Finite Group Databasebeta

← #36369589

Subgroups — #36369589

Order 22144.

390 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 72 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
1C2×D173⋊M16C211yesmaximal, characteristic, complemented
2C2×#19149739C212yesmaximal, complemented
3#26012463C211yesmaximal, characteristic, complemented
4#26012371C212yesmaximal, complemented
5C2×#19149652C211yesmaximal, characteristic
6C2×#14583836—21no—
7D173⋊M16—24no—
8C2×#14583835C2212yes—
9C2×#14583835C2212yes—
10C2×D173⋊C8—21no—
11#19149739—24no—
12#19149736—21no—
13#19149736—21no—
14C22×D692C411yescharacteristic
15C22×C22⋊D173C411yescharacteristic
16C22×C22⋊D173C2211yescharacteristic
17C22×C4×D173C2211yescharacteristic
18#19149652—21no—
19#19149652—21no—
20#19149652—21no—
21#19149652—21no—
22C2×#14583805C2211yescharacteristic
23C2×#14583803C411yescharacteristic
24C2×C22⋊C4×C173C411yescharacteristic
25#14583836—24no—
26#14583835—22no—
27#14583835—22no—
28#14583835—22no—
29#14583835D414yescomplemented
30D173⋊C8—24no—
31C2×#13063443—22no—
32C2×#13063443—22no—
33C23×D173C2×C411yescharacteristic
34C23×D173C2×C411yescharacteristic
35C2×D692—21no—
36C2×D692—21no—
37C2×D692—42no—
38C2×C22⋊D173—21no—
39C2×C22⋊D173—21no—
40C2×C22⋊D173—21no—
41C2×C22⋊D173—21no—
42C2×C22⋊D173—22no—
43C2×C22⋊D173—22no—
44C2×C22⋊D173—42no—
45C22×Dic173C2×C411yescharacteristic
46C22×Dic173C2311yescharacteristic
47C2×C4×D173—22no—
48C2×C4×D173—22no—
49C2×C4×D173D411yescharacteristic
50C2×C4×D173D411yescharacteristic
51C2×C4×D173D411yescharacteristic
52C2×C4×D173D411yescharacteristic
53#14583805—21no—
54#14583805—21no—
55#14583803—21no—
56#14583803—21no—
57C24×C173C2×C411yescharacteristic
58C22×C4×C173C2×C411yescharacteristic
59C22×C4×C173C2×C411yescharacteristic
60C22⋊C4×C173—21no—
61C22⋊C4×C173—21no—
62#13063443—24no—
63#13063443—42no—
64C22×D173—22no—
65C22×D173—22no—
66C22×D173—21no—
67C22×D173—22no—
68C22×D173—41no—
69C22×D173—41no—
70C22×D173—41no—
71C22×D173C22⋊C411yescharacteristic
72C22×D173C22⋊C411yescharacteristic
73D692—42no—
74D692—42no—
75C22⋊D173—42no—
76C22⋊D173—42no—
77C22⋊D173—41no—
78C22⋊D173—42no—
79C22⋊D173—41no—
80C2×Dic173—21no—
81C2×Dic173—41no—
82C2×Dic173C2×D412yes—
83C2×Dic173C2×D411yescharacteristic
84C2×Dic173C2×D411yescharacteristic
85C2×Dic173#4112yes—
86C4×D173—21no—
87C4×D173—21no—
88C4×D173—22no—
89C4×D173—22no—
90C4×D173—21no—
91C4×D173—21no—
92C23×C173—21no—
93C23×C173—22no—
94C23×C173—41no—
95C23×C173—41no—
96C23×C173C22×C411yescharacteristic
97C2×C4×C173—22no—
98C2×C4×C173—21no—
99C2×C4×C173—41no—
100C2×C4×C173C22⋊C411yescharacteristic
101C2×C4×C173C22⋊C411yescharacteristic
102C22×C173—21no—
103C22×C173—21no—
104C22×C173—41no—
105C22×C173—41no—
106C22×C173—41no—
107C22×C173—82no—
108C22×C173C2×C22⋊C411yescharacteristic
109C22×C173C4×D412yes—
110C2×D173—21no—
111C2×D173—21no—
112C2×D173—21no—
113C2×D173—21no—
114C2×D173—21no—
115C2×D173—44no—
116C2×D173—41no—
117C2×D173—41no—
118C2×D173—41no—
119C2×D173—82no—
120C2×D173#10012yescomplemented
121C4×C173—21no—
122C4×C173—21no—
123C4×C173—41no—
124Dic173—41no—
125Dic173#12011yescharacteristic
126Dic173#12112yes—
127Dic173#12311yescharacteristic
128C2×C173—22no—
129C2×C173—81no—
130C2×C173#38611yescharacteristic
131C2×C173C2×#10011yescharacteristic
132C2×C173#41311yescharacteristic
133D173—22no—
134D173—41no—
135D173—81no—
136C173#190011yescharacteristic, complemented, Sylow
137#1900—1731nomaximal, Sylow
138C22×M16—1731no—
139C2×#121—1731no—
140C2×#100—1732no—
141#407—1731no—
142#343—1732no—
143C22×D4—1731no—
144C22×D4—1731no—
145C22×D4—1731no—
146C23×C4—1731no—
147C2×M16—3461no—
148C2×M16—3464no—
149C2×M16—1732no—
150C2×M16—1732no—
151C22×C8—3461no—
152#121—3461no—
153#121—3461no—
154#121—3461no—
155#121—3461no—
156C2×C4⋊C4—1731no—
157C2×C22⋊C4—1731no—
158C2×C22⋊C4#1062192411yescharacteristic
159#100—3464no—
160C22⋊C8—3461no—
161C22⋊C8—3461no—
162C24—1731no—
163C24—1731no—
164C24C2×#1062192411yescharacteristic
165C2×D4—3461no—
166C2×D4—3461no—
167C2×D4—3461no—
168C2×D4—3461no—
169C2×D4—3462no—
170C2×D4—3461no—
171C2×D4—3461no—
172C2×D4—3462no—
173C2×D4—6922no—
174C2×D4—6922no—
175C22×C4—3462no—
176C22×C4—3462no—
177C22×C4—1731no—
178C22×C4—1731no—
179C22×C4—1731no—
180C22×C4—1731no—
181C22×C4—1731no—
182C22×C4—1731no—
183C22×C4C2×#1062192411yescharacteristic
184C22×C4C2×#1062192411yescharacteristic
185M16—3464no—
186M16—3462no—
187M16—3462no—
188M16—3462no—
189M16—1734no—
190C2×C8—3464no—
191C2×C8—3462no—
192C2×C8—3462no—
193C4⋊C4—3461no—
194C4⋊C4—3461no—
195C22⋊C4—21no—
196C22⋊C4—21no—
197C22⋊C4—3461no—
198C22⋊C4—3461no—
199C23—21no—
200C23—22no—
201C23—41no—
202C23—41no—
203C23—3462no—
204C23—3462no—
205C23—3462no—
206C23—3461no—
207C23—1731no—
208C23—1731no—
209C23—6921no—
210C23—6921no—
211C23—6921no—
212C23C22×#1062192411yescharacteristic
213D4—6922no—
214D4—6922no—
215D4—6921no—
216D4—6922no—
217D4—6921no—
218D4—6922no—
219D4—6922no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—41no—
223C2×C4—3461no—
224C2×C4—3461no—
225C2×C4—3462no—
226C2×C4—3462no—
227C2×C4—3461no—
228C2×C4—3461no—
229C2×C4—3461no—
230C2×C4—1731no—
231C2×C4—1731no—
232C2×C4—1732no—
233C2×C4—1732no—
234C2×C4—6921no—
235C2×C4C22⋊#1062192411yescharacteristic
236C2×C4C22⋊#1062192411yescharacteristic
237C8—3464no—
238C8—6922no—
239C22—21no—
240C22—21no—
241C22—41no—
242C22—41no—
243C22—41no—
244C22—82no—
245C22—3461no—
246C22—3461no—
247C22—3461no—
248C22—3461no—
249C22—3461no—
250C22—13842no—
251C22—1732no—
252C22—6924no—
253C22—6921no—
254C22—6921no—
255C22—6921no—
256C22C2×C22⋊#1062192411yescharacteristic
257C22D4×#1062192412yes—
258C4—21no—
259C4—21no—
260C4—41no—
261C4—1731no—
262C4—1732no—
263C4—1731no—
264C4—6921no—
265C2—22no—
266C2—81no—
267C2—3462no—
268C2—13841no—
269C2—6921no—
270C2#2601243911yescharacteristic
271C2D173⋊#10011yescharacteristic
272C2#2601247711yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC23×C173C22×C4
FrattiniC23C22×#10621924
FittingC2×C22⋊C4×C173C4
socleC22×C173C2×C22⋊C4
prime radicalC2×#19149652C2
cosocleC22×Dic173C23
hypercenterC2×C22⋊C4#10621924
hypocenterC173#1900
LCSG ▷ C23×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×D173⋊M16C2semidirect
2. C2×#19149739C2semidirect
3. #26012463C2semidirect
4. #26012371C2semidirect
29. #14583835D4semidirect
120. C2×D173#100semidirect
136. C173#1900semidirect

Maximal subgroups

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

subgroupindex
C2×D173⋊M162
C2×#191497392
#260124632
#260123712
C2×#191496522
#1900173