Finite Group Databasebeta

← C2×#20292792 #28345673

Subgroups — C2×#20292792 #28345673

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×#13133819C211yesmaximal, characteristic
2C22×#13133808C211yesmaximal, characteristic, complemented
3C22×#13133807C211yesmaximal, characteristic, complemented
4C2×#16812089C211yesmaximal, characteristic, complemented
5#20292792C218yesmaximal, direct product factor
6C2×#16812029C211yesmaximal, characteristic, complemented
7C2×(C2×D107)⋊C8C211yesmaximal, characteristic, complemented
8C2×#103×C107C211yesmaximal, characteristic, complemented
9C2×#13133819—24no—
10C2×#13133819—22no—
11C2×#13133819—24no—
12C2×#13133819C2214yes—
13C22×Dic214C2211yescharacteristic
14C22×D428—21no—
15C22×D428C2211yescharacteristic
16C22×C4×D107—21no—
17C22×C4×D107C2211yescharacteristic
18C2×#13133808—24no—
19C2×#13133808C2214yescomplemented
20C2×#13133807—24no—
21C2×#13133807C2214yescomplemented
22C22×C107⋊C8C2211yescharacteristic
23#16812089—22no—
24#16812089C2214yescomplemented
25C2×C4⋊Dic107—21no—
26C2×(C2×D107)⋊C4—21no—
27#16812029C2214yescomplemented
28(C2×D107)⋊C8C2214yescomplemented
29C22×Q8×C107C2211yescharacteristic
30C22×C8×C107C2211yescharacteristic
31C2×C4⋊C4×C107C2211yescharacteristic
32#103×C107C2214yescomplemented
33C23×D107—21no—
34C23×D107D411yescharacteristic
35#13133819—216no—
36#13133819—28no—
37#13133819—28no—
38C2×Dic214—22no—
39C2×Dic214—22no—
40C2×Dic214—21no—
41C2×Dic214C2312yes—
42C2×D428—28no—
43C2×D428—24no—
44C2×D428—21no—
45C2×D428—22no—
46C2×D428D414yes—
47C2×D428C2312yes—
48C22×Dic107—21no—
49C22×Dic107D411yescharacteristic
50C2×C4×D107—28no—
51C2×C4×D107—22no—
52C2×C4×D107—22no—
53C2×C4×D107—24no—
54C2×C4×D107D412yes—
55C2×C4×D107C2312yes—
56#13133808—28no—
57#13133807—28no—
58C2×C107⋊C8—22no—
59C2×C107⋊C8C2312yes—
60C4⋊Dic107—24no—
61(C2×D107)⋊C4—24no—
62C2×Q8×C107—22no—
63C2×Q8×C107—21no—
64C2×Q8×C107D414yes—
65C2×Q8×C107C2312yes—
66C22×C4×C107—21no—
67C22×C4×C107—21no—
68C22×C4×C107C2311yescharacteristic
69C2×C8×C107—22no—
70C2×C8×C107C2312yes—
71C4⋊C4×C107—21no—
72C4⋊C4×C107C2312yes—
73C23×C107C2×D411yescharacteristic
74C22×D107—22no—
75C22×D107—24no—
76C22×D107—22no—
77C22×D107—22no—
78C22×D107—22no—
79C22×D107—42no—
80C22×D107—42no—
81C22×D107C2×D412yes—
82Q8×C107—24no—
83Q8×C107—22no—
84Q8×C107C2×D414yes—
85C2×C4×C107—24no—
86C2×C4×C107—22no—
87C2×C4×C107—22no—
88C2×C4×C107—42no—
89C2×C4×C107C2×D411yescharacteristic
90C2×C4×C107C2×D412yes—
91C2×C4×C107C2×D412yes—
92C2×C4×C107C2411yescharacteristic
93C2×Dic107—22no—
94C2×Dic107—22no—
95C2×Dic107—42no—
96C2×Dic107C2×D412yes—
97D428—216no—
98D428—24no—
99D428—22no—
100D428C2×D414yes—
101C4×D107—216no—
102C4×D107—24no—
103C4×D107—22no—
104C4×D107—22no—
105Dic214—22no—
106Dic214—22no—
107Dic214—42no—
108C8×C107—22no—
109C107⋊C8—22no—
110C22×C107#12012yes—
111C22×C107#13512yes—
112C22×C107#13712yes—
113C22×C107C22×D411yescharacteristic
114C2×D107—22no—
115C2×D107—24no—
116C2×D107—22no—
117C2×D107—22no—
118C2×D107—42no—
119C2×D107—42no—
120C2×D107—44no—
121C2×D107—48no—
122C4×C107—24no—
123C4×C107—42no—
124C4×C107#12012yes—
125C4×C107C22×D411yescharacteristic
126C4×C107C22×D411yescharacteristic
127Dic107—22no—
128Dic107—42no—
129C2×C107#45214yescomplemented
130C2×C107C2×#12011yescharacteristic
131C2×C107C2×#13511yescharacteristic
132C2×C107C2×#13711yescharacteristic
133D107—42no—
134D107—44no—
135C2×#452—1071nomaximal, Sylow
136C107C2×#45211yescharacteristic, complemented, Sylow
137C22×#41—1071no—
138C22×Q16—1071no—
139C22×SD16—1071no—
140C2×C22⋊Q8—1071no—
141#452—1078no—
142C2×#103D10711yescharacteristic, complemented
143C2×#102—1071no—
144C2×C22⋊C8—1071no—
145C2×#41—1074no—
146C2×#41—2142no—
147C2×#41—2144no—
148C2×#41—2144no—
149C22×Q8—1071no—
150C22×Q8C2×D10711yescharacteristic
151C22×D4—1071no—
152C22×D4—2141no—
153C23×C4—1071no—
154C23×C4—2141no—
155C2×Q16—1074no—
156C2×Q16—2144no—
157C2×SD16—1074no—
158C2×SD16—2144no—
159C22×C8—1071no—
160C22×C8C2×D10711yescharacteristic
161C22⋊Q8—1074no—
162C22⋊Q8—2142no—
163C2×C4⋊C4—2141no—
164C2×C4⋊C4C2×D10711yescharacteristic
165C2×C22⋊C4—2141no—
166#103C2×D10714yescomplemented
167#102—1074no—
168C22⋊C8—1074no—
169C24—1071no—
170C24—2141no—
171#41—2148no—
172#41—21416no—
173#41—2148no—
174C2×Q8—22no—
175C2×Q8—21no—
176C2×Q8—1072no—
177C2×Q8—2142no—
178C2×Q8—2142no—
179C2×Q8—2141no—
180C2×Q8D42814yes—
181C2×Q8C22×D10712yes—
182C2×D4—1074no—
183C2×D4—1072no—
184C2×D4—2144no—
185C2×D4—2148no—
186C2×D4—2141no—
187C2×D4—2142no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—1071no—
191C22×C4—1072no—
192C22×C4—1072no—
193C22×C4—2144no—
194C22×C4—2148no—
195C22×C4—2141no—
196C22×C4—2142no—
197C22×C4—2142no—
198C22×C4C22×D10711yescharacteristic
199Q16—2148no—
200SD16—2148no—
201C2×C8—22no—
202C2×C8—1072no—
203C2×C8—2142no—
204C2×C8C22×D10712yes—
205C4⋊C4—21no—
206C4⋊C4—2144no—
207C4⋊C4C22×D10712yes—
208C22⋊C4—2144no—
209C23—1072no—
210C23—4282no—
211C23—4282no—
212C23—2144no—
213C23—2142no—
214C23—2142no—
215C23—2142no—
216C23—2142no—
217C23D4×D10711yescharacteristic
218Q8—24no—
219Q8—22no—
220Q8—4282no—
221Q8—2142no—
222Q8—2142no—
223Q8C2×D42814yes—
224D4—1074no—
225D4—21416no—
226D4—2144no—
227D4—2142no—
228C2×C4—24no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—42no—
232C2×C4—1072no—
233C2×C4—4282no—
234C2×C4—21416no—
235C2×C4—2142no—
236C2×C4—2142no—
237C2×C4—2144no—
238C2×C4—2142no—
239C2×C4—2142no—
240C2×C4D4×D10711yescharacteristic
241C2×C4D4×D10712yes—
242C2×C4C2×D42812yes—
243C2×C4C23×D10711yescharacteristic
244C8—22no—
245C8—2142no—
246C22—4282no—
247C22—4282no—
248C22—4288no—
249C22—4284no—
250C22—2142no—
251C22—2142no—
252C22—2144no—
253C22—2142no—
254C22#1681207912yes—
255C22#1681213012yes—
256C22#1681214112yes—
257C22C2×D4×D10711yescharacteristic
258C4—24no—
259C4—42no—
260C4—4282no—
261C4—2142no—
262C4#1681207912yes—
263C4C2×D4×D10711yescharacteristic
264C4C22×D42811yescharacteristic
265C2—4284no—
266C2—4282no—
267C2#2029279214yesdirect product factor
268C2C2×#1681207911yescharacteristic
269C2C2×#1681213011yescharacteristic
270C2C2×#1681214111yescharacteristic
subgroupG/H
centerC23D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×#103×C107C2
socleC23×C107C2×D4
prime radicalC22×#13133819C2
hypercenterC2×#103D107
hypocenterC107C2×#452
LCSG ▷ C2×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×#13133808C2semidirect
3. C22×#13133807C2semidirect
4. C2×#16812089C2semidirect
5. #20292792C2direct (N × H)
6. C2×#16812029C2semidirect
7. C2×(C2×D107)⋊C8C2semidirect
8. C2×#103×C107C2semidirect
19. C2×#13133808C22semidirect
21. C2×#13133807C22semidirect
24. #16812089C22semidirect
27. #16812029C22semidirect
28. (C2×D107)⋊C8C22semidirect
32. #103×C107C22semidirect
129. C2×C107#452semidirect
136. C107C2×#452semidirect
142. C2×#103D107semidirect
166. #103C2×D107semidirect
267. C2#20292792direct (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×#131338082
C22×#131338072
C2×#168120892
#202927922
C2×#168120292
C2×(C2×D107)⋊C82
C2×#103×C1072
C2×#452107