Finite Group Databasebeta

← C2×#20292878 #28346055

Subgroups — C2×#20292878 #28346055

Order 13696.

596 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C22×#13133803C211yesmaximal, characteristic, complemented
2C22×#13133802C211yesmaximal, characteristic, complemented
3C2×#16812083C211yesmaximal, characteristic, complemented
4#20292878C218yesmaximal, direct product factor
5C2×#16812040C211yesmaximal, characteristic, complemented
6C2×#16812030C211yesmaximal, characteristic, complemented
7C2×#16812029C211yesmaximal, characteristic, complemented
8C2×#121×C107C211yesmaximal, characteristic, complemented
9C22×D428C2211yescharacteristic
10C22×C22⋊D107—21no—
11C2×#13133803—24no—
12C2×#13133803C2214yescomplemented
13C2×#13133802—24no—
14C2×#13133802C2214yescomplemented
15C22×C107⋊C8C2211yescharacteristic
16C22×C107⋊C8C2211yescharacteristic
17#16812083—22no—
18#16812083C2214yescomplemented
19C2×C4⋊Dic107C2211yescharacteristic
20C2×(C2×D107)⋊C4—21no—
21#16812040C2214yescomplemented
22#16812030C2214yescomplemented
23#16812029C2214yescomplemented
24C22×D4×C107—21no—
25C22×D4×C107C2211yescharacteristic
26C23×C4×C107C2211yescharacteristic
27#121×C107—22no—
28#121×C107C2214yescomplemented
29C2×C4⋊C4×C107C2211yescharacteristic
30C2×C22⋊C4×C107—21no—
31C23×D107—21no—
32C2×D428—22no—
33C2×D428—22no—
34C2×D428—21no—
35C2×D428C2312yes—
36C2×C22⋊D107—24no—
37C2×C22⋊D107—44no—
38C22×Dic107—21no—
39#13133803—28no—
40#13133802—28no—
41C2×C107⋊C8—22no—
42C2×C107⋊C8D414yescomplemented
43C2×C107⋊C8C2312yes—
44C2×C107⋊C8C2312yes—
45C4⋊Dic107—21no—
46C4⋊Dic107C2312yes—
47(C2×D107)⋊C4—24no—
48C24×C107—21no—
49C24×C107D411yescharacteristic
50C2×D4×C107—24no—
51C2×D4×C107—22no—
52C2×D4×C107—21no—
53C2×D4×C107—22no—
54C2×D4×C107—44no—
55C2×D4×C107C2312yes—
56C22×C4×C107—22no—
57C22×C4×C107—22no—
58C22×C4×C107—21no—
59C22×C4×C107D411yescharacteristic
60C22×C4×C107D412yes—
61C22×C4×C107C2312yes—
62C22×C4×C107C2311yescharacteristic
63C4⋊C4×C107—21no—
64C4⋊C4×C107C2312yes—
65C22⋊C4×C107—24no—
66C23×C107—22no—
67C23×C107—22no—
68C23×C107—22no—
69C23×C107—22no—
70C23×C107—42no—
71C23×C107—42no—
72C23×C107—42no—
73C23×C107C2×D412yes—
74C23×C107C2×D411yescharacteristic
75C22×D107—22no—
76C22×D107—42no—
77C22×D107—42no—
78C22×D107—42no—
79D4×C107—22no—
80D4×C107—22no—
81D4×C107—48no—
82D4×C107—42no—
83C2×C4×C107—22no—
84C2×C4×C107—24no—
85C2×C4×C107—22no—
86C2×C4×C107—22no—
87C2×C4×C107—22no—
88C2×C4×C107—42no—
89C2×C4×C107C2×D412yes—
90C2×C4×C107C2×D412yes—
91C2×C4×C107C2×D411yescharacteristic
92C2×C4×C107#4112yes—
93C2×C4×C107C2411yescharacteristic
94C22⋊D107—48no—
95C2×Dic107—22no—
96C2×Dic107—42no—
97D428—22no—
98D428—22no—
99D428—42no—
100C107⋊C8—22no—
101C107⋊C8C2×D414yescomplemented
102C22×C107—22no—
103C22×C107—22no—
104C22×C107—22no—
105C22×C107—44no—
106C22×C107—42no—
107C22×C107—42no—
108C22×C107—42no—
109C22×C107—84no—
110C22×C107#12112yes—
111C22×C107AGL(1,ℤ/8)12yes—
112C22×C107AGL(1,ℤ/8)12yes—
113C22×C107C22×D411yescharacteristic
114C2×D107—42no—
115C2×D107—42no—
116C2×D107—42no—
117C2×D107—84no—
118C4×C107—22no—
119C4×C107—42no—
120C4×C107#12112yes—
121C4×C107C22×D411yescharacteristic
122C4×C107C2×#4111yescharacteristic
123Dic107—42no—
124C2×C107—42no—
125C2×C107—82no—
126C2×C107#46914yescomplemented
127C2×C107C2×#12111yescharacteristic
128C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
129C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
130D107—82no—
131C2×#469—1071nomaximal, Sylow
132C107C2×#46911yescharacteristic, complemented, Sylow
133C22×D8—1071no—
134C22×M16—1071no—
135C2×#121—1071no—
136C2×#121D10711yescharacteristic, complemented
137#469—1078no—
138C2×#106—1071no—
139C2×#102—1071no—
140C2×#102—1071no—
141C22×D4—21no—
142C22×D4—1071no—
143C22×D4—2141no—
144C22×D4C2×D10711yescharacteristic
145C23×C4C2×D10711yescharacteristic
146C2×D8—1074no—
147C2×D8—2144no—
148C2×M16—1074no—
149C2×M16—2144no—
150C22×C8—1071no—
151C22×C8—1071no—
152#121—22no—
153#121—1074no—
154#121—2142no—
155#121C2×D10714yescomplemented
156C2×C4⋊C4—1071no—
157C2×C4⋊C4C2×D10711yescharacteristic
158C2×C22⋊C4—21no—
159C2×C22⋊C4—2141no—
160#106—1074no—
161#102—1074no—
162#102—1074no—
163C24—21no—
164C24—2141no—
165C24C22⋊D10711yescharacteristic
166C2×D4—24no—
167C2×D4—22no—
168C2×D4—21no—
169C2×D4—22no—
170C2×D4—44no—
171C2×D4—1072no—
172C2×D4—4284no—
173C2×D4—2144no—
174C2×D4—2142no—
175C2×D4—2141no—
176C2×D4—2142no—
177C2×D4C22×D10712yes—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—21no—
181C22×C4—2141no—
182C22×C4C22⋊D10712yes—
183C22×C4C22⋊D10711yescharacteristic
184C22×C4C22×D10712yes—
185C22×C4C22×D10711yescharacteristic
186D8—2148no—
187M16—2148no—
188C2×C8—1074no—
189C2×C8—1072no—
190C2×C8—1072no—
191C2×C8—2142no—
192C4⋊C4—21no—
193C4⋊C4—1072no—
194C4⋊C4—2141no—
195C4⋊C4C22×D10712yes—
196C22⋊C4—24no—
197C22⋊C4—2144no—
198C23—22no—
199C23—22no—
200C23—22no—
201C23—22no—
202C23—42no—
203C23—42no—
204C23—42no—
205C23—4282no—
206C23—4282no—
207C23—4282no—
208C23—2142no—
209C23C2×C22⋊D10712yes—
210C23C2×C22⋊D10711yescharacteristic
211D4—22no—
212D4—22no—
213D4—48no—
214D4—42no—
215D4—4288no—
216D4—4282no—
217D4—2142no—
218D4—2142no—
219C2×C4—22no—
220C2×C4—24no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—42no—
225C2×C4—4282no—
226C2×C4—2142no—
227C2×C4D4×D10712yes—
228C2×C4C2×C22⋊D10712yes—
229C2×C4C2×C22⋊D10711yescharacteristic
230C2×C4#1313381712yes—
231C2×C4C23×D10711yescharacteristic
232C8—1074no—
233C8—2142no—
234C22—22no—
235C22—22no—
236C22—22no—
237C22—44no—
238C22—42no—
239C22—42no—
240C22—42no—
241C22—84no—
242C22—8564no—
243C22—4282no—
244C22—4282no—
245C22—4282no—
246C22#1681208112yes—
247C22#1681213412yes—
248C22#1681213512yes—
249C22C22×C22⋊D10711yescharacteristic
250C4—22no—
251C4—42no—
252C4—4282no—
253C4#1681208112yes—
254C4C2×D4×D10711yescharacteristic
255C4C2×#1313381711yescharacteristic
256C2—42no—
257C2—82no—
258C2—8562no—
259C2#2029287814yesdirect product factor
260C2C2×#1681208111yescharacteristic
261C2C2×#1681213411yescharacteristic
262C2C2×#1681213511yescharacteristic
subgroupG/H
centerC23C2×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×#121×C107C2
socleC23×C107C2×D4
hypercenterC2×#121D107
hypocenterC107C2×#469
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C22×#13133803C2semidirect
2. C22×#13133802C2semidirect
3. C2×#16812083C2semidirect
4. #20292878C2direct (N × H)
5. C2×#16812040C2semidirect
6. C2×#16812030C2semidirect
7. C2×#16812029C2semidirect
8. C2×#121×C107C2semidirect
12. C2×#13133803C22semidirect
14. C2×#13133802C22semidirect
18. #16812083C22semidirect
21. #16812040C22semidirect
22. #16812030C22semidirect
23. #16812029C22semidirect
28. #121×C107C22semidirect
42. C2×C107⋊C8D4semidirect
101. C107⋊C8C2×D4semidirect
126. C2×C107#469semidirect
132. C107C2×#469semidirect
136. C2×#121D107semidirect
155. #121C2×D107semidirect
259. C2#20292878direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131338032
C22×#131338022
C2×#168120832
#202928782
C2×#168120402
C2×#168120302
C2×#168120292
C2×#121×C1072
C2×#469107