Finite Group Databasebeta

← #28346081

Subgroups — #28346081

Order 13696.

548 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 143 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×M16×D107C211yesmaximal, characteristic, complemented
3#20293139C211yesmaximal, characteristic, complemented
4C2×#16812084C211yesmaximal, characteristic, complemented
5#20292881C212yesmaximal, complemented
6#20292874C212yesmaximal, complemented
7#20292864C212yesmaximal, complemented
8#20292857C212yesmaximal, complemented
9#20292631C211yesmaximal, characteristic, complemented
10#20292626C211yesmaximal, characteristic, complemented
11#428×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2211yescharacteristic
13C22×D428—21no—
14C22×D428C2211yescharacteristic
15C22×C4×D107C2211yescharacteristic
16#16812137—24no—
17C2×#13133807C2212yescomplemented
18C2×D856C2212yescomplemented
19C2×#13133802C2211yescharacteristic
20C2×#13133801C2212yescomplemented
21M16×D107—24no—
22C2×C8×D107C2212yescomplemented
23#16812093—21no—
24#16812089C2211yescharacteristic, complemented
25#16812089C2211yescharacteristic, complemented
26#16812084—21no—
27#16812084—21no—
28#16812084C2212yes—
29#16812084C2212yes—
30#16812084C2211yescharacteristic, complemented
31#16812084C2211yescharacteristic, complemented
32#16812079—21no—
33C4×D428—21no—
34C4×D428—21no—
35C2×(C2×D107)⋊C4—21no—
36#16812032C2212yescomplemented
37#16812029C2212yescomplemented
38#16812029C2212yescomplemented
39#16812029C2212yescomplemented
40C2×M16×C107C2211yescharacteristic
41#117×C107C2211yescharacteristic
42C2×C4⋊C4×C107C2211yescharacteristic
43#107×C107C2212yescomplemented
44#106×C107C2212yescomplemented
45C23×D107—21no—
46C23×D107D411yescharacteristic
47#13133818—22no—
48#13133818—22no—
49C2×Dic214C2311yescharacteristic
50C2×D428—22no—
51C2×D428—22no—
52C2×D428—21no—
53C2×D428—21no—
54C2×D428—22no—
55C2×D428—22no—
56C2×D428—21no—
57C2×D428—44no—
58C2×D428C2312yes—
59C2×D428C2311yescharacteristic
60C2×C22⋊D107—21no—
61C22×Dic107D411yescharacteristic
62C2×C4×D107—22no—
63C2×C4×D107—22no—
64C2×C4×D107—21no—
65C2×C4×D107D411yescharacteristic
66C2×C4×D107D411yescharacteristic
67C2×C4×D107C2311yescharacteristic
68C2×C4×D107C2311yescharacteristic
69#13133807—24no—
70D856—24no—
71#13133802—22no—
72#13133801—24no—
73C2×C107⋊C8C2312yes—
74C8×D107—24no—
75C4⋊Dic107—21no—
76C4⋊Dic107—21no—
77(C2×D107)⋊C4—21no—
78(C2×D107)⋊C4—21no—
79(C2×D107)⋊C4—22no—
80(C2×D107)⋊C4—22no—
81(C2×D107)⋊C4—21no—
82(C2×D107)⋊C4—21no—
83C22×C4×C107—21no—
84C22×C4×C107C2311yescharacteristic
85M16×C107D414yescomplemented
86C2×C8×C107C2312yes—
87C4⋊C4×C107—21no—
88C4⋊C4×C107C2312yes—
89C4⋊C4×C107C2311yescharacteristic
90C4⋊C4×C107C2311yescharacteristic
91C22⋊C4×C107—21no—
92C42×C107—21no—
93C23×C107C2×D411yescharacteristic
94C22×D107—22no—
95C22×D107—22no—
96C22×D107—21no—
97C22×D107—22no—
98C22×D107—41no—
99C22×D107—42no—
100C22×D107—42no—
101C22×D107—42no—
102C22×D107C2×D411yescharacteristic
103C22×D107C2×D411yescharacteristic
104C2×C4×C107—21no—
105C2×C4×C107—21no—
106C2×C4×C107—22no—
107C2×C4×C107—42no—
108C2×C4×C107C2×D411yescharacteristic
109C2×C4×C107C2×D412yes—
110C2×C4×C107#4112yes—
111C2×C4×C107C2411yescharacteristic
112C22⋊D107—42no—
113C2×Dic107—22no—
114C2×Dic107—21no—
115C2×Dic107C2×D411yescharacteristic
116C2×Dic107C2×D411yescharacteristic
117D428—21no—
118D428—22no—
119D428—22no—
120D428—21no—
121D428—42no—
122D428—44no—
123D428—42no—
124D428—44no—
125D428—42no—
126C4×D107—22no—
127C4×D107—22no—
128C4×D107—21no—
129C4×D107—21no—
130C4×D107—21no—
131C4×D107—21no—
132C4×D107—42no—
133Dic214—21no—
134Dic214—21no—
135C8×C107C2×D414yescomplemented
136C107⋊C8—22no—
137C22×C107—21no—
138C22×C107—21no—
139C22×C107#12112yes—
140C22×C107C22×D411yescharacteristic
141C2×D107—21no—
142C2×D107—21no—
143C2×D107—21no—
144C2×D107—21no—
145C2×D107—21no—
146C2×D107—44no—
147C2×D107—41no—
148C2×D107—42no—
149C2×D107—42no—
150C2×D107—41no—
151C2×D107—41no—
152C2×D107—42no—
153C2×D107—84no—
154C2×D107AGL(1,ℤ/8)12yescomplemented
155C4×C107—42no—
156C4×C107—41no—
157C4×C107—41no—
158C4×C107#12112yes—
159C4×C107C22×D411yescharacteristic
160C4×C107C2×#4111yescharacteristic
161Dic107—21no—
162Dic107—21no—
163Dic107—41no—
164C2×C107—22no—
165C2×C107C2×#12111yescharacteristic
166C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
167C2×C107#57511yescharacteristic
168D107—22no—
169D107—41no—
170D107—81no—
171D107—82no—
172#3055—1071nomaximal, Sylow
173C107#305511yescharacteristic, complemented, Sylow
174C2×AGL(1,ℤ/8)—1071no—
175C22×M16—1071no—
176#525—1071no—
177C2×#121—1071no—
178#469—1072no—
179#468—1072no—
180#466—1072no—
181#465—1072no—
182#428D10711yescharacteristic, complemented
183#418—1071no—
184#417—1071no—
185C2×#41—1071no—
186C22×D4—1071no—
187C22×D4—2141no—
188C23×C4—1071no—
189AGL(1,ℤ/8)—2144no—
190C2×SD16—1072no—
191C2×D8—1072no—
192C2×M16—1071no—
193C2×M16—1072no—
194C2×M16—2144no—
195C2×M16C2×D10711yescharacteristic
196C22×C8—1072no—
197#123—2141no—
198C22⋊Q8—1071no—
199C22⋊Q8—1071no—
200#121—1072no—
201#121—1072no—
202#121—1071no—
203#121—1071no—
204#121—2141no—
205#121—2141no—
206#120—2141no—
207C4×D4—2141no—
208C4×D4—2141no—
209#117C2×D10711yescharacteristic
210C2×C4⋊C4C2×D10711yescharacteristic
211C2×C22⋊C4—2141no—
212#107C2×D10712yescomplemented
213#106C2×D10712yescomplemented
214#103—1072no—
215#102—1072no—
216#102—1072no—
217#102—1072no—
218C24—1071no—
219C24—2141no—
220#41—2142no—
221#41—2142no—
222C2×Q8—1071no—
223C2×D4—1071no—
224C2×D4—1072no—
225C2×D4—4284no—
226C2×D4—2141no—
227C2×D4—2141no—
228C2×D4—2142no—
229C2×D4—2142no—
230C2×D4—2141no—
231C2×D4—2141no—
232C2×D4—2142no—
233C2×D4—2142no—
234C22×C4—21no—
235C22×C4—1071no—
236C22×C4—1071no—
237C22×C4—1071no—
238C22×C4—1071no—
239C22×C4—1071no—
240C22×C4—2141no—
241C22×C4—2142no—
242C22×C4—2142no—
243C22×C4C22×D10711yescharacteristic
244SD16—2144no—
245D8—2144no—
246M16—2142no—
247M16—2144no—
248M16D42814yescomplemented
249C2×C8—1072no—
250C2×C8—2144no—
251C2×C8C22×D10712yes—
252C4⋊C4—21no—
253C4⋊C4—2141no—
254C4⋊C4—2141no—
255C4⋊C4C22×D10711yescharacteristic
256C4⋊C4C22×D10711yescharacteristic
257C4⋊C4C22×D10712yes—
258C22⋊C4—21no—
259C22⋊C4—2142no—
260C22⋊C4—2142no—
261C22⋊C4—2141no—
262C22⋊C4—2141no—
263C22⋊C4—2141no—
264C22⋊C4—2141no—
265C42—21no—
266C23—1071no—
267C23—1071no—
268C23—4282no—
269C23—4282no—
270C23—4281no—
271C23—4282no—
272C23—2142no—
273C23—2141no—
274C23—2142no—
275C23—2142no—
276C23D4×D10711yescharacteristic
277Q8—2141no—
278Q8—2141no—
279D4—4284no—
280D4—4282no—
281D4—4284no—
282D4—4282no—
283D4—4282no—
284D4—4282no—
285D4—2141no—
286D4—2142no—
287D4—2142no—
288D4—2141no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—42no—
293C2×C4—1071no—
294C2×C4—1071no—
295C2×C4—4282no—
296C2×C4—2141no—
297C2×C4—2142no—
298C2×C4—2142no—
299C2×C4—2142no—
300C2×C4—2141no—
301C2×C4—2141no—
302C2×C4—2141no—
303C2×C4—2141no—
304C2×C4D4×D10711yescharacteristic
305C2×C4C2×D42812yes—
306C2×C4#1313381912yes—
307C2×C4C23×D10711yescharacteristic
308C8—2142no—
309C8C2×D42814yescomplemented
310C22—21no—
311C22—21no—
312C22—8564no—
313C22—1072no—
314C22—4284no—
315C22—4281no—
316C22—4282no—
317C22—4282no—
318C22—4281no—
319C22—4281no—
320C22—4282no—
321C22—2141no—
322C22—2141no—
323C22—2141no—
324C22—2141no—
325C22—2141no—
326C22#1681208412yes—
327C22C2×D4×D10711yescharacteristic
328C4—42no—
329C4—41no—
330C4—41no—
331C4—4281no—
332C4—2141no—
333C4—2141no—
334C4#1681208412yes—
335C4C22×D42811yescharacteristic
336C4C2×#1313381911yescharacteristic
337C2—22no—
338C2—8562no—
339C2—8561no—
340C2—4281no—
341C2—2142no—
342C2C2×#1681208411yescharacteristic
343C2AGL(1,ℤ/8)×D10711yescharacteristic
344C2#2029342611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#428×C107C2
socleC22×C107C22×D4
hypercenter#428D107
hypocenterC107#3055
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×M16×D107C2semidirect
3. #20293139C2semidirect
4. C2×#16812084C2semidirect
5. #20292881C2semidirect
6. #20292874C2semidirect
7. #20292864C2semidirect
8. #20292857C2semidirect
9. #20292631C2semidirect
10. #20292626C2semidirect
11. #428×C107C2semidirect
17. C2×#13133807C22semidirect
18. C2×D856C22semidirect
20. C2×#13133801C22semidirect
22. C2×C8×D107C22semidirect
24. #16812089C22semidirect
25. #16812089C22semidirect
30. #16812084C22semidirect
31. #16812084C22semidirect
36. #16812032C22semidirect
37. #16812029C22semidirect
38. #16812029C22semidirect
39. #16812029C22semidirect
43. #107×C107C22semidirect
44. #106×C107C22semidirect
85. M16×C107D4semidirect
135. C8×C107C2×D4semidirect
154. C2×D107AGL(1,ℤ/8)semidirect
173. C107#3055semidirect
182. #428D107semidirect
212. #107C2×D107semidirect
213. #106C2×D107semidirect
248. M16D428semidirect
309. C8C2×D428semidirect

Maximal subgroups

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

subgroupindex
C2×#168121372
C2×M16×D1072
#202931392
C2×#168120842
#202928812
#202928742
#202928642
#202928572
#202926312
#202926262
#428×C1072
#3055107