dic346—42no—
56C2×Dic173—22no—
57C2×Dic173—22no—
58C2×Dic173—22no—
59C2×Dic173—21no—
60C2×Dic173—21no—
61C2×Dic173—22no—
62C2×Dic173—41no—
63C2×Dic173—41no—
64C2×Dic173—41no—
65C2×Dic173—41no—
66C2×Dic173C22⋊C411yescharacteristic
67C2×Dic173C22⋊C411yescharacteristic
68C2×Dic173C2×D411yescharacteristic
69C2×Dic173C2×D411yescharacteristic
70C23×C173—22no—
71C23×C173—22no—
72C23×C173—41no—
73C23×C173C22⋊C411yescharacteristic
74C23×C173C22⋊C411yescharacteristic
75C23×C173C22×C411yescharacteristic
76C2×C4×C173—21no—
77C2×C4×C173—21no—
78C2×C4×C173—22no—
79C2×C4×C173—41no—
80C22×C173—21no—
81C22×C173—21no—
82C22×C173—21no—
83C22×C173—21no—
84C22×C173—21no—
85C22×C173—21no—
86C22×C173—21no—
87C22×C173—44no—
88C22×C173#10112yes—
89C22×C173C2×C22⋊C411yescharacteristic
90C22×C173#13112yes—
91C4×C173—41no—
92C4×C173—41no—
93Dic173—21no—
94Dic173—21no—
95Dic173—21no—
96Dic173—41no—
97Dic173—41no—
98Dic173#9912yes—
99C2×C173—22no—
100C2×C173—22no—
101C2×C173—41no—
102C2×C173#40811yescharacteristic
103C2×C173C2×#9911yescharacteristic
104C2×C173C2×#10111yescharacteristic
105C173#147211yescharacteristic, complemented, Sylow
106#1472—1731nomaximal, Sylow
107C2×C22⋊Q8—1731no—
108#407—1731no—
109C2×C22⋊C8—1731no—
110#324—1732no—
111#324—1732no—
112C22×Q8—1731no—
113C23×C4—1731no—
114C2×M16—3461no—
115C22×C8—3461no—
116C22⋊Q8—3461no—
117C22⋊Q8—3461no—
118C22⋊Q8—3461no—
119C22⋊Q8—3461no—
120C2×C4⋊C4—1731no—
121C2×C4⋊C4—1731no—
122C2×C4⋊C4—1731no—
123C2×C22⋊C4—1731no—
124C2×C22⋊C4#1062192411yescharacteristic
125C22⋊C8—3461no—
126C22⋊C8—3461no—
127C22⋊C8—1732no—
128C22⋊C8—1732no—
129C24C2×#1062192411yescharacteristic
130C2×Q8—3461no—
131C2×Q8—3461no—
132C2×Q8—6922no—
133C22×C4—1731no—
134C22×C4—1731no—
135C22×C4—1731no—
136C22×C4—1731no—
137C22×C4—1731no—
138C22×C4—1731no—
139C22×C4—1731no—
140C22×C4—1731no—
141C22×C4—6922no—
142C22×C4C2×#1062192411yescharacteristic
143C22×C4C2×#1062192411yescharacteristic
144M16—6922no—
145C2×C8—3462no—
146C2×C8—3462no—
147C2×C8—6922no—
148C4⋊C4—3461no—
149C4⋊C4—3461no—
150C4⋊C4—3461no—
151C4⋊C4—3461no—
152C4⋊C4—3461no—
153C4⋊C4—3461no—
154C22⋊C4—21no—
155C22⋊C4—21no—
156C22⋊C4—3461no—
157C22⋊C4—3461no—
158C23—22no—
159C23—22no—
160C23—41no—
161C23C22⋊#1062192411yescharacteristic
162C23C22⋊#1062192411yescharacteristic
163C23C22×#1062192411yescharacteristic
164Q8—6922no—
165Q8—6922no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—22no—
169C2×C4—41no—
170C2×C4—3462no—
171C2×C4—3462no—
172C2×C4—3462no—
173C2×C4—3461no—
174C2×C4—3461no—
175C2×C4—3462no—
176C2×C4—1731no—
177C2×C4—1731no—
178C2×C4—1731no—
179C2×C4—1731no—
180C2×C4—6921no—
181C2×C4—6921no—
182C2×C4—6921no—
183C2×C4—6921no—
184C8—6922no—
185C8—6922no—
186C22—21no—
187C22—21no—
188C22—21no—
189C22—21no—
190C22—21no—
191C22—21no—
192C22—21no—
193C22—44no—
194C22#1914974012yes—
195C22C2×C22⋊#1062192411yescharacteristic
196C22#1914976612yes—
197C4—41no—
198C4—41no—
199C4—3461no—
200C4—3461no—
201C4—3461no—
202C4—1732no—
203C4—6921no—
204C4—6921no—
205C2—22no—
206C2—22no—
207C2—41no—
208C2#2601246411yescharacteristic
209C2D173⋊#9911yescharacteristic
210C2C2×#1914974011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC23×C173C22×C4
FrattiniC23C22×#10621924
FittingC2×C22⋊C4×C173C4
socleC22×C173C2×C22⋊C4
prime radicalC24×C173C2×C4
cosocleC22×Dic173C23
hypercenterC2×C22⋊C4#10621924
hypocenterC173#1472
LCSG ▷ C23×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C24 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. #26012337C2semidirect
4. #26012336C2semidirect
105. C173#1472semidirect

Maximal subgroups

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

subgroupindex
#260124612
C2×C22⋊#130634432
#260123372
#260123362
C2×#191496562
#1472173