Finite Group Databasebeta

← #28346103

Subgroups — #28346103

Order 13696.

492 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 133 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×#16812117C211yesmaximal, characteristic, complemented
3#20293201C211yesmaximal, characteristic, complemented
4#20293196C211yesmaximal, characteristic
5#20292876C214yesmaximal, complemented
6#20292859C214yesmaximal, complemented
7#20292663C211yesmaximal, characteristic, complemented
8#20292633C211yesmaximal, characteristic, complemented
9#419×C107C211yesmaximal, characteristic, complemented
10C2×#13133818C2211yescharacteristic
11C22×D428C2211yescharacteristic
12C2×#13133805—24no—
13C2×#13133805C2212yes—
14C2×#13133805C2212yes—
15#16812117—24no—
16C2×#13133801C2212yescomplemented
17C22×C107⋊C8C2211yescharacteristic
18C2×C8×D107C2212yescomplemented
19#16812107—21no—
20#16812106—21no—
21#16812098—21no—
22#16812097—21no—
23#16812092—22no—
24#16812085C2214yescomplemented
25#16812084C2214yes—
26#16812079—22no—
27#16812068C2211yescharacteristic
28#16812042C2212yescomplemented
29#16812040C2212yescomplemented
30#16812031C2214yes—
31C22×Q8×C107C2211yescharacteristic
32C2×M16×C107C2211yescharacteristic
33#117×C107C2211yescharacteristic
34#103×C107C2214yescomplemented
35C23×D107—21no—
36#13133818—22no—
37#13133818—22no—
38C2×Dic214D411yescharacteristic
39C2×D428—22no—
40C2×D428—22no—
41C2×D428—21no—
42C2×D428—24no—
43C2×D428D411yescharacteristic
44C2×D428C2312yes—
45C2×C22⋊D107D412yes—
46C2×C4×D107C2312yes—
47#13133805—24no—
48#13133805—24no—
49#13133801—24no—
50C2×C107⋊C8D414yescomplemented
51C2×C107⋊C8C2311yescharacteristic
52C2×C107⋊C8C2311yescharacteristic
53C8×D107—24no—
54C4⋊Dic107C2312yes—
55Dic107⋊C4—24no—
56C22⋊Dic107—21no—
57(C2×D107)⋊C4—24no—
58(C2×D107)⋊C4—24no—
59C4×Dic107—21no—
60C2×Q8×C107—21no—
61C2×Q8×C107—22no—
62C2×Q8×C107—22no—
63C2×Q8×C107C2312yes—
64C22×C4×C107—21no—
65C22×C4×C107C2311yescharacteristic
66M16×C107—22no—
67C2×C8×C107C2312yes—
68C4⋊C4×C107C2312yes—
69C22⋊C4×C107—21no—
70C42×C107—21no—
71C23×C107C2×D411yescharacteristic
72C22×D107—22no—
73C22×D107—42no—
74C22×D107—82no—
75C22×D107C2×D412yes—
76Q8×C107—22no—
77Q8×C107—22no—
78Q8×C107—42no—
79C2×C4×C107—22no—
80C2×C4×C107—22no—
81C2×C4×C107—42no—
82C2×C4×C107C2×D411yescharacteristic
83C2×C4×C107C2×D412yes—
84C2×C4×C107#4112yes—
85C2×C4×C107C2411yescharacteristic
86C22⋊D107—24no—
87C2×Dic107—22no—
88C2×Dic107C2×D412yes—
89D428—22no—
90D428—22no—
91D428—22no—
92D428—42no—
93D428—84no—
94C4×D107—22no—
95C4×D107—22no—
96Dic214—22no—
97C8×C107—22no—
98C107⋊C8C2×D412yes—
99C107⋊C8C2×D412yes—
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#427—1071no—
129#419D10711yescharacteristic, complemented
130#418—1071no—
131C2×#41—1071no—
132C22×Q8C2×D10711yescharacteristic
133C22×D4—1071no—
134C2×SD16—1072no—
135C2×SD16—1072no—
136C2×SD16—2144no—
137#131—2144no—
138C2×M16—1072no—
139C2×M16C2×D10711yescharacteristic
140C22×C8—1071no—
141C22×C8—1072no—
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#117—1071no—
151#117C2×D10711yescharacteristic
152#107—1072no—
153#106—1072no—
154#103C2×D10714yescomplemented
155#102—1074no—
156C24—2141no—
157#41—2142no—
158#41—2142no—
159C2×Q8—21no—
160C2×Q8—22no—
161C2×Q8—22no—
162C2×Q8—1071no—
163C2×Q8C22×D10712yes—
164C2×D4—1072no—
165C2×D4—1071no—
166C2×D4—1072no—
167C2×D4—2144no—
168C2×D4—2141no—
169C2×D4—2142no—
170C2×D4—2142no—
171C22×C4—21no—
172C22×C4—1072no—
173C22×C4C22×D10711yescharacteristic
174SD16—2144no—
175SD16—2144no—
176M16—22no—
177M16—2144no—
178C2×C8—1074no—
179C2×C8—1071no—
180C2×C8—1071no—
181C2×C8—2144no—
182C2×C8C22×D10712yes—
183C4⋊C4—1072no—
184C4⋊C4—2144no—
185C4⋊C4C22×D10712yes—
186C22⋊C4—21no—
187C22⋊C4—2144no—
188C22⋊C4—2144no—
189C22⋊C4—2141no—
190C42—21no—
191C42—2141no—
192C23—8562no—
193C23—1072no—
194C23—4282no—
195C23—2142no—
196C23D4×D10711yescharacteristic
197Q8—22no—
198Q8—22no—
199Q8—42no—
200Q8—2142no—
201D4—8564no—
202D4—4282no—
203D4—2144no—
204D4—2142no—
205D4—2142no—
206D4—2142no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—42no—
210C2×C4—1072no—
211C2×C4—2142no—
212C2×C4—2142no—
213C2×C4—2142no—
214C2×C4D4×D10711yescharacteristic
215C2×C4D4×D10712yes—
216C2×C4#1313381812yes—
217C2×C4C23×D10711yescharacteristic
218C8—22no—
219C8—1072no—
220C8—1072no—
221C22—22no—
222C22—8564no—
223C22—4284no—
224C22—4282no—
225C22—2144no—
226C22—2142no—
227C22#1681208212yes—
228C22C2×D4×D10711yescharacteristic
229C4—42no—
230C4—42no—
231C4—4282no—
232C4—2142no—
233C4#1681208212yes—
234C4C2×D4×D10711yescharacteristic
235C4C2×#1313381811yescharacteristic
236C2—22no—
237C2—8562no—
238C2—4282no—
239C2C2×#1681208211yescharacteristic
240C2#2029343612yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#419×C107C2
socleC22×C107C22×D4
prime radical#20293196C2
hypercenter#419D107
hypocenterC107#3057
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. C22×#13133805C2semidirect
2. C2×#16812117C2semidirect
3. #20293201C2semidirect
5. #20292876C2semidirect
6. #20292859C2semidirect
7. #20292663C2semidirect
8. #20292633C2semidirect
9. #419×C107C2semidirect
16. C2×#13133801C22semidirect
18. C2×C8×D107C22semidirect
24. #16812085C22semidirect
28. #16812042C22semidirect
29. #16812040C22semidirect
34. #103×C107C22semidirect
50. C2×C107⋊C8D4semidirect
121. C107#3057semidirect
129. #419D107semidirect
154. #103C2×D107semidirect

Maximal subgroups

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

subgroupindex
C22×#131338052
C2×#168121172
#202932012
#202931962
#202928762
#202928592
#202926632
#202926332
#419×C1072
#3057107