Finite Group Databasebeta

← #28348256

Subgroups — #28348256

Order 13696.

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

normal subgroups only

This group has 460 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#20293438C212yesmaximal, complemented
2#20293437C212yesmaximal, complemented
3#20292999C211yesmaximal, characteristic, complemented
4#20292889C212yesmaximal, complemented
5#20292888C212yesmaximal, complemented
6#20292811C212yesmaximal, complemented
7#20292794C211yesmaximal, characteristic, complemented
8#20292741C212yesmaximal, complemented
9#453×C107C211yesmaximal, characteristic, complemented
10#16812156C2212yes—
11#16812154C2211yescharacteristic, complemented
12#16812142C2212yescomplemented
13#16812139—22no—
14#16812139C2212yescomplemented
15#16812138—22no—
16#16812135—22no—
17#16812135C2212yes—
18#16812133—22no—
19#16812129—22no—
20#16812129C2212yes—
21#16812128—22no—
22C2×#13133806—22no—
23C2×#13133806C2212yescomplemented
24C2×#13133806C2212yescomplemented
25SD16×D107—22no—
26#16812117C2212yescomplemented
27#16812116C2212yescomplemented
28#16812109C2211yescharacteristic, complemented
29#16812044C2212yescomplemented
30#16812037C2212yescomplemented
31#16812027C2212yescomplemented
32#16812026C2211yescharacteristic, complemented
33C4×C107⋊C8C2211yescharacteristic, complemented
34H(5,𝔽2)×C107C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C107C2212yes—
36#127×C107C2211yescharacteristic
37#104×C107C2212yescomplemented
38#100×C107C2211yescharacteristic, complemented
39#13133818—22no—
40#13133818C2312yes—
41#13133817—22no—
42#13133817—22no—
43#13133817—42no—
44C2×Dic214—22no—
45C2×Dic214C2312yes—
46Q8×D107—22no—
47C2×C22⋊D107—41no—
48D4×D107—22no—
49Dic428—22no—
50#13133808—22no—
51#13133808D412yescomplemented
52#13133807—22no—
53#13133806—22no—
54#13133806—22no—
55#13133806—22no—
56#13133806—22no—
57#13133806—22no—
58#13133806—22no—
59#13133806D412yescomplemented
60#13133805D412yescomplemented
61#13133803—22no—
62#13133803D412yescomplemented
63#13133802—22no—
64#13133802C2312yes—
65#13133801—22no—
66C2×C107⋊C8—22no—
67C2×C107⋊C8C2312yes—
68C8×D107—22no—
69C4⋊Dic107—41no—
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—41no—
85Q8×C107C2×D412yes—
86D4×C107—22no—
87D4×C107—22no—
88D4×C107—22no—
89D4×C107—42no—
90D4×C107—82no—
91D4×C107C2×D412yes—
92C2×C4×C107—21no—
93C2×C4×C107—22no—
94C2×C4×C107—41no—
95C2×C4×C107C2411yescharacteristic
96C22⋊D107—22no—
97C22⋊D107—42no—
98C22⋊D107—42no—
99C2×Dic107—22no—
100C2×Dic107—41no—
101C2×Dic107—42no—
102D428C2×D412yes—
103C4×D107—22no—
104C4×D107—42no—
105Dic214—22no—
106Dic214—22no—
107Dic214—42no—
108Dic214C2×D412yes—
109C8×C107—22no—
110C107⋊C8—22no—
111C107⋊C8C2×D412yes—
112C107⋊C8C2×D412yes—
113C22×C107—22no—
114C22×C107—41no—
115C22×C107—41no—
116C22×C107—81no—
117C22×C107—81no—
118C22×C107—161no—
119C22×C107H(5,𝔽2)11yescharacteristic
120C2×D107—22no—
121C2×D107—161no—
122C4×C107—22no—
123C4×C107—22no—
124C4×C107C22×D412yes—
125Dic107—22no—
126Dic107—42no—
127C2×C107—21no—
128C2×C107—42no—
129C2×C107—81no—
130C2×C107—81no—
131C2×C107D4211yescharacteristic
132D107—42no—
133#3291—1071nomaximal, Sylow
134C107#329111yescharacteristic, complemented, Sylow
135#577—1072no—
136#577—1072no—
137#492—1071no—
138#471—1072no—
139#471—1072no—
140#456—1072no—
141#453—1071no—
142#453D10711yescharacteristic, complemented
143#443—1072no—
1442−1+4—1072no—
145H(5,𝔽2)—1071no—
146H(5,𝔽2)C2×D10711yescharacteristic, complemented
147#137—1072no—
148#137—1072no—
149#137—2142no—
150#137—2142no—
151AGL(1,ℤ/8)—1072no—
152AGL(1,ℤ/8)—2142no—
153AGL(1,ℤ/8)—2142no—
154AGL(1,ℤ/8)C2×D10712yes—
155#135—1072no—
156#135—2142no—
157#135—2142no—
158C2×SD16—1072no—
159C2×SD16—1072no—
160C2×SD16—2142no—
161C2×SD16—2142no—
162#131—1072no—
163#131—1072no—
164C4⋊Q8—1071no—
165#127C2×D10711yescharacteristic
166#108—1072no—
167#104—1072no—
168#104C2×D10712yescomplemented
169#101—1072no—
170#100—1071no—
171#100C2×D10711yescharacteristic, complemented
172C4×C8—1071no—
173#41—22no—
174#41—1072no—
175#41—4282no—
176#41—2142no—
177#41—2142no—
178#41—2142no—
179#41C22×D10712yes—
180C2×Q8—1072no—
181C2×Q8—2142no—
182C2×Q8—2142no—
183C2×D4—22no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—4281no—
187C2×D4—2142no—
188C2×D4C22×D10711yescharacteristic
189C2×D4C22×D10711yescharacteristic
190Q16—1072no—
191Q16—2142no—
192Q16—2142no—
193SD16—22no—
194SD16—1072no—
195SD16—1072no—
196SD16—2142no—
197SD16—2142no—
198SD16—2142no—
199SD16—2142no—
200SD16—2142no—
201SD16—2142no—
202SD16—2142no—
203D8—22no—
204D8—1072no—
205D8—2142no—
206M16—1072no—
207M16—2142no—
208M16—2142no—
209M16C22×D10712yes—
210C2×C8—1072no—
211C2×C8—2142no—
212C2×C8—2142no—
213C4⋊C4—4281no—
214C42C22×D10711yescharacteristic
215C23—21no—
216C23—21no—
217C23—41no—
218C23—4281no—
219Q8—1072no—
220Q8—4282no—
221Q8—2142no—
222Q8—2142no—
223Q8D4×D10712yes—
224D4—22no—
225D4—22no—
226D4—22no—
227D4—42no—
228D4—82no—
229D4—1072no—
230D4—4282no—
231D4—4282no—
232D4—2142no—
233D4D4×D10712yes—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—41no—
237C2×C4—4282no—
238C2×C4—4282no—
239C2×C4—4281no—
240C2×C4—2142no—
241C2×C4—2142no—
242C2×C4C23×D10711yescharacteristic
243C8—22no—
244C8—1072no—
245C8—1072no—
246C8—2142no—
247C22—22no—
248C22—41no—
249C22—41no—
250C22—81no—
251C22—81no—
252C22—161no—
253C22—17121no—
254C22—2142no—
255C22#1681215411yescharacteristic
256C4—22no—
257C4—22no—
258C4—4282no—
259C4—2142no—
260C4C2×D4×D10712yes—
261C2—21no—
262C2—42no—
263C2—81no—
264C2—81no—
265C2—4282no—
266C2#2029325911yescharacteristic
subgroupG/H
centerC2#20293259
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#453×C107C2
socleC2×C107D42
hypercenter#453D107
hypocenterC107#3291
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. #20293438C2semidirect
2. #20293437C2semidirect
3. #20292999C2semidirect
4. #20292889C2semidirect
5. #20292888C2semidirect
6. #20292811C2semidirect
7. #20292794C2semidirect
8. #20292741C2semidirect
9. #453×C107C2semidirect
11. #16812154C22semidirect
12. #16812142C22semidirect
14. #16812139C22semidirect
23. C2×#13133806C22semidirect
24. C2×#13133806C22semidirect
26. #16812117C22semidirect
27. #16812116C22semidirect
28. #16812109C22semidirect
29. #16812044C22semidirect
30. #16812037C22semidirect
31. #16812027C22semidirect
32. #16812026C22semidirect
33. C4×C107⋊C8C22semidirect
34. H(5,𝔽2)×C107C22semidirect
37. #104×C107C22semidirect
38. #100×C107C22semidirect
51. #13133808D4semidirect
59. #13133806D4semidirect
60. #13133805D4semidirect
62. #13133803D4semidirect
134. C107#3291semidirect
142. #453D107semidirect
146. H(5,𝔽2)C2×D107semidirect
168. #104C2×D107semidirect
171. #100C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202934382
#202934372
#202929992
#202928892
#202928882
#202928112
#202927942
#202927412
#453×C1072
#3291107