Finite Group Databasebeta

← #28348230

Subgroups — #28348230

Order 13696.

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#20293436C212yesmaximal, complemented
2#20293430C212yesmaximal, complemented
3#20293017C211yesmaximal, characteristic, complemented
4#20292896C212yesmaximal, complemented
5#20292892C212yesmaximal, complemented
6#20292799C211yesmaximal, characteristic, complemented
7#20292796C212yesmaximal, complemented
8#20292746C212yesmaximal, complemented
9#455×C107C211yesmaximal, characteristic, complemented
10#16812157C2211yescharacteristic, complemented
11#16812155C2212yescomplemented
12#16812141—22no—
13#16812137C2212yescomplemented
14#16812136—22no—
15#16812135—22no—
16#16812135C2212yescomplemented
17#16812134—22no—
18#16812134C2212yes—
19#16812132—22no—
20#16812129—22no—
21#16812129C2212yes—
22C2×#13133805C2212yescomplemented
23SD16×D107—22no—
24C2×#13133803—22no—
25C2×#13133803C2212yescomplemented
26#16812117C2212yescomplemented
27#16812116C2212yescomplemented
28#16812106C2211yescharacteristic, complemented
29#16812044C2212yescomplemented
30#16812037C2212yescomplemented
31#16812028C2211yescharacteristic, complemented
32#16812025C2212yescomplemented
33#16812020C2211yescharacteristic, complemented
34H(5,𝔽2)×C107C2211yescharacteristic
35#137×C107C2212yescomplemented
36#124×C107C2211yescharacteristic, complemented
37#104×C107C2212yescomplemented
38#100×C107C2211yescharacteristic, complemented
39#13133819—22no—
40#13133819—22no—
41#13133818—22no—
42#13133818—41no—
43#13133818C2312yes—
44Q8×D107—22no—
45C2×D428—22no—
46C2×D428—41no—
47C2×D428C2312yes—
48D4×D107—22no—
49D4×D107—42no—
50#13133808D412yescomplemented
51#13133807—22no—
52#13133806—22no—
53#13133806D412yescomplemented
54#13133805—22no—
55#13133805—22no—
56#13133805—22no—
57#13133805D412yescomplemented
58D856—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×Q8×C107C2311yescharacteristic
73C2×D4×C107—22no—
74C2×D4×C107—41no—
75C2×D4×C107C2311yescharacteristic
76Q16×C107—22no—
77SD16×C107—22no—
78M16×C107C2312yes—
79C22⋊C4×C107—41no—
80C42×C107C2311yescharacteristic
81C23×C107—21no—
82C23×C107—41no—
83C22×D107—22no—
84C22×D107—42no—
85Q8×C107—22no—
86Q8×C107C2×D412yes—
87D4×C107—22no—
88D4×C107—22no—
89D4×C107—42no—
90D4×C107C2×D412yes—
91C2×C4×C107—22no—
92C2×C4×C107—21no—
93C2×C4×C107—21no—
94C2×C4×C107—41no—
95C2×C4×C107C2411yescharacteristic
96C22⋊D107—22no—
97C22⋊D107—42no—
98D428—22no—
99D428—22no—
100D428—42no—
101D428—41no—
102D428—82no—
103D428C2×D412yes—
104C4×D107—22no—
105C4×D107—42no—
106C4×D107—42no—
107Dic214—41no—
108Dic214C2×D412yes—
109C8×C107—22no—
110C107⋊C8—22no—
111C107⋊C8C2×D412yes—
112C107⋊C8C2×D412yes—
113C22×C107—22no—
114C22×C107—41no—
115C22×C107—81no—
116C22×C107—161no—
117C22×C107H(5,𝔽2)11yescharacteristic
118C2×D107—22no—
119C2×D107—42no—
120C2×D107—82no—
121C2×D107—162no—
122C4×C107—22no—
123C4×C107—41no—
124C4×C107—41no—
125C4×C107C22×D412yes—
126Dic107—22no—
127C2×C107—21no—
128C2×C107—42no—
129C2×C107—81no—
130C2×C107D4211yescharacteristic
131D107—42no—
132D107—82no—
133#3288—1071nomaximal, Sylow
134C107#328811yescharacteristic, complemented, Sylow
135#577—1072no—
136#576—1072no—
137#496—1071no—
138#472—1072no—
139#471—1072no—
140#455D10711yescharacteristic, complemented
141#454—1071no—
142#453—1072no—
143#444—1072no—
1442−1+4—1071no—
145H(5,𝔽2)—1072no—
146H(5,𝔽2)C2×D10711yescharacteristic
147#137—2142no—
148#137C2×D10712yescomplemented
149AGL(1,ℤ/8)—1072no—
150AGL(1,ℤ/8)—1072no—
151AGL(1,ℤ/8)—1072no—
152AGL(1,ℤ/8)—2142no—
153AGL(1,ℤ/8)—2142no—
154AGL(1,ℤ/8)—2142no—
155#135—1072no—
156#135—2142no—
157#135—2142no—
158C2×SD16—1072no—
159C2×SD16—2142no—
160C2×D8—1072no—
161C2×D8—2142no—
162#131—1072no—
163#131—1072no—
164#127—1071no—
165#124C2×D10711yescharacteristic, complemented
166#108—1072no—
167#104—1072no—
168#104C2×D10712yescomplemented
169#101—1071no—
170#100—1072no—
171#100C2×D10711yescharacteristic, complemented
172#97—1071no—
173#41—22no—
174#41—1072no—
175#41—4281no—
176#41—2142no—
177#41—2142no—
178#41—2142no—
179#41C22×D10712yes—
180C2×Q8—2142no—
181C2×Q8C22×D10711yescharacteristic
182C2×D4—22no—
183C2×D4—41no—
184C2×D4—1072no—
185C2×D4—4281no—
186C2×D4—4282no—
187C2×D4—2142no—
188C2×D4—2142no—
189C2×D4C22×D10711yescharacteristic
190Q16—22no—
191Q16—1072no—
192SD16—22no—
193SD16—1072no—
194SD16—1072no—
195SD16—2142no—
196SD16—2142no—
197SD16—2142no—
198SD16—2142no—
199SD16—2142no—
200D8—1072no—
201D8—2142no—
202D8—2142no—
203D8—2142no—
204D8—2142no—
205D8—2142no—
206M16—1072no—
207M16—2142no—
208M16—2142no—
209M16C22×D10712yes—
210C2×C8—1072no—
211C2×C8—2142no—
212C2×C8—2142no—
213C22⋊C4—41no—
214C42C22×D10711yescharacteristic
215C23—21no—
216C23—41no—
217C23—4282no—
218C23—2142no—
219Q8—22no—
220Q8—1072no—
221Q8—4281no—
222Q8D4×D10712yes—
223D4—22no—
224D4—22no—
225D4—42no—
226D4—8562no—
227D4—1072no—
228D4—4282no—
229D4—4281no—
230D4—4282no—
231D4—2142no—
232D4—2142no—
233D4—2142no—
234D4D4×D10712yes—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—41no—
239C2×C4—4282no—
240C2×C4—4282no—
241C2×C4—2142no—
242C2×C4C23×D10711yescharacteristic
243C8—22no—
244C8—1072no—
245C8—1072no—
246C8—2142no—
247C22—22no—
248C22—41no—
249C22—81no—
250C22—161no—
251C22—8562no—
252C22—4282no—
253C22—17122no—
254C22—2142no—
255C22#1681215411yescharacteristic
256C4—22no—
257C4—41no—
258C4—41no—
259C4—2142no—
260C4C2×D4×D10712yes—
261C2—21no—
262C2—42no—
263C2—81no—
264C2—8562no—
265C2—4282no—
266C2#2029325911yescharacteristic
subgroupG/H
centerC2#20293259
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#455×C107C2
socleC2×C107D42
hypercenter#455D107
hypocenterC107#3288
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C2×C4 ◁ #455

Complements

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

NH ≅ G/Ntype
1. #20293436C2semidirect
2. #20293430C2semidirect
3. #20293017C2semidirect
4. #20292896C2semidirect
5. #20292892C2semidirect
6. #20292799C2semidirect
7. #20292796C2semidirect
8. #20292746C2semidirect
9. #455×C107C2semidirect
10. #16812157C22semidirect
11. #16812155C22semidirect
13. #16812137C22semidirect
16. #16812135C22semidirect
22. C2×#13133805C22semidirect
25. C2×#13133803C22semidirect
26. #16812117C22semidirect
27. #16812116C22semidirect
28. #16812106C22semidirect
29. #16812044C22semidirect
30. #16812037C22semidirect
31. #16812028C22semidirect
32. #16812025C22semidirect
33. #16812020C22semidirect
35. #137×C107C22semidirect
36. #124×C107C22semidirect
37. #104×C107C22semidirect
38. #100×C107C22semidirect
50. #13133808D4semidirect
53. #13133806D4semidirect
57. #13133805D4semidirect
63. #13133803D4semidirect
134. C107#3288semidirect
140. #455D107semidirect
148. #137C2×D107semidirect
165. #124C2×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
#202934362
#202934302
#202930172
#202928962
#202928922
#202927992
#202927962
#202927462
#455×C1072
#3288107