Finite Group Databasebeta

← #27182901

Subgroups — #27182901

Order 12928.

464 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 117 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#20265986C212yesmaximal, complemented
2#20265980C212yesmaximal, complemented
3#20265567C211yesmaximal, characteristic, complemented
4#20265446C212yesmaximal, complemented
5#20265442C212yesmaximal, complemented
6#20265349C211yesmaximal, characteristic, complemented
7#20265346C212yesmaximal, complemented
8#20265296C212yesmaximal, complemented
9#455×C101C211yesmaximal, characteristic, complemented
10#15692303C2211yescharacteristic, complemented
11#15692301C2212yescomplemented
12#15692287—22no—
13#15692283C2212yescomplemented
14#15692282—22no—
15#15692281—22no—
16#15692281C2212yescomplemented
17#15692280—22no—
18#15692280C2212yes—
19#15692278—22no—
20#15692275—22no—
21#15692275C2212yes—
22C2×#13108061C2212yescomplemented
23SD16×D101—22no—
24C2×#13108059—22no—
25C2×#13108059C2212yescomplemented
26#15692263C2212yescomplemented
27#15692262C2212yescomplemented
28#15692252C2211yescharacteristic, complemented
29#15692190C2212yescomplemented
30#15692183C2212yescomplemented
31#15692174C2211yescharacteristic, complemented
32#15692171C2212yescomplemented
33#15692166C2211yescharacteristic, complemented
34H(5,𝔽2)×C101C2211yescharacteristic
35#137×C101C2212yescomplemented
36#124×C101C2211yescharacteristic, complemented
37#104×C101C2212yescomplemented
38#100×C101C2211yescharacteristic, complemented
39#13108075—22no—
40#13108075—22no—
41#13108074—22no—
42#13108074—41no—
43#13108074C2312yes—
44Q8×D101—22no—
45C2×D404—22no—
46C2×D404—41no—
47C2×D404C2312yes—
48D4×D101—22no—
49D4×D101—42no—
50#13108064D412yescomplemented
51#13108063—22no—
52#13108062—22no—
53#13108062D412yescomplemented
54#13108061—22no—
55#13108061—22no—
56#13108061—22no—
57#13108061D412yescomplemented
58D808—22no—
59#13108059—22no—
60#13108059—22no—
61#13108059—22no—
62#13108059—22no—
63#13108059D412yescomplemented
64#13108058—22no—
65#13108058C2312yes—
66#13108057—22no—
67C2×#11717579—22no—
68C2×#11717579C2312yes—
69C8×D101—22no—
70#41×C101—22no—
71#41×C101C2312yes—
72C2×Q8×C101C2311yescharacteristic
73C2×D4×C101—22no—
74C2×D4×C101—41no—
75C2×D4×C101C2311yescharacteristic
76Q16×C101—22no—
77SD16×C101—22no—
78M16×C101C2312yes—
79C22⋊C4×C101—41no—
80C42×C101C2311yescharacteristic
81C23×C101—21no—
82C23×C101—41no—
83C22×D101—22no—
84C22×D101—42no—
85Q8×C101—22no—
86Q8×C101C2×D412yes—
87D4×C101—22no—
88D4×C101—22no—
89D4×C101—42no—
90D4×C101C2×D412yes—
91C2×C4×C101—22no—
92C2×C4×C101—21no—
93C2×C4×C101—21no—
94C2×C4×C101—41no—
95C2×C4×C101C2411yescharacteristic
96C22⋊D101—22no—
97C22⋊D101—42no—
98D404—22no—
99D404—22no—
100D404—42no—
101D404—41no—
102D404—82no—
103D404C2×D412yes—
104C4×D101—22no—
105C4×D101—42no—
106C4×D101—42no—
107Dic202—41no—
108Dic202C2×D412yes—
109C8×C101—22no—
110#11717579—22no—
111#11717579C2×D412yes—
112#11717579C2×D412yes—
113C22×C101—22no—
114C22×C101—41no—
115C22×C101—81no—
116C22×C101—161no—
117C22×C101H(5,𝔽2)11yescharacteristic
118C2×D101—22no—
119C2×D101—42no—
120C2×D101—82no—
121C2×D101—162no—
122C4×C101—22no—
123C4×C101—41no—
124C4×C101—41no—
125C4×C101C22×D412yes—
126Dic101—22no—
127C2×C101—21no—
128C2×C101—42no—
129C2×C101—81no—
130C2×C101D4211yescharacteristic
131D101—42no—
132D101—82no—
133#3288—1011nomaximal, Sylow
134C101#328811yescharacteristic, complemented, Sylow
135#577—1012no—
136#576—1012no—
137#496—1011no—
138#472—1012no—
139#471—1012no—
140#455D10111yescharacteristic, complemented
141#454—1011no—
142#453—1012no—
143#444—1012no—
1442−1+4—1011no—
145H(5,𝔽2)—1012no—
146H(5,𝔽2)C2×D10111yescharacteristic
147#137—2022no—
148#137C2×D10112yescomplemented
149AGL(1,ℤ/8)—1012no—
150AGL(1,ℤ/8)—1012no—
151AGL(1,ℤ/8)—1012no—
152AGL(1,ℤ/8)—2022no—
153AGL(1,ℤ/8)—2022no—
154AGL(1,ℤ/8)—2022no—
155#135—1012no—
156#135—2022no—
157#135—2022no—
158C2×SD16—1012no—
159C2×SD16—2022no—
160C2×D8—1012no—
161C2×D8—2022no—
162#131—1012no—
163#131—1012no—
164#127—1011no—
165#124C2×D10111yescharacteristic, complemented
166#108—1012no—
167#104—1012no—
168#104C2×D10112yescomplemented
169#101—1011no—
170#100—1012no—
171#100C2×D10111yescharacteristic, complemented
172#97—1011no—
173#41—22no—
174#41—1012no—
175#41—4041no—
176#41—2022no—
177#41—2022no—
178#41—2022no—
179#41C22×D10112yes—
180C2×Q8—2022no—
181C2×Q8C22×D10111yescharacteristic
182C2×D4—22no—
183C2×D4—41no—
184C2×D4—1012no—
185C2×D4—4041no—
186C2×D4—4042no—
187C2×D4—2022no—
188C2×D4—2022no—
189C2×D4C22×D10111yescharacteristic
190Q16—22no—
191Q16—1012no—
192SD16—22no—
193SD16—1012no—
194SD16—1012no—
195SD16—2022no—
196SD16—2022no—
197SD16—2022no—
198SD16—2022no—
199SD16—2022no—
200D8—1012no—
201D8—2022no—
202D8—2022no—
203D8—2022no—
204D8—2022no—
205D8—2022no—
206M16—1012no—
207M16—2022no—
208M16—2022no—
209M16C22×D10112yes—
210C2×C8—1012no—
211C2×C8—2022no—
212C2×C8—2022no—
213C22⋊C4—41no—
214C42C22×D10111yescharacteristic
215C23—21no—
216C23—41no—
217C23—4042no—
218C23—2022no—
219Q8—22no—
220Q8—1012no—
221Q8—4041no—
222Q8D4×D10112yes—
223D4—22no—
224D4—22no—
225D4—42no—
226D4—8082no—
227D4—1012no—
228D4—4042no—
229D4—4041no—
230D4—4042no—
231D4—2022no—
232D4—2022no—
233D4—2022no—
234D4D4×D10112yes—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—41no—
239C2×C4—4042no—
240C2×C4—4042no—
241C2×C4—2022no—
242C2×C4C23×D10111yescharacteristic
243C8—22no—
244C8—1012no—
245C8—1012no—
246C8—2022no—
247C22—22no—
248C22—41no—
249C22—81no—
250C22—161no—
251C22—8082no—
252C22—16162no—
253C22—4042no—
254C22—2022no—
255C22#1569230011yescharacteristic
256C4—22no—
257C4—41no—
258C4—41no—
259C4—2022no—
260C4C2×D4×D10112yes—
261C2—21no—
262C2—42no—
263C2—81no—
264C2—8082no—
265C2—4042no—
266C2#2026580911yescharacteristic
subgroupG/H
centerC2#20265809
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#455×C101C2
socleC2×C101D42
hypercenter#455D101
hypocenterC101#3288
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C2×C4 ◁ #455

Complements

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

NH ≅ G/Ntype
1. #20265986C2semidirect
2. #20265980C2semidirect
3. #20265567C2semidirect
4. #20265446C2semidirect
5. #20265442C2semidirect
6. #20265349C2semidirect
7. #20265346C2semidirect
8. #20265296C2semidirect
9. #455×C101C2semidirect
10. #15692303C22semidirect
11. #15692301C22semidirect
13. #15692283C22semidirect
16. #15692281C22semidirect
22. C2×#13108061C22semidirect
25. C2×#13108059C22semidirect
26. #15692263C22semidirect
27. #15692262C22semidirect
28. #15692252C22semidirect
29. #15692190C22semidirect
30. #15692183C22semidirect
31. #15692174C22semidirect
32. #15692171C22semidirect
33. #15692166C22semidirect
35. #137×C101C22semidirect
36. #124×C101C22semidirect
37. #104×C101C22semidirect
38. #100×C101C22semidirect
50. #13108064D4semidirect
53. #13108062D4semidirect
57. #13108061D4semidirect
63. #13108059D4semidirect
134. C101#3288semidirect
140. #455D101semidirect
148. #137C2×D101semidirect
165. #124C2×D101semidirect
168. #104C2×D101semidirect
171. #100C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202659862
#202659802
#202655672
#202654462
#202654422
#202653492
#202653462
#202652962
#455×C1012
#3288101