Finite Group Databasebeta

← #28345371

Subgroups — #28345371

Order 13696.

458 conjugacy classes of subgroups; only the 217 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 217 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
1#20293394C212yesmaximal, complemented
2#20292749C214yesmaximal, complemented
3#20292744C214yesmaximal, complemented
4#20292676C211yesmaximal, characteristic, complemented
5C2×#16812036C212yesmaximal, complemented
6M16×Dic107C211yesmaximal, characteristic, complemented
7#575×C107C211yesmaximal, characteristic
8#16812116—24no—
9#16812116C2218yescomplemented
10C2×#13133802—24no—
11C2×#13133802C2212yes—
12C2×#13133802C2211yescharacteristic
13C2×C4×Dic107C2211yescharacteristic
14#16812045C2214yescomplemented
15#16812036C2214yescomplemented
16#16812036C2214yescomplemented
17#16812019C2212yescomplemented
18C8×Dic107C2212yescomplemented
19C2×#41×C107C2212yes—
20#137×C107C414yescomplemented
21AGL(1,ℤ/8)×C107C414yescomplemented
22#135×C107C2214yes—
23C2×M16×C107C2211yescharacteristic
24C22×Dic107—21no—
25#13133802—22no—
26#13133802—28no—
27#13133802D414yes—
28#13133802C2314yes—
29C2×C107⋊C8—22no—
30C2×C107⋊C8—28no—
31C2×C107⋊C8C2312yes—
32C4×Dic107—21no—
33C4×Dic107C2311yescharacteristic
34C4×Dic107C2311yescharacteristic
35#41×C107—22no—
36#41×C107—22no—
37#41×C107C2×C414yes—
38#41×C107C2314yes—
39C2×Q8×C107C2×C412yes—
40C2×D4×C107—22no—
41C2×D4×C107C2×C412yes—
42C22×C4×C107—22no—
43C22×C4×C107C2311yescharacteristic
44Q16×C107C2×C414yescomplemented
45SD16×C107C2×C418yescomplemented
46D8×C107C2×C414yescomplemented
47M16×C107C2×C414yes—
48C2×C8×C107C2312yes—
49C23×C107—22no—
50C23×C107#4111yescharacteristic
51Q8×C107—22no—
52Q8×C107C22×C414yes—
53D4×C107—24no—
54D4×C107—24no—
55D4×C107—22no—
56D4×C107C22×C414yes—
57C2×C4×C107—22no—
58C2×C4×C107—24no—
59C2×C4×C107—24no—
60C2×C4×C107C22×C412yes—
61C2×C4×C107C2×D412yes—
62C2×C4×C107#4111yescharacteristic
63C2×C4×C107C2411yescharacteristic
64C2×Dic107—21no—
65C2×Dic107—21no—
66C2×Dic107—42no—
67C8×C107C22×C414yes—
68C107⋊C8—24no—
69C107⋊C8C2×D414yes—
70C22×C107—24no—
71C22×C107—41no—
72C22×C107—82no—
73C22×C107C4×D412yes—
74C22×C107C2×#4111yescharacteristic
75C4×C107—24no—
76C4×C107C4×D412yes—
77C4×C107C23×C411yescharacteristic
78C4×C107C22×D411yescharacteristic
79Dic107—22no—
80Dic107—41no—
81C2×C107—21no—
82C2×C107—22no—
83C2×C107—44no—
84C2×C107C2×C4×D411yescharacteristic
85#2956—1071nomaximal, Sylow
86C107#295611yescharacteristic, complemented, Sylow
87#575D10711yescharacteristic
88#568—1072no—
89#444—1074no—
90#443—1074no—
91#430—1071no—
92C2×#104—1072no—
93C4×M16—1071no—
94C2×#41C2×D10712yes—
95#137Dic10714yescomplemented
96AGL(1,ℤ/8)Dic10714yescomplemented
97#135C2×D10714yes—
98#131—1078no—
99#131—2144no—
100C2×M16—1071no—
101C2×M16—1072no—
102C2×M16—2144no—
103C2×M16C2×D10711yescharacteristic
104C2×C42—1071no—
105#108—1074no—
106#104—1074no—
107#104—1074no—
108#97—1072no—
109C4×C8—1072no—
110#41—22no—
111#41—22no—
112#41C2×Dic10714yes—
113#41C22×D10714yes—
114C2×Q8C2×Dic10712yes—
115C2×D4—22no—
116C2×D4C2×Dic10712yes—
117C22×C4—22no—
118C22×C4—2141no—
119C22×C4C22×D10711yescharacteristic
120Q16C2×Dic10714yescomplemented
121SD16C2×Dic10718yescomplemented
122D8C2×Dic10714yescomplemented
123M16—1074no—
124M16—1074no—
125M16—2148no—
126M16—2142no—
127M16C2×Dic10714yes—
128C2×C8—1072no—
129C2×C8—2148no—
130C2×C8—2142no—
131C2×C8C22×D10712yes—
132C42—1071no—
133C42—1071no—
134C42—2141no—
135C23—22no—
136C23#1313381711yescharacteristic
137Q8—22no—
138Q8C22×Dic10714yes—
139D4—24no—
140D4—24no—
141D4—22no—
142D4C22×Dic10714yes—
143C2×C4—22no—
144C2×C4—24no—
145C2×C4—24no—
146C2×C4—4282no—
147C2×C4—2141no—
148C2×C4—2141no—
149C2×C4C22×Dic10712yes—
150C2×C4D4×D10712yes—
151C2×C4#1313381711yescharacteristic
152C2×C4C23×D10711yescharacteristic
153C8—1074no—
154C8—2144no—
155C8C22×Dic10714yes—
156C22—24no—
157C22—41no—
158C22—82no—
159C22D4×Dic10712yes—
160C22C2×#1313381711yescharacteristic
161C4—24no—
162C4—4281no—
163C4—2142no—
164C4D4×Dic10712yes—
165C4C23×Dic10711yescharacteristic
166C4C2×D4×D10711yescharacteristic
167C2—21no—
168C2—22no—
169C2—44no—
170C2C2×D4×Dic10711yescharacteristic
subgroupG/H
centerC4C2×D4×D107
commutatorC4×C107C23×C4
FrattiniC2×C4C23×D107
Fitting, prime radical#575×C107C2
socleC2×C107C2×C4×D4
cosocleC2×C4×C107C24
hypercenter#575D107
hypocenterC107#2956
LCSG ▷ C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C4 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #20293394C2semidirect
2. #20292749C2semidirect
3. #20292744C2semidirect
4. #20292676C2semidirect
5. C2×#16812036C2semidirect
6. M16×Dic107C2semidirect
9. #16812116C22semidirect
14. #16812045C22semidirect
15. #16812036C22semidirect
16. #16812036C22semidirect
17. #16812019C22semidirect
18. C8×Dic107C22semidirect
20. #137×C107C4semidirect
21. AGL(1,ℤ/8)×C107C4semidirect
44. Q16×C107C2×C4semidirect
45. SD16×C107C2×C4semidirect
46. D8×C107C2×C4semidirect
86. C107#2956semidirect
95. #137Dic107semidirect
96. AGL(1,ℤ/8)Dic107semidirect
120. Q16C2×Dic107semidirect
121. SD16C2×Dic107semidirect
122. D8C2×Dic107semidirect

Maximal subgroups

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

subgroupindex
#202933942
#202927492
#202927442
#202926762
C2×#168120362
M16×Dic1072
#575×C1072
#2956107