Finite Group Databasebeta

← #28335581

Subgroups — #28335581

Order 13696.

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

normal subgroups only

This group has 218 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#20293362C211yesmaximal, characteristic
2#20293001C211yesmaximal, characteristic
3#20292919C211yesmaximal, characteristic, complemented
4#20292370C211yesmaximal, characteristic
5Q8⋊(C107⋊C8)C211yesmaximal, characteristic, complemented
6#20292357C211yesmaximal, characteristic, complemented
7#477×C107C211yesmaximal, characteristic, complemented
8C2×#13133805—22no—
9#16812106C2211yescharacteristic
10#16812098—21no—
11#16812097—21no—
12#16812094—22no—
13C4×Dic214C2211yescharacteristic
14C4×D428—21no—
15#16812071—21no—
16#16812042—21no—
17C4⋊(C107⋊C8)C2211yescharacteristic
18#16812032—21no—
19C4×C107⋊C8C2211yescharacteristic
20C4⋊Q8×C107C2211yescharacteristic
21C4×Q8×C107C2211yescharacteristic
22#107×C107—21no—
23C4⋊C8×C107C2211yescharacteristic
24#103×C107—21no—
25C2×Dic214D411yescharacteristic
26C2×D428—21no—
27C2×D428—21no—
28C2×D428D412yes—
29C2×C4×D107—22no—
30#13133805—44no—
31C2×C107⋊C8—21no—
32C2×C107⋊C8—21no—
33C4⋊Dic107D411yescharacteristic
34Dic107⋊C4—21no—
35(C2×D107)⋊C4—22no—
36(C2×D107)⋊C4—22no—
37(C2×D107)⋊C4—22no—
38C4×Dic107—21no—
39C2×Q8×C107—21no—
40C2×Q8×C107D411yescharacteristic
41C2×C8×C107—21no—
42C4⋊C4×C107—21no—
43C4⋊C4×C107—21no—
44C4⋊C4×C107—21no—
45C4⋊C4×C107D411yescharacteristic
46C42×C107—21no—
47C42×C107C2311yescharacteristic
48C22×D107—22no—
49Q8×C107—21no—
50Q8×C107—21no—
51Q8×C107—42no—
52C2×C4×C107—21no—
53C2×C4×C107—21no—
54C2×C4×C107—41no—
55C2×C4×C107C2×D411yescharacteristic
56C2×C4×C107C2×D411yescharacteristic
57C2×C4×C107C2×D411yescharacteristic
58C2×Dic107—21no—
59C2×Dic107—21no—
60D428—42no—
61D428—41no—
62D428—41no—
63D428—41no—
64D428—41no—
65C4×D107—82no—
66Dic214—21no—
67Dic214—21no—
68C8×C107—41no—
69C107⋊C8—22no—
70C107⋊C8—41no—
71C22×C107#12011yescharacteristic
72C2×D107—42no—
73C2×D107—42no—
74C2×D107—42no—
75C4×C107—21no—
76C4×C107—41no—
77C4×C107—41no—
78C4×C107—81no—
79C4×C107#13511yescharacteristic
80C4×C107#13511yescharacteristic
81C4×C107#13711yescharacteristic
82C4×C107#13711yescharacteristic
83Dic107—41no—
84Dic107—41no—
85C2×C107#45211yescharacteristic
86C2×C107#45211yescharacteristic
87C2×C107#45311yescharacteristic
88D107—82no—
89#1630—1071nomaximal, Sylow
90C107#163011yescharacteristic, complemented, Sylow
91#559—1071no—
92#492—1071no—
93#477—1071no—
94#477D10711yescharacteristic, complemented
95#331—1071no—
96Q8⋊C8—1071no—
97Q8⋊C8—1071no—
98C2×SD16—2142no—
99C4⋊Q8C2×D10711yescharacteristic
100#127—1071no—
101#124—2141no—
102#124—2141no—
103#123—2142no—
104C4×Q8—1071no—
105C4×Q8C2×D10711yescharacteristic
106C4×D4—2141no—
107C4×D4—2141no—
108#107—21no—
109#107—2141no—
110C4⋊C8—1071no—
111C4⋊C8C2×D10711yescharacteristic
112#103—21no—
113#103—2141no—
114C4×C8—1071no—
115C2×Q8—21no—
116C2×Q8—1071no—
117C2×Q8C22⋊D10711yescharacteristic
118C2×D4—1072no—
119C2×D4—2141no—
120C2×D4—2141no—
121C22×C4—2142no—
122SD16—4284no—
123C2×C8—21no—
124C2×C8—2141no—
125C2×C8—2141no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—1071no—
130C4⋊C4—2141no—
131C4⋊C4C22⋊D10711yescharacteristic
132C22⋊C4—2142no—
133C22⋊C4—2142no—
134C22⋊C4—2142no—
135C42—21no—
136C42—2141no—
137C42C22×D10711yescharacteristic
138C23—2142no—
139Q8—21no—
140Q8—21no—
141Q8—42no—
142Q8—2141no—
143Q8—2141no—
144D4—4282no—
145D4—4281no—
146D4—4281no—
147D4—4281no—
148D4—4281no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—8562no—
153C2×C4—2141no—
154C2×C4—2141no—
155C2×C4D4×D10711yescharacteristic
156C2×C4D4×D10711yescharacteristic
157C2×C4C2×C22⋊D10711yescharacteristic
158C8—41no—
159C8—4281no—
160C8—2142no—
161C22—4282no—
162C22—4282no—
163C22—4282no—
164C22#1681207711yescharacteristic
165C4—21no—
166C4—41no—
167C4—41no—
168C4—81no—
169C4—4281no—
170C4—4281no—
171C4#1681212911yescharacteristic
172C4#1681213211yescharacteristic
173C4#1681214011yescharacteristic
174C4#1681214111yescharacteristic
175C2—8562no—
176C2#2029278711yescharacteristic
177C2#2029278811yescharacteristic
178C2#2029279311yescharacteristic
subgroupG/H
centerC22#16812077
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
Fitting#477×C107C2
socleC22×C107#120
prime radical#16812106C22
hypercenter#477D107
hypocenterC107#1630
LCSG ▷ C42×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
3. #20292919C2semidirect
5. Q8⋊(C107⋊C8)C2semidirect
6. #20292357C2semidirect
7. #477×C107C2semidirect
90. C107#1630semidirect
94. #477D107semidirect

Maximal subgroups

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

subgroupindex
#202933622
#202930012
#202929192
#202923702
Q8⋊(C107⋊C8)2
#202923572
#477×C1072
#1630107