Finite Group Databasebeta

← #36364012

Subgroups — #36364012

Order 22144.

656 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 141 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
2C2×#19149703C212yesmaximal, complemented
3#26012040C211yesmaximal, characteristic, complemented
4#127×D173C211yesmaximal, characteristic, complemented
5#26011881C211yesmaximal, characteristic, complemented
6#26011880C211yesmaximal, characteristic, complemented
7#26011878C211yesmaximal, characteristic, complemented
8#26011866C212yesmaximal, complemented
9#26011862C212yesmaximal, complemented
10#26011443C211yesmaximal, characteristic, complemented
11#496×C173C211yesmaximal, characteristic, complemented
12C2×#14583827C2211yescharacteristic
13C2×#14583825C2211yescharacteristic
14C2×D4×D173—22no—
15C2×D4×D173C2212yes—
16#19149706—28no—
17#19149703—28no—
18C2×#14583815C2212yescomplemented
19C2×#14583814C2212yescomplemented
20C2×#14583813C2212yescomplemented
21C2×D1384C2212yescomplemented
22C2×#14583811C2212yescomplemented
23C2×#14583811C2212yescomplemented
24C2×#14583809C2212yes—
25#19149677C2211yescharacteristic, complemented
26#19149676C2211yescharacteristic
27#19149675—22no—
28#19149675C2212yes—
29#19149675C2211yescharacteristic, complemented
30#19149667C2211yescharacteristic, complemented
31#19149652—22no—
32#19149639—22no—
33C42×D173C2211yescharacteristic
34#19149590C2211yescharacteristic, complemented
35C8×Dic173C2212yescomplemented
36C2×SD16×C173C2212yescomplemented
37C2×D8×C173C2212yescomplemented
38#127×C173C2211yescharacteristic
39#124×C173C2211yescharacteristic, complemented
40#97×C173C2211yescharacteristic, complemented
41C23×D173—22no—
42#14583827—24no—
43#14583825—24no—
44C2×Dic346C2311yescharacteristic
45C2×D692—21no—
46C2×D692—22no—
47C2×D692C2311yescharacteristic
48C2×D692C2312yes—
49C2×C22⋊D173—21no—
50C2×C22⋊D173—22no—
51C2×C22⋊D173—22no—
52C2×C22⋊D173—24no—
53D4×D173—24no—
54D4×D173—24no—
55D4×D173—48no—
56C22×Dic173—21no—
57C2×C4×D173—21no—
58C2×C4×D173D412yes—
59C2×C4×D173C2311yescharacteristic
60#14583815—24no—
61#14583814—24no—
62#14583813—24no—
63D1384—24no—
64#14583811—24no—
65#14583811—24no—
66#14583809D418yescomplemented
67C2×#13063436C2312yes—
68#14583805—22no—
69#14583803—22no—
70C4×Dic173D412yes—
71C4×Dic173C2311yescharacteristic
72C2×Q8×C173C2311yescharacteristic
73C2×D4×C173—22no—
74C2×D4×C173C2311yescharacteristic
75C2×D4×C173C2312yes—
76SD16×C173—24no—
77D8×C173—24no—
78C2×C8×C173C2312yes—
79C22⋊C4×C173—22no—
80C42×C173C2311yescharacteristic
81C22×D173—21no—
82C22×D173—22no—
83C22×D173—48no—
84C22×D173—44no—
85C22×D173C2×D411yescharacteristic
86Dic346—22no—
87D692—22no—
88D692—22no—
89D692—22no—
90D692—44no—
91D692—42no—
92C22⋊D173—42no—
93C22⋊D173—44no—
94C22⋊D173—48no—
95C22⋊D173—44no—
96C2×Dic173—21no—
97C2×Dic173—42no—
98C2×Dic173C2×D411yescharacteristic
99C2×Dic173C2×D412yes—
100C4×D173—24no—
101C4×D173—42no—
102C4×D173C2×D414yes—
103#13063436C2×D414yescomplemented
104C23×C173—21no—
105C23×C173—22no—
106Q8×C173—22no—
107D4×C173—22no—
108D4×C173—22no—
109D4×C173—22no—
110D4×C173—44no—
111C2×C4×C173—21no—
112C2×C4×C173C2×D412yes—
113C2×C4×C173C2411yescharacteristic
114C8×C173C2×D414yescomplemented
115C22×C173—41no—
116C22×C173—42no—
117C22×C173—44no—
118C22×C173—42no—
119C22×C173C22×D411yescharacteristic
120C2×D173—22no—
121C2×D173—41no—
122C2×D173—42no—
123C2×D173—44no—
124C2×D173—42no—
125C2×D173—88no—
126C2×D173#12712yes—
127C4×C173—22no—
128C4×C173—41no—
129C4×C173C22×D412yes—
130Dic173—41no—
131Dic173#12712yes—
132Dic173AGL(1,ℤ/8)14yescomplemented
133C2×C173—82no—
134C2×C173—81no—
135C2×C173C2×#12711yescharacteristic
136C2×C173C2×AGL(1,ℤ/8)12yes—
137D173—22no—
138D173—81no—
139D173—82no—
140C173#315111yescharacteristic, complemented, Sylow
141#3151—1731nomaximal, Sylow
142C2×AGL(1,ℤ/8)—1732no—
143C2×AGL(1,ℤ/8)—1732no—
144#532—1731no—
145C2×#127—1731no—
146#496—1731no—
147#496—1731no—
148#496—1731no—
149#496D17311yescharacteristic, complemented
150#493—1732no—
151#492—1732no—
152C4×M16—1731no—
153C2×#41—1731no—
154C2×#41—1731no—
155C22×D4—3462no—
156C22×D4—1732no—
157AGL(1,ℤ/8)—3468no—
158AGL(1,ℤ/8)—3468no—
159C2×SD16—1732no—
160C2×SD16—1732no—
161C2×SD16—1732no—
162C2×SD16C2×D17312yescomplemented
163C2×D8—1732no—
164C2×D8—1732no—
165C2×D8—1732no—
166C2×D8C2×D17312yescomplemented
167C2×M16—1732no—
168C4⋊Q8—1731no—
169#127—3462no—
170#127—1732no—
171#127—1731no—
172#127—1731no—
173#127C2×D17311yescharacteristic
174#124—1731no—
175#124C2×D17311yescharacteristic, complemented
176#121—3462no—
177C4×D4—3462no—
178C2×C42—1731no—
179#97—1731no—
180#97C2×D17311yescharacteristic, complemented
181C4×C8—1732no—
182C24—3462no—
183#41—3464no—
184#41—3464no—
185C2×Q8—1731no—
186C2×Q8C22×D17311yescharacteristic
187C2×D4—22no—
188C2×D4—3461no—
189C2×D4—3461no—
190C2×D4—3464no—
191C2×D4—3464no—
192C2×D4—3462no—
193C2×D4—3462no—
194C2×D4—3464no—
195C2×D4—3462no—
196C2×D4—1731no—
197C2×D4—1732no—
198C2×D4—6928no—
199C2×D4C22×D17311yescharacteristic
200C2×D4C22×D17312yes—
201C22×C4—3461no—
202C22×C4—3461no—
203C22×C4—1732no—
204C22×C4—1731no—
205SD16—24no—
206SD16—3464no—
207SD16—3464no—
208SD16—3464no—
209D8—24no—
210D8—3464no—
211D8—3464no—
212D8—3464no—
213M16—1738no—
214C2×C8—1732no—
215C2×C8C22×D17312yes—
216C4⋊C4—3462no—
217C22⋊C4—22no—
218C22⋊C4—3462no—
219C42—1732no—
220C42—1731no—
221C42C22×D17311yescharacteristic
222C23—21no—
223C23—22no—
224C23—3461no—
225C23—3462no—
226C23—1731no—
227C23—6928no—
228C23—6924no—
229Q8—22no—
230Q8—3462no—
231D4—22no—
232D4—22no—
233D4—22no—
234D4—44no—
235D4—3462no—
236D4—3462no—
237D4—3462no—
238D4—6924no—
239D4—6922no—
240D4—6924no—
241D4—6924no—
242D4—6928no—
243D4—6922no—
244C2×C4—21no—
245C2×C4—3464no—
246C2×C4—3461no—
247C2×C4—1732no—
248C2×C4—1731no—
249C2×C4—1734no—
250C2×C4—6922no—
251C2×C4—6922no—
252C2×C4D4×D17312yes—
253C2×C4C23×D17311yescharacteristic
254C8—1734no—
255C8D4×D17314yescomplemented
256C22—44no—
257C22—42no—
258C22—41no—
259C22—42no—
260C22—3462no—
261C22—13848no—
262C22—1732no—
263C22—6924no—
264C22—6922no—
265C22—6921no—
266C22—6922no—
267C22C2×D4×D17311yescharacteristic
268C4—22no—
269C4—41no—
270C4—1734no—
271C4—1732no—
272C4—6921no—
273C4C2×D4×D17312yes—
274C2—81no—
275C2—82no—
276C2—3462no—
277C2—13842no—
278C2—13841no—
279C2#127×D17311yescharacteristic
280C2AGL(1,ℤ/8)×D17312yes—
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#496×C173C2
socleC22×C173C22×D4
hypercenter#496D173
hypocenterC173#3151
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#19149706C2semidirect
2. C2×#19149703C2semidirect
3. #26012040C2semidirect
4. #127×D173C2semidirect
5. #26011881C2semidirect
6. #26011880C2semidirect
7. #26011878C2semidirect
8. #26011866C2semidirect
9. #26011862C2semidirect
10. #26011443C2semidirect
11. #496×C173C2semidirect
18. C2×#14583815C22semidirect
19. C2×#14583814C22semidirect
20. C2×#14583813C22semidirect
21. C2×D1384C22semidirect
22. C2×#14583811C22semidirect
23. C2×#14583811C22semidirect
25. #19149677C22semidirect
29. #19149675C22semidirect
30. #19149667C22semidirect
34. #19149590C22semidirect
35. C8×Dic173C22semidirect
36. C2×SD16×C173C22semidirect
37. C2×D8×C173C22semidirect
39. #124×C173C22semidirect
40. #97×C173C22semidirect
66. #14583809D4semidirect
103. #13063436C2×D4semidirect
114. C8×C173C2×D4semidirect
132. Dic173AGL(1,ℤ/8)semidirect
140. C173#3151semidirect
149. #496D173semidirect
162. C2×SD16C2×D173semidirect
166. C2×D8C2×D173semidirect
175. #124C2×D173semidirect
180. #97C2×D173semidirect
255. C8D4×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497062
C2×#191497032
#260120402
#127×D1732
#260118812
#260118802
#260118782
#260118662
#260118622
#260114432
#496×C1732
#3151173