Finite Group Databasebeta

← #36365162

Subgroups — #36365162

Order 22144.

546 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 125 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×#19149706C212yesmaximal, complemented
2#26012144C211yesmaximal, characteristic, complemented
3#26012126C211yesmaximal, characteristic, complemented
4#26011734C212yesmaximal, complemented
5#26011693C211yesmaximal, characteristic, complemented
6#26011683C211yesmaximal, characteristic, complemented
7#26011640C212yesmaximal, complemented
8#26011635C212yesmaximal, complemented
9C4×D1384C211yesmaximal, characteristic, complemented
10#26011570C211yesmaximal, characteristic, complemented
11#492×C173C211yesmaximal, characteristic, complemented
12C2×#14583827C2212yescomplemented
13C2×D4×D173—21no—
14C2×D4×D173C2212yes—
15#19149706—28no—
16C2×#14583813C2212yes—
17C2×D1384C2211yescharacteristic
18C2×#14583811C2212yescomplemented
19C2×#14583809C2212yescomplemented
20#19149664—22no—
21#19149655C2212yescomplemented
22#19149654—21no—
23#19149650—21no—
24#19149650C2212yes—
25#19149647—22no—
26C4×D692C2211yescharacteristic
27C4×D692C2211yescharacteristic
28C4×D692C2211yescharacteristic, complemented
29C4⋊C4×D173—21no—
30#19149613C2211yescharacteristic, complemented
31C4⋊#13063436C2211yescharacteristic, complemented
32#19149603C2212yescomplemented
33#19149601C2211yescharacteristic
34#19149601C2211yescharacteristic, complemented
35#19149600C2212yescomplemented
36#19149591C2212yescomplemented
37C2×SD16×C173C2212yescomplemented
38C2×SD16×C173C2212yescomplemented
39C4⋊Q8×C173C2211yescharacteristic, complemented
40#127×C173C2211yescharacteristic
41C4×C8×C173C2211yescharacteristic, complemented
42C23×D173—22no—
43#14583827—24no—
44#14583827—42no—
45C2×D692—22no—
46C2×D692D411yescharacteristic
47C2×D692C2311yescharacteristic
48C2×D692C2311yescharacteristic
49C2×C22⋊D173—22no—
50C2×C22⋊D173—22no—
51D4×D173—24no—
52D4×D173—42no—
53D4×D173—44no—
54C2×C4×D173—22no—
55C2×C4×D173—21no—
56C2×C4×D173—21no—
57C2×C4×D173C2312yes—
58#14583813—24no—
59D1384D414yescomplemented
60#14583811—24no—
61#14583809—24no—
62C2×#13063436C2312yes—
63C4⋊Dic173D411yescharacteristic
64C4⋊Dic173C2311yescharacteristic
65C4⋊Dic173C2311yescharacteristic
66#14583805—22no—
67C22⋊Dic173—22no—
68#14583803—22no—
69#14583803—21no—
70#14583803—21no—
71#14583803D412yes—
72C2×Q8×C173C2312yes—
73C2×D4×C173—21no—
74C2×D4×C173—21no—
75C2×D4×C173C2312yes—
76SD16×C173—24no—
77SD16×C173—42no—
78C2×C8×C173C2311yescharacteristic
79C2×C8×C173C2311yescharacteristic
80C4⋊C4×C173—21no—
81C4⋊C4×C173—21no—
82C42×C173C2311yescharacteristic
83C22×D173—21no—
84C22×D173—21no—
85C22×D173—44no—
86C22×D173—44no—
87C22×D173—82no—
88C22×D173C2×D412yes—
89D692—22no—
90D692—21no—
91D692—21no—
92D692—44no—
93D692C2×D412yes—
94D692C2×D412yescomplemented
95C22⋊D173—44no—
96C22⋊D173—44no—
97C2×Dic173—21no—
98C2×Dic173—21no—
99C2×Dic173C2×D412yes—
100C4×D173—22no—
101C4×D173—22no—
102C4×D173—44no—
103C4×D173—42no—
104C4×D173—42no—
105#13063436—22no—
106C23×C173—22no—
107Q8×C173—22no—
108Q8×C173—22no—
109D4×C173—22no—
110D4×C173—22no—
111D4×C173—42no—
112D4×C173—81no—
113C2×C4×C173—22no—
114C2×C4×C173C2×D411yescharacteristic
115C2×C4×C173C2×D411yescharacteristic
116C2×C4×C173C2411yescharacteristic
117C8×C173—21no—
118C8×C173C2×D412yescomplemented
119C22×C173—42no—
120C22×C173—42no—
121C22×C173—42no—
122C22×C173C22×D411yescharacteristic
123C2×D173—22no—
124C2×D173—22no—
125C2×D173—22no—
126C2×D173—22no—
127C2×D173—22no—
128C2×D173—41no—
129C2×D173—41no—
130C2×D173—41no—
131C2×D173—41no—
132C2×D173—88no—
133C4×C173—21no—
134C4×C173—42no—
135C4×C173AGL(1,ℤ/8)12yescomplemented
136C4×C173C22×D411yescharacteristic
137C4×C173H(5,𝔽2)11yescharacteristic
138Dic173—22no—
139Dic173—41no—
140Dic173—41no—
141C2×C173—82no—
142C2×C173D4211yescharacteristic
143C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
144C2×C173#57711yescharacteristic
145D173—42no—
146D173—42no—
147D173—42no—
148C173#327211yescharacteristic, complemented, Sylow
149#3272—1731nomaximal, Sylow
150C2×AGL(1,ℤ/8)—1732no—
151#547—1731no—
152D42—1731no—
153#492D17311yescharacteristic, complemented
154#468—1732no—
155#461—1731no—
156#459—1731no—
157#450—1732no—
158#449—1732no—
159C4×D8—1731no—
160#436—1731no—
161C2×#41—1732no—
162C22×D4—3461no—
163C22×D4—1732no—
164AGL(1,ℤ/8)—3468no—
165C2×SD16—1732no—
166C2×SD16C2×D17312yescomplemented
167C2×SD16C2×D17312yescomplemented
168C2×D8—1732no—
169C2×D8—1731no—
170C2×M16—1732no—
171C4⋊Q8C2×D17311yescharacteristic, complemented
172#127C2×D17311yescharacteristic
173#123—3462no—
174C22⋊Q8—1732no—
175#121—3461no—
176#121—3461no—
177#121—1732no—
178#120—3462no—
179C4×D4—1731no—
180C4×D4—1731no—
181C4×D4—1731no—
182C2×C4⋊C4—3461no—
183#107—1731no—
184C4⋊C8—1731no—
185#103—1732no—
186#102—1732no—
187#102—1731no—
188#102—1731no—
189C22⋊C8—1732no—
190C4×C8C2×D17311yescharacteristic, complemented
191C24—3462no—
192#41—3464no—
193#41—6922no—
194C2×Q8C22×D17312yes—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—3462no—
198C2×D4—3464no—
199C2×D4—3462no—
200C2×D4—3462no—
201C2×D4—1731no—
202C2×D4—1731no—
203C2×D4—1731no—
204C2×D4—6922no—
205C2×D4—6924no—
206C2×D4C22×D17312yes—
207C22×C4—3462no—
208C22×C4—3461no—
209C22×C4—3461no—
210C22×C4—1732no—
211SD16—24no—
212SD16—42no—
213SD16—3464no—
214D8—3464no—
215D8—1734no—
216M16—3464no—
217C2×C8—1732no—
218C2×C8C22×D17311yescharacteristic
219C2×C8C22×D17311yescharacteristic
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—3462no—
223C4⋊C4—1731no—
224C4⋊C4—1731no—
225C4⋊C4—1731no—
226C22⋊C4—3462no—
227C22⋊C4—3462no—
228C22⋊C4—3461no—
229C22⋊C4—3461no—
230C22⋊C4—1732no—
231C42C22×D17311yescharacteristic
232C23—22no—
233C23—3461no—
234C23—3461no—
235C23—13842no—
236C23—1732no—
237C23—6924no—
238C23—6924no—
239Q8—22no—
240Q8—22no—
241D4—22no—
242D4—22no—
243D4—42no—
244D4—81no—
245D4—3462no—
246D4—3461no—
247D4—3461no—
248D4—1732no—
249D4—1732no—
250D4—6924no—
251D4—6924no—
252D4—6924no—
253C2×C4—22no—
254C2×C4—3462no—
255C2×C4—3462no—
256C2×C4—3461no—
257C2×C4—3461no—
258C2×C4—1732no—
259C2×C4—6924no—
260C2×C4—6922no—
261C2×C4—6922no—
262C2×C4D4×D17311yescharacteristic
263C2×C4D4×D17311yescharacteristic
264C2×C4C23×D17311yescharacteristic
265C8—21no—
266C8—3462no—
267C8D4×D17312yescomplemented
268C22—42no—
269C22—42no—
270C22—42no—
271C22—3462no—
272C22—3462no—
273C22—3462no—
274C22—3462no—
275C22—3462no—
276C22—13848no—
277C22—6921no—
278C22—6921no—
279C22—6921no—
280C22—6921no—
281C22C2×D4×D17311yescharacteristic
282C4—21no—
283C4—42no—
284C4—3462no—
285C4—6921no—
286C4—6921no—
287C4#1914970612yescomplemented
288C4C2×D4×D17311yescharacteristic
289C4#1914972411yescharacteristic
290C2—82no—
291C2—6922no—
292C2—6922no—
293C2—6922no—
294C2#2601212611yescharacteristic
295C2C2×#1914970611yescharacteristic
296C2#2601229711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#492×C173C2
socleC22×C173C22×D4
hypercenter#492D173
hypocenterC173#3272
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. C2×#19149706C2semidirect
2. #26012144C2semidirect
3. #26012126C2semidirect
4. #26011734C2semidirect
5. #26011693C2semidirect
6. #26011683C2semidirect
7. #26011640C2semidirect
8. #26011635C2semidirect
9. C4×D1384C2semidirect
10. #26011570C2semidirect
11. #492×C173C2semidirect
12. C2×#14583827C22semidirect
18. C2×#14583811C22semidirect
19. C2×#14583809C22semidirect
21. #19149655C22semidirect
28. C4×D692C22semidirect
30. #19149613C22semidirect
31. C4⋊#13063436C22semidirect
32. #19149603C22semidirect
34. #19149601C22semidirect
35. #19149600C22semidirect
36. #19149591C22semidirect
37. C2×SD16×C173C22semidirect
38. C2×SD16×C173C22semidirect
39. C4⋊Q8×C173C22semidirect
41. C4×C8×C173C22semidirect
59. D1384D4semidirect
94. D692C2×D4semidirect
118. C8×C173C2×D4semidirect
135. C4×C173AGL(1,ℤ/8)semidirect
148. C173#3272semidirect
153. #492D173semidirect
166. C2×SD16C2×D173semidirect
167. C2×SD16C2×D173semidirect
171. C4⋊Q8C2×D173semidirect
190. C4×C8C2×D173semidirect
267. C8D4×D173semidirect
287. C4#19149706semidirect

Maximal subgroups

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

subgroupindex
C2×#191497062
#260121442
#260121262
#260117342
#260116932
#260116832
#260116402
#260116352
C4×D13842
#260115702
#492×C1732
#3272173