Finite Group Databasebeta

← #28348257

Subgroups — #28348257

Order 13696.

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#20293431C212yesmaximal, complemented
2#20293430C212yesmaximal, complemented
3#20293003C211yesmaximal, characteristic, complemented
4#20292896C212yesmaximal, complemented
5#20292895C212yesmaximal, complemented
6#20292796C212yesmaximal, complemented
7#20292794C211yesmaximal, characteristic, complemented
8#20292741C212yesmaximal, complemented
9#453×C107C211yesmaximal, characteristic, complemented
10#16812155C2212yescomplemented
11#16812154C2211yescharacteristic, complemented
12#16812137C2212yescomplemented
13#16812136—22no—
14#16812135—22no—
15#16812135C2212yescomplemented
16#16812134—22no—
17#16812134C2212yescomplemented
18#16812133—22no—
19#16812130—22no—
20#16812129—22no—
21#16812129C2212yes—
22C2×#13133803—22no—
23C2×#13133803C2212yescomplemented
24C2×#13133803C2212yescomplemented
25D8×D107—22no—
26#16812117C2212yescomplemented
27#16812116C2212yescomplemented
28#16812106C2211yescharacteristic, complemented
29#16812044C2212yescomplemented
30#16812037C2212yescomplemented
31#16812026C2211yescharacteristic, complemented
32#16812025C2212yescomplemented
33C4×C107⋊C8C2211yescharacteristic, complemented
34H(5,𝔽2)×C107C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C107C2212yescomplemented
36#127×C107C2211yescharacteristic, complemented
37#104×C107C2212yescomplemented
38#100×C107C2211yescharacteristic, complemented
39#13133819—22no—
40#13133818—22no—
41#13133818C2312yes—
42#13133817—22no—
43C2×D428—22no—
44C2×D428—41no—
45C2×D428C2312yes—
46C2×C22⋊D107—41no—
47D4×D107—22no—
48D4×D107—22no—
49D4×D107—42no—
50#13133808D412yescomplemented
51#13133807—22no—
52#13133806—22no—
53#13133806D412yescomplemented
54#13133805—22no—
55#13133805D412yescomplemented
56D856—22no—
57#13133803—22no—
58#13133803—22no—
59#13133803—22no—
60#13133803—22no—
61#13133803—22no—
62#13133803—22no—
63#13133803D412yescomplemented
64#13133802—22no—
65#13133802C2312yes—
66#13133801—22no—
67C2×C107⋊C8—22no—
68C2×C107⋊C8C2312yes—
69C8×D107—22no—
70#41×C107—22no—
71#41×C107C2312yes—
72C2×D4×C107—22no—
73C2×D4×C107—41no—
74C2×D4×C107—41no—
75C2×D4×C107C2311yescharacteristic
76C2×D4×C107C2311yescharacteristic
77SD16×C107—22no—
78D8×C107—22no—
79M16×C107C2312yes—
80C42×C107C2311yescharacteristic
81C23×C107—21no—
82C23×C107—21no—
83C23×C107—41no—
84C22×D107—22no—
85C22×D107—41no—
86C22×D107—42no—
87Q8×C107C2×D412yes—
88D4×C107—22no—
89D4×C107—22no—
90D4×C107—22no—
91D4×C107—42no—
92D4×C107—82no—
93D4×C107C2×D412yes—
94C2×C4×C107—21no—
95C2×C4×C107—22no—
96C2×C4×C107—41no—
97C2×C4×C107C2411yescharacteristic
98C22⋊D107—22no—
99C22⋊D107—42no—
100C22⋊D107—42no—
101C2×Dic107—41no—
102D428—22no—
103D428—22no—
104D428—42no—
105D428—82no—
106D428C2×D412yes—
107C4×D107—22no—
108C4×D107—42no—
109Dic214C2×D412yes—
110C8×C107—22no—
111C107⋊C8—22no—
112C107⋊C8C2×D412yes—
113C107⋊C8C2×D412yes—
114C22×C107—22no—
115C22×C107—41no—
116C22×C107—41no—
117C22×C107—81no—
118C22×C107—81no—
119C22×C107—161no—
120C22×C107H(5,𝔽2)11yescharacteristic
121C2×D107—22no—
122C2×D107—42no—
123C2×D107—82no—
124C2×D107—161no—
125C2×D107—162no—
126C4×C107—22no—
127C4×C107—22no—
128C4×C107C22×D412yes—
129Dic107—22no—
130C2×C107—21no—
131C2×C107—42no—
132C2×C107—81no—
133C2×C107—81no—
134C2×C107D4211yescharacteristic
135D107—42no—
136D107—82no—
137#3292—1071nomaximal, Sylow
138C107#329211yescharacteristic, complemented, Sylow
139#576—1072no—
140#576—1072no—
141#493—1071no—
142#472—1072no—
143#472—1072no—
144#453—1071no—
145#453—1072no—
146#453D10711yescharacteristic, complemented
147#443—1072no—
148H(5,𝔽2)—1072no—
149H(5,𝔽2)—1071no—
150H(5,𝔽2)C2×D10711yescharacteristic, complemented
151AGL(1,ℤ/8)—1072no—
152AGL(1,ℤ/8)—1072no—
153AGL(1,ℤ/8)—1072no—
154AGL(1,ℤ/8)—2142no—
155AGL(1,ℤ/8)—2142no—
156AGL(1,ℤ/8)—2142no—
157AGL(1,ℤ/8)—2142no—
158AGL(1,ℤ/8)C2×D10712yescomplemented
159#135—1072no—
160#135—2142no—
161#135—2142no—
162C2×D8—1072no—
163C2×D8—1072no—
164C2×D8—2142no—
165C2×D8—2142no—
166#131—1072no—
167#131—1072no—
168#127—1071no—
169#127C2×D10711yescharacteristic, complemented
170#108—1072no—
171#104—1072no—
172#104C2×D10712yescomplemented
173#100—1071no—
174#100—1072no—
175#100C2×D10711yescharacteristic, complemented
176C4×C8—1071no—
177#41—22no—
178#41—1072no—
179#41—2142no—
180#41—2142no—
181#41—2142no—
182#41C22×D10712yes—
183C2×D4—22no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—1072no—
187C2×D4—4281no—
188C2×D4—4282no—
189C2×D4—4281no—
190C2×D4—2142no—
191C2×D4—2142no—
192C2×D4—2142no—
193C2×D4C22×D10711yescharacteristic
194C2×D4C22×D10711yescharacteristic
195Q16—1072no—
196SD16—22no—
197SD16—1072no—
198SD16—1072no—
199SD16—2142no—
200SD16—2142no—
201SD16—2142no—
202D8—22no—
203D8—1072no—
204D8—2142no—
205D8—2142no—
206D8—2142no—
207D8—2142no—
208D8—2142no—
209D8—2142no—
210D8—2142no—
211M16—1072no—
212M16—2142no—
213M16—2142no—
214M16C22×D10712yes—
215C2×C8—1072no—
216C2×C8—2142no—
217C2×C8—2142no—
218C42C22×D10711yescharacteristic
219C23—21no—
220C23—21no—
221C23—41no—
222C23—4282no—
223C23—4281no—
224C23—2142no—
225Q8—1072no—
226Q8D4×D10712yes—
227D4—22no—
228D4—22no—
229D4—22no—
230D4—42no—
231D4—82no—
232D4—8562no—
233D4—1072no—
234D4—4282no—
235D4—4282no—
236D4—4282no—
237D4—2142no—
238D4—2142no—
239D4—2142no—
240D4D4×D10712yes—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—4282no—
245C2×C4—4281no—
246C2×C4—2142no—
247C2×C4C23×D10711yescharacteristic
248C8—22no—
249C8—1072no—
250C8—1072no—
251C8—2142no—
252C22—22no—
253C22—41no—
254C22—41no—
255C22—81no—
256C22—81no—
257C22—161no—
258C22—8562no—
259C22—4282no—
260C22—17122no—
261C22—17121no—
262C22—2142no—
263C22#1681215411yescharacteristic
264C4—22no—
265C4—22no—
266C4—2142no—
267C4C2×D4×D10712yes—
268C2—21no—
269C2—42no—
270C2—81no—
271C2—81no—
272C2—8562no—
273C2—4282no—
274C2#2029325911yescharacteristic
subgroupG/H
centerC2#20293259
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#453×C107C2
socleC2×C107D42
hypercenter#453D107
hypocenterC107#3292
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #20293431C2semidirect
2. #20293430C2semidirect
3. #20293003C2semidirect
4. #20292896C2semidirect
5. #20292895C2semidirect
6. #20292796C2semidirect
7. #20292794C2semidirect
8. #20292741C2semidirect
9. #453×C107C2semidirect
10. #16812155C22semidirect
11. #16812154C22semidirect
12. #16812137C22semidirect
15. #16812135C22semidirect
17. #16812134C22semidirect
23. C2×#13133803C22semidirect
24. C2×#13133803C22semidirect
26. #16812117C22semidirect
27. #16812116C22semidirect
28. #16812106C22semidirect
29. #16812044C22semidirect
30. #16812037C22semidirect
31. #16812026C22semidirect
32. #16812025C22semidirect
33. C4×C107⋊C8C22semidirect
34. H(5,𝔽2)×C107C22semidirect
35. AGL(1,ℤ/8)×C107C22semidirect
36. #127×C107C22semidirect
37. #104×C107C22semidirect
38. #100×C107C22semidirect
50. #13133808D4semidirect
53. #13133806D4semidirect
55. #13133805D4semidirect
63. #13133803D4semidirect
138. C107#3292semidirect
146. #453D107semidirect
150. H(5,𝔽2)C2×D107semidirect
158. AGL(1,ℤ/8)C2×D107semidirect
169. #127C2×D107semidirect
172. #104C2×D107semidirect
175. #100C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202934312
#202934302
#202930032
#202928962
#202928952
#202927962
#202927942
#202927412
#453×C1072
#3292107