Finite Group Databasebeta

← C2×#20292780 #28345645

Subgroups — C2×#20292780 #28345645

Order 13696.

996 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 996 conjugacy classes of subgroups — too many to draw. Only its 181 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
1C23×D428C211yesmaximal, characteristic
2C22×#13133807C212yesmaximal, complemented
3C2×#16812088C211yesmaximal, characteristic, complemented
4#20292780C218yesmaximal, direct product factor
5C2×#16812031C212yesmaximal, complemented
6C2×C22⋊C8×C107C211yesmaximal, characteristic, complemented
7C24×D107—21no—
8C22×Dic214C2211yescharacteristic, complemented
9C22×D428—28no—
10C22×D428—21no—
11C22×D428—22no—
12C22×D428C2214yes—
13C22×D428C2212yes—
14C2×#13133807—28no—
15C2×#13133807C2218yescomplemented
16#16812088—22no—
17#16812088C2214yescomplemented
18C2×C4⋊Dic107C2211yescharacteristic, complemented
19C2×Dic107⋊C4—21no—
20C2×C22⋊Dic107—21no—
21#16812031C2218yescomplemented
22C23×C4×C107C2211yescharacteristic
23C22×C8×C107C2212yescomplemented
24C22⋊C8×C107C2214yescomplemented
25C23×D107—28no—
26C23×D107—24no—
27C23×D107—22no—
28C23×D107—44no—
29C23×D107—44no—
30C2×Dic214—24no—
31C2×Dic214—21no—
32C2×Dic214C2312yescomplemented
33C2×D428—28no—
34C2×D428—216no—
35C2×D428—24no—
36C2×D428—24no—
37C2×D428—22no—
38C2×D428—216no—
39C2×D428D418yes—
40C2×D428C2314yes—
41C22×Dic107—21no—
42C22×Dic107—21no—
43#13133807—216no—
44C4⋊Dic107—21no—
45C4⋊Dic107C2312yescomplemented
46Dic107⋊C4—24no—
47C22⋊Dic107—24no—
48C24×C107D411yescharacteristic
49C22×C4×C107—24no—
50C22×C4×C107D411yescharacteristic
51C22×C4×C107D412yes—
52C22×C4×C107C2312yes—
53C22×C4×C107C2311yescharacteristic
54C2×C8×C107—24no—
55C2×C8×C107C2314yescomplemented
56C23×C107—22no—
57C23×C107—22no—
58C23×C107SD1614yescomplemented
59C23×C107C2×D412yes—
60C23×C107C2×D411yescharacteristic
61C22×D107—216no—
62C22×D107—28no—
63C22×D107—24no—
64C22×D107—48no—
65C22×D107—432no—
66C22×D107—44no—
67C22×D107—48no—
68C22×D107—44no—
69C2×C4×C107—22no—
70C2×C4×C107—24no—
71C2×C4×C107—24no—
72C2×C4×C107C2×D414yes—
73C2×C4×C107C2×D412yes—
74C2×C4×C107C2×D411yescharacteristic
75C2×C4×C107C2411yescharacteristic
76C2×Dic107—22no—
77C2×Dic107—22no—
78C2×Dic107—42no—
79C2×Dic107—42no—
80D428—216no—
81D428—28no—
82D428—24no—
83D428C2×D418yes—
84Dic214—24no—
85Dic214—42no—
86C8×C107—24no—
87C22×C107—22no—
88C22×C107—22no—
89C22×C107—28no—
90C22×C107#12012yes—
91C22×C107C2×SD1612yes—
92C22×C107C2×SD1614yescomplemented
93C22×C107AGL(1,ℤ/8)12yes—
94C22×C107C22×D411yescharacteristic
95C2×D107—28no—
96C2×D107—432no—
97C2×D107—44no—
98C2×D107—44no—
99C2×D107—416no—
100C4×C107—22no—
101C4×C107#12012yes—
102C4×C107C22×D412yes—
103Dic107—42no—
104Dic107—42no—
105C2×C107—24no—
106C2×C107#45014yescomplemented
107C2×C107C2×#12011yescharacteristic
108C2×C107C22×SD1611yescharacteristic
109C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
110D107—48no—
111C2×#450—1071nomaximal, Sylow
112C107C2×#45011yescharacteristic, complemented, Sylow
113C23×D4—1071no—
114C22×SD16—1072no—
115C2×C22⋊Q8—1071no—
116#450—1078no—
117C2×#102—1072no—
118C2×C22⋊C8D10711yescharacteristic, complemented
119C25—2141no—
120C22×Q8—1071no—
121C22×D4—1072no—
122C22×D4—1074no—
123C22×D4—2148no—
124C22×D4—2142no—
125C22×D4—2141no—
126C23×C4C2×D10711yescharacteristic
127C2×SD16—1078no—
128C2×SD16—2148no—
129C22×C8C2×D10712yescomplemented
130C22⋊Q8—1074no—
131C22⋊Q8—2142no—
132C2×C4⋊C4—1071no—
133C2×C4⋊C4—2141no—
134C2×C22⋊C4—2141no—
135#102—1078no—
136C22⋊C8C2×D10714yescomplemented
137C24—4284no—
138C24—4284no—
139C24—2148no—
140C24—2144no—
141C24—2142no—
142C24D42811yescharacteristic
143C2×Q8—1072no—
144C2×Q8—2144no—
145C2×Q8—2141no—
146C2×D4—1078no—
147C2×D4—1074no—
148C2×D4—21416no—
149C2×D4—2148no—
150C2×D4—21416no—
151C2×D4—2144no—
152C2×D4—2144no—
153C2×D4—2142no—
154C22×C4—24no—
155C22×C4—2141no—
156C22×C4—2141no—
157C22×C4D42812yes—
158C22×C4D42811yescharacteristic
159C22×C4C22×D10712yes—
160C22×C4C22×D10711yescharacteristic
161SD16—21416no—
162C2×C8—24no—
163C2×C8C22×D10714yescomplemented
164C4⋊C4—1072no—
165C4⋊C4—2141no—
166C4⋊C4—2144no—
167C22⋊C4—2144no—
168C23—22no—
169C23—22no—
170C23—4288no—
171C23—42832no—
172C23—4284no—
173C23—4288no—
174C23—4284no—
175C23—21416no—
176C23—2148no—
177C23—2144no—
178C23#1313380714yescomplemented
179C23C2×D42812yes—
180C23C2×D42811yescharacteristic
181Q8—4282no—
182Q8—2144no—
183D4—1078no—
184D4—21416no—
185D4—2148no—
186D4—2144no—
187C2×C4—22no—
188C2×C4—24no—
189C2×C4—24no—
190C2×C4—4282no—
191C2×C4—4282no—
192C2×C4—2142no—
193C2×C4—2142no—
194C2×C4D4×D10714yes—
195C2×C4C2×D42812yes—
196C2×C4C2×D42811yescharacteristic
197C2×C4C23×D10711yescharacteristic
198C8—24no—
199C22—22no—
200C22—28no—
201C22—22no—
202C22—42832no—
203C22—4284no—
204C22—4284no—
205C22—42816no—
206C22—2148no—
207C22#1681207912yes—
208C22C2×#1313380712yes—
209C22C2×#1313380714yescomplemented
210C22#1681213712yes—
211C22C22×D42811yescharacteristic
212C4—22no—
213C4—4282no—
214C4—4282no—
215C4#1681207912yes—
216C4C2×D4×D10712yes—
217C2—24no—
218C2—4288no—
219C2#2029278014yesdirect product factor
220C2C2×#1681207911yescharacteristic
221C2C22×#1313380711yescharacteristic
222C2C2×#1681213711yescharacteristic
subgroupG/H
centerC23C2×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×C22⋊C8×C107C2
socleC23×C107C2×D4
prime radicalC23×D428C2
hypercenterC2×C22⋊C8D107
hypocenterC107C2×#450
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C22×#13133807C2semidirect
3. C2×#16812088C2semidirect
4. #20292780C2direct (N × H)
5. C2×#16812031C2semidirect
6. C2×C22⋊C8×C107C2semidirect
8. C22×Dic214C22semidirect
15. C2×#13133807C22semidirect
17. #16812088C22semidirect
18. C2×C4⋊Dic107C22semidirect
21. #16812031C22semidirect
23. C22×C8×C107C22semidirect
24. C22⋊C8×C107C22semidirect
32. C2×Dic214C23semidirect
45. C4⋊Dic107C23semidirect
55. C2×C8×C107C23semidirect
58. C23×C107SD16semidirect
92. C22×C107C2×SD16semidirect
106. C2×C107#450semidirect
112. C107C2×#450semidirect
118. C2×C22⋊C8D107semidirect
129. C22×C8C2×D107semidirect
136. C22⋊C8C2×D107semidirect
163. C2×C8C22×D107semidirect
178. C23#13133807semidirect
209. C22C2×#13133807semidirect
219. C2#20292780direct (N × H)

Maximal subgroups

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

subgroupindex
C23×D4282
C22×#131338072
C2×#168120882
#202927802
C2×#168120312
C2×C22⋊C8×C1072
C2×#450107