Finite Group Databasebeta

← #28335630

Subgroups — #28335630

Order 13696.

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

normal subgroups only

This group has 194 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#20292919C211yesmaximal, characteristic, complemented
3#20292914C211yesmaximal, characteristic
4#20292373C211yesmaximal, characteristic, complemented
5#20292359C211yesmaximal, characteristic, complemented
6#20292356C211yesmaximal, characteristic
7#487×C107C211yesmaximal, characteristic, complemented
8#16812104—22no—
9#16812098—21no—
10#16812089—21no—
11Q8×Dic107—21no—
12C4×Dic214C2211yescharacteristic
13C4×D428C2211yescharacteristic
14#16812067—21no—
15#16812042—21no—
16#16812040—21no—
17C4⋊(C107⋊C8)C2211yescharacteristic
18C4⋊(C107⋊C8)C2211yescharacteristic
19#16812032—21no—
20#16812029—21no—
21C4⋊Q8×C107C2211yescharacteristic
22C4⋊Q8×C107C2211yescharacteristic
23#103×C107—22no—
24C4×C8×C107C2211yescharacteristic
25C2×Dic214D411yescharacteristic
26C2×D428D411yescharacteristic
27C2×C4×D107—21no—
28C2×C107⋊C8—21no—
29C2×C107⋊C8—21no—
30C4⋊Dic107—22no—
31C4⋊Dic107D411yescharacteristic
32C4⋊Dic107D411yescharacteristic
33Dic107⋊C4—21no—
34(C2×D107)⋊C4—22no—
35(C2×D107)⋊C4—22no—
36(C2×D107)⋊C4—21no—
37C4×Dic107—22no—
38C4×Dic107—21no—
39C2×Q8×C107—21no—
40C2×Q8×C107—21no—
41C2×C8×C107—21no—
42C4⋊C4×C107—21no—
43C4⋊C4×C107—21no—
44C4⋊C4×C107—21no—
45C4⋊C4×C107D412yes—
46C42×C107C2311yescharacteristic
47C22×D107—21no—
48Q8×C107—42no—
49Q8×C107—41no—
50Q8×C107—41no—
51C2×C4×C107—22no—
52C2×C4×C107—41no—
53C2×C4×C107C2×D411yescharacteristic
54C2×C4×C107C2×D411yescharacteristic
55C2×C4×C107C2×D411yescharacteristic
56C2×Dic107—21no—
57C2×Dic107—21no—
58C2×Dic107—21no—
59D428—21no—
60D428—21no—
61C4×D107—41no—
62C4×D107—41no—
63Dic214—21no—
64Dic214—21no—
65C8×C107—41no—
66C107⋊C8—41no—
67C107⋊C8—41no—
68C22×C107#12011yescharacteristic
69C2×D107—41no—
70C2×D107—41no—
71C2×D107—41no—
72C4×C107—21no—
73C4×C107—42no—
74C4×C107—81no—
75C4×C107#13511yescharacteristic
76C4×C107#13511yescharacteristic
77C4×C107#13711yescharacteristic
78C4×C107#13711yescharacteristic
79Dic107—41no—
80Dic107—41no—
81Dic107—41no—
82C2×C107#45211yescharacteristic
83C2×C107#45211yescharacteristic
84C2×C107#45411yescharacteristic
85D107—81no—
86#1638—1071nomaximal, Sylow
87C107#163811yescharacteristic, complemented, Sylow
88#556—1071no—
89#487D10711yescharacteristic, complemented
90#477—1071no—
91#476—1071no—
92#332—1071no—
93Q8⋊C8—1071no—
94D4⋊C8—1071no—
95C4⋊Q8C2×D10711yescharacteristic
96C4⋊Q8C2×D10711yescharacteristic
97#126—2142no—
98#124—2141no—
99C22⋊Q8—2141no—
100C4×Q8—1071no—
101C4×Q8—2141no—
102C4×D4—1071no—
103#117—2141no—
104#107—2141no—
105#106—2141no—
106C4⋊C8—1071no—
107C4⋊C8—1071no—
108#103—22no—
109#103—2141no—
110#102—2141no—
111C4×C8C2×D10711yescharacteristic
112C2×Q8—21no—
113C2×Q8—21no—
114C2×Q8—1071no—
115C2×D4—1071no—
116C22×C4—2141no—
117C2×C8—21no—
118C2×C8—2141no—
119C2×C8—2141no—
120C4⋊C4—21no—
121C4⋊C4—21no—
122C4⋊C4—21no—
123C4⋊C4—1071no—
124C4⋊C4—1071no—
125C4⋊C4—2142no—
126C4⋊C4—2141no—
127C4⋊C4C22⋊D10712yes—
128C22⋊C4—2142no—
129C22⋊C4—2142no—
130C22⋊C4—2141no—
131C42—2142no—
132C42—2141no—
133C42C22×D10711yescharacteristic
134C23—2141no—
135Q8—42no—
136Q8—41no—
137Q8—41no—
138Q8—2141no—
139Q8—2141no—
140D4—2141no—
141D4—2141no—
142C2×C4—22no—
143C2×C4—41no—
144C2×C4—4281no—
145C2×C4—4281no—
146C2×C4—2141no—
147C2×C4—2141no—
148C2×C4—2141no—
149C2×C4D4×D10711yescharacteristic
150C2×C4D4×D10711yescharacteristic
151C2×C4C2×C22⋊D10711yescharacteristic
152C8—41no—
153C8—4281no—
154C8—4281no—
155C22—4281no—
156C22—4281no—
157C22—4281no—
158C22#1681207711yescharacteristic
159C4—21no—
160C4—42no—
161C4—81no—
162C4—4281no—
163C4—4281no—
164C4—4281no—
165C4#1681213011yescharacteristic
166C4#1681213211yescharacteristic
167C4#1681213811yescharacteristic
168C4#1681214111yescharacteristic
169C2—8561no—
170C2#2029278611yescharacteristic
171C2#2029278811yescharacteristic
172C2#2029279911yescharacteristic
subgroupG/H
centerC22#16812077
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
Fitting#487×C107C2
socleC22×C107#120
prime radicalC2×D428D4
hypercenter#487D107
hypocenterC107#1638
LCSG ▷ C42×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
2. #20292919C2semidirect
4. #20292373C2semidirect
5. #20292359C2semidirect
7. #487×C107C2semidirect
87. C107#1638semidirect
89. #487D107semidirect

Maximal subgroups

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

subgroupindex
#202933512
#202929192
#202929142
#202923732
#202923592
#202923562
#487×C1072
#1638107