Finite Group Databasebeta

← #36362807

Subgroups — #36362807

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×#14583825C211yesmaximal, characteristic, complemented
2C2×#19149712C211yesmaximal, characteristic, complemented
3C2×#19149705C211yesmaximal, characteristic, complemented
4#26012003C211yesmaximal, characteristic, complemented
5#26011654C214yesmaximal, complemented
6#26011637C214yesmaximal, complemented
7#26011499C211yesmaximal, characteristic, complemented
8#26011454C211yesmaximal, characteristic, complemented
9#418×C173C211yesmaximal, characteristic, complemented
10C2×#14583826C2211yescharacteristic
11C2×#14583825—24no—
12C2×#14583825—24no—
13C2×#14583825—22no—
14C2×#14583825C2214yes—
15C22×Dic346C2211yescharacteristic
16C22×C22⋊D173—21no—
17C23×Dic173—21no—
18C22×C4×D173C2211yescharacteristic
19#19149712—24no—
20#19149705—24no—
21C2×Dic692C2212yescomplemented
22C2×#14583815C2212yescomplemented
23C2×#14583814C2212yescomplemented
24C2×#14583811C2212yescomplemented
25C2×#14583810C2211yescharacteristic
26#19149663—21no—
27#19149659C2212yescomplemented
28#19149654C2212yescomplemented
29#19149649—21no—
30C4×D692—22no—
31#19149602C2214yescomplemented
32#19149591C2214yescomplemented
33C22×D4×C173C2211yescharacteristic
34C2×M16×C173C2211yescharacteristic
35#117×C173C2211yescharacteristic
36#102×C173C2214yescomplemented
37C23×D173D411yescharacteristic
38#14583826—22no—
39#14583826—22no—
40#14583825—216no—
41#14583825—28no—
42#14583825—28no—
43C2×Dic346—21no—
44C2×Dic346—22no—
45C2×Dic346D414yescomplemented
46C2×Dic346C2311yescharacteristic
47C2×Dic346C2312yes—
48C2×D692—22no—
49C2×D692C2311yescharacteristic
50C2×C22⋊D173—28no—
51C2×C22⋊D173—24no—
52C2×C22⋊D173—21no—
53C22×Dic173—28no—
54C22×Dic173—24no—
55C22×Dic173—21no—
56C22×Dic173D411yescharacteristic
57C2×C4×D173—22no—
58C2×C4×D173—22no—
59C2×C4×D173—21no—
60C2×C4×D173D412yes—
61C2×C4×D173C2312yes—
62Dic692—24no—
63#14583815—24no—
64#14583814—24no—
65#14583811—24no—
66#14583810—22no—
67C2×#13063436C2312yes—
68C4⋊Dic173—22no—
69#14583803—22no—
70#14583803—22no—
71C24×C173—21no—
72C2×D4×C173—21no—
73C2×D4×C173—22no—
74C2×D4×C173D414yescomplemented
75C2×D4×C173C2312yes—
76C22×C4×C173C2311yescharacteristic
77M16×C173—22no—
78C2×C8×C173C2312yes—
79C4⋊C4×C173C2312yes—
80C22⋊C4×C173—21no—
81C42×C173—21no—
82C22×D173—22no—
83C22×D173—21no—
84C22×D173—42no—
85C22×D173C2×D412yes—
86Dic346—24no—
87Dic346—22no—
88Dic346—21no—
89Dic346—21no—
90Dic346C2×D414yes—
91D692—21no—
92D692—21no—
93D692—44no—
94C22⋊D173—216no—
95C22⋊D173—42no—
96C2×Dic173—216no—
97C2×Dic173—22no—
98C2×Dic173—24no—
99C2×Dic173—22no—
100C2×Dic173—21no—
101C2×Dic173C2×D412yes—
102C4×D173—24no—
103C4×D173—22no—
104C4×D173—22no—
105C4×D173—42no—
106#13063436—22no—
107C23×C173—24no—
108C23×C173—22no—
109C23×C173—44no—
110C23×C173C2×D411yescharacteristic
111D4×C173—24no—
112D4×C173—22no—
113D4×C173C2×D414yes—
114C2×C4×C173—22no—
115C2×C4×C173C2×D411yescharacteristic
116C2×C4×C173C2×D412yes—
117C2×C4×C173C2×D412yes—
118C2×C4×C173C2411yescharacteristic
119C8×C173—22no—
120C22×C173—22no—
121C22×C173—24no—
122C22×C173—48no—
123C22×C173—44no—
124C22×C173#12012yes—
125C22×C173C22×D411yescharacteristic
126C2×D173—22no—
127C2×D173—22no—
128C2×D173—22no—
129C2×D173—44no—
130C2×D173—41no—
131C2×D173—42no—
132C4×C173—42no—
133C4×C173#12012yes—
134C4×C173C22×D411yescharacteristic
135C4×C173C22×D411yescharacteristic
136Dic173—22no—
137Dic173—24no—
138Dic173—41no—
139C2×C173—22no—
140C2×C173—44no—
141C2×C173C2×#12011yescharacteristic
142C2×C173#57512yes—
143D173—42no—
144D173—81no—
145C173#300511yescharacteristic, complemented, Sylow
146#3005—1731nomaximal, Sylow
147C22×#41—1731no—
148C2×#137—1731no—
149C2×AGL(1,ℤ/8)—1731no—
150#525—1731no—
151#452—1734no—
152#449—1734no—
153#419—1731no—
154#418D17311yescharacteristic, complemented
155#407—1731no—
156C2×#41—3464no—
157C2×#41—3464no—
158C2×#41—3462no—
159C2×#41—1734no—
160C2×#41—1731no—
161C22×Q8—1731no—
162C22×D4—3461no—
163C22×D4C2×D17311yescharacteristic
164C23×C4—3461no—
165C23×C4—1731no—
166#137—3464no—
167AGL(1,ℤ/8)—3464no—
168C2×Q16—1732no—
169C2×SD16—1732no—
170C2×SD16—1732no—
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#117C2×D17311yescharacteristic
180#103—1734no—
181#102C2×D17314yescomplemented
182C22⋊C8—1734no—
183C24—21no—
184C24—1731no—
185#41—34616no—
186#41—3468no—
187#41—3468no—
188#41—3462no—
189#41—3462no—
190C2×Q8—3461no—
191C2×Q8—3462no—
192C2×Q8—1734no—
193C2×Q8—1731no—
194C2×Q8—1732no—
195C2×D4—21no—
196C2×D4—22no—
197C2×D4—3462no—
198C2×D4—3468no—
199C2×D4—3464no—
200C2×D4—3461no—
201C2×D4—1731no—
202C2×D4D69214yescomplemented
203C2×D4C22×D17312yes—
204C22×C4—3468no—
205C22×C4—3461no—
206C22×C4—3464no—
207C22×C4—3462no—
208C22×C4—3462no—
209C22×C4—3461no—
210C22×C4—1731no—
211C22×C4—1732no—
212C22×C4—1732no—
213C22×C4C22×D17311yescharacteristic
214Q16—3464no—
215SD16—3464no—
216SD16—3464no—
217D8—3464no—
218M16—22no—
219M16—3462no—
220C2×C8—1732no—
221C2×C8C22×D17312yes—
222C4⋊C4—3462no—
223C4⋊C4C22×D17312yes—
224C22⋊C4—21no—
225C22⋊C4—3462no—
226C22⋊C4—3462no—
227C42—21no—
228C23—24no—
229C23—22no—
230C23—44no—
231C23—3462no—
232C23—3461no—
233C23—1732no—
234C23—6922no—
235C23D4×D17311yescharacteristic
236Q8—3464no—
237Q8—3462no—
238Q8—3461no—
239Q8—3461no—
240Q8—1734no—
241D4—24no—
242D4—22no—
243D4—34616no—
244D4—3461no—
245D4—3461no—
246D4—6924no—
247D4—6922no—
248D4C2×D69214yes—
249C2×C4—22no—
250C2×C4—34616no—
251C2×C4—3462no—
252C2×C4—3464no—
253C2×C4—3462no—
254C2×C4—3464no—
255C2×C4—3462no—
256C2×C4—3462no—
257C2×C4—3461no—
258C2×C4—1732no—
259C2×C4—6922no—
260C2×C4D4×D17311yescharacteristic
261C2×C4D4×D17312yes—
262C2×C4C2×D69212yes—
263C2×C4C23×D17311yescharacteristic
264C8—22no—
265C8—3462no—
266C22—22no—
267C22—24no—
268C22—48no—
269C22—44no—
270C22—3462no—
271C22—3462no—
272C22—3462no—
273C22—6924no—
274C22—6921no—
275C22—6922no—
276C22#1914964912yes—
277C22C2×D4×D17311yescharacteristic
278C4—42no—
279C4—3462no—
280C4—3464no—
281C4—6921no—
282C4#1914964912yes—
283C4C2×D4×D17311yescharacteristic
284C4C22×D69211yescharacteristic
285C2—22no—
286C2—44no—
287C2—13841no—
288C2—6922no—
289C2C2×#1914964911yescharacteristic
290C2#2601228712yes—
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#418×C173C2
socleC22×C173C22×D4
hypercenter#418D173
hypocenterC173#3005
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C22×#14583825C2semidirect
2. C2×#19149712C2semidirect
3. C2×#19149705C2semidirect
4. #26012003C2semidirect
5. #26011654C2semidirect
6. #26011637C2semidirect
7. #26011499C2semidirect
8. #26011454C2semidirect
9. #418×C173C2semidirect
21. C2×Dic692C22semidirect
22. C2×#14583815C22semidirect
23. C2×#14583814C22semidirect
24. C2×#14583811C22semidirect
27. #19149659C22semidirect
28. #19149654C22semidirect
31. #19149602C22semidirect
32. #19149591C22semidirect
36. #102×C173C22semidirect
45. C2×Dic346D4semidirect
74. C2×D4×C173D4semidirect
145. C173#3005semidirect
154. #418D173semidirect
181. #102C2×D173semidirect
202. C2×D4D692semidirect

Maximal subgroups

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

subgroupindex
C22×#145838252
C2×#191497122
C2×#191497052
#260120032
#260116542
#260116372
#260114992
#260114542
#418×C1732
#3005173