Finite Group Databasebeta

← #16832627

Subgroups — #16832627

Order 3712.

364 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 113 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×#11745281C211yesmaximal, characteristic, complemented
2#135×D29C211yesmaximal, characteristic, complemented
3#14271605C212yesmaximal, complemented
4#14271604C211yesmaximal, characteristic, complemented
5#14271601C211yesmaximal, characteristic, complemented
6#14271600C212yesmaximal, complemented
7#14271598C211yesmaximal, characteristic, complemented
8#14271596C212yesmaximal, complemented
9#14271594C212yesmaximal, complemented
10M32×D29C211yesmaximal, characteristic, complemented
11C29×#510C211yesmaximal, characteristic, complemented
12#41×D29—21no—
13C2×#90809—21no—
14C29×#135C2211yescharacteristic
15C2×Q16×C29C2211yescharacteristic, complemented
16#11745312—21no—
17C2×#90784—21no—
18#11745281—21no—
19#11745281—21no—
20#11745281C2212yes—
21#11745281C2212yescomplemented
22#11745281C2211yescharacteristic, complemented
23Q16×D29C2211yescharacteristic, complemented
24#11745278—21no—
25SD16×D29—21no—
26#11745274C2211yescharacteristic, complemented
27D8×D29C2211yescharacteristic, complemented
28#11745266C2211yescharacteristic
29C2×D232C2211yescharacteristic, complemented
30C2×C8×D29C2211yescharacteristic
31C29×Q32C2212yescomplemented
32C29×SD32C2212yescomplemented
33C29×M32C2211yescharacteristic, complemented
34#11745177C2212yescomplemented
35#11745176C2212yescomplemented
36#11745176C2212yescomplemented
37#11745174C2212yescomplemented
38#11745160C2211yescharacteristic, complemented
39#11745148C2212yescomplemented
40D464C2212yescomplemented
41#11745146C2212yescomplemented
42C16×D29C2212yescomplemented
43#41×C29—21no—
44C2×Q8×C29—21no—
45#90809—22no—
46#90809—22no—
47#90809—21no—
48#90809—41no—
49#90809—41no—
50Q8×D29—21no—
51#90807—21no—
52#90807—41no—
53D4×D29—21no—
54D4×D29—41no—
55#90805—21no—
56#90805—41no—
57C2×D116—21no—
58C2×D116—41no—
59C2×C4×D29—41no—
60C2×C4×D29—41no—
61C2×C4×D29D411yescharacteristic
62Q16×C29—21no—
63Q16×C29C2312yes—
64Q16×C29C2311yescharacteristic
65SD16×C29—21no—
66D8×C29C2311yescharacteristic
67C2×C8×C29C2311yescharacteristic
68#90785—21no—
69#90784—22no—
70#90784—22no—
71#90784—21no—
72#90783—21no—
73#90782—21no—
74C2×#6597D411yescharacteristic
75Dic116C2311yescharacteristic
76D232—21no—
77D232C2312yes—
78D232C2311yescharacteristic
79#90773—21no—
80C8×D29D411yescharacteristic
81C8×D29D411yescharacteristic
82C8×D29C2311yescharacteristic
83C8×D29C2311yescharacteristic
84C16×C29C2312yes—
85#90768C2312yes—
86C22×D29—41no—
87C22×D29—41no—
88C22×D29D811yescharacteristic
89Q8×C29—22no—
90Q8×C29—21no—
91Q8×C29—41no—
92D4×C29—21no—
93D4×C29—41no—
94C2×C4×C29—41no—
95C2×C4×C29—41no—
96C2×C4×C29C2×D411yescharacteristic
97C22⋊D29—41no—
98C22⋊D29—41no—
99C22⋊D29—41no—
100C2×Dic29—41no—
101C2×Dic29D811yescharacteristic
102D116—21no—
103D116—22no—
104D116—42no—
105D116—42no—
106D116—41no—
107D116—41no—
108C4×D29—41no—
109C4×D29—41no—
110C4×D29—41no—
111C4×D29—42no—
112C4×D29—42no—
113C4×D29D811yescharacteristic
114C4×D29D811yescharacteristic
115C4×D29C2×D411yescharacteristic
116C4×D29C2×D411yescharacteristic
117Dic58—21no—
118Dic58—41no—
119C8×C29C2×D411yescharacteristic
120C8×C29C2411yescharacteristic
121#6597C2×D411yescharacteristic
122#6597C2×D411yescharacteristic
123#3414—291nomaximal, Sylow
124C22×C29—41no—
125C22×C29C2×D811yescharacteristic
126C2×D29—41no—
127C2×D29—42no—
128C2×D29—41no—
129C2×D29—82no—
130C2×D29—82no—
131C2×D29C2×D811yescharacteristic
132C2×D29C2×D811yescharacteristic
133C4×C29—41no—
134C4×C29—42no—
135C4×C29C2×D811yescharacteristic
136C4×C29C22×D411yescharacteristic
137Dic29—41no—
138Dic29C2×D811yescharacteristic
139Dic29C2×D811yescharacteristic
140C2×#135—291no—
141C2×#135—291no—
142#510—291no—
143#510—292no—
144#510D2911yescharacteristic, complemented
145#509—291no—
146#509—292no—
147#509—291no—
148#508—292no—
149#508—292no—
150C2×M32—291no—
151C2×C29—21no—
152C2×C29—81no—
153C2×C29C22×D811yescharacteristic
154D29—21no—
155D29—21no—
156D29—82no—
157D29—81no—
158C2×#41—581no—
159C2×#41—581no—
160#135—292no—
161#135—291no—
162#135—292no—
163#135—291no—
164#135—291no—
165#135—581no—
166#135—581no—
167#135—581no—
168#135—581no—
169#135C2×D2911yescharacteristic
170C2×Q16—291no—
171C2×Q16C2×D2911yescharacteristic, complemented
172C2×SD16—581no—
173C2×SD16—581no—
174C2×D8—291no—
175C2×D8—291no—
176C22×C8—291no—
177Q32—292no—
178Q32C2×D2912yescomplemented
179SD32—292no—
180SD32—292no—
181SD32—292no—
182SD32C2×D2912yescomplemented
183D16—292no—
184D16—292no—
185M32—291no—
186M32—292no—
187M32C2×D2911yescharacteristic, complemented
188C2×C16—292no—
189C29#341411yescharacteristic, complemented, Sylow
190#41—21no—
191#41—581no—
192#41—582no—
193#41—581no—
194#41—582no—
195#41—581no—
196#41—1161no—
197#41—1161no—
198#41—1161no—
199#41—1161no—
200C2×Q8—21no—
201C2×Q8—581no—
202C2×D4—581no—
203C2×D4—581no—
204C2×D4—1161no—
205C2×D4—1161no—
206C22×C4—291no—
207C22×C4—1161no—
208C22×C4—1161no—
209Q16—21no—
210Q16—291no—
211Q16—581no—
212Q16C22×D2911yescharacteristic
213Q16C22×D2912yes—
214SD16—21no—
215SD16—581no—
216SD16—582no—
217SD16—581no—
218SD16—582no—
219SD16—581no—
220D8—292no—
221D8—291no—
222D8—581no—
223D8—581no—
224D8C22×D2911yescharacteristic
225C2×C8—291no—
226C2×C8—291no—
227C2×C8—291no—
228C2×C8—291no—
229C2×C8—291no—
230C2×C8C22×D2911yescharacteristic
231C16—292no—
232C16C22×D2912yes—
233C23—291no—
234C23—1161no—
235C23—1161no—
236Q8—21no—
237Q8—22no—
238Q8—41no—
239Q8—581no—
240Q8—1161no—
241D4—21no—
242D4—41no—
243D4—581no—
244D4—582no—
245D4—1161no—
246D4—1162no—
247D4—1162no—
248D4—1161no—
249D4—1161no—
250D4—1161no—
251D4—1161no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—291no—
255C2×C4—291no—
256C2×C4—291no—
257C2×C4—291no—
258C2×C4—291no—
259C2×C4—1161no—
260C2×C4—1162no—
261C2×C4—1162no—
262C2×C4—1161no—
263C2×C4—1161no—
264C2×C4—1161no—
265C2×C4D4×D2911yescharacteristic
266C8—291no—
267C8—291no—
268C8D4×D2911yescharacteristic
269C8C23×D2911yescharacteristic
270C22—41no—
271C22—291no—
272C22—291no—
273C22—1161no—
274C22—1162no—
275C22—1161no—
276C22—2322no—
277C22—2322no—
278C22D8×D2911yescharacteristic
279C4—41no—
280C4—42no—
281C4—291no—
282C4—291no—
283C4—1161no—
284C4D8×D2911yescharacteristic
285C4C2×D4×D2911yescharacteristic
286C2—21no—
287C2—81no—
288C2—581no—
289C2—581no—
290C2—2322no—
291C2—2321no—
292C2C2×D8×D2911yescharacteristic
subgroupG/H
centerC2C2×D8×D29
commutator, cosocleC8×C29C24
FrattiniC8C23×D29
FittingC29×#510C2
socleC2×C29C22×D8
hypercenter#510D29
hypocenterC29#3414
LCSG ▷ C8×C29 ▷ C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #510

Complements

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

NH ≅ G/Ntype
1. C2×#11745281C2semidirect
2. #135×D29C2semidirect
3. #14271605C2semidirect
4. #14271604C2semidirect
5. #14271601C2semidirect
6. #14271600C2semidirect
7. #14271598C2semidirect
8. #14271596C2semidirect
9. #14271594C2semidirect
10. M32×D29C2semidirect
11. C29×#510C2semidirect
15. C2×Q16×C29C22semidirect
21. #11745281C22semidirect
22. #11745281C22semidirect
23. Q16×D29C22semidirect
26. #11745274C22semidirect
27. D8×D29C22semidirect
29. C2×D232C22semidirect
31. C29×Q32C22semidirect
32. C29×SD32C22semidirect
33. C29×M32C22semidirect
34. #11745177C22semidirect
35. #11745176C22semidirect
36. #11745176C22semidirect
37. #11745174C22semidirect
38. #11745160C22semidirect
39. #11745148C22semidirect
40. D464C22semidirect
41. #11745146C22semidirect
42. C16×D29C22semidirect
144. #510D29semidirect
171. C2×Q16C2×D29semidirect
178. Q32C2×D29semidirect
182. SD32C2×D29semidirect
187. M32C2×D29semidirect
189. C29#3414semidirect

Maximal subgroups

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

subgroupindex
C2×#117452812
#135×D292
#142716052
#142716042
#142716012
#142716002
#142715982
#142715962
#142715942
M32×D292
C29×#5102
#341429