Finite Group Databasebeta

← #28348070

Subgroups — #28348070

Order 13696.

520 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 121 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
2#20293270C212yesmaximal, complemented
3#20292870C212yesmaximal, complemented
4#20292839C212yesmaximal, complemented
5#20292788C212yesmaximal, complemented
6#20292759C212yesmaximal, complemented
7C4×D856C211yesmaximal, characteristic, complemented
8#20292706C211yesmaximal, characteristic, complemented
9#495×C107C211yesmaximal, characteristic, complemented
10C2×#13133819C2212yes—
11C2×D4×D107C2212yescomplemented
12#16812136—28no—
13C2×#13133805C2212yescomplemented
14C2×D856C2211yescharacteristic
15C2×#13133803C2212yes—
16C2×#13133801C2212yescomplemented
17#16812091—22no—
18#16812085C2212yescomplemented
19#16812082—22no—
20#16812080C2212yes—
21#16812079—22no—
22C4×D428C2211yescharacteristic
23C4×D428C2212yescomplemented
24C22⋊C4×D107—22no—
25#16812043C2211yescharacteristic, complemented
26C4⋊(C107⋊C8)C2211yescharacteristic, complemented
27#16812033C2212yescomplemented
28#16812031C2212yescomplemented
29#16812030C2212yescomplemented
30(C2×D107)⋊C8C2212yescomplemented
31C2×Q16×C107C2211yescharacteristic, complemented
32C2×SD16×C107C2212yescomplemented
33C2×D8×C107C2211yescharacteristic
34#124×C107C2212yescomplemented
35C4×C8×C107C2211yescharacteristic, complemented
36C23×D107—22no—
37#13133819—24no—
38#13133819—42no—
39C2×D428—22no—
40C2×D428D411yescharacteristic
41C2×D428C2312yes—
42C2×C22⋊D107—22no—
43C2×C22⋊D107—22no—
44D4×D107—24no—
45D4×D107—42no—
46C2×C4×D107—22no—
47C2×C4×D107—22no—
48C2×C4×D107C2312yes—
49#13133805—24no—
50D856D414yescomplemented
51#13133803—24no—
52#13133801—24no—
53C2×C107⋊C8C2312yes—
54C4⋊Dic107D411yescharacteristic
55C4⋊Dic107C2312yes—
56Dic107⋊C4—22no—
57C22⋊Dic107—22no—
58(C2×D107)⋊C4—22no—
59(C2×D107)⋊C4—22no—
60(C2×D107)⋊C4D412yes—
61C2×Q8×C107C2312yes—
62C2×D4×C107C2312yes—
63Q16×C107—41no—
64SD16×C107—24no—
65D8×C107—41no—
66C2×C8×C107C2311yescharacteristic
67C2×C8×C107C2311yescharacteristic
68C22⋊C4×C107—22no—
69C22⋊C4×C107—22no—
70C42×C107C2311yescharacteristic
71C23×C107—22no—
72C22×D107—22no—
73C22×D107—44no—
74C22×D107—44no—
75C22×D107—82no—
76C22×D107C2×D412yes—
77Q8×C107—22no—
78Q8×C107—22no—
79D4×C107—22no—
80D4×C107—22no—
81C2×C4×C107—22no—
82C2×C4×C107C2×D411yescharacteristic
83C2×C4×C107C2×D411yescharacteristic
84C2×C4×C107C2411yescharacteristic
85C22⋊D107—44no—
86C22⋊D107—82no—
87C2×Dic107—22no—
88C2×Dic107C2×D412yes—
89D428—22no—
90D428—41no—
91D428—44no—
92D428C2×D414yes—
93C4×D107—22no—
94C4×D107—22no—
95C4×D107—44no—
96C4×D107—82no—
97C8×C107—21no—
98C8×C107C2×D412yes—
99C107⋊C8—22no—
100C22×C107—42no—
101C22×C107—42no—
102C22×C107—42no—
103C22×C107C22×D411yescharacteristic
104C2×D107—22no—
105C2×D107—24no—
106C2×D107—22no—
107C2×D107—22no—
108C2×D107—42no—
109C2×D107—42no—
110C2×D107—88no—
111C4×C107—21no—
112C4×C107—21no—
113C4×C107—42no—
114C4×C107C22×D411yescharacteristic
115C4×C107H(5,𝔽2)11yescharacteristic
116Dic107—22no—
117Dic107—42no—
118C2×C107—82no—
119C2×C107D4211yescharacteristic
120C2×C107#57511yescharacteristic
121C2×C107#57611yescharacteristic
122D107—42no—
123D107—44no—
124#3273—1071nomaximal, Sylow
125C107#327311yescharacteristic, complemented, Sylow
126C2×AGL(1,ℤ/8)—1072no—
127#546—1072no—
128#495D10711yescharacteristic, complemented
129#468—1072no—
130#463—1072no—
131#452—1072no—
132#447—1072no—
133C4×D8—1071no—
134#436—1071no—
135C2×#41—1072no—
136C22×D4—1072no—
137AGL(1,ℤ/8)—2148no—
138C2×Q16C2×D10711yescharacteristic, complemented
139C2×SD16—1072no—
140C2×SD16C2×D10712yescomplemented
141C2×D8—1072no—
142C2×D8—1071no—
143C2×D8C2×D10711yescharacteristic
144C2×M16—1072no—
145#124C2×D10712yescomplemented
146#123—2142no—
147C22⋊Q8—1072no—
148#121—1072no—
149#121—2142no—
150#120—2142no—
151C4×D4—1071no—
152C4×D4—1072no—
153C2×C22⋊C4—2142no—
154#107—1071no—
155C4⋊C8—1071no—
156#103—1072no—
157#102—1072no—
158#102—1072no—
159C22⋊C8—1072no—
160C4×C8C2×D10711yescharacteristic, complemented
161C24—2142no—
162#41—4282no—
163#41—2144no—
164C2×Q8C22×D10712yes—
165C2×D4—1071no—
166C2×D4—1072no—
167C2×D4—4282no—
168C2×D4—2142no—
169C2×D4—2142no—
170C2×D4—2144no—
171C2×D4—2142no—
172C2×D4C22×D10712yes—
173C22×C4—1072no—
174C22×C4—2142no—
175C22×C4—2142no—
176Q16—41no—
177SD16—24no—
178SD16—2144no—
179D8—41no—
180D8—1074no—
181D8—2144no—
182M16—2144no—
183C2×C8—1072no—
184C2×C8C22×D10711yescharacteristic
185C2×C8C22×D10711yescharacteristic
186C4⋊C4—1071no—
187C4⋊C4—1072no—
188C4⋊C4—2142no—
189C22⋊C4—22no—
190C22⋊C4—22no—
191C22⋊C4—1072no—
192C22⋊C4—2142no—
193C22⋊C4—2142no—
194C22⋊C4—2142no—
195C42C22×D10711yescharacteristic
196C23—22no—
197C23—8562no—
198C23—1072no—
199C23—4284no—
200C23—4284no—
201C23—2142no—
202Q8—22no—
203Q8—22no—
204D4—22no—
205D4—22no—
206D4—8562no—
207D4—1074no—
208D4—4281no—
209D4—4284no—
210D4—4284no—
211D4—2142no—
212C2×C4—22no—
213C2×C4—8562no—
214C2×C4—1072no—
215C2×C4—4284no—
216C2×C4—2142no—
217C2×C4—2142no—
218C2×C4—2142no—
219C2×C4D4×D10711yescharacteristic
220C2×C4D4×D10711yescharacteristic
221C2×C4C23×D10711yescharacteristic
222C8—21no—
223C8—2142no—
224C8D4×D10712yes—
225C22—42no—
226C22—42no—
227C22—42no—
228C22—8568no—
229C22—4282no—
230C22—4282no—
231C22—2142no—
232C22—2142no—
233C22—2144no—
234C22—2142no—
235C22C2×D4×D10711yescharacteristic
236C4—21no—
237C4—21no—
238C4—42no—
239C4—4282no—
240C4—2142no—
241C4C2×D4×D10711yescharacteristic
242C4#1681215411yescharacteristic
243C2—82no—
244C2—4282no—
245C2—4284no—
246C2#2029326211yescharacteristic
247C2#2029342411yescharacteristic
248C2#2029342911yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#495×C107C2
socleC22×C107C22×D4
hypercenter#495D107
hypocenterC107#3273
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. #20293270C2semidirect
3. #20292870C2semidirect
4. #20292839C2semidirect
5. #20292788C2semidirect
6. #20292759C2semidirect
7. C4×D856C2semidirect
8. #20292706C2semidirect
9. #495×C107C2semidirect
11. C2×D4×D107C22semidirect
13. C2×#13133805C22semidirect
16. C2×#13133801C22semidirect
18. #16812085C22semidirect
23. C4×D428C22semidirect
25. #16812043C22semidirect
26. C4⋊(C107⋊C8)C22semidirect
27. #16812033C22semidirect
28. #16812031C22semidirect
29. #16812030C22semidirect
30. (C2×D107)⋊C8C22semidirect
31. C2×Q16×C107C22semidirect
32. C2×SD16×C107C22semidirect
34. #124×C107C22semidirect
35. C4×C8×C107C22semidirect
50. D856D4semidirect
125. C107#3273semidirect
128. #495D107semidirect
138. C2×Q16C2×D107semidirect
140. C2×SD16C2×D107semidirect
145. #124C2×D107semidirect
160. 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
#202932702
#202928702
#202928392
#202927882
#202927592
C4×D8562
#202927062
#495×C1072
#3273107