Finite Group Databasebeta

← #29505357

Subgroups — #29505357

Order 15296.

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#21439953C211yesmaximal, characteristic, complemented
2#21439953C211yesmaximal, characteristic, complemented
3#21439947C211yesmaximal, characteristic, complemented
4#21439946C211yesmaximal, characteristic, complemented
5#21439942C211yesmaximal, characteristic, complemented
6#21439941C211yesmaximal, characteristic, complemented
7#21439937C211yesmaximal, characteristic, complemented
8#21439932C211yesmaximal, characteristic, complemented
9#21439932C211yesmaximal, characteristic, complemented
10#21439931C211yesmaximal, characteristic, complemented
11#21439929C211yesmaximal, characteristic, complemented
12D4×Dic239C211yesmaximal, characteristic, complemented
13(C22⋊D239)⋊C4C211yesmaximal, characteristic, complemented
14C2×C22⋊Dic239C211yesmaximal, characteristic, complemented
15#120×C239C211yesmaximal, characteristic, complemented
16C2×Dic478C2211yescharacteristic, complemented
17C2×C22⋊D239C2211yescharacteristic
18C2×C22⋊D239C2211yescharacteristic
19C2×C22⋊D239C2211yescharacteristic
20C2×C22⋊D239C2211yescharacteristic, complemented
21C22×Dic239C2211yescharacteristic
22C22×Dic239C2211yescharacteristic
23C22×Dic239C2211yescharacteristic, complemented
24C2×C4×D239C2211yescharacteristic, complemented
25C4⋊Dic239C2211yescharacteristic, complemented
26C4⋊Dic239C2211yescharacteristic, complemented
27Dic239⋊C4C2211yescharacteristic, complemented
28Dic239⋊C4C2211yescharacteristic, complemented
29Dic239⋊C4C2211yescharacteristic, complemented
30Dic239⋊C4C2211yescharacteristic, complemented
31C22⋊Dic239C2211yescharacteristic
32C22⋊Dic239C2211yescharacteristic
33C22⋊Dic239C2211yescharacteristic
34C22⋊Dic239C2211yescharacteristic
35C22⋊Dic239C2211yescharacteristic
36C22⋊Dic239C2211yescharacteristic
37C22⋊Dic239C2211yescharacteristic
38C22⋊Dic239C2211yescharacteristic, complemented
39C22⋊Dic239C2211yescharacteristic, complemented
40(C2×D239)⋊C4C2211yescharacteristic, complemented
41(C2×D239)⋊C4C2211yescharacteristic, complemented
42C4×Dic239C2211yescharacteristic, complemented
43C4×Dic239C2211yescharacteristic, complemented
44C24×C239C2211yescharacteristic
45C2×D4×C239C2211yescharacteristic
46C2×D4×C239C2211yescharacteristic
47C2×D4×C239C2211yescharacteristic
48C22⋊C4×C239C2211yescharacteristic
49C22⋊C4×C239C2211yescharacteristic
50C22⋊C4×C239C2211yescharacteristic
51C22×D239C2311yescharacteristic
52Dic478—41no—
53C22⋊D239—22no—
54C22⋊D239—41no—
55C22⋊D239—41no—
56C22⋊D239—41no—
57C2×Dic239—41no—
58C2×Dic239—41no—
59C2×Dic239—41no—
60C2×Dic239C2311yescharacteristic
61C2×Dic239C2311yescharacteristic
62C2×Dic239C2311yescharacteristic
63C2×Dic239C2311yescharacteristic
64C2×Dic239C2311yescharacteristic
65C2×Dic239C2311yescharacteristic
66C2×Dic239C2311yescharacteristic
67C4×D239—41no—
68C23×C239—22no—
69C23×C239—41no—
70C23×C239—41no—
71C23×C239C2311yescharacteristic
72C23×C239C2311yescharacteristic
73C23×C239C2311yescharacteristic
74C23×C239C2311yescharacteristic
75D4×C239—22no—
76D4×C239—41no—
77D4×C239—41no—
78C2×C4×C239C2311yescharacteristic
79C2×C4×C239C2311yescharacteristic
80C2×C4×C239C2311yescharacteristic
81C22×C239—21no—
82C22×C239—21no—
83C22×C239—21no—
84C22×C239—21no—
85C22×C239—21no—
86C22×C239—21no—
87C22×C239—21no—
88C22×C239—21no—
89C22×C239—21no—
90C22×C239—21no—
91C22×C239—21no—
92C22×C239—44no—
93C22×C239#4112yes—
94C22×C239C2411yescharacteristic
95C2×D239—21no—
96C2×D239—21no—
97C2×D239—21no—
98C4×C239—21no—
99C4×C239—21no—
100C4×C239—21no—
101Dic239—21no—
102Dic239—21no—
103Dic239—21no—
104Dic239—21no—
105Dic239—21no—
106Dic239—21no—
107Dic239—21no—
108C2×C239—22no—
109C2×C239—41no—
110C2×C239—41no—
111C2×C239—41no—
112C2×C239C2×#4111yescharacteristic
113C2×C239H(5,𝔽2)11yescharacteristic
114C2×C239H(5,𝔽2)11yescharacteristic
115D239—41no—
116C239#53811yescharacteristic, complemented, Sylow
117#538—2391nomaximal, Sylow
118#126—2391no—
119#126—2391no—
120#124—2391no—
121#124—2391no—
122#123—2391no—
123#123—2391no—
124C22⋊Q8—2391no—
125#121—2391no—
126#121—2391no—
127#121—2391no—
128#120—2391no—
129#120D23911yescharacteristic, complemented
130C4×D4—2391no—
131C4×D4—2391no—
132C2×C22⋊C4—2391no—
133C24C2×D23911yescharacteristic
134C2×Q8—2391no—
135C2×D4—2391no—
136C2×D4—2391no—
137C2×D4—2391no—
138C2×D4—2391no—
139C2×D4C2×D23911yescharacteristic
140C2×D4C2×D23911yescharacteristic
141C2×D4C2×D23911yescharacteristic
142C22×C4—2391no—
143C22×C4—2391no—
144C22×C4—2391no—
145C22×C4—2391no—
146C4⋊C4—2391no—
147C4⋊C4—2391no—
148C4⋊C4—2391no—
149C4⋊C4—2391no—
150C4⋊C4—2391no—
151C4⋊C4—2391no—
152C22⋊C4—2391no—
153C22⋊C4—2391no—
154C22⋊C4—2391no—
155C22⋊C4—2391no—
156C22⋊C4—2391no—
157C22⋊C4—2391no—
158C22⋊C4—2391no—
159C22⋊C4—2391no—
160C22⋊C4—2391no—
161C22⋊C4—2391no—
162C22⋊C4—2391no—
163C22⋊C4C2×D23911yescharacteristic
164C22⋊C4C2×D23911yescharacteristic
165C22⋊C4C2×D23911yescharacteristic
166C42—2391no—
167C42—2391no—
168C23—22no—
169C23—41no—
170C23—41no—
171C23—2391no—
172C23C22×D23911yescharacteristic
173C23C22×D23911yescharacteristic
174C23C22×D23911yescharacteristic
175C23C22×D23911yescharacteristic
176Q8—9561no—
177D4—22no—
178D4—41no—
179D4—41no—
180D4—9561no—
181D4—9561no—
182D4—9561no—
183D4—4782no—
184C2×C4—9561no—
185C2×C4—9561no—
186C2×C4—9561no—
187C2×C4—9561no—
188C2×C4—2391no—
189C2×C4—2391no—
190C2×C4—2391no—
191C2×C4—2391no—
192C2×C4—2391no—
193C2×C4—2391no—
194C2×C4—2391no—
195C2×C4C22×D23911yescharacteristic
196C2×C4C22×D23911yescharacteristic
197C2×C4C22×D23911yescharacteristic
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—4781no—
211C22—4781no—
212C22—4781no—
213C22#1683854712yes—
214C22C23×D23911yescharacteristic
215C4—21no—
216C4—21no—
217C4—21no—
218C4—4781no—
219C4—4781no—
220C4—4781no—
221C4—4781no—
222C4—4781no—
223C4—4781no—
224C4—4781no—
225C2—22no—
226C2—41no—
227C2—41no—
228C2—41no—
229C2—9561no—
230C2C2×#1683854711yescharacteristic
231C2#2144000511yescharacteristic
232C2#2144000511yescharacteristic
subgroupG/H
center, FrattiniC22C23×D239
commutator, socle, cosocleC22×C239C24
Fitting#120×C239C2
hypercenter#120D239
hypocenterC239#538
LCSG ▷ C22×C239 ▷ C239
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. #21439953C2semidirect
2. #21439953C2semidirect
3. #21439947C2semidirect
4. #21439946C2semidirect
5. #21439942C2semidirect
6. #21439941C2semidirect
7. #21439937C2semidirect
8. #21439932C2semidirect
9. #21439932C2semidirect
10. #21439931C2semidirect
11. #21439929C2semidirect
12. D4×Dic239C2semidirect
13. (C22⋊D239)⋊C4C2semidirect
14. C2×C22⋊Dic239C2semidirect
15. #120×C239C2semidirect
16. C2×Dic478C22semidirect
20. C2×C22⋊D239C22semidirect
23. C22×Dic239C22semidirect
24. C2×C4×D239C22semidirect
25. C4⋊Dic239C22semidirect
26. C4⋊Dic239C22semidirect
27. Dic239⋊C4C22semidirect
28. Dic239⋊C4C22semidirect
29. Dic239⋊C4C22semidirect
30. Dic239⋊C4C22semidirect
38. C22⋊Dic239C22semidirect
39. C22⋊Dic239C22semidirect
40. (C2×D239)⋊C4C22semidirect
41. (C2×D239)⋊C4C22semidirect
42. C4×Dic239C22semidirect
43. C4×Dic239C22semidirect
116. C239#538semidirect
129. #120D239semidirect

Maximal subgroups

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

subgroupindex
#214399532
#214399472
#214399462
#214399422
#214399412
#214399372
#214399322
#214399312
#214399292
D4×Dic2392
(C22⋊D239)⋊C42
C2×C22⋊Dic2392
#120×C2392
#538239