Finite Group Databasebeta

← #28346923

Subgroups — #28346923

Order 13696.

572 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16812142C211yesmaximal, characteristic, complemented
2C2×#16812139C211yesmaximal, characteristic, complemented
3C2×#16812135C211yesmaximal, characteristic, complemented
4#20293173C211yesmaximal, characteristic, complemented
5C2×#16812096C211yesmaximal, characteristic, complemented
6#20293018C212yesmaximal, complemented
7#20293015C212yesmaximal, complemented
8#20293010C212yesmaximal, complemented
9#20293008C212yesmaximal, complemented
10M16×Dic107C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2211yescharacteristic
13C22×Dic214C2211yescharacteristic
14#16812142—24no—
15#16812139—24no—
16#16812135—24no—
17C2×Dic428C2212yescomplemented
18C2×#13133808C2212yescomplemented
19C2×#13133807C2212yescomplemented
20C2×#13133806C2212yes—
21C2×#13133806C2212yescomplemented
22C2×#13133803C2212yes—
23C2×#13133802C2211yescharacteristic
24#16812107C2211yescharacteristic, complemented
25#16812105C2211yescharacteristic, complemented
26#16812098C2211yescharacteristic, complemented
27#16812096—22no—
28#16812096C2212yes—
29#16812096C2212yes—
30#16812096C2211yescharacteristic, complemented
31#16812081—22no—
32C4×C22⋊D107—22no—
33C2×C22⋊Dic107—22no—
34C2×C4×Dic107C2211yescharacteristic
35#16812019C2212yescomplemented
36C8×Dic107C2212yescomplemented
37C2×#41×C107C2211yescharacteristic
38C22×D4×C107C2211yescharacteristic
39AGL(1,ℤ/8)×C107—24no—
40C2×SD16×C107C2212yes—
41C2×D8×C107C2212yes—
42C2×M16×C107C2211yescharacteristic
43#13133818—22no—
44#13133818—22no—
45C2×Dic214—22no—
46C2×Dic214—22no—
47C2×Dic214—21no—
48C2×Dic214C2312yes—
49C2×Dic214C2311yescharacteristic
50C2×D428C2311yescharacteristic
51C2×C22⋊D107—21no—
52C2×C22⋊D107—22no—
53C22×Dic107—21no—
54C22×Dic107D412yes—
55C2×C4×D107—21no—
56Dic428—24no—
57#13133808—24no—
58#13133807—24no—
59#13133806—24no—
60#13133806—24no—
61#13133803—24no—
62#13133802D414yescomplemented
63C2×C107⋊C8C2312yes—
64Dic107⋊C4—22no—
65C22⋊Dic107—24no—
66C22⋊Dic107—24no—
67C22⋊Dic107—22no—
68(C2×D107)⋊C4—22no—
69C4×Dic107D412yes—
70C4×Dic107C2311yescharacteristic
71C4×Dic107C2311yescharacteristic
72C24×C107—21no—
73#41×C107—22no—
74#41×C107—22no—
75C2×Q8×C107C2311yescharacteristic
76C2×D4×C107—21no—
77C2×D4×C107—22no—
78C2×D4×C107—22no—
79C2×D4×C107—21no—
80C2×D4×C107C2312yes—
81C2×D4×C107C2311yescharacteristic
82C22×C4×C107—21no—
83C22×C4×C107C2311yescharacteristic
84SD16×C107—24no—
85D8×C107—24no—
86M16×C107D414yescomplemented
87C2×C8×C107C2312yes—
88C23×C107—21no—
89C23×C107—22no—
90C23×C107—44no—
91C23×C107—42no—
92C23×C107C2×D411yescharacteristic
93C22×D107—21no—
94Q8×C107—21no—
95Q8×C107—21no—
96D4×C107—21no—
97D4×C107—22no—
98D4×C107—22no—
99D4×C107—21no—
100D4×C107—42no—
101D4×C107—42no—
102C2×C4×C107—21no—
103C2×C4×C107—42no—
104C2×C4×C107C2×D411yescharacteristic
105C2×C4×C107C2×D412yes—
106C2×C4×C107C2×D412yes—
107C2×C4×C107C2411yescharacteristic
108C22⋊D107—44no—
109C22⋊D107—42no—
110C2×Dic107—22no—
111C2×Dic107—21no—
112C2×Dic107—42no—
113C2×Dic107—42no—
114C2×Dic107C2×D412yes—
115C2×Dic107C2×D412yes—
116D428—21no—
117D428—21no—
118C4×D107—42no—
119Dic214—21no—
120Dic214—22no—
121Dic214—22no—
122Dic214—21no—
123Dic214—42no—
124C8×C107C2×D414yes—
125C107⋊C8C2×D414yes—
126C22×C107—22no—
127C22×C107—41no—
128C22×C107—42no—
129C22×C107—42no—
130C22×C107—44no—
131C22×C107—84no—
132C22×C107#12712yes—
133C22×C107C22×D411yescharacteristic
134C2×D107—41no—
135C2×D107—42no—
136C4×C107—41no—
137C4×C107#12712yes—
138C4×C107C22×D411yescharacteristic
139C4×C107C22×D411yescharacteristic
140Dic107—22no—
141Dic107—22no—
142Dic107—42no—
143Dic107—41no—
144C2×C107—22no—
145C2×C107—81no—
146C2×C107—82no—
147C2×C107C2×#12711yescharacteristic
148C2×C107#57512yes—
149D107—81no—
150#3153—1071nomaximal, Sylow
151C107#315311yescharacteristic, complemented, Sylow
152C2×#137—1071no—
153C2×#137—1071no—
154C2×AGL(1,ℤ/8)—1071no—
155C2×AGL(1,ℤ/8)D10711yescharacteristic, complemented
156#532—1071no—
157C2×#124—1071no—
158#497—1072no—
159#496—1072no—
160#495—1072no—
161#495—1072no—
162C4×M16—1071no—
163C2×#41—1071no—
164C2×#41C2×D10711yescharacteristic
165C22×Q8—1071no—
166C22×D4C2×D10711yescharacteristic
167#137—2144no—
168#137—2144no—
169AGL(1,ℤ/8)—24no—
170AGL(1,ℤ/8)—2144no—
171C2×Q16—1072no—
172C2×Q16—1072no—
173C2×SD16—1072no—
174C2×SD16—1072no—
175C2×SD16—1072no—
176C2×SD16C2×D10712yes—
177C2×D8—1072no—
178C2×D8C2×D10712yes—
179C2×M16—1071no—
180C2×M16C2×D10711yescharacteristic
181C4⋊Q8—1071no—
182#127—1071no—
183#124—1071no—
184#124—1072no—
185#124—1072no—
186#124—1071no—
187#124—2142no—
188#121—2142no—
189C4×D4—2142no—
190C2×C22⋊C4—2142no—
191C2×C42—1071no—
192#97—1072no—
193C4×C8—1072no—
194C24—21no—
195#41—22no—
196#41—22no—
197#41—2142no—
198#41—2142no—
199C2×Q8—1072no—
200C2×Q8—1071no—
201C2×Q8—2142no—
202C2×Q8—2142no—
203C2×Q8—2141no—
204C2×Q8C22×D10711yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—1071no—
210C2×D4—2142no—
211C2×D4—2141no—
212C2×D4C22×D10712yes—
213C2×D4C22×D10711yescharacteristic
214C22×C4—21no—
215C22×C4—1072no—
216C22×C4—2141no—
217C22×C4—2141no—
218C22×C4C22×D10711yescharacteristic
219Q16—2144no—
220Q16—2144no—
221SD16—24no—
222SD16—2144no—
223SD16—2144no—
224SD16—2144no—
225D8—24no—
226D8—2144no—
227M16—1074no—
228M16C22⋊D10714yescomplemented
229C2×C8—1072no—
230C2×C8C22×D10712yes—
231C4⋊C4—2142no—
232C22⋊C4—2142no—
233C22⋊C4—2144no—
234C22⋊C4—2144no—
235C22⋊C4—2142no—
236C42—1072no—
237C42—1071no—
238C42—1071no—
239C23—22no—
240C23—21no—
241C23—44no—
242C23—42no—
243C23—2141no—
244C23D4×D10711yescharacteristic
245Q8—21no—
246Q8—21no—
247Q8—4282no—
248Q8—2141no—
249Q8—2142no—
250Q8—2142no—
251Q8—2141no—
252D4—22no—
253D4—21no—
254D4—21no—
255D4—22no—
256D4—42no—
257D4—42no—
258D4—4282no—
259D4—4284no—
260D4—2141no—
261D4—2141no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—1072no—
265C2×C4—1072no—
266C2×C4—4282no—
267C2×C4—4282no—
268C2×C4—4282no—
269C2×C4—2142no—
270C2×C4—2141no—
271C2×C4D4×D10711yescharacteristic
272C2×C4D4×D10712yes—
273C2×C4C2×C22⋊D10712yes—
274C2×C4C23×D10711yescharacteristic
275C8—1074no—
276C8C2×C22⋊D10714yes—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—42no—
281C22—44no—
282C22—84no—
283C22—4281no—
284C22—4282no—
285C22#1681210512yes—
286C22C2×D4×D10711yescharacteristic
287C4—41no—
288C4—4281no—
289C4—4282no—
290C4—2142no—
291C4—2142no—
292C4#1681210512yes—
293C4C2×D4×D10711yescharacteristic
294C4C22×C22⋊D10711yescharacteristic
295C2—22no—
296C2—82no—
297C2—81no—
298C2—8561no—
299C2C2×#1681210511yescharacteristic
300C2#2029342312yes—
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#3153
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×#16812142C2semidirect
2. C2×#16812139C2semidirect
3. C2×#16812135C2semidirect
4. #20293173C2semidirect
5. C2×#16812096C2semidirect
6. #20293018C2semidirect
7. #20293015C2semidirect
8. #20293010C2semidirect
9. #20293008C2semidirect
10. M16×Dic107C2semidirect
11. C2×AGL(1,ℤ/8)×C107C2semidirect
17. C2×Dic428C22semidirect
18. C2×#13133808C22semidirect
19. C2×#13133807C22semidirect
21. C2×#13133806C22semidirect
24. #16812107C22semidirect
25. #16812105C22semidirect
26. #16812098C22semidirect
30. #16812096C22semidirect
35. #16812019C22semidirect
36. C8×Dic107C22semidirect
62. #13133802D4semidirect
86. M16×C107D4semidirect
151. C107#3153semidirect
155. C2×AGL(1,ℤ/8)D107semidirect
228. M16C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121422
C2×#168121392
C2×#168121352
#202931732
C2×#168120962
#202930182
#202930152
#202930102
#202930082
M16×Dic1072
C2×AGL(1,ℤ/8)×C1072
#3153107