Finite Group Databasebeta

← #36369590

Subgroups — #36369590

Order 22144.

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

normal subgroups only

This group has 338 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
1C22×#14583835C211yesmaximal, characteristic
2C2×#19149740C212yesmaximal, complemented
3#26012461C211yesmaximal, characteristic
4#26012371C212yesmaximal, complemented
5C2×#19149656C211yesmaximal, characteristic
6C2×#14583835—24no—
7C2×#14583835—21no—
8C2×#14583835C2212yes—
9C2×#14583835C2212yes—
10C22×#13063443—21no—
11#19149740—24no—
12C22⋊#13063443—21no—
13C22⋊#13063443—21no—
14C22×Dic346C2211yescharacteristic
15C23×Dic173C2211yescharacteristic
16#19149656—21no—
17#19149656—21no—
18#19149656—21no—
19#19149656—21no—
20C2×C4⋊Dic173C411yescharacteristic
21C2×#14583805C411yescharacteristic
22C2×#14583805C2211yescharacteristic
23C2×C22⋊Dic173C411yescharacteristic
24C2×C22⋊C4×C173C411yescharacteristic
25#14583835—24no—
26#14583835—22no—
27#14583835—22no—
28#14583835—22no—
29#14583835D414yes—
30C2×#13063443—24no—
31C2×#13063443—22no—
32C2×#13063443—22no—
33C2×Dic346—21no—
34C2×Dic346—21no—
35C2×Dic346—22no—
36C2×Dic346—22no—
37C22×Dic173—22no—
38C22×Dic173—22no—
39C22×Dic173C2×C411yescharacteristic
40C22×Dic173C2×C411yescharacteristic
41C22×Dic173C2×C411yescharacteristic
42C22×Dic173D411yescharacteristic
43C22×Dic173D411yescharacteristic
44C22×Dic173D411yescharacteristic
45C22×Dic173D411yescharacteristic
46C22×Dic173C2311yescharacteristic
47C4⋊Dic173—21no—
48C4⋊Dic173—21no—
49#14583805—21no—
50#14583805—21no—
51#14583805—21no—
52#14583805—21no—
53C22⋊Dic173—21no—
54C22⋊Dic173—21no—
55C24×C173C2×C411yescharacteristic
56C22×C4×C173C2×C411yescharacteristic
57C22×C4×C173C2×C411yescharacteristic
58C22⋊C4×C173—21no—
59C22⋊C4×C173—21no—
60#13063443—24no—
61#13063443—42no—
62Dic346—41no—
63Dic346—42no—
64Dic346—41no—
65C2×Dic173—22no—
66C2×Dic173—21no—
67C2×Dic173—21no—
68C2×Dic173—22no—
69C2×Dic173—22no—
70C2×Dic173—21no—
71C2×Dic173—21no—
72C2×Dic173—21no—
73C2×Dic173—21no—
74C2×Dic173—22no—
75C2×Dic173—41no—
76C2×Dic173C22⋊C411yescharacteristic
77C2×Dic173C22⋊C411yescharacteristic
78C2×Dic173C2×D412yes—
79C2×Dic173C2×D411yescharacteristic
80C2×Dic173C2×D411yescharacteristic
81C2×Dic173#4112yes—
82C23×C173—22no—
83C23×C173—22no—
84C23×C173—41no—
85C23×C173C22⋊C411yescharacteristic
86C23×C173C22⋊C411yescharacteristic
87C23×C173C22×C411yescharacteristic
88C2×C4×C173—21no—
89C2×C4×C173—21no—
90C2×C4×C173—22no—
91C2×C4×C173—41no—
92C22×C173—21no—
93C22×C173—21no—
94C22×C173—21no—
95C22×C173—21no—
96C22×C173—21no—
97C22×C173—21no—
98C22×C173—21no—
99C22×C173—44no—
100C22×C173#10112yescomplemented
101C22×C173C2×C22⋊C411yescharacteristic
102C22×C173C4×D412yes—
103C4×C173—41no—
104C4×C173—41no—
105Dic173—21no—
106Dic173—21no—
107Dic173—41no—
108Dic173—41no—
109Dic173#12011yescharacteristic
110Dic173#12112yes—
111Dic173#12311yescharacteristic
112C2×C173—22no—
113C2×C173—22no—
114C2×C173—41no—
115C2×C173#38611yescharacteristic
116C2×C173C2×#10111yescharacteristic
117C2×C173#41311yescharacteristic
118C173#190111yescharacteristic, complemented, Sylow
119#1901—1731nomaximal, Sylow
120C22×M16—1731no—
121C2×C22⋊Q8—1731no—
122C2×#101—1732no—
123#407—1731no—
124#343—1732no—
125C22×Q8—1731no—
126C23×C4—1731no—
127C2×M16—3461no—
128C2×M16—3464no—
129C2×M16—1732no—
130C2×M16—1732no—
131C22×C8—3461no—
132C22⋊Q8—3461no—
133C22⋊Q8—3461no—
134C22⋊Q8—3461no—
135C22⋊Q8—3461no—
136C2×C4⋊C4—1731no—
137C2×C4⋊C4—1731no—
138C2×C4⋊C4—1731no—
139C2×C22⋊C4—1731no—
140C2×C22⋊C4#1062192411yescharacteristic
141#101—3464no—
142C22⋊C8—3461no—
143C22⋊C8—3461no—
144C24C2×#1062192411yescharacteristic
145C2×Q8—3462no—
146C2×Q8—3461no—
147C2×Q8—3461no—
148C2×Q8—3462no—
149C22×C4—3462no—
150C22×C4—3462no—
151C22×C4—1731no—
152C22×C4—1731no—
153C22×C4—1731no—
154C22×C4—1731no—
155C22×C4—1731no—
156C22×C4—1731no—
157C22×C4—1731no—
158C22×C4—1731no—
159C22×C4C2×#1062192411yescharacteristic
160C22×C4C2×#1062192411yescharacteristic
161M16—3464no—
162M16—3462no—
163M16—3462no—
164M16—3462no—
165M16—1734no—
166C2×C8—3464no—
167C2×C8—3462no—
168C2×C8—3462no—
169C4⋊C4—3461no—
170C4⋊C4—3461no—
171C4⋊C4—3461no—
172C4⋊C4—3461no—
173C4⋊C4—3461no—
174C4⋊C4—3461no—
175C22⋊C4—21no—
176C22⋊C4—21no—
177C22⋊C4—3461no—
178C22⋊C4—3461no—
179C23—22no—
180C23—22no—
181C23—41no—
182C23C22⋊#1062192411yescharacteristic
183C23C22⋊#1062192411yescharacteristic
184C23C22×#1062192411yescharacteristic
185Q8—6921no—
186Q8—6922no—
187Q8—6921no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—41no—
192C2×C4—3462no—
193C2×C4—3461no—
194C2×C4—3461no—
195C2×C4—3462no—
196C2×C4—3462no—
197C2×C4—3461no—
198C2×C4—3461no—
199C2×C4—3461no—
200C2×C4—3461no—
201C2×C4—3462no—
202C2×C4—1731no—
203C2×C4—1731no—
204C2×C4—1731no—
205C2×C4—1731no—
206C2×C4—1732no—
207C2×C4—1732no—
208C2×C4—6921no—
209C8—3464no—
210C8—6922no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—21no—
217C22—21no—
218C22—44no—
219C22#1914974012yescomplemented
220C22C2×C22⋊#1062192411yescharacteristic
221C22D4×#1062192412yes—
222C4—41no—
223C4—41no—
224C4—3461no—
225C4—3461no—
226C4—1731no—
227C4—1732no—
228C4—1731no—
229C4—6921no—
230C4—6921no—
231C2—22no—
232C2—22no—
233C2—41no—
234C2#2601243911yescharacteristic
235C2C2×#1914974011yescharacteristic
236C2#2601247811yescharacteristic
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#1901
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
2. C2×#19149740C2semidirect
4. #26012371C2semidirect
100. C22×C173#101semidirect
118. C173#1901semidirect
219. C22#19149740semidirect

Maximal subgroups

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

subgroupindex
C22×#145838352
C2×#191497402
#260124612
#260123712
C2×#191496562
#1901173