Finite Group Databasebeta

← #36378372

Subgroups — #36378372

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#26014985C211yesmaximal, characteristic, complemented
2#26014985C211yesmaximal, characteristic, complemented
3#26014979C211yesmaximal, characteristic, complemented
4#26014978C211yesmaximal, characteristic, complemented
5#26014974C211yesmaximal, characteristic, complemented
6#26014973C211yesmaximal, characteristic, complemented
7#26014969C211yesmaximal, characteristic, complemented
8#26014964C211yesmaximal, characteristic, complemented
9#26014964C211yesmaximal, characteristic, complemented
10#26014963C211yesmaximal, characteristic, complemented
11#26014961C211yesmaximal, characteristic, complemented
12D4×Dic353C211yesmaximal, characteristic, complemented
13#26014952C211yesmaximal, characteristic, complemented
14C2×C22⋊Dic353C211yesmaximal, characteristic, complemented
15#120×C353C211yesmaximal, characteristic, complemented
16C2×Dic706C2211yescharacteristic, complemented
17C2×C22⋊D353C2211yescharacteristic
18C2×C22⋊D353C2211yescharacteristic
19C2×C22⋊D353C2211yescharacteristic
20C2×C22⋊D353C2211yescharacteristic, complemented
21C22×Dic353C2211yescharacteristic
22C22×Dic353C2211yescharacteristic
23C22×Dic353C2211yescharacteristic, complemented
24C2×C4×D353C2211yescharacteristic, complemented
25C4⋊Dic353C2211yescharacteristic, complemented
26C4⋊Dic353C2211yescharacteristic, complemented
27#19150716C2211yescharacteristic, complemented
28#19150716C2211yescharacteristic, complemented
29#19150716C2211yescharacteristic, complemented
30#19150716C2211yescharacteristic, complemented
31C22⋊Dic353C2211yescharacteristic
32C22⋊Dic353C2211yescharacteristic
33C22⋊Dic353C2211yescharacteristic
34C22⋊Dic353C2211yescharacteristic
35C22⋊Dic353C2211yescharacteristic
36C22⋊Dic353C2211yescharacteristic
37C22⋊Dic353C2211yescharacteristic
38C22⋊Dic353C2211yescharacteristic, complemented
39C22⋊Dic353C2211yescharacteristic, complemented
40#19150714C2211yescharacteristic, complemented
41#19150714C2211yescharacteristic, complemented
42C4×Dic353C2211yescharacteristic, complemented
43C4×Dic353C2211yescharacteristic, complemented
44C24×C353C2211yescharacteristic
45C2×D4×C353C2211yescharacteristic
46C2×D4×C353C2211yescharacteristic
47C2×D4×C353C2211yescharacteristic
48C22⋊C4×C353C2211yescharacteristic
49C22⋊C4×C353C2211yescharacteristic
50C22⋊C4×C353C2211yescharacteristic
51C22×D353C2311yescharacteristic
52Dic706—41no—
53C22⋊D353—22no—
54C22⋊D353—41no—
55C22⋊D353—41no—
56C22⋊D353—41no—
57C2×Dic353—41no—
58C2×Dic353—41no—
59C2×Dic353—41no—
60C2×Dic353C2311yescharacteristic
61C2×Dic353C2311yescharacteristic
62C2×Dic353C2311yescharacteristic
63C2×Dic353C2311yescharacteristic
64C2×Dic353C2311yescharacteristic
65C2×Dic353C2311yescharacteristic
66C2×Dic353C2311yescharacteristic
67C4×D353—41no—
68C23×C353—22no—
69C23×C353—41no—
70C23×C353—41no—
71C23×C353C2311yescharacteristic
72C23×C353C2311yescharacteristic
73C23×C353C2311yescharacteristic
74C23×C353C2311yescharacteristic
75D4×C353—22no—
76D4×C353—41no—
77D4×C353—41no—
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—21no—
87C22×C353—21no—
88C22×C353—21no—
89C22×C353—21no—
90C22×C353—21no—
91C22×C353—21no—
92C22×C353—44no—
93C22×C353#4112yes—
94C22×C353C2411yescharacteristic
95C2×D353—21no—
96C2×D353—21no—
97C2×D353—21no—
98C4×C353—21no—
99C4×C353—21no—
100C4×C353—21no—
101Dic353—21no—
102Dic353—21no—
103Dic353—21no—
104Dic353—21no—
105Dic353—21no—
106Dic353—21no—
107Dic353—21no—
108C2×C353—22no—
109C2×C353—41no—
110C2×C353—41no—
111C2×C353—41no—
112C2×C353C2×#4111yescharacteristic
113C2×C353H(5,𝔽2)11yescharacteristic
114C2×C353H(5,𝔽2)11yescharacteristic
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#120D35311yescharacteristic, complemented
130C4×D4—3531no—
131C4×D4—3531no—
132C2×C22⋊C4—3531no—
133C24C2×D35311yescharacteristic
134C2×Q8—3531no—
135C2×D4—3531no—
136C2×D4—3531no—
137C2×D4—3531no—
138C2×D4—3531no—
139C2×D4C2×D35311yescharacteristic
140C2×D4C2×D35311yescharacteristic
141C2×D4C2×D35311yescharacteristic
142C22×C4—3531no—
143C22×C4—3531no—
144C22×C4—3531no—
145C22×C4—3531no—
146C4⋊C4—3531no—
147C4⋊C4—3531no—
148C4⋊C4—3531no—
149C4⋊C4—3531no—
150C4⋊C4—3531no—
151C4⋊C4—3531no—
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⋊C4C2×D35311yescharacteristic
164C22⋊C4C2×D35311yescharacteristic
165C22⋊C4C2×D35311yescharacteristic
166C42—3531no—
167C42—3531no—
168C23—22no—
169C23—41no—
170C23—41no—
171C23—3531no—
172C23C22×D35311yescharacteristic
173C23C22×D35311yescharacteristic
174C23C22×D35311yescharacteristic
175C23C22×D35311yescharacteristic
176Q8—14121no—
177D4—22no—
178D4—41no—
179D4—41no—
180D4—14121no—
181D4—14121no—
182D4—14121no—
183D4—7062no—
184C2×C4—3531no—
185C2×C4—3531no—
186C2×C4—3531no—
187C2×C4—3531no—
188C2×C4—3531no—
189C2×C4—3531no—
190C2×C4—3531no—
191C2×C4—14121no—
192C2×C4—14121no—
193C2×C4—14121no—
194C2×C4—14121no—
195C2×C4C22×D35311yescharacteristic
196C2×C4C22×D35311yescharacteristic
197C2×C4C22×D35311yescharacteristic
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—21no—
208C22—21no—
209C22—44no—
210C22—7061no—
211C22—7061no—
212C22—7061no—
213C22#1915073612yes—
214C22C23×D35311yescharacteristic
215C4—21no—
216C4—21no—
217C4—21no—
218C4—7061no—
219C4—7061no—
220C4—7061no—
221C4—7061no—
222C4—7061no—
223C4—7061no—
224C4—7061no—
225C2—22no—
226C2—41no—
227C2—41no—
228C2—41no—
229C2—14121no—
230C2C2×#1915073611yescharacteristic
231C2#2601503711yescharacteristic
232C2#2601503711yescharacteristic
subgroupG/H
center, FrattiniC22C23×D353
commutator, socle, cosocleC22×C353C24
Fitting#120×C353C2
hypercenter#120D353
hypocenterC353#538
LCSG ▷ C22×C353 ▷ C353
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. #26014985C2semidirect
2. #26014985C2semidirect
3. #26014979C2semidirect
4. #26014978C2semidirect
5. #26014974C2semidirect
6. #26014973C2semidirect
7. #26014969C2semidirect
8. #26014964C2semidirect
9. #26014964C2semidirect
10. #26014963C2semidirect
11. #26014961C2semidirect
12. D4×Dic353C2semidirect
13. #26014952C2semidirect
14. C2×C22⋊Dic353C2semidirect
15. #120×C353C2semidirect
16. C2×Dic706C22semidirect
20. C2×C22⋊D353C22semidirect
23. C22×Dic353C22semidirect
24. C2×C4×D353C22semidirect
25. C4⋊Dic353C22semidirect
26. C4⋊Dic353C22semidirect
27. #19150716C22semidirect
28. #19150716C22semidirect
29. #19150716C22semidirect
30. #19150716C22semidirect
38. C22⋊Dic353C22semidirect
39. C22⋊Dic353C22semidirect
40. #19150714C22semidirect
41. #19150714C22semidirect
42. C4×Dic353C22semidirect
43. C4×Dic353C22semidirect
116. C353#538semidirect
129. #120D353semidirect

Maximal subgroups

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

subgroupindex
#260149852
#260149792
#260149782
#260149742
#260149732
#260149692
#260149642
#260149632
#260149612
D4×Dic3532
#260149522
C2×C22⋊Dic3532
#120×C3532
#538353