Finite Group Databasebeta

← #28346102

Subgroups — #28346102

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×#13133805C211yesmaximal, characteristic, complemented
2C2×#16812116C211yesmaximal, characteristic, complemented
3#20293191C211yesmaximal, characteristic
4#20292875C214yesmaximal, complemented
5#20292858C214yesmaximal, complemented
6#20292663C211yesmaximal, characteristic, complemented
7#20292636C211yesmaximal, characteristic, complemented
8#20292631C211yesmaximal, characteristic, complemented
9#536×C107C211yesmaximal, characteristic, complemented
10C22×D428C2211yescharacteristic
11C2×#13133805—24no—
12C2×#13133805C2212yes—
13C2×#13133805C2212yes—
14#16812116—24no—
15C2×#13133802C2211yescharacteristic
16C2×#13133802C2212yescomplemented
17C22×C107⋊C8C2211yescharacteristic
18C22×C107⋊C8C2212yescomplemented
19#16812105—21no—
20#16812098—21no—
21#16812083C2214yes—
22#16812077—22no—
23#16812068C2211yescharacteristic
24#16812042C2212yescomplemented
25#16812040C2212yescomplemented
26#16812033C2214yescomplemented
27#16812029C2214yes—
28C2×#41×C107C2211yescharacteristic
29C22×Q8×C107C2211yescharacteristic
30C4⋊Q8×C107—21no—
31#124×C107—21no—
32#123×C107—22no—
33C22⋊Q8×C107C2214yescomplemented
34#117×C107C2211yescharacteristic
35C23×D107—21no—
36C2×D428—22no—
37C2×D428—22no—
38C2×D428—21no—
39C2×D428C2312yes—
40C2×C22⋊D107—24no—
41#13133805—24no—
42#13133805—24no—
43#13133802—22no—
44#13133802—24no—
45C2×C107⋊C8—24no—
46C2×C107⋊C8D414yescomplemented
47C2×C107⋊C8C2312yes—
48C2×C107⋊C8C2311yescharacteristic
49C2×C107⋊C8C2311yescharacteristic
50C4⋊Dic107C2312yes—
51C22⋊Dic107—21no—
52(C2×D107)⋊C4—24no—
53C4×Dic107—21no—
54#41×C107—22no—
55#41×C107—22no—
56C2×Q8×C107—21no—
57C2×Q8×C107—22no—
58C2×Q8×C107—22no—
59C2×Q8×C107D411yescharacteristic
60C2×Q8×C107C2312yes—
61C2×D4×C107D412yes—
62C2×D4×C107D411yescharacteristic
63C22×C4×C107—21no—
64C22×C4×C107C2312yes—
65C22×C4×C107C2311yescharacteristic
66C4⋊C4×C107—24no—
67C4⋊C4×C107C2312yes—
68C22⋊C4×C107—21no—
69C22⋊C4×C107—24no—
70C42×C107—21no—
71C23×C107C2×D412yes—
72C23×C107C2×D411yescharacteristic
73C22×D107—22no—
74C22×D107—42no—
75C22×D107—82no—
76Q8×C107—22no—
77Q8×C107—22no—
78Q8×C107—22no—
79Q8×C107—42no—
80D4×C107—24no—
81D4×C107—22no—
82C2×C4×C107—22no—
83C2×C4×C107—22no—
84C2×C4×C107—22no—
85C2×C4×C107—22no—
86C2×C4×C107—42no—
87C2×C4×C107C2×D412yes—
88C2×C4×C107C2×D412yes—
89C2×C4×C107C2×D411yescharacteristic
90C2×C4×C107#4112yes—
91C2×C4×C107C2411yescharacteristic
92C22⋊D107—84no—
93C2×Dic107—22no—
94D428—22no—
95D428—22no—
96D428—42no—
97C107⋊C8—22no—
98C107⋊C8C2×D412yes—
99C107⋊C8C2×D412yes—
100C22×C107—22no—
101C22×C107—22no—
102C22×C107—24no—
103C22×C107#12112yes—
104C22×C107C22×D411yescharacteristic
105C2×D107—42no—
106C2×D107—44no—
107C2×D107—84no—
108C4×C107—22no—
109C4×C107—42no—
110C4×C107—42no—
111C4×C107#12112yes—
112C4×C107C22×D411yescharacteristic
113C4×C107C2×#4111yescharacteristic
114Dic107—42no—
115C2×C107—22no—
116C2×C107—42no—
117C2×C107C2×#12111yescharacteristic
118C2×C107#57712yes—
119D107—82no—
120#3057—1071nomaximal, Sylow
121C107#305711yescharacteristic, complemented, Sylow
122C22×SD16—1071no—
123C2×#131—1071no—
124#536D10711yescharacteristic, complemented
125#535—1071no—
126#468—1074no—
127#465—1074no—
128#427—1071no—
129#419—1071no—
130#418—1071no—
131C2×#41C2×D10711yescharacteristic
132C22×Q8C2×D10711yescharacteristic
133C22×D4—1071no—
134C2×SD16—1072no—
135C2×SD16—1072no—
136C2×SD16—2144no—
137#131—2144no—
138C2×M16—1071no—
139C2×M16—1072no—
140C22×C8—1071no—
141C22×C8—1072no—
142C4⋊Q8—21no—
143#127—2141no—
144#124—21no—
145#124—2141no—
146#123—22no—
147C22⋊Q8C2×D10714yescomplemented
148#121—1074no—
149#120—2142no—
150#117—1071no—
151#117C2×D10711yescharacteristic
152#107—1072no—
153#106—1072no—
154#103—1074no—
155#102—1074no—
156C24—2141no—
157#41—22no—
158#41—22no—
159C2×Q8—21no—
160C2×Q8—22no—
161C2×Q8—22no—
162C2×Q8C22⋊D10711yescharacteristic
163C2×Q8C22×D10712yes—
164C2×D4—1072no—
165C2×D4—2144no—
166C2×D4—2142no—
167C2×D4—2141no—
168C2×D4—2142no—
169C2×D4C22⋊D10712yes—
170C2×D4C22⋊D10711yescharacteristic
171C22×C4—21no—
172C22×C4C22×D10712yes—
173C22×C4C22×D10711yescharacteristic
174SD16—2144no—
175SD16—2144no—
176M16—2144no—
177M16—2142no—
178C2×C8—1074no—
179C2×C8—1071no—
180C2×C8—1071no—
181C2×C8—1072no—
182C2×C8—2144no—
183C4⋊C4—24no—
184C4⋊C4—1072no—
185C4⋊C4C22×D10712yes—
186C22⋊C4—21no—
187C22⋊C4—24no—
188C22⋊C4—2141no—
189C22⋊C4—2144no—
190C42—21no—
191C42—2141no—
192C23—8562no—
193C23—4282no—
194C23—2142no—
195C23C2×C22⋊D10712yes—
196C23C2×C22⋊D10711yescharacteristic
197Q8—22no—
198Q8—22no—
199Q8—22no—
200Q8—42no—
201D4—24no—
202D4—22no—
203D4—8564no—
204D4—4282no—
205D4—2142no—
206D4—2142no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—42no—
212C2×C4—2142no—
213C2×C4D4×D10712yes—
214C2×C4C2×C22⋊D10712yes—
215C2×C4C2×C22⋊D10711yescharacteristic
216C2×C4#1313381712yes—
217C2×C4C23×D10711yescharacteristic
218C8—1072no—
219C8—1072no—
220C8—2142no—
221C22—22no—
222C22—24no—
223C22—22no—
224C22—8564no—
225C22—4284no—
226C22—4282no—
227C22#1681208112yes—
228C22C22×C22⋊D10711yescharacteristic
229C4—22no—
230C4—42no—
231C4—42no—
232C4—4282no—
233C4#1681208112yes—
234C4C2×D4×D10711yescharacteristic
235C4C2×#1313381711yescharacteristic
236C2—22no—
237C2—42no—
238C2—8562no—
239C2C2×#1681208111yescharacteristic
240C2#2029343512yes—
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#536×C107C2
socleC22×C107C22×D4
prime radical#20293191C2
hypercenter#536D107
hypocenterC107#3057
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C22×#13133805C2semidirect
2. C2×#16812116C2semidirect
4. #20292875C2semidirect
5. #20292858C2semidirect
6. #20292663C2semidirect
7. #20292636C2semidirect
8. #20292631C2semidirect
9. #536×C107C2semidirect
16. C2×#13133802C22semidirect
18. C22×C107⋊C8C22semidirect
24. #16812042C22semidirect
25. #16812040C22semidirect
26. #16812033C22semidirect
33. C22⋊Q8×C107C22semidirect
46. C2×C107⋊C8D4semidirect
121. C107#3057semidirect
124. #536D107semidirect
147. C22⋊Q8C2×D107semidirect

Maximal subgroups

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

subgroupindex
C22×#131338052
C2×#168121162
#202931912
#202928752
#202928582
#202926632
#202926362
#202926312
#536×C1072
#3057107