Finite Group Databasebeta

← #28346078

Subgroups — #28346078

Order 13696.

548 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 135 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×#16812135C211yesmaximal, characteristic, complemented
2C2×M16×D107C211yesmaximal, characteristic, complemented
3#20293139C211yesmaximal, characteristic, complemented
4C2×#16812080C211yesmaximal, characteristic, complemented
5#20292879C212yesmaximal, complemented
6#20292872C212yesmaximal, complemented
7#20292862C212yesmaximal, complemented
8#20292855C212yesmaximal, complemented
9#20292670C211yesmaximal, characteristic, complemented
10#20292629C211yesmaximal, characteristic, complemented
11#418×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2211yescharacteristic
13C22×C22⋊D107—21no—
14C22×C4×D107C2211yescharacteristic
15#16812135—24no—
16C2×#13133806C2212yescomplemented
17C2×#13133803C2212yescomplemented
18C2×#13133802C2211yescharacteristic
19C2×#13133801C2212yescomplemented
20M16×D107—24no—
21C2×C8×D107C2212yescomplemented
22#16812093—21no—
23#16812089C2211yescharacteristic, complemented
24#16812089C2211yescharacteristic, complemented
25#16812084C2211yescharacteristic, complemented
26#16812084C2211yescharacteristic, complemented
27#16812080—21no—
28#16812080—21no—
29#16812080C2212yes—
30#16812080C2212yes—
31#16812079—21no—
32C4×D428—21no—
33C4×D428—21no—
34C2×C4⋊Dic107C2211yescharacteristic
35C2×C22⋊Dic107—21no—
36#16812042C2212yescomplemented
37#16812040C2212yescomplemented
38#16812034C2212yescomplemented
39#16812031C2212yescomplemented
40C22×D4×C107C2211yescharacteristic
41C2×M16×C107C2211yescharacteristic
42#117×C107C2211yescharacteristic
43#102×C107C2212yescomplemented
44#102×C107C2212yescomplemented
45C23×D107D411yescharacteristic
46#13133818—22no—
47#13133818—22no—
48C2×Dic214C2311yescharacteristic
49C2×D428—21no—
50C2×D428—21no—
51C2×D428C2311yescharacteristic
52C2×C22⋊D107—21no—
53C2×C22⋊D107—22no—
54C2×C22⋊D107—22no—
55C2×C22⋊D107—44no—
56C22×Dic107—21no—
57C22×Dic107D411yescharacteristic
58C2×C4×D107—22no—
59C2×C4×D107—22no—
60C2×C4×D107—21no—
61C2×C4×D107D411yescharacteristic
62C2×C4×D107D411yescharacteristic
63C2×C4×D107C2311yescharacteristic
64C2×C4×D107C2311yescharacteristic
65#13133806—24no—
66#13133803—24no—
67#13133802D414yescomplemented
68#13133801—24no—
69C2×C107⋊C8C2312yes—
70C8×D107—24no—
71C4⋊Dic107—21no—
72C4⋊Dic107—21no—
73C4⋊Dic107—21no—
74C4⋊Dic107C2312yes—
75C22⋊Dic107—22no—
76C22⋊Dic107—22no—
77(C2×D107)⋊C4—21no—
78(C2×D107)⋊C4—21no—
79(C2×D107)⋊C4—21no—
80(C2×D107)⋊C4—21no—
81C24×C107—21no—
82C2×D4×C107—22no—
83C2×D4×C107—21no—
84C2×D4×C107—22no—
85C2×D4×C107C2312yes—
86C22×C4×C107C2311yescharacteristic
87M16×C107—22no—
88C2×C8×C107C2312yes—
89C4⋊C4×C107C2311yescharacteristic
90C4⋊C4×C107C2311yescharacteristic
91C22⋊C4×C107—21no—
92C42×C107—21no—
93C23×C107—22no—
94C23×C107—42no—
95C23×C107—42no—
96C23×C107—42no—
97C23×C107C2×D411yescharacteristic
98C22×D107—22no—
99C22×D107—22no—
100C22×D107—21no—
101C22×D107—41no—
102C22×D107C2×D411yescharacteristic
103C22×D107C2×D411yescharacteristic
104D4×C107—22no—
105D4×C107—22no—
106D4×C107—42no—
107C2×C4×C107—21no—
108C2×C4×C107—21no—
109C2×C4×C107C2×D411yescharacteristic
110C2×C4×C107C2×D412yes—
111C2×C4×C107#4112yes—
112C2×C4×C107C2411yescharacteristic
113C22⋊D107—44no—
114C22⋊D107—42no—
115C22⋊D107—44no—
116C2×Dic107—22no—
117C2×Dic107—21no—
118C2×Dic107—22no—
119C2×Dic107—42no—
120C2×Dic107C2×D411yescharacteristic
121C2×Dic107C2×D411yescharacteristic
122D428—21no—
123D428—21no—
124D428—42no—
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×C107—22no—
136C107⋊C8C2×D414yescomplemented
137C22×C107—21no—
138C22×C107—21no—
139C22×C107—42no—
140C22×C107—42no—
141C22×C107—42no—
142C22×C107—84no—
143C22×C107#12112yes—
144C22×C107C22×D411yescharacteristic
145C2×D107—21no—
146C2×D107—21no—
147C2×D107—21no—
148C2×D107—21no—
149C2×D107—21no—
150C2×D107—44no—
151C2×D107—41no—
152C2×D107—41no—
153C2×D107—41no—
154C2×D107AGL(1,ℤ/8)12yescomplemented
155C4×C107—41no—
156C4×C107—41no—
157C4×C107#12112yes—
158C4×C107C22×D411yescharacteristic
159C4×C107C2×#4111yescharacteristic
160Dic107—21no—
161Dic107—21no—
162Dic107—41no—
163Dic107—42no—
164C2×C107—22no—
165C2×C107—82no—
166C2×C107C2×#12111yescharacteristic
167C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
168C2×C107#57511yescharacteristic
169D107—22no—
170D107—41no—
171D107—81no—
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#428—1071no—
183#418D10711yescharacteristic, complemented
184#417—1071no—
185C2×#41—1071no—
186C22×D4—2141no—
187C22×D4C2×D10711yescharacteristic
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⋊C4—1071no—
211C2×C22⋊C4—2141no—
212#107—1072no—
213#106—1072no—
214#103—1072no—
215#102—1072no—
216#102C2×D10712yescomplemented
217#102C2×D10712yescomplemented
218C24—21no—
219C24—1071no—
220#41—2142no—
221#41—2142no—
222C2×Q8—1071no—
223C2×D4—22no—
224C2×D4—21no—
225C2×D4—22no—
226C2×D4—1071no—
227C2×D4—4284no—
228C2×D4—2141no—
229C2×D4—2141no—
230C2×D4—2142no—
231C2×D4—2142no—
232C2×D4—2141no—
233C2×D4C22×D10712yes—
234C22×C4—1071no—
235C22×C4—1071no—
236C22×C4—1071no—
237C22×C4—1071no—
238C22×C4—1071no—
239C22×C4—2141no—
240C22×C4—2141no—
241C22×C4—2142no—
242C22×C4—2142no—
243C22×C4C22×D10711yescharacteristic
244SD16—2144no—
245D8—2144no—
246M16—22no—
247M16—1074no—
248M16—2144no—
249C2×C8—1072no—
250C2×C8—2144no—
251C2×C8C22×D10712yes—
252C4⋊C4—1072no—
253C4⋊C4—2141no—
254C4⋊C4—2141no—
255C4⋊C4—2141no—
256C4⋊C4C22×D10711yescharacteristic
257C4⋊C4C22×D10711yescharacteristic
258C22⋊C4—21no—
259C22⋊C4—2141no—
260C22⋊C4—2141no—
261C22⋊C4—2141no—
262C22⋊C4—2141no—
263C22⋊C4—2142no—
264C22⋊C4—2142no—
265C42—21no—
266C23—22no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—1071no—
271C23—1071no—
272C23—4281no—
273C23—2142no—
274C23—2141no—
275C23—2142no—
276C23D4×D10711yescharacteristic
277Q8—2141no—
278Q8—2141no—
279D4—22no—
280D4—22no—
281D4—42no—
282D4—4282no—
283D4—4282no—
284D4—4284no—
285D4—4284no—
286D4—4282no—
287D4—2141no—
288D4—2141no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—1071no—
292C2×C4—1071no—
293C2×C4—4282no—
294C2×C4—4282no—
295C2×C4—2142no—
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×C4D4×D10712yes—
306C2×C4#1313381812yes—
307C2×C4C23×D10711yescharacteristic
308C8—22no—
309C8—1074no—
310C22—21no—
311C22—21no—
312C22—42no—
313C22—42no—
314C22—42no—
315C22—84no—
316C22—1072no—
317C22—4284no—
318C22—4281no—
319C22—4281no—
320C22—4281no—
321C22—2141no—
322C22—2141no—
323C22—2141no—
324C22—2141no—
325C22—2141no—
326C22#1681208212yes—
327C22C2×D4×D10711yescharacteristic
328C4—41no—
329C4—41no—
330C4—4282no—
331C4—4281no—
332C4—2141no—
333C4—2141no—
334C4#1681208212yes—
335C4C2×D4×D10711yescharacteristic
336C4C2×#1313381811yescharacteristic
337C2—22no—
338C2—82no—
339C2—8561no—
340C2—4281no—
341C2—2142no—
342C2C2×#1681208211yescharacteristic
343C2AGL(1,ℤ/8)×D10711yescharacteristic
344C2#2029342311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#418×C107C2
socleC22×C107C22×D4
hypercenter#418D107
hypocenterC107#3055
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C2×#16812135C2semidirect
2. C2×M16×D107C2semidirect
3. #20293139C2semidirect
4. C2×#16812080C2semidirect
5. #20292879C2semidirect
6. #20292872C2semidirect
7. #20292862C2semidirect
8. #20292855C2semidirect
9. #20292670C2semidirect
10. #20292629C2semidirect
11. #418×C107C2semidirect
16. C2×#13133806C22semidirect
17. C2×#13133803C22semidirect
19. C2×#13133801C22semidirect
21. C2×C8×D107C22semidirect
23. #16812089C22semidirect
24. #16812089C22semidirect
25. #16812084C22semidirect
26. #16812084C22semidirect
36. #16812042C22semidirect
37. #16812040C22semidirect
38. #16812034C22semidirect
39. #16812031C22semidirect
43. #102×C107C22semidirect
44. #102×C107C22semidirect
67. #13133802D4semidirect
136. C107⋊C8C2×D4semidirect
154. C2×D107AGL(1,ℤ/8)semidirect
173. C107#3055semidirect
183. #418D107semidirect
216. #102C2×D107semidirect
217. #102C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121352
C2×M16×D1072
#202931392
C2×#168120802
#202928792
#202928722
#202928622
#202928552
#202926702
#202926292
#418×C1072
#3055107