Finite Group Databasebeta

← #28346142

Subgroups — #28346142

Order 13696.

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

normal subgroups only

This group has 504 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×#16812137C211yesmaximal, characteristic, complemented
2C2×#16812117C211yesmaximal, characteristic, complemented
3#20293208C211yesmaximal, characteristic, complemented
4#20293196C211yesmaximal, characteristic, complemented
5#20292881C212yesmaximal, complemented
6#20292874C212yesmaximal, complemented
7#20292864C212yesmaximal, complemented
8#20292857C212yesmaximal, complemented
9#20292621C211yesmaximal, characteristic, complemented
10C2×#16812029C211yesmaximal, characteristic, complemented
11#428×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2211yescharacteristic
13C2×#13133818C2211yescharacteristic
14C22×D428C2211yescharacteristic
15#16812137—24no—
16C2×#13133807C2212yes—
17C2×D856C2212yes—
18#16812117—24no—
19C2×#13133801C2212yescomplemented
20C22×C107⋊C8C2211yescharacteristic
21C2×C8×D107C2212yescomplemented
22#16812106—21no—
23#16812097—21no—
24#16812089—21no—
25#16812089C2212yescomplemented
26#16812084—21no—
27#16812084C2212yes—
28#16812084C2212yes—
29#16812084C2212yescomplemented
30#16812083—22no—
31#16812079—22no—
32#16812032C2211yescharacteristic, complemented
33#16812032C2211yescharacteristic, complemented
34#16812029C2212yes—
35#16812029C2212yes—
36#16812029C2211yescharacteristic, complemented
37#16812029C2211yescharacteristic, complemented
38C2×M16×C107C2211yescharacteristic
39#117×C107C2211yescharacteristic
40C2×C4⋊C4×C107C2211yescharacteristic
41#107×C107C2212yescomplemented
42#106×C107C2212yescomplemented
43C23×D107—21no—
44#13133818—22no—
45#13133818—22no—
46#13133818—22no—
47#13133818—22no—
48C2×Dic214D411yescharacteristic
49C2×Dic214C2311yescharacteristic
50C2×D428—22no—
51C2×D428—22no—
52C2×D428—21no—
53C2×D428—22no—
54C2×D428—22no—
55C2×D428—22no—
56C2×D428D411yescharacteristic
57C2×D428C2312yes—
58C2×D428C2311yescharacteristic
59C2×C22⋊D107—21no—
60C2×C22⋊D107D412yes—
61C2×C4×D107—21no—
62C2×C4×D107C2312yes—
63#13133807—24no—
64D856—24no—
65#13133801—24no—
66C2×C107⋊C8—22no—
67C2×C107⋊C8C2311yescharacteristic
68C2×C107⋊C8C2311yescharacteristic
69C8×D107—24no—
70C4⋊Dic107—22no—
71(C2×D107)⋊C4—22no—
72(C2×D107)⋊C4—22no—
73(C2×D107)⋊C4—22no—
74(C2×D107)⋊C4—22no—
75C22×C4×C107—21no—
76C22×C4×C107C2311yescharacteristic
77M16×C107D414yescomplemented
78C2×C8×C107C2312yes—
79C4⋊C4×C107—21no—
80C4⋊C4×C107C2311yescharacteristic
81C4⋊C4×C107C2311yescharacteristic
82C4⋊C4×C107C2312yes—
83C22⋊C4×C107—21no—
84C42×C107—21no—
85C23×C107C2×D411yescharacteristic
86C22×D107—21no—
87C22×D107—22no—
88C22×D107—42no—
89C22×D107—81no—
90C22×D107—81no—
91C22×D107C2×D412yes—
92C2×C4×C107—21no—
93C2×C4×C107—21no—
94C2×C4×C107—22no—
95C2×C4×C107—81no—
96C2×C4×C107C2×D411yescharacteristic
97C2×C4×C107C2×D412yes—
98C2×C4×C107#4112yes—
99C2×C4×C107C2411yescharacteristic
100C22⋊D107—24no—
101C22⋊D107—42no—
102C2×Dic107—21no—
103C2×Dic107C2×D412yes—
104D428—22no—
105D428—21no—
106D428—22no—
107D428—22no—
108D428—21no—
109D428—42no—
110D428—82no—
111D428—82no—
112D428—82no—
113C4×D107—22no—
114C4×D107—22no—
115C4×D107—42no—
116Dic214—22no—
117Dic214—21no—
118Dic214—21no—
119C8×C107C2×D414yes—
120C107⋊C8—21no—
121C107⋊C8—21no—
122C22×C107—21no—
123C22×C107—21no—
124C22×C107#12112yes—
125C22×C107C22×D411yescharacteristic
126C2×D107—22no—
127C2×D107—22no—
128C2×D107—22no—
129C2×D107—41no—
130C2×D107—42no—
131C2×D107—41no—
132C2×D107—42no—
133C2×D107—42no—
134C2×D107—41no—
135C2×D107—84no—
136C4×C107—41no—
137C4×C107—42no—
138C4×C107—41no—
139C4×C107#12112yes—
140C4×C107C22×D411yescharacteristic
141C4×C107C2×#4111yescharacteristic
142Dic107—22no—
143Dic107—41no—
144C2×C107—22no—
145C2×C107C2×#12111yescharacteristic
146C2×C107#57611yescharacteristic
147C2×C107#57711yescharacteristic
148D107—42no—
149D107—81no—
150D107—82no—
151#3062—1071nomaximal, Sylow
152C107#306211yescharacteristic, complemented, Sylow
153C2×AGL(1,ℤ/8)—1071no—
154C2×#131—1071no—
155#537—1071no—
156#535—1071no—
157#469—1072no—
158#468—1072no—
159#466—1072no—
160#465—1072no—
161#428D10711yescharacteristic, complemented
162#416—1071no—
163C2×#102—1071no—
164C2×#41—1071no—
165C2×#41—1071no—
166C22×D4—1071no—
167AGL(1,ℤ/8)—2144no—
168C2×SD16—1072no—
169C2×D8—1072no—
170#131—2144no—
171C2×M16—1072no—
172C2×M16C2×D10711yescharacteristic
173C22×C8—1071no—
174C22×C8—1072no—
175#127—2141no—
176#124—2141no—
177C22⋊Q8—1072no—
178C22⋊Q8—2141no—
179#121—1072no—
180#121—1072no—
181#121—1072no—
182#121—2141no—
183#121—2142no—
184#120—2142no—
185#117C2×D10711yescharacteristic
186C2×C4⋊C4C2×D10711yescharacteristic
187#107C2×D10712yescomplemented
188#106C2×D10712yescomplemented
189#103—1071no—
190#103—1071no—
191#102—1072no—
192#102—1072no—
193#102—1071no—
194#102—1071no—
195C24—2141no—
196#41—2142no—
197#41—2142no—
198#41—2142no—
199#41—2142no—
200C2×Q8—1071no—
201C2×Q8—1071no—
202C2×D4—1072no—
203C2×D4—1071no—
204C2×D4—1071no—
205C2×D4—1072no—
206C2×D4—2142no—
207C2×D4—2142no—
208C2×D4—2142no—
209C2×D4—2141no—
210C2×D4—2141no—
211C2×D4—2142no—
212C2×D4—2142no—
213C22×C4—21no—
214C22×C4—1072no—
215C22×C4—2141no—
216C22×C4C22×D10711yescharacteristic
217SD16—2144no—
218D8—2144no—
219M16—2144no—
220M16D42814yescomplemented
221C2×C8—1071no—
222C2×C8—1071no—
223C2×C8—2142no—
224C2×C8—2144no—
225C2×C8C22×D10712yes—
226C4⋊C4—21no—
227C4⋊C4—2142no—
228C4⋊C4C22×D10712yes—
229C4⋊C4C22×D10711yescharacteristic
230C4⋊C4C22×D10711yescharacteristic
231C22⋊C4—21no—
232C22⋊C4—2142no—
233C22⋊C4—2142no—
234C22⋊C4—2142no—
235C22⋊C4—2142no—
236C42—21no—
237C23—8561no—
238C23—8561no—
239C23—1072no—
240C23—4282no—
241C23—2141no—
242C23—2142no—
243C23D4×D10711yescharacteristic
244Q8—2141no—
245Q8—2141no—
246Q8—2142no—
247D4—8562no—
248D4—8562no—
249D4—8562no—
250D4—4282no—
251D4—4282no—
252D4—2144no—
253D4—2142no—
254D4—2141no—
255D4—2142no—
256D4—2142no—
257D4—2141no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—81no—
262C2×C4—1072no—
263C2×C4—4282no—
264C2×C4—2141no—
265C2×C4—2142no—
266C2×C4—2142no—
267C2×C4D4×D10711yescharacteristic
268C2×C4C2×D42812yes—
269C2×C4#1313381912yes—
270C2×C4C23×D10711yescharacteristic
271C8—2141no—
272C8—2141no—
273C8C2×D42814yes—
274C22—21no—
275C22—21no—
276C22—8564no—
277C22—4281no—
278C22—4282no—
279C22—4281no—
280C22—4282no—
281C22—4282no—
282C22—4281no—
283C22—2142no—
284C22—2142no—
285C22—2142no—
286C22#1681208412yes—
287C22C2×D4×D10711yescharacteristic
288C4—42no—
289C4—41no—
290C4—41no—
291C4—4281no—
292C4—2142no—
293C4#1681208412yes—
294C4C22×D42811yescharacteristic
295C4C2×#1313381911yescharacteristic
296C2—22no—
297C2—8562no—
298C2—8561no—
299C2—4282no—
300C2C2×#1681208411yescharacteristic
301C2#2029343111yescharacteristic
302C2#2029343611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#428×C107C2
socleC22×C107C22×D4
hypercenter#428D107
hypocenterC107#3062
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. C2×#16812137C2semidirect
2. C2×#16812117C2semidirect
3. #20293208C2semidirect
4. #20293196C2semidirect
5. #20292881C2semidirect
6. #20292874C2semidirect
7. #20292864C2semidirect
8. #20292857C2semidirect
9. #20292621C2semidirect
10. C2×#16812029C2semidirect
11. #428×C107C2semidirect
19. C2×#13133801C22semidirect
21. C2×C8×D107C22semidirect
25. #16812089C22semidirect
29. #16812084C22semidirect
32. #16812032C22semidirect
33. #16812032C22semidirect
36. #16812029C22semidirect
37. #16812029C22semidirect
41. #107×C107C22semidirect
42. #106×C107C22semidirect
77. M16×C107D4semidirect
152. C107#3062semidirect
161. #428D107semidirect
187. #107C2×D107semidirect
188. #106C2×D107semidirect
220. M16D428semidirect

Maximal subgroups

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

subgroupindex
C2×#168121372
C2×#168121172
#202932082
#202931962
#202928812
#202928742
#202928642
#202928572
#202926212
C2×#168120292
#428×C1072
#3062107