Finite Group Databasebeta

← #28335645

Subgroups — #28335645

Order 13696.

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

normal subgroups only

This group has 212 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#20292975C211yesmaximal, characteristic
3#20292909C211yesmaximal, characteristic, complemented
4#20292370C211yesmaximal, characteristic
5Q8⋊(C107⋊C8)C211yesmaximal, characteristic, complemented
6#20292357C211yesmaximal, characteristic, complemented
7#462×C107C211yesmaximal, characteristic, complemented
8#16812106C2211yescharacteristic
9#16812098—21no—
10#16812097—21no—
11#16812094—22no—
12C4×Dic214C2211yescharacteristic
13C4×D428—21no—
14#16812071—21no—
15#16812040—21no—
16C4⋊(C107⋊C8)C2211yescharacteristic
17#16812032—21no—
18#16812029—22no—
19C4×C107⋊C8C2211yescharacteristic
20C2×Q16×C107—21no—
21C4⋊Q8×C107C2211yescharacteristic
22C4×Q8×C107C2211yescharacteristic
23C4⋊C8×C107C2211yescharacteristic
24#103×C107—21no—
25C2×Dic214D411yescharacteristic
26C2×D428—21no—
27C2×D428—21no—
28C2×D428D412yes—
29C2×C4×D107—22no—
30C2×C107⋊C8—21no—
31C2×C107⋊C8—21no—
32C4⋊Dic107D411yescharacteristic
33Dic107⋊C4—21no—
34(C2×D107)⋊C4—22no—
35(C2×D107)⋊C4—22no—
36(C2×D107)⋊C4—22no—
37C4×Dic107—21no—
38C2×Q8×C107—21no—
39C2×Q8×C107D411yescharacteristic
40Q16×C107—42no—
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—41no—
52Q8×C107—41no—
53C2×C4×C107—21no—
54C2×C4×C107—21no—
55C2×C4×C107—41no—
56C2×C4×C107C2×D411yescharacteristic
57C2×C4×C107C2×D411yescharacteristic
58C2×C4×C107C2×D411yescharacteristic
59C2×Dic107—21no—
60C2×Dic107—21no—
61D428—42no—
62D428—41no—
63D428—41no—
64D428—41no—
65D428—41no—
66C4×D107—82no—
67Dic214—21no—
68Dic214—21no—
69C8×C107—41no—
70C107⋊C8—41no—
71C107⋊C8—41no—
72C22×C107#12011yescharacteristic
73C2×D107—42no—
74C2×D107—42no—
75C2×D107—42no—
76C4×C107—21no—
77C4×C107—41no—
78C4×C107—41no—
79C4×C107—81no—
80C4×C107#13511yescharacteristic
81C4×C107#13511yescharacteristic
82C4×C107AGL(1,ℤ/8)11yescharacteristic
83C4×C107#13711yescharacteristic
84Dic107—41no—
85Dic107—41no—
86C2×C107#44911yescharacteristic
87C2×C107#45211yescharacteristic
88C2×C107#45511yescharacteristic
89D107—82no—
90#1640—1071nomaximal, Sylow
91C107#164011yescharacteristic, complemented, Sylow
92#559—1071no—
93#486—1071no—
94#475—1071no—
95#462D10711yescharacteristic, complemented
96#331—1071no—
97Q8⋊C8—1071no—
98Q8⋊C8—1071no—
99C2×Q16—21no—
100C4⋊Q8C2×D10711yescharacteristic
101#127—1071no—
102#124—2141no—
103#124—2141no—
104#123—2142no—
105C4×Q8—1071no—
106C4×Q8C2×D10711yescharacteristic
107C4×D4—2141no—
108C4×D4—2141no—
109#106—2141no—
110C4⋊C8—1071no—
111C4⋊C8C2×D10711yescharacteristic
112#103—21no—
113#103—2141no—
114#102—2142no—
115C4×C8—1071no—
116C2×Q8—21no—
117C2×Q8—1071no—
118C2×Q8C22⋊D10711yescharacteristic
119C2×D4—1072no—
120C2×D4—2141no—
121C2×D4—2141no—
122C22×C4—2142no—
123Q16—42no—
124C2×C8—21no—
125C2×C8—2141no—
126C2×C8—2141no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—1071no—
131C4⋊C4—2141no—
132C4⋊C4C22⋊D10711yescharacteristic
133C22⋊C4—2142no—
134C22⋊C4—2142no—
135C22⋊C4—2142no—
136C42—21no—
137C42—2141no—
138C42C22×D10711yescharacteristic
139C23—2142no—
140Q8—21no—
141Q8—21no—
142Q8—41no—
143Q8—41no—
144Q8—2141no—
145Q8—2141no—
146D4—4282no—
147D4—4281no—
148D4—4281no—
149D4—4281no—
150D4—4281no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—41no—
154C2×C4—8562no—
155C2×C4—2141no—
156C2×C4—2141no—
157C2×C4D4×D10711yescharacteristic
158C2×C4D4×D10711yescharacteristic
159C2×C4C2×C22⋊D10711yescharacteristic
160C8—41no—
161C8—4281no—
162C8—4281no—
163C22—4282no—
164C22—4282no—
165C22—4282no—
166C22#1681207711yescharacteristic
167C4—21no—
168C4—41no—
169C4—41no—
170C4—81no—
171C4—4281no—
172C4—4281no—
173C4#1681212911yescharacteristic
174C4#1681213211yescharacteristic
175C4#1681213511yescharacteristic
176C4#1681214111yescharacteristic
177C2—8562no—
178C2#2029277011yescharacteristic
179C2#2029278811yescharacteristic
180C2#2029280311yescharacteristic
subgroupG/H
centerC22#16812077
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
Fitting#462×C107C2
socleC22×C107#120
prime radical#16812106C22
hypercenter#462D107
hypocenterC107#1640
LCSG ▷ C42×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
3. #20292909C2semidirect
5. Q8⋊(C107⋊C8)C2semidirect
6. #20292357C2semidirect
7. #462×C107C2semidirect
91. C107#1640semidirect
95. #462D107semidirect

Maximal subgroups

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

subgroupindex
#202933622
#202929752
#202929092
#202923702
Q8⋊(C107⋊C8)2
#202923572
#462×C1072
#1640107