Finite Group Databasebeta

← #28348071

Subgroups — #28348071

Order 13696.

520 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 129 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#20293272C212yesmaximal, complemented
3#20293011C211yesmaximal, characteristic, complemented
4#20292874C212yesmaximal, complemented
5#20292842C212yesmaximal, complemented
6#20292790C212yesmaximal, complemented
7#20292762C212yesmaximal, complemented
8#20292706C211yesmaximal, characteristic, complemented
9C4×D8×C107C211yesmaximal, characteristic, complemented
10C2×#13133817C2212yes—
11C2×D4×D107C2212yescomplemented
12#16812133—28no—
13C2×Dic428C2211yescharacteristic, complemented
14C2×#13133807C2212yescomplemented
15C2×#13133806C2212yescomplemented
16C2×D856C2211yescharacteristic
17C2×#13133803C2212yes—
18C2×#13133801C2212yescomplemented
19#16812097C2212yescomplemented
20#16812093—22no—
21#16812089C2212yescomplemented
22#16812084C2212yes—
23#16812083—22no—
24#16812079—22no—
25C4×D428C2211yescharacteristic
26C2×(C2×D107)⋊C4—22no—
27C4⋊(C107⋊C8)C2211yescharacteristic, complemented
28#16812032C2212yescomplemented
29#16812029C2212yescomplemented
30(C2×D107)⋊C8C2212yescomplemented
31C2×D8×C107C2211yescharacteristic
32C4×D4×C107C2212yescomplemented
33#107×C107C2211yescharacteristic, complemented
34#102×C107C2212yescomplemented
35C4×C8×C107C2211yescharacteristic, complemented
36C23×D107—22no—
37#13133817—24no—
38#13133817—42no—
39C2×Dic214C2312yes—
40C2×D428—22no—
41C2×D428D411yescharacteristic
42C2×D428C2312yes—
43C2×C22⋊D107—22no—
44C2×C22⋊D107—22no—
45D4×D107—24no—
46D4×D107—42no—
47C22×Dic107—22no—
48C2×C4×D107C2312yes—
49Dic428—41no—
50#13133807—24no—
51#13133806—24no—
52D856—41no—
53#13133803—24no—
54#13133801—24no—
55C2×C107⋊C8C2312yes—
56C4⋊Dic107—22no—
57C4⋊Dic107D411yescharacteristic
58(C2×D107)⋊C4—22no—
59(C2×D107)⋊C4—22no—
60(C2×D107)⋊C4—22no—
61(C2×D107)⋊C4—22no—
62(C2×D107)⋊C4D412yes—
63C2×D4×C107C2312yes—
64C22×C4×C107—22no—
65D8×C107D414yescomplemented
66C2×C8×C107C2311yescharacteristic
67C2×C8×C107C2311yescharacteristic
68C4⋊C4×C107C2312yes—
69C22⋊C4×C107—22no—
70C42×C107C2311yescharacteristic
71C23×C107—22no—
72C22×D107—22no—
73C22×D107—44no—
74C22×D107—44no—
75C22×D107—82no—
76C22×D107C2×D412yes—
77D4×C107—22no—
78D4×C107C2×D414yes—
79C2×C4×C107—22no—
80C2×C4×C107—82no—
81C2×C4×C107C2×D411yescharacteristic
82C2×C4×C107C2×D411yescharacteristic
83C2×C4×C107C2411yescharacteristic
84C22⋊D107—44no—
85C22⋊D107—44no—
86C2×Dic107—22no—
87C2×Dic107—44no—
88C2×Dic107C2×D412yes—
89D428—22no—
90D428—22no—
91D428—41no—
92D428—82no—
93C4×D107—22no—
94C4×D107—22no—
95Dic214—22no—
96Dic214—22no—
97C8×C107—21no—
98C8×C107C2×D412yes—
99C107⋊C8—22no—
100C22×C107—24no—
101C22×C107—42no—
102C22×C107—42no—
103C22×C107C22×D411yescharacteristic
104C2×D107—22no—
105C2×D107—22no—
106C2×D107—22no—
107C2×D107—42no—
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—44no—
119C2×C107D4211yescharacteristic
120C2×C107#57511yescharacteristic
121C2×C107#57611yescharacteristic
122D107—42no—
123D107—82no—
124#3273—1071nomaximal, Sylow
125C107#327311yescharacteristic, complemented, Sylow
126C2×AGL(1,ℤ/8)—1072no—
127#546—1072no—
128#495—1071no—
129#468—1072no—
130#463—1072no—
131#452—1072no—
132#447—1072no—
133C4×D8D10711yescharacteristic, complemented
134#436—1071no—
135C2×#41—1072no—
136C22×D4—1072no—
137AGL(1,ℤ/8)—2148no—
138C2×Q16—1071no—
139C2×SD16—1072no—
140C2×SD16—1072no—
141C2×D8—1072no—
142C2×D8—1071no—
143C2×D8C2×D10711yescharacteristic
144C2×M16—1072no—
145#124—1072no—
146#123—2142no—
147C22⋊Q8—1072no—
148#121—1072no—
149#121—2142no—
150#120—2142no—
151C4×D4—1071no—
152C4×D4C2×D10712yescomplemented
153C2×C22⋊C4—2142no—
154#107C2×D10711yescharacteristic, complemented
155C4⋊C8—1071no—
156#103—1072no—
157#102—1072no—
158#102C2×D10712yescomplemented
159C22⋊C8—1072no—
160C4×C8C2×D10711yescharacteristic, complemented
161C24—2142no—
162#41—4282no—
163#41—2144no—
164C2×Q8—1072no—
165C2×D4—1071no—
166C2×D4—1072no—
167C2×D4—4282no—
168C2×D4—2142no—
169C2×D4—2142no—
170C2×D4—2142no—
171C2×D4—2144no—
172C2×D4C22×D10712yes—
173C22×C4—22no—
174C22×C4—1072no—
175C22×C4—2142no—
176Q16—4281no—
177SD16—2144no—
178SD16—2144no—
179D8—4281no—
180D8—2144no—
181D8D42814yescomplemented
182M16—2144no—
183C2×C8—1072no—
184C2×C8C22×D10711yescharacteristic
185C2×C8C22×D10711yescharacteristic
186C4⋊C4—1071no—
187C4⋊C4—2142no—
188C4⋊C4C22×D10712yes—
189C22⋊C4—22no—
190C22⋊C4—1072no—
191C22⋊C4—2142no—
192C22⋊C4—2142no—
193C22⋊C4—2142no—
194C22⋊C4—2142no—
195C42C22×D10711yescharacteristic
196C23—22no—
197C23—8562no—
198C23—1072no—
199C23—4284no—
200C23—4284no—
201C23—2142no—
202Q8—2142no—
203Q8—2142no—
204D4—22no—
205D4—8562no—
206D4—4281no—
207D4—4284no—
208D4—4284no—
209D4—2142no—
210D4—2142no—
211D4C2×D42814yes—
212C2×C4—22no—
213C2×C4—82no—
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—
224C8C2×D42812yes—
225C22—24no—
226C22—42no—
227C22—42no—
228C22—8568no—
229C22—4282no—
230C22—4282no—
231C22—4282no—
232C22—2142no—
233C22—2142no—
234C22—2142no—
235C22C2×D4×D10711yescharacteristic
236C4—21no—
237C4—21no—
238C4—42no—
239C4—4282no—
240C4—2142no—
241C4C22×D42811yescharacteristic
242C4#1681215511yescharacteristic
243C2—44no—
244C2—8562no—
245C2—4282no—
246C2D4×D42811yescharacteristic
247C2#2029342411yescharacteristic
248C2#2029342811yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC4×D8×C107C2
socleC22×C107C22×D4
hypercenterC4×D8D107
hypocenterC107#3273
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. C2×#16812133C2semidirect
2. #20293272C2semidirect
3. #20293011C2semidirect
4. #20292874C2semidirect
5. #20292842C2semidirect
6. #20292790C2semidirect
7. #20292762C2semidirect
8. #20292706C2semidirect
9. C4×D8×C107C2semidirect
11. C2×D4×D107C22semidirect
13. C2×Dic428C22semidirect
14. C2×#13133807C22semidirect
15. C2×#13133806C22semidirect
18. C2×#13133801C22semidirect
19. #16812097C22semidirect
21. #16812089C22semidirect
27. C4⋊(C107⋊C8)C22semidirect
28. #16812032C22semidirect
29. #16812029C22semidirect
30. (C2×D107)⋊C8C22semidirect
32. C4×D4×C107C22semidirect
33. #107×C107C22semidirect
34. #102×C107C22semidirect
35. C4×C8×C107C22semidirect
65. D8×C107D4semidirect
125. C107#3273semidirect
133. C4×D8D107semidirect
152. C4×D4C2×D107semidirect
154. #107C2×D107semidirect
158. #102C2×D107semidirect
160. C4×C8C2×D107semidirect
181. D8D428semidirect

Maximal subgroups

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

subgroupindex
C2×#168121332
#202932722
#202930112
#202928742
#202928422
#202927902
#202927622
#202927062
C4×D8×C1072
#3273107