Finite Group Databasebeta

← #28335678

Subgroups — #28335678

Order 13696.

208 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 57 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#20293334C211yesmaximal, characteristic
2#20292978C211yesmaximal, characteristic, complemented
3#20292907C211yesmaximal, characteristic
4#20292838C211yesmaximal, characteristic, complemented
5#20292375C211yesmaximal, characteristic, complemented
6#20292359C211yesmaximal, characteristic, complemented
7D4⋊C8×C107C211yesmaximal, characteristic
8C2×Dic428—21no—
9#16812109C2211yescharacteristic
10#16812109C2211yescharacteristic
11#16812100—21no—
12#16812088—21no—
13#16812086—22no—
14C4×Dic214C2211yescharacteristic
15D4×Dic107—21no—
16C2×C4⋊Dic107—21no—
17#16812042—21no—
18C4⋊(C107⋊C8)C2211yescharacteristic
19#16812034—22no—
20#16812034—21no—
21#16812030—21no—
22C4×D4×C107C2211yescharacteristic
23C4⋊C8×C107C2211yescharacteristic
24C4×C8×C107C2211yescharacteristic
25C2×Dic214—21no—
26C2×Dic214D411yescharacteristic
27C2×Dic214D412yes—
28C22×Dic107—22no—
29Dic428—42no—
30C2×C107⋊C8—21no—
31C4⋊Dic107—22no—
32C4⋊Dic107—21no—
33C4⋊Dic107—21no—
34C4⋊Dic107—21no—
35C4⋊Dic107—21no—
36C4⋊Dic107D411yescharacteristic
37Dic107⋊C4—22no—
38Dic107⋊C4—21no—
39C22⋊Dic107—22no—
40C4×Dic107—21no—
41C2×D4×C107D411yescharacteristic
42C22×C4×C107—21no—
43C2×C8×C107—21no—
44C2×C8×C107—21no—
45C4⋊C4×C107D411yescharacteristic
46C22⋊C4×C107—21no—
47C42×C107C2311yescharacteristic
48C23×C107—21no—
49D4×C107—21no—
50D4×C107Q1612yescomplemented
51C2×C4×C107—21no—
52C2×C4×C107—41no—
53C2×C4×C107—41no—
54C2×C4×C107C2×D411yescharacteristic
55C2×C4×C107C2×D411yescharacteristic
56C2×C4×C107C2×D411yescharacteristic
57C2×Dic107—21no—
58C2×Dic107—21no—
59C2×Dic107—22no—
60C2×Dic107—41no—
61C2×Dic107—82no—
62Dic214—21no—
63Dic214—21no—
64Dic214—42no—
65Dic214—41no—
66Dic214—41no—
67C8×C107—41no—
68C8×C107—41no—
69C107⋊C8—41no—
70C22×C107—22no—
71C22×C107—41no—
72C22×C107—41no—
73C22×C107#12011yescharacteristic
74C4×C107—21no—
75C4×C107—41no—
76C4×C107C2×Q1611yescharacteristic
77C4×C107#13511yescharacteristic
78C4×C107AGL(1,ℤ/8)11yescharacteristic
79C4×C107#13711yescharacteristic
80Dic107—41no—
81Dic107—42no—
82Dic107—41no—
83Dic107—81no—
84C2×C107—42no—
85C2×C107#44911yescharacteristic
86C2×C107#45111yescharacteristic
87C2×C107#45511yescharacteristic
88#1644—1071nomaximal, Sylow
89C107#164411yescharacteristic, complemented, Sylow
90#554—1071no—
91#487—1071no—
92#474—1071no—
93#462—1071no—
94#333—1071no—
95Q8⋊C8—1071no—
96D4⋊C8D10711yescharacteristic
97C2×Q16—2141no—
98C4⋊Q8—1071no—
99C4⋊Q8—1071no—
100#125—2141no—
101C22⋊Q8—2142no—
102C22⋊Q8—2141no—
103C4×Q8—1071no—
104C4×D4—2141no—
105C4×D4C2×D10711yescharacteristic
106C2×C4⋊C4—2141no—
107#107—2141no—
108C4⋊C8—1071no—
109C4⋊C8C2×D10711yescharacteristic
110#103—2141no—
111#103—2142no—
112#102—2141no—
113C4×C8C2×D10711yescharacteristic
114C2×Q8—1072no—
115C2×Q8—1071no—
116C2×Q8—2141no—
117C2×D4D42811yescharacteristic
118C22×C4—21no—
119C22×C4—2142no—
120Q16—4282no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—2141no—
124C4⋊C4—1071no—
125C4⋊C4—2142no—
126C4⋊C4—2142no—
127C4⋊C4—2141no—
128C4⋊C4—2141no—
129C4⋊C4—2141no—
130C4⋊C4—2141no—
131C4⋊C4—2141no—
132C4⋊C4D42811yescharacteristic
133C22⋊C4—21no—
134C22⋊C4—2142no—
135C42—2141no—
136C42C22×D10711yescharacteristic
137C23—21no—
138Q8—4282no—
139Q8—4281no—
140Q8—4281no—
141Q8—2141no—
142Q8—2141no—
143D4—21no—
144D4Dic42812yescomplemented
145C2×C4—21no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—8562no—
149C2×C4—4281no—
150C2×C4—2141no—
151C2×C4—2141no—
152C2×C4—2142no—
153C2×C4D4×D10711yescharacteristic
154C2×C4D4×D10711yescharacteristic
155C2×C4C2×D42811yescharacteristic
156C8—41no—
157C8—41no—
158C8—4281no—
159C22—22no—
160C22—41no—
161C22—41no—
162C22#1681207911yescharacteristic
163C4—21no—
164C4—41no—
165C4—8561no—
166C4—4281no—
167C4—4281no—
168C4—4282no—
169C4C2×Dic42811yescharacteristic
170C4#1681213011yescharacteristic
171C4#1681213311yescharacteristic
172C4#1681214211yescharacteristic
173C2—42no—
174C2#2029277311yescharacteristic
175C2#2029278511yescharacteristic
176C2#2029280611yescharacteristic
subgroupG/H
centerC22#16812079
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
FittingD4⋊C8×C107C2
socleC22×C107#120
prime radicalC2×D4×C107D4
hypercenterD4⋊C8D107
hypocenterC107#1644
LCSG ▷ C42×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #20292978C2semidirect
4. #20292838C2semidirect
5. #20292375C2semidirect
6. #20292359C2semidirect
50. D4×C107Q16semidirect
89. C107#1644semidirect
144. D4Dic428semidirect

Maximal subgroups

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

subgroupindex
#202933342
#202929782
#202929072
#202928382
#202923752
#202923592
D4⋊C8×C1072
#1644107