Finite Group Databasebeta

← #28335592

Subgroups — #28335592

Order 13696.

214 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 55 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#20293338C211yesmaximal, characteristic
2#20292919C211yesmaximal, characteristic, complemented
3#20292914C211yesmaximal, characteristic
4#20292376C211yesmaximal, characteristic, complemented
5#20292359C211yesmaximal, characteristic, complemented
6#20292356C211yesmaximal, characteristic
7#494×C107C211yesmaximal, characteristic, complemented
8#16812100—21no—
9#16812089—21no—
10#16812085—22no—
11C4×Dic214C2211yescharacteristic
12C4×D428C2211yescharacteristic
13#16812071—21no—
14C4⋊C4×D107—21no—
15#16812042—21no—
16#16812040—21no—
17C4⋊(C107⋊C8)C2211yescharacteristic
18C4⋊(C107⋊C8)C2211yescharacteristic
19#16812032—21no—
20#16812029—21no—
21C2×Q16×C107—22no—
22C4⋊Q8×C107C2211yescharacteristic
23C4⋊Q8×C107C2211yescharacteristic
24C4×C8×C107C2211yescharacteristic
25C2×Dic214D411yescharacteristic
26C2×D428D411yescharacteristic
27C2×C4×D107—21no—
28C2×C4×D107—22no—
29C2×C107⋊C8—21no—
30C2×C107⋊C8—21no—
31C4⋊Dic107—22no—
32C4⋊Dic107D411yescharacteristic
33C4⋊Dic107D411yescharacteristic
34Dic107⋊C4—22no—
35Dic107⋊C4—21no—
36(C2×D107)⋊C4—22no—
37(C2×D107)⋊C4—21no—
38C4×Dic107—21no—
39C2×Q8×C107—21no—
40C2×Q8×C107D412yes—
41Q16×C107—44no—
42C2×C8×C107—21no—
43C4⋊C4×C107—21no—
44C4⋊C4×C107—21no—
45C4⋊C4×C107—21no—
46C4⋊C4×C107—21no—
47C42×C107C2311yescharacteristic
48C22×D107—21no—
49Q8×C107—42no—
50Q8×C107—42no—
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—
60D428Q1612yescomplemented
61C4×D107—41no—
62C4×D107—41no—
63C4×D107—82no—
64Dic214—21no—
65Dic214—21no—
66C8×C107—22no—
67C107⋊C8—41no—
68C107⋊C8—41no—
69C22×C107#12011yescharacteristic
70C2×D107—22no—
71C2×D107—41no—
72C2×D107—41no—
73C4×C107—21no—
74C4×C107—42no—
75C4×C107—81no—
76C4×C107C2×Q1611yescharacteristic
77C4×C107#13511yescharacteristic
78C4×C107#13711yescharacteristic
79C4×C107#13711yescharacteristic
80Dic107—41no—
81Dic107—41no—
82Dic107—41no—
83C2×C107#45111yescharacteristic
84C2×C107#45211yescharacteristic
85C2×C107#45311yescharacteristic
86D107—42no—
87#1632—1071nomaximal, Sylow
88C107#163211yescharacteristic, complemented, Sylow
89#554—1071no—
90#494D10711yescharacteristic, complemented
91#477—1071no—
92#476—1071no—
93#333—1071no—
94Q8⋊C8—1071no—
95D4⋊C8—1071no—
96C2×Q16—22no—
97C4⋊Q8C2×D10711yescharacteristic
98C4⋊Q8C2×D10711yescharacteristic
99#125—2141no—
100C22⋊Q8—2141no—
101C22⋊Q8—2142no—
102C4×Q8—1071no—
103C4×D4—1071no—
104C4×D4—2141no—
105C2×C4⋊C4—2141no—
106#107—2141no—
107#106—2141no—
108C4⋊C8—1071no—
109C4⋊C8—1071no—
110#103—2141no—
111#102—2141no—
112C4×C8C2×D10711yescharacteristic
113C2×Q8—21no—
114C2×Q8—1071no—
115C2×Q8C22⋊D10712yes—
116C2×D4—1071no—
117C22×C4—2142no—
118C22×C4—2141no—
119Q16—44no—
120C2×C8—21no—
121C2×C8—2141no—
122C2×C8—2141no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—1071no—
128C4⋊C4—1071no—
129C4⋊C4—2142no—
130C4⋊C4—2142no—
131C4⋊C4—2141no—
132C22⋊C4—2142no—
133C22⋊C4—2141no—
134C42—2141no—
135C42C22×D10711yescharacteristic
136C23—2141no—
137Q8—42no—
138Q8—42no—
139Q8—2141no—
140Q8—2141no—
141D4—1072no—
142D4—2141no—
143C2×C4—22no—
144C2×C4—41no—
145C2×C4—8562no—
146C2×C4—4281no—
147C2×C4—4281no—
148C2×C4—2141no—
149C2×C4—2141no—
150C2×C4—2141no—
151C2×C4D4×D10711yescharacteristic
152C2×C4D4×D10711yescharacteristic
153C2×C4C2×C22⋊D10711yescharacteristic
154C8—22no—
155C8—4281no—
156C8—4281no—
157C22—4281no—
158C22—4281no—
159C22—2142no—
160C22#1681207711yescharacteristic
161C4—21no—
162C4—42no—
163C4—81no—
164C4—4281no—
165C4—4281no—
166C4—4281no—
167C4Q16×D10711yescharacteristic
168C4#1681213211yescharacteristic
169C4#1681214111yescharacteristic
170C4#1681214111yescharacteristic
171C2—4282no—
172C2#2029278111yescharacteristic
173C2#2029278811yescharacteristic
174C2#2029279411yescharacteristic
subgroupG/H
centerC22#16812077
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
Fitting#494×C107C2
socleC22×C107#120
prime radicalC2×D428D4
hypercenter#494D107
hypocenterC107#1632
LCSG ▷ C42×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #494

Complements

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

NH ≅ G/Ntype
2. #20292919C2semidirect
4. #20292376C2semidirect
5. #20292359C2semidirect
7. #494×C107C2semidirect
60. D428Q16semidirect
88. C107#1632semidirect
90. #494D107semidirect

Maximal subgroups

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

subgroupindex
#202933382
#202929192
#202929142
#202923762
#202923592
#202923562
#494×C1072
#1632107