Finite Group Databasebeta

← #28345984

Subgroups — #28345984

Order 13696.

490 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C22×#13133805C211yesmaximal, characteristic, complemented
2#20293191C211yesmaximal, characteristic
3#20293105C211yesmaximal, characteristic, complemented
4#20292851C214yesmaximal, complemented
5#20292825C214yesmaximal, complemented
6#20292655C211yesmaximal, characteristic, complemented
7#20292633C211yesmaximal, characteristic, complemented
8#20292623C211yesmaximal, characteristic, complemented
9C2×#137×C107C211yesmaximal, characteristic, complemented
10C22×D428C2211yescharacteristic
11C2×#13133805—24no—
12C2×#13133805C2214yes—
13C22×C107⋊C8C2211yescharacteristic
14#16812105C2212yes—
15#16812098C2212yes—
16#16812083—22no—
17#16812077—22no—
18Q8×Dic107—22no—
19Q8×Dic107C2214yescomplemented
20#16812068—22no—
21#16812068C2211yescharacteristic
22#16812068C2211yescharacteristic
23Dic107⋊C8C2214yescomplemented
24#16812033C2212yescomplemented
25#16812031C2214yes—
26#16812030C2212yescomplemented
27C2×#41×C107C2211yescharacteristic
28C22×Q8×C107C2211yescharacteristic
29#137×C107—24no—
30C2×Q16×C107C2212yescomplemented
31C2×SD16×C107C2212yescomplemented
32C2×M16×C107C2211yescharacteristic
33C23×D107—21no—
34C2×D428—22no—
35C2×D428—22no—
36C2×D428—21no—
37C2×D428C2312yes—
38C2×C22⋊D107—24no—
39#13133805—28no—
40C2×C107⋊C8—22no—
41C2×C107⋊C8C2312yes—
42C4⋊Dic107—24no—
43C4⋊Dic107D412yes—
44C4⋊Dic107C2312yes—
45C22⋊Dic107—21no—
46C22⋊Dic107D412yes—
47(C2×D107)⋊C4—24no—
48C4×Dic107—21no—
49C4×Dic107—24no—
50C4×Dic107C2312yes—
51#41×C107—22no—
52#41×C107—22no—
53C2×Q8×C107—21no—
54C2×Q8×C107—22no—
55C2×Q8×C107D414yescomplemented
56C2×Q8×C107C2311yescharacteristic
57C2×Q8×C107C2312yes—
58C2×D4×C107—21no—
59C2×D4×C107C2311yescharacteristic
60C22×C4×C107—21no—
61C22×C4×C107—21no—
62C22×C4×C107C2311yescharacteristic
63Q16×C107—24no—
64SD16×C107—24no—
65M16×C107—22no—
66C2×C8×C107C2312yes—
67C23×C107—21no—
68C23×C107C2×D411yescharacteristic
69C22×D107—22no—
70C22×D107—42no—
71C22×D107—82no—
72Q8×C107—24no—
73Q8×C107—22no—
74Q8×C107—21no—
75Q8×C107—21no—
76Q8×C107C2×D414yes—
77D4×C107—21no—
78D4×C107—21no—
79D4×C107—42no—
80C2×C4×C107—24no—
81C2×C4×C107—22no—
82C2×C4×C107—21no—
83C2×C4×C107—42no—
84C2×C4×C107C2×D411yescharacteristic
85C2×C4×C107C2×D412yes—
86C2×C4×C107#4112yes—
87C2×C4×C107C2411yescharacteristic
88C22⋊D107—84no—
89C2×Dic107—22no—
90C2×Dic107C2×D414yes—
91D428—22no—
92D428—22no—
93D428—42no—
94C8×C107—22no—
95C107⋊C8—22no—
96C22×C107—22no—
97C22×C107—41no—
98C22×C107—42no—
99C22×C107#12112yes—
100C22×C107C22×D411yescharacteristic
101C2×D107—42no—
102C2×D107—44no—
103C2×D107—84no—
104C4×C107—24no—
105C4×C107—41no—
106C4×C107#12112yes—
107C4×C107C22×D411yescharacteristic
108C4×C107C2×#4111yescharacteristic
109Dic107—24no—
110Dic107—42no—
111C2×C107—22no—
112C2×C107—81no—
113C2×C107C2×#12111yescharacteristic
114C2×C107#57712yes—
115D107—82no—
116#3043—1071nomaximal, Sylow
117C107#304311yescharacteristic, complemented, Sylow
118C2×#137D10711yescharacteristic, complemented
119C22×SD16—1071no—
120#535—1071no—
121#519—1071no—
122#464—1074no—
123#460—1074no—
124#424—1071no—
125#418—1071no—
126#416—1071no—
127C2×#41C2×D10711yescharacteristic
128C22×Q8C2×D10711yescharacteristic
129C22×D4—1071no—
130#137—24no—
131C2×Q16C2×D10712yescomplemented
132C2×SD16—1074no—
133C2×SD16—2144no—
134C2×SD16C2×D10712yescomplemented
135C2×M16C2×D10711yescharacteristic
136C22×C8—1071no—
137#127—1072no—
138#124—1072no—
139#121—2142no—
140#120—2142no—
141C4×Q8—1074no—
142C4×Q8—2142no—
143#117—1071no—
144#117—1071no—
145#117—2142no—
146C4⋊C8—1074no—
147#103—1072no—
148#102—1072no—
149#102—1074no—
150C24—2141no—
151#41—22no—
152#41—22no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8C22⋊D10714yescomplemented
156C2×Q8C22×D10711yescharacteristic
157C2×Q8C22×D10712yes—
158C2×D4—21no—
159C2×D4—1072no—
160C2×D4—2144no—
161C2×D4—2141no—
162C2×D4—2142no—
163C2×D4—2142no—
164C2×D4C22×D10711yescharacteristic
165C22×C4—21no—
166C22×C4—21no—
167C22×C4C22×D10711yescharacteristic
168Q16—24no—
169SD16—24no—
170SD16—2148no—
171M16—22no—
172C2×C8—1072no—
173C2×C8—2142no—
174C2×C8C22×D10712yes—
175C4⋊C4—1072no—
176C4⋊C4—1072no—
177C4⋊C4—2144no—
178C22⋊C4—1072no—
179C22⋊C4—2144no—
180C22⋊C4—2141no—
181C42—1072no—
182C42—2144no—
183C42—2141no—
184C23—21no—
185C23—8562no—
186C23—4282no—
187C23—2142no—
188C23D4×D10711yescharacteristic
189Q8—24no—
190Q8—22no—
191Q8—21no—
192Q8—21no—
193Q8C2×C22⋊D10714yes—
194D4—21no—
195D4—21no—
196D4—42no—
197D4—8564no—
198D4—4282no—
199D4—2142no—
200D4—2142no—
201C2×C4—24no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—42no—
205C2×C4—1074no—
206C2×C4—2142no—
207C2×C4D4×D10711yescharacteristic
208C2×C4C2×C22⋊D10712yes—
209C2×C4#1313381712yes—
210C2×C4C23×D10711yescharacteristic
211C8—22no—
212C8—2142no—
213C22—22no—
214C22—42no—
215C22—41no—
216C22—8564no—
217C22—4284no—
218C22—4282no—
219C22#1681208112yes—
220C22C2×D4×D10711yescharacteristic
221C4—24no—
222C4—41no—
223C4—4282no—
224C4—2144no—
225C4#1681208112yes—
226C4C22×C22⋊D10711yescharacteristic
227C4C2×#1313381711yescharacteristic
228C2—22no—
229C2—81no—
230C2—8562no—
231C2C2×#1681208111yescharacteristic
232C2#2029343612yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×#137×C107C2
socleC22×C107C22×D4
prime radical#20293191C2
hypercenterC2×#137D107
hypocenterC107#3043
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C22×#13133805C2semidirect
3. #20293105C2semidirect
4. #20292851C2semidirect
5. #20292825C2semidirect
6. #20292655C2semidirect
7. #20292633C2semidirect
8. #20292623C2semidirect
9. C2×#137×C107C2semidirect
19. Q8×Dic107C22semidirect
23. Dic107⋊C8C22semidirect
24. #16812033C22semidirect
26. #16812030C22semidirect
30. C2×Q16×C107C22semidirect
31. C2×SD16×C107C22semidirect
55. C2×Q8×C107D4semidirect
117. C107#3043semidirect
118. C2×#137D107semidirect
131. C2×Q16C2×D107semidirect
134. C2×SD16C2×D107semidirect
155. C2×Q8C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C22×#131338052
#202931912
#202931052
#202928512
#202928252
#202926552
#202926332
#202926232
C2×#137×C1072
#3043107