Finite Group Databasebeta

← #36362792

Subgroups — #36362792

Order 22144.

994 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 153 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×D4×D173C211yesmaximal, characteristic, complemented
2C2×#19149707C211yesmaximal, characteristic, complemented
3C2×#19149705C211yesmaximal, characteristic, complemented
4#26012003C211yesmaximal, characteristic, complemented
5#26011643C214yesmaximal, complemented
6#26011626C214yesmaximal, complemented
7#26011495C211yesmaximal, characteristic, complemented
8#26011454C211yesmaximal, characteristic, complemented
9#418×C173C211yesmaximal, characteristic, complemented
10C24×D173—21no—
11C2×#14583826C2211yescharacteristic, complemented
12C22×D692C2211yescharacteristic
13C22×C22⋊D173—21no—
14C2×D4×D173—24no—
15C2×D4×D173—24no—
16C2×D4×D173—22no—
17C2×D4×D173C2214yes—
18C22×C4×D173C2211yescharacteristic
19#19149707—24no—
20#19149705—24no—
21C2×#14583815C2212yescomplemented
22C2×#14583814C2212yescomplemented
23C2×D1384C2212yescomplemented
24C2×#14583811C2212yescomplemented
25C2×#14583810C2211yescharacteristic, complemented
26#19149663—21no—
27#19149659C2212yescomplemented
28#19149654C2212yescomplemented
29#19149649—21no—
30C4×D692—22no—
31#19149599C2214yescomplemented
32#19149591C2214yescomplemented
33C22×D4×C173C2211yescharacteristic
34C2×M16×C173C2211yescharacteristic, complemented
35#117×C173C2211yescharacteristic, complemented
36#102×C173C2214yescomplemented
37C23×D173—28no—
38C23×D173—24no—
39C23×D173—21no—
40C23×D173—48no—
41C23×D173D411yescharacteristic
42#14583826—22no—
43#14583826—22no—
44C2×Dic346C2311yescharacteristic, complemented
45C2×D692—22no—
46C2×D692—21no—
47C2×D692—22no—
48C2×D692D414yescomplemented
49C2×D692C2312yes—
50C2×D692C2311yescharacteristic, complemented
51C2×C22⋊D173—28no—
52C2×C22⋊D173—24no—
53C2×C22⋊D173—21no—
54D4×D173—216no—
55D4×D173—28no—
56D4×D173—28no—
57C22×Dic173D411yescharacteristic
58C2×C4×D173—22no—
59C2×C4×D173—22no—
60C2×C4×D173—21no—
61C2×C4×D173D412yes—
62C2×C4×D173C2312yes—
63#14583815—24no—
64#14583814—24no—
65D1384—24no—
66#14583811—24no—
67#14583810—22no—
68C2×#13063436C2312yescomplemented
69C4⋊Dic173—22no—
70#14583803—22no—
71#14583803—22no—
72C24×C173—21no—
73C2×D4×C173—21no—
74C2×D4×C173—22no—
75C2×D4×C173D414yescomplemented
76C2×D4×C173C2312yes—
77C22×C4×C173C2311yescharacteristic
78M16×C173—22no—
79C2×C8×C173C2312yescomplemented
80C4⋊C4×C173C2312yescomplemented
81C22⋊C4×C173—21no—
82C42×C173—21no—
83C22×D173—22no—
84C22×D173—24no—
85C22×D173—216no—
86C22×D173—24no—
87C22×D173—22no—
88C22×D173—21no—
89C22×D173—432no—
90C22×D173—48no—
91C22×D173—44no—
92C22×D173—48no—
93C22×D173C2×D412yes—
94Dic346—21no—
95Dic346—21no—
96D692—24no—
97D692—22no—
98D692—21no—
99D692—21no—
100D692—44no—
101D692C2×D414yes—
102C22⋊D173—216no—
103C22⋊D173—42no—
104C2×Dic173—22no—
105C2×Dic173—21no—
106C2×Dic173C2×D412yes—
107C4×D173—24no—
108C4×D173—22no—
109C4×D173—22no—
110C4×D173—42no—
111#13063436—22no—
112C23×C173—24no—
113C23×C173—22no—
114C23×C173—44no—
115C23×C173C2×D411yescharacteristic
116D4×C173—24no—
117D4×C173—22no—
118D4×C173C2×D414yes—
119C2×C4×C173—22no—
120C2×C4×C173C2×D411yescharacteristic
121C2×C4×C173C2×D412yes—
122C2×C4×C173C2×D412yes—
123C2×C4×C173C2411yescharacteristic
124C8×C173—22no—
125C22×C173—22no—
126C22×C173—24no—
127C22×C173—48no—
128C22×C173—44no—
129C22×C173#12012yes—
130C22×C173C22×D411yescharacteristic
131C2×D173—28no—
132C2×D173—22no—
133C2×D173—22no—
134C2×D173—24no—
135C2×D173—432no—
136C2×D173—48no—
137C2×D173—44no—
138C2×D173—41no—
139C2×D173—42no—
140C2×D173AGL(1,ℤ/8)14yescomplemented
141C4×C173—42no—
142C4×C173#12012yes—
143C4×C173C22×D411yescharacteristic
144C4×C173C22×D411yescharacteristic
145Dic173—22no—
146Dic173—41no—
147C2×C173—22no—
148C2×C173—44no—
149C2×C173C2×#12011yescharacteristic
150C2×C173C2×AGL(1,ℤ/8)12yes—
151D173—24no—
152D173—44no—
153D173—81no—
154C173#300311yescharacteristic, complemented, Sylow
155#3003—1731nomaximal, Sylow
156C23×D4—1731no—
157C2×AGL(1,ℤ/8)—1731no—
158C2×AGL(1,ℤ/8)—1731no—
159#525—1731no—
160#450—1734no—
161#447—1734no—
162#418—1731no—
163#418D17311yescharacteristic, complemented
164#407—1731no—
165C25—3461no—
166C2×#41—1731no—
167C22×D4—3464no—
168C22×D4—3464no—
169C22×D4—3462no—
170C22×D4—3461no—
171C22×D4—1734no—
172C22×D4—1731no—
173C22×D4C2×D17311yescharacteristic
174C23×C4—1731no—
175AGL(1,ℤ/8)—3464no—
176AGL(1,ℤ/8)—3464no—
177C2×SD16—1732no—
178C2×SD16—1732no—
179C2×D8—1732no—
180C2×D8—1732no—
181C2×M16—1731no—
182C2×M16C2×D17311yescharacteristic, complemented
183#123—3461no—
184C22⋊Q8—1732no—
185#121—1732no—
186#120—3461no—
187C4×D4—3462no—
188#117C2×D17311yescharacteristic, complemented
189#102—1734no—
190#102C2×D17314yescomplemented
191C22⋊C8—1734no—
192C24—21no—
193C24—3468no—
194C24—3464no—
195C24—3461no—
196C24—1731no—
197C24—6928no—
198#41—3462no—
199#41—3462no—
200C2×Q8—1731no—
201C2×D4—21no—
202C2×D4—22no—
203C2×D4—3462no—
204C2×D4—34616no—
205C2×D4—3468no—
206C2×D4—3468no—
207C2×D4—3468no—
208C2×D4—3461no—
209C2×D4—3464no—
210C2×D4—3461no—
211C2×D4—3462no—
212C2×D4—1734no—
213C2×D4—1731no—
214C2×D4—1732no—
215C2×D4D69214yescomplemented
216C2×D4C22×D17312yes—
217C22×C4—3462no—
218C22×C4—3462no—
219C22×C4—3461no—
220C22×C4—1731no—
221C22×C4—1732no—
222C22×C4—1732no—
223C22×C4C22×D17311yescharacteristic
224SD16—3464no—
225SD16—3464no—
226D8—3464no—
227D8—3464no—
228M16—22no—
229M16—3462no—
230C2×C8—1732no—
231C2×C8C22×D17312yescomplemented
232C4⋊C4—3462no—
233C4⋊C4C22×D17312yescomplemented
234C22⋊C4—21no—
235C22⋊C4—3462no—
236C22⋊C4—3462no—
237C42—21no—
238C23—24no—
239C23—22no—
240C23—44no—
241C23—3464no—
242C23—34616no—
243C23—3462no—
244C23—3464no—
245C23—3462no—
246C23—3461no—
247C23—1732no—
248C23—6924no—
249C23—69232no—
250C23—6928no—
251C23—6928no—
252C23D4×D17311yescharacteristic
253Q8—3461no—
254Q8—3461no—
255D4—24no—
256D4—22no—
257D4—34616no—
258D4—3464no—
259D4—3461no—
260D4—3462no—
261D4—3461no—
262D4—1734no—
263D4—6924no—
264D4—6922no—
265D4C2×D69214yes—
266C2×C4—22no—
267C2×C4—3462no—
268C2×C4—3464no—
269C2×C4—3462no—
270C2×C4—3462no—
271C2×C4—3461no—
272C2×C4—1732no—
273C2×C4—6922no—
274C2×C4D4×D17311yescharacteristic
275C2×C4D4×D17312yes—
276C2×C4C2×D69212yes—
277C2×C4C23×D17311yescharacteristic
278C8—22no—
279C8—3462no—
280C22—22no—
281C22—24no—
282C22—48no—
283C22—44no—
284C22—3468no—
285C22—3462no—
286C22—3462no—
287C22—3464no—
288C22—1734no—
289C22—6928no—
290C22—69232no—
291C22—6921no—
292C22—6922no—
293C22—6924no—
294C22#1914964912yes—
295C22C2×D4×D17311yescharacteristic
296C4—42no—
297C4—3462no—
298C4—6921no—
299C4#1914964912yes—
300C4C2×D4×D17311yescharacteristic
301C4C22×D69211yescharacteristic
302C2—22no—
303C2—44no—
304C2—3464no—
305C2—13841no—
306C2—6924no—
307C2C2×#1914964911yescharacteristic
308C2AGL(1,ℤ/8)×D17312yes—
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#418×C173C2
socleC22×C173C22×D4
hypercenter#418D173
hypocenterC173#3003
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×D4×D173C2semidirect
2. C2×#19149707C2semidirect
3. C2×#19149705C2semidirect
4. #26012003C2semidirect
5. #26011643C2semidirect
6. #26011626C2semidirect
7. #26011495C2semidirect
8. #26011454C2semidirect
9. #418×C173C2semidirect
11. C2×#14583826C22semidirect
21. C2×#14583815C22semidirect
22. C2×#14583814C22semidirect
23. C2×D1384C22semidirect
24. C2×#14583811C22semidirect
25. C2×#14583810C22semidirect
27. #19149659C22semidirect
28. #19149654C22semidirect
31. #19149599C22semidirect
32. #19149591C22semidirect
34. C2×M16×C173C22semidirect
35. #117×C173C22semidirect
36. #102×C173C22semidirect
44. C2×Dic346C23semidirect
48. C2×D692D4semidirect
50. C2×D692C23semidirect
68. C2×#13063436C23semidirect
75. C2×D4×C173D4semidirect
79. C2×C8×C173C23semidirect
80. C4⋊C4×C173C23semidirect
140. C2×D173AGL(1,ℤ/8)semidirect
154. C173#3003semidirect
163. #418D173semidirect
182. C2×M16C2×D173semidirect
188. #117C2×D173semidirect
190. #102C2×D173semidirect
215. C2×D4D692semidirect
231. C2×C8C22×D173semidirect
233. C4⋊C4C22×D173semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1732
C2×#191497072
C2×#191497052
#260120032
#260116432
#260116262
#260114952
#260114542
#418×C1732
#3003173