Finite Group Databasebeta

← #28345939

Subgroups — #28345939

Order 13696.

584 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 145 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×#16812134C211yesmaximal, characteristic, complemented
2C2×#16812105C211yesmaximal, characteristic, complemented
3#41×Dic107C211yesmaximal, characteristic, complemented
4#20292825C214yesmaximal, complemented
5#20292820C214yesmaximal, complemented
6#20292650C211yesmaximal, characteristic, complemented
7#20292633C211yesmaximal, characteristic, complemented
8#20292632C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C107C211yesmaximal, characteristic, complemented
10C22×D428C2211yescharacteristic
11C22×C22⋊D107—22no—
12#16812134—24no—
13C2×#13133805C2212yescomplemented
14C2×#13133803C2212yescomplemented
15C2×#13133802C2211yescharacteristic
16#16812105—21no—
17#16812105—21no—
18#16812105C2214yes—
19Q8×Dic107C2212yescomplemented
20D4×Dic107—22no—
21D4×Dic107C2212yescomplemented
22#16812068—21no—
23#16812068C2211yescharacteristic
24C2×C4×Dic107—21no—
25C2×C4×Dic107C2211yescharacteristic
26Dic107⋊C8C2214yescomplemented
27#16812031C2214yescomplemented
28#16812030C2214yescomplemented
29C2×#41×C107C2211yescharacteristic
30C22×D4×C107C2211yescharacteristic
31AGL(1,ℤ/8)×C107—24no—
32C2×SD16×C107C2212yescomplemented
33C2×D8×C107C2212yescomplemented
34C2×M16×C107C2211yescharacteristic
35C23×D107—21no—
36C2×D428—22no—
37C2×D428—22no—
38C2×D428—21no—
39C2×D428C2312yes—
40C2×C22⋊D107—24no—
41C2×C22⋊D107—24no—
42C2×C22⋊D107—48no—
43C22×Dic107—22no—
44C22×Dic107D412yes—
45#13133805—24no—
46#13133803—24no—
47#13133802—22no—
48C2×C107⋊C8C2312yes—
49C4⋊Dic107—22no—
50C4⋊Dic107C2312yes—
51C22⋊Dic107—21no—
52C22⋊Dic107—22no—
53C4×Dic107—21no—
54C4×Dic107—22no—
55C4×Dic107D411yescharacteristic
56C4×Dic107D411yescharacteristic
57C4×Dic107C2312yes—
58C24×C107—21no—
59#41×C107—22no—
60#41×C107D414yescomplemented
61C2×Q8×C107C2311yescharacteristic
62C2×D4×C107—21no—
63C2×D4×C107—22no—
64C2×D4×C107—21no—
65C2×D4×C107—22no—
66C2×D4×C107C2312yes—
67C2×D4×C107C2311yescharacteristic
68C22×C4×C107—21no—
69C22×C4×C107C2311yescharacteristic
70SD16×C107—24no—
71D8×C107—24no—
72M16×C107—22no—
73C2×C8×C107C2312yes—
74C23×C107—21no—
75C23×C107—22no—
76C23×C107—44no—
77C23×C107—42no—
78C23×C107C2×D411yescharacteristic
79C22×D107—22no—
80C22×D107—44no—
81C22×D107—42no—
82Q8×C107—21no—
83Q8×C107C2×D412yescomplemented
84D4×C107—24no—
85D4×C107—22no—
86D4×C107—21no—
87D4×C107—22no—
88D4×C107—42no—
89D4×C107C2×D412yescomplemented
90C2×C4×C107—21no—
91C2×C4×C107—24no—
92C2×C4×C107C2×D411yescharacteristic
93C2×C4×C107C2×D412yes—
94C2×C4×C107#4112yes—
95C2×C4×C107C2411yescharacteristic
96C22⋊D107—48no—
97C22⋊D107—48no—
98C2×Dic107—24no—
99C2×Dic107—22no—
100C2×Dic107—44no—
101C2×Dic107C2×D412yes—
102C2×Dic107C2×D412yes—
103D428—22no—
104D428—22no—
105D428—42no—
106C8×C107—22no—
107C107⋊C8—22no—
108C22×C107—22no—
109C22×C107—22no—
110C22×C107—42no—
111C22×C107—42no—
112C22×C107—44no—
113C22×C107—84no—
114C22×C107#12112yes—
115C22×C107C22×D411yescharacteristic
116C2×D107—42no—
117C2×D107—44no—
118C2×D107—84no—
119C4×C107—22no—
120C4×C107#12112yes—
121C4×C107C22×D411yescharacteristic
122C4×C107C2×#4111yescharacteristic
123Dic107—22no—
124Dic107—42no—
125Dic107AGL(1,ℤ/8)14yescomplemented
126C2×C107—22no—
127C2×C107—42no—
128C2×C107—82no—
129C2×C107C2×#12111yescharacteristic
130C2×C107C2×AGL(1,ℤ/8)12yes—
131D107—82no—
132#3038—1071nomaximal, Sylow
133C107#303811yescharacteristic, complemented, Sylow
134C2×AGL(1,ℤ/8)—1071no—
135C2×AGL(1,ℤ/8)D10711yescharacteristic, complemented
136C2×#127—1071no—
137C4×#41—1071no—
138#460—1074no—
139#459—1074no—
140#423—1071no—
141#418—1071no—
142#418—1071no—
143C2×#41C2×D10711yescharacteristic
144C22×D4—1071no—
145C22×D4—2142no—
146C22×D4C2×D10711yescharacteristic
147AGL(1,ℤ/8)—24no—
148AGL(1,ℤ/8)—2144no—
149C2×SD16—1072no—
150C2×SD16C2×D10712yescomplemented
151C2×D8—1072no—
152C2×D8C2×D10712yescomplemented
153C2×M16—1071no—
154C2×M16C2×D10711yescharacteristic
155#127—1074no—
156#127—2141no—
157#127—2141no—
158C4×Q8—1072no—
159C4×D4—1072no—
160C4×D4—2142no—
161#117—1071no—
162#117—2141no—
163C2×C42—1071no—
164C2×C42—2141no—
165C4⋊C8—1074no—
166#102—1074no—
167#102—1074no—
168C24—21no—
169C24—2141no—
170#41—22no—
171#41C22⋊D10714yescomplemented
172C2×Q8C22×D10711yescharacteristic
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—22no—
176C2×D4—22no—
177C2×D4—1072no—
178C2×D4—4288no—
179C2×D4—2144no—
180C2×D4—2144no—
181C2×D4—2141no—
182C2×D4—2142no—
183C2×D4—2142no—
184C2×D4C22×D10712yes—
185C2×D4C22×D10711yescharacteristic
186C22×C4—21no—
187C22×C4—1072no—
188C22×C4—2142no—
189C22×C4C22×D10711yescharacteristic
190SD16—24no—
191SD16—2144no—
192D8—24no—
193D8—2144no—
194M16—22no—
195M16—2142no—
196C2×C8—1072no—
197C2×C8C22×D10712yes—
198C4⋊C4—1072no—
199C4⋊C4—2142no—
200C22⋊C4—2142no—
201C22⋊C4—2141no—
202C42—1071no—
203C42—1071no—
204C42—1072no—
205C42—2142no—
206C42—2141no—
207C23—21no—
208C23—22no—
209C23—44no—
210C23—42no—
211C23—4284no—
212C23—4282no—
213C23—2142no—
214C23D4×D10711yescharacteristic
215Q8—21no—
216Q8C2×C22⋊D10712yescomplemented
217D4—24no—
218D4—22no—
219D4—21no—
220D4—22no—
221D4—42no—
222D4—4288no—
223D4—4288no—
224D4—4282no—
225D4—2142no—
226D4—2142no—
227D4C2×C22⋊D10712yescomplemented
228C2×C4—21no—
229C2×C4—24no—
230C2×C4—1072no—
231C2×C4—1072no—
232C2×C4—4284no—
233C2×C4—2144no—
234C2×C4—2142no—
235C2×C4D4×D10711yescharacteristic
236C2×C4C2×C22⋊D10712yes—
237C2×C4#1313381712yes—
238C2×C4C23×D10711yescharacteristic
239C8—22no—
240C8—2142no—
241C22—22no—
242C22—22no—
243C22—42no—
244C22—44no—
245C22—42no—
246C22—84no—
247C22—8564no—
248C22—4284no—
249C22—4282no—
250C22#1681208112yes—
251C22C2×D4×D10711yescharacteristic
252C4—22no—
253C4—1074no—
254C4—4282no—
255C4—2142no—
256C4#1681208112yes—
257C4C22×C22⋊D10711yescharacteristic
258C4C2×#1313381711yescharacteristic
259C2—22no—
260C2—42no—
261C2—82no—
262C2—8562no—
263C2C2×#1681208111yescharacteristic
264C2AGL(1,ℤ/8)×D10712yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×AGL(1,ℤ/8)×C107C2
socleC22×C107C22×D4
hypercenterC2×AGL(1,ℤ/8)D107
hypocenterC107#3038
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×#16812134C2semidirect
2. C2×#16812105C2semidirect
3. #41×Dic107C2semidirect
4. #20292825C2semidirect
5. #20292820C2semidirect
6. #20292650C2semidirect
7. #20292633C2semidirect
8. #20292632C2semidirect
9. C2×AGL(1,ℤ/8)×C107C2semidirect
13. C2×#13133805C22semidirect
14. C2×#13133803C22semidirect
19. Q8×Dic107C22semidirect
21. D4×Dic107C22semidirect
26. Dic107⋊C8C22semidirect
27. #16812031C22semidirect
28. #16812030C22semidirect
32. C2×SD16×C107C22semidirect
33. C2×D8×C107C22semidirect
60. #41×C107D4semidirect
83. Q8×C107C2×D4semidirect
89. D4×C107C2×D4semidirect
125. Dic107AGL(1,ℤ/8)semidirect
133. C107#3038semidirect
135. C2×AGL(1,ℤ/8)D107semidirect
150. C2×SD16C2×D107semidirect
152. C2×D8C2×D107semidirect
171. #41C22⋊D107semidirect
216. Q8C2×C22⋊D107semidirect
227. D4C2×C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121342
C2×#168121052
#41×Dic1072
#202928252
#202928202
#202926502
#202926332
#202926322
C2×AGL(1,ℤ/8)×C1072
#3038107