Finite Group Databasebeta

← #29505449

Subgroups — #29505449

Order 15296.

248 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 97 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#21439955C211yesmaximal, characteristic, complemented
2#21439955C211yesmaximal, characteristic, complemented
3#21439949C211yesmaximal, characteristic, complemented
4#21439945C211yesmaximal, characteristic, complemented
5#21439945C211yesmaximal, characteristic, complemented
6#21439944C211yesmaximal, characteristic, complemented
7#21439936C211yesmaximal, characteristic, complemented
8#21439930C211yesmaximal, characteristic
9#21439922C211yesmaximal, characteristic, complemented
10C4×C22⋊D239C211yesmaximal, characteristic, complemented
11#21439918C211yesmaximal, characteristic, complemented
12#21439916C211yesmaximal, characteristic, complemented
13C2×(C2×D239)⋊C4C211yesmaximal, characteristic, complemented
14C22⋊C4×D239C211yesmaximal, characteristic, complemented
15C22⋊Q8×C239C211yesmaximal, characteristic, complemented
16C23×D239C2211yescharacteristic
17C2×D956C2211yescharacteristic
18C2×D956C2211yescharacteristic
19C2×C22⋊D239C2211yescharacteristic
20C22×Dic239C2211yescharacteristic
21C2×C4×D239C2211yescharacteristic
22C2×C4×D239C2211yescharacteristic
23C2×C4×D239C2211yescharacteristic, complemented
24C4⋊Dic239C2211yescharacteristic, complemented
25C4⋊Dic239C2211yescharacteristic, complemented
26Dic239⋊C4C2211yescharacteristic, complemented
27Dic239⋊C4C2211yescharacteristic, complemented
28Dic239⋊C4C2211yescharacteristic, complemented
29C22⋊Dic239C2211yescharacteristic
30(C2×D239)⋊C4C2211yescharacteristic
31(C2×D239)⋊C4C2211yescharacteristic
32(C2×D239)⋊C4C2211yescharacteristic
33(C2×D239)⋊C4C2211yescharacteristic
34(C2×D239)⋊C4C2211yescharacteristic
35(C2×D239)⋊C4C2211yescharacteristic
36(C2×D239)⋊C4C2211yescharacteristic
37(C2×D239)⋊C4C2211yescharacteristic
38(C2×D239)⋊C4C2211yescharacteristic, complemented
39(C2×D239)⋊C4C2211yescharacteristic, complemented
40(C2×D239)⋊C4C2211yescharacteristic, complemented
41C4×Dic239C2211yescharacteristic, complemented
42C4×Dic239C2211yescharacteristic, complemented
43C4×Dic239C2211yescharacteristic, complemented
44C2×Q8×C239C2211yescharacteristic, complemented
45C22×C4×C239C2211yescharacteristic
46C4⋊C4×C239C2211yescharacteristic, complemented
47C4⋊C4×C239C2211yescharacteristic, complemented
48C4⋊C4×C239C2211yescharacteristic, complemented
49C22⋊C4×C239C2211yescharacteristic
50C22⋊C4×C239C2211yescharacteristic
51C22×D239—22no—
52C22×D239—22no—
53C22×D239—41no—
54C22×D239C2311yescharacteristic
55C22×D239C2311yescharacteristic
56C22×D239C2311yescharacteristic
57D956—22no—
58D956—41no—
59C22⋊D239—22no—
60C2×Dic239—22no—
61C2×Dic239C2311yescharacteristic
62C2×Dic239C2311yescharacteristic
63C2×Dic239C2311yescharacteristic
64C2×Dic239C2311yescharacteristic
65C2×Dic239C2311yescharacteristic
66C4×D239—22no—
67C4×D239—41no—
68C4×D239—41no—
69C23×C239C2311yescharacteristic
70Q8×C239—41no—
71C2×C4×C239—41no—
72C2×C4×C239C2311yescharacteristic
73C2×C4×C239C2311yescharacteristic
74C2×C4×C239C2311yescharacteristic
75C2×C4×C239C2311yescharacteristic
76C2×C4×C239C2311yescharacteristic
77C2×C4×C239C2311yescharacteristic
78C22×C239—21no—
79C22×C239—21no—
80C22×C239#4112yes—
81C22×C239C2411yescharacteristic
82C2×D239—21no—
83C2×D239—21no—
84C2×D239—21no—
85C2×D239—21no—
86C2×D239—21no—
87C2×D239—21no—
88C2×D239—21no—
89C2×D239—21no—
90C2×D239—44no—
91C2×D239#4112yes—
92C4×C239—21no—
93C4×C239—21no—
94C4×C239—21no—
95C4×C239—21no—
96C4×C239—21no—
97C4×C239—21no—
98Dic239—21no—
99Dic239—21no—
100Dic239—21no—
101Dic239—21no—
102Dic239—21no—
103C2×C239—22no—
104C2×C239C2×#4111yescharacteristic
105C2×C239C2×#4111yescharacteristic
106C2×C239H(5,𝔽2)11yescharacteristic
107D239—22no—
108D239—41no—
109D239—41no—
110C239#55111yescharacteristic, complemented, Sylow
111#551—2391nomaximal, Sylow
112#126—2391no—
113#126—2391no—
114#124—2391no—
115#123—2391no—
116#123—2391no—
117#123—2391no—
118C22⋊Q8—2391no—
119C22⋊Q8D23911yescharacteristic, complemented
120#120—2391no—
121C4×D4—2391no—
122C4×D4—2391no—
123#117—2391no—
124#117—2391no—
125C2×C22⋊C4—2391no—
126C2×C22⋊C4—2391no—
127C24—2391no—
128C2×Q8C2×D23911yescharacteristic, complemented
129C2×D4—2391no—
130C2×D4—2391no—
131C2×D4—2391no—
132C22×C4—2391no—
133C22×C4—2391no—
134C22×C4—2391no—
135C22×C4—2391no—
136C22×C4C2×D23911yescharacteristic
137C4⋊C4—2391no—
138C4⋊C4—2391no—
139C4⋊C4—2391no—
140C4⋊C4—2391no—
141C4⋊C4—2391no—
142C4⋊C4C2×D23911yescharacteristic, complemented
143C4⋊C4C2×D23911yescharacteristic, complemented
144C4⋊C4C2×D23911yescharacteristic, complemented
145C22⋊C4—2391no—
146C22⋊C4—2391no—
147C22⋊C4—2391no—
148C22⋊C4—2391no—
149C22⋊C4—2391no—
150C22⋊C4—2391no—
151C22⋊C4—2391no—
152C22⋊C4—2391no—
153C22⋊C4—2391no—
154C22⋊C4—2391no—
155C22⋊C4—2391no—
156C22⋊C4—2391no—
157C22⋊C4C2×D23911yescharacteristic
158C22⋊C4C2×D23911yescharacteristic
159C42—2391no—
160C42—2391no—
161C42—2391no—
162C23—9561no—
163C23—4782no—
164C23—4782no—
165C23—2391no—
166C23—2391no—
167C23—2391no—
168C23C22×D23911yescharacteristic
169Q8—41no—
170D4—9561no—
171D4—4782no—
172D4—4782no—
173C2×C4—41no—
174C2×C4—9561no—
175C2×C4—9561no—
176C2×C4—4782no—
177C2×C4—4782no—
178C2×C4—2391no—
179C2×C4—2391no—
180C2×C4—2391no—
181C2×C4—2391no—
182C2×C4—2391no—
183C2×C4C22×D23911yescharacteristic
184C2×C4C22×D23911yescharacteristic
185C2×C4C22×D23911yescharacteristic
186C2×C4C22×D23911yescharacteristic
187C2×C4C22×D23911yescharacteristic
188C2×C4C22×D23911yescharacteristic
189C22—21no—
190C22—21no—
191C22—9564no—
192C22—4781no—
193C22—4781no—
194C22—4781no—
195C22—4781no—
196C22—4781no—
197C22—4781no—
198C22—4781no—
199C22—4781no—
200C22—2392no—
201C22#1683854912yes—
202C22C23×D23911yescharacteristic
203C4—21no—
204C4—21no—
205C4—21no—
206C4—21no—
207C4—21no—
208C4—21no—
209C4—4781no—
210C4—4781no—
211C4—4781no—
212C4—4781no—
213C4—4781no—
214C2—22no—
215C2—9561no—
216C2—9561no—
217C2—4782no—
218C2#41×D23911yescharacteristic
219C2C2×#1683854911yescharacteristic
220C2#2144000511yescharacteristic
subgroupG/H
center, FrattiniC22C23×D239
commutator, socle, cosocleC22×C239C24
FittingC22⋊Q8×C239C2
prime radical#21439930C2
hypercenterC22⋊Q8D239
hypocenterC239#551
LCSG ▷ C22×C239 ▷ C239
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #21439955C2semidirect
2. #21439955C2semidirect
3. #21439949C2semidirect
4. #21439945C2semidirect
5. #21439945C2semidirect
6. #21439944C2semidirect
7. #21439936C2semidirect
9. #21439922C2semidirect
10. C4×C22⋊D239C2semidirect
11. #21439918C2semidirect
12. #21439916C2semidirect
13. C2×(C2×D239)⋊C4C2semidirect
14. C22⋊C4×D239C2semidirect
15. C22⋊Q8×C239C2semidirect
23. C2×C4×D239C22semidirect
24. C4⋊Dic239C22semidirect
25. C4⋊Dic239C22semidirect
26. Dic239⋊C4C22semidirect
27. Dic239⋊C4C22semidirect
28. Dic239⋊C4C22semidirect
38. (C2×D239)⋊C4C22semidirect
39. (C2×D239)⋊C4C22semidirect
40. (C2×D239)⋊C4C22semidirect
41. C4×Dic239C22semidirect
42. C4×Dic239C22semidirect
43. C4×Dic239C22semidirect
44. C2×Q8×C239C22semidirect
46. C4⋊C4×C239C22semidirect
47. C4⋊C4×C239C22semidirect
48. C4⋊C4×C239C22semidirect
110. C239#551semidirect
119. C22⋊Q8D239semidirect
128. C2×Q8C2×D239semidirect
142. C4⋊C4C2×D239semidirect
143. C4⋊C4C2×D239semidirect
144. C4⋊C4C2×D239semidirect

Maximal subgroups

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

subgroupindex
#214399552
#214399492
#214399452
#214399442
#214399362
#214399302
#214399222
C4×C22⋊D2392
#214399182
#214399162
C2×(C2×D239)⋊C42
C22⋊C4×D2392
C22⋊Q8×C2392
#551239