Finite Group Databasebeta

← #28346074

Subgroups — #28346074

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×M16×D107C211yesmaximal, characteristic, complemented
2#20293134C211yesmaximal, characteristic, complemented
3C2×#16812080C211yesmaximal, characteristic, complemented
4#20292877C212yesmaximal, complemented
5#20292870C212yesmaximal, complemented
6#20292860C212yesmaximal, complemented
7#20292853C212yesmaximal, complemented
8#20292669C211yesmaximal, characteristic, complemented
9#20292632C211yesmaximal, characteristic, complemented
10#20292628C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C107C211yesmaximal, characteristic, complemented
12C22×C22⋊D107—21no—
13C22×C4×D107C2211yescharacteristic
14C2×#13133802C2211yescharacteristic
15C2×#13133801C2212yescomplemented
16M16×D107—24no—
17C2×C8×D107C2212yescomplemented
18#16812090—21no—
19#16812085C2211yescharacteristic, complemented
20#16812085C2211yescharacteristic, complemented
21#16812080—21no—
22#16812080—21no—
23#16812080C2212yes—
24#16812080C2212yes—
25#16812080C2211yescharacteristic, complemented
26#16812080C2211yescharacteristic, complemented
27#16812077—21no—
28C4×C22⋊D107—21no—
29C4×C22⋊D107—21no—
30#16812068C2211yescharacteristic
31C2×C4⋊Dic107C2211yescharacteristic
32C2×C22⋊Dic107—21no—
33#16812043C2212yescomplemented
34#16812041C2212yescomplemented
35#16812033C2212yescomplemented
36#16812030C2212yescomplemented
37#16812030C2212yescomplemented
38#16812030C2212yescomplemented
39C2×#41×C107C2211yescharacteristic
40C22×D4×C107C2211yescharacteristic
41AGL(1,ℤ/8)×C107—24no—
42C2×SD16×C107C2212yescomplemented
43C2×D8×C107C2212yescomplemented
44C2×M16×C107C2211yescharacteristic
45C23×D107D411yescharacteristic
46C2×C22⋊D107—21no—
47C2×C22⋊D107—21no—
48C2×C22⋊D107—22no—
49C2×C22⋊D107—22no—
50C2×C22⋊D107—44no—
51C22×Dic107—21no—
52C22×Dic107D411yescharacteristic
53C2×C4×D107—22no—
54C2×C4×D107—22no—
55C2×C4×D107D411yescharacteristic
56C2×C4×D107D411yescharacteristic
57C2×C4×D107C2311yescharacteristic
58C2×C4×D107C2311yescharacteristic
59#13133802—22no—
60#13133801—24no—
61C2×C107⋊C8C2312yes—
62C8×D107—24no—
63C4⋊Dic107—21no—
64C4⋊Dic107C2312yes—
65C4⋊Dic107C2311yescharacteristic
66C4⋊Dic107C2311yescharacteristic
67Dic107⋊C4—21no—
68Dic107⋊C4—21no—
69C22⋊Dic107—21no—
70C22⋊Dic107—22no—
71C22⋊Dic107—22no—
72C22⋊Dic107—21no—
73C22⋊Dic107—21no—
74(C2×D107)⋊C4—21no—
75(C2×D107)⋊C4—21no—
76C4×Dic107—21no—
77C24×C107—21no—
78#41×C107—22no—
79#41×C107—22no—
80C2×Q8×C107C2311yescharacteristic
81C2×D4×C107—21no—
82C2×D4×C107—22no—
83C2×D4×C107—21no—
84C2×D4×C107—22no—
85C2×D4×C107C2312yes—
86C2×D4×C107C2311yescharacteristic
87C22×C4×C107—21no—
88C22×C4×C107C2311yescharacteristic
89SD16×C107—24no—
90D8×C107—24no—
91M16×C107D414yescomplemented
92C2×C8×C107C2312yes—
93C23×C107—21no—
94C23×C107—22no—
95C23×C107—42no—
96C23×C107—42no—
97C23×C107—42no—
98C23×C107C2×D411yescharacteristic
99C22×D107—22no—
100C22×D107—22no—
101C22×D107—41no—
102C22×D107C2×D411yescharacteristic
103C22×D107C2×D411yescharacteristic
104Q8×C107—21no—
105Q8×C107—21no—
106D4×C107—21no—
107D4×C107—22no—
108D4×C107—22no—
109D4×C107—21no—
110D4×C107—42no—
111D4×C107—42no—
112C2×C4×C107—21no—
113C2×C4×C107—42no—
114C2×C4×C107C2×D411yescharacteristic
115C2×C4×C107C2×D412yes—
116C2×C4×C107#4112yes—
117C2×C4×C107C2411yescharacteristic
118C22⋊D107—42no—
119C22⋊D107—44no—
120C22⋊D107—42no—
121C22⋊D107—44no—
122C2×Dic107—22no—
123C2×Dic107—21no—
124C2×Dic107—21no—
125C2×Dic107—22no—
126C2×Dic107—42no—
127C2×Dic107C2×D411yescharacteristic
128C2×Dic107C2×D411yescharacteristic
129C4×D107—22no—
130C4×D107—22no—
131C4×D107—21no—
132C4×D107—21no—
133C4×D107—21no—
134C4×D107—21no—
135C8×C107C2×D414yescomplemented
136C107⋊C8—22no—
137C22×C107—21no—
138C22×C107—21no—
139C22×C107—41no—
140C22×C107—42no—
141C22×C107—42no—
142C22×C107—41no—
143C22×C107—41no—
144C22×C107—42no—
145C22×C107—84no—
146C22×C107#12112yes—
147C22×C107C22×D411yescharacteristic
148C2×D107—21no—
149C2×D107—21no—
150C2×D107—21no—
151C2×D107—21no—
152C2×D107—21no—
153C2×D107—44no—
154C2×D107AGL(1,ℤ/8)12yescomplemented
155C4×C107—41no—
156C4×C107#12112yes—
157C4×C107C22×D411yescharacteristic
158C4×C107C2×#4111yescharacteristic
159Dic107—21no—
160Dic107—21no—
161Dic107—42no—
162Dic107—41no—
163Dic107—41no—
164C2×C107—22no—
165C2×C107—81no—
166C2×C107—82no—
167C2×C107C2×#12111yescharacteristic
168C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
169C2×C107#57511yescharacteristic
170D107—22no—
171D107—41no—
172#3055—1071nomaximal, Sylow
173C107#305511yescharacteristic, complemented, Sylow
174C2×AGL(1,ℤ/8)D10711yescharacteristic, complemented
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#418—1071no—
184#417—1071no—
185C2×#41C2×D10711yescharacteristic
186C22×D4—2141no—
187C22×D4C2×D10711yescharacteristic
188C23×C4—1071no—
189AGL(1,ℤ/8)—24no—
190C2×SD16C2×D10712yescomplemented
191C2×D8C2×D10712yescomplemented
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#117—1071no—
210C2×C4⋊C4—1071no—
211C2×C22⋊C4—2141no—
212#107—1072no—
213#106—1072no—
214#103—1072no—
215#102—1072no—
216#102—1072no—
217#102—1072no—
218C24—21no—
219C24—1071no—
220#41—22no—
221#41—22no—
222C2×Q8C22×D10711yescharacteristic
223C2×D4—21no—
224C2×D4—21no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—4284no—
228C2×D4—2141no—
229C2×D4—2141no—
230C2×D4—2142no—
231C2×D4—2142no—
232C2×D4C22×D10711yescharacteristic
233C2×D4C22×D10712yes—
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—24no—
245D8—24no—
246M16—2142no—
247M16—2144no—
248M16C22⋊D10714yescomplemented
249C2×C8—1072no—
250C2×C8—2144no—
251C2×C8C22×D10712yes—
252C4⋊C4—1071no—
253C4⋊C4—1071no—
254C4⋊C4—1072no—
255C4⋊C4—2141no—
256C4⋊C4—2141no—
257C4⋊C4—2141no—
258C22⋊C4—2141no—
259C22⋊C4—2141no—
260C22⋊C4—2142no—
261C22⋊C4—2142no—
262C22⋊C4—2141no—
263C22⋊C4—2141no—
264C22⋊C4—2141no—
265C42—2141no—
266C23—21no—
267C23—22no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—1071no—
272C23—1071no—
273C23—4281no—
274C23—2142no—
275C23—2142no—
276C23D4×D10711yescharacteristic
277Q8—21no—
278Q8—21no—
279D4—21no—
280D4—22no—
281D4—22no—
282D4—21no—
283D4—42no—
284D4—42no—
285D4—4284no—
286D4—4282no—
287D4—4284no—
288D4—4282no—
289C2×C4—21no—
290C2×C4—42no—
291C2×C4—1071no—
292C2×C4—1071no—
293C2×C4—4282no—
294C2×C4—2141no—
295C2×C4—2141no—
296C2×C4—2142no—
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×C22⋊D10712yes—
306C2×C4#1313381712yes—
307C2×C4C23×D10711yescharacteristic
308C8—2142no—
309C8C2×C22⋊D10714yescomplemented
310C22—21no—
311C22—21no—
312C22—42no—
313C22—41no—
314C22—41no—
315C22—42no—
316C22—41no—
317C22—42no—
318C22—84no—
319C22—1072no—
320C22—4284no—
321C22—2141no—
322C22—2141no—
323C22—2141no—
324C22—2141no—
325C22—2141no—
326C22#1681208012yes—
327C22C2×D4×D10711yescharacteristic
328C4—41no—
329C4—4282no—
330C4—4281no—
331C4—4281no—
332C4—2141no—
333C4—2141no—
334C4#1681208012yes—
335C4C22×C22⋊D10711yescharacteristic
336C4C2×#1313381711yescharacteristic
337C2—22no—
338C2—82no—
339C2—81no—
340C2—4281no—
341C2—2142no—
342C2C2×#1681208011yescharacteristic
343C2AGL(1,ℤ/8)×D10711yescharacteristic
344C2#2029342311yescharacteristic
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#3055
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×M16×D107C2semidirect
2. #20293134C2semidirect
3. C2×#16812080C2semidirect
4. #20292877C2semidirect
5. #20292870C2semidirect
6. #20292860C2semidirect
7. #20292853C2semidirect
8. #20292669C2semidirect
9. #20292632C2semidirect
10. #20292628C2semidirect
11. C2×AGL(1,ℤ/8)×C107C2semidirect
15. C2×#13133801C22semidirect
17. C2×C8×D107C22semidirect
19. #16812085C22semidirect
20. #16812085C22semidirect
25. #16812080C22semidirect
26. #16812080C22semidirect
33. #16812043C22semidirect
34. #16812041C22semidirect
35. #16812033C22semidirect
36. #16812030C22semidirect
37. #16812030C22semidirect
38. #16812030C22semidirect
42. C2×SD16×C107C22semidirect
43. C2×D8×C107C22semidirect
91. M16×C107D4semidirect
135. C8×C107C2×D4semidirect
154. C2×D107AGL(1,ℤ/8)semidirect
173. C107#3055semidirect
174. C2×AGL(1,ℤ/8)D107semidirect
190. C2×SD16C2×D107semidirect
191. C2×D8C2×D107semidirect
248. M16C22⋊D107semidirect
309. C8C2×C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C2×M16×D1072
#202931342
C2×#168120802
#202928772
#202928702
#202928602
#202928532
#202926692
#202926322
#202926282
C2×AGL(1,ℤ/8)×C1072
#3055107