Finite Group Databasebeta

← #28348229

Subgroups — #28348229

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#20293435C212yesmaximal, complemented
2#20293431C212yesmaximal, complemented
3#20293017C211yesmaximal, characteristic, complemented
4#20292895C212yesmaximal, complemented
5#20292893C212yesmaximal, complemented
6#20292804C211yesmaximal, characteristic, complemented
7#20292796C212yesmaximal, complemented
8#20292746C212yesmaximal, complemented
9#454×C107C211yesmaximal, characteristic, complemented
10#16812155C2212yescomplemented
11#16812154C2211yescharacteristic
12#16812140—22no—
13#16812140C2212yescomplemented
14#16812137C2212yes—
15#16812136—22no—
16#16812134—22no—
17#16812134C2212yescomplemented
18#16812133—22no—
19#16812130—22no—
20#16812129—22no—
21#16812129C2212yes—
22C2×#13133805—22no—
23C2×#13133805C2212yescomplemented
24C2×#13133803C2212yescomplemented
25D8×D107—22no—
26#16812117C2212yescomplemented
27#16812116C2212yescomplemented
28#16812106C2211yescharacteristic, complemented
29#16812044C2212yescomplemented
30#16812037C2212yescomplemented
31#16812026C2211yescharacteristic, complemented
32#16812025C2212yescomplemented
33#16812020C2211yescharacteristic, complemented
342−1+4×C107C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C107C2212yescomplemented
36#124×C107C2211yescharacteristic, complemented
37#104×C107C2212yescomplemented
38#101×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#13133808—22no—
51#13133808D412yescomplemented
52#13133807—22no—
53#13133806D412yescomplemented
54#13133805—22no—
55#13133805—22no—
56#13133805—22no—
57#13133805—22no—
58#13133805D412yescomplemented
59D856—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×C107—41no—
72#41×C107C2312yes—
73C2×Q8×C107—22no—
74C2×Q8×C107C2311yescharacteristic
75C2×D4×C107C2311yescharacteristic
76SD16×C107—22no—
77D8×C107—22no—
78M16×C107C2312yes—
79C22⋊C4×C107—41no—
80C42×C107C2311yescharacteristic
81C23×C107—21no—
82C22×D107—22no—
83C22×D107—41no—
84C22×D107—42no—
85Q8×C107—22no—
86Q8×C107—41no—
87Q8×C107C2×D412yes—
88D4×C107—22no—
89D4×C107—22no—
90D4×C107—41no—
91D4×C107C2×D412yes—
92C2×C4×C107—22no—
93C2×C4×C107—21no—
94C2×C4×C107—21no—
95C2×C4×C107—42no—
96C2×C4×C107C2411yescharacteristic
97C22⋊D107—22no—
98C22⋊D107—42no—
99C22⋊D107—42no—
100C2×Dic107—41no—
101D428—22no—
102D428—22no—
103D428—42no—
104D428—82no—
105D428C2×D412yes—
106C4×D107—22no—
107C4×D107—42no—
108Dic214C2×D412yes—
109C8×C107—22no—
110C107⋊C8—22no—
111C107⋊C8C2×D412yes—
112C107⋊C8C2×D412yes—
113C22×C107—22no—
114C22×C107—41no—
115C22×C107—81no—
116C22×C107H(5,𝔽2)11yescharacteristic
117C2×D107—22no—
118C2×D107—42no—
119C2×D107—82no—
120C2×D107—161no—
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#455—1071no—
141#454D10711yescharacteristic, complemented
142#453—1072no—
143#444—1072no—
1442−1+4C2×D10711yescharacteristic, complemented
145H(5,𝔽2)—1072no—
146H(5,𝔽2)—1071no—
147#137—1072no—
148#137—2142no—
149AGL(1,ℤ/8)—1072no—
150AGL(1,ℤ/8)—1072no—
151AGL(1,ℤ/8)—2142no—
152AGL(1,ℤ/8)—2142no—
153AGL(1,ℤ/8)—2142no—
154AGL(1,ℤ/8)C2×D10712yescomplemented
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#101C2×D10711yescharacteristic, complemented
170#100—1071no—
171#100—1072no—
172#97—1071no—
173#41—22no—
174#41—41no—
175#41—1072no—
176#41—2142no—
177#41—2142no—
178#41—2142no—
179#41C22×D10712yes—
180C2×Q8—22no—
181C2×Q8C22×D10711yescharacteristic
182C2×D4—1072no—
183C2×D4—4281no—
184C2×D4—4282no—
185C2×D4—4281no—
186C2×D4—2142no—
187C2×D4—2142no—
188C2×D4—2142no—
189C2×D4C22×D10711yescharacteristic
190Q16—1072no—
191Q16—2142no—
192SD16—22no—
193SD16—1072no—
194SD16—1072no—
195SD16—2142no—
196SD16—2142no—
197SD16—2142no—
198SD16—2142no—
199SD16—2142no—
200D8—22no—
201D8—1072no—
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—4282no—
217C23—4281no—
218C23—2142no—
219Q8—22no—
220Q8—41no—
221Q8—1072no—
222Q8D4×D10712yes—
223D4—22no—
224D4—22no—
225D4—41no—
226D4—8562no—
227D4—1072no—
228D4—4282no—
229D4—4282no—
230D4—4282no—
231D4—2142no—
232D4—2142no—
233D4—2142no—
234D4D4×D10712yes—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—42no—
239C2×C4—4282no—
240C2×C4—4281no—
241C2×C4—2142no—
242C2×C4C23×D10711yescharacteristic
243C8—22no—
244C8—1072no—
245C8—1072no—
246C8—2142no—
247C22—22no—
248C22—41no—
249C22—81no—
250C22—8562no—
251C22—4282no—
252C22—17122no—
253C22—17121no—
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#454×C107C2
socleC2×C107D42
hypercenter#454D107
hypocenterC107#3288
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C2×C4 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #20293435C2semidirect
2. #20293431C2semidirect
3. #20293017C2semidirect
4. #20292895C2semidirect
5. #20292893C2semidirect
6. #20292804C2semidirect
7. #20292796C2semidirect
8. #20292746C2semidirect
9. #454×C107C2semidirect
10. #16812155C22semidirect
13. #16812140C22semidirect
17. #16812134C22semidirect
23. C2×#13133805C22semidirect
24. C2×#13133803C22semidirect
26. #16812117C22semidirect
27. #16812116C22semidirect
28. #16812106C22semidirect
29. #16812044C22semidirect
30. #16812037C22semidirect
31. #16812026C22semidirect
32. #16812025C22semidirect
33. #16812020C22semidirect
34. 2−1+4×C107C22semidirect
35. AGL(1,ℤ/8)×C107C22semidirect
36. #124×C107C22semidirect
37. #104×C107C22semidirect
38. #101×C107C22semidirect
51. #13133808D4semidirect
53. #13133806D4semidirect
58. #13133805D4semidirect
63. #13133803D4semidirect
134. C107#3288semidirect
141. #454D107semidirect
144. 2−1+4C2×D107semidirect
154. AGL(1,ℤ/8)C2×D107semidirect
165. #124C2×D107semidirect
168. #104C2×D107semidirect
169. #101C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202934352
#202934312
#202930172
#202928952
#202928932
#202928042
#202927962
#202927462
#454×C1072
#3288107