Finite Group Databasebeta

← #28345688

Subgroups — #28345688

Order 13696.

994 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 173 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×#16812135C212yesmaximal, complemented
2#20293138C211yesmaximal, characteristic, complemented
3#20292778C214yesmaximal, complemented
4#20292761C214yesmaximal, complemented
5#20292632C212yesmaximal, complemented
6#20292591C211yesmaximal, characteristic, complemented
7C23×D4×C107C211yesmaximal, characteristic, complemented
8C2×#13133818C2211yescharacteristic, complemented
9#16812135—28no—
10C2×#13133806C2214yescomplemented
11C2×#13133803C2214yescomplemented
12C2×#13133802C2212yescomplemented
13#16812092—21no—
14#16812088C2212yescomplemented
15#16812083C2212yescomplemented
16#16812078—21no—
17C4×C22⋊D107—22no—
18#16812068C2211yescharacteristic, complemented
19#16812030C2218yescomplemented
20C22⋊(C107⋊C8)C2214yescomplemented
21C25×C107—21no—
22C22×D4×C107—28no—
23C22×D4×C107—22no—
24C22×D4×C107—21no—
25C22×D4×C107C2214yes—
26C22×D4×C107C2212yes—
27C23×C4×C107C2211yescharacteristic
28#13133818—24no—
29C2×Dic214C2311yescharacteristic, complemented
30C2×D428C2311yescharacteristic, complemented
31C2×C22⋊D107—22no—
32C2×C22⋊D107—21no—
33C2×C4×D107—21no—
34#13133806—28no—
35#13133803—28no—
36#13133802—24no—
37C2×C107⋊C8C2314yescomplemented
38C4⋊Dic107C2312yescomplemented
39Dic107⋊C4—22no—
40C22⋊Dic107—21no—
41C22⋊Dic107—22no—
42(C2×D107)⋊C4—22no—
43C4×Dic107—21no—
44C24×C107—28no—
45C24×C107—24no—
46C24×C107—22no—
47C24×C107—48no—
48C24×C107D411yescharacteristic
49C2×D4×C107—28no—
50C2×D4×C107—216no—
51C2×D4×C107—216no—
52C2×D4×C107—24no—
53C2×D4×C107—22no—
54C2×D4×C107—24no—
55C2×D4×C107D418yescomplemented
56C2×D4×C107C2314yes—
57C22×C4×C107—24no—
58C22×C4×C107D412yes—
59C22×C4×C107D411yescharacteristic
60C22×C4×C107C2312yes—
61C22×C4×C107C2311yescharacteristic
62C23×C107—22no—
63C23×C107—24no—
64C23×C107—216no—
65C23×C107—28no—
66C23×C107—24no—
67C23×C107—432no—
68C23×C107—48no—
69C23×C107—48no—
70C23×C107—48no—
71C23×C107C2×D411yescharacteristic
72C23×C107C2×D412yes—
73C22×D107—21no—
74D4×C107—216no—
75D4×C107—28no—
76D4×C107—24no—
77D4×C107C2×D418yes—
78C2×C4×C107—22no—
79C2×C4×C107—24no—
80C2×C4×C107—24no—
81C2×C4×C107C2×D412yes—
82C2×C4×C107C2×D411yescharacteristic
83C2×C4×C107C2×D414yes—
84C2×C4×C107C2411yescharacteristic
85C22⋊D107—42no—
86C22⋊D107—44no—
87C2×Dic107—22no—
88C2×Dic107—21no—
89D428—22no—
90C4×D107—42no—
91Dic214—22no—
92C107⋊C8—24no—
93C22×C107—28no—
94C22×C107—22no—
95C22×C107—22no—
96C22×C107—22no—
97C22×C107—28no—
98C22×C107—416no—
99C22×C107—432no—
100C22×C107—48no—
101C22×C107#12012yes—
102C22×C107AGL(1,ℤ/8)14yescomplemented
103C22×C107C22×D411yescharacteristic
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×C107—48no—
114C2×C107C2×#12011yescharacteristic
115C2×C107C2×AGL(1,ℤ/8)12yes—
116D107—81no—
117#3003—1071nomaximal, Sylow
118C107#300311yescharacteristic, complemented, Sylow
119C23×D4D10711yescharacteristic, complemented
120C2×AGL(1,ℤ/8)—1072no—
121#525—1071no—
122#450—1074no—
123#447—1074no—
124#418—1072no—
125#407—1071no—
126C25—21no—
127C2×#41—1071no—
128C22×D4—28no—
129C22×D4—22no—
130C22×D4—21no—
131C22×D4C2×D10714yes—
132C22×D4C2×D10712yes—
133C23×C4C2×D10711yescharacteristic
134AGL(1,ℤ/8)—2148no—
135C2×SD16—1074no—
136C2×D8—1074no—
137C2×M16—1072no—
138#123—2141no—
139C22⋊Q8—1072no—
140#121—1072no—
141#120—2141no—
142C4×D4—2142no—
143#117—1071no—
144#102—1078no—
145C22⋊C8—1074no—
146C24—28no—
147C24—24no—
148C24—22no—
149C24—48no—
150C24C22⋊D10711yescharacteristic
151#41—2144no—
152C2×Q8—1071no—
153C2×D4—28no—
154C2×D4—216no—
155C2×D4—216no—
156C2×D4—24no—
157C2×D4—22no—
158C2×D4—24no—
159C2×D4—1071no—
160C2×D4—2142no—
161C2×D4—2141no—
162C2×D4C22⋊D10718yescomplemented
163C2×D4C22×D10714yes—
164C22×C4—24no—
165C22×C4—2141no—
166C22×C4C22⋊D10712yes—
167C22×C4C22⋊D10711yescharacteristic
168C22×C4C22×D10712yes—
169C22×C4C22×D10711yescharacteristic
170SD16—2148no—
171D8—2148no—
172M16—2144no—
173C2×C8—1074no—
174C4⋊C4—1072no—
175C4⋊C4—2142no—
176C22⋊C4—2141no—
177C22⋊C4—2142no—
178C22⋊C4—2142no—
179C42—2141no—
180C23—22no—
181C23—24no—
182C23—216no—
183C23—28no—
184C23—24no—
185C23—432no—
186C23—48no—
187C23—48no—
188C23—48no—
189C23—2141no—
190C23C2×C22⋊D10711yescharacteristic
191C23C2×C22⋊D10712yes—
192Q8—2142no—
193D4—216no—
194D4—28no—
195D4—24no—
196D4—4282no—
197D4—4284no—
198D4—2142no—
199D4C2×C22⋊D10718yes—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—4282no—
204C2×C4—2142no—
205C2×C4—2141no—
206C2×C4C2×C22⋊D10712yes—
207C2×C4C2×C22⋊D10711yescharacteristic
208C2×C4C2×C22⋊D10714yes—
209C2×C4C23×D10711yescharacteristic
210C8—2144no—
211C22—28no—
212C22—22no—
213C22—22no—
214C22—22no—
215C22—28no—
216C22—416no—
217C22—432no—
218C22—48no—
219C22—4282no—
220C22—4281no—
221C22#1681207812yes—
222C22#1681213514yescomplemented
223C22C22×C22⋊D10711yescharacteristic
224C4—22no—
225C4—4282no—
226C4—4281no—
227C4#1681207812yes—
228C4C22×C22⋊D10712yes—
229C2—22no—
230C2—24no—
231C2—48no—
232C2—8561no—
233C2C2×#1681207811yescharacteristic
234C2C2×#1681213512yes—
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC23×D4×C107C2
socleC22×C107C22×D4
hypercenterC23×D4D107
hypocenterC107#3003
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C2×#16812135C2semidirect
2. #20293138C2semidirect
3. #20292778C2semidirect
4. #20292761C2semidirect
5. #20292632C2semidirect
6. #20292591C2semidirect
7. C23×D4×C107C2semidirect
8. C2×#13133818C22semidirect
10. C2×#13133806C22semidirect
11. C2×#13133803C22semidirect
12. C2×#13133802C22semidirect
14. #16812088C22semidirect
15. #16812083C22semidirect
18. #16812068C22semidirect
19. #16812030C22semidirect
20. C22⋊(C107⋊C8)C22semidirect
29. C2×Dic214C23semidirect
30. C2×D428C23semidirect
37. C2×C107⋊C8C23semidirect
38. C4⋊Dic107C23semidirect
55. C2×D4×C107D4semidirect
102. C22×C107AGL(1,ℤ/8)semidirect
118. C107#3003semidirect
119. C23×D4D107semidirect
162. C2×D4C22⋊D107semidirect
222. C22#16812135semidirect

Maximal subgroups

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

subgroupindex
C2×#168121352
#202931382
#202927782
#202927612
#202926322
#202925912
C23×D4×C1072
#3003107