Finite Group Databasebeta

← #28335579

Subgroups — #28335579

Order 13696.

314 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 59 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
1D4×D428C211yesmaximal, characteristic
2#20293000C211yesmaximal, characteristic, complemented
3#20292918C211yesmaximal, characteristic, complemented
4#20292917C211yesmaximal, characteristic, complemented
5#20292370C211yesmaximal, characteristic, complemented
6#20292356C211yesmaximal, characteristic, complemented
7D4⋊C8×C107C211yesmaximal, characteristic, complemented
8C22×D428—21no—
9C2×D4×D107—21no—
10C2×#13133807—22no—
11#16812109C2211yescharacteristic, complemented
12#16812106C2211yescharacteristic
13#16812084—21no—
14#16812083—21no—
15#16812079—22no—
16C4×D428C2211yescharacteristic
17#16812041—21no—
18#16812040—21no—
19C4⋊(C107⋊C8)C2211yescharacteristic, complemented
20#16812031—21no—
21#16812030—21no—
22C4×D4×C107C2211yescharacteristic
23C4⋊C8×C107C2211yescharacteristic, complemented
24C4×C8×C107C2211yescharacteristic, complemented
25C23×D107—22no—
26C2×Dic214—21no—
27C2×D428—22no—
28C2×D428—21no—
29C2×D428—21no—
30C2×D428—42no—
31C2×D428—42no—
32C2×D428D411yescharacteristic
33C2×D428D412yes—
34C2×C22⋊D107—22no—
35D4×D107—42no—
36D4×D107—42no—
37C2×C4×D107—21no—
38#13133807—44no—
39C2×C107⋊C8—21no—
40C4⋊Dic107—21no—
41C4⋊Dic107—21no—
42C4⋊Dic107D411yescharacteristic
43(C2×D107)⋊C4—22no—
44(C2×D107)⋊C4—21no—
45C2×D4×C107D411yescharacteristic
46C22×C4×C107—21no—
47C2×C8×C107—21no—
48C2×C8×C107—21no—
49C4⋊C4×C107D411yescharacteristic
50C22⋊C4×C107—21no—
51C42×C107C2311yescharacteristic
52C23×C107—21no—
53C22×D107—21no—
54C22×D107—22no—
55C22×D107—44no—
56C22×D107—44no—
57C22×D107—82no—
58D4×C107—21no—
59D4×C107SD1612yescomplemented
60C2×C4×C107—21no—
61C2×C4×C107—41no—
62C2×C4×C107—41no—
63C2×C4×C107C2×D411yescharacteristic
64C2×C4×C107C2×D411yescharacteristic
65C2×C4×C107C2×D411yescharacteristic
66C22⋊D107—44no—
67C2×Dic107—21no—
68C2×Dic107—41no—
69D428—21no—
70D428—42no—
71D428—44no—
72D428—41no—
73D428—41no—
74D428—41no—
75D428—41no—
76D428SD1612yescomplemented
77C4×D107—41no—
78C4×D107—41no—
79Dic214—42no—
80C8×C107—22no—
81C8×C107—41no—
82C107⋊C8—41no—
83C22×C107—22no—
84C22×C107—41no—
85C22×C107—41no—
86C22×C107#12011yescharacteristic
87C2×D107—22no—
88C2×D107—41no—
89C2×D107—41no—
90C2×D107—42no—
91C2×D107—42no—
92C2×D107—42no—
93C2×D107—88no—
94C4×C107—21no—
95C4×C107—41no—
96C4×C107C2×SD1611yescharacteristic
97C4×C107C2×SD1611yescharacteristic
98C4×C107AGL(1,ℤ/8)11yescharacteristic
99C4×C107AGL(1,ℤ/8)11yescharacteristic
100Dic107—41no—
101Dic107—81no—
102C2×C107—42no—
103C2×C107#45011yescharacteristic
104C2×C107#45011yescharacteristic
105C2×C107#45311yescharacteristic
106D107—42no—
107D107—82no—
108#1629—1071nomaximal, Sylow
109C107#162911yescharacteristic, complemented, Sylow
110D42—1071no—
111#492—1071no—
112#476—1071no—
113#476—1071no—
114#331—1071no—
115D4⋊C8—1071no—
116D4⋊C8D10711yescharacteristic, complemented
117C22×D4—2141no—
118C22×D4—2141no—
119C2×SD16—2142no—
120C4⋊Q8—1071no—
121#127—1071no—
122#121—2141no—
123#121—2141no—
124#120—2142no—
125C4×D4—1071no—
126C4×D4C2×D10711yescharacteristic
127#106—2141no—
128#106—2141no—
129C4⋊C8—1071no—
130C4⋊C8C2×D10711yescharacteristic, complemented
131#102—2141no—
132#102—2141no—
133C4×C8C2×D10711yescharacteristic, complemented
134C24—2142no—
135C2×Q8—2141no—
136C2×D4—1071no—
137C2×D4—1072no—
138C2×D4—4282no—
139C2×D4—4282no—
140C2×D4—4282no—
141C2×D4—4282no—
142C2×D4—2142no—
143C2×D4—2142no—
144C2×D4—2141no—
145C2×D4—2141no—
146C2×D4D42811yescharacteristic
147C22×C4—21no—
148C22×C4—2141no—
149SD16—4284no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—2141no—
153C4⋊C4—1071no—
154C4⋊C4—2141no—
155C4⋊C4—2141no—
156C4⋊C4D42811yescharacteristic
157C22⋊C4—21no—
158C22⋊C4—2142no—
159C22⋊C4—2141no—
160C42C22×D10711yescharacteristic
161C23—21no—
162C23—8562no—
163C23—4284no—
164C23—4284no—
165C23—2142no—
166C23—2141no—
167Q8—4282no—
168D4—21no—
169D4—1072no—
170D4—4282no—
171D4—4284no—
172D4—4284no—
173D4—4281no—
174D4—4281no—
175D4—4281no—
176D4—4281no—
177D4—2141no—
178D4#1313380712yescomplemented
179C2×C4—21no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—4281no—
183C2×C4—4281no—
184C2×C4—4281no—
185C2×C4—2141no—
186C2×C4D4×D10711yescharacteristic
187C2×C4D4×D10711yescharacteristic
188C2×C4C2×D42811yescharacteristic
189C8—22no—
190C8—41no—
191C8—4281no—
192C22—22no—
193C22—41no—
194C22—41no—
195C22—8568no—
196C22—4282no—
197C22—4282no—
198C22—4282no—
199C22—4281no—
200C22—4281no—
201C22—2142no—
202C22#1681207911yescharacteristic
203C4—21no—
204C4—41no—
205C4—8561no—
206C4—4281no—
207C4SD16×D10711yescharacteristic
208C4C2×#1313380711yescharacteristic
209C4#1681213311yescharacteristic
210C4#1681213711yescharacteristic
211C2—42no—
212C2—8562no—
213C2—4282no—
214C2#2029277911yescharacteristic
215C2#2029278011yescharacteristic
216C2#2029279611yescharacteristic
subgroupG/H
centerC22#16812079
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
FittingD4⋊C8×C107C2
socleC22×C107#120
prime radicalD4×D428C2
hypercenterD4⋊C8D107
hypocenterC107#1629
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. #20292918C2semidirect
4. #20292917C2semidirect
5. #20292370C2semidirect
6. #20292356C2semidirect
7. D4⋊C8×C107C2semidirect
11. #16812109C22semidirect
19. C4⋊(C107⋊C8)C22semidirect
23. C4⋊C8×C107C22semidirect
24. C4×C8×C107C22semidirect
59. D4×C107SD16semidirect
76. D428SD16semidirect
109. C107#1629semidirect
116. D4⋊C8D107semidirect
130. C4⋊C8C2×D107semidirect
133. C4×C8C2×D107semidirect
178. D4#13133807semidirect

Maximal subgroups

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

subgroupindex
D4×D4282
#202930002
#202929182
#202929172
#202923702
#202923562
D4⋊C8×C1072
#1629107