Finite Group Databasebeta

← #28344921

Subgroups — #28344921

Order 13696.

800 conjugacy classes of subgroups; only the 249 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 249 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×#16812154C211yesmaximal, characteristic
2#20293102C211yesmaximal, characteristic, complemented
3#20292600C214yesmaximal, complemented
4C2×#16812023C218yesmaximal, complemented
5#518×C107C211yesmaximal, characteristic, complemented
6#16812154—24no—
7#16812154—24no—
8C2×#13133818C2212yes—
9C2×#13133817C414yescomplemented
10C22×C22⋊D107C2214yes—
11C2×D4×D107C414yescomplemented
12D4×Dic107—24no—
13#16812068C2212yescomplemented
14C2×C22⋊Dic107C2214yescomplemented
15#16812023C22116yescomplemented
16C22×D4×C107C2211yescharacteristic
17C4×D4×C107—24no—
18#117×C107C2212yescomplemented
19C2×C22⋊C4×C107C2214yescomplemented
20C23×D107C2×C414yescomplemented
21#13133818—24no—
22#13133818—24no—
23#13133817—216no—
24C2×Dic214C2×C412yescomplemented
25C2×D428C2×C412yescomplemented
26C2×C22⋊D107—28no—
27C2×C22⋊D107—28no—
28C2×C22⋊D107C2×C418yescomplemented
29C2×C22⋊D107C2314yes—
30D4×D107—216no—
31C22×Dic107—24no—
32C22×Dic107C2×C414yescomplemented
33C2×C4×D107C2×C414yescomplemented
34C4⋊Dic107—22no—
35C22⋊Dic107—24no—
36C22⋊Dic107C2314yes—
37C4×Dic107—22no—
38C24×C107C2312yes—
39C2×D4×C107C2×C414yes—
40C2×D4×C107D414yes—
41C2×D4×C107D414yes—
42C22×C4×C107—24no—
43C22×C4×C107C2311yescharacteristic
44C4⋊C4×C107—22no—
45C22⋊C4×C107—24no—
46C22⋊C4×C107C2314yes—
47C42×C107—22no—
48C23×C107—82no—
49C23×C107C22×C414yes—
50C23×C107C2×D414yes—
51C23×C107C2×D414yes—
52C23×C107C2411yescharacteristic
53C22×D107—28no—
54C22×D107—84no—
55C22×D107C22×C414yes—
56D4×C107C22⋊C4116yes—
57C2×C4×C107—24no—
58C2×C4×C107—84no—
59C2×C4×C107C22×C412yes—
60C2×C4×C107C2×D412yes—
61C2×C4×C107C2×D412yes—
62C22⋊D107—216no—
63C22⋊D107—24no—
64C22⋊D107—24no—
65C2×Dic107—28no—
66C2×Dic107—24no—
67C2×Dic107—84no—
68C2×Dic107C22×C414yes—
69D428—24no—
70C4×D107—28no—
71Dic214—24no—
72C22×C107—44no—
73C22×C107—44no—
74C22×C107—44no—
75C22×C107—41no—
76C22×C107C2×C22⋊C418yes—
77C22×C107C23×C411yescharacteristic
78C22×C107C22×D411yescharacteristic
79C22×C107C22×D411yescharacteristic
80C2×D107—24no—
81C2×D107—44no—
82C2×D107—88no—
83C4×C107—44no—
84C4×C107C2×C22⋊C414yes—
85Dic107—24no—
86Dic107—44no—
87C2×C107—21no—
88C2×C107—21no—
89C2×C107—21no—
90C2×C107—28no—
91C2×C107C22×C22⋊C411yescharacteristic
92D107—44no—
93#2883—1071nomaximal, Sylow
94C107#288311yescharacteristic, complemented, Sylow
95C2×H(5,𝔽2)—1071no—
96#518—1071no—
97#518D10711yescharacteristic, complemented
98#410—1074no—
99C2×#99—1078no—
100H(5,𝔽2)—2144no—
101H(5,𝔽2)—2144no—
102C2×#41—1074no—
103C2×#41—1072no—
104C22×D4—1074no—
105C22×D4—1074no—
106C22×D4C2×D10711yescharacteristic
107C4×D4—24no—
108C4×D4—2144no—
109#117—1072no—
110#117C2×D10712yescomplemented
111C2×C22⋊C4—1074no—
112C2×C22⋊C4C2×D10714yescomplemented
113#99—10716no—
114C24—1074no—
115C24C22×D10712yes—
116#41—21416no—
117#41—2144no—
118#41—2144no—
119C2×Q8—1072no—
120C2×D4—1078no—
121C2×D4—1074no—
122C2×D4—1072no—
123C2×D4—2148no—
124C2×D4—2148no—
125C2×D4—21416no—
126C2×D4C4×D10714yes—
127C2×D4D42814yes—
128C2×D4C22⋊D10714yes—
129C22×C4—24no—
130C22×C4—1074no—
131C22×C4—1074no—
132C22×C4—2144no—
133C22×C4C22×D10711yescharacteristic
134C4⋊C4—22no—
135C4⋊C4—2142no—
136C22⋊C4—24no—
137C22⋊C4—1074no—
138C22⋊C4—2144no—
139C22⋊C4C22×D10714yes—
140C42—22no—
141C42—2142no—
142C23—82no—
143C23—8564no—
144C23—1074no—
145C23—2148no—
146C23C2×C4×D10714yes—
147C23C2×C22⋊D10714yes—
148C23C2×D42814yes—
149C23C23×D10711yescharacteristic
150Q8—2144no—
151D4—21416no—
152D4—2144no—
153D4—2144no—
154D4—2144no—
155D4(C2×D107)⋊C4116yes—
156C2×C4—24no—
157C2×C4—84no—
158C2×C4—8564no—
159C2×C4—1074no—
160C2×C4—2144no—
161C2×C4—2148no—
162C2×C4—2148no—
163C2×C4C2×C4×D10712yes—
164C2×C4C2×C22⋊D10712yes—
165C2×C4C2×D42812yes—
166C22—44no—
167C22—44no—
168C22—41no—
169C22—44no—
170C22—8568no—
171C22—4284no—
172C22—2144no—
173C22C2×(C2×D107)⋊C418yes—
174C22C22×C4×D10711yescharacteristic
175C22C22×C22⋊D10711yescharacteristic
176C22C22×D42811yescharacteristic
177C4—44no—
178C4—4284no—
179C4—2144no—
180C4C2×(C2×D107)⋊C414yes—
181C2—21no—
182C2—21no—
183C2—21no—
184C2—28no—
185C2—4284no—
186C2C22×(C2×D107)⋊C411yescharacteristic
subgroupG/H
centerC2C22×(C2×D107)⋊C4
commutatorC22×C107C23×C4
FrattiniC23C23×D107
Fitting#518×C107C2
socleC2×C107C22×C22⋊C4
prime radicalC2×#16812154C2
cosocleC23×C107C24
hypercenter#518D107
hypocenterC107#2883
LCSG ▷ C22×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22×D4 ◁ #518

Complements

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

NH ≅ G/Ntype
2. #20293102C2semidirect
3. #20292600C2semidirect
4. C2×#16812023C2semidirect
5. #518×C107C2semidirect
9. C2×#13133817C4semidirect
11. C2×D4×D107C4semidirect
13. #16812068C22semidirect
14. C2×C22⋊Dic107C22semidirect
15. #16812023C22semidirect
18. #117×C107C22semidirect
19. C2×C22⋊C4×C107C22semidirect
20. C23×D107C2×C4semidirect
24. C2×Dic214C2×C4semidirect
25. C2×D428C2×C4semidirect
28. C2×C22⋊D107C2×C4semidirect
32. C22×Dic107C2×C4semidirect
33. C2×C4×D107C2×C4semidirect
94. C107#2883semidirect
97. #518D107semidirect
110. #117C2×D107semidirect
112. C2×C22⋊C4C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121542
#202931022
#202926002
C2×#168120232
#518×C1072
#2883107