Finite Group Databasebeta

← #28346927

Subgroups — #28346927

Order 13696.

572 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 165 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×#16812142C212yesmaximal, complemented
2C2×#16812137C212yesmaximal, complemented
3#20293177C211yesmaximal, characteristic, complemented
4C2×#16812097C211yesmaximal, characteristic, complemented
5#20293021C212yesmaximal, complemented
6#20293016C212yesmaximal, complemented
7#20293011C214yesmaximal, complemented
8C4×M16×C107C211yesmaximal, characteristic, complemented
9C2×#13133818C2212yes—
10C22×Dic214C2211yescharacteristic
11C22×D428C2211yescharacteristic
12#16812142—28no—
13#16812137—28no—
14C2×Dic428C2214yescomplemented
15C2×#13133807C2214yes—
16C2×#13133807C2214yescomplemented
17C2×D856C2214yes—
18#16812109C2211yescharacteristic, complemented
19#16812106C2211yescharacteristic, complemented
20#16812097—22no—
21#16812097C2212yes—
22#16812097C2212yes—
23#16812097C2212yescomplemented
24#16812083—22no—
25C4×D428—22no—
26C2×(C2×D107)⋊C4—22no—
27C2×M16×C107C2212yes—
28C2×C42×C107C2211yescharacteristic
29#97×C107C2212yescomplemented
30C4×C8×C107C2212yescomplemented
31C23×D107—21no—
32#13133818—24no—
33#13133818—24no—
34C2×Dic214—24no—
35C2×Dic214—21no—
36C2×Dic214C2312yes—
37C2×Dic214C2312yes—
38C2×D428—24no—
39C2×D428—22no—
40C2×D428—21no—
41C2×D428C2312yes—
42C2×D428C2312yes—
43C2×C22⋊D107—22no—
44C22×Dic107—21no—
45C2×C4×D107—22no—
46Dic428—28no—
47#13133807—28no—
48#13133807—28no—
49D856—28no—
50C4⋊Dic107—22no—
51(C2×D107)⋊C4—24no—
52(C2×D107)⋊C4—24no—
53(C2×D107)⋊C4—24no—
54C22×C4×C107D412yes—
55C22×C4×C107C2311yescharacteristic
56M16×C107D418yescomplemented
57C2×C8×C107C2314yes—
58C42×C107D412yes—
59C42×C107C2311yescharacteristic
60C42×C107C2311yescharacteristic
61C23×C107C2×D411yescharacteristic
62C22×D107—22no—
63C22×D107—22no—
64C22×D107—44no—
65C22×D107—42no—
66C2×C4×C107—42no—
67C2×C4×C107C2×D412yes—
68C2×C4×C107C2×D412yes—
69C2×C4×C107C2×D411yescharacteristic
70C2×C4×C107C2×D414yes—
71C2×C4×C107C2411yescharacteristic
72C22⋊D107—44no—
73C2×Dic107—22no—
74C2×Dic107—22no—
75C2×Dic107—42no—
76D428—22no—
77D428—24no—
78D428—22no—
79D428—44no—
80D428—42no—
81C4×D107—44no—
82Dic214—22no—
83Dic214—24no—
84Dic214—22no—
85Dic214—42no—
86C8×C107C2×D418yes—
87C22×C107—22no—
88C22×C107#12712yes—
89C22×C107C22×D411yescharacteristic
90C2×D107—42no—
91C2×D107—42no—
92C2×D107—44no—
93C2×D107—44no—
94C2×D107—84no—
95C4×C107—22no—
96C4×C107—22no—
97C4×C107#12712yes—
98C4×C107C22×D412yes—
99Dic107—42no—
100Dic107—42no—
101C2×C107—22no—
102C2×C107C2×#12711yescharacteristic
103C2×C107#57512yes—
104D107—82no—
105D107—82no—
106#3153—1071nomaximal, Sylow
107C107#315311yescharacteristic, complemented, Sylow
108C2×#137—1072no—
109C2×AGL(1,ℤ/8)—1072no—
110#532—1071no—
111C2×#124—1071no—
112#497—1072no—
113#496—1072no—
114#495—1074no—
115C4×M16D10711yescharacteristic, complemented
116C2×#41—1072no—
117C22×Q8—1071no—
118C22×D4—1071no—
119#137—2148no—
120AGL(1,ℤ/8)—2148no—
121C2×Q16—1074no—
122C2×SD16—1074no—
123C2×SD16—1074no—
124C2×D8—1074no—
125C2×M16C2×D10712yes—
126C4⋊Q8—1071no—
127#127—1071no—
128#124—1072no—
129#124—1072no—
130#124—1072no—
131#124—2142no—
132#121—2142no—
133C4×D4—2142no—
134C2×C22⋊C4—2142no—
135C2×C42C2×D10711yescharacteristic
136#97C2×D10712yescomplemented
137C4×C8C2×D10712yescomplemented
138C24—2141no—
139#41—2144no—
140#41—2144no—
141C2×Q8—1072no—
142C2×Q8—1072no—
143C2×Q8—2144no—
144C2×Q8—2141no—
145C2×D4—1072no—
146C2×D4—1072no—
147C2×D4—2142no—
148C2×D4—2142no—
149C2×D4—2141no—
150C2×D4—2144no—
151C22×C4—2141no—
152C22×C4—2142no—
153C22×C4D42812yes—
154C22×C4C22×D10711yescharacteristic
155Q16—2148no—
156SD16—2148no—
157SD16—2148no—
158D8—2148no—
159M16D42818yescomplemented
160C2×C8C22×D10714yes—
161C4⋊C4—2142no—
162C22⋊C4—2144no—
163C22⋊C4—2144no—
164C22⋊C4—2144no—
165C42D42812yes—
166C42C22×D10711yescharacteristic
167C42C22×D10711yescharacteristic
168C23—4284no—
169C23—4282no—
170C23—2142no—
171C23—2142no—
172C23C2×D42811yescharacteristic
173Q8—4282no—
174Q8—2142no—
175Q8—2144no—
176Q8—2142no—
177D4—4284no—
178D4—4284no—
179D4—4282no—
180D4—2142no—
181D4—2144no—
182D4—2142no—
183C2×C4—42no—
184C2×C4—4282no—
185C2×C4—4284no—
186C2×C4—2142no—
187C2×C4—2142no—
188C2×C4C2×D42812yes—
189C2×C4C2×D42812yes—
190C2×C4C2×D42811yescharacteristic
191C2×C4C2×D42814yes—
192C2×C4C23×D10711yescharacteristic
193C8C2×D42818yes—
194C22—22no—
195C22—8564no—
196C22—4282no—
197C22—4282no—
198C22—4284no—
199C22—4284no—
200C22#1681210612yes—
201C22C22×D42811yescharacteristic
202C4—22no—
203C4—22no—
204C4—4282no—
205C4—4282no—
206C4#1681210612yes—
207C4C22×D42812yes—
208C2—22no—
209C2—8562no—
210C2—8562no—
211C2C2×#1681210611yescharacteristic
212C2#2029342712yes—
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC4×M16×C107C2
socleC22×C107C22×D4
hypercenterC4×M16D107
hypocenterC107#3153
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×M16

Complements

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

NH ≅ G/Ntype
1. C2×#16812142C2semidirect
2. C2×#16812137C2semidirect
3. #20293177C2semidirect
4. C2×#16812097C2semidirect
5. #20293021C2semidirect
6. #20293016C2semidirect
7. #20293011C2semidirect
8. C4×M16×C107C2semidirect
14. C2×Dic428C22semidirect
16. C2×#13133807C22semidirect
18. #16812109C22semidirect
19. #16812106C22semidirect
23. #16812097C22semidirect
29. #97×C107C22semidirect
30. C4×C8×C107C22semidirect
56. M16×C107D4semidirect
107. C107#3153semidirect
115. C4×M16D107semidirect
136. #97C2×D107semidirect
137. C4×C8C2×D107semidirect
159. M16D428semidirect

Maximal subgroups

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

subgroupindex
C2×#168121422
C2×#168121372
#202931772
C2×#168120972
#202930212
#202930162
#202930112
C4×M16×C1072
#3153107