Finite Group Databasebeta

← #16829974

Subgroups — #16829974

Order 3712.

596 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 137 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×#11745276C212yesmaximal, complemented
2C2×#11745281C211yesmaximal, characteristic, complemented
3C2×D8×D29C211yesmaximal, characteristic, complemented
4#14271731C212yesmaximal, complemented
5#14271553C212yesmaximal, complemented
6#14271549C212yesmaximal, complemented
7#14271548C211yesmaximal, characteristic, complemented
8#14271540C211yesmaximal, characteristic, complemented
9#14271538C211yesmaximal, characteristic, complemented
10#14271117C211yesmaximal, characteristic, complemented
11C29×#495C211yesmaximal, characteristic, complemented
12C2×#90809C2212yes—
13C2×D4×D29C2212yes—
14C2×Q16×C29C2211yescharacteristic, complemented
15C2×SD16×C29C2212yescomplemented
16C2×D8×C29C2211yescharacteristic
17C29×#124C2212yescomplemented
18#11745309C2212yescomplemented
19C2×#90784C2212yescomplemented
20C2×#90784C2212yescomplemented
21#11745302C2212yescomplemented
22C2×#90782C2212yes—
23C2×#90782C2212yescomplemented
24#11745281—24no—
25#11745276—28no—
26D8×D29—24no—
27C2×D232C2211yescharacteristic
28C2×D232C2211yescharacteristic, complemented
29C2×D232C2212yescomplemented
30C2×#90772C2211yescharacteristic
31C2×C8×D29C2211yescharacteristic
32#11745246—24no—
33#11745244—24no—
34#11745236C2212yescomplemented
35#11745234C2211yescharacteristic
36C4×C8×C29C2211yescharacteristic, complemented
37#11745163C2211yescharacteristic, complemented
38C8×Dic29C2211yescharacteristic, complemented
39#11745151C2211yescharacteristic, complemented
40C23×D29—22no—
41C2×Q8×C29C2312yes—
42C2×D4×C29C2312yes—
43C2×C22⋊D29—22no—
44C2×C22⋊D29—24no—
45#90809—24no—
46#90809—24no—
47D4×D29—24no—
48D4×D29—24no—
49C2×D116—22no—
50C2×D116—24no—
51C2×D116C2312yes—
52C2×D116C2312yes—
53C2×C4×D29—22no—
54C2×C4×D29C2311yescharacteristic
55Q16×C29—22no—
56SD16×C29—24no—
57D8×C29—22no—
58C2×C8×C29C2311yescharacteristic
59C2×C8×C29C2311yescharacteristic
60C22⋊C4×C29—24no—
61C42×C29C2311yescharacteristic
62#90784—24no—
63#90784—24no—
64#90782—24no—
65#90782—24no—
66#90781—24no—
67#90781D412yes—
68#90779D412yes—
69C4×Dic29C2311yescharacteristic
70C2×#6597C2311yescharacteristic
71C2×#6597C2311yescharacteristic
72D232—22no—
73D232—22no—
74D232—24no—
75#90772D414yescomplemented
76C8×D29D414yescomplemented
77C23×C29—22no—
78C22×D29—22no—
79C22×D29—22no—
80C22×D29—44no—
81C22×D29—84no—
82C22×D29C2×D411yescharacteristic
83Q8×C29—22no—
84Q8×C29—22no—
85D4×C29—22no—
86D4×C29—22no—
87C2×C4×C29—22no—
88C2×C4×C29C2×D412yes—
89C2×C4×C29C2411yescharacteristic
90C22⋊D29—44no—
91C22⋊D29—84no—
92C2×Dic29C2×D412yes—
93C2×Dic29C2×D411yescharacteristic
94D116—22no—
95D116—22no—
96D116—22no—
97D116—22no—
98D116—44no—
99D116—84no—
100C4×D29—44no—
101C4×D29C2×D412yes—
102C4×D29C2×D412yes—
103C8×C29C2×D412yes—
104C8×C29C2×D412yes—
105#6597C2×D412yes—
106#6597C2×D412yes—
107#3155—291nomaximal, Sylow
108C22×C29—42no—
109C22×C29—44no—
110C22×C29C22×D411yescharacteristic
111C2×D29—22no—
112C2×D29—42no—
113C2×D29—42no—
114C2×D29—44no—
115C2×D29—44no—
116C2×D29—88no—
117C2×D29#12712yes—
118C4×C29—22no—
119C4×C29—42no—
120C4×C29C22×D411yescharacteristic
121C4×C29C22×D411yescharacteristic
122Dic29—22no—
123Dic29#12712yes—
124C2×AGL(1,ℤ/8)—292no—
125C2×#135—291no—
126C22×D8—291no—
127#535—292no—
128#496—292no—
129#496—292no—
130#495—291no—
131#495D2911yescharacteristic, complemented
132#493—291no—
133#493—291no—
134#405—291no—
135C2×C29—82no—
136C2×C29C2×#12711yescharacteristic
137C2×C29#57612yes—
138D29—22no—
139D29—82no—
140D29—82no—
141C2×#41—292no—
142C22×D4—292no—
143AGL(1,ℤ/8)—588no—
144#135—584no—
145C2×Q16C2×D2911yescharacteristic, complemented
146C2×SD16—292no—
147C2×SD16—292no—
148C2×SD16C2×D2912yescomplemented
149C2×D8—292no—
150C2×D8—292no—
151C2×D8—291no—
152C2×D8—291no—
153C2×D8—292no—
154C2×D8—584no—
155C2×D8C2×D2911yescharacteristic
156C2×M16—291no—
157C22×C8—291no—
158#127—292no—
159#127—292no—
160#124—292no—
161#124C2×D2912yescomplemented
162#121—584no—
163#120—584no—
164#117—291no—
165#97—291no—
166#97—291no—
167C4×C8—291no—
168C4×C8C2×D2911yescharacteristic, complemented
169C29#315511yescharacteristic, complemented, Sylow
170C24—582no—
171#41—584no—
172#41—584no—
173C2×Q8C22×D2912yes—
174C2×D4—292no—
175C2×D4—292no—
176C2×D4—582no—
177C2×D4—584no—
178C2×D4—584no—
179C2×D4—582no—
180C2×D4—584no—
181C2×D4—584no—
182C2×D4C22×D2912yes—
183C22×C4—291no—
184C22×C4—582no—
185Q16—22no—
186SD16—24no—
187SD16—584no—
188SD16—584no—
189D8—22no—
190D8—584no—
191D8—584no—
192D8—582no—
193D8—582no—
194D8—584no—
195M16—294no—
196C2×C8—291no—
197C2×C8—291no—
198C2×C8—294no—
199C2×C8C22×D2911yescharacteristic
200C2×C8C22×D2911yescharacteristic
201C4⋊C4—292no—
202C22⋊C4—24no—
203C22⋊C4—292no—
204C22⋊C4—584no—
205C42—291no—
206C42C22×D2911yescharacteristic
207C23—22no—
208C23—291no—
209C23—582no—
210C23—582no—
211C23—1164no—
212C23—2324no—
213Q8—22no—
214Q8—22no—
215D4—22no—
216D4—22no—
217D4—582no—
218D4—582no—
219D4—582no—
220D4—582no—
221D4—1164no—
222D4—1164no—
223D4—2324no—
224D4—2324no—
225C2×C4—22no—
226C2×C4—291no—
227C2×C4—292no—
228C2×C4—292no—
229C2×C4—292no—
230C2×C4—1164no—
231C2×C4D4×D2912yes—
232C2×C4C23×D2911yescharacteristic
233C8—292no—
234C8—292no—
235C8D4×D2912yes—
236C8D4×D2912yes—
237C22—42no—
238C22—44no—
239C22—292no—
240C22—582no—
241C22—1162no—
242C22—1162no—
243C22—1164no—
244C22—1164no—
245C22—2328no—
246C22C2×D4×D2911yescharacteristic
247C4—22no—
248C4—42no—
249C4—292no—
250C4—582no—
251C4C2×D4×D2911yescharacteristic
252C4C2×D4×D2911yescharacteristic
253C2—82no—
254C2—582no—
255C2—2322no—
256C2—2322no—
257C2#127×D2911yescharacteristic
258C2#1427196512yes—
subgroupG/H
centerC22C2×D4×D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#495C2
socleC22×C29C22×D4
hypercenter#495D29
hypocenterC29#3155
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. C2×#11745276C2semidirect
2. C2×#11745281C2semidirect
3. C2×D8×D29C2semidirect
4. #14271731C2semidirect
5. #14271553C2semidirect
6. #14271549C2semidirect
7. #14271548C2semidirect
8. #14271540C2semidirect
9. #14271538C2semidirect
10. #14271117C2semidirect
11. C29×#495C2semidirect
14. C2×Q16×C29C22semidirect
15. C2×SD16×C29C22semidirect
17. C29×#124C22semidirect
18. #11745309C22semidirect
19. C2×#90784C22semidirect
20. C2×#90784C22semidirect
21. #11745302C22semidirect
23. C2×#90782C22semidirect
28. C2×D232C22semidirect
29. C2×D232C22semidirect
34. #11745236C22semidirect
36. C4×C8×C29C22semidirect
37. #11745163C22semidirect
38. C8×Dic29C22semidirect
39. #11745151C22semidirect
75. #90772D4semidirect
76. C8×D29D4semidirect
131. #495D29semidirect
145. C2×Q16C2×D29semidirect
148. C2×SD16C2×D29semidirect
161. #124C2×D29semidirect
168. C4×C8C2×D29semidirect
169. C29#3155semidirect

Maximal subgroups

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

subgroupindex
C2×#117452762
C2×#117452812
C2×D8×D292
#142717312
#142715532
#142715492
#142715482
#142715402
#142715382
#142711172
C29×#4952
#315529