Finite Group Databasebeta

← #49170551

Subgroups — #49170551

Order 37504.

464 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#34036935C212yesmaximal, complemented
2#34036930C212yesmaximal, complemented
3#34036396C212yesmaximal, complemented
4#34036389C212yesmaximal, complemented
5#34036306C211yesmaximal, characteristic, complemented
6#34036301C211yesmaximal, characteristic, complemented
7#34036296C212yesmaximal, complemented
8#34036252C212yesmaximal, complemented
9#496×C293C211yesmaximal, characteristic, complemented
10#22627672C2211yescharacteristic, complemented
11#22627669C2211yescharacteristic
12#22627669C2212yescomplemented
13#22627655C2212yescomplemented
14#22627653—22no—
15#22627651—22no—
16#22627650C2212yes—
17#22627650C2212yescomplemented
18#22627650C2212yescomplemented
19#22627648—22no—
20#22627648—22no—
21#22627646C2212yes—
22#22627645—22no—
23#22627643—22no—
24SD16×D293—22no—
25D8×D293—22no—
26#22627630C2212yescomplemented
27#22627630C2212yescomplemented
28#22627560C2212yescomplemented
29#22627552C2212yescomplemented
30#22627552C2212yescomplemented
31#22627543C2211yescharacteristic, complemented
32#22627541C2212yescomplemented
33#22627541C2211yescharacteristic, complemented
34C2×SD16×C293C2212yescomplemented
35C2×D8×C293C2212yescomplemented
36#127×C293C2211yescharacteristic, complemented
37#124×C293C2211yescharacteristic, complemented
38#97×C293C2211yescharacteristic, complemented
39#17989803—22no—
40#17989802—41no—
41#17989802C2312yes—
42#17989802C2312yes—
43#17989801—22no—
44#17989801—22no—
45#17989801—22no—
46Q8×D293—22no—
47C2×C22⋊D293—41no—
48C2×C22⋊D293—42no—
49D4×D293—22no—
50D4×D293—22no—
51D4×D293—22no—
52Dic1172D412yescomplemented
53#17989792—22no—
54#17989791D412yescomplemented
55#17989791D412yescomplemented
56#17989790—22no—
57#17989790—22no—
58#17989790—22no—
59#17989789—22no—
60D2344D412yescomplemented
61#17989787—22no—
62#17989787—22no—
63#17989787—22no—
64#17989786C2312yes—
65#17989786C2312yes—
66#17989785—22no—
67#17989785—22no—
68C8×D293—22no—
69C8×D293—22no—
70C2×Q8×C293C2311yescharacteristic
71C2×D4×C293—41no—
72C2×D4×C293C2312yes—
73C2×D4×C293C2311yescharacteristic
74SD16×C293—22no—
75SD16×C293—22no—
76D8×C293—22no—
77D8×C293—22no—
78C2×C8×C293C2312yes—
79C22⋊C4×C293—41no—
80C42×C293C2311yescharacteristic
81C22×D293—42no—
82C22×D293—41no—
83Dic586—41no—
84Dic586C2×D412yes—
85Dic586C2×D412yes—
86D1172—41no—
87D1172C2×D412yes—
88D1172C2×D412yes—
89C22⋊D293—22no—
90C22⋊D293—22no—
91C22⋊D293—42no—
92C22⋊D293—42no—
93C22⋊D293—42no—
94C2×Dic293—42no—
95C2×Dic293—41no—
96C4×D293—22no—
97C4×D293—22no—
98C4×D293—42no—
99#14556873—22no—
100#14556873—22no—
101C23×C293—21no—
102C23×C293—22no—
103Q8×C293—22no—
104D4×C293—22no—
105D4×C293—22no—
106D4×C293—22no—
107D4×C293—82no—
108C2×C4×C293—21no—
109C2×C4×C293—21no—
110C2×C4×C293C2411yescharacteristic
111C8×C293C2×D412yes—
112C8×C293C2×D412yes—
113C22×C293—41no—
114C22×C293—42no—
115C22×C293—82no—
116C22×C293—81no—
117C22×C293H(5,𝔽2)11yescharacteristic
118C2×D293—22no—
119C2×D293—22no—
120C2×D293—162no—
121C2×D293—161no—
122C4×C293—41no—
123C4×C293—41no—
124C4×C293C22×D412yes—
125Dic293—22no—
126Dic293—22no—
127C2×C293—21no—
128C2×C293—82no—
129C2×C293—81no—
130C2×C293D4211yescharacteristic
131D293—42no—
132D293—42no—
133C293#328811yescharacteristic, complemented, Sylow
134#3288—2931nomaximal, Sylow
135#577—2932no—
136#576—2932no—
137#496D29311yescharacteristic, complemented
138#472—2932no—
139#471—2932no—
140#455—2931no—
141#454—2931no—
142#453—2932no—
143#444—2932no—
1442−1+4—2931no—
145H(5,𝔽2)—2931no—
146H(5,𝔽2)—2932no—
147#137—2932no—
148#137—5862no—
149AGL(1,ℤ/8)—2932no—
150AGL(1,ℤ/8)—2932no—
151AGL(1,ℤ/8)—2932no—
152AGL(1,ℤ/8)—5862no—
153AGL(1,ℤ/8)—5862no—
154AGL(1,ℤ/8)—5862no—
155#135—2932no—
156#135—5862no—
157#135—5862no—
158C2×SD16—5862no—
159C2×SD16C2×D29312yescomplemented
160C2×D8—5862no—
161C2×D8C2×D29312yescomplemented
162#131—2932no—
163#131—2932no—
164#127C2×D29311yescharacteristic, complemented
165#124C2×D29311yescharacteristic, complemented
166#108—2932no—
167#104—2932no—
168#104—2932no—
169#101—2931no—
170#100—2932no—
171#100—2931no—
172#97C2×D29311yescharacteristic, complemented
173#41—2932no—
174#41—2932no—
175#41—5862no—
176#41—5862no—
177#41—5862no—
178#41—5862no—
179#41—11721no—
180C2×Q8—5862no—
181C2×Q8C22×D29311yescharacteristic
182C2×D4—41no—
183C2×D4—5862no—
184C2×D4—5862no—
185C2×D4—5862no—
186C2×D4—11721no—
187C2×D4—11722no—
188C2×D4C22×D29311yescharacteristic
189C2×D4C22×D29312yes—
190Q16—2932no—
191Q16—5862no—
192SD16—22no—
193SD16—22no—
194SD16—2932no—
195SD16—2932no—
196SD16—5862no—
197SD16—5862no—
198SD16—5862no—
199SD16—5862no—
200D8—22no—
201D8—22no—
202D8—2932no—
203D8—5862no—
204D8—5862no—
205D8—5862no—
206M16—2932no—
207M16—2932no—
208M16—5862no—
209M16—5862no—
210C2×C8—5862no—
211C2×C8—5862no—
212C2×C8C22×D29312yes—
213C22⋊C4—41no—
214C42C22×D29311yescharacteristic
215C23—21no—
216C23—22no—
217C23—11722no—
218C23—11721no—
219Q8—22no—
220Q8—2932no—
221Q8—2932no—
222Q8—11721no—
223D4—22no—
224D4—22no—
225D4—22no—
226D4—82no—
227D4—2932no—
228D4—2932no—
229D4—5862no—
230D4—5862no—
231D4—11721no—
232D4—11722no—
233D4—11722no—
234D4—11722no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—5862no—
238C2×C4—5862no—
239C2×C4—11722no—
240C2×C4—11721no—
241C2×C4—11722no—
242C2×C4C23×D29311yescharacteristic
243C8—5862no—
244C8—5862no—
245C8D4×D29312yes—
246C8D4×D29312yes—
247C22—41no—
248C22—42no—
249C22—82no—
250C22—81no—
251C22—5862no—
252C22—5862no—
253C22—46881no—
254C22—46882no—
255C22#2262766911yescharacteristic
256C4—41no—
257C4—41no—
258C4—5862no—
259C4—5862no—
260C4C2×D4×D29312yes—
261C2—21no—
262C2—82no—
263C2—81no—
264C2—11722no—
265C2—11722no—
266C2#3403676411yescharacteristic
subgroupG/H
centerC2#34036764
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#496×C293C2
socleC2×C293D42
hypercenter#496D293
hypocenterC293#3288
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C2 ◁ C2×C4 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #34036935C2semidirect
2. #34036930C2semidirect
3. #34036396C2semidirect
4. #34036389C2semidirect
5. #34036306C2semidirect
6. #34036301C2semidirect
7. #34036296C2semidirect
8. #34036252C2semidirect
9. #496×C293C2semidirect
10. #22627672C22semidirect
12. #22627669C22semidirect
13. #22627655C22semidirect
17. #22627650C22semidirect
18. #22627650C22semidirect
26. #22627630C22semidirect
27. #22627630C22semidirect
28. #22627560C22semidirect
29. #22627552C22semidirect
30. #22627552C22semidirect
31. #22627543C22semidirect
32. #22627541C22semidirect
33. #22627541C22semidirect
34. C2×SD16×C293C22semidirect
35. C2×D8×C293C22semidirect
36. #127×C293C22semidirect
37. #124×C293C22semidirect
38. #97×C293C22semidirect
52. Dic1172D4semidirect
54. #17989791D4semidirect
55. #17989791D4semidirect
60. D2344D4semidirect
133. C293#3288semidirect
137. #496D293semidirect
159. C2×SD16C2×D293semidirect
161. C2×D8C2×D293semidirect
164. #127C2×D293semidirect
165. #124C2×D293semidirect
172. #97C2×D293semidirect

Maximal subgroups

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

subgroupindex
#340369352
#340369302
#340363962
#340363892
#340363062
#340363012
#340362962
#340362522
#496×C2932
#3288293