Finite Group Databasebeta

← #28345835

Subgroups — #28345835

Order 13696.

736 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 145 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#20293463C211yesmaximal, characteristic, complemented
2C2×#16812137C212yesmaximal, complemented
3#20293191C211yesmaximal, characteristic, complemented
4#20292802C214yesmaximal, complemented
5#20292797C214yesmaximal, complemented
6#20292643C212yesmaximal, complemented
7#413×C107C211yesmaximal, characteristic, complemented
8#16812157C2212yescomplemented
9#16812156—22no—
10#16812155—22no—
11#16812154—22no—
12#16812154C2212yescomplemented
13C2×#13133818—22no—
14C2×#13133818C2212yes—
15#41×D107—24no—
16C22×D428C2211yescharacteristic, complemented
17#16812137—24no—
18#16812137C2218yescomplemented
19C2×#13133807—22no—
20C2×D856—22no—
21#16812105C2212yescomplemented
22#16812098C2212yescomplemented
23#16812083—22no—
24#16812077—22no—
25#16812068C2211yescharacteristic, complemented
26#16812035C2218yescomplemented
27C2×#41×C107C2211yescharacteristic
28C2×M16×C107C2212yescomplemented
29#101×C107C2212yescomplemented
30#100×C107C2212yescomplemented
31C23×D107—21no—
32#13133819—24no—
33#13133818—24no—
34#13133818—24no—
35#13133818—22no—
36#13133818—24no—
37#13133818—22no—
38#13133818D414yescomplemented
39#13133818C2314yes—
40#13133817—28no—
41#13133817—24no—
42C2×Dic214—22no—
43C2×Dic214D412yescomplemented
44Q8×D107—24no—
45C2×D428—22no—
46C2×D428—22no—
47C2×D428—22no—
48C2×D428—21no—
49C2×D428D412yescomplemented
50C2×D428C2312yes—
51C2×C22⋊D107—24no—
52C2×C22⋊D107—22no—
53C2×C22⋊D107—24no—
54D4×D107—28no—
55D4×D107—24no—
56C2×C4×D107—24no—
57C2×C4×D107—22no—
58#13133807—28no—
59D856—28no—
60C4⋊Dic107—21no—
61C22⋊Dic107—21no—
62(C2×D107)⋊C4—24no—
63C4×Dic107C2312yes—
64#41×C107—22no—
65#41×C107—22no—
66C2×Q8×C107C2311yescharacteristic
67C2×D4×C107D412yes—
68C2×D4×C107C2311yescharacteristic
69C22×C4×C107D412yes—
70C22×C4×C107C2311yescharacteristic
71M16×C107—22no—
72M16×C107C2314yes—
73C2×C8×C107—22no—
74C23×C107C2×D412yes—
75C23×C107C2×D411yescharacteristic
76C22×D107—24no—
77C22×D107—22no—
78C22×D107—22no—
79C22×D107—42no—
80C22×D107—82no—
81Q8×C107—22no—
82D4×C107—24no—
83D4×C107—22no—
84C2×C4×C107—22no—
85C2×C4×C107—22no—
86C2×C4×C107C2×D412yes—
87C2×C4×C107C2×D412yes—
88C2×C4×C107C2×D412yes—
89C2×C4×C107C2×D411yescharacteristic
90C2×C4×C107C2411yescharacteristic
91C22⋊D107—28no—
92C22⋊D107—24no—
93C22⋊D107—24no—
94C22⋊D107—84no—
95C2×Dic107—24no—
96C2×Dic107—22no—
97C2×Dic107—22no—
98D428—24no—
99D428—22no—
100D428—24no—
101D428—41no—
102D428—41no—
103D428C2×D414yes—
104C4×D107—28no—
105C4×D107—24no—
106C4×D107—24no—
107Dic214—24no—
108Dic214—22no—
109Dic214C2×D414yes—
110C8×C107—24no—
111C22×C107—22no—
112C22×C107—41no—
113C22×C107—42no—
114C22×C107#12012yes—
115C22×C107C22×D411yescharacteristic
116C2×D107—24no—
117C2×D107—44no—
118C2×D107—42no—
119C2×D107—82no—
120C2×D107—84no—
121C2×D107—84no—
122C4×C107—22no—
123C4×C107#12011yescharacteristic
124C4×C107#12011yescharacteristic
125C4×C107C22×D412yes—
126Dic107—24no—
127Dic107—42no—
128C2×C107—21no—
129C2×C107—22no—
130C2×C107—42no—
131C2×C107C2×#12011yescharacteristic
132D107—44no—
133D107—82no—
134#3021—1071nomaximal, Sylow
135C107#302111yescharacteristic, complemented, Sylow
136#585—1071no—
137C2×AGL(1,ℤ/8)—1072no—
138#535—1071no—
139#454—1074no—
140#453—1074no—
141#421—1072no—
142#413D10711yescharacteristic, complemented
1432−1+4—1072no—
1442−1+4—2142no—
145H(5,𝔽2)—1072no—
146H(5,𝔽2)—2142no—
147H(5,𝔽2)—2142no—
148C2×#41—1072no—
149C2×#41—2144no—
150C2×#41—2142no—
151C2×#41C2×D10711yescharacteristic
152C22×D4—1071no—
153AGL(1,ℤ/8)—1078no—
154AGL(1,ℤ/8)—2144no—
155C2×SD16—2142no—
156C2×D8—2142no—
157C2×M16C2×D10712yescomplemented
158#127—1072no—
159#124—1072no—
160#121—2142no—
161#120—2142no—
162#117—1071no—
163#104—1078no—
164#101C2×D10712yescomplemented
165#100C2×D10712yescomplemented
166C24—2141no—
167#41—22no—
168#41—22no—
169#41—1074no—
170#41—1074no—
171#41—2144no—
172#41—2144no—
173#41—2144no—
174#41—2148no—
175#41—2144no—
176#41—2142no—
177#41—2142no—
178#41—2144no—
179C2×Q8—1072no—
180C2×Q8—2142no—
181C2×Q8—2144no—
182C2×Q8C22×D10711yescharacteristic
183C2×D4—1072no—
184C2×D4—1072no—
185C2×D4—2142no—
186C2×D4—2144no—
187C2×D4—2148no—
188C2×D4—2144no—
189C2×D4—2142no—
190C2×D4—2144no—
191C2×D4—2142no—
192C2×D4—2142no—
193C2×D4—2141no—
194C2×D4D42812yes—
195C2×D4C22×D10711yescharacteristic
196C22×C4—2144no—
197C22×C4—2142no—
198C22×C4D42812yes—
199C22×C4C22×D10711yescharacteristic
200SD16—2148no—
201D8—2148no—
202M16—22no—
203M16C22×D10714yes—
204C2×C8—22no—
205C4⋊C4—2141no—
206C22⋊C4—2141no—
207C22⋊C4—2144no—
208C42—1072no—
209C23—8562no—
210C23—4282no—
211C23—2144no—
212C23—2142no—
213C23—2142no—
214C23C2×D42812yes—
215C23C2×D42811yescharacteristic
216Q8—22no—
217Q8—1074no—
218Q8—2144no—
219Q8—2142no—
220D4—24no—
221D4—22no—
222D4—8564no—
223D4—1074no—
224D4—4281no—
225D4—4281no—
226D4—2144no—
227D4—2148no—
228D4—2144no—
229D4—2144no—
230D4—2142no—
231D4—2144no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—2148no—
235C2×C4—2142no—
236C2×C4—2144no—
237C2×C4—2142no—
238C2×C4—2144no—
239C2×C4—2144no—
240C2×C4D4×D10712yes—
241C2×C4D4×D10712yes—
242C2×C4C2×D42812yes—
243C2×C4C2×D42811yescharacteristic
244C2×C4C23×D10711yescharacteristic
245C8—24no—
246C22—22no—
247C22—41no—
248C22—42no—
249C22—8564no—
250C22—8562no—
251C22—8564no—
252C22—4284no—
253C22—4282no—
254C22—2144no—
255C22#1681207912yes—
256C22C22×D42811yescharacteristic
257C4—22no—
258C4—4282no—
259C4—2144no—
260C4#1681207911yescharacteristic
261C4#1681207911yescharacteristic
262C4C2×D4×D10712yes—
263C2—21no—
264C2—22no—
265C2—42no—
266C2—8562no—
267C2—4284no—
268C2C2×#1681207911yescharacteristic
subgroupG/H
centerC2C2×#16812079
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#413×C107C2
socleC2×C107C2×#120
hypercenter#413D107
hypocenterC107#3021
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
1. #20293463C2semidirect
2. C2×#16812137C2semidirect
3. #20293191C2semidirect
4. #20292802C2semidirect
5. #20292797C2semidirect
6. #20292643C2semidirect
7. #413×C107C2semidirect
8. #16812157C22semidirect
12. #16812154C22semidirect
16. C22×D428C22semidirect
18. #16812137C22semidirect
21. #16812105C22semidirect
22. #16812098C22semidirect
25. #16812068C22semidirect
26. #16812035C22semidirect
28. C2×M16×C107C22semidirect
29. #101×C107C22semidirect
30. #100×C107C22semidirect
38. #13133818D4semidirect
43. C2×Dic214D4semidirect
49. C2×D428D4semidirect
135. C107#3021semidirect
142. #413D107semidirect
157. C2×M16C2×D107semidirect
164. #101C2×D107semidirect
165. #100C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202934632
C2×#168121372
#202931912
#202928022
#202927972
#202926432
#413×C1072
#3021107