Finite Group Databasebeta

← #36369541

Subgroups — #36369541

Order 22144.

408 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 92 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×#14583836C211yesmaximal, characteristic, complemented
2C2×#19149736C211yesmaximal, characteristic, complemented
3C2×C22⋊#13063443C212yesmaximal, complemented
4#26012357C212yesmaximal, complemented
5C2×#19149639C211yesmaximal, characteristic
6C2×#14583836—24no—
7C2×#14583836—22no—
8C22×#13063443C2212yes—
9C22×#13063443C2212yes—
10#19149736—22no—
11C22⋊#13063443—24no—
12C22×C22⋊D173C411yescharacteristic
13C23×Dic173C2211yescharacteristic
14C22×C4×D173C2211yescharacteristic
15#19149639—22no—
16#19149639—22no—
17C2×#14583805C411yescharacteristic
18C2×#14583803C411yescharacteristic
19C2×C4×Dic173C2211yescharacteristic
20C2×C22⋊C4×C173C411yescharacteristic
21#14583836—28no—
22C2×#13063443—22no—
23C2×#13063443—22no—
24C2×#13063443—22no—
25C2×#13063443—22no—
26C2×#13063443—22no—
27C2×#13063443D414yescomplemented
28C23×D173C2×C411yescharacteristic
29C2×C22⋊D173—22no—
30C2×C22⋊D173—22no—
31C2×C22⋊D173—22no—
32C22×Dic173—21no—
33C22×Dic173—21no—
34C22×Dic173—22no—
35C22×Dic173—22no—
36C22×Dic173C2×C411yescharacteristic
37C22×Dic173C2×C411yescharacteristic
38C22×Dic173C2311yescharacteristic
39C2×C4×D173—22no—
40C2×C4×D173—22no—
41C2×C4×D173D412yes—
42C2×C4×D173D412yes—
43#14583805—22no—
44#14583803—22no—
45C4×Dic173—21no—
46C4×Dic173—21no—
47C24×C173C2×C411yescharacteristic
48C22×C4×C173C2×C411yescharacteristic
49C22×C4×C173C2×C411yescharacteristic
50C22⋊C4×C173—22no—
51#13063443—24no—
52#13063443—42no—
53C22×D173—22no—
54C22×D173—21no—
55C22×D173—21no—
56C22×D173—22no—
57C22×D173C22⋊C412yes—
58C22⋊D173—28no—
59C2×Dic173—21no—
60C2×Dic173—21no—
61C2×Dic173—21no—
62C2×Dic173—22no—
63C2×Dic173—41no—
64C2×Dic173—42no—
65C2×Dic173—41no—
66C2×Dic173M1612yescomplemented
67C2×Dic173C2×D412yes—
68C2×Dic173C2×D411yescharacteristic
69C2×Dic173C2×D411yescharacteristic
70C2×Dic173#4112yes—
71C4×D173—22no—
72C4×D173—24no—
73C4×D173—22no—
74C23×C173—21no—
75C23×C173—22no—
76C23×C173—21no—
77C23×C173—22no—
78C23×C173M1612yescomplemented
79C23×C173C22×C411yescharacteristic
80C2×C4×C173—22no—
81C2×C4×C173—21no—
82C2×C4×C173—22no—
83C2×C4×C173C22⋊C412yes—
84C22×C173—22no—
85C22×C173—24no—
86C22×C173—41no—
87C22×C173—41no—
88C22×C173—42no—
89C22×C173C2×C22⋊C411yescharacteristic
90C22×C173C4×D412yes—
91C22×C173C2×M1611yescharacteristic
92C22×C173C2×M1611yescharacteristic
93C22×C173#13112yes—
94C2×D173—22no—
95C2×D173—22no—
96C2×D173—22no—
97C2×D173—42no—
98C2×D173—42no—
99C4×C173—22no—
100C4×C173—41no—
101Dic173—22no—
102Dic173—41no—
103Dic173#12011yescharacteristic
104Dic173#12112yes—
105Dic173#12311yescharacteristic
106C2×C173—42no—
107C2×C173#38611yescharacteristic
108C2×C173#40711yescharacteristic
109C2×C173#40811yescharacteristic
110C2×C173#43512yes—
111C2×C173#43612yes—
112D173—42no—
113C173#187011yescharacteristic, complemented, Sylow
114#1870—1731nomaximal, Sylow
115C22×M16—1731no—
116C2×C4×D4—1731no—
117C2×C22⋊C8—1732no—
118C2×C22⋊C8—1731no—
119#336—1732no—
120C22×D4—1731no—
121C23×C4—1731no—
122C23×C4—1731no—
123C2×M16—3462no—
124C2×M16—3464no—
125C22×C8—1732no—
126C22×C8—1732no—
127C4×D4—3462no—
128C4×D4—3462no—
129C2×C4⋊C4—1731no—
130C2×C22⋊C4—1731no—
131C2×C22⋊C4#1062192411yescharacteristic
132C2×C42—1731no—
133C22⋊C8—3462no—
134C22⋊C8—3464no—
135C24—1731no—
136C24C2×#1062192411yescharacteristic
137C2×D4—3462no—
138C2×D4—3462no—
139C2×D4—3462no—
140C22×C4—3462no—
141C22×C4—3462no—
142C22×C4—3462no—
143C22×C4—3461no—
144C22×C4—3461no—
145C22×C4—3462no—
146C22×C4—1731no—
147C22×C4—1731no—
148C22×C4—1732no—
149C22×C4—1732no—
150C22×C4—1731no—
151C22×C4C2×#1062192411yescharacteristic
152C22×C4C2×#1062192411yescharacteristic
153M16—3468no—
154C2×C8—3462no—
155C2×C8—3462no—
156C2×C8—3462no—
157C2×C8—3462no—
158C2×C8—3462no—
159C2×C8—1734no—
160C4⋊C4—3462no—
161C22⋊C4—22no—
162C22⋊C4—3462no—
163C42—3461no—
164C42—3461no—
165C23—21no—
166C23—22no—
167C23—21no—
168C23—22no—
169C23—3462no—
170C23—3461no—
171C23—3461no—
172C23—3462no—
173C23—1732no—
174C23#1458383612yescomplemented
175C23C22×#1062192411yescharacteristic
176D4—3468no—
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—3462no—
181C2×C4—3464no—
182C2×C4—3462no—
183C2×C4—3461no—
184C2×C4—3461no—
185C2×C4—3461no—
186C2×C4—3462no—
187C2×C4—1732no—
188C2×C4—1731no—
189C2×C4—1731no—
190C2×C4—1732no—
191C2×C4—1732no—
192C2×C4—6921no—
193C2×C4—6922no—
194C2×C4—6921no—
195C2×C4C22⋊#1062192412yes—
196C8—3464no—
197C8—6922no—
198C22—22no—
199C22—24no—
200C22—41no—
201C22—41no—
202C22—42no—
203C22—3462no—
204C22—3462no—
205C22—3462no—
206C22—6922no—
207C22—6922no—
208C22C2×C22⋊#1062192411yescharacteristic
209C22D4×#1062192412yes—
210C22D173⋊M1611yescharacteristic
211C22C2×#1458383611yescharacteristic
212C22#1914976612yes—
213C4—22no—
214C4—41no—
215C4—3462no—
216C4—1731no—
217C4—1732no—
218C4—1731no—
219C4—6921no—
220C2—42no—
221C2—6922no—
222C2#2601243911yescharacteristic
223C2#2601246211yescharacteristic
224C2#2601246511yescharacteristic
225C2#2601252012yes—
226C2#2601252412yes—
subgroupG/H
center, FrattiniC23C22×#10621924
commutator, socleC23×C173C22×C4
FittingC2×C22⋊C4×C173C4
prime radicalC22×C22⋊D173C4
cosocleC22×Dic173C23
hypercenterC2×C22⋊C4#10621924
hypocenterC173#1870
LCSG ▷ C23×C173 ▷ C173
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#14583836C2semidirect
2. C2×#19149736C2semidirect
3. C2×C22⋊#13063443C2semidirect
4. #26012357C2semidirect
27. C2×#13063443D4semidirect
66. C2×Dic173M16semidirect
78. C23×C173M16semidirect
113. C173#1870semidirect
174. C23#14583836semidirect

Maximal subgroups

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

subgroupindex
C22×#145838362
C2×#191497362
C2×C22⋊#130634432
#260123572
C2×#191496392
#1870173