Finite Group Databasebeta

← #36378525

Subgroups — #36378525

Order 22592.

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#26014988C211yesmaximal, characteristic, complemented
2#26014987C211yesmaximal, characteristic, complemented
3#26014985C211yesmaximal, characteristic, complemented
4#26014977C211yesmaximal, characteristic, complemented
5#26014976C211yesmaximal, characteristic, complemented
6#26014975C211yesmaximal, characteristic, complemented
7#26014966C211yesmaximal, characteristic, complemented
8#26014965C211yesmaximal, characteristic, complemented
9#26014964C211yesmaximal, characteristic, complemented
10#26014964C211yesmaximal, characteristic, complemented
11#26014963C211yesmaximal, characteristic, complemented
12#26014962C211yesmaximal, characteristic, complemented
13#26014960C211yesmaximal, characteristic, complemented
14#26014960C211yesmaximal, characteristic, complemented
15#121×C353C211yesmaximal, characteristic, complemented
16C23×D353C2211yescharacteristic
17C2×D1412C2211yescharacteristic
18C2×D1412C2211yescharacteristic
19C2×C22⋊D353C2211yescharacteristic
20C2×C22⋊D353C2211yescharacteristic
21C2×C22⋊D353C2211yescharacteristic
22C2×C22⋊D353C2211yescharacteristic
23C2×C22⋊D353C2211yescharacteristic
24C2×C22⋊D353C2211yescharacteristic, complemented
25C2×C22⋊D353C2211yescharacteristic, complemented
26C22×Dic353C2211yescharacteristic, complemented
27C2×C4×D353C2211yescharacteristic, complemented
28C4⋊Dic353C2211yescharacteristic, complemented
29C4⋊Dic353C2211yescharacteristic, complemented
30#19150716C2211yescharacteristic, complemented
31#19150716C2211yescharacteristic, complemented
32#19150716C2211yescharacteristic, complemented
33C22⋊Dic353C2211yescharacteristic
34C22⋊Dic353C2211yescharacteristic
35C22⋊Dic353C2211yescharacteristic, complemented
36#19150714C2211yescharacteristic
37#19150714C2211yescharacteristic
38#19150714C2211yescharacteristic
39#19150714C2211yescharacteristic
40#19150714C2211yescharacteristic
41#19150714C2211yescharacteristic
42#19150714C2211yescharacteristic, complemented
43C4×Dic353C2211yescharacteristic, complemented
44C2×D4×C353C2211yescharacteristic
45C2×D4×C353C2211yescharacteristic
46C2×D4×C353C2211yescharacteristic, complemented
47C22×C4×C353C2211yescharacteristic, complemented
48C4⋊C4×C353C2211yescharacteristic, complemented
49C22⋊C4×C353C2211yescharacteristic
50C22⋊C4×C353C2211yescharacteristic, complemented
51C22×D353—41no—
52C22×D353—41no—
53C22×D353—41no—
54C22×D353C2311yescharacteristic
55C22×D353C2311yescharacteristic
56C22×D353C2311yescharacteristic
57D1412—41no—
58D1412—41no—
59C22⋊D353—41no—
60C22⋊D353—41no—
61C22⋊D353—41no—
62C22⋊D353—41no—
63C22⋊D353—41no—
64C22⋊D353—41no—
65C22⋊D353—41no—
66C2×Dic353—41no—
67C2×Dic353C2311yescharacteristic
68C2×Dic353C2311yescharacteristic
69C2×Dic353C2311yescharacteristic
70C2×Dic353C2311yescharacteristic
71C2×Dic353C2311yescharacteristic
72C4×D353—41no—
73C23×C353C2311yescharacteristic
74C23×C353C2311yescharacteristic
75C23×C353C2311yescharacteristic
76D4×C353—41no—
77D4×C353—41no—
78D4×C353—41no—
79C2×C4×C353—41no—
80C2×C4×C353C2311yescharacteristic
81C2×C4×C353C2311yescharacteristic
82C2×C4×C353C2311yescharacteristic
83C2×C4×C353C2311yescharacteristic
84C22×C353—21no—
85C22×C353—21no—
86C22×C353—21no—
87C22×C353—21no—
88C22×C353—21no—
89C22×C353—21no—
90C22×C353—21no—
91C22×C353—21no—
92C22×C353—21no—
93C22×C353C2411yescharacteristic
94C2×D353—21no—
95C2×D353—21no—
96C2×D353—21no—
97C2×D353—21no—
98C2×D353—21no—
99C2×D353—21no—
100C2×D353—21no—
101C2×D353—21no—
102C2×D353—21no—
103C2×D353—44no—
104C4×C353—21no—
105C4×C353—21no—
106C4×C353—21no—
107C4×C353—21no—
108Dic353—21no—
109Dic353—21no—
110Dic353—21no—
111Dic353—21no—
112Dic353—21no—
113C2×C353—41no—
114C2×C353—41no—
115C2×C353—41no—
116C2×C353H(5,𝔽2)11yescharacteristic
117C2×C353H(5,𝔽2)11yescharacteristic
118C2×C353H(5,𝔽2)11yescharacteristic
119D353—41no—
120D353—41no—
121D353—41no—
122C353#56011yescharacteristic, complemented, Sylow
123#560—3531nomaximal, Sylow
124#127—3531no—
125#126—3531no—
126#126—3531no—
127#123—3531no—
128#123—3531no—
129#123—3531no—
130#121—3531no—
131#121—3531no—
132#121—3531no—
133#121—3531no—
134#121—3531no—
135#121D35311yescharacteristic, complemented
136#120—3531no—
137#120—3531no—
138#120—3531no—
139C24—3531no—
140C2×D4—3531no—
141C2×D4—3531no—
142C2×D4—3531no—
143C2×D4—3531no—
144C2×D4—3531no—
145C2×D4—3531no—
146C2×D4—3531no—
147C2×D4—3531no—
148C2×D4—3531no—
149C2×D4C2×D35311yescharacteristic
150C2×D4C2×D35311yescharacteristic
151C2×D4C2×D35311yescharacteristic, complemented
152C22×C4—3531no—
153C22×C4—3531no—
154C22×C4C2×D35311yescharacteristic, complemented
155C4⋊C4—3531no—
156C4⋊C4—3531no—
157C4⋊C4—3531no—
158C4⋊C4—3531no—
159C4⋊C4—3531no—
160C4⋊C4C2×D35311yescharacteristic, complemented
161C22⋊C4—3531no—
162C22⋊C4—3531no—
163C22⋊C4—3531no—
164C22⋊C4—3531no—
165C22⋊C4—3531no—
166C22⋊C4—3531no—
167C22⋊C4—3531no—
168C22⋊C4—3531no—
169C22⋊C4—3531no—
170C22⋊C4—3531no—
171C22⋊C4C2×D35311yescharacteristic
172C22⋊C4C2×D35311yescharacteristic, complemented
173C42—3531no—
174C23—3531no—
175C23—3531no—
176C23—3531no—
177C23—14121no—
178C23—14121no—
179C23—14121no—
180C23C22×D35311yescharacteristic
181C23C22×D35311yescharacteristic
182C23C22×D35311yescharacteristic
183D4—41no—
184D4—41no—
185D4—41no—
186D4—14121no—
187D4—14121no—
188D4—14121no—
189D4—14121no—
190D4—14121no—
191D4—14121no—
192D4—14121no—
193D4—14121no—
194D4—14121no—
195C2×C4—41no—
196C2×C4—3531no—
197C2×C4—3531no—
198C2×C4—3531no—
199C2×C4—3531no—
200C2×C4—3531no—
201C2×C4—14121no—
202C2×C4—14121no—
203C2×C4C22×D35311yescharacteristic
204C2×C4C22×D35311yescharacteristic
205C2×C4C22×D35311yescharacteristic
206C2×C4C22×D35311yescharacteristic
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—14124no—
217C22—7061no—
218C22—7061no—
219C22—7061no—
220C22—7061no—
221C22—7061no—
222C22—7061no—
223C22—7061no—
224C22—7061no—
225C22—7061no—
226C22C23×D35311yescharacteristic
227C4—21no—
228C4—21no—
229C4—21no—
230C4—21no—
231C4—7061no—
232C4—7061no—
233C4—7061no—
234C4—7061no—
235C4—7061no—
236C2—41no—
237C2—41no—
238C2—41no—
239C2—14121no—
240C2—14121no—
241C2—14121no—
242C2#2601503711yescharacteristic
243C2#2601503711yescharacteristic
244C2#2601503811yescharacteristic
subgroupG/H
center, FrattiniC22C23×D353
commutator, socle, cosocleC22×C353C24
Fitting#121×C353C2
hypercenter#121D353
hypocenterC353#560
LCSG ▷ C22×C353 ▷ C353
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #26014988C2semidirect
2. #26014987C2semidirect
3. #26014985C2semidirect
4. #26014977C2semidirect
5. #26014976C2semidirect
6. #26014975C2semidirect
7. #26014966C2semidirect
8. #26014965C2semidirect
9. #26014964C2semidirect
10. #26014964C2semidirect
11. #26014963C2semidirect
12. #26014962C2semidirect
13. #26014960C2semidirect
14. #26014960C2semidirect
15. #121×C353C2semidirect
24. C2×C22⋊D353C22semidirect
25. C2×C22⋊D353C22semidirect
26. C22×Dic353C22semidirect
27. C2×C4×D353C22semidirect
28. C4⋊Dic353C22semidirect
29. C4⋊Dic353C22semidirect
30. #19150716C22semidirect
31. #19150716C22semidirect
32. #19150716C22semidirect
35. C22⋊Dic353C22semidirect
42. #19150714C22semidirect
43. C4×Dic353C22semidirect
46. C2×D4×C353C22semidirect
47. C22×C4×C353C22semidirect
48. C4⋊C4×C353C22semidirect
50. C22⋊C4×C353C22semidirect
122. C353#560semidirect
135. #121D353semidirect
151. C2×D4C2×D353semidirect
154. C22×C4C2×D353semidirect
160. C4⋊C4C2×D353semidirect
172. C22⋊C4C2×D353semidirect

Maximal subgroups

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

subgroupindex
#260149882
#260149872
#260149852
#260149772
#260149762
#260149752
#260149662
#260149652
#260149642
#260149632
#260149622
#260149602
#121×C3532
#560353