Finite Group Databasebeta

← #28335591

Subgroups — #28335591

Order 13696.

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

normal subgroups only

This group has 256 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#20293298C211yesmaximal, characteristic
2#20293000C211yesmaximal, characteristic, complemented
3#20292923C211yesmaximal, characteristic, complemented
4#20292917C211yesmaximal, characteristic, complemented
5#20292370C211yesmaximal, characteristic, complemented
6#20292359C211yesmaximal, characteristic, complemented
7D4⋊C8×C107C211yesmaximal, characteristic, complemented
8C2×#13133818—21no—
9C2×#13133807—22no—
10#16812109C2211yescharacteristic
11#16812106C2211yescharacteristic
12#16812105—21no—
13#16812083—21no—
14#16812082—22no—
15C4×Dic214C2211yescharacteristic
16#16812071—21no—
17#16812043—21no—
18#16812040—21no—
19C4⋊(C107⋊C8)C2211yescharacteristic
20#16812034—21no—
21#16812030—21no—
22C4×D4×C107C2211yescharacteristic
23C4⋊C8×C107C2211yescharacteristic
24C4×C8×C107C2211yescharacteristic
25#13133818—42no—
26#13133818—42no—
27C2×Dic214—21no—
28C2×Dic214D411yescharacteristic
29C2×D428—21no—
30C2×D428—21no—
31C2×D428D412yes—
32C2×C22⋊D107—22no—
33C2×C22⋊D107—22no—
34C2×C4×D107—22no—
35#13133807—44no—
36C2×C107⋊C8—21no—
37C4⋊Dic107—21no—
38C4⋊Dic107—21no—
39C4⋊Dic107D411yescharacteristic
40Dic107⋊C4—21no—
41(C2×D107)⋊C4—22no—
42C4×Dic107—21no—
43C2×D4×C107D411yescharacteristic
44C22×C4×C107—21no—
45C2×C8×C107—21no—
46C2×C8×C107—21no—
47C4⋊C4×C107D411yescharacteristic
48C22⋊C4×C107—21no—
49C42×C107C2311yescharacteristic
50C23×C107—21no—
51C22×D107—22no—
52D4×C107—21no—
53D4×C107—21no—
54C2×C4×C107—21no—
55C2×C4×C107—41no—
56C2×C4×C107—41no—
57C2×C4×C107C2×D411yescharacteristic
58C2×C4×C107C2×D411yescharacteristic
59C2×C4×C107C2×D411yescharacteristic
60C22⋊D107—44no—
61C22⋊D107—82no—
62C2×Dic107—21no—
63C2×Dic107—21no—
64C2×Dic107—41no—
65D428—42no—
66D428—41no—
67D428—41no—
68D428—41no—
69D428—41no—
70C4×D107—44no—
71Dic214—21no—
72Dic214—42no—
73Dic214SD1612yescomplemented
74C8×C107—22no—
75C8×C107—41no—
76C107⋊C8—41no—
77C22×C107—41no—
78C22×C107—41no—
79C22×C107—41no—
80C22×C107#12011yescharacteristic
81C2×D107—42no—
82C2×D107—42no—
83C2×D107—42no—
84C4×C107—21no—
85C4×C107—41no—
86C4×C107C2×SD1611yescharacteristic
87C4×C107#13511yescharacteristic
88C4×C107AGL(1,ℤ/8)11yescharacteristic
89C4×C107#13711yescharacteristic
90Dic107—22no—
91Dic107—41no—
92Dic107—81no—
93C2×C107—81no—
94C2×C107#45011yescharacteristic
95C2×C107#45211yescharacteristic
96C2×C107#45311yescharacteristic
97D107—82no—
98#1631—1071nomaximal, Sylow
99C107#163111yescharacteristic, complemented, Sylow
100#550—1071no—
101#492—1071no—
102#477—1071no—
103#476—1071no—
104#331—1071no—
105Q8⋊C8—1071no—
106D4⋊C8D10711yescharacteristic, complemented
107C2×#41—2141no—
108C2×SD16—2142no—
109C4⋊Q8—1071no—
110#127—1071no—
111#127—2141no—
112#121—2142no—
113#121—2141no—
114C4×Q8—1071no—
115C4×D4—2141no—
116C4×D4C2×D10711yescharacteristic
117#107—2141no—
118#106—2141no—
119C4⋊C8—1071no—
120C4⋊C8C2×D10711yescharacteristic
121#103—2141no—
122#102—2141no—
123C4×C8C2×D10711yescharacteristic
124#41—4282no—
125#41—4282no—
126C2×Q8—1071no—
127C2×Q8—2141no—
128C2×D4—1072no—
129C2×D4—2142no—
130C2×D4—2142no—
131C2×D4—2141no—
132C2×D4—2141no—
133C2×D4D42811yescharacteristic
134C22×C4—21no—
135C22×C4—2142no—
136SD16—4284no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—2141no—
140C4⋊C4—1071no—
141C4⋊C4—2141no—
142C4⋊C4—2141no—
143C4⋊C4—2141no—
144C4⋊C4D42811yescharacteristic
145C22⋊C4—21no—
146C22⋊C4—2142no—
147C42—2141no—
148C42C22×D10711yescharacteristic
149C23—21no—
150C23—2142no—
151Q8—1072no—
152Q8—4282no—
153Q8—2141no—
154D4—21no—
155D4—21no—
156D4—8562no—
157D4—4282no—
158D4—4284no—
159D4—4281no—
160D4—4281no—
161D4—4281no—
162D4—4281no—
163C2×C4—21no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—4284no—
167C2×C4—4281no—
168C2×C4—2141no—
169C2×C4—2141no—
170C2×C4D4×D10711yescharacteristic
171C2×C4D4×D10711yescharacteristic
172C2×C4C2×D42811yescharacteristic
173C8—22no—
174C8—41no—
175C8—4281no—
176C22—41no—
177C22—41no—
178C22—41no—
179C22—4282no—
180C22—4282no—
181C22—4282no—
182C22#1681207911yescharacteristic
183C4—21no—
184C4—41no—
185C4—8561no—
186C4—4281no—
187C4—2142no—
188C4SD16×D10711yescharacteristic
189C4#1681213111yescharacteristic
190C4#1681213311yescharacteristic
191C4#1681214211yescharacteristic
192C2—81no—
193C2—8562no—
194C2#2029277911yescharacteristic
195C2#2029279111yescharacteristic
196C2#2029279611yescharacteristic
subgroupG/H
centerC22#16812079
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
FittingD4⋊C8×C107C2
socleC22×C107#120
prime radical#20293298C2
hypercenterD4⋊C8D107
hypocenterC107#1631
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. #20293000C2semidirect
3. #20292923C2semidirect
4. #20292917C2semidirect
5. #20292370C2semidirect
6. #20292359C2semidirect
7. D4⋊C8×C107C2semidirect
73. Dic214SD16semidirect
99. C107#1631semidirect
106. D4⋊C8D107semidirect

Maximal subgroups

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

subgroupindex
#202932982
#202930002
#202929232
#202929172
#202923702
#202923592
D4⋊C8×C1072
#1631107