Finite Group Databasebeta

← C2×#20292627 #28344972

Subgroups — C2×#20292627 #28344972

Order 13696.

816 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 229 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×#13133819C211yesmaximal, characteristic
2C22×#13133801C211yesmaximal, characteristic, complemented
3C2×C4⋊C4×D107C211yesmaximal, characteristic, complemented
4#20292627C218yesmaximal, direct product factor
5C2×#16812033C211yesmaximal, characteristic, complemented
6C2×#16812031C211yesmaximal, characteristic, complemented
7C2×#16812029C211yesmaximal, characteristic, complemented
8C2×#103×C107C211yesmaximal, characteristic, complemented
9C2×#13133819—24no—
10C2×#13133819—22no—
11C2×#13133819C418yescomplemented
12C2×#13133819C2214yes—
13C22×D428—21no—
14C22×D428C2211yescharacteristic
15C22×C4×D107—21no—
16C22×C4×D107—21no—
17C22×C4×D107C2211yescharacteristic
18C2×#13133801—24no—
19C2×#13133801C2214yescomplemented
20C22×C107⋊C8C2211yescharacteristic
21C2×C4⋊Dic107C2211yescharacteristic
22C2×Dic107⋊C4—21no—
23C4⋊C4×D107—22no—
24C4⋊C4×D107C2214yescomplemented
25#16812033C2214yescomplemented
26#16812031C2214yescomplemented
27#16812029C2214yescomplemented
28C22×Q8×C107C2211yescharacteristic
29C22×C8×C107C2211yescharacteristic
30C2×C4⋊C4×C107C2211yescharacteristic
31#103×C107C2214yescomplemented
32C23×D107—21no—
33C23×D107D411yescharacteristic
34#13133819—216no—
35#13133819—28no—
36#13133819C2×C4116yescomplemented
37C2×D428—28no—
38C2×D428—24no—
39C2×D428—21no—
40C2×D428—22no—
41C2×D428C2×C414yescomplemented
42C2×D428C2312yes—
43C22×Dic107—21no—
44C22×Dic107D411yescharacteristic
45C2×C4×D107—28no—
46C2×C4×D107—24no—
47C2×C4×D107—24no—
48C2×C4×D107—44no—
49C2×C4×D107C2×C414yes—
50C2×C4×D107D412yes—
51C2×C4×D107D414yes—
52C2×C4×D107C2312yes—
53#13133801—28no—
54C2×C107⋊C8—22no—
55C2×C107⋊C8C2312yes—
56C4⋊Dic107—21no—
57C4⋊Dic107C2312yes—
58Dic107⋊C4—24no—
59C2×Q8×C107—22no—
60C2×Q8×C107—21no—
61C2×Q8×C107C2×C414yescomplemented
62C2×Q8×C107C2312yes—
63C22×C4×C107—21no—
64C22×C4×C107—21no—
65C22×C4×C107C2311yescharacteristic
66C2×C8×C107—22no—
67C2×C8×C107C2312yes—
68C4⋊C4×C107—21no—
69C4⋊C4×C107C2312yes—
70C23×C107C2×D411yescharacteristic
71C22×D107—22no—
72C22×D107—22no—
73C22×D107—24no—
74C22×D107—22no—
75C22×D107—42no—
76C22×D107—42no—
77C22×D107C22⋊C414yes—
78C22×D107C2×D412yes—
79Q8×C107—24no—
80Q8×C107—22no—
81Q8×C107C22×C414yescomplemented
82C2×C4×C107—24no—
83C2×C4×C107—22no—
84C2×C4×C107—22no—
85C2×C4×C107—42no—
86C2×C4×C107C22×C412yes—
87C2×C4×C107C2×D411yescharacteristic
88C2×C4×C107C2×D412yes—
89C2×C4×C107C2411yescharacteristic
90C2×Dic107—22no—
91C2×Dic107—42no—
92C2×Dic107C22⋊C414yes—
93C2×Dic107C2×D412yes—
94D428—216no—
95D428—24no—
96D428—22no—
97D428C22×C414yescomplemented
98C4×D107—216no—
99C4×D107—48no—
100C4×D107C22⋊C418yes—
101C4×D107C22×C414yes—
102C4×D107C2×D414yes—
103C8×C107—22no—
104C107⋊C8—22no—
105C22×C107C2×C22⋊C412yes—
106C22×C107AGL(1,ℤ/8)12yes—
107C22×C107#13712yes—
108C22×C107C22×D411yescharacteristic
109C2×D107—22no—
110C2×D107—22no—
111C2×D107—28no—
112C2×D107—24no—
113C2×D107—42no—
114C2×D107—42no—
115C2×D107—48no—
116C2×D107C2×C22⋊C414yes—
117C4×C107—24no—
118C4×C107—42no—
119C4×C107C2×C22⋊C412yes—
120C4×C107C23×C411yescharacteristic
121C4×C107C22×D411yescharacteristic
122Dic107—42no—
123Dic107C2×C22⋊C414yes—
124C2×C107#41714yescomplemented
125C2×C107C22×C22⋊C411yescharacteristic
126C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
127C2×C107C2×#13711yescharacteristic
128D107—24no—
129D107—44no—
130C2×#417—1071nomaximal, Sylow
131C107C2×#41711yescharacteristic, complemented, Sylow
132C22×#41—1071no—
133C22×M16—1071no—
134C22×C4⋊C4—1071no—
135#417—1078no—
136C2×#103—1071no—
137C2×#103D10711yescharacteristic, complemented
138C2×#102—1071no—
139C2×#102—1071no—
140C2×#41—1078no—
141C2×#41—1074no—
142C2×#41—2142no—
143C2×#41—2144no—
144C22×Q8C2×D10711yescharacteristic
145C22×D4—1071no—
146C22×D4—2141no—
147C23×C4—1071no—
148C23×C4—2141no—
149C23×C4—2141no—
150C2×M16—1074no—
151C2×M16—2144no—
152C22×C8—1071no—
153C22×C8C2×D10711yescharacteristic
154C2×C4⋊C4—1071no—
155C2×C4⋊C4—1074no—
156C2×C4⋊C4—2142no—
157C2×C4⋊C4—2141no—
158C2×C4⋊C4C2×D10711yescharacteristic
159#103—1074no—
160#103C2×D10714yescomplemented
161#102—1074no—
162#102—1074no—
163C24—1071no—
164C24—2141no—
165#41—10716no—
166#41—21416no—
167#41—2148no—
168C2×Q8—22no—
169C2×Q8—21no—
170C2×Q8C4×D10714yescomplemented
171C2×Q8C22×D10712yes—
172C2×D4—1074no—
173C2×D4—1072no—
174C2×D4—2148no—
175C2×D4—2144no—
176C2×D4—2141no—
177C2×D4—2142no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—1071no—
181C22×C4—1074no—
182C22×C4—1072no—
183C22×C4—1074no—
184C22×C4—1072no—
185C22×C4—4284no—
186C22×C4—2148no—
187C22×C4—2144no—
188C22×C4—2144no—
189C22×C4—2141no—
190C22×C4C22×D10711yescharacteristic
191M16—2148no—
192C2×C8—22no—
193C2×C8—1072no—
194C2×C8—2142no—
195C2×C8C22×D10712yes—
196C4⋊C4—21no—
197C4⋊C4—1072no—
198C4⋊C4—2144no—
199C4⋊C4—2141no—
200C4⋊C4C22×D10712yes—
201C23—1074no—
202C23—1072no—
203C23—4282no—
204C23—4282no—
205C23—2144no—
206C23—2142no—
207C23—2142no—
208C23—2142no—
209C23D4×D10711yescharacteristic
210Q8—24no—
211Q8—22no—
212Q8C2×C4×D10714yescomplemented
213D4—1074no—
214D4—21416no—
215D4—2144no—
216D4—2142no—
217C2×C4—24no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—42no—
221C2×C4—1074no—
222C2×C4—1072no—
223C2×C4—1078no—
224C2×C4—1074no—
225C2×C4—1074no—
226C2×C4—4288no—
227C2×C4—4282no—
228C2×C4—21416no—
229C2×C4—2142no—
230C2×C4C2×C4×D10712yes—
231C2×C4D4×D10711yescharacteristic
232C2×C4D4×D10712yes—
233C2×C4C23×D10711yescharacteristic
234C8—22no—
235C8—2142no—
236C22—1074no—
237C22—4288no—
238C22—4282no—
239C22—4282no—
240C22—2144no—
241C22—2148no—
242C22—2142no—
243C22—2142no—
244C22C22⋊C4×D10712yes—
245C22#1681213612yes—
246C22#1681214112yes—
247C22C2×D4×D10711yescharacteristic
248C4—24no—
249C4—42no—
250C4—1074no—
251C4—4282no—
252C4C22⋊C4×D10712yes—
253C4C22×C4×D10711yescharacteristic
254C4C2×D4×D10711yescharacteristic
255C2—4284no—
256C2—2144no—
257C2#2029262714yesdirect product factor
258C2C2×C22⋊C4×D10711yescharacteristic
259C2C2×#1681213611yescharacteristic
260C2C2×#1681214111yescharacteristic
subgroupG/H
centerC23D4×D107
commutatorC4×C107C23×C4
FrattiniC2×C4C23×D107
FittingC2×#103×C107C2
socleC23×C107C2×D4
prime radicalC22×#13133819C2
cosocleC2×C4×C107C24
hypercenterC2×#103D107
hypocenterC107C2×#417
LCSG ▷ C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
2. C22×#13133801C2semidirect
3. C2×C4⋊C4×D107C2semidirect
4. #20292627C2direct (N × H)
5. C2×#16812033C2semidirect
6. C2×#16812031C2semidirect
7. C2×#16812029C2semidirect
8. C2×#103×C107C2semidirect
11. C2×#13133819C4semidirect
19. C2×#13133801C22semidirect
24. C4⋊C4×D107C22semidirect
25. #16812033C22semidirect
26. #16812031C22semidirect
27. #16812029C22semidirect
31. #103×C107C22semidirect
36. #13133819C2×C4semidirect
41. C2×D428C2×C4semidirect
61. C2×Q8×C107C2×C4semidirect
81. Q8×C107C22×C4semidirect
97. D428C22×C4semidirect
124. C2×C107#417semidirect
131. C107C2×#417semidirect
137. C2×#103D107semidirect
160. #103C2×D107semidirect
170. C2×Q8C4×D107semidirect
212. Q8C2×C4×D107semidirect
257. C2#20292627direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131338192
C22×#131338012
C2×C4⋊C4×D1072
#202926272
C2×#168120332
C2×#168120312
C2×#168120292
C2×#103×C1072
C2×#417107