Finite Group Databasebeta

← #28345963

Subgroups — #28345963

Order 13696.

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

normal subgroups only

This group has 526 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
1C2×#16812137C211yesmaximal, characteristic, complemented
2C22×#13133803C211yesmaximal, characteristic, complemented
3#20293194C211yesmaximal, characteristic, complemented
4#20293101C211yesmaximal, characteristic, complemented
5#20292841C214yesmaximal, complemented
6#20292821C214yesmaximal, complemented
7#20292655C211yesmaximal, characteristic, complemented
8#20292621C211yesmaximal, characteristic, complemented
9#418×C107C211yesmaximal, characteristic, complemented
10C2×#13133818C2211yescharacteristic
11C22×D428C2211yescharacteristic
12#16812137—24no—
13C2×#13133807C2212yescomplemented
14C2×D856C2212yescomplemented
15C2×#13133803—24no—
16C2×#13133803C2214yes—
17C22×C107⋊C8C2211yescharacteristic
18#16812105C2212yescomplemented
19#16812096C2212yescomplemented
20#16812080—22no—
21#16812078—22no—
22C4×D428—22no—
23#16812071C2214yescomplemented
24#16812068C2211yescharacteristic
25C22⋊C4×D107—22no—
26Dic107⋊C8C2214yescomplemented
27#16812032C2212yescomplemented
28#16812029C2212yescomplemented
29C22×D4×C107C2211yescharacteristic
30C2×M16×C107C2211yescharacteristic
31#117×C107C2211yescharacteristic
32#102×C107C2214yescomplemented
33C23×D107—21no—
34#13133818—22no—
35#13133818—22no—
36C2×Dic214C2311yescharacteristic
37C2×D428—21no—
38C2×D428—22no—
39C2×D428D414yescomplemented
40C2×D428C2311yescharacteristic
41C2×D428C2312yes—
42C2×C22⋊D107—21no—
43C2×C22⋊D107—24no—
44C2×C4×D107—21no—
45C2×C4×D107—24no—
46#13133807—24no—
47D856—24no—
48#13133803—28no—
49C2×C107⋊C8—22no—
50C2×C107⋊C8C2312yes—
51C4⋊Dic107D412yes—
52C22⋊Dic107—24no—
53C22⋊Dic107D412yes—
54(C2×D107)⋊C4—24no—
55C4×Dic107C2312yes—
56C24×C107—21no—
57C2×D4×C107—22no—
58C2×D4×C107—21no—
59C2×D4×C107—22no—
60C2×D4×C107C2312yes—
61C22×C4×C107C2311yescharacteristic
62M16×C107—22no—
63C2×C8×C107C2312yes—
64C4⋊C4×C107C2312yes—
65C22⋊C4×C107—21no—
66C42×C107—21no—
67C23×C107—22no—
68C23×C107—42no—
69C23×C107—82no—
70C23×C107C2×D411yescharacteristic
71C22×D107—22no—
72C22×D107—24no—
73C22×D107—21no—
74C22×D107—82no—
75D4×C107—22no—
76D4×C107—22no—
77D4×C107—42no—
78C2×C4×C107—22no—
79C2×C4×C107C2×D411yescharacteristic
80C2×C4×C107C2×D412yes—
81C2×C4×C107#4112yes—
82C2×C4×C107C2411yescharacteristic
83C22⋊D107—42no—
84C22⋊D107—84no—
85C2×Dic107—21no—
86C2×Dic107C2×D414yes—
87D428—24no—
88D428—21no—
89D428—22no—
90D428—21no—
91D428C2×D414yes—
92C4×D107—42no—
93C4×D107—84no—
94Dic214—21no—
95Dic214—21no—
96C8×C107—22no—
97C107⋊C8—22no—
98C22×C107—22no—
99C22×C107—42no—
100C22×C107—44no—
101C22×C107—84no—
102C22×C107#12112yes—
103C22×C107C22×D411yescharacteristic
104C2×D107—24no—
105C2×D107—44no—
106C2×D107—41no—
107C2×D107—42no—
108C2×D107—84no—
109C4×C107—42no—
110C4×C107#12112yes—
111C4×C107C22×D411yescharacteristic
112C4×C107C2×#4111yescharacteristic
113Dic107—24no—
114Dic107—41no—
115C2×C107—22no—
116C2×C107—82no—
117C2×C107C2×#12111yescharacteristic
118C2×C107#57612yes—
119D107—44no—
120D107—81no—
121#3041—1071nomaximal, Sylow
122C107#304111yescharacteristic, complemented, Sylow
123C2×AGL(1,ℤ/8)—1071no—
124C22×D8—1071no—
125#535—1071no—
126#518—1071no—
127#463—1074no—
128#459—1074no—
129#424—1071no—
130#418D10711yescharacteristic, complemented
131#416—1071no—
132C2×#41—1071no—
133C22×D4—1071no—
134C22×D4C2×D10711yescharacteristic
135AGL(1,ℤ/8)—2144no—
136C2×SD16—1072no—
137C2×D8—1074no—
138C2×D8—1072no—
139C2×D8—2144no—
140C2×M16C2×D10711yescharacteristic
141C22×C8—1071no—
142#127—1072no—
143#124—1072no—
144#121—2142no—
145#120—2142no—
146C4×D4—1074no—
147C4×D4—2142no—
148#117—1071no—
149#117C2×D10711yescharacteristic
150C2×C22⋊C4—2142no—
151C4⋊C8—1074no—
152#103—1072no—
153#102—1072no—
154#102C2×D10714yescomplemented
155C24—21no—
156C24—2141no—
157#41—2142no—
158#41—2142no—
159C2×Q8—1071no—
160C2×D4—22no—
161C2×D4—21no—
162C2×D4—22no—
163C2×D4—1074no—
164C2×D4—1072no—
165C2×D4—1071no—
166C2×D4—2144no—
167C2×D4—2141no—
168C2×D4—2141no—
169C2×D4—2142no—
170C2×D4C22×D10712yes—
171C22×C4—2144no—
172C22×C4—2141no—
173C22×C4C22×D10711yescharacteristic
174SD16—2144no—
175D8—2144no—
176D8—2148no—
177M16—22no—
178C2×C8—1072no—
179C2×C8—2142no—
180C2×C8C22×D10712yes—
181C4⋊C4—1072no—
182C4⋊C4C22×D10712yes—
183C22⋊C4—21no—
184C22⋊C4—1072no—
185C22⋊C4—2144no—
186C22⋊C4—2144no—
187C42—21no—
188C42—1072no—
189C23—22no—
190C23—42no—
191C23—82no—
192C23—8562no—
193C23—2144no—
194C23—2141no—
195C23—2142no—
196C23D4×D10711yescharacteristic
197Q8—2141no—
198Q8—2141no—
199D4—22no—
200D4—22no—
201D4—42no—
202D4—8564no—
203D4—1074no—
204D4—4282no—
205D4—2144no—
206D4—2141no—
207D4—2141no—
208D4—2142no—
209C2×C4—22no—
210C2×C4—8564no—
211C2×C4—1074no—
212C2×C4—4282no—
213C2×C4—2141no—
214C2×C4D4×D10711yescharacteristic
215C2×C4D4×D10712yes—
216C2×C4#1313381812yes—
217C2×C4C23×D10711yescharacteristic
218C8—22no—
219C8—2142no—
220C22—22no—
221C22—44no—
222C22—42no—
223C22—84no—
224C22—8564no—
225C22—4282no—
226C22—4281no—
227C22—4284no—
228C22—2144no—
229C22#1681208212yes—
230C22C2×D4×D10711yescharacteristic
231C4—42no—
232C4—4281no—
233C4—2144no—
234C4#1681208212yes—
235C4C2×D4×D10711yescharacteristic
236C4C2×#1313381811yescharacteristic
237C2—22no—
238C2—82no—
239C2—8561no—
240C2—4284no—
241C2C2×#1681208211yescharacteristic
242C2#2029343112yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#418×C107C2
socleC22×C107C22×D4
hypercenter#418D107
hypocenterC107#3041
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C2×#16812137C2semidirect
2. C22×#13133803C2semidirect
3. #20293194C2semidirect
4. #20293101C2semidirect
5. #20292841C2semidirect
6. #20292821C2semidirect
7. #20292655C2semidirect
8. #20292621C2semidirect
9. #418×C107C2semidirect
13. C2×#13133807C22semidirect
14. C2×D856C22semidirect
18. #16812105C22semidirect
19. #16812096C22semidirect
23. #16812071C22semidirect
26. Dic107⋊C8C22semidirect
27. #16812032C22semidirect
28. #16812029C22semidirect
32. #102×C107C22semidirect
39. C2×D428D4semidirect
122. C107#3041semidirect
130. #418D107semidirect
154. #102C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121372
C22×#131338032
#202931942
#202931012
#202928412
#202928212
#202926552
#202926212
#418×C1072
#3041107