Finite Group Databasebeta

← #28345961

Subgroups — #28345961

Order 13696.

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

normal subgroups only

This group has 526 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
1C2×#16812135C211yesmaximal, characteristic, complemented
2#20293194C211yesmaximal, characteristic, complemented
3#20293102C211yesmaximal, characteristic, complemented
4#20292840C214yesmaximal, complemented
5#20292820C214yesmaximal, complemented
6#20292653C211yesmaximal, characteristic, complemented
7#20292632C211yesmaximal, characteristic, complemented
8#20292624C211yesmaximal, characteristic, complemented
9C22×D8×C107C211yesmaximal, characteristic, complemented
10C2×#13133818C2211yescharacteristic
11#16812135—24no—
12C2×#13133806C2212yescomplemented
13C2×#13133803C2212yescomplemented
14C2×#13133802C2211yescharacteristic
15#16812105C2212yescomplemented
16#16812096C2212yescomplemented
17#16812080—22no—
18#16812078—22no—
19D4×Dic107—22no—
20D4×Dic107C2214yescomplemented
21#16812068C2211yescharacteristic
22#16812068C2211yescharacteristic
23C2×C22⋊Dic107—22no—
24Dic107⋊C8C2214yescomplemented
25#16812034C2212yescomplemented
26#16812031C2212yescomplemented
27#16812030C2214yescomplemented
28C22×D4×C107C2211yescharacteristic
29C22×D4×C107C2211yescharacteristic
30C2×D8×C107—24no—
31C2×D8×C107C2214yes—
32C22×C8×C107C2211yescharacteristic
33#13133818—22no—
34#13133818—22no—
35C2×Dic214C2311yescharacteristic
36C2×D428C2311yescharacteristic
37C2×C22⋊D107—21no—
38C2×C22⋊D107—24no—
39C22×Dic107—24no—
40C2×C4×D107—21no—
41#13133806—24no—
42#13133803—24no—
43#13133802—22no—
44C2×C107⋊C8C2312yes—
45C4⋊Dic107D412yes—
46C4⋊Dic107C2312yes—
47C22⋊Dic107—21no—
48C22⋊Dic107—24no—
49C22⋊Dic107—24no—
50C22⋊Dic107D412yes—
51C4×Dic107—21no—
52C4×Dic107C2312yes—
53C24×C107—21no—
54C24×C107—21no—
55C2×D4×C107—21no—
56C2×D4×C107—22no—
57C2×D4×C107—21no—
58C2×D4×C107—22no—
59C2×D4×C107—22no—
60C2×D4×C107D414yescomplemented
61C2×D4×C107C2312yes—
62C2×D4×C107C2312yes—
63C22×C4×C107C2311yescharacteristic
64D8×C107—28no—
65C2×C8×C107—22no—
66C2×C8×C107C2312yes—
67C23×C107—22no—
68C23×C107—24no—
69C23×C107—22no—
70C23×C107—42no—
71C23×C107—82no—
72C23×C107—82no—
73C23×C107C2×D411yescharacteristic
74C22×D107—21no—
75D4×C107—24no—
76D4×C107—22no—
77D4×C107—22no—
78D4×C107—22no—
79D4×C107—42no—
80D4×C107C2×D414yes—
81C2×C4×C107C2×D411yescharacteristic
82C2×C4×C107C2×D412yes—
83C2×C4×C107#4112yes—
84C2×C4×C107C2411yescharacteristic
85C22⋊D107—42no—
86C22⋊D107—84no—
87C2×Dic107—22no—
88C2×Dic107—21no—
89C2×Dic107—84no—
90C2×Dic107C2×D414yes—
91D428—21no—
92D428—21no—
93C4×D107—42no—
94Dic214—21no—
95Dic214—21no—
96C8×C107—22no—
97C107⋊C8—22no—
98C22×C107—22no—
99C22×C107—24no—
100C22×C107—44no—
101C22×C107—42no—
102C22×C107—44no—
103C22×C107—84no—
104C22×C107—84no—
105C22×C107#12112yes—
106C22×C107C22×D411yescharacteristic
107C2×D107—41no—
108C2×D107—42no—
109C4×C107#12112yes—
110C4×C107C22×D411yescharacteristic
111C4×C107C2×#4111yescharacteristic
112Dic107—24no—
113Dic107—41no—
114Dic107—42no—
115C2×C107—22no—
116C2×C107—44no—
117C2×C107—82no—
118C2×C107C2×#12111yescharacteristic
119C2×C107#57612yes—
120D107—81no—
121#3041—1071nomaximal, Sylow
122C107#304111yescharacteristic, complemented, Sylow
123C2×AGL(1,ℤ/8)—1071no—
124C22×D8D10711yescharacteristic, complemented
125#535—1071no—
126#518—1071no—
127#463—1074no—
128#459—1074no—
129#424—1071no—
130#418—1071no—
131#416—1071no—
132C2×#41—1071no—
133C22×D4C2×D10711yescharacteristic
134C22×D4C2×D10711yescharacteristic
135AGL(1,ℤ/8)—2144no—
136C2×SD16—1072no—
137C2×D8—24no—
138C2×D8—1072no—
139C2×D8C2×D10714yes—
140C2×M16—1071no—
141C22×C8C2×D10711yescharacteristic
142#127—1072no—
143#124—1072no—
144#121—2142no—
145#120—2142no—
146C4×D4—1074no—
147C4×D4—2142no—
148#117—1071no—
149#117—1071no—
150C2×C22⋊C4—2142no—
151C4⋊C8—1074no—
152#103—1072no—
153#102—1074no—
154#102—1072no—
155C24—21no—
156C24—21no—
157#41—2142no—
158#41—2142no—
159C2×Q8—1071no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—22no—
163C2×D4—22no—
164C2×D4—22no—
165C2×D4—1071no—
166C2×D4—2144no—
167C2×D4—2141no—
168C2×D4C22⋊D10714yescomplemented
169C2×D4C22×D10712yes—
170C2×D4C22×D10712yes—
171C22×C4—2144no—
172C22×C4—2141no—
173C22×C4C22×D10711yescharacteristic
174SD16—2144no—
175D8—28no—
176D8—2144no—
177M16—2142no—
178C2×C8—22no—
179C2×C8—1072no—
180C2×C8C22×D10712yes—
181C4⋊C4—1072no—
182C4⋊C4—1072no—
183C22⋊C4—1072no—
184C22⋊C4—2144no—
185C22⋊C4—2144no—
186C22⋊C4—2141no—
187C42—1072no—
188C42—2141no—
189C23—22no—
190C23—24no—
191C23—22no—
192C23—42no—
193C23—82no—
194C23—82no—
195C23—2141no—
196C23D4×D10711yescharacteristic
197Q8—2141no—
198Q8—2141no—
199D4—24no—
200D4—22no—
201D4—22no—
202D4—22no—
203D4—42no—
204D4—8564no—
205D4—4282no—
206D4—2141no—
207D4—2141no—
208D4C2×C22⋊D10714yes—
209C2×C4—8564no—
210C2×C4—1074no—
211C2×C4—4282no—
212C2×C4—2142no—
213C2×C4—2141no—
214C2×C4D4×D10711yescharacteristic
215C2×C4C2×C22⋊D10712yes—
216C2×C4#1313381712yes—
217C2×C4C23×D10711yescharacteristic
218C8—22no—
219C8—2142no—
220C22—22no—
221C22—24no—
222C22—44no—
223C22—44no—
224C22—42no—
225C22—84no—
226C22—84no—
227C22—4282no—
228C22—4281no—
229C22#1681208112yes—
230C22C2×D4×D10711yescharacteristic
231C4—4282no—
232C4—4281no—
233C4—2144no—
234C4#1681208112yes—
235C4C22×C22⋊D10711yescharacteristic
236C4C2×#1313381711yescharacteristic
237C2—22no—
238C2—44no—
239C2—82no—
240C2—8561no—
241C2C2×#1681208111yescharacteristic
242C2#2029342812yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC22×D8×C107C2
socleC22×C107C22×D4
hypercenterC22×D8D107
hypocenterC107#3041
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C2×#16812135C2semidirect
2. #20293194C2semidirect
3. #20293102C2semidirect
4. #20292840C2semidirect
5. #20292820C2semidirect
6. #20292653C2semidirect
7. #20292632C2semidirect
8. #20292624C2semidirect
9. C22×D8×C107C2semidirect
12. C2×#13133806C22semidirect
13. C2×#13133803C22semidirect
15. #16812105C22semidirect
16. #16812096C22semidirect
20. D4×Dic107C22semidirect
24. Dic107⋊C8C22semidirect
25. #16812034C22semidirect
26. #16812031C22semidirect
27. #16812030C22semidirect
60. C2×D4×C107D4semidirect
122. C107#3041semidirect
124. C22×D8D107semidirect
168. C2×D4C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121352
#202931942
#202931022
#202928402
#202928202
#202926532
#202926322
#202926242
C22×D8×C1072
#3041107