Finite Group Databasebeta

← #36362803

Subgroups — #36362803

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×#14583826C211yesmaximal, characteristic, complemented
2C2×#19149709C211yesmaximal, characteristic, complemented
3C2×#19149704C211yesmaximal, characteristic, complemented
4#26011651C214yesmaximal, complemented
5#26011634C214yesmaximal, complemented
6#26011499C211yesmaximal, characteristic, complemented
7#26011495C211yesmaximal, characteristic, complemented
8#26011455C211yesmaximal, characteristic, complemented
9#525×C173C211yesmaximal, characteristic, complemented
10C2×#14583826—22no—
11C2×#14583826—24no—
12C2×#14583826—24no—
13C2×#14583826C2214yes—
14C22×Dic346C2211yescharacteristic
15C22×D692C2211yescharacteristic
16C22×C22⋊D173—21no—
17C22×C4×D173—21no—
18#19149709—24no—
19#19149704—24no—
20C2×#14583816C2212yescomplemented
21C2×#14583814C2212yescomplemented
22C2×#14583813C2212yescomplemented
23C2×#14583811C2212yescomplemented
24C2×#14583810C2211yescharacteristic
25C2×#14583810C2211yescharacteristic
26#19149602C2214yescomplemented
27#19149599C2214yescomplemented
28C22⋊#13063436C2214yescomplemented
29C2×#41×C173C2211yescharacteristic
30C23×C4×C173C2211yescharacteristic
31#123×C173—21no—
32C22⋊Q8×C173C2212yescomplemented
33#121×C173C2212yescomplemented
34#120×C173—21no—
35C4×D4×C173—22no—
36#117×C173C2211yescharacteristic
37C23×D173—21no—
38#14583826—216no—
39#14583826—28no—
40#14583826—28no—
41C2×Dic346—22no—
42C2×Dic346—21no—
43C2×Dic346D414yescomplemented
44C2×Dic346C2312yes—
45C2×D692—22no—
46C2×D692—21no—
47C2×D692D414yescomplemented
48C2×D692C2312yes—
49C2×C22⋊D173—28no—
50C2×C22⋊D173—24no—
51C22×Dic173—21no—
52C2×C4×D173—28no—
53C2×C4×D173—24no—
54#14583816—24no—
55#14583814—24no—
56#14583813—24no—
57#14583811—24no—
58#14583810—22no—
59#14583810—22no—
60C2×#13063436C2312yes—
61C2×#13063436C2312yes—
62C24×C173D411yescharacteristic
63#41×C173—22no—
64#41×C173—22no—
65C2×Q8×C173C2311yescharacteristic
66C2×D4×C173—21no—
67C2×D4×C173—22no—
68C2×D4×C173C2311yescharacteristic
69C22×C4×C173—22no—
70C22×C4×C173—22no—
71C22×C4×C173—21no—
72C22×C4×C173D411yescharacteristic
73C22×C4×C173D412yes—
74C22×C4×C173C2312yes—
75C22×C4×C173C2311yescharacteristic
76C4⋊C4×C173—22no—
77C4⋊C4×C173C2312yes—
78C22⋊C4×C173—21no—
79C22⋊C4×C173—22no—
80C22⋊C4×C173—22no—
81C42×C173—21no—
82C22×D173—24no—
83C22×D173—22no—
84C22×D173—44no—
85Dic346—24no—
86Dic346—22no—
87Dic346C2×D414yes—
88D692—24no—
89D692—22no—
90D692C2×D414yes—
91C22⋊D173—216no—
92C2×Dic173—24no—
93C2×Dic173—22no—
94C4×D173—216no—
95#13063436—22no—
96#13063436—22no—
97C23×C173—22no—
98C23×C173—21no—
99C23×C173—42no—
100C23×C173C2×D412yes—
101C23×C173C2×D411yescharacteristic
102Q8×C173—21no—
103Q8×C173—21no—
104D4×C173—21no—
105D4×C173—21no—
106D4×C173—44no—
107D4×C173—42no—
108C2×C4×C173—22no—
109C2×C4×C173—24no—
110C2×C4×C173—22no—
111C2×C4×C173—22no—
112C2×C4×C173—22no—
113C2×C4×C173—21no—
114C2×C4×C173—42no—
115C2×C4×C173C2×D412yes—
116C2×C4×C173C2×D412yes—
117C2×C4×C173C2×D412yes—
118C2×C4×C173C2×D411yescharacteristic
119C2×C4×C173C2411yescharacteristic
120C22×C173—22no—
121C22×C173—22no—
122C22×C173—22no—
123C22×C173—22no—
124C22×C173—44no—
125C22×C173—41no—
126C22×C173—42no—
127C22×C173#12012yes—
128C22×C173C22×D411yescharacteristic
129C2×D173—24no—
130C2×D173—48no—
131C2×D173—44no—
132C4×C173—22no—
133C4×C173—41no—
134C4×C173—42no—
135C4×C173#12012yes—
136C4×C173C22×D411yescharacteristic
137C4×C173C22×D411yescharacteristic
138Dic173—24no—
139C2×C173—22no—
140C2×C173—42no—
141C2×C173—81no—
142C2×C173C2×#12011yescharacteristic
143C2×C173#57512yes—
144D173—44no—
145C173#300511yescharacteristic, complemented, Sylow
146#3005—1731nomaximal, Sylow
147C22×#41—1731no—
148C2×#137—1731no—
149C2×AGL(1,ℤ/8)—1731no—
150#525D17311yescharacteristic, complemented
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×Q8—1731no—
162C22×D4—3461no—
163C22×D4—1731no—
164C23×C4—3461no—
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—21no—
175C22⋊Q8C2×D17312yescomplemented
176#121C2×D17312yescomplemented
177#120—21no—
178C4×D4—22no—
179#117C2×D17311yescharacteristic
180#103—1734no—
181#102—1734no—
182C22⋊C8—1734no—
183C24—3461no—
184C24C22⋊D17311yescharacteristic
185#41—22no—
186#41—22no—
187#41—34616no—
188#41—3468no—
189#41—3468no—
190C2×Q8—3462no—
191C2×Q8—3461no—
192C2×Q8—1734no—
193C2×Q8—1732no—
194C2×Q8C22×D17311yescharacteristic
195C2×D4—22no—
196C2×D4—21no—
197C2×D4—3468no—
198C2×D4—3464no—
199C2×D4—3462no—
200C2×D4—3461no—
201C2×D4—1734no—
202C2×D4—1732no—
203C2×D4C22×D17311yescharacteristic
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—3468no—
208C22×C4—3461no—
209C22×C4—3464no—
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—22no—
223C4⋊C4C22×D17312yes—
224C22⋊C4—21no—
225C22⋊C4—22no—
226C22⋊C4—22no—
227C42—21no—
228C23—22no—
229C23—21no—
230C23—42no—
231C23—3464no—
232C23—3462no—
233C23—6924no—
234C23C2×C22⋊D17312yes—
235C23C2×C22⋊D17311yescharacteristic
236Q8—21no—
237Q8—21no—
238Q8—3464no—
239Q8—3462no—
240Q8—1734no—
241D4—21no—
242D4—21no—
243D4—44no—
244D4—42no—
245D4—34616no—
246D4—3464no—
247D4—3462no—
248D4—1734no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—42no—
256C2×C4—34616no—
257C2×C4—3464no—
258C2×C4—3462no—
259C2×C4D4×D17312yes—
260C2×C4D4×D17312yes—
261C2×C4C2×C22⋊D17312yes—
262C2×C4C2×C22⋊D17311yescharacteristic
263C2×C4C23×D17311yescharacteristic
264C8—3462no—
265C8—3462no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—44no—
271C22—41no—
272C22—42no—
273C22—3464no—
274C22—6928no—
275C22—6924no—
276C22#1914964712yes—
277C22C22×C22⋊D17311yescharacteristic
278C4—22no—
279C4—41no—
280C4—42no—
281C4—3464no—
282C4#1914964712yes—
283C4C2×D4×D17311yescharacteristic
284C4C2×D4×D17311yescharacteristic
285C2—22no—
286C2—42no—
287C2—81no—
288C2—6924no—
289C2C2×#1914964711yescharacteristic
290C2#2601228912yes—
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#525×C173C2
socleC22×C173C22×D4
hypercenter#525D173
hypocenterC173#3005
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C22×#14583826C2semidirect
2. C2×#19149709C2semidirect
3. C2×#19149704C2semidirect
4. #26011651C2semidirect
5. #26011634C2semidirect
6. #26011499C2semidirect
7. #26011495C2semidirect
8. #26011455C2semidirect
9. #525×C173C2semidirect
20. C2×#14583816C22semidirect
21. C2×#14583814C22semidirect
22. C2×#14583813C22semidirect
23. C2×#14583811C22semidirect
26. #19149602C22semidirect
27. #19149599C22semidirect
28. C22⋊#13063436C22semidirect
32. C22⋊Q8×C173C22semidirect
33. #121×C173C22semidirect
43. C2×Dic346D4semidirect
47. C2×D692D4semidirect
145. C173#3005semidirect
150. #525D173semidirect
175. C22⋊Q8C2×D173semidirect
176. #121C2×D173semidirect

Maximal subgroups

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

subgroupindex
C22×#145838262
C2×#191497092
C2×#191497042
#260116512
#260116342
#260114992
#260114952
#260114552
#525×C1732
#3005173