Finite Group Databasebeta

← #28345832

Subgroups — #28345832

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×#16812135C212yesmaximal, complemented
3#20292799C214yesmaximal, complemented
4#20292794C214yesmaximal, complemented
5#20292645C212yesmaximal, complemented
6#20292611C211yesmaximal, characteristic, complemented
7#535×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—
16#16812135—24no—
17#16812135C2218yescomplemented
18C2×#13133806—22no—
19C2×#13133803—22no—
20C2×#13133802C2212yescomplemented
21#16812037C2218yescomplemented
22#16812028C2212yescomplemented
23#16812026C2212yescomplemented
24C2×#41×C107C2211yescharacteristic
25C22×D4×C107C2211yescharacteristic, complemented
26#127×C107C2212yescomplemented
27#124×C107C2212yescomplemented
28#121×C107—22no—
29#120×C107—22no—
30#117×C107C2211yescharacteristic, complemented
31#13133819—24no—
32#13133818—24no—
33#13133818—24no—
34#13133818—22no—
35#13133818—24no—
36#13133818—22no—
37#13133818D414yescomplemented
38#13133818C2314yes—
39#13133817—28no—
40#13133817—24no—
41C2×Dic214—22no—
42C2×Dic214D412yescomplemented
43Q8×D107—24no—
44C2×D428—22no—
45C2×D428D412yescomplemented
46C2×C22⋊D107—24no—
47C2×C22⋊D107—22no—
48D4×D107—28no—
49D4×D107—24no—
50C2×C4×D107—24no—
51C2×C4×D107—22no—
52#13133806—28no—
53#13133803—28no—
54#13133802—22no—
55#13133802C2314yes—
56C2×C107⋊C8—22no—
57C24×C107—21no—
58#41×C107—22no—
59#41×C107—22no—
60C2×Q8×C107C2311yescharacteristic
61C2×D4×C107—24no—
62C2×D4×C107—22no—
63C2×D4×C107—22no—
64C2×D4×C107—21no—
65C2×D4×C107D412yes—
66C2×D4×C107C2312yes—
67C2×D4×C107C2311yescharacteristic
68C22×C4×C107D412yes—
69C22×C4×C107C2311yescharacteristic
70C4⋊C4×C107—21no—
71C22⋊C4×C107—21no—
72C22⋊C4×C107—24no—
73C42×C107C2312yes—
74C23×C107—22no—
75C23×C107—42no—
76C23×C107—82no—
77C23×C107C2×D412yes—
78C23×C107C2×D411yescharacteristic
79C22×D107—24no—
80C22×D107—22no—
81Q8×C107—22no—
82D4×C107—24no—
83D4×C107—22no—
84D4×C107—24no—
85D4×C107—41no—
86D4×C107—41no—
87D4×C107—84no—
88C2×C4×C107—22no—
89C2×C4×C107—22no—
90C2×C4×C107—22no—
91C2×C4×C107C2×D412yes—
92C2×C4×C107C2×D412yes—
93C2×C4×C107C2×D412yes—
94C2×C4×C107C2×D411yescharacteristic
95C2×C4×C107C2411yescharacteristic
96C22⋊D107—28no—
97C22⋊D107—24no—
98C22⋊D107—24no—
99C2×Dic107—24no—
100C2×Dic107—22no—
101D428—24no—
102D428—22no—
103D428C2×D414yes—
104C4×D107—28no—
105C4×D107—24no—
106C4×D107—24no—
107Dic214—24no—
108Dic214—22no—
109Dic214C2×D414yes—
110C107⋊C8—24no—
111C22×C107—22no—
112C22×C107—41no—
113C22×C107—42no—
114C22×C107—44no—
115C22×C107—42no—
116C22×C107—84no—
117C22×C107#12012yes—
118C22×C107C22×D411yescharacteristic
119C2×D107—24no—
120C2×D107—82no—
121C2×D107—84no—
122C4×C107—22no—
123C4×C107—42no—
124C4×C107#12011yescharacteristic
125C4×C107#12011yescharacteristic
126C4×C107C22×D412yes—
127Dic107—24no—
128C2×C107—21no—
129C2×C107—22no—
130C2×C107—42no—
131C2×C107—82no—
132C2×C107C2×#12011yescharacteristic
133D107—44no—
134#3021—1071nomaximal, Sylow
135C107#302111yescharacteristic, complemented, Sylow
136#585—1071no—
137C2×AGL(1,ℤ/8)—1072no—
138#535D10711yescharacteristic, complemented
139#454—1074no—
140#453—1074no—
141#421—1072no—
142#413—1071no—
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×D4C2×D10711yescharacteristic, complemented
153AGL(1,ℤ/8)—1078no—
154AGL(1,ℤ/8)—2144no—
155C2×SD16—2142no—
156C2×D8—2142no—
157C2×M16—1072no—
158#127C2×D10712yescomplemented
159#124C2×D10712yescomplemented
160#121—22no—
161#120—22no—
162#117C2×D10711yescharacteristic, complemented
163#104—1078no—
164#101—1072no—
165#100—1072no—
166C24—21no—
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—24no—
184C2×D4—22no—
185C2×D4—22no—
186C2×D4—21no—
187C2×D4—1072no—
188C2×D4—2142no—
189C2×D4—2144no—
190C2×D4—2148no—
191C2×D4—2142no—
192C2×D4—2144no—
193C2×D4C22⋊D10712yes—
194C2×D4C22×D10712yes—
195C2×D4C22×D10711yescharacteristic
196C22×C4—2144no—
197C22×C4—2142no—
198C22×C4C22⋊D10712yes—
199C22×C4C22×D10711yescharacteristic
200SD16—2148no—
201D8—2148no—
202M16—1074no—
203M16—2142no—
204C2×C8—2142no—
205C4⋊C4—21no—
206C22⋊C4—21no—
207C22⋊C4—24no—
208C42C22×D10712yes—
209C23—22no—
210C23—42no—
211C23—82no—
212C23—2144no—
213C23—2142no—
214C23C2×C22⋊D10712yes—
215C23C2×C22⋊D10711yescharacteristic
216Q8—22no—
217Q8—1074no—
218Q8—2144no—
219Q8—2142no—
220D4—24no—
221D4—22no—
222D4—24no—
223D4—41no—
224D4—41no—
225D4—84no—
226D4—1074no—
227D4—2144no—
228D4—2148no—
229D4—2144no—
230D4—2144no—
231D4—2142no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—2148no—
236C2×C4—2144no—
237C2×C4—2142no—
238C2×C4—2144no—
239C2×C4—2144no—
240C2×C4D4×D10712yes—
241C2×C4D4×D10712yes—
242C2×C4C2×C22⋊D10712yes—
243C2×C4C2×C22⋊D10711yescharacteristic
244C2×C4C23×D10711yescharacteristic
245C8—2144no—
246C22—22no—
247C22—41no—
248C22—42no—
249C22—44no—
250C22—42no—
251C22—84no—
252C22—8562no—
253C22—8564no—
254C22—2144no—
255C22#1681207712yes—
256C22C22×C22⋊D10711yescharacteristic
257C4—22no—
258C4—42no—
259C4—2144no—
260C4#1681207711yescharacteristic
261C4#1681207711yescharacteristic
262C4C2×D4×D10712yes—
263C2—21no—
264C2—22no—
265C2—42no—
266C2—82no—
267C2—4284no—
268C2C2×#1681207711yescharacteristic
subgroupG/H
centerC2C2×#16812077
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#535×C107C2
socleC2×C107C2×#120
hypercenter#535D107
hypocenterC107#3021
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22×C4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #20293463C2semidirect
2. C2×#16812135C2semidirect
3. #20292799C2semidirect
4. #20292794C2semidirect
5. #20292645C2semidirect
6. #20292611C2semidirect
7. #535×C107C2semidirect
8. #16812157C22semidirect
12. #16812154C22semidirect
17. #16812135C22semidirect
20. C2×#13133802C22semidirect
21. #16812037C22semidirect
22. #16812028C22semidirect
23. #16812026C22semidirect
25. C22×D4×C107C22semidirect
26. #127×C107C22semidirect
27. #124×C107C22semidirect
30. #117×C107C22semidirect
37. #13133818D4semidirect
42. C2×Dic214D4semidirect
45. C2×D428D4semidirect
135. C107#3021semidirect
138. #535D107semidirect
152. C22×D4C2×D107semidirect
158. #127C2×D107semidirect
159. #124C2×D107semidirect
162. #117C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202934632
C2×#168121352
#202927992
#202927942
#202926452
#202926112
#535×C1072
#3021107