Finite Group Databasebeta

← #28374798

Subgroups — #28374798

Order 13952.

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#20295832C212yesmaximal, complemented
2#20295827C212yesmaximal, complemented
3#20295293C212yesmaximal, complemented
4#20295286C212yesmaximal, complemented
5#20295203C211yesmaximal, characteristic, complemented
6#20295198C211yesmaximal, characteristic, complemented
7#20295193C212yesmaximal, complemented
8#20295149C212yesmaximal, complemented
9#496×C109C211yesmaximal, characteristic, complemented
10#16812747C2211yescharacteristic, complemented
11#16812744C2211yescharacteristic
12#16812744C2212yescomplemented
13#16812730C2212yescomplemented
14#16812728—22no—
15#16812726—22no—
16#16812725C2212yes—
17#16812725C2212yescomplemented
18#16812725C2212yescomplemented
19#16812723—22no—
20#16812723—22no—
21#16812721C2212yes—
22#16812720—22no—
23#16812718—22no—
24SD16×D109—22no—
25D8×D109—22no—
26#16812705C2212yescomplemented
27#16812705C2212yescomplemented
28#16812635C2212yescomplemented
29#16812627C2212yescomplemented
30#16812627C2212yescomplemented
31#16812618C2211yescharacteristic, complemented
32#16812616C2212yescomplemented
33#16812616C2211yescharacteristic, complemented
34C2×SD16×C109C2212yescomplemented
35C2×D8×C109C2212yescomplemented
36#127×C109C2211yescharacteristic, complemented
37#124×C109C2211yescharacteristic, complemented
38#97×C109C2211yescharacteristic, complemented
39#13182030—22no—
40#13182029—41no—
41#13182029C2312yes—
42#13182029C2312yes—
43#13182028—22no—
44#13182028—22no—
45#13182028—22no—
46Q8×D109—22no—
47C2×C22⋊D109—41no—
48C2×C22⋊D109—42no—
49D4×D109—22no—
50D4×D109—22no—
51D4×D109—22no—
52Dic436D412yescomplemented
53#13182019—22no—
54#13182018D412yescomplemented
55#13182018D412yescomplemented
56#13182017—22no—
57#13182017—22no—
58#13182017—22no—
59#13182016—22no—
60D872D412yescomplemented
61#13182014—22no—
62#13182014—22no—
63#13182014—22no—
64#13182013C2312yes—
65#13182013C2312yes—
66#13182012—22no—
67#13182012—22no—
68C8×D109—22no—
69C8×D109—22no—
70C2×Q8×C109C2311yescharacteristic
71C2×D4×C109—41no—
72C2×D4×C109C2311yescharacteristic
73C2×D4×C109C2312yes—
74SD16×C109—22no—
75SD16×C109—22no—
76D8×C109—22no—
77D8×C109—22no—
78C2×C8×C109C2312yes—
79C22⋊C4×C109—41no—
80C42×C109C2311yescharacteristic
81C23×C109—21no—
82C23×C109—22no—
83C22×D109—42no—
84C22×D109—41no—
85Q8×C109—22no—
86D4×C109—22no—
87D4×C109—22no—
88D4×C109—22no—
89D4×C109—82no—
90C2×C4×C109—21no—
91C2×C4×C109—21no—
92C2×C4×C109C2411yescharacteristic
93C22⋊D109—22no—
94C22⋊D109—22no—
95C22⋊D109—42no—
96C22⋊D109—42no—
97C22⋊D109—42no—
98C2×Dic109—42no—
99C2×Dic109—41no—
100D436—41no—
101D436C2×D412yes—
102D436C2×D412yes—
103C4×D109—22no—
104C4×D109—22no—
105C4×D109—42no—
106Dic218—41no—
107Dic218C2×D412yes—
108Dic218C2×D412yes—
109C8×C109C2×D412yes—
110C8×C109C2×D412yes—
111#11724784—22no—
112#11724784—22no—
113C22×C109—41no—
114C22×C109—42no—
115C22×C109—82no—
116C22×C109—81no—
117C22×C109H(5,𝔽2)11yescharacteristic
118C2×D109—22no—
119C2×D109—22no—
120C2×D109—162no—
121C2×D109—161no—
122C4×C109—41no—
123C4×C109—41no—
124C4×C109C22×D412yes—
125Dic109—22no—
126Dic109—22no—
127C2×C109—21no—
128C2×C109—82no—
129C2×C109—81no—
130C2×C109D4211yescharacteristic
131D109—42no—
132D109—42no—
133#3288—1091nomaximal, Sylow
134C109#328811yescharacteristic, complemented, Sylow
135#577—1092no—
136#576—1092no—
137#496D10911yescharacteristic, complemented
138#472—1092no—
139#471—1092no—
140#455—1091no—
141#454—1091no—
142#453—1092no—
143#444—1092no—
1442−1+4—1091no—
145H(5,𝔽2)—1091no—
146H(5,𝔽2)—1092no—
147#137—1092no—
148#137—2182no—
149AGL(1,ℤ/8)—1092no—
150AGL(1,ℤ/8)—1092no—
151AGL(1,ℤ/8)—1092no—
152AGL(1,ℤ/8)—2182no—
153AGL(1,ℤ/8)—2182no—
154AGL(1,ℤ/8)—2182no—
155#135—1092no—
156#135—2182no—
157#135—2182no—
158C2×SD16—2182no—
159C2×SD16C2×D10912yescomplemented
160C2×D8—2182no—
161C2×D8C2×D10912yescomplemented
162#131—1092no—
163#131—1092no—
164#127C2×D10911yescharacteristic, complemented
165#124C2×D10911yescharacteristic, complemented
166#108—1092no—
167#104—1092no—
168#104—1092no—
169#101—1091no—
170#100—1092no—
171#100—1091no—
172#97C2×D10911yescharacteristic, complemented
173#41—1092no—
174#41—1092no—
175#41—4361no—
176#41—2182no—
177#41—2182no—
178#41—2182no—
179#41—2182no—
180C2×Q8—2182no—
181C2×Q8C22×D10911yescharacteristic
182C2×D4—41no—
183C2×D4—4361no—
184C2×D4—4362no—
185C2×D4—2182no—
186C2×D4—2182no—
187C2×D4—2182no—
188C2×D4C22×D10911yescharacteristic
189C2×D4C22×D10912yes—
190Q16—1092no—
191Q16—2182no—
192SD16—22no—
193SD16—22no—
194SD16—1092no—
195SD16—1092no—
196SD16—2182no—
197SD16—2182no—
198SD16—2182no—
199SD16—2182no—
200D8—22no—
201D8—22no—
202D8—1092no—
203D8—2182no—
204D8—2182no—
205D8—2182no—
206M16—1092no—
207M16—1092no—
208M16—2182no—
209M16—2182no—
210C2×C8—2182no—
211C2×C8—2182no—
212C2×C8C22×D10912yes—
213C22⋊C4—41no—
214C42C22×D10911yescharacteristic
215C23—21no—
216C23—22no—
217C23—4362no—
218C23—4361no—
219Q8—22no—
220Q8—1092no—
221Q8—1092no—
222Q8—4361no—
223D4—22no—
224D4—22no—
225D4—22no—
226D4—82no—
227D4—1092no—
228D4—1092no—
229D4—4361no—
230D4—4362no—
231D4—4362no—
232D4—4362no—
233D4—2182no—
234D4—2182no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—4362no—
238C2×C4—4361no—
239C2×C4—4362no—
240C2×C4—2182no—
241C2×C4—2182no—
242C2×C4C23×D10911yescharacteristic
243C8—2182no—
244C8—2182no—
245C8D4×D10912yes—
246C8D4×D10912yes—
247C22—41no—
248C22—42no—
249C22—82no—
250C22—81no—
251C22—17441no—
252C22—17442no—
253C22—2182no—
254C22—2182no—
255C22#1681274411yescharacteristic
256C4—41no—
257C4—41no—
258C4—2182no—
259C4—2182no—
260C4C2×D4×D10912yes—
261C2—21no—
262C2—82no—
263C2—81no—
264C2—4362no—
265C2—4362no—
266C2#2029566111yescharacteristic
subgroupG/H
centerC2#20295661
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#496×C109C2
socleC2×C109D42
hypercenter#496D109
hypocenterC109#3288
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C2×C4 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #20295832C2semidirect
2. #20295827C2semidirect
3. #20295293C2semidirect
4. #20295286C2semidirect
5. #20295203C2semidirect
6. #20295198C2semidirect
7. #20295193C2semidirect
8. #20295149C2semidirect
9. #496×C109C2semidirect
10. #16812747C22semidirect
12. #16812744C22semidirect
13. #16812730C22semidirect
17. #16812725C22semidirect
18. #16812725C22semidirect
26. #16812705C22semidirect
27. #16812705C22semidirect
28. #16812635C22semidirect
29. #16812627C22semidirect
30. #16812627C22semidirect
31. #16812618C22semidirect
32. #16812616C22semidirect
33. #16812616C22semidirect
34. C2×SD16×C109C22semidirect
35. C2×D8×C109C22semidirect
36. #127×C109C22semidirect
37. #124×C109C22semidirect
38. #97×C109C22semidirect
52. Dic436D4semidirect
54. #13182018D4semidirect
55. #13182018D4semidirect
60. D872D4semidirect
134. C109#3288semidirect
137. #496D109semidirect
159. C2×SD16C2×D109semidirect
161. C2×D8C2×D109semidirect
164. #127C2×D109semidirect
165. #124C2×D109semidirect
172. #97C2×D109semidirect

Maximal subgroups

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

subgroupindex
#202958322
#202958272
#202952932
#202952862
#202952032
#202951982
#202951932
#202951492
#496×C1092
#3288109