Finite Group Databasebeta

← #28345310

Subgroups — #28345310

Order 13696.

528 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 175 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
1C2×D8×D107C211yesmaximal, characteristic
2#20293099C212yesmaximal, complemented
3#20292740C211yesmaximal, characteristic, complemented
4#20292739C211yesmaximal, characteristic, complemented
5#20292738C211yesmaximal, characteristic, complemented
6C4×#13133803C212yesmaximal, complemented
7#20292707C212yesmaximal, complemented
8#20292662C211yesmaximal, characteristic, complemented
9#20292630C212yesmaximal, complemented
10#20292584C211yesmaximal, characteristic, complemented
11#442×C107C211yesmaximal, characteristic, complemented
12C2×D4×D107C2212yes—
13C2×D856C2211yescharacteristic
14C2×#13133803C2212yes—
15D8×D107C418yescomplemented
16C2×#13133801C2211yescharacteristic
17C2×C8×D107C2211yescharacteristic
18C4×D428C2212yescomplemented
19D4×Dic107C2212yescomplemented
20#16812071C2212yescomplemented
21C4×C22⋊D107—22no—
22(C22⋊D107)⋊C4—22no—
23#16812067C2212yes—
24#16812066C2211yescharacteristic
25C2×(C2×D107)⋊C4—22no—
26C22⋊C4×D107—22no—
27#16812042C2212yescomplemented
28#16812041C2211yescharacteristic, complemented
29#16812031C2212yescomplemented
30#16812030C2212yescomplemented
31#16812029C2212yescomplemented
32#16812019C2211yescharacteristic, complemented
33C4×C107⋊C8C2211yescharacteristic, complemented
34C8×Dic107C2211yescharacteristic, complemented
35C2×D8×C107C2211yescharacteristic
36C4×D4×C107C2212yescomplemented
37#106×C107C2211yescharacteristic, complemented
38#102×C107C2212yescomplemented
39#97×C107C2211yescharacteristic, complemented
40C23×D107—22no—
41C2×D428C2312yes—
42C2×C22⋊D107—22no—
43D4×D107—24no—
44D4×D107C2×C418yescomplemented
45C22×Dic107—22no—
46C2×C4×D107—22no—
47C2×C4×D107—22no—
48C2×C4×D107C2311yescharacteristic
49D856C2×C414yescomplemented
50#13133803C2×C418yescomplemented
51#13133801—22no—
52C2×C107⋊C8C2311yescharacteristic
53C2×C107⋊C8C2311yescharacteristic
54C8×D107C2×C414yescomplemented
55C4⋊Dic107C2312yes—
56Dic107⋊C4D412yes—
57C22⋊Dic107—22no—
58(C2×D107)⋊C4—22no—
59(C2×D107)⋊C4—22no—
60(C2×D107)⋊C4—22no—
61(C2×D107)⋊C4D412yes—
62C4×Dic107—22no—
63C4×Dic107C2311yescharacteristic
64C2×D4×C107C2312yes—
65C22×C4×C107—22no—
66D8×C107C2×C414yescomplemented
67C2×C8×C107C2311yescharacteristic
68C2×C8×C107C2311yescharacteristic
69C4⋊C4×C107C2312yes—
70C22⋊C4×C107—22no—
71C42×C107C2311yescharacteristic
72C23×C107—22no—
73C22×D107—22no—
74C22×D107—28no—
75C22×D107—82no—
76C22×D107—82no—
77C22×D107C2×D411yescharacteristic
78D4×C107—22no—
79D4×C107C22×C414yes—
80C2×C4×C107—22no—
81C2×C4×C107—82no—
82C2×C4×C107C2×D412yes—
83C2×C4×C107C2411yescharacteristic
84C22⋊D107—28no—
85C2×Dic107—22no—
86C2×Dic107—82no—
87C2×Dic107C2×D412yes—
88C2×Dic107C2×D411yescharacteristic
89D428—22no—
90D428C22×C414yes—
91C4×D107—82no—
92C4×D107—82no—
93C4×D107C22×C412yes—
94C4×D107#4112yes—
95C8×C107—21no—
96C8×C107C22×C412yes—
97C107⋊C8—21no—
98C107⋊C8C22×C412yes—
99C22×C107—24no—
100C22×C107—42no—
101C22×C107—42no—
102C22×C107C22×D411yescharacteristic
103C2×D107—22no—
104C2×D107—24no—
105C2×D107—42no—
106C2×D107—42no—
107C2×D107—88no—
108C2×D107C4×D412yes—
109C4×C107—22no—
110C4×C107—42no—
111C4×C107C23×C411yescharacteristic
112C4×C107C2×#4111yescharacteristic
113Dic107—22no—
114Dic107—42no—
115Dic107C4×D412yes—
116C2×C107—44no—
117C2×C107C2×C4×D411yescharacteristic
118C2×C107#57612yes—
119D107—22no—
120D107—44no—
121#2948—1071nomaximal, Sylow
122C107#294811yescharacteristic, complemented, Sylow
123C22×D8—1071no—
124#518—1072no—
125#442—1071no—
126#442—1071no—
127#442—1071no—
128#442D10711yescharacteristic, complemented
129C4×D8—1072no—
130C4×D8—1072no—
131#427—1071no—
132#418—1072no—
133#405—1071no—
134C22×D4—1072no—
135C2×D8—1072no—
136C2×D8—1078no—
137C2×D8—1071no—
138C2×D8C2×D10711yescharacteristic
139C2×M16—1071no—
140C22×C8—1071no—
141C4×D4—1072no—
142C4×D4—1072no—
143C4×D4—1072no—
144C4×D4—2142no—
145C4×D4—2142no—
146C4×D4C2×D10712yescomplemented
147#117—1071no—
148#117—1072no—
149C2×C22⋊C4—2142no—
150C2×C22⋊C4—2142no—
151#107—1072no—
152#106—1071no—
153#106C2×D10711yescharacteristic, complemented
154#102—1072no—
155#102—1072no—
156#102—1072no—
157#102C2×D10712yescomplemented
158#97—1071no—
159#97C2×D10711yescharacteristic, complemented
160C4×C8—1071no—
161C4×C8—1071no—
162C24—2142no—
163C2×D4—1078no—
164C2×D4—1072no—
165C2×D4—2144no—
166C2×D4—2142no—
167C2×D4C22×D10712yes—
168C22×C4—22no—
169C22×C4—1071no—
170C22×C4—2142no—
171C22×C4—2142no—
172C22×C4—2142no—
173D8—1078no—
174D8—1074no—
175D8C4×D10714yescomplemented
176M16—2142no—
177C2×C8—1071no—
178C2×C8—1074no—
179C2×C8—1071no—
180C2×C8C22×D10711yescharacteristic
181C2×C8C22×D10711yescharacteristic
182C4⋊C4—1072no—
183C4⋊C4—1072no—
184C4⋊C4C22×D10712yes—
185C22⋊C4—22no—
186C22⋊C4—1072no—
187C22⋊C4—2142no—
188C22⋊C4—2142no—
189C22⋊C4—2142no—
190C22⋊C4—2142no—
191C42—1071no—
192C42—2142no—
193C42C22×D10711yescharacteristic
194C23—22no—
195C23—8562no—
196C23—8562no—
197C23—1071no—
198C23—2148no—
199C23—2142no—
200D4—22no—
201D4—1074no—
202D4—2148no—
203D4—2142no—
204D4C2×C4×D10714yes—
205C2×C4—22no—
206C2×C4—82no—
207C2×C4—8562no—
208C2×C4—8562no—
209C2×C4—8562no—
210C2×C4—1071no—
211C2×C4—1072no—
212C2×C4—1072no—
213C2×C4—1072no—
214C2×C4—2142no—
215C2×C4D4×D10712yes—
216C2×C4C23×D10711yescharacteristic
217C8—21no—
218C8—1072no—
219C8—2141no—
220C8C2×C4×D10712yes—
221C22—24no—
222C22—42no—
223C22—42no—
224C22—8568no—
225C22—1072no—
226C22—4282no—
227C22—4282no—
228C22—2144no—
229C22—2142no—
230C22C2×D4×D10711yescharacteristic
231C4—22no—
232C4—42no—
233C4—1072no—
234C4—4282no—
235C4—2142no—
236C4C22×C4×D10711yescharacteristic
237C4#41×D10711yescharacteristic
238C2—44no—
239C2—4284no—
240C2—2142no—
241C2C4×D4×D10711yescharacteristic
242C2#2029343112yes—
subgroupG/H
centerC22C2×D4×D107
commutatorC4×C107C23×C4
FrattiniC2×C4C23×D107
Fitting#442×C107C2
socleC22×C107C22×D4
prime radicalC2×D8×D107C2
cosocleC2×C4×C107C24
hypercenter#442D107
hypocenterC107#2948
LCSG ▷ C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
2. #20293099C2semidirect
3. #20292740C2semidirect
4. #20292739C2semidirect
5. #20292738C2semidirect
6. C4×#13133803C2semidirect
7. #20292707C2semidirect
8. #20292662C2semidirect
9. #20292630C2semidirect
10. #20292584C2semidirect
11. #442×C107C2semidirect
15. D8×D107C4semidirect
18. C4×D428C22semidirect
19. D4×Dic107C22semidirect
20. #16812071C22semidirect
27. #16812042C22semidirect
28. #16812041C22semidirect
29. #16812031C22semidirect
30. #16812030C22semidirect
31. #16812029C22semidirect
32. #16812019C22semidirect
33. C4×C107⋊C8C22semidirect
34. C8×Dic107C22semidirect
36. C4×D4×C107C22semidirect
37. #106×C107C22semidirect
38. #102×C107C22semidirect
39. #97×C107C22semidirect
44. D4×D107C2×C4semidirect
49. D856C2×C4semidirect
50. #13133803C2×C4semidirect
54. C8×D107C2×C4semidirect
66. D8×C107C2×C4semidirect
122. C107#2948semidirect
128. #442D107semidirect
146. C4×D4C2×D107semidirect
153. #106C2×D107semidirect
157. #102C2×D107semidirect
159. #97C2×D107semidirect
175. D8C4×D107semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D1072
#202930992
#202927402
#202927392
#202927382
C4×#131338032
#202927072
#202926622
#202926302
#202925842
#442×C1072
#2948107