Finite Group Databasebeta

← #36362804

Subgroups — #36362804

Order 22144.

758 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 149 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×#14583827C211yesmaximal, characteristic, complemented
2C2×#19149704C211yesmaximal, characteristic, complemented
3#26011998C211yesmaximal, characteristic, complemented
4#26011652C214yesmaximal, complemented
5#26011635C214yesmaximal, complemented
6#26011500C211yesmaximal, characteristic, complemented
7#26011497C211yesmaximal, characteristic, complemented
8#26011454C211yesmaximal, characteristic, complemented
9C2×#137×C173C211yesmaximal, characteristic, complemented
10C2×#14583827—24no—
11C2×#14583827—24no—
12C2×#14583827—22no—
13C2×#14583827C2214yes—
14C22×D692—21no—
15C22×D692C2211yescharacteristic
16C22×C4×D173—21no—
17C22×C4×D173C2211yescharacteristic
18#19149704—24no—
19C2×#14583813C2212yescomplemented
20C2×#14583811C2212yescomplemented
21C2×#14583810C2211yescharacteristic
22#19149660—21no—
23#19149655C2212yescomplemented
24#19149650C2212yescomplemented
25#19149647—21no—
26C4×C22⋊D173—22no—
27#19149638C2211yescharacteristic
28#19149603C2214yescomplemented
29#19149601C2214yescomplemented
30#19149591C2214yescomplemented
31C2×#41×C173C2211yescharacteristic
32C22×Q8×C173C2211yescharacteristic
33#137×C173—24no—
34C2×Q16×C173C2212yescomplemented
35C2×SD16×C173C2212yescomplemented
36C2×M16×C173C2211yescharacteristic
37C23×D173—21no—
38C23×D173D411yescharacteristic
39#14583827—216no—
40#14583827—28no—
41#14583827—28no—
42C2×D692—28no—
43C2×D692—24no—
44C2×D692—21no—
45C2×D692—22no—
46C2×D692D414yescomplemented
47C2×D692C2312yes—
48C2×C22⋊D173—22no—
49C22×Dic173D411yescharacteristic
50C2×C4×D173—28no—
51C2×C4×D173—24no—
52C2×C4×D173—22no—
53C2×C4×D173—22no—
54C2×C4×D173D412yes—
55C2×C4×D173C2312yes—
56#14583813—24no—
57#14583811—24no—
58#14583810—22no—
59C2×#13063436C2312yes—
60C4⋊Dic173C2312yes—
61#14583805—22no—
62C22⋊Dic173—22no—
63C22⋊Dic173—21no—
64#14583803—22no—
65C4×Dic173—21no—
66#41×C173—22no—
67#41×C173—22no—
68C2×Q8×C173—22no—
69C2×Q8×C173—21no—
70C2×Q8×C173D414yescomplemented
71C2×Q8×C173C2311yescharacteristic
72C2×Q8×C173C2312yes—
73C2×D4×C173—21no—
74C2×D4×C173C2311yescharacteristic
75C22×C4×C173—21no—
76C22×C4×C173—21no—
77C22×C4×C173C2311yescharacteristic
78Q16×C173—24no—
79SD16×C173—24no—
80M16×C173—22no—
81C2×C8×C173C2312yes—
82C22×D173—22no—
83C22×D173—24no—
84C22×D173—22no—
85C22×D173—42no—
86C22×D173—44no—
87C22×D173C2×D412yes—
88D692—216no—
89D692—24no—
90D692—22no—
91D692C2×D414yes—
92C22⋊D173—44no—
93C2×Dic173—22no—
94C2×Dic173—22no—
95C2×Dic173C2×D412yes—
96C4×D173—216no—
97C4×D173—24no—
98C4×D173—22no—
99C4×D173—22no—
100#13063436—22no—
101C23×C173—21no—
102C23×C173C2×D411yescharacteristic
103Q8×C173—24no—
104Q8×C173—22no—
105Q8×C173—21no—
106Q8×C173—21no—
107Q8×C173C2×D414yes—
108D4×C173—21no—
109D4×C173—21no—
110D4×C173—42no—
111C2×C4×C173—24no—
112C2×C4×C173—22no—
113C2×C4×C173—21no—
114C2×C4×C173—42no—
115C2×C4×C173C2×D411yescharacteristic
116C2×C4×C173C2×D412yes—
117C2×C4×C173C2×D412yes—
118C2×C4×C173C2411yescharacteristic
119C8×C173—22no—
120C22×C173—22no—
121C22×C173—41no—
122C22×C173—42no—
123C22×C173#12012yes—
124C22×C173C22×D411yescharacteristic
125C2×D173—22no—
126C2×D173—24no—
127C2×D173—22no—
128C2×D173—22no—
129C2×D173—44no—
130C2×D173—48no—
131C2×D173—44no—
132C4×C173—24no—
133C4×C173—41no—
134C4×C173#12012yes—
135C4×C173C22×D411yescharacteristic
136C4×C173C22×D411yescharacteristic
137Dic173—22no—
138Dic173—42no—
139C2×C173—22no—
140C2×C173—81no—
141C2×C173C2×#12011yescharacteristic
142C2×C173#57512yes—
143D173—42no—
144D173—44no—
145C173#300511yescharacteristic, complemented, Sylow
146#3005—1731nomaximal, Sylow
147C22×#41—1731no—
148C2×#137D17311yescharacteristic, complemented
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—3464no—
157C2×#41—3464no—
158C2×#41—3462no—
159C2×#41—1734no—
160C2×#41C2×D17311yescharacteristic
161C22×Q8C2×D17311yescharacteristic
162C22×D4—3461no—
163C22×D4—1731no—
164C23×C4—3461no—
165C23×C4—1731no—
166#137—24no—
167AGL(1,ℤ/8)—3464no—
168C2×Q16C2×D17312yescomplemented
169C2×SD16—1732no—
170C2×SD16C2×D17312yescomplemented
171C2×D8—1732no—
172C2×M16—1731no—
173C2×M16C2×D17311yescharacteristic
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—3461no—
184C24—1731no—
185#41—22no—
186#41—22no—
187#41—34616no—
188#41—3468no—
189#41—3468no—
190C2×Q8—22no—
191C2×Q8—21no—
192C2×Q8C22⋊D17314yescomplemented
193C2×Q8C22×D17311yescharacteristic
194C2×Q8C22×D17312yes—
195C2×D4—21no—
196C2×D4—3468no—
197C2×D4—3464no—
198C2×D4—3462no—
199C2×D4—3461no—
200C2×D4—3462no—
201C2×D4—1734no—
202C2×D4—1732no—
203C2×D4C22×D17311yescharacteristic
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—3468no—
207C22×C4—3464no—
208C22×C4—3462no—
209C22×C4—3462no—
210C22×C4—1731no—
211C22×C4—1732no—
212C22×C4—1732no—
213C22×C4C22×D17311yescharacteristic
214Q16—24no—
215SD16—24no—
216SD16—3464no—
217D8—3464no—
218M16—22no—
219M16—3462no—
220C2×C8—1732no—
221C2×C8C22×D17312yes—
222C4⋊C4—3462no—
223C4⋊C4—1732no—
224C22⋊C4—3462no—
225C22⋊C4—3461no—
226C22⋊C4—3462no—
227C42—3461no—
228C23—21no—
229C23—3462no—
230C23—3464no—
231C23—3462no—
232C23—1732no—
233C23—6922no—
234C23—6924no—
235C23D4×D17311yescharacteristic
236Q8—24no—
237Q8—22no—
238Q8—21no—
239Q8—21no—
240Q8C2×C22⋊D17314yes—
241D4—21no—
242D4—21no—
243D4—42no—
244D4—34616no—
245D4—3464no—
246D4—3462no—
247D4—1734no—
248D4—6924no—
249C2×C4—24no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—42no—
253C2×C4—34616no—
254C2×C4—3462no—
255C2×C4—3464no—
256C2×C4—3462no—
257C2×C4—3462no—
258C2×C4—3462no—
259C2×C4—1732no—
260C2×C4D4×D17311yescharacteristic
261C2×C4D4×D17312yes—
262C2×C4C2×C22⋊D17312yes—
263C2×C4C23×D17311yescharacteristic
264C8—22no—
265C8—3462no—
266C22—22no—
267C22—41no—
268C22—42no—
269C22—3462no—
270C22—3464no—
271C22—3462no—
272C22—3462no—
273C22—6924no—
274C22—6928no—
275C22—6924no—
276C22#1914964712yes—
277C22C2×D4×D17311yescharacteristic
278C4—24no—
279C4—41no—
280C4—3462no—
281C4—6922no—
282C4#1914964712yes—
283C4C2×D4×D17311yescharacteristic
284C4C22×C22⋊D17311yescharacteristic
285C2—22no—
286C2—81no—
287C2—6922no—
288C2—6924no—
289C2C2×#1914964711yescharacteristic
290C2#2601229012yes—
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC2×#137×C173C2
socleC22×C173C22×D4
hypercenterC2×#137D173
hypocenterC173#3005
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C22×#14583827C2semidirect
2. C2×#19149704C2semidirect
3. #26011998C2semidirect
4. #26011652C2semidirect
5. #26011635C2semidirect
6. #26011500C2semidirect
7. #26011497C2semidirect
8. #26011454C2semidirect
9. C2×#137×C173C2semidirect
19. C2×#14583813C22semidirect
20. C2×#14583811C22semidirect
23. #19149655C22semidirect
24. #19149650C22semidirect
28. #19149603C22semidirect
29. #19149601C22semidirect
30. #19149591C22semidirect
34. C2×Q16×C173C22semidirect
35. C2×SD16×C173C22semidirect
46. C2×D692D4semidirect
70. C2×Q8×C173D4semidirect
145. C173#3005semidirect
148. C2×#137D173semidirect
168. C2×Q16C2×D173semidirect
170. C2×SD16C2×D173semidirect
192. C2×Q8C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
C22×#145838272
C2×#191497042
#260119982
#260116522
#260116352
#260115002
#260114972
#260114542
C2×#137×C1732
#3005173