Finite Group Databasebeta

← #28346929

Subgroups — #28346929

Order 13696.

572 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16812139C212yesmaximal, complemented
2C2×#16812134C212yesmaximal, complemented
3C2×#16812097C211yesmaximal, characteristic, complemented
4#20293019C212yesmaximal, complemented
5#20293014C212yesmaximal, complemented
6#20293009C214yesmaximal, complemented
7C4×#13133802C211yesmaximal, characteristic, complemented
8#532×C107C211yesmaximal, characteristic, complemented
9C22×Dic214C2211yescharacteristic
10C22×D428C2211yescharacteristic
11#16812139—28no—
12#16812134—28no—
13C2×#13133808C2214yescomplemented
14C2×#13133806C2214yescomplemented
15C2×#13133805C2214yes—
16C2×#13133803C2214yes—
17C2×#13133802C2212yes—
18#16812097—22no—
19#16812097C2212yes—
20#16812097C2212yes—
21C2×(C2×D107)⋊C4—22no—
22#16812020C2212yescomplemented
23C4×C107⋊C8C2212yescomplemented
24C2×#41×C107C2212yes—
25C4⋊Q8×C107C2211yescharacteristic, complemented
26#127×C107C2211yescharacteristic, complemented
27#124×C107C2212yescomplemented
28#121×C107—22no—
29C4×D4×C107—22no—
30C2×C42×C107C2211yescharacteristic
31C23×D107—21no—
32C2×Dic214—24no—
33C2×Dic214—21no—
34C2×Dic214C2312yes—
35C2×D428—24no—
36C2×D428—21no—
37C2×D428C2312yes—
38C22×Dic107—21no—
39#13133808—28no—
40#13133806—28no—
41#13133805—28no—
42#13133803—28no—
43#13133802D418yescomplemented
44C2×C107⋊C8C2314yes—
45(C2×D107)⋊C4—24no—
46(C2×D107)⋊C4—24no—
47#41×C107—24no—
48#41×C107—24no—
49C2×Q8×C107C2312yes—
50C2×D4×C107—22no—
51C2×D4×C107—22no—
52C2×D4×C107C2312yes—
53C22×C4×C107—22no—
54C22×C4×C107D412yes—
55C22×C4×C107C2311yescharacteristic
56C4⋊C4×C107—22no—
57C22⋊C4×C107—24no—
58C42×C107D412yes—
59C42×C107C2311yescharacteristic
60C42×C107C2311yescharacteristic
61C23×C107—22no—
62C23×C107C2×D411yescharacteristic
63C22×D107—22no—
64C22×D107—44no—
65C22×D107—42no—
66Q8×C107—22no—
67Q8×C107—22no—
68D4×C107—22no—
69D4×C107—22no—
70D4×C107—44no—
71D4×C107—44no—
72C2×C4×C107—22no—
73C2×C4×C107—42no—
74C2×C4×C107—44no—
75C2×C4×C107C2×D414yes—
76C2×C4×C107C2×D412yes—
77C2×C4×C107C2×D412yes—
78C2×C4×C107C2×D411yescharacteristic
79C2×C4×C107C2411yescharacteristic
80C2×Dic107—22no—
81C2×Dic107—42no—
82D428—24no—
83D428—42no—
84Dic214—24no—
85Dic214—42no—
86C107⋊C8C2×D418yes—
87C22×C107—22no—
88C22×C107—42no—
89C22×C107—44no—
90C22×C107#12712yes—
91C22×C107C22×D411yescharacteristic
92C2×D107—42no—
93C2×D107—44no—
94C2×D107—84no—
95C4×C107—22no—
96C4×C107—22no—
97C4×C107—42no—
98C4×C107#12712yes—
99C4×C107C22×D412yes—
100Dic107—42no—
101C2×C107—22no—
102C2×C107—82no—
103C2×C107C2×#12711yescharacteristic
104C2×C107#57512yes—
105D107—82no—
106#3153—1071nomaximal, Sylow
107C107#315311yescharacteristic, complemented, Sylow
108C2×#137—1072no—
109C2×AGL(1,ℤ/8)—1072no—
110#532D10711yescharacteristic, complemented
111C2×#124—1071no—
112#497—1072no—
113#496—1072no—
114#495—1074no—
115C4×M16—1071no—
116C2×#41C2×D10712yes—
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×M16—1072no—
126C4⋊Q8C2×D10711yescharacteristic, complemented
127#127C2×D10711yescharacteristic, complemented
128#124—1072no—
129#124—1072no—
130#124—2142no—
131#124C2×D10712yescomplemented
132#121—22no—
133C4×D4—22no—
134C2×C22⋊C4—2142no—
135C2×C42C2×D10711yescharacteristic
136#97—1072no—
137C4×C8—1072no—
138C24—2141no—
139#41—24no—
140#41—24no—
141C2×Q8—1072no—
142C2×Q8—2144no—
143C2×Q8—2141no—
144C2×Q8C22×D10712yes—
145C2×D4—22no—
146C2×D4—22no—
147C2×D4—1072no—
148C2×D4—2144no—
149C2×D4—2141no—
150C2×D4C22×D10712yes—
151C22×C4—22no—
152C22×C4—2141no—
153C22×C4C22⋊D10712yes—
154C22×C4C22×D10711yescharacteristic
155Q16—2148no—
156SD16—2148no—
157SD16—2148no—
158D8—2148no—
159M16—1078no—
160C2×C8—1074no—
161C4⋊C4—22no—
162C22⋊C4—24no—
163C22⋊C4—2144no—
164C22⋊C4—2144no—
165C42C22⋊D10712yes—
166C42C22×D10711yescharacteristic
167C42C22×D10711yescharacteristic
168C23—22no—
169C23—4284no—
170C23—4282no—
171C23—2142no—
172C23C2×C22⋊D10711yescharacteristic
173Q8—22no—
174Q8—22no—
175Q8—4282no—
176Q8—2144no—
177D4—22no—
178D4—22no—
179D4—44no—
180D4—44no—
181D4—4282no—
182D4—2144no—
183C2×C4—22no—
184C2×C4—42no—
185C2×C4—44no—
186C2×C4—4282no—
187C2×C4—2142no—
188C2×C4D4×D10714yes—
189C2×C4C2×C22⋊D10712yes—
190C2×C4C2×C22⋊D10712yes—
191C2×C4C2×C22⋊D10711yescharacteristic
192C2×C4C23×D10711yescharacteristic
193C8—1078no—
194C22—22no—
195C22—42no—
196C22—44no—
197C22—8564no—
198C22—4282no—
199C22—4284no—
200C22#1681210512yes—
201C22C22×C22⋊D10711yescharacteristic
202C4—22no—
203C4—22no—
204C4—42no—
205C4—4282no—
206C4#1681210512yes—
207C4C2×D4×D10712yes—
208C2—22no—
209C2—82no—
210C2—8562no—
211C2C2×#1681210511yescharacteristic
212C2#2029342512yes—
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#532×C107C2
socleC22×C107C22×D4
hypercenter#532D107
hypocenterC107#3153
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. C2×#16812139C2semidirect
2. C2×#16812134C2semidirect
3. C2×#16812097C2semidirect
4. #20293019C2semidirect
5. #20293014C2semidirect
6. #20293009C2semidirect
7. C4×#13133802C2semidirect
8. #532×C107C2semidirect
13. C2×#13133808C22semidirect
14. C2×#13133806C22semidirect
22. #16812020C22semidirect
23. C4×C107⋊C8C22semidirect
25. C4⋊Q8×C107C22semidirect
26. #127×C107C22semidirect
27. #124×C107C22semidirect
43. #13133802D4semidirect
107. C107#3153semidirect
110. #532D107semidirect
126. C4⋊Q8C2×D107semidirect
127. #127C2×D107semidirect
131. #124C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121392
C2×#168121342
C2×#168120972
#202930192
#202930142
#202930092
C4×#131338022
#532×C1072
#3153107