Finite Group Databasebeta

← #28348286

Subgroups — #28348286

Order 13696.

462 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 139 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#20293334C211yesmaximal, characteristic, complemented
2D4×D428C211yesmaximal, characteristic
3#20292974C211yesmaximal, characteristic, complemented
4#20292946C212yesmaximal, complemented
5#20292918C211yesmaximal, characteristic, complemented
6#20292856C212yesmaximal, complemented
7#20292828C211yesmaximal, characteristic, complemented
8#20292780C212yesmaximal, complemented
9C4×#13133807C211yesmaximal, characteristic, complemented
10C2×#16812031C212yesmaximal, complemented
11C8×D4×C107C211yesmaximal, characteristic, complemented
12C22×D428C2212yes—
13C2×D4×D107—21no—
14C2×#13133807C2211yescharacteristic
15C2×#13133807C2212yescomplemented
16#16812109C2211yescharacteristic, complemented
17#16812106C2211yescharacteristic
18#16812100—21no—
19#16812088C2212yescomplemented
20#16812086—22no—
21#16812084—21no—
22#16812083C2212yes—
23#16812079—22no—
24C4×Dic214C2211yescharacteristic, complemented
25C4×D428C2211yescharacteristic
26D4×Dic107—21no—
27C2×C4⋊Dic107C2212yescomplemented
28#16812041C2212yescomplemented
29#16812041C2211yescharacteristic, complemented
30#16812034C2211yescharacteristic, complemented
31#16812031C2211yescharacteristic
32#16812031C2212yes—
33#16812031C2212yescomplemented
34#16812031C2212yescomplemented
35#16812031C2212yescomplemented
36C22×C8×C107C2212yescomplemented
37C4×D4×C107C2211yescharacteristic
38C4⋊C8×C107C2211yescharacteristic, complemented
39C22⋊C8×C107C2212yescomplemented
40C4×C8×C107C2211yescharacteristic, complemented
41C23×D107—22no—
42C2×Dic214—21no—
43C2×Dic214C2311yescharacteristic
44C2×D428—24no—
45C2×D428—22no—
46C2×D428—21no—
47C2×D428—21no—
48C2×D428—42no—
49C2×D428C2312yes—
50C2×D428C2311yescharacteristic
51C2×C22⋊D107—22no—
52D4×D107—44no—
53C22×Dic107—22no—
54C2×C4×D107—21no—
55#13133807—22no—
56#13133807—42no—
57C4⋊Dic107—22no—
58C4⋊Dic107—21no—
59C4⋊Dic107C2311yescharacteristic
60C4⋊Dic107C2311yescharacteristic
61C4⋊Dic107C2312yes—
62Dic107⋊C4—21no—
63Dic107⋊C4—22no—
64C22⋊Dic107—22no—
65(C2×D107)⋊C4—21no—
66(C2×D107)⋊C4—22no—
67C4×Dic107—21no—
68C2×D4×C107D411yescharacteristic
69C22×C4×C107C2312yes—
70C2×C8×C107—42no—
71C2×C8×C107C2311yescharacteristic
72C2×C8×C107C2311yescharacteristic
73C2×C8×C107C2312yes—
74C4⋊C4×C107D411yescharacteristic
75C22⋊C4×C107D412yes—
76C42×C107C2311yescharacteristic
77C23×C107C2×D412yes—
78C22×D107—21no—
79C22×D107—22no—
80C22×D107—44no—
81C22×D107—44no—
82C22×D107—82no—
83D4×C107SD1614yescomplemented
84C2×C4×C107—22no—
85C2×C4×C107—22no—
86C2×C4×C107C2×D411yescharacteristic
87C2×C4×C107C2×D412yes—
88C2×C4×C107C2×D411yescharacteristic
89C2×C4×C107C2411yescharacteristic
90C22⋊D107—44no—
91C2×Dic107—21no—
92C2×Dic107—21no—
93C2×Dic107—21no—
94C2×Dic107—22no—
95C2×Dic107—82no—
96D428—22no—
97D428—21no—
98D428—22no—
99D428—44no—
100D428—42no—
101D428—81no—
102D428#4112yes—
103C4×D107—42no—
104Dic214—21no—
105Dic214—21no—
106Dic214—81no—
107C8×C107—21no—
108C8×C107—21no—
109C8×C107—22no—
110C22×C107—22no—
111C22×C107—22no—
112C22×C107C2×SD1614yes—
113C22×C107C22×D411yescharacteristic
114C2×D107—22no—
115C2×D107—41no—
116C2×D107—41no—
117C2×D107—42no—
118C2×D107—42no—
119C2×D107—42no—
120C2×D107—88no—
121C4×C107—22no—
122C4×C107—21no—
123C4×C107C2×SD1612yes—
124C4×C107C2×#4111yescharacteristic
125C4×C107H(5,𝔽2)11yescharacteristic
126Dic107—41no—
127Dic107—41no—
128Dic107—42no—
129Dic107—41no—
130C2×C107—24no—
131C2×C107#54611yescharacteristic
132C2×C107C22×SD1611yescharacteristic
133C2×C107#57611yescharacteristic
134D107—42no—
135D107—82no—
136#3295—1071nomaximal, Sylow
137C107#329511yescharacteristic, complemented, Sylow
138#554—1071no—
139D42—1071no—
140#486—1071no—
141#481—1072no—
142#476—1071no—
143#465—1072no—
144#460—1071no—
145#450—1072no—
146C4×SD16—1071no—
147C8×D4D10711yescharacteristic, complemented
148C2×#102—1072no—
149C22×D4—1072no—
150C22×D4—2141no—
151C2×SD16—1072no—
152C2×SD16—1071no—
153C22×C8C2×D10712yescomplemented
154C4⋊Q8—1071no—
155#127—1071no—
156#125—2141no—
157C22⋊Q8—1072no—
158C22⋊Q8—2142no—
159#121—1072no—
160#121—2141no—
161#120—2142no—
162C4×Q8—1071no—
163C4×D4—1071no—
164C4×D4—2141no—
165C4×D4C2×D10711yescharacteristic
166C2×C4⋊C4—1072no—
167#106—1071no—
168#106—1072no—
169C4⋊C8C2×D10711yescharacteristic, complemented
170#103—1071no—
171#102—1071no—
172#102—1072no—
173#102—1072no—
174#102—1072no—
175#102—1072no—
176C22⋊C8C2×D10712yescomplemented
177C4×C8C2×D10711yescharacteristic, complemented
178C24—2142no—
179C2×Q8—1071no—
180C2×Q8—2141no—
181C2×D4—1071no—
182C2×D4—1072no—
183C2×D4—4282no—
184C2×D4—4284no—
185C2×D4—2144no—
186C2×D4—2142no—
187C2×D4—2141no—
188C2×D4—2142no—
189C2×D4—2141no—
190C2×D4D42811yescharacteristic
191C22×C4—2142no—
192C22×C4—2141no—
193C22×C4C22×D10712yes—
194SD16—4282no—
195SD16—2142no—
196C2×C8—42no—
197C2×C8C22×D10711yescharacteristic
198C2×C8C22×D10711yescharacteristic
199C2×C8C22×D10712yes—
200C4⋊C4—1071no—
201C4⋊C4—1072no—
202C4⋊C4—1071no—
203C4⋊C4—2142no—
204C4⋊C4—2141no—
205C4⋊C4—2141no—
206C4⋊C4—2142no—
207C4⋊C4D42811yescharacteristic
208C22⋊C4—2142no—
209C22⋊C4—2141no—
210C22⋊C4—2142no—
211C22⋊C4D42812yes—
212C42—2141no—
213C42C22×D10711yescharacteristic
214C23—8562no—
215C23—4284no—
216C23—4284no—
217C23—2142no—
218C23—2141no—
219C23C2×D42812yes—
220Q8—8561no—
221Q8—2141no—
222Q8—2141no—
223D4—8561no—
224D4—1072no—
225D4—4284no—
226D4—4284no—
227D4—4282no—
228D4—2142no—
229D4—2142no—
230D4—2141no—
231D4#1313380714yescomplemented
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—8562no—
235C2×C4—4282no—
236C2×C4—2141no—
237C2×C4—2141no—
238C2×C4—2141no—
239C2×C4—2142no—
240C2×C4C2×D42811yescharacteristic
241C2×C4C2×D42812yes—
242C2×C4C2×D42811yescharacteristic
243C2×C4C23×D10711yescharacteristic
244C8—21no—
245C8—21no—
246C8—22no—
247C22—22no—
248C22—22no—
249C22—8568no—
250C22—4282no—
251C22—4282no—
252C22—4282no—
253C22—4281no—
254C22—4281no—
255C22—2142no—
256C22C2×#1313380714yes—
257C22C22×D42811yescharacteristic
258C4—21no—
259C4—22no—
260C4—4281no—
261C4—4281no—
262C4—4282no—
263C4—4281no—
264C4C2×#1313380712yes—
265C4#41×D10711yescharacteristic
266C4#1681215411yescharacteristic
267C2—24no—
268C2—8562no—
269C2—4282no—
270C2#2029327211yescharacteristic
271C2C22×#1313380711yescharacteristic
272C2#2029343211yescharacteristic
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC8×D4×C107C2
socleC22×C107C22×D4
prime radicalD4×D428C2
hypercenterC8×D4D107
hypocenterC107#3295
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #20293334C2semidirect
3. #20292974C2semidirect
4. #20292946C2semidirect
5. #20292918C2semidirect
6. #20292856C2semidirect
7. #20292828C2semidirect
8. #20292780C2semidirect
9. C4×#13133807C2semidirect
10. C2×#16812031C2semidirect
11. C8×D4×C107C2semidirect
15. C2×#13133807C22semidirect
16. #16812109C22semidirect
19. #16812088C22semidirect
24. C4×Dic214C22semidirect
27. C2×C4⋊Dic107C22semidirect
28. #16812041C22semidirect
29. #16812041C22semidirect
30. #16812034C22semidirect
33. #16812031C22semidirect
34. #16812031C22semidirect
35. #16812031C22semidirect
36. C22×C8×C107C22semidirect
38. C4⋊C8×C107C22semidirect
39. C22⋊C8×C107C22semidirect
40. C4×C8×C107C22semidirect
83. D4×C107SD16semidirect
137. C107#3295semidirect
147. C8×D4D107semidirect
153. C22×C8C2×D107semidirect
169. C4⋊C8C2×D107semidirect
176. C22⋊C8C2×D107semidirect
177. C4×C8C2×D107semidirect
231. D4#13133807semidirect

Maximal subgroups

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

subgroupindex
#202933342
D4×D4282
#202929742
#202929462
#202929182
#202928562
#202928282
#202927802
C4×#131338072
C2×#168120312
C8×D4×C1072
#3295107