Finite Group Databasebeta

← #36362805

Subgroups — #36362805

Order 22144.

758 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 165 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×#19149710C211yesmaximal, characteristic, complemented
2C2×#19149705C211yesmaximal, characteristic, complemented
3#26012002C211yesmaximal, characteristic, complemented
4#26011653C214yesmaximal, complemented
5#26011636C214yesmaximal, complemented
6#26011500C211yesmaximal, characteristic, complemented
7#26011496C211yesmaximal, characteristic, complemented
8#26011455C211yesmaximal, characteristic, complemented
9C22×#41×C173C211yesmaximal, characteristic, complemented
10C2×#14583826C2211yescharacteristic
11#19149710—24no—
12#19149705—24no—
13C2×#14583816C2212yescomplemented
14C2×#14583814C2212yescomplemented
15C2×#14583813C2212yescomplemented
16C2×#14583811C2212yescomplemented
17C2×#14583810C2211yescharacteristic
18C2×#14583810C2211yescharacteristic
19#19149662—21no—
20#19149658C2212yescomplemented
21#19149653C2212yescomplemented
22#19149648—21no—
23C4×C22⋊D173—22no—
24#19149638C2211yescharacteristic
25#19149603C2214yescomplemented
26#19149600C2214yescomplemented
27C22⋊#13063436C2214yescomplemented
28C2×#41×C173—24no—
29C2×#41×C173—24no—
30C2×#41×C173—22no—
31C2×#41×C173C2214yes—
32C22×Q8×C173C2211yescharacteristic
33C22×D4×C173—21no—
34C22×D4×C173C2211yescharacteristic
35C23×C4×C173—21no—
36C23×C4×C173C2211yescharacteristic
37#14583826—22no—
38#14583826—22no—
39C2×Dic346C2311yescharacteristic
40C2×D692C2311yescharacteristic
41C2×C22⋊D173—22no—
42C2×C22⋊D173—21no—
43C2×C4×D173—21no—
44#14583816—24no—
45#14583814—24no—
46#14583813—24no—
47#14583811—24no—
48#14583810—22no—
49#14583810—22no—
50C2×#13063436C2312yes—
51C2×#13063436C2312yes—
52C4⋊Dic173C2312yes—
53#14583805—22no—
54C22⋊Dic173—21no—
55C22⋊Dic173—22no—
56#14583803—22no—
57C4×Dic173—21no—
58C24×C173—21no—
59C24×C173D411yescharacteristic
60#41×C173—216no—
61#41×C173—28no—
62#41×C173—28no—
63C2×Q8×C173—22no—
64C2×Q8×C173—21no—
65C2×Q8×C173D414yescomplemented
66C2×Q8×C173C2312yes—
67C2×D4×C173—28no—
68C2×D4×C173—24no—
69C2×D4×C173—21no—
70C2×D4×C173—22no—
71C2×D4×C173D414yescomplemented
72C2×D4×C173C2312yes—
73C22×C4×C173—28no—
74C22×C4×C173—21no—
75C22×C4×C173—24no—
76C22×C4×C173—22no—
77C22×C4×C173—22no—
78C22×C4×C173D411yescharacteristic
79C22×C4×C173D412yes—
80C22×C4×C173C2312yes—
81C22×C4×C173C2311yescharacteristic
82C22×D173—21no—
83Dic346—21no—
84Dic346—21no—
85D692—21no—
86D692—21no—
87C22⋊D173—44no—
88C22⋊D173—42no—
89C2×Dic173—21no—
90C2×Dic173—22no—
91C4×D173—42no—
92#13063436—22no—
93#13063436—22no—
94C23×C173—22no—
95C23×C173—24no—
96C23×C173—22no—
97C23×C173—42no—
98C23×C173—44no—
99C23×C173C2×D412yes—
100C23×C173C2×D411yescharacteristic
101Q8×C173—24no—
102Q8×C173—22no—
103Q8×C173C2×D414yes—
104D4×C173—216no—
105D4×C173—24no—
106D4×C173—22no—
107D4×C173C2×D414yes—
108C2×C4×C173—216no—
109C2×C4×C173—22no—
110C2×C4×C173—24no—
111C2×C4×C173—22no—
112C2×C4×C173—24no—
113C2×C4×C173—22no—
114C2×C4×C173—22no—
115C2×C4×C173C2×D412yes—
116C2×C4×C173C2×D411yescharacteristic
117C2×C4×C173C2×D412yes—
118C2×C4×C173C2×D412yes—
119C2×C4×C173C2411yescharacteristic
120C22×C173—22no—
121C22×C173—22no—
122C22×C173—24no—
123C22×C173—22no—
124C22×C173—22no—
125C22×C173—44no—
126C22×C173—48no—
127C22×C173—44no—
128C22×C173#12012yes—
129C22×C173C22×D411yescharacteristic
130C2×D173—41no—
131C2×D173—42no—
132C4×C173—22no—
133C4×C173—24no—
134C4×C173#12012yes—
135C4×C173C22×D411yescharacteristic
136C4×C173C22×D411yescharacteristic
137Dic173—41no—
138Dic173—42no—
139C2×C173—22no—
140C2×C173—42no—
141C2×C173—44no—
142C2×C173C2×#12011yescharacteristic
143C2×C173#57512yes—
144D173—81no—
145C173#300511yescharacteristic, complemented, Sylow
146#3005—1731nomaximal, Sylow
147C22×#41D17311yescharacteristic, complemented
148C2×#137—1731no—
149C2×AGL(1,ℤ/8)—1731no—
150#525—1731no—
151#452—1734no—
152#449—1734no—
153#419—1731no—
154#418—1731no—
155#407—1731no—
156C2×#41—24no—
157C2×#41—24no—
158C2×#41—22no—
159C2×#41—1731no—
160C2×#41C2×D17314yes—
161C22×Q8C2×D17311yescharacteristic
162C22×D4—21no—
163C22×D4C2×D17311yescharacteristic
164C23×C4—21no—
165C23×C4C2×D17311yescharacteristic
166#137—3464no—
167AGL(1,ℤ/8)—3464no—
168C2×Q16—1732no—
169C2×SD16—1732no—
170C2×SD16—1732no—
171C2×D8—1732no—
172C2×M16—1731no—
173C2×M16—1731no—
174#123—3461no—
175C22⋊Q8—1732no—
176#121—1732no—
177#120—3461no—
178C4×D4—3462no—
179#117—1731no—
180#103—1734no—
181#102—1734no—
182C22⋊C8—1734no—
183C24—21no—
184C24C22⋊D17311yescharacteristic
185#41—216no—
186#41—28no—
187#41—28no—
188#41—3462no—
189#41—3462no—
190C2×Q8—22no—
191C2×Q8—21no—
192C2×Q8—1731no—
193C2×Q8C22⋊D17314yescomplemented
194C2×Q8C22×D17312yes—
195C2×D4—28no—
196C2×D4—24no—
197C2×D4—21no—
198C2×D4—22no—
199C2×D4—3462no—
200C2×D4—3461no—
201C2×D4—1731no—
202C2×D4C22⋊D17314yescomplemented
203C2×D4C22×D17312yes—
204C22×C4—28no—
205C22×C4—21no—
206C22×C4—24no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—3461no—
210C22×C4C22⋊D17312yes—
211C22×C4C22⋊D17311yescharacteristic
212C22×C4C22×D17312yes—
213C22×C4C22×D17311yescharacteristic
214Q16—3464no—
215SD16—3464no—
216SD16—3464no—
217D8—3464no—
218M16—3462no—
219M16—3462no—
220C2×C8—1732no—
221C2×C8—1732no—
222C4⋊C4—3462no—
223C4⋊C4—1732no—
224C22⋊C4—3461no—
225C22⋊C4—3462no—
226C22⋊C4—3462no—
227C42—3461no—
228C23—22no—
229C23—24no—
230C23—22no—
231C23—42no—
232C23—44no—
233C23—3461no—
234C23C2×C22⋊D17312yes—
235C23C2×C22⋊D17311yescharacteristic
236Q8—24no—
237Q8—22no—
238Q8—3461no—
239Q8—3461no—
240Q8C2×C22⋊D17314yes—
241D4—216no—
242D4—24no—
243D4—22no—
244D4—3461no—
245D4—3461no—
246D4—6924no—
247D4—6922no—
248D4C2×C22⋊D17314yes—
249C2×C4—216no—
250C2×C4—22no—
251C2×C4—24no—
252C2×C4—22no—
253C2×C4—24no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—3462no—
257C2×C4—3461no—
258C2×C4—6922no—
259C2×C4C2×C22⋊D17312yes—
260C2×C4C2×C22⋊D17311yescharacteristic
261C2×C4C2×C22⋊D17312yes—
262C2×C4C2×C22⋊D17312yes—
263C2×C4C23×D17311yescharacteristic
264C8—3462no—
265C8—3462no—
266C22—22no—
267C22—22no—
268C22—24no—
269C22—22no—
270C22—22no—
271C22—44no—
272C22—48no—
273C22—44no—
274C22—6921no—
275C22—6922no—
276C22#1914964812yes—
277C22C22×C22⋊D17311yescharacteristic
278C4—22no—
279C4—24no—
280C4—6921no—
281C4—6922no—
282C4#1914964812yes—
283C4C22×C22⋊D17311yescharacteristic
284C4C22×C22⋊D17311yescharacteristic
285C2—22no—
286C2—42no—
287C2—44no—
288C2—13841no—
289C2C2×#1914964811yescharacteristic
290C2#2601228912yes—
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC22×#41×C173C2
socleC22×C173C22×D4
hypercenterC22×#41D173
hypocenterC173#3005
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#19149710C2semidirect
2. C2×#19149705C2semidirect
3. #26012002C2semidirect
4. #26011653C2semidirect
5. #26011636C2semidirect
6. #26011500C2semidirect
7. #26011496C2semidirect
8. #26011455C2semidirect
9. C22×#41×C173C2semidirect
13. C2×#14583816C22semidirect
14. C2×#14583814C22semidirect
15. C2×#14583813C22semidirect
16. C2×#14583811C22semidirect
20. #19149658C22semidirect
21. #19149653C22semidirect
25. #19149603C22semidirect
26. #19149600C22semidirect
27. C22⋊#13063436C22semidirect
65. C2×Q8×C173D4semidirect
71. C2×D4×C173D4semidirect
145. C173#3005semidirect
147. C22×#41D173semidirect
193. C2×Q8C22⋊D173semidirect
202. C2×D4C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497102
C2×#191497052
#260120022
#260116532
#260116362
#260115002
#260114962
#260114552
C22×#41×C1732
#3005173