Finite Group Databasebeta

← #28346943

Subgroups — #28346943

Order 13696.

596 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 137 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×#16812132C211yesmaximal, characteristic, complemented
3C2×D8×D107C211yesmaximal, characteristic, complemented
4#20293195C212yesmaximal, complemented
5#20293017C212yesmaximal, complemented
6#20293013C212yesmaximal, complemented
7#20293012C211yesmaximal, characteristic, complemented
8#20293004C211yesmaximal, characteristic, complemented
9#20293002C211yesmaximal, characteristic, complemented
10#20292581C211yesmaximal, characteristic, complemented
11#495×C107C211yesmaximal, characteristic, complemented
12C2×#13133819C2212yes—
13C2×D4×D107C2212yes—
14#16812136—28no—
15#16812132—24no—
16C2×#13133805C2212yescomplemented
17C2×#13133805C2212yescomplemented
18C2×D856C2211yescharacteristic
19C2×D856C2211yescharacteristic, complemented
20C2×D856C2212yescomplemented
21C2×#13133803C2212yes—
22C2×#13133803C2212yescomplemented
23D8×D107—24no—
24C2×#13133801C2211yescharacteristic
25C2×C8×D107C2211yescharacteristic
26#16812106C2212yescomplemented
27#16812105C2212yescomplemented
28#16812098C2212yescomplemented
29#16812082—24no—
30#16812079—24no—
31#16812066C2211yescharacteristic
32#16812020C2211yescharacteristic, complemented
33#16812019C2211yescharacteristic, complemented
34C8×Dic107C2211yescharacteristic, complemented
35C2×Q16×C107C2211yescharacteristic, complemented
36C2×SD16×C107C2212yescomplemented
37C2×D8×C107C2211yescharacteristic
38#124×C107C2212yescomplemented
39C4×C8×C107C2211yescharacteristic, complemented
40C23×D107—22no—
41#13133819—24no—
42#13133819—24no—
43C2×D428—22no—
44C2×D428—24no—
45C2×D428C2312yes—
46C2×D428C2312yes—
47C2×C22⋊D107—22no—
48C2×C22⋊D107—24no—
49D4×D107—24no—
50D4×D107—24no—
51C2×C4×D107—22no—
52C2×C4×D107C2311yescharacteristic
53#13133805—24no—
54#13133805—24no—
55D856—22no—
56D856—22no—
57D856—24no—
58#13133803—24no—
59#13133803—24no—
60#13133801D414yescomplemented
61C2×C107⋊C8C2311yescharacteristic
62C2×C107⋊C8C2311yescharacteristic
63C8×D107D414yescomplemented
64Dic107⋊C4D412yes—
65(C2×D107)⋊C4—24no—
66(C2×D107)⋊C4D412yes—
67C4×Dic107C2311yescharacteristic
68C2×Q8×C107C2312yes—
69C2×D4×C107C2312yes—
70Q16×C107—22no—
71SD16×C107—24no—
72D8×C107—22no—
73C2×C8×C107C2311yescharacteristic
74C2×C8×C107C2311yescharacteristic
75C22⋊C4×C107—24no—
76C42×C107C2311yescharacteristic
77C23×C107—22no—
78C22×D107—22no—
79C22×D107—22no—
80C22×D107—44no—
81C22×D107—84no—
82C22×D107C2×D411yescharacteristic
83Q8×C107—22no—
84Q8×C107—22no—
85D4×C107—22no—
86D4×C107—22no—
87C2×C4×C107—22no—
88C2×C4×C107C2×D412yes—
89C2×C4×C107C2411yescharacteristic
90C22⋊D107—44no—
91C22⋊D107—84no—
92C2×Dic107C2×D412yes—
93C2×Dic107C2×D411yescharacteristic
94D428—22no—
95D428—22no—
96D428—22no—
97D428—22no—
98D428—44no—
99D428—84no—
100C4×D107—44no—
101C4×D107C2×D412yes—
102C4×D107C2×D412yes—
103C8×C107C2×D412yes—
104C8×C107C2×D412yes—
105C107⋊C8C2×D412yes—
106C107⋊C8C2×D412yes—
107C22×C107—42no—
108C22×C107—44no—
109C22×C107C22×D411yescharacteristic
110C2×D107—22no—
111C2×D107—42no—
112C2×D107—42no—
113C2×D107—44no—
114C2×D107—44no—
115C2×D107—88no—
116C2×D107#12712yes—
117C4×C107—22no—
118C4×C107—42no—
119C4×C107C22×D411yescharacteristic
120C4×C107C22×D411yescharacteristic
121Dic107—22no—
122Dic107#12712yes—
123C2×C107—82no—
124C2×C107C2×#12711yescharacteristic
125C2×C107#57612yes—
126D107—22no—
127D107—82no—
128D107—82no—
129#3155—1071nomaximal, Sylow
130C107#315511yescharacteristic, complemented, Sylow
131C2×AGL(1,ℤ/8)—1072no—
132C2×#135—1071no—
133C22×D8—1071no—
134#535—1072no—
135#496—1072no—
136#496—1072no—
137#495—1071no—
138#495D10711yescharacteristic, complemented
139#493—1071no—
140#493—1071no—
141#405—1071no—
142C2×#41—1072no—
143C22×D4—1072no—
144AGL(1,ℤ/8)—2148no—
145#135—2144no—
146C2×Q16C2×D10711yescharacteristic, complemented
147C2×SD16—1072no—
148C2×SD16—1072no—
149C2×SD16C2×D10712yescomplemented
150C2×D8—1072no—
151C2×D8—1072no—
152C2×D8—1071no—
153C2×D8—1071no—
154C2×D8—1072no—
155C2×D8—2144no—
156C2×D8C2×D10711yescharacteristic
157C2×M16—1071no—
158C22×C8—1071no—
159#127—1072no—
160#127—1072no—
161#124—1072no—
162#124C2×D10712yescomplemented
163#121—2144no—
164#120—2144no—
165#117—1071no—
166#97—1071no—
167#97—1071no—
168C4×C8—1071no—
169C4×C8C2×D10711yescharacteristic, complemented
170C24—2142no—
171#41—2144no—
172#41—2144no—
173C2×Q8C22×D10712yes—
174C2×D4—1072no—
175C2×D4—1072no—
176C2×D4—2142no—
177C2×D4—2144no—
178C2×D4—2144no—
179C2×D4—2144no—
180C2×D4—2142no—
181C2×D4—2144no—
182C2×D4C22×D10712yes—
183C22×C4—1071no—
184C22×C4—2142no—
185Q16—22no—
186SD16—24no—
187SD16—2144no—
188SD16—2144no—
189D8—22no—
190D8—2144no—
191D8—2144no—
192D8—2142no—
193D8—2142no—
194D8—2144no—
195M16—1074no—
196C2×C8—1071no—
197C2×C8—1071no—
198C2×C8—1074no—
199C2×C8C22×D10711yescharacteristic
200C2×C8C22×D10711yescharacteristic
201C4⋊C4—1072no—
202C22⋊C4—24no—
203C22⋊C4—1072no—
204C22⋊C4—2144no—
205C42—1071no—
206C42C22×D10711yescharacteristic
207C23—22no—
208C23—8564no—
209C23—1071no—
210C23—4284no—
211C23—2142no—
212C23—2142no—
213Q8—22no—
214Q8—22no—
215D4—22no—
216D4—22no—
217D4—8564no—
218D4—8564no—
219D4—4284no—
220D4—4284no—
221D4—2142no—
222D4—2142no—
223D4—2142no—
224D4—2142no—
225C2×C4—22no—
226C2×C4—1071no—
227C2×C4—1072no—
228C2×C4—1072no—
229C2×C4—1072no—
230C2×C4—4284no—
231C2×C4D4×D10712yes—
232C2×C4C23×D10711yescharacteristic
233C8—1072no—
234C8—1072no—
235C8D4×D10712yes—
236C8D4×D10712yes—
237C22—42no—
238C22—44no—
239C22—8568no—
240C22—1072no—
241C22—4282no—
242C22—4282no—
243C22—4284no—
244C22—4284no—
245C22—2142no—
246C22C2×D4×D10711yescharacteristic
247C4—22no—
248C4—42no—
249C4—1072no—
250C4—2142no—
251C4C2×D4×D10711yescharacteristic
252C4C2×D4×D10711yescharacteristic
253C2—82no—
254C2—8562no—
255C2—8562no—
256C2—2142no—
257C2#127×D10711yescharacteristic
258C2#2029342912yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#495×C107C2
socleC22×C107C22×D4
hypercenter#495D107
hypocenterC107#3155
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. C2×#16812136C2semidirect
2. C2×#16812132C2semidirect
3. C2×D8×D107C2semidirect
4. #20293195C2semidirect
5. #20293017C2semidirect
6. #20293013C2semidirect
7. #20293012C2semidirect
8. #20293004C2semidirect
9. #20293002C2semidirect
10. #20292581C2semidirect
11. #495×C107C2semidirect
16. C2×#13133805C22semidirect
17. C2×#13133805C22semidirect
19. C2×D856C22semidirect
20. C2×D856C22semidirect
22. C2×#13133803C22semidirect
26. #16812106C22semidirect
27. #16812105C22semidirect
28. #16812098C22semidirect
32. #16812020C22semidirect
33. #16812019C22semidirect
34. C8×Dic107C22semidirect
35. C2×Q16×C107C22semidirect
36. C2×SD16×C107C22semidirect
38. #124×C107C22semidirect
39. C4×C8×C107C22semidirect
60. #13133801D4semidirect
63. C8×D107D4semidirect
130. C107#3155semidirect
138. #495D107semidirect
146. C2×Q16C2×D107semidirect
149. C2×SD16C2×D107semidirect
162. #124C2×D107semidirect
169. C4×C8C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121362
C2×#168121322
C2×D8×D1072
#202931952
#202930172
#202930132
#202930122
#202930042
#202930022
#202925812
#495×C1072
#3155107