Finite Group Databasebeta

← #28335714

Subgroups — #28335714

Order 13696.

200 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 53 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#20293351C211yesmaximal, characteristic
2#20292909C211yesmaximal, characteristic, complemented
3#20292829C211yesmaximal, characteristic
4#20292373C211yesmaximal, characteristic, complemented
5#20292359C211yesmaximal, characteristic, complemented
6#20292356C211yesmaximal, characteristic
7#500×C107C211yesmaximal, characteristic, complemented
8C2×#13133805—21no—
9#16812104—22no—
10#16812098—21no—
11#16812089—21no—
12Q8×Dic107—21no—
13C4×Dic214C2211yescharacteristic
14C4×D428C2211yescharacteristic
15#16812067—21no—
16#16812040—21no—
17C4⋊(C107⋊C8)C2211yescharacteristic
18C4⋊(C107⋊C8)C2211yescharacteristic
19#16812033—21no—
20#16812032—21no—
21C4⋊Q8×C107C2211yescharacteristic
22C4⋊Q8×C107C2211yescharacteristic
23#107×C107—22no—
24C4×C8×C107C2211yescharacteristic
25C2×Dic214D411yescharacteristic
26C2×D428D411yescharacteristic
27C2×C4×D107—21no—
28#13133805—42no—
29C2×C107⋊C8—21no—
30C2×C107⋊C8—21no—
31C4⋊Dic107—22no—
32C4⋊Dic107D411yescharacteristic
33C4⋊Dic107D411yescharacteristic
34Dic107⋊C4—21no—
35(C2×D107)⋊C4—22no—
36(C2×D107)⋊C4—22no—
37(C2×D107)⋊C4—21no—
38C4×Dic107—22no—
39C4×Dic107—21no—
40C2×Q8×C107—21no—
41C2×Q8×C107—21no—
42C2×C8×C107—21no—
43C4⋊C4×C107—21no—
44C4⋊C4×C107—21no—
45C4⋊C4×C107—21no—
46C4⋊C4×C107D412yes—
47C42×C107C2311yescharacteristic
48C22×D107—21no—
49Q8×C107—41no—
50Q8×C107—41no—
51Q8×C107—41no—
52Q8×C107—41no—
53C2×C4×C107—22no—
54C2×C4×C107—41no—
55C2×C4×C107C2×D411yescharacteristic
56C2×C4×C107C2×D411yescharacteristic
57C2×C4×C107C2×D411yescharacteristic
58C2×Dic107—21no—
59C2×Dic107—21no—
60C2×Dic107—21no—
61D428—21no—
62D428—21no—
63C4×D107—41no—
64C4×D107—41no—
65Dic214—21no—
66Dic214—21no—
67C8×C107—22no—
68C107⋊C8—41no—
69C107⋊C8—41no—
70C22×C107#12011yescharacteristic
71C2×D107—41no—
72C2×D107—41no—
73C2×D107—41no—
74C4×C107—21no—
75C4×C107—42no—
76C4×C107—81no—
77C4×C107#13511yescharacteristic
78C4×C107#13511yescharacteristic
79C4×C107AGL(1,ℤ/8)11yescharacteristic
80C4×C107#13711yescharacteristic
81Dic107—41no—
82Dic107—41no—
83Dic107—41no—
84C2×C107#44911yescharacteristic
85C2×C107#45211yescharacteristic
86C2×C107#45611yescharacteristic
87D107—81no—
88#1650—1071nomaximal, Sylow
89C107#165011yescharacteristic, complemented, Sylow
90#556—1071no—
91#500D10711yescharacteristic, complemented
92#475—1071no—
93#461—1071no—
94#332—1071no—
95Q8⋊C8—1071no—
96D4⋊C8—1071no—
97C2×SD16—2141no—
98C4⋊Q8C2×D10711yescharacteristic
99C4⋊Q8C2×D10711yescharacteristic
100#126—2142no—
101#124—2141no—
102C22⋊Q8—2141no—
103C4×Q8—1071no—
104C4×Q8—2141no—
105C4×D4—1071no—
106#117—2141no—
107#107—22no—
108#106—2141no—
109C4⋊C8—1071no—
110C4⋊C8—1071no—
111#103—2141no—
112#103—2141no—
113C4×C8C2×D10711yescharacteristic
114C2×Q8—21no—
115C2×Q8—21no—
116C2×Q8—1071no—
117C2×D4—1071no—
118C22×C4—2141no—
119SD16—4282no—
120C2×C8—21no—
121C2×C8—2141no—
122C2×C8—2141no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—1071no—
127C4⋊C4—1071no—
128C4⋊C4—2142no—
129C4⋊C4—2141no—
130C4⋊C4C22⋊D10712yes—
131C22⋊C4—2142no—
132C22⋊C4—2142no—
133C22⋊C4—2141no—
134C42—2142no—
135C42—2141no—
136C42C22×D10711yescharacteristic
137C23—2141no—
138Q8—41no—
139Q8—41no—
140Q8—41no—
141Q8—41no—
142Q8—2141no—
143Q8—2141no—
144D4—2141no—
145D4—2141no—
146C2×C4—22no—
147C2×C4—41no—
148C2×C4—4281no—
149C2×C4—4281no—
150C2×C4—2141no—
151C2×C4—2141no—
152C2×C4—2141no—
153C2×C4D4×D10711yescharacteristic
154C2×C4D4×D10711yescharacteristic
155C2×C4C2×C22⋊D10711yescharacteristic
156C8—22no—
157C8—4281no—
158C8—4281no—
159C22—4281no—
160C22—4281no—
161C22—4281no—
162C22#1681207711yescharacteristic
163C4—21no—
164C4—42no—
165C4—81no—
166C4—4281no—
167C4—4281no—
168C4—4281no—
169C4#1681212811yescharacteristic
170C4#1681213211yescharacteristic
171C4#1681213311yescharacteristic
172C4#1681214111yescharacteristic
173C2—8561no—
174C2#2029276911yescharacteristic
175C2#2029278811yescharacteristic
176C2#2029280911yescharacteristic
subgroupG/H
centerC22#16812077
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
Fitting#500×C107C2
socleC22×C107#120
prime radicalC2×D428D4
hypercenter#500D107
hypocenterC107#1650
LCSG ▷ C42×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
2. #20292909C2semidirect
4. #20292373C2semidirect
5. #20292359C2semidirect
7. #500×C107C2semidirect
89. C107#1650semidirect
91. #500D107semidirect

Maximal subgroups

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

subgroupindex
#202933512
#202929092
#202928292
#202923732
#202923592
#202923562
#500×C1072
#1650107