Finite Group Databasebeta

← #28348080

Subgroups — #28348080

Order 13696.

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

normal subgroups only

This group has 556 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
1C2×#16812133C212yesmaximal, complemented
2#20293297C211yesmaximal, characteristic, complemented
3#20293262C211yesmaximal, characteristic, complemented
4#20292877C212yesmaximal, complemented
5#20292824C211yesmaximal, characteristic, complemented
6#20292819C211yesmaximal, characteristic, complemented
7#20292769C212yesmaximal, complemented
8#20292759C212yesmaximal, complemented
9C4×#13133807C211yesmaximal, characteristic, complemented
10#20292706C211yesmaximal, characteristic, complemented
11#493×C107C211yesmaximal, characteristic, complemented
12C2×#13133817C2212yescomplemented
13C2×D4×D107—21no—
14C2×D4×D107C2212yes—
15#16812133—28no—
16C2×#13133807C2211yescharacteristic
17C2×#13133806C2212yescomplemented
18C2×#13133803C2212yes—
19C2×#13133801C2212yescomplemented
20#16812105—21no—
21#16812081—22no—
22#16812080—21no—
23#16812080C2212yes—
24#16812080C2212yescomplemented
25#16812077—22no—
26C4×Dic214C2211yescharacteristic, complemented
27C4×D428C2211yescharacteristic
28C4×D428C2211yescharacteristic
29D4×Dic107—21no—
30#16812041C2211yescharacteristic, complemented
31C4⋊(C107⋊C8)C2211yescharacteristic, complemented
32#16812034C2211yescharacteristic, complemented
33#16812031C2211yescharacteristic
34#16812030C2212yescomplemented
35#16812030C2212yescomplemented
36(C2×D107)⋊C8C2212yescomplemented
37C2×D8×C107C2212yescomplemented
38C2×D8×C107C2212yescomplemented
39#127×C107C2211yescharacteristic
40#127×C107C2211yescharacteristic, complemented
41C4×C8×C107C2211yescharacteristic, complemented
42C23×D107—22no—
43#13133817—24no—
44#13133817—42no—
45C2×Dic214C2311yescharacteristic
46C2×D428D411yescharacteristic
47C2×D428C2311yescharacteristic
48C2×C22⋊D107—22no—
49C2×C22⋊D107—22no—
50C2×C22⋊D107—22no—
51C2×C22⋊D107—22no—
52D4×D107—24no—
53D4×D107—42no—
54D4×D107—44no—
55C22×Dic107—22no—
56C2×C4×D107—21no—
57C2×C4×D107C2312yes—
58#13133807D414yescomplemented
59#13133806—24no—
60#13133803—24no—
61#13133801—24no—
62C2×C107⋊C8C2312yes—
63C4⋊Dic107D411yescharacteristic
64C4⋊Dic107C2311yescharacteristic
65C4⋊Dic107C2311yescharacteristic
66Dic107⋊C4—21no—
67C22⋊Dic107—22no—
68C22⋊Dic107—22no—
69(C2×D107)⋊C4—21no—
70(C2×D107)⋊C4D412yes—
71C4×Dic107—21no—
72C2×D4×C107—21no—
73C2×D4×C107—21no—
74C2×D4×C107—21no—
75C2×D4×C107—21no—
76C2×D4×C107C2312yes—
77C2×D4×C107C2312yes—
78D8×C107—24no—
79D8×C107—42no—
80C2×C8×C107C2311yescharacteristic
81C2×C8×C107C2311yescharacteristic
82C42×C107C2311yescharacteristic
83C23×C107—22no—
84C23×C107—22no—
85C22×D107—21no—
86C22×D107—44no—
87C22×D107—44no—
88C22×D107—82no—
89C22×D107C2×D412yes—
90D4×C107—22no—
91D4×C107—22no—
92D4×C107—22no—
93D4×C107—22no—
94D4×C107—42no—
95D4×C107—42no—
96D4×C107—81no—
97D4×C107—81no—
98C2×C4×C107C2×D411yescharacteristic
99C2×C4×C107C2×D411yescharacteristic
100C2×C4×C107C2411yescharacteristic
101C22⋊D107—44no—
102C22⋊D107—44no—
103C22⋊D107—44no—
104C22⋊D107—82no—
105C2×Dic107—21no—
106C2×Dic107—21no—
107C2×Dic107—21no—
108C2×Dic107—44no—
109C2×Dic107C2×D412yes—
110D428—22no—
111D428—21no—
112D428C2×D412yes—
113C4×D107—22no—
114C4×D107—22no—
115C4×D107—42no—
116Dic214—21no—
117Dic214C2×D412yescomplemented
118C8×C107—21no—
119C8×C107C2×D412yescomplemented
120C107⋊C8—22no—
121C22×C107—42no—
122C22×C107—42no—
123C22×C107—42no—
124C22×C107—42no—
125C22×C107—42no—
126C22×C107—42no—
127C22×C107C22×D411yescharacteristic
128C2×D107—22no—
129C2×D107—22no—
130C2×D107—22no—
131C2×D107—22no—
132C2×D107—41no—
133C2×D107—41no—
134C2×D107—88no—
135C4×C107—21no—
136C4×C107AGL(1,ℤ/8)12yescomplemented
137C4×C107C22×D411yescharacteristic
138C4×C107H(5,𝔽2)11yescharacteristic
139Dic107—22no—
140Dic107—22no—
141Dic107—41no—
142Dic107—41no—
143C2×C107—82no—
144C2×C107—82no—
145C2×C107D4211yescharacteristic
146C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
147C2×C107#57611yescharacteristic
148D107—42no—
149D107—42no—
150#3274—1071nomaximal, Sylow
151C107#327411yescharacteristic, complemented, Sylow
152C2×AGL(1,ℤ/8)—1072no—
153#550—1071no—
154D42—1071no—
155#493D10711yescharacteristic, complemented
156#469—1072no—
157#460—1071no—
158#459—1071no—
159#449—1072no—
160#447—1072no—
161C4×SD16—1071no—
162#436—1071no—
163C2×#41—1072no—
164C22×D4—1072no—
165C22×D4—2141no—
166AGL(1,ℤ/8)—2148no—
167C2×SD16—1072no—
168C2×SD16—1071no—
169C2×D8—1072no—
170C2×D8C2×D10712yescomplemented
171C2×D8C2×D10712yescomplemented
172C2×M16—1072no—
173#127—2141no—
174#127C2×D10711yescharacteristic
175#127C2×D10711yescharacteristic, complemented
176#121—1072no—
177#121—1072no—
178#121—2142no—
179#121—2141no—
180#120—2142no—
181C4×Q8—1071no—
182C4×D4—1071no—
183C4×D4—1071no—
184C4×D4—2141no—
185#106—1071no—
186C4⋊C8—1071no—
187#103—1071no—
188#102—1072no—
189#102—1072no—
190#102—1071no—
191C22⋊C8—1072no—
192C4×C8C2×D10711yescharacteristic, complemented
193C24—2142no—
194#41—4282no—
195#41—2144no—
196C2×Q8—1071no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—1071no—
202C2×D4—1071no—
203C2×D4—4284no—
204C2×D4—4282no—
205C2×D4—2142no—
206C2×D4—2144no—
207C2×D4—2142no—
208C2×D4—2142no—
209C2×D4—2142no—
210C2×D4C22×D10712yes—
211C2×D4C22×D10712yes—
212C22×C4—1072no—
213C22×C4—2142no—
214C22×C4—2141no—
215SD16—1074no—
216SD16—2144no—
217D8—24no—
218D8—42no—
219D8—2144no—
220M16—2144no—
221C2×C8—1072no—
222C2×C8C22×D10711yescharacteristic
223C2×C8C22×D10711yescharacteristic
224C4⋊C4—1071no—
225C4⋊C4—1071no—
226C4⋊C4—1071no—
227C4⋊C4—2141no—
228C22⋊C4—1072no—
229C22⋊C4—2142no—
230C22⋊C4—2142no—
231C22⋊C4—2141no—
232C42—2141no—
233C42C22×D10711yescharacteristic
234C23—22no—
235C23—22no—
236C23—8562no—
237C23—1072no—
238C23—4284no—
239C23—4284no—
240C23—2141no—
241Q8—1072no—
242Q8—2141no—
243D4—22no—
244D4—22no—
245D4—22no—
246D4—22no—
247D4—42no—
248D4—42no—
249D4—81no—
250D4—81no—
251D4—8562no—
252D4—1072no—
253D4—4284no—
254D4—4284no—
255D4—4284no—
256D4—2142no—
257D4—2141no—
258C2×C4—1072no—
259C2×C4—4282no—
260C2×C4—4284no—
261C2×C4—2141no—
262C2×C4—2141no—
263C2×C4—2141no—
264C2×C4—2142no—
265C2×C4—2142no—
266C2×C4D4×D10711yescharacteristic
267C2×C4D4×D10711yescharacteristic
268C2×C4C23×D10711yescharacteristic
269C8—21no—
270C8—2142no—
271C8D4×D10712yescomplemented
272C22—42no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—8568no—
279C22—4281no—
280C22—4281no—
281C22—2142no—
282C22—2142no—
283C22—2142no—
284C22—2142no—
285C22C2×D4×D10711yescharacteristic
286C4—21no—
287C4—4281no—
288C4—4281no—
289C4—2142no—
290C4—2142no—
291C4#1681213312yescomplemented
292C4C2×D4×D10711yescharacteristic
293C4#1681215411yescharacteristic
294C2—82no—
295C2—82no—
296C2—4282no—
297C2—4282no—
298C2#2029326211yescharacteristic
299C2C2×#1681213311yescharacteristic
300C2#2029342811yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#493×C107C2
socleC22×C107C22×D4
hypercenter#493D107
hypocenterC107#3274
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #493

Complements

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

NH ≅ G/Ntype
1. C2×#16812133C2semidirect
2. #20293297C2semidirect
3. #20293262C2semidirect
4. #20292877C2semidirect
5. #20292824C2semidirect
6. #20292819C2semidirect
7. #20292769C2semidirect
8. #20292759C2semidirect
9. C4×#13133807C2semidirect
10. #20292706C2semidirect
11. #493×C107C2semidirect
12. C2×#13133817C22semidirect
17. C2×#13133806C22semidirect
19. C2×#13133801C22semidirect
24. #16812080C22semidirect
26. C4×Dic214C22semidirect
30. #16812041C22semidirect
31. C4⋊(C107⋊C8)C22semidirect
32. #16812034C22semidirect
34. #16812030C22semidirect
35. #16812030C22semidirect
36. (C2×D107)⋊C8C22semidirect
37. C2×D8×C107C22semidirect
38. C2×D8×C107C22semidirect
40. #127×C107C22semidirect
41. C4×C8×C107C22semidirect
58. #13133807D4semidirect
117. Dic214C2×D4semidirect
119. C8×C107C2×D4semidirect
136. C4×C107AGL(1,ℤ/8)semidirect
151. C107#3274semidirect
155. #493D107semidirect
170. C2×D8C2×D107semidirect
171. C2×D8C2×D107semidirect
175. #127C2×D107semidirect
192. C4×C8C2×D107semidirect
271. C8D4×D107semidirect
291. C4#16812133semidirect

Maximal subgroups

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

subgroupindex
C2×#168121332
#202932972
#202932622
#202928772
#202928242
#202928192
#202927692
#202927592
C4×#131338072
#202927062
#493×C1072
#3274107