Finite Group Databasebeta

← #16829941

Subgroups — #16829941

Order 3712.

656 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 141 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×#11745273C212yesmaximal, complemented
3#14271712C211yesmaximal, characteristic, complemented
4#127×D29C211yesmaximal, characteristic, complemented
5#14271553C211yesmaximal, characteristic, complemented
6#14271552C211yesmaximal, characteristic, complemented
7#14271550C211yesmaximal, characteristic, complemented
8#14271538C212yesmaximal, complemented
9#14271534C212yesmaximal, complemented
10#14271115C211yesmaximal, characteristic, complemented
11C29×#496C211yesmaximal, characteristic, complemented
12C2×#90809C2211yescharacteristic
13C2×#90807C2211yescharacteristic
14C2×D4×D29—22no—
15C2×D4×D29C2212yes—
16C2×SD16×C29C2212yescomplemented
17C2×D8×C29C2212yescomplemented
18C29×#127C2211yescharacteristic
19C29×#124C2211yescharacteristic, complemented
20#11745306C2211yescharacteristic, complemented
21C2×#90784C2212yescomplemented
22#11745302—22no—
23#11745302C2212yes—
24#11745302C2211yescharacteristic, complemented
25C2×#90783C2212yescomplemented
26C2×#90782C2212yescomplemented
27C2×#90782C2212yescomplemented
28#11745276—28no—
29#11745273—28no—
30C2×D232C2212yescomplemented
31C2×#90773C2212yescomplemented
32C2×#90772C2212yes—
33#11745246—22no—
34#11745243—22no—
35#11745237C2211yescharacteristic, complemented
36#11745236C2211yescharacteristic
37C42×D29C2211yescharacteristic
38C29×#97C2211yescharacteristic, complemented
39C8×Dic29C2212yescomplemented
40#11745151C2211yescharacteristic, complemented
41C23×D29—22no—
42C2×Q8×C29C2311yescharacteristic
43C2×D4×C29—22no—
44C2×D4×C29C2311yescharacteristic
45C2×D4×C29C2312yes—
46C2×C22⋊D29—21no—
47C2×C22⋊D29—22no—
48C2×C22⋊D29—24no—
49C2×C22⋊D29—22no—
50C22×Dic29—21no—
51#90809—24no—
52#90807—24no—
53D4×D29—24no—
54D4×D29—24no—
55D4×D29—48no—
56C2×D116—21no—
57C2×D116—22no—
58C2×D116C2311yescharacteristic
59C2×D116C2312yes—
60C2×C4×D29—21no—
61C2×C4×D29D412yes—
62C2×C4×D29C2311yescharacteristic
63C2×Dic58C2311yescharacteristic
64SD16×C29—24no—
65D8×C29—24no—
66C2×C8×C29C2312yes—
67C22⋊C4×C29—22no—
68C42×C29C2311yescharacteristic
69#90784—24no—
70#90783—24no—
71#90782—24no—
72#90782—24no—
73#90781—22no—
74#90779—22no—
75C4×Dic29D412yes—
76C4×Dic29C2311yescharacteristic
77C2×#6597C2312yes—
78D232—24no—
79#90773—24no—
80#90772D418yescomplemented
81C23×C29—21no—
82C23×C29—22no—
83C22×D29—21no—
84C22×D29—22no—
85C22×D29—48no—
86C22×D29—44no—
87C22×D29C2×D411yescharacteristic
88Q8×C29—22no—
89D4×C29—22no—
90D4×C29—22no—
91D4×C29—22no—
92D4×C29—44no—
93C2×C4×C29—21no—
94C2×C4×C29C2×D412yes—
95C2×C4×C29C2411yescharacteristic
96C22⋊D29—42no—
97C22⋊D29—44no—
98C22⋊D29—48no—
99C22⋊D29—44no—
100C2×Dic29—21no—
101C2×Dic29—42no—
102C2×Dic29C2×D411yescharacteristic
103C2×Dic29C2×D412yes—
104D116—22no—
105D116—22no—
106D116—22no—
107D116—44no—
108D116—42no—
109C4×D29—24no—
110C4×D29—42no—
111C4×D29C2×D414yes—
112Dic58—22no—
113C8×C29C2×D414yescomplemented
114#6597C2×D414yescomplemented
115#3151—291nomaximal, Sylow
116C22×C29—41no—
117C22×C29—42no—
118C22×C29—44no—
119C22×C29—42no—
120C22×C29C22×D411yescharacteristic
121C2×D29—22no—
122C2×D29—41no—
123C2×D29—42no—
124C2×D29—44no—
125C2×D29—42no—
126C2×D29—88no—
127C2×D29#12712yes—
128C4×C29—22no—
129C4×C29—41no—
130C4×C29C22×D412yes—
131Dic29—41no—
132Dic29#12712yes—
133Dic29AGL(1,ℤ/8)14yescomplemented
134C2×AGL(1,ℤ/8)—292no—
135C2×AGL(1,ℤ/8)—292no—
136#532—291no—
137C2×#127—291no—
138#496—291no—
139#496—291no—
140#496—291no—
141#496D2911yescharacteristic, complemented
142#493—292no—
143#492—292no—
144C4×M16—291no—
145C2×C29—82no—
146C2×C29—81no—
147C2×C29C2×#12711yescharacteristic
148C2×C29C2×AGL(1,ℤ/8)12yes—
149D29—22no—
150D29—81no—
151D29—82no—
152C2×#41—291no—
153C2×#41—291no—
154C22×D4—292no—
155C22×D4—582no—
156AGL(1,ℤ/8)—588no—
157AGL(1,ℤ/8)—588no—
158C2×SD16—292no—
159C2×SD16—292no—
160C2×SD16—292no—
161C2×SD16C2×D2912yescomplemented
162C2×D8—292no—
163C2×D8—292no—
164C2×D8—292no—
165C2×D8C2×D2912yescomplemented
166C2×M16—292no—
167C4⋊Q8—291no—
168#127—292no—
169#127—291no—
170#127—291no—
171#127—582no—
172#127C2×D2911yescharacteristic
173#124—291no—
174#124C2×D2911yescharacteristic, complemented
175#121—582no—
176C4×D4—582no—
177C2×C42—291no—
178#97—291no—
179#97C2×D2911yescharacteristic, complemented
180C4×C8—292no—
181C29#315111yescharacteristic, complemented, Sylow
182C24—582no—
183#41—584no—
184#41—584no—
185C2×Q8—291no—
186C2×Q8C22×D2911yescharacteristic
187C2×D4—22no—
188C2×D4—291no—
189C2×D4—292no—
190C2×D4—581no—
191C2×D4—584no—
192C2×D4—584no—
193C2×D4—582no—
194C2×D4—584no—
195C2×D4—582no—
196C2×D4—581no—
197C2×D4—582no—
198C2×D4—1168no—
199C2×D4C22×D2911yescharacteristic
200C2×D4C22×D2912yes—
201C22×C4—291no—
202C22×C4—292no—
203C22×C4—581no—
204C22×C4—581no—
205SD16—24no—
206SD16—584no—
207SD16—584no—
208SD16—584no—
209D8—24no—
210D8—584no—
211D8—584no—
212D8—584no—
213M16—298no—
214C2×C8—292no—
215C2×C8C22×D2912yes—
216C4⋊C4—582no—
217C22⋊C4—22no—
218C22⋊C4—582no—
219C42—292no—
220C42—291no—
221C42C22×D2911yescharacteristic
222C23—21no—
223C23—22no—
224C23—291no—
225C23—581no—
226C23—582no—
227C23—1164no—
228C23—1168no—
229Q8—22no—
230Q8—582no—
231D4—22no—
232D4—22no—
233D4—22no—
234D4—44no—
235D4—582no—
236D4—582no—
237D4—582no—
238D4—1164no—
239D4—1162no—
240D4—1164no—
241D4—1168no—
242D4—1164no—
243D4—1162no—
244C2×C4—21no—
245C2×C4—292no—
246C2×C4—291no—
247C2×C4—294no—
248C2×C4—584no—
249C2×C4—581no—
250C2×C4—1162no—
251C2×C4—1162no—
252C2×C4D4×D2912yes—
253C2×C4C23×D2911yescharacteristic
254C8—294no—
255C8D4×D2914yescomplemented
256C22—44no—
257C22—42no—
258C22—41no—
259C22—42no—
260C22—292no—
261C22—582no—
262C22—1164no—
263C22—1162no—
264C22—1161no—
265C22—1162no—
266C22—2328no—
267C22C2×D4×D2911yescharacteristic
268C4—22no—
269C4—41no—
270C4—294no—
271C4—292no—
272C4—1161no—
273C4C2×D4×D2912yes—
274C2—82no—
275C2—81no—
276C2—582no—
277C2—2322no—
278C2—2321no—
279C2#127×D2911yescharacteristic
280C2AGL(1,ℤ/8)×D2912yes—
subgroupG/H
centerC22C2×D4×D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#496C2
socleC22×C29C22×D4
hypercenter#496D29
hypocenterC29#3151
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#11745276C2semidirect
2. C2×#11745273C2semidirect
3. #14271712C2semidirect
4. #127×D29C2semidirect
5. #14271553C2semidirect
6. #14271552C2semidirect
7. #14271550C2semidirect
8. #14271538C2semidirect
9. #14271534C2semidirect
10. #14271115C2semidirect
11. C29×#496C2semidirect
16. C2×SD16×C29C22semidirect
17. C2×D8×C29C22semidirect
19. C29×#124C22semidirect
20. #11745306C22semidirect
21. C2×#90784C22semidirect
24. #11745302C22semidirect
25. C2×#90783C22semidirect
26. C2×#90782C22semidirect
27. C2×#90782C22semidirect
30. C2×D232C22semidirect
31. C2×#90773C22semidirect
35. #11745237C22semidirect
38. C29×#97C22semidirect
39. C8×Dic29C22semidirect
40. #11745151C22semidirect
80. #90772D4semidirect
113. C8×C29C2×D4semidirect
114. #6597C2×D4semidirect
133. Dic29AGL(1,ℤ/8)semidirect
141. #496D29semidirect
161. C2×SD16C2×D29semidirect
165. C2×D8C2×D29semidirect
174. #124C2×D29semidirect
179. #97C2×D29semidirect
181. C29#3151semidirect
255. C8D4×D29semidirect

Maximal subgroups

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

subgroupindex
C2×#117452762
C2×#117452732
#142717122
#127×D292
#142715532
#142715522
#142715502
#142715382
#142715342
#142711152
C29×#4962
#315129