Finite Group Databasebeta

← #36362789

Subgroups — #36362789

Order 22144.

994 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C23×D692C211yesmaximal, characteristic, complemented
2C2×#19149704C212yesmaximal, complemented
3#26011641C214yesmaximal, complemented
4#26011624C214yesmaximal, complemented
5#26011495C212yesmaximal, complemented
6#26011455C211yesmaximal, characteristic, complemented
7#525×C173C211yesmaximal, characteristic, complemented
8C24×D173—21no—
9C22×D692—22no—
10C22×D692—21no—
11C22×D692—28no—
12C22×D692C2214yes—
13C22×D692C2212yes—
14#19149704—28no—
15C2×#14583813C2214yescomplemented
16C2×#14583811C2214yescomplemented
17C2×#14583810C2212yescomplemented
18#19149599C2218yescomplemented
19C22⋊#13063436C2214yescomplemented
20C2×#41×C173C2211yescharacteristic, complemented
21C23×C4×C173C2211yescharacteristic
22#123×C173—21no—
23C22⋊Q8×C173C2212yescomplemented
24#121×C173C2212yescomplemented
25#120×C173—21no—
26C4×D4×C173—22no—
27#117×C173C2211yescharacteristic, complemented
28C23×D173—28no—
29C23×D173—22no—
30C23×D173—24no—
31C23×D173—48no—
32C2×D692—28no—
33C2×D692—216no—
34C2×D692—216no—
35C2×D692—24no—
36C2×D692—24no—
37C2×D692—22no—
38C2×D692D418yescomplemented
39C2×D692C2314yes—
40#14583813—28no—
41#14583811—28no—
42#14583810—24no—
43C2×#13063436C2314yescomplemented
44C24×C173D411yescharacteristic
45#41×C173—24no—
46C2×Q8×C173C2311yescharacteristic, complemented
47C2×D4×C173—21no—
48C2×D4×C173—22no—
49C2×D4×C173C2311yescharacteristic, complemented
50C22×C4×C173—24no—
51C22×C4×C173—21no—
52C22×C4×C173D412yes—
53C22×C4×C173D411yescharacteristic
54C22×C4×C173C2312yes—
55C22×C4×C173C2311yescharacteristic
56C4⋊C4×C173—22no—
57C4⋊C4×C173C2312yescomplemented
58C22⋊C4×C173—21no—
59C22⋊C4×C173—22no—
60C22⋊C4×C173—22no—
61C42×C173—21no—
62C22×D173—216no—
63C22×D173—28no—
64C22×D173—24no—
65C22×D173—432no—
66C22×D173—48no—
67C22×D173—48no—
68C22×D173—48no—
69D692—216no—
70D692—28no—
71D692—24no—
72D692C2×D418yes—
73#13063436—24no—
74C23×C173—22no—
75C23×C173—24no—
76C23×C173—21no—
77C23×C173C2×D411yescharacteristic
78C23×C173C2×D412yes—
79Q8×C173—22no—
80D4×C173—22no—
81D4×C173—42no—
82D4×C173—44no—
83C2×C4×C173—22no—
84C2×C4×C173—24no—
85C2×C4×C173—24no—
86C2×C4×C173—22no—
87C2×C4×C173—21no—
88C2×C4×C173—42no—
89C2×C4×C173C2×D414yes—
90C2×C4×C173C2×D412yes—
91C2×C4×C173C2×D411yescharacteristic
92C2×C4×C173C2411yescharacteristic
93C22×C173—28no—
94C22×C173—22no—
95C22×C173—22no—
96C22×C173—22no—
97C22×C173—41no—
98C22×C173—42no—
99C22×C173#12012yes—
100C22×C173AGL(1,ℤ/8)14yescomplemented
101C22×C173C22×D411yescharacteristic
102C2×D173—28no—
103C2×D173—416no—
104C2×D173—432no—
105C2×D173—48no—
106C4×C173—22no—
107C4×C173—41no—
108C4×C173—42no—
109C4×C173#12012yes—
110C4×C173C22×D412yes—
111C2×C173—22no—
112C2×C173—24no—
113C2×C173—81no—
114C2×C173C2×#12011yescharacteristic
115C2×C173C2×AGL(1,ℤ/8)12yes—
116D173—48no—
117C173#300311yescharacteristic, complemented, Sylow
118#3003—1731nomaximal, Sylow
119C23×D4—1731no—
120C2×AGL(1,ℤ/8)—1732no—
121#525D17311yescharacteristic, complemented
122#450—1734no—
123#447—1734no—
124#418—1732no—
125#407—1731no—
126C25—3461no—
127C2×#41C2×D17311yescharacteristic, complemented
128C22×D4—3468no—
129C22×D4—3462no—
130C22×D4—3461no—
131C22×D4—1734no—
132C22×D4—1732no—
133C23×C4C2×D17311yescharacteristic
134AGL(1,ℤ/8)—3468no—
135C2×SD16—1734no—
136C2×D8—1734no—
137C2×M16—1732no—
138#123—21no—
139C22⋊Q8C2×D17312yescomplemented
140#121C2×D17312yescomplemented
141#120—21no—
142C4×D4—22no—
143#117C2×D17311yescharacteristic, complemented
144#102—1738no—
145C22⋊C8—1734no—
146C24—3468no—
147C24—3464no—
148C24—3462no—
149C24—6928no—
150C24C22⋊D17311yescharacteristic
151#41—24no—
152C2×Q8C22×D17311yescharacteristic, complemented
153C2×D4—22no—
154C2×D4—21no—
155C2×D4—3468no—
156C2×D4—34616no—
157C2×D4—34616no—
158C2×D4—3464no—
159C2×D4—3464no—
160C2×D4—3462no—
161C2×D4—1738no—
162C2×D4—1734no—
163C2×D4C22×D17311yescharacteristic, complemented
164C22×C4—24no—
165C22×C4—21no—
166C22×C4C22⋊D17312yes—
167C22×C4C22⋊D17311yescharacteristic
168C22×C4C22×D17312yes—
169C22×C4C22×D17311yescharacteristic
170SD16—3468no—
171D8—3468no—
172M16—3464no—
173C2×C8—1734no—
174C4⋊C4—22no—
175C4⋊C4C22×D17312yescomplemented
176C22⋊C4—21no—
177C22⋊C4—22no—
178C22⋊C4—22no—
179C42—21no—
180C23—22no—
181C23—24no—
182C23—21no—
183C23—34616no—
184C23—3468no—
185C23—3464no—
186C23—69232no—
187C23—6928no—
188C23—6928no—
189C23—6928no—
190C23C2×C22⋊D17311yescharacteristic
191C23C2×C22⋊D17312yes—
192Q8—22no—
193D4—22no—
194D4—42no—
195D4—44no—
196D4—34616no—
197D4—3468no—
198D4—3464no—
199D4—1738no—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—42no—
206C2×C4D4×D17314yes—
207C2×C4C2×C22⋊D17312yes—
208C2×C4C2×C22⋊D17311yescharacteristic
209C2×C4C23×D17311yescharacteristic
210C8—3464no—
211C22—28no—
212C22—22no—
213C22—22no—
214C22—22no—
215C22—41no—
216C22—42no—
217C22—3468no—
218C22—69216no—
219C22—69232no—
220C22—6928no—
221C22#1914964712yes—
222C22#1914970414yescomplemented
223C22C22×C22⋊D17311yescharacteristic
224C4—22no—
225C4—41no—
226C4—42no—
227C4#1914964712yes—
228C4C2×D4×D17312yes—
229C2—22no—
230C2—24no—
231C2—81no—
232C2—6928no—
233C2C2×#1914964711yescharacteristic
234C2C2×#1914970412yes—
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#525×C173C2
socleC22×C173C22×D4
hypercenter#525D173
hypocenterC173#3003
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. C23×D692C2semidirect
2. C2×#19149704C2semidirect
3. #26011641C2semidirect
4. #26011624C2semidirect
5. #26011495C2semidirect
6. #26011455C2semidirect
7. #525×C173C2semidirect
15. C2×#14583813C22semidirect
16. C2×#14583811C22semidirect
17. C2×#14583810C22semidirect
18. #19149599C22semidirect
19. C22⋊#13063436C22semidirect
20. C2×#41×C173C22semidirect
23. C22⋊Q8×C173C22semidirect
24. #121×C173C22semidirect
27. #117×C173C22semidirect
38. C2×D692D4semidirect
43. C2×#13063436C23semidirect
46. C2×Q8×C173C23semidirect
49. C2×D4×C173C23semidirect
57. C4⋊C4×C173C23semidirect
100. C22×C173AGL(1,ℤ/8)semidirect
117. C173#3003semidirect
121. #525D173semidirect
127. C2×#41C2×D173semidirect
139. C22⋊Q8C2×D173semidirect
140. #121C2×D173semidirect
143. #117C2×D173semidirect
152. C2×Q8C22×D173semidirect
163. C2×D4C22×D173semidirect
175. C4⋊C4C22×D173semidirect
222. C22#19149704semidirect

Maximal subgroups

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

subgroupindex
C23×D6922
C2×#191497042
#260116412
#260116242
#260114952
#260114552
#525×C1732
#3003173