Finite Group Databasebeta

← #28345767

Subgroups — #28345767

Order 13696.

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

normal subgroups only

This group has 764 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×#16812155C211yesmaximal, characteristic, complemented
2C2×#16812137C211yesmaximal, characteristic, complemented
3C2×#16812129C211yesmaximal, characteristic, complemented
4#20293208C211yesmaximal, characteristic, complemented
5#20292792C212yesmaximal, complemented
6#20292779C212yesmaximal, complemented
7#20292775C212yesmaximal, complemented
8#20292762C212yesmaximal, complemented
9C2×#16812029C211yesmaximal, characteristic, complemented
10#20292595C211yesmaximal, characteristic, complemented
11#417×C107C211yesmaximal, characteristic, complemented
12#16812155—24no—
13#16812155—24no—
14C2×#13133819C2212yes—
15C2×#13133818—21no—
16C2×#13133818C2211yescharacteristic
17C2×#13133818C2211yescharacteristic, complemented
18C22×D428—21no—
19C22×D428C2211yescharacteristic
20C2×D4×D107—22no—
21C2×D4×D107C2212yes—
22#16812137—24no—
23#16812129—24no—
24C2×#13133808C2211yescharacteristic, complemented
25C2×#13133807C2212yescomplemented
26C2×#13133806C2211yescharacteristic, complemented
27C2×#13133805C2211yescharacteristic, complemented
28C2×D856C2212yescomplemented
29C2×#13133803C2211yescharacteristic, complemented
30C22×C107⋊C8C2211yescharacteristic, complemented
31#16812089—21no—
32#16812089C2212yescomplemented
33#16812084—21no—
34#16812084C2212yescomplemented
35#16812083—22no—
36#16812029C2212yescomplemented
37#16812029C2212yescomplemented
38(C2×D107)⋊C8C2212yescomplemented
39(C2×D107)⋊C8C2212yescomplemented
40C2×#41×C107C2211yescharacteristic
41C2×M16×C107C2211yescharacteristic, complemented
42C2×C4⋊C4×C107C2211yescharacteristic, complemented
43#103×C107C2212yescomplemented
44#102×C107C2212yescomplemented
45C23×D107—21no—
46C23×D107—22no—
47#13133819—24no—
48#13133819—24no—
49#13133818—28no—
50#13133818—22no—
51#13133818—22no—
52#13133818—22no—
53#13133818—22no—
54C2×Dic214D411yescharacteristic
55C2×Dic214C2311yescharacteristic
56C2×D428—28no—
57C2×D428—22no—
58C2×D428—21no—
59C2×D428—22no—
60C2×D428—22no—
61C2×D428D411yescharacteristic
62C2×D428D414yescomplemented
63C2×D428C2311yescharacteristic
64C2×D428C2312yes—
65C2×C22⋊D107—21no—
66C2×C22⋊D107—22no—
67C2×C22⋊D107D412yes—
68D4×D107—216no—
69D4×D107—24no—
70D4×D107—24no—
71C2×C4×D107—22no—
72C2×C4×D107—21no—
73C2×C4×D107C2312yes—
74#13133808—22no—
75#13133807—24no—
76#13133806—22no—
77#13133805—22no—
78D856—24no—
79#13133803—22no—
80C2×C107⋊C8—22no—
81C2×C107⋊C8C2311yescharacteristic
82C2×C107⋊C8C2311yescharacteristic
83C4⋊Dic107—22no—
84(C2×D107)⋊C4—22no—
85(C2×D107)⋊C4—22no—
86#41×C107—22no—
87#41×C107D414yescomplemented
88C2×Q8×C107C2311yescharacteristic
89C2×D4×C107—21no—
90C2×D4×C107C2311yescharacteristic
91C22×C4×C107—21no—
92C22×C4×C107—21no—
93C22×C4×C107C2311yescharacteristic
94M16×C107—22no—
95C2×C8×C107C2312yes—
96C4⋊C4×C107—21no—
97C4⋊C4×C107C2312yes—
98C23×C107—21no—
99C23×C107C2×D411yescharacteristic
100C22×D107—22no—
101C22×D107—28no—
102C22×D107—24no—
103C22×D107—21no—
104C22×D107—82no—
105C22×D107—82no—
106C22×D107—81no—
107C22×D107—81no—
108C22×D107C2×D412yes—
109Q8×C107—21no—
110Q8×C107C2×D412yes—
111D4×C107—24no—
112D4×C107—21no—
113D4×C107C2×D412yes—
114C2×C4×C107—21no—
115C2×C4×C107—24no—
116C2×C4×C107—22no—
117C2×C4×C107—81no—
118C2×C4×C107C2×D411yescharacteristic
119C2×C4×C107C2×D412yes—
120C2×C4×C107C2×D412yes—
121C2×C4×C107C2411yescharacteristic
122C22⋊D107—28no—
123C22⋊D107—24no—
124C22⋊D107—42no—
125C2×Dic107—21no—
126C2×Dic107C2×D412yes—
127D428—28no—
128D428—22no—
129D428—24no—
130D428—22no—
131D428—21no—
132D428—21no—
133D428—82no—
134D428C2×D414yes—
135C4×D107—28no—
136C4×D107—22no—
137C4×D107—22no—
138C4×D107—42no—
139Dic214—22no—
140Dic214—21no—
141Dic214—21no—
142C8×C107—22no—
143C107⋊C8—21no—
144C107⋊C8—21no—
145C22×C107—21no—
146C22×C107—21no—
147C22×C107—22no—
148C22×C107—41no—
149C22×C107—41no—
150C22×C107#12012yes—
151C22×C107C22×D411yescharacteristic
152C2×D107—22no—
153C2×D107—24no—
154C2×D107—22no—
155C2×D107—22no—
156C2×D107—42no—
157C2×D107—42no—
158C2×D107—41no—
159C2×D107—41no—
160C2×D107—41no—
161C2×D107—84no—
162C2×D107—88no—
163C4×C107—22no—
164C4×C107—42no—
165C4×C107#12012yes—
166C4×C107C22×D411yescharacteristic
167C4×C107C22×D411yescharacteristic
168Dic107—22no—
169Dic107—41no—
170C2×C107—22no—
171C2×C107—42no—
172C2×C107C2×#12011yescharacteristic
173C2×C107#57611yescharacteristic
174C2×C107#57711yescharacteristic
175D107—42no—
176D107—44no—
177D107—81no—
178#3012—1071nomaximal, Sylow
179C107#301211yescharacteristic, complemented, Sylow
180C2×H(5,𝔽2)—1071no—
181C2×AGL(1,ℤ/8)—1071no—
182C2×#135—1071no—
183#537—1071no—
184#452—1072no—
185#450—1072no—
186#449—1072no—
187#447—1072no—
188#417D10711yescharacteristic, complemented
189C2×#102—1071no—
190#408—1071no—
191H(5,𝔽2)—2144no—
192H(5,𝔽2)—2144no—
193C2×#41—1071no—
194C2×#41—1072no—
195C2×#41—1071no—
196C2×#41—2141no—
197C2×#41C2×D10711yescharacteristic
198C22×D4—1072no—
199C22×D4—1071no—
200C22×D4—2141no—
201C22×D4—2142no—
202AGL(1,ℤ/8)—2144no—
203#135—2144no—
204C2×Q16—1071no—
205C2×SD16—1071no—
206C2×SD16—1071no—
207C2×SD16—1072no—
208C2×D8—1071no—
209C2×D8—1072no—
210C2×M16C2×D10711yescharacteristic, complemented
211C22×C8—1071no—
212C22⋊Q8—1072no—
213C22⋊Q8—2141no—
214#121—1072no—
215#121—2141no—
216#121—2142no—
217C2×C4⋊C4C2×D10711yescharacteristic, complemented
218#103C2×D10712yescomplemented
219#102—1072no—
220#102—1072no—
221#102C2×D10712yescomplemented
222C22⋊C8—1072no—
223C22⋊C8—1072no—
224C24—2142no—
225C24—2141no—
226#41—22no—
227#41—2148no—
228#41—2142no—
229#41—2142no—
230#41—2144no—
231#41—2144no—
232#41—2142no—
233#41—2142no—
234#41D42814yescomplemented
235C2×Q8—1071no—
236C2×Q8—1071no—
237C2×Q8C22×D10711yescharacteristic
238C2×D4—21no—
239C2×D4—1072no—
240C2×D4—1074no—
241C2×D4—1071no—
242C2×D4—1072no—
243C2×D4—1071no—
244C2×D4—2142no—
245C2×D4—2142no—
246C2×D4—2148no—
247C2×D4—2142no—
248C2×D4—2144no—
249C2×D4—21416no—
250C2×D4—2141no—
251C2×D4—2144no—
252C2×D4—2141no—
253C2×D4—2142no—
254C2×D4C22×D10711yescharacteristic
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—1072no—
258C22×C4—2142no—
259C22×C4—2141no—
260C22×C4C22×D10711yescharacteristic
261Q16—2142no—
262SD16—2144no—
263SD16—2142no—
264SD16—2142no—
265D8—2144no—
266D8—2142no—
267M16—22no—
268C2×C8—1071no—
269C2×C8—1071no—
270C2×C8—2142no—
271C2×C8C22×D10712yes—
272C4⋊C4—21no—
273C4⋊C4—2142no—
274C4⋊C4C22×D10712yes—
275C22⋊C4—2142no—
276C22⋊C4—2142no—
277C23—21no—
278C23—8561no—
279C23—8561no—
280C23—8562no—
281C23—8562no—
282C23—1072no—
283C23—2148no—
284C23—2144no—
285C23—2141no—
286C23—2142no—
287C23D4×D10711yescharacteristic
288Q8—21no—
289Q8—2141no—
290Q8—2141no—
291Q8—2142no—
292Q8C2×D42812yes—
293D4—24no—
294D4—21no—
295D4—8562no—
296D4—1074no—
297D4—4282no—
298D4—2148no—
299D4—2148no—
300D4—2144no—
301D4—2144no—
302D4—2142no—
303D4—2141no—
304D4—2141no—
305D4—2142no—
306D4C2×D42812yes—
307C2×C4—21no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—81no—
311C2×C4—1072no—
312C2×C4—4282no—
313C2×C4—2148no—
314C2×C4—2141no—
315C2×C4—2142no—
316C2×C4—2142no—
317C2×C4D4×D10711yescharacteristic
318C2×C4D4×D10712yes—
319C2×C4C2×D42812yes—
320C2×C4C23×D10711yescharacteristic
321C8—22no—
322C8—2141no—
323C8—2141no—
324C22—21no—
325C22—21no—
326C22—22no—
327C22—41no—
328C22—41no—
329C22—8564no—
330C22—8568no—
331C22—4281no—
332C22—4281no—
333C22—4281no—
334C22—4282no—
335C22—4282no—
336C22—2142no—
337C22—2142no—
338C22—2144no—
339C22—2142no—
340C22#1681207912yes—
341C22C2×D4×D10711yescharacteristic
342C4—22no—
343C4—42no—
344C4—4281no—
345C4—2142no—
346C4#1681207912yes—
347C4C2×D4×D10711yescharacteristic
348C4C22×D42811yescharacteristic
349C2—22no—
350C2—42no—
351C2—8561no—
352C2—4284no—
353C2—4282no—
354C2C2×#1681207911yescharacteristic
355C2#2029343111yescharacteristic
356C2#2029343611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#417×C107C2
socleC22×C107C22×D4
hypercenter#417D107
hypocenterC107#3012
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#16812155C2semidirect
2. C2×#16812137C2semidirect
3. C2×#16812129C2semidirect
4. #20293208C2semidirect
5. #20292792C2semidirect
6. #20292779C2semidirect
7. #20292775C2semidirect
8. #20292762C2semidirect
9. C2×#16812029C2semidirect
10. #20292595C2semidirect
11. #417×C107C2semidirect
17. C2×#13133818C22semidirect
24. C2×#13133808C22semidirect
25. C2×#13133807C22semidirect
26. C2×#13133806C22semidirect
27. C2×#13133805C22semidirect
28. C2×D856C22semidirect
29. C2×#13133803C22semidirect
30. C22×C107⋊C8C22semidirect
32. #16812089C22semidirect
34. #16812084C22semidirect
36. #16812029C22semidirect
37. #16812029C22semidirect
38. (C2×D107)⋊C8C22semidirect
39. (C2×D107)⋊C8C22semidirect
41. C2×M16×C107C22semidirect
42. C2×C4⋊C4×C107C22semidirect
43. #103×C107C22semidirect
44. #102×C107C22semidirect
62. C2×D428D4semidirect
87. #41×C107D4semidirect
179. C107#3012semidirect
188. #417D107semidirect
210. C2×M16C2×D107semidirect
217. C2×C4⋊C4C2×D107semidirect
218. #103C2×D107semidirect
221. #102C2×D107semidirect
234. #41D428semidirect

Maximal subgroups

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

subgroupindex
C2×#168121552
C2×#168121372
C2×#168121292
#202932082
#202927922
#202927792
#202927752
#202927622
C2×#168120292
#202925952
#417×C1072
#3012107