Finite Group Databasebeta

← #28346913

Subgroups — #28346913

Order 13696.

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

normal subgroups only

This group has 656 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×#16812137C214yesmaximal, complemented
2#20293177C211yesmaximal, characteristic, complemented
3C2×#16812106C211yesmaximal, characteristic, complemented
4#20293016C214yesmaximal, complemented
5#20293004C212yesmaximal, complemented
6#20293000C212yesmaximal, complemented
7C4×M16×C107C211yesmaximal, characteristic, complemented
8C2×#13133818C2212yes—
9C22×D428—22no—
10C22×D428C2212yes—
11#16812137—216no—
12C2×#13133807C2218yescomplemented
13C2×D856C2218yescomplemented
14#16812109C2211yescharacteristic, complemented
15#16812106—22no—
16#16812106C2212yes—
17#16812106C2212yes—
18#16812106C2211yescharacteristic, complemented
19#16812097C2212yescomplemented
20#16812083—22no—
21C4×D428—22no—
22C2×M16×C107C2212yes—
23C2×C42×C107C2211yescharacteristic
24#97×C107C2212yescomplemented
25C4×C8×C107C2212yescomplemented
26C23×D107—22no—
27#13133818—28no—
28C2×Dic214C2312yes—
29C2×D428—28no—
30C2×D428—22no—
31C2×D428—22no—
32C2×D428—24no—
33C2×D428—24no—
34C2×D428—48no—
35C2×D428C2312yes—
36C2×D428C2314yes—
37C2×C22⋊D107—22no—
38C2×C4×D107—22no—
39#13133807—216no—
40D856—216no—
41C4⋊Dic107—22no—
42(C2×D107)⋊C4—24no—
43C22×C4×C107D412yes—
44C22×C4×C107C2311yescharacteristic
45M16×C107D418yescomplemented
46C2×C8×C107C2314yes—
47C42×C107D412yes—
48C42×C107C2311yescharacteristic
49C42×C107C2311yescharacteristic
50C23×C107C2×D411yescharacteristic
51C22×D107—22no—
52C22×D107—24no—
53C22×D107—48no—
54C22×D107—44no—
55C2×C4×C107—24no—
56C2×C4×C107C2×D412yes—
57C2×C4×C107C2×D411yescharacteristic
58C2×C4×C107C2×D414yes—
59C2×C4×C107C2×D412yes—
60C2×C4×C107C2411yescharacteristic
61C22⋊D107—44no—
62C2×Dic107—22no—
63D428—24no—
64D428—28no—
65D428—48no—
66D428—44no—
67D428—44no—
68D428—48no—
69C4×D107—44no—
70Dic214—24no—
71C8×C107C2×D418yescomplemented
72C22×C107—22no—
73C22×C107#12712yes—
74C22×C107C22×D411yescharacteristic
75C2×D107—42no—
76C2×D107—44no—
77C2×D107—48no—
78C2×D107—44no—
79C2×D107—88no—
80C4×C107—22no—
81C4×C107#12712yes—
82C4×C107AGL(1,ℤ/8)14yescomplemented
83C4×C107C22×D412yes—
84Dic107—42no—
85C2×C107—22no—
86C2×C107C2×#12711yescharacteristic
87C2×C107C2×AGL(1,ℤ/8)12yes—
88D107—82no—
89D107—84no—
90#3151—1071nomaximal, Sylow
91C107#315111yescharacteristic, complemented, Sylow
92C2×AGL(1,ℤ/8)—1074no—
93#532—1071no—
94C2×#127—1071no—
95#496—1074no—
96#493—1072no—
97#492—1072no—
98C4×M16D10711yescharacteristic, complemented
99C2×#41—1072no—
100C22×D4—1072no—
101C22×D4—2142no—
102AGL(1,ℤ/8)—21416no—
103C2×SD16—1078no—
104C2×D8—1078no—
105C2×M16C2×D10712yes—
106C4⋊Q8—1071no—
107#127—1072no—
108#127—1072no—
109#127—1071no—
110#127—2142no—
111#124—1072no—
112#121—2142no—
113C4×D4—2142no—
114C2×C42C2×D10711yescharacteristic
115#97C2×D10712yescomplemented
116C4×C8C2×D10712yescomplemented
117C24—2142no—
118#41—2148no—
119C2×Q8—1072no—
120C2×D4—1072no—
121C2×D4—1074no—
122C2×D4—4288no—
123C2×D4—2142no—
124C2×D4—2142no—
125C2×D4—2144no—
126C2×D4—2144no—
127C2×D4—2142no—
128C2×D4—2148no—
129C22×C4—2142no—
130C22×C4D42812yes—
131C22×C4C22×D10711yescharacteristic
132SD16—21416no—
133D8—21416no—
134M16D42818yescomplemented
135C2×C8C22×D10714yes—
136C4⋊C4—2142no—
137C22⋊C4—2144no—
138C42D42812yes—
139C42C22×D10711yescharacteristic
140C42C22×D10711yescharacteristic
141C23—4288no—
142C23—4284no—
143C23—2142no—
144C23—2144no—
145C23C2×D42811yescharacteristic
146Q8—2144no—
147D4—4288no—
148D4—4284no—
149D4—4284no—
150D4—4288no—
151D4—4284no—
152D4—2144no—
153D4—2148no—
154C2×C4—24no—
155C2×C4—4284no—
156C2×C4—2142no—
157C2×C4C2×D42812yes—
158C2×C4C2×D42811yescharacteristic
159C2×C4C2×D42814yes—
160C2×C4C2×D42812yes—
161C2×C4C23×D10711yescharacteristic
162C8C2×D42818yescomplemented
163C22—22no—
164C22—8568no—
165C22—4288no—
166C22—4284no—
167C22—4282no—
168C22—4284no—
169C22#1681210612yes—
170C22C22×D42811yescharacteristic
171C4—22no—
172C4—4282no—
173C4#1681210612yes—
174C4#1681213714yescomplemented
175C4C22×D42812yes—
176C2—22no—
177C2—8564no—
178C2—8562no—
179C2C2×#1681210611yescharacteristic
180C2C2×#1681213712yes—
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC4×M16×C107C2
socleC22×C107C22×D4
hypercenterC4×M16D107
hypocenterC107#3151
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×#16812137C2semidirect
2. #20293177C2semidirect
3. C2×#16812106C2semidirect
4. #20293016C2semidirect
5. #20293004C2semidirect
6. #20293000C2semidirect
7. C4×M16×C107C2semidirect
12. C2×#13133807C22semidirect
13. C2×D856C22semidirect
14. #16812109C22semidirect
18. #16812106C22semidirect
19. #16812097C22semidirect
24. #97×C107C22semidirect
25. C4×C8×C107C22semidirect
45. M16×C107D4semidirect
71. C8×C107C2×D4semidirect
82. C4×C107AGL(1,ℤ/8)semidirect
91. C107#3151semidirect
98. C4×M16D107semidirect
115. #97C2×D107semidirect
116. C4×C8C2×D107semidirect
134. M16D428semidirect
162. C8C2×D428semidirect
174. C4#16812137semidirect

Maximal subgroups

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

subgroupindex
C2×#168121372
#202931772
C2×#168121062
#202930162
#202930042
#202930002
C4×M16×C1072
#3151107