Finite Group Databasebeta

← #28374828

Subgroups — #28374828

Order 13952.

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

normal subgroups only

This group has 476 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#20295830C212yesmaximal, complemented
2#20295829C212yesmaximal, complemented
3#20295402C211yesmaximal, characteristic, complemented
4#20295295C212yesmaximal, complemented
5#20295294C212yesmaximal, complemented
6#20295195C212yesmaximal, complemented
7#20295193C211yesmaximal, characteristic, complemented
8#20295140C212yesmaximal, complemented
9#453×C109C211yesmaximal, characteristic, complemented
10#16812745C2212yescomplemented
11#16812744C2211yescharacteristic, complemented
12#16812727C2212yescomplemented
13#16812726—22no—
14#16812725—22no—
15#16812725C2212yescomplemented
16#16812724—22no—
17#16812724C2212yescomplemented
18#16812723—22no—
19#16812720—22no—
20#16812719—22no—
21#16812719C2212yes—
22C2×#13182014—22no—
23C2×#13182014C2212yescomplemented
24C2×#13182014C2212yescomplemented
25D8×D109—22no—
26#16812707C2212yescomplemented
27#16812706C2212yescomplemented
28#16812696C2211yescharacteristic, complemented
29#16812634C2212yescomplemented
30#16812627C2212yescomplemented
31#16812616C2211yescharacteristic, complemented
32#16812615C2212yescomplemented
33C4×#11724784C2211yescharacteristic, complemented
34H(5,𝔽2)×C109C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C109C2212yescomplemented
36#127×C109C2211yescharacteristic, complemented
37#104×C109C2212yescomplemented
38#100×C109C2211yescharacteristic, complemented
39#13182030—22no—
40#13182029—22no—
41#13182029C2312yes—
42#13182028—22no—
43C2×D436—22no—
44C2×D436—41no—
45C2×D436C2312yes—
46C2×C22⋊D109—41no—
47D4×D109—22no—
48D4×D109—22no—
49D4×D109—42no—
50#13182019D412yescomplemented
51#13182018—22no—
52#13182017—22no—
53#13182017D412yescomplemented
54#13182016—22no—
55#13182016D412yescomplemented
56D872—22no—
57#13182014—22no—
58#13182014—22no—
59#13182014—22no—
60#13182014—22no—
61#13182014—22no—
62#13182014—22no—
63#13182014D412yescomplemented
64#13182013—22no—
65#13182013C2312yes—
66#13182012—22no—
67C2×#11724784—22no—
68C2×#11724784C2312yes—
69C8×D109—22no—
70#41×C109—22no—
71#41×C109C2312yes—
72C2×D4×C109—22no—
73C2×D4×C109—41no—
74C2×D4×C109—41no—
75C2×D4×C109C2311yescharacteristic
76C2×D4×C109C2311yescharacteristic
77SD16×C109—22no—
78D8×C109—22no—
79M16×C109C2312yes—
80C42×C109C2311yescharacteristic
81C23×C109—21no—
82C23×C109—21no—
83C23×C109—41no—
84C22×D109—22no—
85C22×D109—41no—
86C22×D109—42no—
87Q8×C109C2×D412yes—
88D4×C109—22no—
89D4×C109—22no—
90D4×C109—22no—
91D4×C109—42no—
92D4×C109—82no—
93D4×C109C2×D412yes—
94C2×C4×C109—21no—
95C2×C4×C109—22no—
96C2×C4×C109—41no—
97C2×C4×C109C2411yescharacteristic
98C22⋊D109—22no—
99C22⋊D109—42no—
100C22⋊D109—42no—
101C2×Dic109—41no—
102D436—22no—
103D436—22no—
104D436—42no—
105D436—82no—
106D436C2×D412yes—
107C4×D109—22no—
108C4×D109—42no—
109Dic218C2×D412yes—
110C8×C109—22no—
111#11724784—22no—
112#11724784C2×D412yes—
113#11724784C2×D412yes—
114C22×C109—22no—
115C22×C109—41no—
116C22×C109—41no—
117C22×C109—81no—
118C22×C109—81no—
119C22×C109—161no—
120C22×C109H(5,𝔽2)11yescharacteristic
121C2×D109—22no—
122C2×D109—42no—
123C2×D109—82no—
124C2×D109—161no—
125C2×D109—162no—
126C4×C109—22no—
127C4×C109—22no—
128C4×C109C22×D412yes—
129Dic109—22no—
130C2×C109—21no—
131C2×C109—42no—
132C2×C109—81no—
133C2×C109—81no—
134C2×C109D4211yescharacteristic
135D109—42no—
136D109—82no—
137#3292—1091nomaximal, Sylow
138C109#329211yescharacteristic, complemented, Sylow
139#576—1092no—
140#576—1092no—
141#493—1091no—
142#472—1092no—
143#472—1092no—
144#453—1091no—
145#453—1092no—
146#453D10911yescharacteristic, complemented
147#443—1092no—
148H(5,𝔽2)—1092no—
149H(5,𝔽2)—1091no—
150H(5,𝔽2)C2×D10911yescharacteristic, complemented
151AGL(1,ℤ/8)—1092no—
152AGL(1,ℤ/8)—1092no—
153AGL(1,ℤ/8)—1092no—
154AGL(1,ℤ/8)—2182no—
155AGL(1,ℤ/8)—2182no—
156AGL(1,ℤ/8)—2182no—
157AGL(1,ℤ/8)—2182no—
158AGL(1,ℤ/8)C2×D10912yescomplemented
159#135—1092no—
160#135—2182no—
161#135—2182no—
162C2×D8—1092no—
163C2×D8—1092no—
164C2×D8—2182no—
165C2×D8—2182no—
166#131—1092no—
167#131—1092no—
168#127—1091no—
169#127C2×D10911yescharacteristic, complemented
170#108—1092no—
171#104—1092no—
172#104C2×D10912yescomplemented
173#100—1091no—
174#100—1092no—
175#100C2×D10911yescharacteristic, complemented
176C4×C8—1091no—
177#41—22no—
178#41—1092no—
179#41—2182no—
180#41—2182no—
181#41—2182no—
182#41C22×D10912yes—
183C2×D4—22no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—1092no—
187C2×D4—4361no—
188C2×D4—4362no—
189C2×D4—4361no—
190C2×D4—2182no—
191C2×D4—2182no—
192C2×D4—2182no—
193C2×D4C22×D10911yescharacteristic
194C2×D4C22×D10911yescharacteristic
195Q16—1092no—
196SD16—22no—
197SD16—1092no—
198SD16—1092no—
199SD16—2182no—
200SD16—2182no—
201SD16—2182no—
202D8—22no—
203D8—1092no—
204D8—2182no—
205D8—2182no—
206D8—2182no—
207D8—2182no—
208D8—2182no—
209D8—2182no—
210D8—2182no—
211M16—1092no—
212M16—2182no—
213M16—2182no—
214M16C22×D10912yes—
215C2×C8—1092no—
216C2×C8—2182no—
217C2×C8—2182no—
218C42C22×D10911yescharacteristic
219C23—21no—
220C23—21no—
221C23—41no—
222C23—4362no—
223C23—4361no—
224C23—2182no—
225Q8—1092no—
226Q8D4×D10912yes—
227D4—22no—
228D4—22no—
229D4—22no—
230D4—42no—
231D4—82no—
232D4—8722no—
233D4—1092no—
234D4—4362no—
235D4—4362no—
236D4—4362no—
237D4—2182no—
238D4—2182no—
239D4—2182no—
240D4D4×D10912yes—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—4362no—
245C2×C4—4361no—
246C2×C4—2182no—
247C2×C4C23×D10911yescharacteristic
248C8—22no—
249C8—1092no—
250C8—1092no—
251C8—2182no—
252C22—22no—
253C22—41no—
254C22—41no—
255C22—81no—
256C22—81no—
257C22—161no—
258C22—8722no—
259C22—4362no—
260C22—17442no—
261C22—17441no—
262C22—2182no—
263C22#1681274411yescharacteristic
264C4—22no—
265C4—22no—
266C4—2182no—
267C4C2×D4×D10912yes—
268C2—21no—
269C2—42no—
270C2—81no—
271C2—81no—
272C2—8722no—
273C2—4362no—
274C2#2029565811yescharacteristic
subgroupG/H
centerC2#20295658
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#453×C109C2
socleC2×C109D42
hypercenter#453D109
hypocenterC109#3292
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #20295830C2semidirect
2. #20295829C2semidirect
3. #20295402C2semidirect
4. #20295295C2semidirect
5. #20295294C2semidirect
6. #20295195C2semidirect
7. #20295193C2semidirect
8. #20295140C2semidirect
9. #453×C109C2semidirect
10. #16812745C22semidirect
11. #16812744C22semidirect
12. #16812727C22semidirect
15. #16812725C22semidirect
17. #16812724C22semidirect
23. C2×#13182014C22semidirect
24. C2×#13182014C22semidirect
26. #16812707C22semidirect
27. #16812706C22semidirect
28. #16812696C22semidirect
29. #16812634C22semidirect
30. #16812627C22semidirect
31. #16812616C22semidirect
32. #16812615C22semidirect
33. C4×#11724784C22semidirect
34. H(5,𝔽2)×C109C22semidirect
35. AGL(1,ℤ/8)×C109C22semidirect
36. #127×C109C22semidirect
37. #104×C109C22semidirect
38. #100×C109C22semidirect
50. #13182019D4semidirect
53. #13182017D4semidirect
55. #13182016D4semidirect
63. #13182014D4semidirect
138. C109#3292semidirect
146. #453D109semidirect
150. H(5,𝔽2)C2×D109semidirect
158. AGL(1,ℤ/8)C2×D109semidirect
169. #127C2×D109semidirect
172. #104C2×D109semidirect
175. #100C2×D109semidirect

Maximal subgroups

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

subgroupindex
#202958302
#202958292
#202954022
#202952952
#202952942
#202951952
#202951932
#202951402
#453×C1092
#3292109