Finite Group Databasebeta

← C2×#20292787 #28345668

Subgroups — C2×#20292787 #28345668

Order 13696.

760 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 165 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×#13133818C211yesmaximal, characteristic
2C22×#13133808C211yesmaximal, characteristic, complemented
3C22×#13133805C211yesmaximal, characteristic, complemented
4#20292787C218yesmaximal, direct product factor
5C2×#16812032C211yesmaximal, characteristic, complemented
6C2×#16812029C211yesmaximal, characteristic, complemented
7C2×C22⋊(C107⋊C8)C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8×C107C211yesmaximal, characteristic, complemented
9C2×#13133818—24no—
10C2×#13133818—24no—
11C2×#13133818—22no—
12C2×#13133818C2214yes—
13C22×Dic214C2211yescharacteristic
14C22×D428C2211yescharacteristic
15C22×C22⋊D107—21no—
16C22×C4×D107—21no—
17C2×#13133808—24no—
18C2×#13133808C2214yescomplemented
19C2×#13133805—24no—
20C2×#13133805C2214yescomplemented
21C22×C107⋊C8C2211yescharacteristic
22C22×C107⋊C8C2211yescharacteristic
23#16812032C2214yescomplemented
24#16812029C2214yescomplemented
25C22⋊(C107⋊C8)C2214yescomplemented
26C22×Q8×C107C2211yescharacteristic
27C23×C4×C107C2211yescharacteristic
28C22⋊Q8×C107—22no—
29C22⋊Q8×C107C2214yescomplemented
30C2×C4⋊C4×C107—21no—
31C2×C4⋊C4×C107C2211yescharacteristic
32C2×C22⋊C4×C107—21no—
33C23×D107—21no—
34#13133818—216no—
35#13133818—28no—
36#13133818—28no—
37C2×Dic214—22no—
38C2×Dic214—21no—
39C2×Dic214D414yes—
40C2×Dic214C2312yes—
41C2×D428—22no—
42C2×D428—21no—
43C2×D428D414yes—
44C2×D428C2312yes—
45C2×C22⋊D107—28no—
46C2×C22⋊D107—24no—
47C22×Dic107—21no—
48C2×C4×D107—28no—
49C2×C4×D107—24no—
50#13133808—28no—
51#13133805—28no—
52C2×C107⋊C8—22no—
53C2×C107⋊C8—22no—
54C2×C107⋊C8C2312yes—
55C2×C107⋊C8C2312yes—
56C24×C107D411yescharacteristic
57C2×Q8×C107—21no—
58C2×Q8×C107—22no—
59C2×Q8×C107—22no—
60C2×Q8×C107C2312yes—
61C22×C4×C107—22no—
62C22×C4×C107—22no—
63C22×C4×C107—21no—
64C22×C4×C107—21no—
65C22×C4×C107D411yescharacteristic
66C22×C4×C107D412yes—
67C22×C4×C107C2312yes—
68C22×C4×C107C2311yescharacteristic
69C4⋊C4×C107—24no—
70C4⋊C4×C107—21no—
71C4⋊C4×C107C2312yes—
72C22⋊C4×C107—24no—
73C23×C107—22no—
74C23×C107—22no—
75C23×C107—22no—
76C23×C107C2×D412yes—
77C23×C107C2×D411yescharacteristic
78C22×D107—24no—
79C22×D107—22no—
80C22×D107—42no—
81C22×D107—42no—
82Q8×C107—22no—
83Q8×C107—22no—
84Q8×C107—42no—
85C2×C4×C107—22no—
86C2×C4×C107—24no—
87C2×C4×C107—22no—
88C2×C4×C107—22no—
89C2×C4×C107—22no—
90C2×C4×C107—22no—
91C2×C4×C107—42no—
92C2×C4×C107—42no—
93C2×C4×C107C2×D412yes—
94C2×C4×C107C2×D412yes—
95C2×C4×C107C2×D412yes—
96C2×C4×C107C2×D411yescharacteristic
97C2×C4×C107C2411yescharacteristic
98C22⋊D107—216no—
99C2×Dic107—24no—
100C2×Dic107—22no—
101D428—24no—
102D428—22no—
103D428C2×D414yes—
104C4×D107—216no—
105Dic214—24no—
106Dic214—22no—
107Dic214C2×D414yes—
108C107⋊C8—22no—
109C107⋊C8—22no—
110C22×C107—22no—
111C22×C107—22no—
112C22×C107—22no—
113C22×C107—44no—
114C22×C107#12012yes—
115C22×C107#13512yes—
116C22×C107#13712yes—
117C22×C107C22×D411yescharacteristic
118C2×D107—24no—
119C2×D107—42no—
120C2×D107—42no—
121C2×D107—48no—
122C4×C107—22no—
123C4×C107—42no—
124C4×C107—42no—
125C4×C107#12012yes—
126C4×C107C22×D411yescharacteristic
127C4×C107C22×D411yescharacteristic
128Dic107—24no—
129C2×C107—42no—
130C2×C107#45214yescomplemented
131C2×C107C2×#12011yescharacteristic
132C2×C107C2×#13511yescharacteristic
133C2×C107C2×#13711yescharacteristic
134D107—44no—
135C2×#452—1071nomaximal, Sylow
136C107C2×#45211yescharacteristic, complemented, Sylow
137C22×#41—1071no—
138C22×Q16—1071no—
139C22×SD16—1071no—
140C2×C22⋊Q8D10711yescharacteristic, complemented
141#452—1078no—
142C2×#103—1071no—
143C2×#102—1071no—
144C2×C22⋊C8—1071no—
145C2×#41—1074no—
146C2×#41—2144no—
147C2×#41—2144no—
148C2×#41—2142no—
149C22×Q8—1071no—
150C22×Q8C2×D10711yescharacteristic
151C22×D4—1071no—
152C22×D4—2141no—
153C23×C4—2141no—
154C23×C4C2×D10711yescharacteristic
155C2×Q16—1074no—
156C2×Q16—2144no—
157C2×SD16—1074no—
158C2×SD16—2144no—
159C22×C8—1071no—
160C22×C8—1071no—
161C22⋊Q8—22no—
162C22⋊Q8C2×D10714yescomplemented
163C2×C4⋊C4—21no—
164C2×C4⋊C4C2×D10711yescharacteristic
165C2×C22⋊C4—21no—
166#103—1074no—
167#102—1074no—
168C22⋊C8—1074no—
169C24—2141no—
170C24C22⋊D10711yescharacteristic
171#41—2148no—
172#41—2148no—
173#41—21416no—
174C2×Q8—21no—
175C2×Q8—22no—
176C2×Q8—22no—
177C2×Q8—1074no—
178C2×Q8—1072no—
179C2×Q8—2142no—
180C2×Q8—2141no—
181C2×Q8C22×D10712yes—
182C2×D4—1074no—
183C2×D4—1072no—
184C2×D4—2148no—
185C2×D4—2144no—
186C2×D4—2142no—
187C2×D4—2141no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—2148no—
193C22×C4—2141no—
194C22×C4—2144no—
195C22×C4C22⋊D10712yes—
196C22×C4C22⋊D10711yescharacteristic
197C22×C4C22×D10712yes—
198C22×C4C22×D10711yescharacteristic
199Q16—2148no—
200SD16—2148no—
201C2×C8—1072no—
202C2×C8—1072no—
203C2×C8—2142no—
204C2×C8—2142no—
205C4⋊C4—24no—
206C4⋊C4—21no—
207C4⋊C4C22×D10712yes—
208C22⋊C4—24no—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—4282no—
213C23—4282no—
214C23—2144no—
215C23—2142no—
216C23C2×C22⋊D10712yes—
217C23C2×C22⋊D10711yescharacteristic
218Q8—22no—
219Q8—22no—
220Q8—42no—
221Q8—1074no—
222Q8—2144no—
223Q8—2142no—
224D4—1074no—
225D4—21416no—
226D4—2144no—
227D4—2142no—
228C2×C4—22no—
229C2×C4—24no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—21416no—
237C2×C4—2144no—
238C2×C4—2142no—
239C2×C4D4×D10712yes—
240C2×C4D4×D10712yes—
241C2×C4C2×C22⋊D10712yes—
242C2×C4C2×C22⋊D10711yescharacteristic
243C2×C4C23×D10711yescharacteristic
244C8—2142no—
245C8—2142no—
246C22—22no—
247C22—22no—
248C22—22no—
249C22—44no—
250C22—4282no—
251C22—4282no—
252C22—4288no—
253C22—2144no—
254C22#1681207712yes—
255C22#1681212912yes—
256C22#1681214012yes—
257C22C22×C22⋊D10711yescharacteristic
258C4—22no—
259C4—42no—
260C4—42no—
261C4—2144no—
262C4#1681207712yes—
263C4C2×D4×D10711yescharacteristic
264C4C2×D4×D10711yescharacteristic
265C2—42no—
266C2—4284no—
267C2#2029278714yesdirect product factor
268C2C2×#1681207711yescharacteristic
269C2C2×#1681212911yescharacteristic
270C2C2×#1681214011yescharacteristic
subgroupG/H
centerC23C2×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×C22⋊Q8×C107C2
socleC23×C107C2×D4
prime radicalC22×#13133818C2
hypercenterC2×C22⋊Q8D107
hypocenterC107C2×#452
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C22×#13133808C2semidirect
3. C22×#13133805C2semidirect
4. #20292787C2direct (N × H)
5. C2×#16812032C2semidirect
6. C2×#16812029C2semidirect
7. C2×C22⋊(C107⋊C8)C2semidirect
8. C2×C22⋊Q8×C107C2semidirect
18. C2×#13133808C22semidirect
20. C2×#13133805C22semidirect
23. #16812032C22semidirect
24. #16812029C22semidirect
25. C22⋊(C107⋊C8)C22semidirect
29. C22⋊Q8×C107C22semidirect
130. C2×C107#452semidirect
136. C107C2×#452semidirect
140. C2×C22⋊Q8D107semidirect
162. C22⋊Q8C2×D107semidirect
267. C2#20292787direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131338182
C22×#131338082
C22×#131338052
#202927872
C2×#168120322
C2×#168120292
C2×C22⋊(C107⋊C8)2
C2×C22⋊Q8×C1072
C2×#452107