Finite Group Databasebeta

← #28346101

Subgroups — #28346101

Order 13696.

492 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 141 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×#13133807C211yesmaximal, characteristic, complemented
2C2×#16812115C211yesmaximal, characteristic, complemented
3#20293203C211yesmaximal, characteristic, complemented
4#20293196C211yesmaximal, characteristic
5#20292874C214yesmaximal, complemented
6#20292857C214yesmaximal, complemented
7#20292635C211yesmaximal, characteristic, complemented
8#20292631C211yesmaximal, characteristic, complemented
9#427×C107C211yesmaximal, characteristic, complemented
10C2×#13133818C2211yescharacteristic
11C22×Dic214C2211yescharacteristic
12C22×D428C2211yescharacteristic
13C2×#13133807—24no—
14C2×#13133807C2212yes—
15C2×#13133807C2212yes—
16#16812115—24no—
17C2×#13133802C2211yescharacteristic
18C2×#13133801C2212yescomplemented
19C2×C8×D107C2212yescomplemented
20#16812109—21no—
21#16812106—21no—
22#16812097—21no—
23#16812097—21no—
24#16812093—22no—
25#16812089C2214yescomplemented
26#16812084C2214yes—
27#16812079—22no—
28#16812032C2214yescomplemented
29#16812029C2214yes—
30C22×C8×C107C2211yescharacteristic
31#117×C107C2211yescharacteristic
32#117×C107C2211yescharacteristic
33#107×C107C2212yescomplemented
34#106×C107C2212yescomplemented
35C23×D107—21no—
36#13133818—22no—
37#13133818—22no—
38C2×Dic214—22no—
39C2×Dic214—22no—
40C2×Dic214—21no—
41C2×Dic214D411yescharacteristic
42C2×Dic214C2312yes—
43C2×D428—22no—
44C2×D428—22no—
45C2×D428—21no—
46C2×D428—24no—
47C2×D428D411yescharacteristic
48C2×D428C2312yes—
49C2×C22⋊D107D412yes—
50C22×Dic107—21no—
51C2×C4×D107C2312yes—
52#13133807—24no—
53#13133807—24no—
54#13133802—22no—
55#13133801—24no—
56C2×C107⋊C8C2312yes—
57C8×D107—24no—
58C4⋊Dic107—24no—
59(C2×D107)⋊C4—24no—
60(C2×D107)⋊C4—24no—
61C22×C4×C107C2311yescharacteristic
62C2×C8×C107D414yescomplemented
63C2×C8×C107C2311yescharacteristic
64C2×C8×C107C2311yescharacteristic
65C4⋊C4×C107C2312yes—
66C4⋊C4×C107C2312yes—
67C22⋊C4×C107—21no—
68C22⋊C4×C107—21no—
69C42×C107—21no—
70C42×C107—21no—
71C23×C107C2×D411yescharacteristic
72C22×D107—22no—
73C22×D107—42no—
74C22×D107—82no—
75C22×D107C2×D412yes—
76C2×C4×C107—22no—
77C2×C4×C107—22no—
78C2×C4×C107C2×D411yescharacteristic
79C2×C4×C107C2×D412yes—
80C2×C4×C107#4112yes—
81C2×C4×C107C2411yescharacteristic
82C22⋊D107—24no—
83C2×Dic107—22no—
84C2×Dic107—42no—
85C2×Dic107C2×D412yes—
86D428—22no—
87D428—22no—
88D428—22no—
89D428—42no—
90D428—84no—
91C4×D107—22no—
92C4×D107—22no—
93Dic214—22no—
94Dic214—22no—
95Dic214—22no—
96Dic214—42no—
97C8×C107C2×D412yes—
98C8×C107C2×D412yes—
99C107⋊C8—22no—
100C22×C107—22no—
101C22×C107#12112yes—
102C22×C107C22×D411yescharacteristic
103C2×D107—22no—
104C2×D107—24no—
105C2×D107—42no—
106C2×D107—44no—
107C2×D107—84no—
108C4×C107—42no—
109C4×C107—42no—
110C4×C107#12112yes—
111C4×C107C22×D411yescharacteristic
112C4×C107C2×#4111yescharacteristic
113Dic107—22no—
114Dic107—42no—
115C2×C107—22no—
116C2×C107C2×#12111yescharacteristic
117C2×C107#57712yes—
118D107—42no—
119D107—82no—
120#3057—1071nomaximal, Sylow
121C107#305711yescharacteristic, complemented, Sylow
122C22×SD16—1071no—
123C2×#131—1071no—
124#536—1071no—
125#535—1071no—
126#468—1074no—
127#465—1074no—
128#427D10711yescharacteristic, complemented
129#419—1071no—
130#418—1071no—
131C2×#41—1071no—
132C22×Q8—1071no—
133C22×D4—1071no—
134C2×SD16—1072no—
135C2×SD16—1072no—
136C2×SD16—2144no—
137#131—2144no—
138C2×M16—1071no—
139C2×M16—1072no—
140C22×C8—1072no—
141C22×C8C2×D10711yescharacteristic
142C4⋊Q8—2141no—
143#127—2141no—
144#124—2141no—
145#124—2141no—
146#123—2142no—
147C22⋊Q8—1074no—
148#121—1074no—
149#120—2142no—
150#117C2×D10711yescharacteristic
151#117C2×D10711yescharacteristic
152#107C2×D10712yescomplemented
153#106C2×D10712yescomplemented
154#103—1074no—
155#102—1074no—
156C24—2141no—
157#41—2142no—
158#41—2142no—
159C2×Q8—1071no—
160C2×Q8—1072no—
161C2×Q8—2142no—
162C2×Q8—2142no—
163C2×Q8—2141no—
164C2×D4—1072no—
165C2×D4—1071no—
166C2×D4—1072no—
167C2×D4—2144no—
168C2×D4—2141no—
169C2×D4—2142no—
170C2×D4—2142no—
171C22×C4—1072no—
172C22×C4—2141no—
173C22×C4C22×D10711yescharacteristic
174SD16—2144no—
175SD16—2144no—
176M16—2142no—
177M16—2144no—
178C2×C8—1072no—
179C2×C8—2144no—
180C2×C8D42814yescomplemented
181C2×C8C22×D10711yescharacteristic
182C2×C8C22×D10711yescharacteristic
183C4⋊C4—2144no—
184C4⋊C4C22×D10712yes—
185C4⋊C4C22×D10712yes—
186C22⋊C4—21no—
187C22⋊C4—21no—
188C22⋊C4—2144no—
189C22⋊C4—2144no—
190C42—21no—
191C42—21no—
192C23—8562no—
193C23—1072no—
194C23—4282no—
195C23—2142no—
196C23D4×D10711yescharacteristic
197Q8—4282no—
198Q8—2142no—
199Q8—2142no—
200Q8—2142no—
201D4—8564no—
202D4—4282no—
203D4—2144no—
204D4—2142no—
205D4—2142no—
206D4—2142no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—1072no—
210C2×C4—4282no—
211C2×C4—2142no—
212C2×C4—2142no—
213C2×C4—2142no—
214C2×C4D4×D10711yescharacteristic
215C2×C4C2×D42812yes—
216C2×C4#1313381912yes—
217C2×C4C23×D10711yescharacteristic
218C8—2142no—
219C8C2×D42812yes—
220C8C2×D42812yes—
221C22—22no—
222C22—8564no—
223C22—4282no—
224C22—4284no—
225C22—2142no—
226C22—2144no—
227C22#1681208412yes—
228C22C2×D4×D10711yescharacteristic
229C4—42no—
230C4—42no—
231C4—4282no—
232C4—2142no—
233C4#1681208412yes—
234C4C22×D42811yescharacteristic
235C4C2×#1313381911yescharacteristic
236C2—22no—
237C2—8562no—
238C2—4282no—
239C2C2×#1681208411yescharacteristic
240C2#2029343412yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#427×C107C2
socleC22×C107C22×D4
prime radical#20293196C2
hypercenter#427D107
hypocenterC107#3057
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #427

Complements

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

NH ≅ G/Ntype
1. C22×#13133807C2semidirect
2. C2×#16812115C2semidirect
3. #20293203C2semidirect
5. #20292874C2semidirect
6. #20292857C2semidirect
7. #20292635C2semidirect
8. #20292631C2semidirect
9. #427×C107C2semidirect
18. C2×#13133801C22semidirect
19. C2×C8×D107C22semidirect
25. #16812089C22semidirect
28. #16812032C22semidirect
33. #107×C107C22semidirect
34. #106×C107C22semidirect
62. C2×C8×C107D4semidirect
121. C107#3057semidirect
128. #427D107semidirect
152. #107C2×D107semidirect
153. #106C2×D107semidirect
180. C2×C8D428semidirect

Maximal subgroups

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

subgroupindex
C22×#131338072
C2×#168121152
#202932032
#202931962
#202928742
#202928572
#202926352
#202926312
#427×C1072
#3057107