Finite Group Databasebeta

← #28345953

Subgroups — #28345953

Order 13696.

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

normal subgroups only

This group has 500 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×#16812139C211yesmaximal, characteristic, complemented
2C2×#16812096C211yesmaximal, characteristic, complemented
3#41×Dic107C211yesmaximal, characteristic, complemented
4#20292847C214yesmaximal, complemented
5#20292840C214yesmaximal, complemented
6#20292650C211yesmaximal, characteristic, complemented
7#20292637C211yesmaximal, characteristic, complemented
8#20292632C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C107C211yesmaximal, characteristic
10C22×Dic214C2211yescharacteristic
11#16812139—24no—
12C2×#13133808C2212yescomplemented
13C2×#13133806C2212yescomplemented
14C2×#13133802C2211yescharacteristic
15#16812096—21no—
16#16812096—21no—
17#16812096C2214yes—
18Q8×Dic107C2212yescomplemented
19D4×Dic107—22no—
20D4×Dic107C2212yescomplemented
21#16812068—21no—
22#16812068C2211yescharacteristic
23C2×C22⋊Dic107—22no—
24C2×C4×Dic107—21no—
25C2×C4×Dic107C2211yescharacteristic
26Dic107⋊C8C2214yescomplemented
27#16812034C2214yescomplemented
28#16812030C2214yes—
29C2×#41×C107C2211yescharacteristic
30C22×D4×C107C2211yescharacteristic
31AGL(1,ℤ/8)×C107—24no—
32C2×SD16×C107C2212yes—
33C2×D8×C107C2212yes—
34C2×M16×C107C2211yescharacteristic
35C2×Dic214—22no—
36C2×Dic214—22no—
37C2×Dic214—21no—
38C2×Dic214C2312yes—
39C22×Dic107—21no—
40C22×Dic107—22no—
41C22×Dic107D412yes—
42#13133808—24no—
43#13133806—24no—
44#13133802—22no—
45C2×C107⋊C8C2312yes—
46C4⋊Dic107—22no—
47C4⋊Dic107C2312yes—
48C22⋊Dic107—21no—
49C22⋊Dic107—24no—
50C22⋊Dic107—24no—
51C22⋊Dic107—22no—
52C4×Dic107—21no—
53C4×Dic107—22no—
54C4×Dic107D411yescharacteristic
55C4×Dic107D411yescharacteristic
56C4×Dic107C2312yes—
57C24×C107—21no—
58#41×C107—22no—
59#41×C107D414yes—
60C2×Q8×C107C2311yescharacteristic
61C2×D4×C107—21no—
62C2×D4×C107—22no—
63C2×D4×C107—21no—
64C2×D4×C107—22no—
65C2×D4×C107C2312yes—
66C2×D4×C107C2311yescharacteristic
67C22×C4×C107—21no—
68C22×C4×C107C2311yescharacteristic
69SD16×C107—24no—
70D8×C107—24no—
71M16×C107—22no—
72C2×C8×C107C2312yes—
73C23×C107—21no—
74C23×C107—22no—
75C23×C107—44no—
76C23×C107—42no—
77C23×C107C2×D411yescharacteristic
78Q8×C107—21no—
79Q8×C107C2×D412yes—
80D4×C107—24no—
81D4×C107—22no—
82D4×C107—21no—
83D4×C107—22no—
84D4×C107—42no—
85D4×C107C2×D412yes—
86C2×C4×C107—21no—
87C2×C4×C107—24no—
88C2×C4×C107C2×D411yescharacteristic
89C2×C4×C107C2×D412yes—
90C2×C4×C107#4112yes—
91C2×C4×C107C2411yescharacteristic
92C2×Dic107—22no—
93C2×Dic107—22no—
94C2×Dic107—42no—
95C2×Dic107—44no—
96C2×Dic107—42no—
97C2×Dic107C2×D412yes—
98C2×Dic107C2×D412yes—
99Dic214—22no—
100Dic214—22no—
101Dic214—42no—
102C8×C107—22no—
103C107⋊C8—22no—
104C22×C107—22no—
105C22×C107—22no—
106C22×C107—42no—
107C22×C107—42no—
108C22×C107—44no—
109C22×C107—84no—
110C22×C107#12112yes—
111C22×C107C22×D411yescharacteristic
112C4×C107—22no—
113C4×C107#12112yes—
114C4×C107C22×D411yescharacteristic
115C4×C107C2×#4111yescharacteristic
116Dic107—22no—
117Dic107—22no—
118Dic107—42no—
119Dic107—42no—
120C2×C107—22no—
121C2×C107—42no—
122C2×C107—82no—
123C2×C107C2×#12111yescharacteristic
124C2×C107#57512yes—
125#3040—1071nomaximal, Sylow
126C107#304011yescharacteristic, complemented, Sylow
127C2×#137—1071no—
128C2×AGL(1,ℤ/8)D10711yescharacteristic
129C2×#124—1071no—
130C4×#41—1071no—
131#464—1074no—
132#463—1074no—
133#423—1071no—
134#419—1071no—
135#418—1071no—
136C2×#41C2×D10711yescharacteristic
137C22×Q8—1071no—
138C22×D4C2×D10711yescharacteristic
139#137—2144no—
140AGL(1,ℤ/8)—24no—
141C2×Q16—1072no—
142C2×SD16—1072no—
143C2×SD16C2×D10712yes—
144C2×D8C2×D10712yes—
145C2×M16—1071no—
146C2×M16C2×D10711yescharacteristic
147#124—1074no—
148#124—2141no—
149#124—2141no—
150C4×Q8—1072no—
151C4×D4—1072no—
152C4×D4—2142no—
153#117—1071no—
154#117—2141no—
155C2×C22⋊C4—2142no—
156C2×C42—1071no—
157C2×C42—2141no—
158C4⋊C8—1074no—
159#103—1074no—
160#102—1074no—
161C24—21no—
162#41—22no—
163#41C22⋊D10714yes—
164C2×Q8—1072no—
165C2×Q8—2142no—
166C2×Q8—2142no—
167C2×Q8—2141no—
168C2×Q8C22×D10711yescharacteristic
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—22no—
172C2×D4—22no—
173C2×D4C22×D10712yes—
174C2×D4C22×D10711yescharacteristic
175C22×C4—21no—
176C22×C4—1072no—
177C22×C4—2142no—
178C22×C4—2141no—
179C22×C4C22×D10711yescharacteristic
180Q16—2144no—
181SD16—24no—
182SD16—2144no—
183D8—24no—
184M16—22no—
185M16—2142no—
186C2×C8—1072no—
187C2×C8C22×D10712yes—
188C4⋊C4—1072no—
189C4⋊C4—2142no—
190C22⋊C4—2144no—
191C22⋊C4—2144no—
192C22⋊C4—2142no—
193C22⋊C4—2141no—
194C42—1071no—
195C42—1071no—
196C42—1072no—
197C42—2142no—
198C42—2141no—
199C23—21no—
200C23—22no—
201C23—44no—
202C23—42no—
203C23D4×D10711yescharacteristic
204Q8—21no—
205Q8—4282no—
206Q8—2142no—
207Q8—2142no—
208Q8C2×C22⋊D10712yes—
209D4—24no—
210D4—22no—
211D4—21no—
212D4—22no—
213D4—42no—
214D4C2×C22⋊D10712yes—
215C2×C4—21no—
216C2×C4—24no—
217C2×C4—1072no—
218C2×C4—1072no—
219C2×C4—4282no—
220C2×C4—4284no—
221C2×C4—4282no—
222C2×C4—2142no—
223C2×C4—2142no—
224C2×C4D4×D10711yescharacteristic
225C2×C4C2×C22⋊D10712yes—
226C2×C4#1313381712yes—
227C2×C4C23×D10711yescharacteristic
228C8—22no—
229C8—2142no—
230C22—22no—
231C22—22no—
232C22—42no—
233C22—44no—
234C22—42no—
235C22—84no—
236C22#1681208112yes—
237C22C2×D4×D10711yescharacteristic
238C4—22no—
239C4—4282no—
240C4—4282no—
241C4—2142no—
242C4—2142no—
243C4#1681208112yes—
244C4C22×C22⋊D10711yescharacteristic
245C4C2×#1313381711yescharacteristic
246C2—22no—
247C2—42no—
248C2—82no—
249C2C2×#1681208111yescharacteristic
250C2#2029342312yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting, prime radicalC2×AGL(1,ℤ/8)×C107C2
socleC22×C107C22×D4
hypercenterC2×AGL(1,ℤ/8)D107
hypocenterC107#3040
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#16812139C2semidirect
2. C2×#16812096C2semidirect
3. #41×Dic107C2semidirect
4. #20292847C2semidirect
5. #20292840C2semidirect
6. #20292650C2semidirect
7. #20292637C2semidirect
8. #20292632C2semidirect
12. C2×#13133808C22semidirect
13. C2×#13133806C22semidirect
18. Q8×Dic107C22semidirect
20. D4×Dic107C22semidirect
26. Dic107⋊C8C22semidirect
27. #16812034C22semidirect
126. C107#3040semidirect

Maximal subgroups

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

subgroupindex
C2×#168121392
C2×#168120962
#41×Dic1072
#202928472
#202928402
#202926502
#202926372
#202926322
C2×AGL(1,ℤ/8)×C1072
#3040107