Finite Group Databasebeta

← #28345691

Subgroups — #28345691

Order 13696.

994 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C23×D428C211yesmaximal, characteristic, complemented
2C2×#16812137C212yesmaximal, complemented
3#20293138C211yesmaximal, characteristic, complemented
4#20292780C214yesmaximal, complemented
5#20292763C214yesmaximal, complemented
6#20292633C212yesmaximal, complemented
7#407×C107C211yesmaximal, characteristic, complemented
8C24×D107—21no—
9C2×#13133818C2211yescharacteristic, complemented
10C22×D428—28no—
11C22×D428—22no—
12C22×D428—21no—
13C22×D428C2214yes—
14C22×D428C2212yes—
15#16812137—28no—
16C2×#13133807C2214yescomplemented
17C2×D856C2214yescomplemented
18#16812092—21no—
19#16812088C2212yescomplemented
20#16812083C2212yescomplemented
21#16812078—21no—
22C4×C22⋊D107—22no—
23#16812068C2211yescharacteristic, complemented
24#16812031C2218yescomplemented
25C23×C4×C107C2211yescharacteristic
26C2×M16×C107C2212yescomplemented
27C22⋊C8×C107C2214yescomplemented
28C23×D107—28no—
29C23×D107—24no—
30C23×D107—22no—
31C23×D107—48no—
32#13133818—24no—
33C2×Dic214C2311yescharacteristic, complemented
34C2×D428—28no—
35C2×D428—216no—
36C2×D428—24no—
37C2×D428—24no—
38C2×D428—22no—
39C2×D428—216no—
40C2×D428D418yescomplemented
41C2×D428C2314yes—
42C2×D428C2311yescharacteristic, complemented
43C2×C22⋊D107—22no—
44C2×C22⋊D107—21no—
45C2×C4×D107—21no—
46#13133807—28no—
47D856—28no—
48C4⋊Dic107C2312yescomplemented
49Dic107⋊C4—22no—
50C22⋊Dic107—21no—
51C22⋊Dic107—22no—
52(C2×D107)⋊C4—22no—
53C4×Dic107—21no—
54C24×C107D411yescharacteristic
55C22×C4×C107—24no—
56C22×C4×C107D412yes—
57C22×C4×C107D411yescharacteristic
58C22×C4×C107C2312yes—
59C22×C4×C107C2311yescharacteristic
60M16×C107—24no—
61C2×C8×C107C2314yescomplemented
62C23×C107—22no—
63C23×C107—24no—
64C23×C107C2×D411yescharacteristic
65C23×C107C2×D412yes—
66C22×D107—216no—
67C22×D107—28no—
68C22×D107—24no—
69C22×D107—21no—
70C22×D107—432no—
71C22×D107—48no—
72C22×D107—48no—
73C22×D107—48no—
74C2×C4×C107—22no—
75C2×C4×C107—24no—
76C2×C4×C107—24no—
77C2×C4×C107C2×D414yes—
78C2×C4×C107C2×D412yes—
79C2×C4×C107C2×D411yescharacteristic
80C2×C4×C107C2411yescharacteristic
81C22⋊D107—42no—
82C22⋊D107—44no—
83C2×Dic107—22no—
84C2×Dic107—21no—
85D428—216no—
86D428—28no—
87D428—24no—
88D428—22no—
89D428C2×D418yes—
90C4×D107—42no—
91Dic214—22no—
92C8×C107—24no—
93C22×C107—28no—
94C22×C107—22no—
95C22×C107—22no—
96C22×C107—22no—
97C22×C107#12012yes—
98C22×C107AGL(1,ℤ/8)14yescomplemented
99C22×C107C22×D411yescharacteristic
100C2×D107—28no—
101C2×D107—416no—
102C2×D107—432no—
103C2×D107—48no—
104C2×D107—41no—
105C2×D107—42no—
106C4×C107—22no—
107C4×C107#12012yes—
108C4×C107C22×D412yes—
109Dic107—41no—
110Dic107—42no—
111C2×C107—22no—
112C2×C107—24no—
113C2×C107C2×#12011yescharacteristic
114C2×C107C2×AGL(1,ℤ/8)12yes—
115D107—48no—
116D107—81no—
117#3003—1071nomaximal, Sylow
118C107#300311yescharacteristic, complemented, Sylow
119C23×D4—1071no—
120C2×AGL(1,ℤ/8)—1072no—
121#525—1071no—
122#450—1074no—
123#447—1074no—
124#418—1072no—
125#407D10711yescharacteristic, complemented
126C25—2141no—
127C2×#41—1071no—
128C22×D4—1072no—
129C22×D4—1074no—
130C22×D4—2148no—
131C22×D4—2142no—
132C22×D4—2141no—
133C23×C4C2×D10711yescharacteristic
134AGL(1,ℤ/8)—2148no—
135C2×SD16—1074no—
136C2×D8—1074no—
137C2×M16C2×D10712yescomplemented
138#123—2141no—
139C22⋊Q8—1072no—
140#121—1072no—
141#120—2141no—
142C4×D4—2142no—
143#117—1071no—
144#102—1078no—
145C22⋊C8C2×D10714yescomplemented
146C24—4288no—
147C24—2148no—
148C24—2142no—
149C24—2144no—
150C24D42811yescharacteristic
151#41—2144no—
152C2×Q8—1071no—
153C2×D4—1078no—
154C2×D4—1074no—
155C2×D4—1071no—
156C2×D4—21416no—
157C2×D4—2148no—
158C2×D4—21416no—
159C2×D4—2142no—
160C2×D4—2141no—
161C2×D4—2144no—
162C2×D4—2144no—
163C2×D4—2142no—
164C22×C4—24no—
165C22×C4—2141no—
166C22×C4D42812yes—
167C22×C4D42811yescharacteristic
168C22×C4C22×D10712yes—
169C22×C4C22×D10711yescharacteristic
170SD16—2148no—
171D8—2148no—
172M16—24no—
173C2×C8C22×D10714yescomplemented
174C4⋊C4—1072no—
175C4⋊C4—2142no—
176C22⋊C4—2141no—
177C22⋊C4—2142no—
178C22⋊C4—2142no—
179C42—2141no—
180C23—22no—
181C23—24no—
182C23—42832no—
183C23—4288no—
184C23—4288no—
185C23—4288no—
186C23—21416no—
187C23—2141no—
188C23—2148no—
189C23—2144no—
190C23C2×D42811yescharacteristic
191C23C2×D42812yes—
192Q8—2142no—
193D4—1078no—
194D4—4282no—
195D4—4284no—
196D4—21416no—
197D4—2148no—
198D4—2142no—
199D4—2144no—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—4282no—
204C2×C4—2142no—
205C2×C4—2141no—
206C2×C4D4×D10714yes—
207C2×C4C2×D42812yes—
208C2×C4C2×D42811yescharacteristic
209C2×C4C23×D10711yescharacteristic
210C8—24no—
211C22—28no—
212C22—22no—
213C22—22no—
214C22—22no—
215C22—42816no—
216C22—42832no—
217C22—4282no—
218C22—4281no—
219C22—4288no—
220C22—2148no—
221C22#1681207912yes—
222C22#1681213714yescomplemented
223C22C22×D42811yescharacteristic
224C4—22no—
225C4—4282no—
226C4—4281no—
227C4#1681207912yes—
228C4C2×D4×D10712yes—
229C2—22no—
230C2—24no—
231C2—8561no—
232C2—4288no—
233C2C2×#1681207911yescharacteristic
234C2C2×#1681213712yes—
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#407×C107C2
socleC22×C107C22×D4
hypercenter#407D107
hypocenterC107#3003
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. C23×D428C2semidirect
2. C2×#16812137C2semidirect
3. #20293138C2semidirect
4. #20292780C2semidirect
5. #20292763C2semidirect
6. #20292633C2semidirect
7. #407×C107C2semidirect
9. C2×#13133818C22semidirect
16. C2×#13133807C22semidirect
17. C2×D856C22semidirect
19. #16812088C22semidirect
20. #16812083C22semidirect
23. #16812068C22semidirect
24. #16812031C22semidirect
26. C2×M16×C107C22semidirect
27. C22⋊C8×C107C22semidirect
33. C2×Dic214C23semidirect
40. C2×D428D4semidirect
42. C2×D428C23semidirect
48. C4⋊Dic107C23semidirect
61. C2×C8×C107C23semidirect
98. C22×C107AGL(1,ℤ/8)semidirect
118. C107#3003semidirect
125. #407D107semidirect
137. C2×M16C2×D107semidirect
145. C22⋊C8C2×D107semidirect
173. C2×C8C22×D107semidirect
222. C22#16812137semidirect

Maximal subgroups

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

subgroupindex
C23×D4282
C2×#168121372
#202931382
#202927802
#202927632
#202926332
#407×C1072
#3003107