Finite Group Databasebeta

← #36378370

Subgroups — #36378370

Order 22592.

250 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 95 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#26014987C211yesmaximal, characteristic, complemented
2#26014986C211yesmaximal, characteristic, complemented
3#26014980C211yesmaximal, characteristic, complemented
4#26014978C211yesmaximal, characteristic, complemented
5#26014977C211yesmaximal, characteristic, complemented
6#26014975C211yesmaximal, characteristic, complemented
7#26014969C211yesmaximal, characteristic, complemented
8#26014966C211yesmaximal, characteristic, complemented
9#26014965C211yesmaximal, characteristic, complemented
10#26014964C211yesmaximal, characteristic, complemented
11#26014962C211yesmaximal, characteristic, complemented
12#26014960C211yesmaximal, characteristic, complemented
13C4×C22⋊D353C211yesmaximal, characteristic, complemented
14C2×#19150714C211yesmaximal, characteristic, complemented
15C4×D4×C353C211yesmaximal, characteristic, complemented
16C23×D353C2211yescharacteristic
17C2×Dic706C2211yescharacteristic, complemented
18C2×D1412C2211yescharacteristic
19C2×D1412C2211yescharacteristic
20C2×C22⋊D353C2211yescharacteristic
21C2×C22⋊D353C2211yescharacteristic
22C2×C22⋊D353C2211yescharacteristic
23C2×C22⋊D353C2211yescharacteristic, complemented
24C22×Dic353C2211yescharacteristic
25C2×C4×D353C2211yescharacteristic, complemented
26C4⋊Dic353C2211yescharacteristic, complemented
27#19150716C2211yescharacteristic, complemented
28#19150716C2211yescharacteristic, complemented
29#19150716C2211yescharacteristic, complemented
30#19150716C2211yescharacteristic, complemented
31C22⋊Dic353C2211yescharacteristic
32C22⋊Dic353C2211yescharacteristic, complemented
33#19150714C2211yescharacteristic
34#19150714C2211yescharacteristic
35#19150714C2211yescharacteristic
36#19150714C2211yescharacteristic
37#19150714C2211yescharacteristic
38#19150714C2211yescharacteristic
39#19150714C2211yescharacteristic
40#19150714C2211yescharacteristic
41#19150714C2211yescharacteristic, complemented
42#19150714C2211yescharacteristic, complemented
43C4×Dic353C2211yescharacteristic, complemented
44C2×D4×C353C2211yescharacteristic
45C22×C4×C353C2211yescharacteristic
46C22×C4×C353C2211yescharacteristic, complemented
47C4⋊C4×C353C2211yescharacteristic, complemented
48C22⋊C4×C353C2211yescharacteristic
49C22⋊C4×C353C2211yescharacteristic, complemented
50C42×C353C2211yescharacteristic, complemented
51C22×D353—22no—
52C22×D353—41no—
53C22×D353—41no—
54C22×D353C2311yescharacteristic
55C22×D353C2311yescharacteristic
56C22×D353C2311yescharacteristic
57Dic706—41no—
58D1412—41no—
59D1412—41no—
60C22⋊D353—22no—
61C22⋊D353—41no—
62C22⋊D353—41no—
63C22⋊D353—41no—
64C2×Dic353—41no—
65C2×Dic353C2311yescharacteristic
66C2×Dic353C2311yescharacteristic
67C2×Dic353C2311yescharacteristic
68C2×Dic353C2311yescharacteristic
69C2×Dic353C2311yescharacteristic
70C4×D353—41no—
71C23×C353C2311yescharacteristic
72C23×C353C2311yescharacteristic
73D4×C353—22no—
74C2×C4×C353—41no—
75C2×C4×C353—41no—
76C2×C4×C353C2311yescharacteristic
77C2×C4×C353C2311yescharacteristic
78C2×C4×C353C2311yescharacteristic
79C2×C4×C353C2311yescharacteristic
80C2×C4×C353C2311yescharacteristic
81C22×C353—21no—
82C22×C353—21no—
83C22×C353—21no—
84C22×C353—21no—
85C22×C353—21no—
86C22×C353#4112yes—
87C22×C353C2411yescharacteristic
88C2×D353—21no—
89C2×D353—21no—
90C2×D353—21no—
91C2×D353—21no—
92C2×D353—21no—
93C2×D353—21no—
94C2×D353—21no—
95C2×D353—21no—
96C2×D353—21no—
97C2×D353—44no—
98C4×C353—21no—
99C4×C353—21no—
100C4×C353—21no—
101C4×C353—21no—
102C4×C353—21no—
103Dic353—21no—
104Dic353—21no—
105Dic353—21no—
106Dic353—21no—
107Dic353—21no—
108C2×C353—22no—
109C2×C353—41no—
110C2×C353C2×#4111yescharacteristic
111C2×C353H(5,𝔽2)11yescharacteristic
112C2×C353H(5,𝔽2)11yescharacteristic
113D353—41no—
114D353—41no—
115D353—41no—
116C353#53811yescharacteristic, complemented, Sylow
117#538—3531nomaximal, Sylow
118#126—3531no—
119#126—3531no—
120#124—3531no—
121#124—3531no—
122#123—3531no—
123#123—3531no—
124C22⋊Q8—3531no—
125#121—3531no—
126#121—3531no—
127#121—3531no—
128#120—3531no—
129#120—3531no—
130C4×D4—3531no—
131C4×D4D35311yescharacteristic, complemented
132C2×C22⋊C4—3531no—
133C24—3531no—
134C2×Q8—3531no—
135C2×D4—3531no—
136C2×D4—3531no—
137C2×D4—3531no—
138C2×D4—3531no—
139C2×D4—3531no—
140C2×D4—3531no—
141C2×D4C2×D35311yescharacteristic
142C22×C4—3531no—
143C22×C4—3531no—
144C22×C4C2×D35311yescharacteristic
145C22×C4C2×D35311yescharacteristic, complemented
146C4⋊C4—3531no—
147C4⋊C4—3531no—
148C4⋊C4—3531no—
149C4⋊C4—3531no—
150C4⋊C4—3531no—
151C4⋊C4C2×D35311yescharacteristic, complemented
152C22⋊C4—3531no—
153C22⋊C4—3531no—
154C22⋊C4—3531no—
155C22⋊C4—3531no—
156C22⋊C4—3531no—
157C22⋊C4—3531no—
158C22⋊C4—3531no—
159C22⋊C4—3531no—
160C22⋊C4—3531no—
161C22⋊C4—3531no—
162C22⋊C4—3531no—
163C22⋊C4—3531no—
164C22⋊C4C2×D35311yescharacteristic
165C22⋊C4C2×D35311yescharacteristic, complemented
166C42—3531no—
167C42C2×D35311yescharacteristic, complemented
168C23—3531no—
169C23—3531no—
170C23—3531no—
171C23—14121no—
172C23—14121no—
173C23—7062no—
174C23C22×D35311yescharacteristic
175C23C22×D35311yescharacteristic
176Q8—14121no—
177D4—22no—
178D4—14121no—
179D4—14121no—
180D4—14121no—
181D4—14121no—
182D4—14121no—
183D4—7062no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—3531no—
187C2×C4—3531no—
188C2×C4—3531no—
189C2×C4—3531no—
190C2×C4—3531no—
191C2×C4—14121no—
192C2×C4—14121no—
193C2×C4C22×D35311yescharacteristic
194C2×C4C22×D35311yescharacteristic
195C2×C4C22×D35311yescharacteristic
196C2×C4C22×D35311yescharacteristic
197C2×C4C22×D35311yescharacteristic
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—14124no—
204C22—7061no—
205C22—7061no—
206C22—7061no—
207C22—7061no—
208C22—7061no—
209C22—7061no—
210C22—7061no—
211C22—7061no—
212C22—7061no—
213C22#1915073712yes—
214C22C23×D35311yescharacteristic
215C4—21no—
216C4—21no—
217C4—21no—
218C4—21no—
219C4—21no—
220C4—7061no—
221C4—7061no—
222C4—7061no—
223C4—7061no—
224C4—7061no—
225C2—22no—
226C2—41no—
227C2—14121no—
228C2—14121no—
229C2—14121no—
230C2C2×#1915073711yescharacteristic
231C2#2601503711yescharacteristic
232C2#2601503811yescharacteristic
subgroupG/H
center, FrattiniC22C23×D353
commutator, socle, cosocleC22×C353C24
FittingC4×D4×C353C2
hypercenterC4×D4D353
hypocenterC353#538
LCSG ▷ C22×C353 ▷ C353
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. #26014987C2semidirect
2. #26014986C2semidirect
3. #26014980C2semidirect
4. #26014978C2semidirect
5. #26014977C2semidirect
6. #26014975C2semidirect
7. #26014969C2semidirect
8. #26014966C2semidirect
9. #26014965C2semidirect
10. #26014964C2semidirect
11. #26014962C2semidirect
12. #26014960C2semidirect
13. C4×C22⋊D353C2semidirect
14. C2×#19150714C2semidirect
15. C4×D4×C353C2semidirect
17. C2×Dic706C22semidirect
23. C2×C22⋊D353C22semidirect
25. C2×C4×D353C22semidirect
26. C4⋊Dic353C22semidirect
27. #19150716C22semidirect
28. #19150716C22semidirect
29. #19150716C22semidirect
30. #19150716C22semidirect
32. C22⋊Dic353C22semidirect
41. #19150714C22semidirect
42. #19150714C22semidirect
43. C4×Dic353C22semidirect
46. C22×C4×C353C22semidirect
47. C4⋊C4×C353C22semidirect
49. C22⋊C4×C353C22semidirect
50. C42×C353C22semidirect
116. C353#538semidirect
131. C4×D4D353semidirect
145. C22×C4C2×D353semidirect
151. C4⋊C4C2×D353semidirect
165. C22⋊C4C2×D353semidirect
167. C42C2×D353semidirect

Maximal subgroups

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

subgroupindex
#260149872
#260149862
#260149802
#260149782
#260149772
#260149752
#260149692
#260149662
#260149652
#260149642
#260149622
#260149602
C4×C22⋊D3532
C2×#191507142
C4×D4×C3532
#538353