Finite Group Databasebeta

← #28348255

Subgroups — #28348255

Order 13696.

460 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 125 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#20293439C212yesmaximal, complemented
2#20293434C212yesmaximal, complemented
3#20293000C211yesmaximal, characteristic, complemented
4#20292891C212yesmaximal, complemented
5#20292890C212yesmaximal, complemented
6#20292811C212yesmaximal, complemented
7#20292796C212yesmaximal, complemented
8#20292745C211yesmaximal, characteristic, complemented
9#443×C107C211yesmaximal, characteristic, complemented
10#16812156C2212yes—
11#16812155C2212yescomplemented
12#16812142—22no—
13#16812142C2212yescomplemented
14#16812141—22no—
15#16812139C2212yescomplemented
16#16812137—22no—
17#16812137C2212yes—
18#16812134C2212yes—
19#16812133—22no—
20#16812131—22no—
21#16812131C2211yescharacteristic
22#16812130—22no—
23C2×#13133807—22no—
24C2×#13133807C2212yescomplemented
25C2×#13133807C2212yescomplemented
26SD16×D107—22no—
27#16812115C2212yescomplemented
28#16812109C2211yescharacteristic, complemented
29#16812106C2211yescharacteristic
30#16812045C2211yescharacteristic, complemented
31#16812037C2212yescomplemented
32#16812027C2212yescomplemented
33#16812025C2212yescomplemented
34#135×C107C2211yescharacteristic
35#131×C107C2212yescomplemented
36#108×C107C2211yescharacteristic, complemented
37#104×C107C2212yescomplemented
38C4×C8×C107C2211yescharacteristic, complemented
39#13133819—22no—
40#13133818—22no—
41#13133818—22no—
42#13133818C2312yes—
43#13133817—22no—
44#13133817—42no—
45C2×Dic214—22no—
46C2×Dic214C2312yes—
47Q8×D107—22no—
48C2×D428—22no—
49C2×D428—41no—
50C2×D428C2312yes—
51D4×D107—22no—
52D4×D107—42no—
53Dic428—22no—
54Dic428D411yescharacteristic, complemented
55#13133808—22no—
56#13133807—22no—
57#13133807—22no—
58#13133807—22no—
59#13133807—22no—
60#13133807—22no—
61#13133807—22no—
62#13133807D412yescomplemented
63#13133806—22no—
64#13133805—22no—
65D856—22no—
66D856D411yescharacteristic, complemented
67#13133803—22no—
68#13133802C2312yes—
69#13133801—22no—
70C8×D107—22no—
71C4⋊Dic107—41no—
72#41×C107C2312yes—
73Q16×C107D411yescharacteristic, complemented
74SD16×C107D412yescomplemented
75D8×C107D411yescharacteristic, complemented
76M16×C107—22no—
77M16×C107C2312yes—
78C2×C8×C107—22no—
79C2×C8×C107C2311yescharacteristic
80C2×C8×C107C2311yescharacteristic
81C42×C107C2311yescharacteristic
82C22×D107—22no—
83C22×D107—42no—
84Q8×C107C2×D412yes—
85D4×C107—22no—
86D4×C107C2×D412yes—
87C2×C4×C107—21no—
88C2×C4×C107—22no—
89C2×C4×C107C2411yescharacteristic
90C22⋊D107—22no—
91C22⋊D107—42no—
92C22⋊D107—42no—
93C2×Dic107—22no—
94C2×Dic107—42no—
95D428—22no—
96D428—22no—
97D428—42no—
98D428—82no—
99D428C2×D412yes—
100C4×D107—22no—
101C4×D107—42no—
102C4×D107—42no—
103Dic214—22no—
104Dic214—22no—
105Dic214—42no—
106Dic214C2×D412yes—
107C8×C107—22no—
108C8×C107C2×D411yescharacteristic
109C8×C107C2×D411yescharacteristic
110C8×C107C2×D411yescharacteristic
111C8×C107C2×D411yescharacteristic
112C107⋊C8—22no—
113C22×C107—22no—
114C22×C107H(5,𝔽2)11yescharacteristic
115C2×D107—22no—
116C2×D107—42no—
117C2×D107—82no—
118C2×D107—162no—
119C4×C107—22no—
120C4×C107—22no—
121C4×C107C22×D411yescharacteristic
122C4×C107C22×D411yescharacteristic
123Dic107—22no—
124Dic107—42no—
125C2×C107—21no—
126C2×C107—42no—
127C2×C107D4211yescharacteristic
128D107—42no—
129D107—82no—
130#3291—1071nomaximal, Sylow
131C107#329111yescharacteristic, complemented, Sylow
132#577—1072no—
133#577—1072no—
134#492—1071no—
135#471—1072no—
136#471—1072no—
137#456—1072no—
138#453—1072no—
139#443—1071no—
140#443D10711yescharacteristic, complemented
1412−1+4—1072no—
142H(5,𝔽2)—1072no—
143#137—1072no—
144#137—1072no—
145#137—2142no—
146#137—2142no—
147AGL(1,ℤ/8)—1072no—
148AGL(1,ℤ/8)—1072no—
149AGL(1,ℤ/8)—2142no—
150AGL(1,ℤ/8)—2142no—
151#135—1071no—
152#135—2142no—
153#135—2142no—
154#135C2×D10711yescharacteristic
155C2×SD16—1072no—
156C2×SD16—1072no—
157C2×SD16—2142no—
158C2×SD16—2142no—
159#131—1072no—
160#131C2×D10712yescomplemented
161C4⋊Q8—1071no—
162#127—1071no—
163#108—1071no—
164#108C2×D10711yescharacteristic, complemented
165#104—1072no—
166#104C2×D10712yescomplemented
167#101—1072no—
168#100—1072no—
169C4×C8C2×D10711yescharacteristic, complemented
170#41—1072no—
171#41—4282no—
172#41—2142no—
173#41—2142no—
174#41—2142no—
175#41—2142no—
176#41C22×D10712yes—
177C2×Q8—1072no—
178C2×Q8—2142no—
179C2×Q8—2142no—
180C2×D4—1072no—
181C2×D4—4281no—
182C2×D4—4282no—
183C2×D4—2142no—
184C2×D4—2142no—
185Q16—1071no—
186Q16—2142no—
187Q16—2142no—
188Q16D42811yescharacteristic, complemented
189SD16—1072no—
190SD16—2142no—
191SD16—2142no—
192SD16—2142no—
193SD16—2142no—
194SD16—2142no—
195SD16—2142no—
196SD16—2142no—
197SD16—2142no—
198SD16D42812yescomplemented
199D8—1071no—
200D8—2142no—
201D8—2142no—
202D8D42811yescharacteristic, complemented
203M16—22no—
204M16—1072no—
205M16—2142no—
206M16C22×D10712yes—
207C2×C8—22no—
208C2×C8—2142no—
209C2×C8C22×D10711yescharacteristic
210C2×C8C22×D10711yescharacteristic
211C4⋊C4—4281no—
212C42C22×D10711yescharacteristic
213C23—4282no—
214C23—2142no—
215Q8—1072no—
216Q8—4282no—
217Q8—2142no—
218Q8—2142no—
219Q8C2×D42812yes—
220D4—22no—
221D4—8562no—
222D4—1072no—
223D4—4282no—
224D4—4282no—
225D4—4282no—
226D4—2142no—
227D4—2142no—
228D4—2142no—
229D4C2×D42812yes—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—4282no—
233C2×C4—4282no—
234C2×C4—4282no—
235C2×C4—2142no—
236C2×C4—2142no—
237C2×C4C23×D10711yescharacteristic
238C8—22no—
239C8—2142no—
240C8D4×D10711yescharacteristic
241C8D4×D10711yescharacteristic
242C8C2×D42811yescharacteristic
243C8C2×D42811yescharacteristic
244C22—22no—
245C22—8562no—
246C22—4282no—
247C22—17122no—
248C22—2142no—
249C22#1681215511yescharacteristic
250C4—22no—
251C4—22no—
252C4—4282no—
253C4—2142no—
254C4C2×D4×D10711yescharacteristic
255C4C22×D42811yescharacteristic
256C2—21no—
257C2—42no—
258C2—8562no—
259C2—4282no—
260C2D4×D42811yescharacteristic
subgroupG/H
centerC2D4×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#443×C107C2
socleC2×C107D42
hypercenter#443D107
hypocenterC107#3291
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C2×C4 ◁ #443

Complements

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

NH ≅ G/Ntype
1. #20293439C2semidirect
2. #20293434C2semidirect
3. #20293000C2semidirect
4. #20292891C2semidirect
5. #20292890C2semidirect
6. #20292811C2semidirect
7. #20292796C2semidirect
8. #20292745C2semidirect
9. #443×C107C2semidirect
11. #16812155C22semidirect
13. #16812142C22semidirect
15. #16812139C22semidirect
24. C2×#13133807C22semidirect
25. C2×#13133807C22semidirect
27. #16812115C22semidirect
28. #16812109C22semidirect
30. #16812045C22semidirect
31. #16812037C22semidirect
32. #16812027C22semidirect
33. #16812025C22semidirect
35. #131×C107C22semidirect
36. #108×C107C22semidirect
37. #104×C107C22semidirect
38. C4×C8×C107C22semidirect
54. Dic428D4semidirect
62. #13133807D4semidirect
66. D856D4semidirect
73. Q16×C107D4semidirect
74. SD16×C107D4semidirect
75. D8×C107D4semidirect
131. C107#3291semidirect
140. #443D107semidirect
160. #131C2×D107semidirect
164. #108C2×D107semidirect
166. #104C2×D107semidirect
169. C4×C8C2×D107semidirect
188. Q16D428semidirect
198. SD16D428semidirect
202. D8D428semidirect

Maximal subgroups

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

subgroupindex
#202934392
#202934342
#202930002
#202928912
#202928902
#202928112
#202927962
#202927452
#443×C1072
#3291107