Finite Group Databasebeta

← #28346942

Subgroups — #28346942

Order 13696.

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

normal subgroups only

This group has 596 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×#16812135C212yesmaximal, complemented
2C2×#16812131C211yesmaximal, characteristic, complemented
3#20293194C212yesmaximal, complemented
4#20293015C214yesmaximal, complemented
5#20293010C212yesmaximal, complemented
6#20293002C212yesmaximal, complemented
7#20292582C211yesmaximal, characteristic, complemented
8C22×D8×C107C211yesmaximal, characteristic, complemented
9C2×#13133818C2212yes—
10#16812135—28no—
11#16812131—24no—
12C2×Dic428C2211yescharacteristic, complemented
13C2×#13133807C2212yescomplemented
14C2×#13133806C2214yescomplemented
15C2×D856C2211yescharacteristic, complemented
16C2×#13133803C2214yescomplemented
17C2×#13133802C2211yescharacteristic
18#16812105C2214yescomplemented
19#16812096C2214yescomplemented
20#16812080—24no—
21#16812078—24no—
22#16812068C2211yescharacteristic
23#16812019C2212yescomplemented
24C8×Dic107C2212yescomplemented
25C22×D4×C107C2212yes—
26C2×D8×C107—24no—
27C2×D8×C107C2212yes—
28C2×D8×C107C2212yes—
29C22×C8×C107C2211yescharacteristic
30#13133818—24no—
31#13133818—24no—
32C2×Dic214C2312yes—
33C2×D428C2312yes—
34C2×C22⋊D107—22no—
35C2×C22⋊D107—28no—
36C2×C4×D107—22no—
37Dic428—22no—
38#13133807—24no—
39#13133806—28no—
40D856—22no—
41#13133803—28no—
42#13133802D414yescomplemented
43C2×C107⋊C8C2312yes—
44C4⋊Dic107D412yes—
45C22⋊Dic107—28no—
46C22⋊Dic107D412yes—
47C4×Dic107C2312yes—
48C24×C107—22no—
49C2×D4×C107—24no—
50C2×D4×C107—24no—
51C2×D4×C107—22no—
52C2×D4×C107C2314yes—
53C22×C4×C107C2311yescharacteristic
54D8×C107—24no—
55D8×C107—24no—
56C2×C8×C107D414yescomplemented
57C2×C8×C107C2311yescharacteristic
58C2×C8×C107C2311yescharacteristic
59C23×C107—24no—
60C23×C107—44no—
61C23×C107—84no—
62C23×C107C2×D411yescharacteristic
63C22×D107—22no—
64D4×C107—24no—
65D4×C107—24no—
66D4×C107—44no—
67C2×C4×C107C2×D411yescharacteristic
68C2×C4×C107C2×D412yes—
69C2×C4×C107C2×D412yes—
70C2×C4×C107C2411yescharacteristic
71C22⋊D107—44no—
72C22⋊D107—88no—
73C2×Dic107—22no—
74C2×Dic107C2×D414yes—
75D428—22no—
76D428—22no—
77C4×D107—44no—
78Dic214—22no—
79Dic214—22no—
80C8×C107C2×D412yes—
81C8×C107C2×D412yes—
82C107⋊C8C2×D414yes—
83C22×C107—22no—
84C22×C107—44no—
85C22×C107—48no—
86C22×C107—88no—
87C22×C107#12712yes—
88C22×C107C22×D411yescharacteristic
89C2×D107—42no—
90C2×D107—44no—
91C4×C107#12712yes—
92C4×C107C22×D411yescharacteristic
93C4×C107C22×D411yescharacteristic
94Dic107—24no—
95Dic107—42no—
96C2×C107—22no—
97C2×C107—84no—
98C2×C107C2×#12711yescharacteristic
99C2×C107#57612yes—
100D107—82no—
101#3155—1071nomaximal, Sylow
102C107#315511yescharacteristic, complemented, Sylow
103C2×AGL(1,ℤ/8)—1072no—
104C2×#135—1071no—
105C22×D8D10711yescharacteristic, complemented
106#535—1072no—
107#496—1074no—
108#495—1072no—
109#493—1072no—
110#405—1071no—
111C2×#41—1072no—
112C22×D4C2×D10712yes—
113AGL(1,ℤ/8)—2148no—
114#135—2144no—
115C2×Q16—1071no—
116C2×SD16—1074no—
117C2×SD16—1072no—
118C2×D8—24no—
119C2×D8—1074no—
120C2×D8—1071no—
121C2×D8C2×D10712yes—
122C2×D8C2×D10712yes—
123C2×M16—1071no—
124C22×C8C2×D10711yescharacteristic
125#127—1074no—
126#124—1074no—
127#121—2144no—
128#120—2144no—
129#117—1071no—
130#97—1072no—
131C4×C8—1072no—
132C24—22no—
133#41—2144no—
134#41—2144no—
135C2×Q8—1072no—
136C2×D4—24no—
137C2×D4—22no—
138C2×D4—24no—
139C2×D4—1072no—
140C2×D4—2148no—
141C2×D4—2142no—
142C2×D4C22×D10714yes—
143C22×C4—2142no—
144C22×C4C22×D10711yescharacteristic
145Q16—2142no—
146SD16—2148no—
147SD16—2144no—
148D8—24no—
149D8—24no—
150D8—2148no—
151D8—2142no—
152M16—1074no—
153C2×C8—1072no—
154C2×C8C22⋊D10714yescomplemented
155C2×C8C22×D10711yescharacteristic
156C2×C8C22×D10711yescharacteristic
157C4⋊C4—1072no—
158C22⋊C4—1072no—
159C22⋊C4—2148no—
160C42—1072no—
161C23—24no—
162C23—44no—
163C23—84no—
164C23—2142no—
165C23D4×D10711yescharacteristic
166Q8—2142no—
167Q8—2142no—
168D4—24no—
169D4—24no—
170D4—44no—
171D4—8568no—
172D4—4284no—
173D4—2142no—
174D4—2142no—
175C2×C4—1074no—
176C2×C4—4284no—
177C2×C4—2142no—
178C2×C4D4×D10711yescharacteristic
179C2×C4D4×D10712yes—
180C2×C4C2×C22⋊D10712yes—
181C2×C4C23×D10711yescharacteristic
182C8—1074no—
183C8C2×C22⋊D10712yes—
184C8C2×C22⋊D10712yes—
185C22—22no—
186C22—44no—
187C22—48no—
188C22—88no—
189C22—4282no—
190C22—4284no—
191C22#1681210512yes—
192C22C2×D4×D10711yescharacteristic
193C4—4282no—
194C4—2144no—
195C4#1681210512yes—
196C4C2×D4×D10711yescharacteristic
197C4C22×C22⋊D10711yescharacteristic
198C2—22no—
199C2—84no—
200C2—8562no—
201C2C2×#1681210511yescharacteristic
202C2#2029342812yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC22×D8×C107C2
socleC22×C107C22×D4
hypercenterC22×D8D107
hypocenterC107#3155
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C2×#16812135C2semidirect
2. C2×#16812131C2semidirect
3. #20293194C2semidirect
4. #20293015C2semidirect
5. #20293010C2semidirect
6. #20293002C2semidirect
7. #20292582C2semidirect
8. C22×D8×C107C2semidirect
12. C2×Dic428C22semidirect
13. C2×#13133807C22semidirect
14. C2×#13133806C22semidirect
15. C2×D856C22semidirect
16. C2×#13133803C22semidirect
18. #16812105C22semidirect
19. #16812096C22semidirect
23. #16812019C22semidirect
24. C8×Dic107C22semidirect
42. #13133802D4semidirect
56. C2×C8×C107D4semidirect
102. C107#3155semidirect
105. C22×D8D107semidirect
154. C2×C8C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121352
C2×#168121312
#202931942
#202930152
#202930102
#202930022
#202925822
C22×D8×C1072
#3155107