Finite Group Databasebeta

← #36365172

Subgroups — #36365172

Order 22144.

520 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 121 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#26012134C212yesmaximal, complemented
3#26011734C212yesmaximal, complemented
4#26011703C212yesmaximal, complemented
5#26011652C212yesmaximal, complemented
6#26011623C212yesmaximal, complemented
7C4×D1384C211yesmaximal, characteristic, complemented
8#26011570C211yesmaximal, characteristic, complemented
9#495×C173C211yesmaximal, characteristic, complemented
10C2×#14583827C2212yes—
11C2×D4×D173C2212yescomplemented
12#19149706—28no—
13C2×#14583813C2212yescomplemented
14C2×D1384C2211yescharacteristic
15C2×#14583811C2212yes—
16C2×#14583809C2212yescomplemented
17#19149661—22no—
18#19149655C2212yescomplemented
19#19149652—22no—
20#19149650C2212yes—
21#19149649—22no—
22C4×D692C2211yescharacteristic
23C4×D692C2212yescomplemented
24C22⋊C4×D173—22no—
25#19149613C2211yescharacteristic, complemented
26C4⋊#13063436C2211yescharacteristic, complemented
27#19149603C2212yescomplemented
28#19149601C2212yescomplemented
29#19149600C2212yescomplemented
30#19149591C2212yescomplemented
31C2×Q16×C173C2211yescharacteristic, complemented
32C2×SD16×C173C2212yescomplemented
33C2×D8×C173C2211yescharacteristic
34#124×C173C2212yescomplemented
35C4×C8×C173C2211yescharacteristic, complemented
36C23×D173—22no—
37#14583827—24no—
38#14583827—42no—
39C2×D692—22no—
40C2×D692D411yescharacteristic
41C2×D692C2312yes—
42C2×C22⋊D173—22no—
43C2×C22⋊D173—22no—
44D4×D173—24no—
45D4×D173—42no—
46C2×C4×D173—22no—
47C2×C4×D173—22no—
48C2×C4×D173C2312yes—
49#14583813—24no—
50D1384D414yescomplemented
51#14583811—24no—
52#14583809—24no—
53C2×#13063436C2312yes—
54C4⋊Dic173D411yescharacteristic
55C4⋊Dic173C2312yes—
56#14583805—22no—
57C22⋊Dic173—22no—
58#14583803—22no—
59#14583803—22no—
60#14583803D412yes—
61C2×Q8×C173C2312yes—
62C2×D4×C173C2312yes—
63Q16×C173—41no—
64SD16×C173—24no—
65D8×C173—41no—
66C2×C8×C173C2311yescharacteristic
67C2×C8×C173C2311yescharacteristic
68C22⋊C4×C173—22no—
69C22⋊C4×C173—22no—
70C42×C173C2311yescharacteristic
71C22×D173—22no—
72C22×D173—44no—
73C22×D173—44no—
74C22×D173—82no—
75C22×D173C2×D412yes—
76D692—22no—
77D692—41no—
78D692—44no—
79D692C2×D414yes—
80C22⋊D173—44no—
81C22⋊D173—82no—
82C2×Dic173—22no—
83C2×Dic173C2×D412yes—
84C4×D173—22no—
85C4×D173—22no—
86C4×D173—44no—
87C4×D173—82no—
88#13063436—22no—
89C23×C173—22no—
90Q8×C173—22no—
91Q8×C173—22no—
92D4×C173—22no—
93D4×C173—22no—
94C2×C4×C173—22no—
95C2×C4×C173C2×D411yescharacteristic
96C2×C4×C173C2×D411yescharacteristic
97C2×C4×C173C2411yescharacteristic
98C8×C173—21no—
99C8×C173C2×D412yes—
100C22×C173—42no—
101C22×C173—42no—
102C22×C173—42no—
103C22×C173C22×D411yescharacteristic
104C2×D173—22no—
105C2×D173—24no—
106C2×D173—22no—
107C2×D173—22no—
108C2×D173—42no—
109C2×D173—42no—
110C2×D173—88no—
111C4×C173—21no—
112C4×C173—21no—
113C4×C173—42no—
114C4×C173C22×D411yescharacteristic
115C4×C173H(5,𝔽2)11yescharacteristic
116Dic173—22no—
117Dic173—42no—
118C2×C173—82no—
119C2×C173D4211yescharacteristic
120C2×C173#57511yescharacteristic
121C2×C173#57611yescharacteristic
122D173—42no—
123D173—44no—
124C173#327311yescharacteristic, complemented, Sylow
125#3273—1731nomaximal, Sylow
126C2×AGL(1,ℤ/8)—1732no—
127#546—1732no—
128#495D17311yescharacteristic, complemented
129#468—1732no—
130#463—1732no—
131#452—1732no—
132#447—1732no—
133C4×D8—1731no—
134#436—1731no—
135C2×#41—1732no—
136C22×D4—1732no—
137AGL(1,ℤ/8)—3468no—
138C2×Q16C2×D17311yescharacteristic, complemented
139C2×SD16—1732no—
140C2×SD16C2×D17312yescomplemented
141C2×D8—1732no—
142C2×D8—1731no—
143C2×D8C2×D17311yescharacteristic
144C2×M16—1732no—
145#124C2×D17312yescomplemented
146#123—3462no—
147C22⋊Q8—1732no—
148#121—3462no—
149#121—1732no—
150#120—3462no—
151C4×D4—1732no—
152C4×D4—1731no—
153C2×C22⋊C4—3462no—
154#107—1731no—
155C4⋊C8—1731no—
156#103—1732no—
157#102—1732no—
158#102—1732no—
159C22⋊C8—1732no—
160C4×C8C2×D17311yescharacteristic, complemented
161C24—3462no—
162#41—3464no—
163#41—6922no—
164C2×Q8C22×D17312yes—
165C2×D4—3462no—
166C2×D4—3464no—
167C2×D4—3462no—
168C2×D4—3462no—
169C2×D4—1731no—
170C2×D4—1732no—
171C2×D4—6922no—
172C2×D4C22×D17312yes—
173C22×C4—3462no—
174C22×C4—3462no—
175C22×C4—1732no—
176Q16—41no—
177SD16—24no—
178SD16—3464no—
179D8—41no—
180D8—3464no—
181D8—1734no—
182M16—3464no—
183C2×C8—1732no—
184C2×C8C22×D17311yescharacteristic
185C2×C8C22×D17311yescharacteristic
186C4⋊C4—3462no—
187C4⋊C4—1732no—
188C4⋊C4—1731no—
189C22⋊C4—22no—
190C22⋊C4—22no—
191C22⋊C4—3462no—
192C22⋊C4—3462no—
193C22⋊C4—3462no—
194C22⋊C4—1732no—
195C42C22×D17311yescharacteristic
196C23—22no—
197C23—3462no—
198C23—13842no—
199C23—1732no—
200C23—6924no—
201C23—6924no—
202Q8—22no—
203Q8—22no—
204D4—22no—
205D4—22no—
206D4—3462no—
207D4—13842no—
208D4—1734no—
209D4—6921no—
210D4—6924no—
211D4—6924no—
212C2×C4—22no—
213C2×C4—3462no—
214C2×C4—3462no—
215C2×C4—3462no—
216C2×C4—13842no—
217C2×C4—1732no—
218C2×C4—6924no—
219C2×C4D4×D17311yescharacteristic
220C2×C4D4×D17311yescharacteristic
221C2×C4C23×D17311yescharacteristic
222C8—21no—
223C8—3462no—
224C8D4×D17312yes—
225C22—42no—
226C22—42no—
227C22—42no—
228C22—3462no—
229C22—3462no—
230C22—3462no—
231C22—3464no—
232C22—13848no—
233C22—6922no—
234C22—6922no—
235C22C2×D4×D17311yescharacteristic
236C4—21no—
237C4—21no—
238C4—42no—
239C4—3462no—
240C4—6922no—
241C4C2×D4×D17311yescharacteristic
242C4#1914972411yescharacteristic
243C2—82no—
244C2—6922no—
245C2—6924no—
246C2#2601212611yescharacteristic
247C2#2601228811yescharacteristic
248C2#2601229311yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#495×C173C2
socleC22×C173C22×D4
hypercenter#495D173
hypocenterC173#3273
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. C2×#19149706C2semidirect
2. #26012134C2semidirect
3. #26011734C2semidirect
4. #26011703C2semidirect
5. #26011652C2semidirect
6. #26011623C2semidirect
7. C4×D1384C2semidirect
8. #26011570C2semidirect
9. #495×C173C2semidirect
11. C2×D4×D173C22semidirect
13. C2×#14583813C22semidirect
16. C2×#14583809C22semidirect
18. #19149655C22semidirect
23. C4×D692C22semidirect
25. #19149613C22semidirect
26. C4⋊#13063436C22semidirect
27. #19149603C22semidirect
28. #19149601C22semidirect
29. #19149600C22semidirect
30. #19149591C22semidirect
31. C2×Q16×C173C22semidirect
32. C2×SD16×C173C22semidirect
34. #124×C173C22semidirect
35. C4×C8×C173C22semidirect
50. D1384D4semidirect
124. C173#3273semidirect
128. #495D173semidirect
138. C2×Q16C2×D173semidirect
140. C2×SD16C2×D173semidirect
145. #124C2×D173semidirect
160. C4×C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497062
#260121342
#260117342
#260117032
#260116522
#260116232
C4×D13842
#260115702
#495×C1732
#3273173