Finite Group Databasebeta

← #28345936

Subgroups — #28345936

Order 13696.

584 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 133 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#127×D107C211yesmaximal, characteristic, complemented
3C4×#13133817C211yesmaximal, characteristic, complemented
4#20292828C211yesmaximal, characteristic, complemented
5#20292826C212yesmaximal, complemented
6#20292824C211yesmaximal, characteristic, complemented
7#20292822C211yesmaximal, characteristic, complemented
8#20292820C212yesmaximal, complemented
9#20292649C211yesmaximal, characteristic, complemented
10#20292630C212yesmaximal, complemented
11#459×C107C211yesmaximal, characteristic, complemented
12C2×#13133817C2211yescharacteristic
13C2×D4×D107—21no—
14C2×D4×D107—21no—
15C2×D4×D107C2212yes—
16#16812133—28no—
17C2×#13133807C2212yescomplemented
18C2×#13133806C2212yescomplemented
19C2×#13133803C2212yescomplemented
20C2×#13133801C2212yes—
21#16812106C2211yescharacteristic
22#16812105—21no—
23#16812105—21no—
24#16812105C2212yes—
25C4×Dic214C2211yescharacteristic, complemented
26#16812074C2211yescharacteristic, complemented
27D4×Dic107C2211yescharacteristic, complemented
28C4×C22⋊D107—21no—
29(C22⋊D107)⋊C4—21no—
30#16812067C2211yescharacteristic
31#16812065—21no—
32C2×C4×Dic107—21no—
33C42×D107C2211yescharacteristic
34C4⋊(C107⋊C8)C2211yescharacteristic, complemented
35Dic107⋊C8C2212yescomplemented
36#16812031C2212yescomplemented
37#16812030C2212yescomplemented
38#16812029C2212yescomplemented
39C2×D8×C107C2212yescomplemented
40#127×C107C2211yescharacteristic
41C4×D4×C107C2211yescharacteristic, complemented
42C4⋊C8×C107C2211yescharacteristic, complemented
43#102×C107C2212yescomplemented
44C23×D107—22no—
45#13133817—22no—
46#13133817D414yescomplemented
47C2×Dic214C2311yescharacteristic
48C2×D428—21no—
49C2×D428—21no—
50C2×D428C2312yes—
51C2×C22⋊D107—21no—
52C2×C22⋊D107—22no—
53C2×C22⋊D107—22no—
54C2×C22⋊D107—22no—
55D4×D107—24no—
56D4×D107—24no—
57D4×D107—44no—
58D4×D107—44no—
59C22×Dic107—21no—
60C22×Dic107—21no—
61C2×C4×D107D411yescharacteristic
62C2×C4×D107D411yescharacteristic
63C2×C4×D107C2311yescharacteristic
64#13133807—24no—
65#13133806—24no—
66#13133803—24no—
67#13133801—24no—
68C2×C107⋊C8C2312yes—
69C4⋊Dic107C2311yescharacteristic
70Dic107⋊C4—21no—
71Dic107⋊C4—21no—
72C22⋊Dic107—21no—
73(C2×D107)⋊C4—21no—
74C4×Dic107—21no—
75C4×Dic107—21no—
76C4×Dic107—21no—
77C4×Dic107D411yescharacteristic
78C4×Dic107D411yescharacteristic
79C4×Dic107C2311yescharacteristic
80C2×D4×C107—21no—
81C2×D4×C107—21no—
82C2×D4×C107C2312yes—
83C2×D4×C107C2311yescharacteristic
84C22×C4×C107—21no—
85D8×C107—24no—
86C2×C8×C107C2312yes—
87C4⋊C4×C107C2311yescharacteristic
88C22⋊C4×C107—21no—
89C42×C107C2311yescharacteristic
90C23×C107—21no—
91C23×C107—22no—
92C22×D107—22no—
93C22×D107—44no—
94C22×D107—44no—
95C22×D107—44no—
96C22×D107C2×D411yescharacteristic
97D4×C107—22no—
98D4×C107—21no—
99D4×C107—22no—
100D4×C107—42no—
101D4×C107—42no—
102D4×C107C2×D412yescomplemented
103C2×C4×C107—21no—
104C2×C4×C107—42no—
105C2×C4×C107C2×D411yescharacteristic
106C2×C4×C107C2×D411yescharacteristic
107C2×C4×C107C2411yescharacteristic
108C22⋊D107—24no—
109C22⋊D107—44no—
110C22⋊D107—44no—
111C22⋊D107—44no—
112C2×Dic107—21no—
113C2×Dic107—24no—
114C2×Dic107—21no—
115C2×Dic107—42no—
116C2×Dic107C2×D411yescharacteristic
117C2×Dic107C2×D411yescharacteristic
118C2×Dic107C2×D411yescharacteristic
119D428—22no—
120D428—22no—
121D428—42no—
122D428—42no—
123C4×D107—22no—
124C4×D107—22no—
125C4×D107C2×D412yes—
126C4×D107#4112yes—
127Dic214—21no—
128Dic214C2×D412yescomplemented
129C8×C107—22no—
130C107⋊C8—22no—
131C22×C107—22no—
132C22×C107—41no—
133C22×C107—41no—
134C22×C107—42no—
135C22×C107—42no—
136C22×C107—42no—
137C22×C107C22×D411yescharacteristic
138C2×D107—21no—
139C2×D107—21no—
140C2×D107—42no—
141C2×D107—42no—
142C2×D107—42no—
143C2×D107—88no—
144C2×D107#12112yes—
145C4×C107—21no—
146C4×C107—41no—
147C4×C107AGL(1,ℤ/8)12yescomplemented
148C4×C107C22×D411yescharacteristic
149C4×C107C2×#4111yescharacteristic
150Dic107—22no—
151Dic107—21no—
152Dic107—41no—
153Dic107#12112yes—
154Dic107AGL(1,ℤ/8)12yescomplemented
155C2×C107—42no—
156C2×C107—82no—
157C2×C107C2×#12111yescharacteristic
158C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
159C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
160D107—22no—
161D107—82no—
162#3038—1071nomaximal, Sylow
163C107#303811yescharacteristic, complemented, Sylow
164C2×AGL(1,ℤ/8)—1072no—
165C2×#127—1071no—
166C4×#41—1071no—
167#460—1071no—
168#460—1071no—
169#460—1072no—
170#459—1072no—
171#459—1071no—
172#459D10711yescharacteristic, complemented
173#423—1071no—
174#418—1072no—
175C2×#41—1071no—
176C22×D4—1072no—
177C22×D4—2141no—
178C22×D4—2141no—
179AGL(1,ℤ/8)—2148no—
180C2×SD16—1072no—
181C2×SD16—1072no—
182C2×D8—1072no—
183C2×D8C2×D10712yescomplemented
184C2×M16—1072no—
185#127—1072no—
186#127—1071no—
187#127—2141no—
188#127—2141no—
189#127C2×D10711yescharacteristic
190C4×Q8—1071no—
191C4×Q8—1071no—
192C4×D4—1071no—
193C4×D4—2141no—
194C4×D4—2141no—
195C4×D4C2×D10711yescharacteristic, complemented
196#117—1071no—
197#117—2141no—
198C2×C42—1071no—
199C2×C42—2141no—
200C4⋊C8—1072no—
201C4⋊C8—1071no—
202C4⋊C8C2×D10711yescharacteristic, complemented
203#102—1072no—
204#102—1072no—
205#102—1072no—
206#102C2×D10712yescomplemented
207C24—2142no—
208#41—1074no—
209#41—2142no—
210C2×Q8—1071no—
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—1072no—
214C2×D4—4284no—
215C2×D4—4284no—
216C2×D4—2144no—
217C2×D4—2144no—
218C2×D4—2142no—
219C2×D4—2142no—
220C2×D4—2142no—
221C2×D4—2141no—
222C2×D4—2141no—
223C2×D4—2141no—
224C2×D4C22×D10712yes—
225C2×D4C22×D10711yescharacteristic
226C22×C4—21no—
227C22×C4—1071no—
228C22×C4—1071no—
229C22×C4—1071no—
230C22×C4—2141no—
231C22×C4—2141no—
232SD16—2144no—
233SD16—2144no—
234D8—24no—
235D8—2144no—
236M16—2144no—
237C2×C8—1072no—
238C2×C8C22×D10712yes—
239C4⋊C4—1071no—
240C4⋊C4—2141no—
241C4⋊C4—2141no—
242C4⋊C4C22×D10711yescharacteristic
243C22⋊C4—21no—
244C22⋊C4—2141no—
245C22⋊C4—2141no—
246C42—1071no—
247C42—1071no—
248C42—1071no—
249C42—2141no—
250C42—2141no—
251C42—2141no—
252C42C22×D10711yescharacteristic
253C23—21no—
254C23—22no—
255C23—1071no—
256C23—4284no—
257C23—4284no—
258C23—4284no—
259C23—2142no—
260Q8—1072no—
261Q8—2141no—
262D4—22no—
263D4—21no—
264D4—22no—
265D4—42no—
266D4—42no—
267D4—4282no—
268D4—4284no—
269D4—4284no—
270D4—4284no—
271D4—4282no—
272D4—2144no—
273D4—2142no—
274D4—2142no—
275D4D4×D10712yescomplemented
276C2×C4—21no—
277C2×C4—42no—
278C2×C4—1071no—
279C2×C4—1071no—
280C2×C4—1072no—
281C2×C4—1072no—
282C2×C4—1071no—
283C2×C4—4282no—
284C2×C4—2142no—
285C2×C4—2144no—
286C2×C4—2141no—
287C2×C4—2141no—
288C2×C4—2142no—
289C2×C4D4×D10711yescharacteristic
290C2×C4D4×D10711yescharacteristic
291C2×C4C23×D10711yescharacteristic
292C8—22no—
293C8—2142no—
294C22—22no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—8568no—
301C22—1072no—
302C22—4282no—
303C22—4282no—
304C22—4282no—
305C22—2141no—
306C22—2141no—
307C22C2×D4×D10711yescharacteristic
308C4—21no—
309C4—41no—
310C4—1072no—
311C4—1072no—
312C4—4281no—
313C4—2141no—
314C4—2142no—
315C4#1681213312yescomplemented
316C4C2×D4×D10711yescharacteristic
317C4#41×D10711yescharacteristic
318C2—42no—
319C2—82no—
320C2—8562no—
321C2—2142no—
322C2#121×D10711yescharacteristic
323C2AGL(1,ℤ/8)×D10711yescharacteristic
324C2C2×#1681213311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#459×C107C2
socleC22×C107C22×D4
hypercenter#459D107
hypocenterC107#3038
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#16812133C2semidirect
2. #127×D107C2semidirect
3. C4×#13133817C2semidirect
4. #20292828C2semidirect
5. #20292826C2semidirect
6. #20292824C2semidirect
7. #20292822C2semidirect
8. #20292820C2semidirect
9. #20292649C2semidirect
10. #20292630C2semidirect
11. #459×C107C2semidirect
17. C2×#13133807C22semidirect
18. C2×#13133806C22semidirect
19. C2×#13133803C22semidirect
25. C4×Dic214C22semidirect
26. #16812074C22semidirect
27. D4×Dic107C22semidirect
34. C4⋊(C107⋊C8)C22semidirect
35. Dic107⋊C8C22semidirect
36. #16812031C22semidirect
37. #16812030C22semidirect
38. #16812029C22semidirect
39. C2×D8×C107C22semidirect
41. C4×D4×C107C22semidirect
42. C4⋊C8×C107C22semidirect
43. #102×C107C22semidirect
46. #13133817D4semidirect
102. D4×C107C2×D4semidirect
128. Dic214C2×D4semidirect
147. C4×C107AGL(1,ℤ/8)semidirect
154. Dic107AGL(1,ℤ/8)semidirect
163. C107#3038semidirect
172. #459D107semidirect
183. C2×D8C2×D107semidirect
195. C4×D4C2×D107semidirect
202. C4⋊C8C2×D107semidirect
206. #102C2×D107semidirect
275. D4D4×D107semidirect
315. C4#16812133semidirect

Maximal subgroups

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

subgroupindex
C2×#168121332
#127×D1072
C4×#131338172
#202928282
#202928262
#202928242
#202928222
#202928202
#202926492
#202926302
#459×C1072
#3038107