Finite Group Databasebeta

← #28345363

Subgroups — #28345363

Order 13696.

456 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 177 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#20293427C211yesmaximal, characteristic
2C2×#16812117C212yesmaximal, complemented
3#20292748C214yesmaximal, complemented
4#20292743C214yesmaximal, complemented
5#20292643C212yesmaximal, complemented
6#20292585C211yesmaximal, characteristic, complemented
7#430×C107C211yesmaximal, characteristic, complemented
8C2×#13133818C2212yes—
9#16812142C414yescomplemented
10#16812137C414yescomplemented
11#16812131C2214yes—
12#16812117—22no—
13#16812117—22no—
14#16812117C2214yescomplemented
15#16812117C2214yescomplemented
16C2×#13133801—22no—
17C22×C107⋊C8C2211yescharacteristic
18C2×C8×D107—22no—
19#16812068C2211yescharacteristic
20#16812035C2218yescomplemented
21#16812019C2212yescomplemented
22C8×Dic107C2212yescomplemented
23C2×M16×C107C2212yes—
24C2×M16×C107C2211yescharacteristic
25#108×C107C2214yescomplemented
26#13133818—22no—
27#13133818—22no—
28#13133818C2×C414yes—
29#13133818C2314yes—
30C2×Dic214C2×C412yes—
31C2×D428C2×C412yes—
32C2×C22⋊D107—22no—
33C2×C4×D107—22no—
34Dic428C2×C414yes—
35#13133807C2×C418yes—
36D856C2×C414yes—
37#13133801—24no—
38#13133801—24no—
39C2×C107⋊C8D412yes—
40C2×C107⋊C8D412yes—
41C2×C107⋊C8C2311yescharacteristic
42C2×C107⋊C8C2311yescharacteristic
43C8×D107—24no—
44C8×D107—24no—
45C4⋊Dic107—21no—
46C22⋊Dic107—21no—
47C4×Dic107C2312yes—
48C22×C4×C107C2311yescharacteristic
49M16×C107—22no—
50M16×C107C2×C414yes—
51M16×C107C2314yes—
52C2×C8×C107—22no—
53C2×C8×C107C2312yes—
54C23×C107#4111yescharacteristic
55C22×D107—22no—
56C2×C4×C107C22×C412yes—
57C2×C4×C107C2×D412yes—
58C2×C4×C107#4111yescharacteristic
59C2×C4×C107C2411yescharacteristic
60C22⋊D107—24no—
61C22⋊D107—24no—
62C2×Dic107—22no—
63C2×Dic107—22no—
64D428—22no—
65D428C22×C414yes—
66C4×D107—24no—
67C4×D107—24no—
68Dic214—22no—
69Dic214C22×C414yes—
70C8×C107—24no—
71C8×C107C22×C414yes—
72C107⋊C8C2×D411yescharacteristic
73C107⋊C8C2×D412yes—
74C107⋊C8C2×D411yescharacteristic
75C22×C107—41no—
76C22×C107C4×D412yes—
77C22×C107C2×#4111yescharacteristic
78C2×D107—24no—
79C2×D107—44no—
80C4×C107C4×D412yes—
81C4×C107C23×C411yescharacteristic
82C4×C107C22×D411yescharacteristic
83Dic107—24no—
84Dic107—42no—
85C2×C107—21no—
86C2×C107—22no—
87C2×C107C2×C4×D411yescharacteristic
88D107—44no—
89#2955—1071nomaximal, Sylow
90C107#295511yescharacteristic, complemented, Sylow
91#575—1071no—
92C2×#131—1072no—
93#444—1074no—
94#443—1074no—
95#430D10711yescharacteristic, complemented
96#421—1072no—
97#405—1071no—
98C2×#41—1072no—
99#137—1074no—
100AGL(1,ℤ/8)—1074no—
101#135—1074no—
102#131—1074no—
103#131—1074no—
104#131—2142no—
105#131—2142no—
106C2×M16—2142no—
107C2×M16C2×D10712yes—
108C2×M16C2×D10711yescharacteristic
109C22×C8—1071no—
110C22×C8—2142no—
111#117—1071no—
112#108C2×D10714yescomplemented
113#104—1078no—
114#97—1072no—
115C4×C8—1072no—
116#41—1074no—
117#41—1074no—
118#41—2142no—
119#41—2142no—
120C2×Q8—1072no—
121C2×D4—1072no—
122C2×D4—2142no—
123C22×C4—2142no—
124C22×C4C22×D10711yescharacteristic
125Q16—1074no—
126SD16—1078no—
127D8—1074no—
128M16—22no—
129M16—2144no—
130M16—2144no—
131M16C4×D10714yes—
132M16C22×D10714yes—
133C2×C8—22no—
134C2×C8—1072no—
135C2×C8—1072no—
136C2×C8—1071no—
137C2×C8—1071no—
138C2×C8—2144no—
139C2×C8—2144no—
140C2×C8C22×D10712yes—
141C4⋊C4—2141no—
142C22⋊C4—2141no—
143C42—1072no—
144C23—2142no—
145C23#1313381911yescharacteristic
146Q8—1074no—
147Q8—2142no—
148D4—1074no—
149D4—2144no—
150D4—2144no—
151D4—2142no—
152C2×C4—2142no—
153C2×C4—2142no—
154C2×C4—2144no—
155C2×C4—2144no—
156C2×C4C2×C4×D10712yes—
157C2×C4D4×D10712yes—
158C2×C4#1313381911yescharacteristic
159C2×C4C23×D10711yescharacteristic
160C8—24no—
161C8—1071no—
162C8—1072no—
163C8—1071no—
164C8C2×C4×D10714yes—
165C22—41no—
166C22—4284no—
167C22—2144no—
168C22#1681207112yes—
169C22C2×#1313381911yescharacteristic
170C4—4282no—
171C4—2144no—
172C4#1681207112yes—
173C4C22×C4×D10711yescharacteristic
174C4C2×D4×D10711yescharacteristic
175C2—21no—
176C2—22no—
177C2—4284no—
178C2C2×#1681207111yescharacteristic
subgroupG/H
centerC4C2×D4×D107
commutatorC4×C107C23×C4
FrattiniC2×C4C23×D107
Fitting#430×C107C2
socleC2×C107C2×C4×D4
prime radical#20293427C2
cosocleC2×C4×C107C24
hypercenter#430D107
hypocenterC107#2955
LCSG ▷ C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C4 ◁ C22×C4 ◁ #430

Complements

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

NH ≅ G/Ntype
2. C2×#16812117C2semidirect
3. #20292748C2semidirect
4. #20292743C2semidirect
5. #20292643C2semidirect
6. #20292585C2semidirect
7. #430×C107C2semidirect
9. #16812142C4semidirect
10. #16812137C4semidirect
14. #16812117C22semidirect
15. #16812117C22semidirect
20. #16812035C22semidirect
21. #16812019C22semidirect
22. C8×Dic107C22semidirect
25. #108×C107C22semidirect
90. C107#2955semidirect
95. #430D107semidirect
112. #108C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202934272
C2×#168121172
#202927482
#202927432
#202926432
#202925852
#430×C1072
#2955107