Finite Group Databasebeta

← #28346910

Subgroups — #28346910

Order 13696.

656 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 141 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×#16812136C212yesmaximal, complemented
2C2×#16812133C212yesmaximal, complemented
3#20293176C211yesmaximal, characteristic, complemented
4#127×D107C211yesmaximal, characteristic, complemented
5#20293017C211yesmaximal, characteristic, complemented
6#20293016C211yesmaximal, characteristic, complemented
7#20293014C211yesmaximal, characteristic, complemented
8#20293002C212yesmaximal, complemented
9#20292998C212yesmaximal, complemented
10#20292579C211yesmaximal, characteristic, complemented
11#496×C107C211yesmaximal, characteristic, complemented
12C2×#13133819C2211yescharacteristic
13C2×#13133817C2211yescharacteristic
14C2×D4×D107—22no—
15C2×D4×D107C2212yes—
16#16812136—28no—
17#16812133—28no—
18C2×#13133807C2212yescomplemented
19C2×#13133806C2212yescomplemented
20C2×#13133805C2212yescomplemented
21C2×D856C2212yescomplemented
22C2×#13133803C2212yescomplemented
23C2×#13133803C2212yescomplemented
24C2×#13133801C2212yes—
25#16812107C2211yescharacteristic, complemented
26#16812106C2211yescharacteristic
27#16812105—22no—
28#16812105C2212yes—
29#16812105C2211yescharacteristic, complemented
30#16812097C2211yescharacteristic, complemented
31#16812082—22no—
32(C22⋊D107)⋊C4—22no—
33C42×D107C2211yescharacteristic
34#16812020C2211yescharacteristic, complemented
35C8×Dic107C2212yescomplemented
36C2×SD16×C107C2212yescomplemented
37C2×D8×C107C2212yescomplemented
38#127×C107C2211yescharacteristic
39#124×C107C2211yescharacteristic, complemented
40#97×C107C2211yescharacteristic, complemented
41C23×D107—22no—
42#13133819—24no—
43#13133817—24no—
44C2×Dic214C2311yescharacteristic
45C2×D428—21no—
46C2×D428—22no—
47C2×D428C2311yescharacteristic
48C2×D428C2312yes—
49C2×C22⋊D107—21no—
50C2×C22⋊D107—22no—
51C2×C22⋊D107—22no—
52C2×C22⋊D107—24no—
53D4×D107—24no—
54D4×D107—24no—
55D4×D107—48no—
56C22×Dic107—21no—
57C2×C4×D107—21no—
58C2×C4×D107D412yes—
59C2×C4×D107C2311yescharacteristic
60#13133807—24no—
61#13133806—24no—
62#13133805—24no—
63D856—24no—
64#13133803—24no—
65#13133803—24no—
66#13133801D418yescomplemented
67C2×C107⋊C8C2312yes—
68Dic107⋊C4—22no—
69(C2×D107)⋊C4—22no—
70C4×Dic107D412yes—
71C4×Dic107C2311yescharacteristic
72C2×Q8×C107C2311yescharacteristic
73C2×D4×C107—22no—
74C2×D4×C107C2312yes—
75C2×D4×C107C2311yescharacteristic
76SD16×C107—24no—
77D8×C107—24no—
78C2×C8×C107C2312yes—
79C22⋊C4×C107—22no—
80C42×C107C2311yescharacteristic
81C23×C107—21no—
82C23×C107—22no—
83C22×D107—21no—
84C22×D107—22no—
85C22×D107—48no—
86C22×D107—44no—
87C22×D107C2×D411yescharacteristic
88Q8×C107—22no—
89D4×C107—22no—
90D4×C107—22no—
91D4×C107—22no—
92D4×C107—44no—
93C2×C4×C107—21no—
94C2×C4×C107C2×D412yes—
95C2×C4×C107C2411yescharacteristic
96C22⋊D107—42no—
97C22⋊D107—44no—
98C22⋊D107—48no—
99C22⋊D107—44no—
100C2×Dic107—21no—
101C2×Dic107—42no—
102C2×Dic107C2×D411yescharacteristic
103C2×Dic107C2×D412yes—
104D428—22no—
105D428—22no—
106D428—22no—
107D428—44no—
108D428—42no—
109C4×D107—24no—
110C4×D107—42no—
111C4×D107C2×D414yes—
112Dic214—22no—
113C8×C107C2×D414yescomplemented
114C107⋊C8C2×D414yescomplemented
115C22×C107—41no—
116C22×C107—42no—
117C22×C107—44no—
118C22×C107—42no—
119C22×C107C22×D411yescharacteristic
120C2×D107—22no—
121C2×D107—41no—
122C2×D107—42no—
123C2×D107—44no—
124C2×D107—42no—
125C2×D107—88no—
126C2×D107#12712yes—
127C4×C107—22no—
128C4×C107—41no—
129C4×C107C22×D412yes—
130Dic107—41no—
131Dic107#12712yes—
132Dic107AGL(1,ℤ/8)14yescomplemented
133C2×C107—82no—
134C2×C107—81no—
135C2×C107C2×#12711yescharacteristic
136C2×C107C2×AGL(1,ℤ/8)12yes—
137D107—22no—
138D107—81no—
139D107—82no—
140#3151—1071nomaximal, Sylow
141C107#315111yescharacteristic, complemented, Sylow
142C2×AGL(1,ℤ/8)—1072no—
143C2×AGL(1,ℤ/8)—1072no—
144#532—1071no—
145C2×#127—1071no—
146#496—1071no—
147#496—1071no—
148#496—1071no—
149#496D10711yescharacteristic, complemented
150#493—1072no—
151#492—1072no—
152C4×M16—1071no—
153C2×#41—1071no—
154C2×#41—1071no—
155C22×D4—1072no—
156C22×D4—2142no—
157AGL(1,ℤ/8)—2148no—
158AGL(1,ℤ/8)—2148no—
159C2×SD16—1072no—
160C2×SD16—1072no—
161C2×SD16—1072no—
162C2×SD16C2×D10712yescomplemented
163C2×D8—1072no—
164C2×D8—1072no—
165C2×D8—1072no—
166C2×D8C2×D10712yescomplemented
167C2×M16—1072no—
168C4⋊Q8—1071no—
169#127—1072no—
170#127—1071no—
171#127—1071no—
172#127—2142no—
173#127C2×D10711yescharacteristic
174#124—1071no—
175#124C2×D10711yescharacteristic, complemented
176#121—2142no—
177C4×D4—2142no—
178C2×C42—1071no—
179#97—1071no—
180#97C2×D10711yescharacteristic, complemented
181C4×C8—1072no—
182C24—2142no—
183#41—2144no—
184#41—2144no—
185C2×Q8—1071no—
186C2×Q8C22×D10711yescharacteristic
187C2×D4—22no—
188C2×D4—1071no—
189C2×D4—1072no—
190C2×D4—4288no—
191C2×D4—2141no—
192C2×D4—2144no—
193C2×D4—2144no—
194C2×D4—2142no—
195C2×D4—2144no—
196C2×D4—2142no—
197C2×D4—2141no—
198C2×D4—2142no—
199C2×D4C22×D10711yescharacteristic
200C2×D4C22×D10712yes—
201C22×C4—1071no—
202C22×C4—1072no—
203C22×C4—2141no—
204C22×C4—2141no—
205SD16—24no—
206SD16—2144no—
207SD16—2144no—
208SD16—2144no—
209D8—24no—
210D8—2144no—
211D8—2144no—
212D8—2144no—
213M16—1078no—
214C2×C8—1072no—
215C2×C8C22×D10712yes—
216C4⋊C4—2142no—
217C22⋊C4—22no—
218C22⋊C4—2142no—
219C42—1072no—
220C42—1071no—
221C42C22×D10711yescharacteristic
222C23—21no—
223C23—22no—
224C23—1071no—
225C23—4284no—
226C23—4288no—
227C23—2141no—
228C23—2142no—
229Q8—22no—
230Q8—2142no—
231D4—22no—
232D4—22no—
233D4—22no—
234D4—44no—
235D4—4284no—
236D4—4282no—
237D4—4288no—
238D4—4284no—
239D4—4284no—
240D4—4282no—
241D4—2142no—
242D4—2142no—
243D4—2142no—
244C2×C4—21no—
245C2×C4—1072no—
246C2×C4—1071no—
247C2×C4—1074no—
248C2×C4—4282no—
249C2×C4—4282no—
250C2×C4—2144no—
251C2×C4—2141no—
252C2×C4D4×D10712yes—
253C2×C4C23×D10711yescharacteristic
254C8—1074no—
255C8D4×D10714yescomplemented
256C22—44no—
257C22—42no—
258C22—41no—
259C22—42no—
260C22—8568no—
261C22—1072no—
262C22—4284no—
263C22—4282no—
264C22—4281no—
265C22—4282no—
266C22—2142no—
267C22C2×D4×D10711yescharacteristic
268C4—22no—
269C4—41no—
270C4—1074no—
271C4—1072no—
272C4—4281no—
273C4C2×D4×D10712yes—
274C2—82no—
275C2—81no—
276C2—8562no—
277C2—8561no—
278C2—2142no—
279C2#127×D10711yescharacteristic
280C2AGL(1,ℤ/8)×D10712yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#496×C107C2
socleC22×C107C22×D4
hypercenter#496D107
hypocenterC107#3151
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#16812136C2semidirect
2. C2×#16812133C2semidirect
3. #20293176C2semidirect
4. #127×D107C2semidirect
5. #20293017C2semidirect
6. #20293016C2semidirect
7. #20293014C2semidirect
8. #20293002C2semidirect
9. #20292998C2semidirect
10. #20292579C2semidirect
11. #496×C107C2semidirect
18. C2×#13133807C22semidirect
19. C2×#13133806C22semidirect
20. C2×#13133805C22semidirect
21. C2×D856C22semidirect
22. C2×#13133803C22semidirect
23. C2×#13133803C22semidirect
25. #16812107C22semidirect
29. #16812105C22semidirect
30. #16812097C22semidirect
34. #16812020C22semidirect
35. C8×Dic107C22semidirect
36. C2×SD16×C107C22semidirect
37. C2×D8×C107C22semidirect
39. #124×C107C22semidirect
40. #97×C107C22semidirect
66. #13133801D4semidirect
113. C8×C107C2×D4semidirect
114. C107⋊C8C2×D4semidirect
132. Dic107AGL(1,ℤ/8)semidirect
141. C107#3151semidirect
149. #496D107semidirect
162. C2×SD16C2×D107semidirect
166. C2×D8C2×D107semidirect
175. #124C2×D107semidirect
180. #97C2×D107semidirect
255. C8D4×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121362
C2×#168121332
#202931762
#127×D1072
#202930172
#202930162
#202930142
#202930022
#202929982
#202925792
#496×C1072
#3151107