Finite Group Databasebeta

← #35203961

Subgroups — #35203961

Order 20288.

252 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 91 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
1#23778776C211yesmaximal, characteristic, complemented
2#23778775C211yesmaximal, characteristic, complemented
3#23778773C211yesmaximal, characteristic, complemented
4#23778765C211yesmaximal, characteristic, complemented
5#23778764C211yesmaximal, characteristic, complemented
6#23778763C211yesmaximal, characteristic, complemented
7#23778754C211yesmaximal, characteristic, complemented
8#23778753C211yesmaximal, characteristic, complemented
9#23778752C211yesmaximal, characteristic, complemented
10#23778752C211yesmaximal, characteristic, complemented
11#23778751C211yesmaximal, characteristic, complemented
12#23778750C211yesmaximal, characteristic, complemented
13#23778748C211yesmaximal, characteristic, complemented
14#23778748C211yesmaximal, characteristic, complemented
15#121×C317C211yesmaximal, characteristic, complemented
16C23×D317C2211yescharacteristic
17C2×D1268C2211yescharacteristic
18C2×D1268C2211yescharacteristic
19C2×C22⋊D317C2211yescharacteristic
20C2×C22⋊D317C2211yescharacteristic
21C2×C22⋊D317C2211yescharacteristic
22C2×C22⋊D317C2211yescharacteristic
23C2×C22⋊D317C2211yescharacteristic
24C2×C22⋊D317C2211yescharacteristic, complemented
25C2×C22⋊D317C2211yescharacteristic, complemented
26C22×Dic317C2211yescharacteristic, complemented
27C2×C4×D317C2211yescharacteristic, complemented
28C4⋊Dic317C2211yescharacteristic, complemented
29C4⋊Dic317C2211yescharacteristic, complemented
30#19099553C2211yescharacteristic, complemented
31#19099553C2211yescharacteristic, complemented
32#19099553C2211yescharacteristic, complemented
33C22⋊Dic317C2211yescharacteristic
34C22⋊Dic317C2211yescharacteristic
35C22⋊Dic317C2211yescharacteristic, complemented
36#19099551C2211yescharacteristic
37#19099551C2211yescharacteristic
38#19099551C2211yescharacteristic
39#19099551C2211yescharacteristic
40#19099551C2211yescharacteristic
41#19099551C2211yescharacteristic
42#19099551C2211yescharacteristic, complemented
43C4×Dic317C2211yescharacteristic, complemented
44C2×D4×C317C2211yescharacteristic
45C2×D4×C317C2211yescharacteristic
46C2×D4×C317C2211yescharacteristic, complemented
47C22×C4×C317C2211yescharacteristic, complemented
48C4⋊C4×C317C2211yescharacteristic, complemented
49C22⋊C4×C317C2211yescharacteristic
50C22⋊C4×C317C2211yescharacteristic, complemented
51C22×D317—41no—
52C22×D317—41no—
53C22×D317—41no—
54C22×D317C2311yescharacteristic
55C22×D317C2311yescharacteristic
56C22×D317C2311yescharacteristic
57D1268—41no—
58D1268—41no—
59C22⋊D317—41no—
60C22⋊D317—41no—
61C22⋊D317—41no—
62C22⋊D317—41no—
63C22⋊D317—41no—
64C22⋊D317—41no—
65C22⋊D317—41no—
66C2×Dic317—41no—
67C2×Dic317C2311yescharacteristic
68C2×Dic317C2311yescharacteristic
69C2×Dic317C2311yescharacteristic
70C2×Dic317C2311yescharacteristic
71C2×Dic317C2311yescharacteristic
72C4×D317—41no—
73C23×C317C2311yescharacteristic
74C23×C317C2311yescharacteristic
75C23×C317C2311yescharacteristic
76D4×C317—41no—
77D4×C317—41no—
78D4×C317—41no—
79C2×C4×C317—41no—
80C2×C4×C317C2311yescharacteristic
81C2×C4×C317C2311yescharacteristic
82C2×C4×C317C2311yescharacteristic
83C2×C4×C317C2311yescharacteristic
84C22×C317—21no—
85C22×C317—21no—
86C22×C317—21no—
87C22×C317—21no—
88C22×C317—21no—
89C22×C317—21no—
90C22×C317—21no—
91C22×C317—21no—
92C22×C317—21no—
93C22×C317C2411yescharacteristic
94C2×D317—21no—
95C2×D317—21no—
96C2×D317—21no—
97C2×D317—21no—
98C2×D317—21no—
99C2×D317—21no—
100C2×D317—21no—
101C2×D317—21no—
102C2×D317—21no—
103C2×D317—44no—
104C4×C317—21no—
105C4×C317—21no—
106C4×C317—21no—
107C4×C317—21no—
108Dic317—21no—
109Dic317—21no—
110Dic317—21no—
111Dic317—21no—
112Dic317—21no—
113C2×C317—41no—
114C2×C317—41no—
115C2×C317—41no—
116C2×C317H(5,𝔽2)11yescharacteristic
117C2×C317H(5,𝔽2)11yescharacteristic
118C2×C317H(5,𝔽2)11yescharacteristic
119D317—41no—
120D317—41no—
121D317—41no—
122C317#56011yescharacteristic, complemented, Sylow
123#560—3171nomaximal, Sylow
124#127—3171no—
125#126—3171no—
126#126—3171no—
127#123—3171no—
128#123—3171no—
129#123—3171no—
130#121—3171no—
131#121—3171no—
132#121—3171no—
133#121—3171no—
134#121—3171no—
135#121D31711yescharacteristic, complemented
136#120—3171no—
137#120—3171no—
138#120—3171no—
139C24—3171no—
140C2×D4—3171no—
141C2×D4—3171no—
142C2×D4—3171no—
143C2×D4—3171no—
144C2×D4—3171no—
145C2×D4—3171no—
146C2×D4—3171no—
147C2×D4—3171no—
148C2×D4—3171no—
149C2×D4C2×D31711yescharacteristic
150C2×D4C2×D31711yescharacteristic
151C2×D4C2×D31711yescharacteristic, complemented
152C22×C4—3171no—
153C22×C4—3171no—
154C22×C4C2×D31711yescharacteristic, complemented
155C4⋊C4—3171no—
156C4⋊C4—3171no—
157C4⋊C4—3171no—
158C4⋊C4—3171no—
159C4⋊C4—3171no—
160C4⋊C4C2×D31711yescharacteristic, complemented
161C22⋊C4—3171no—
162C22⋊C4—3171no—
163C22⋊C4—3171no—
164C22⋊C4—3171no—
165C22⋊C4—3171no—
166C22⋊C4—3171no—
167C22⋊C4—3171no—
168C22⋊C4—3171no—
169C22⋊C4—3171no—
170C22⋊C4—3171no—
171C22⋊C4C2×D31711yescharacteristic
172C22⋊C4C2×D31711yescharacteristic, complemented
173C42—3171no—
174C23—3171no—
175C23—3171no—
176C23—3171no—
177C23—12681no—
178C23—12681no—
179C23—12681no—
180C23C22×D31711yescharacteristic
181C23C22×D31711yescharacteristic
182C23C22×D31711yescharacteristic
183D4—41no—
184D4—41no—
185D4—41no—
186D4—12681no—
187D4—12681no—
188D4—12681no—
189D4—12681no—
190D4—12681no—
191D4—12681no—
192D4—12681no—
193D4—12681no—
194D4—12681no—
195C2×C4—41no—
196C2×C4—3171no—
197C2×C4—3171no—
198C2×C4—3171no—
199C2×C4—3171no—
200C2×C4—3171no—
201C2×C4—12681no—
202C2×C4—12681no—
203C2×C4C22×D31711yescharacteristic
204C2×C4C22×D31711yescharacteristic
205C2×C4C22×D31711yescharacteristic
206C2×C4C22×D31711yescharacteristic
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—6341no—
217C22—6341no—
218C22—6341no—
219C22—6341no—
220C22—6341no—
221C22—6341no—
222C22—6341no—
223C22—6341no—
224C22—6341no—
225C22—12684no—
226C22C23×D31711yescharacteristic
227C4—21no—
228C4—21no—
229C4—21no—
230C4—21no—
231C4—6341no—
232C4—6341no—
233C4—6341no—
234C4—6341no—
235C4—6341no—
236C2—41no—
237C2—41no—
238C2—41no—
239C2—12681no—
240C2—12681no—
241C2—12681no—
242C2#2377882511yescharacteristic
243C2#2377882511yescharacteristic
244C2#2377882611yescharacteristic
subgroupG/H
center, FrattiniC22C23×D317
commutator, socle, cosocleC22×C317C24
Fitting#121×C317C2
hypercenter#121D317
hypocenterC317#560
LCSG ▷ C22×C317 ▷ C317
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #23778776C2semidirect
2. #23778775C2semidirect
3. #23778773C2semidirect
4. #23778765C2semidirect
5. #23778764C2semidirect
6. #23778763C2semidirect
7. #23778754C2semidirect
8. #23778753C2semidirect
9. #23778752C2semidirect
10. #23778752C2semidirect
11. #23778751C2semidirect
12. #23778750C2semidirect
13. #23778748C2semidirect
14. #23778748C2semidirect
15. #121×C317C2semidirect
24. C2×C22⋊D317C22semidirect
25. C2×C22⋊D317C22semidirect
26. C22×Dic317C22semidirect
27. C2×C4×D317C22semidirect
28. C4⋊Dic317C22semidirect
29. C4⋊Dic317C22semidirect
30. #19099553C22semidirect
31. #19099553C22semidirect
32. #19099553C22semidirect
35. C22⋊Dic317C22semidirect
42. #19099551C22semidirect
43. C4×Dic317C22semidirect
46. C2×D4×C317C22semidirect
47. C22×C4×C317C22semidirect
48. C4⋊C4×C317C22semidirect
50. C22⋊C4×C317C22semidirect
122. C317#560semidirect
135. #121D317semidirect
151. C2×D4C2×D317semidirect
154. C22×C4C2×D317semidirect
160. C4⋊C4C2×D317semidirect
172. C22⋊C4C2×D317semidirect

Maximal subgroups

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

subgroupindex
#237787762
#237787752
#237787732
#237787652
#237787642
#237787632
#237787542
#237787532
#237787522
#237787512
#237787502
#237787482
#121×C3172
#560317