Finite Group Databasebeta

← #49170580

Subgroups — #49170580

Order 37504.

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

normal subgroups only

This group has 460 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#34036940C212yesmaximal, complemented
2#34036939C212yesmaximal, complemented
3#34036501C211yesmaximal, characteristic, complemented
4#34036391C212yesmaximal, complemented
5#34036390C212yesmaximal, complemented
6#34036313C212yesmaximal, complemented
7#34036296C211yesmaximal, characteristic, complemented
8#34036243C212yesmaximal, complemented
9#453×C293C211yesmaximal, characteristic, complemented
10#22627671C2212yes—
11#22627669C2211yescharacteristic, complemented
12#22627657C2212yescomplemented
13#22627654—22no—
14#22627654C2212yescomplemented
15#22627653—22no—
16#22627650—22no—
17#22627650C2212yes—
18#22627648—22no—
19#22627644—22no—
20#22627644C2212yes—
21#22627643—22no—
22C2×#17989790—22no—
23C2×#17989790C2212yescomplemented
24C2×#17989790C2212yescomplemented
25SD16×D293—22no—
26#22627632C2212yescomplemented
27#22627631C2212yescomplemented
28#22627624C2211yescharacteristic, complemented
29#22627559C2212yescomplemented
30#22627552C2212yescomplemented
31#22627542C2212yescomplemented
32#22627541C2211yescharacteristic, complemented
33C4×#14556873C2211yescharacteristic, complemented
34H(5,𝔽2)×C293C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C293C2212yes—
36#127×C293C2211yescharacteristic
37#104×C293C2212yescomplemented
38#100×C293C2211yescharacteristic, complemented
39#17989802—22no—
40#17989802C2312yes—
41#17989801—22no—
42#17989801—22no—
43#17989801—42no—
44C2×Dic586—22no—
45C2×Dic586C2312yes—
46Q8×D293—22no—
47C2×C22⋊D293—41no—
48D4×D293—22no—
49Dic1172—22no—
50#17989792—22no—
51#17989792D412yescomplemented
52#17989791—22no—
53#17989790—22no—
54#17989790—22no—
55#17989790—22no—
56#17989790—22no—
57#17989790—22no—
58#17989790—22no—
59#17989790D412yescomplemented
60#17989789D412yescomplemented
61#17989787—22no—
62#17989787D412yescomplemented
63#17989786—22no—
64#17989786C2312yes—
65#17989785—22no—
66C2×#14556873—22no—
67C2×#14556873C2312yes—
68C8×D293—22no—
69C4⋊Dic293—41no—
70#41×C293—22no—
71#41×C293C2312yes—
72C2×D4×C293—22no—
73C2×D4×C293—41no—
74C2×D4×C293—41no—
75C2×D4×C293C2311yescharacteristic
76C2×D4×C293C2311yescharacteristic
77SD16×C293—22no—
78D8×C293—22no—
79M16×C293C2312yes—
80C42×C293C2311yescharacteristic
81C22×D293—41no—
82Dic586—22no—
83Dic586—22no—
84Dic586—42no—
85Dic586C2×D412yes—
86D1172C2×D412yes—
87C22⋊D293—22no—
88C22⋊D293—42no—
89C22⋊D293—42no—
90C2×Dic293—22no—
91C2×Dic293—41no—
92C2×Dic293—42no—
93C4×D293—22no—
94C4×D293—42no—
95#14556873—22no—
96#14556873C2×D412yes—
97#14556873C2×D412yes—
98C23×C293—21no—
99C23×C293—21no—
100C23×C293—41no—
101Q8×C293C2×D412yes—
102D4×C293—22no—
103D4×C293—22no—
104D4×C293—22no—
105D4×C293—42no—
106D4×C293—82no—
107D4×C293C2×D412yes—
108C2×C4×C293—21no—
109C2×C4×C293—22no—
110C2×C4×C293—41no—
111C2×C4×C293C2411yescharacteristic
112C8×C293—22no—
113C22×C293—22no—
114C22×C293—41no—
115C22×C293—41no—
116C22×C293—81no—
117C22×C293—81no—
118C22×C293—161no—
119C22×C293H(5,𝔽2)11yescharacteristic
120C2×D293—22no—
121C2×D293—161no—
122C4×C293—22no—
123C4×C293—22no—
124C4×C293C22×D412yes—
125Dic293—22no—
126Dic293—42no—
127C2×C293—21no—
128C2×C293—42no—
129C2×C293—81no—
130C2×C293—81no—
131C2×C293D4211yescharacteristic
132D293—42no—
133C293#329111yescharacteristic, complemented, Sylow
134#3291—2931nomaximal, Sylow
135#577—2932no—
136#577—2932no—
137#492—2931no—
138#471—2932no—
139#471—2932no—
140#456—2932no—
141#453—2931no—
142#453D29311yescharacteristic, complemented
143#443—2932no—
1442−1+4—2932no—
145H(5,𝔽2)—2931no—
146H(5,𝔽2)C2×D29311yescharacteristic, complemented
147#137—2932no—
148#137—2932no—
149#137—5862no—
150#137—5862no—
151AGL(1,ℤ/8)—2932no—
152AGL(1,ℤ/8)—5862no—
153AGL(1,ℤ/8)—5862no—
154AGL(1,ℤ/8)C2×D29312yes—
155#135—2932no—
156#135—5862no—
157#135—5862no—
158C2×SD16—2932no—
159C2×SD16—2932no—
160C2×SD16—5862no—
161C2×SD16—5862no—
162#131—2932no—
163#131—2932no—
164C4⋊Q8—2931no—
165#127C2×D29311yescharacteristic
166#108—2932no—
167#104—2932no—
168#104C2×D29312yescomplemented
169#101—2932no—
170#100—2931no—
171#100C2×D29311yescharacteristic, complemented
172C4×C8—2931no—
173#41—22no—
174#41—2932no—
175#41—5862no—
176#41—5862no—
177#41—5862no—
178#41—11722no—
179#41C22×D29312yes—
180C2×Q8—2932no—
181C2×Q8—5862no—
182C2×Q8—5862no—
183C2×D4—22no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—5862no—
187C2×D4—11721no—
188C2×D4C22×D29311yescharacteristic
189C2×D4C22×D29311yescharacteristic
190Q16—2932no—
191Q16—5862no—
192Q16—5862no—
193SD16—22no—
194SD16—2932no—
195SD16—2932no—
196SD16—5862no—
197SD16—5862no—
198SD16—5862no—
199SD16—5862no—
200SD16—5862no—
201SD16—5862no—
202SD16—5862no—
203D8—22no—
204D8—2932no—
205D8—5862no—
206M16—2932no—
207M16—5862no—
208M16—5862no—
209M16C22×D29312yes—
210C2×C8—2932no—
211C2×C8—5862no—
212C2×C8—5862no—
213C4⋊C4—11721no—
214C42C22×D29311yescharacteristic
215C23—21no—
216C23—21no—
217C23—41no—
218C23—11721no—
219Q8—2932no—
220Q8—5862no—
221Q8—5862no—
222Q8—11722no—
223Q8D4×D29312yes—
224D4—22no—
225D4—22no—
226D4—22no—
227D4—42no—
228D4—82no—
229D4—2932no—
230D4—5862no—
231D4—11722no—
232D4—11722no—
233D4D4×D29312yes—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—41no—
237C2×C4—5862no—
238C2×C4—5862no—
239C2×C4—11722no—
240C2×C4—11721no—
241C2×C4—11722no—
242C2×C4C23×D29311yescharacteristic
243C8—22no—
244C8—2932no—
245C8—2932no—
246C8—5862no—
247C22—22no—
248C22—41no—
249C22—41no—
250C22—81no—
251C22—81no—
252C22—161no—
253C22—5862no—
254C22—46881no—
255C22#2262766911yescharacteristic
256C4—22no—
257C4—22no—
258C4—5862no—
259C4—11722no—
260C4C2×D4×D29312yes—
261C2—21no—
262C2—42no—
263C2—81no—
264C2—81no—
265C2—11722no—
266C2#3403676111yescharacteristic
subgroupG/H
centerC2#34036761
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#453×C293C2
socleC2×C293D42
hypercenter#453D293
hypocenterC293#3291
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #34036940C2semidirect
2. #34036939C2semidirect
3. #34036501C2semidirect
4. #34036391C2semidirect
5. #34036390C2semidirect
6. #34036313C2semidirect
7. #34036296C2semidirect
8. #34036243C2semidirect
9. #453×C293C2semidirect
11. #22627669C22semidirect
12. #22627657C22semidirect
14. #22627654C22semidirect
23. C2×#17989790C22semidirect
24. C2×#17989790C22semidirect
26. #22627632C22semidirect
27. #22627631C22semidirect
28. #22627624C22semidirect
29. #22627559C22semidirect
30. #22627552C22semidirect
31. #22627542C22semidirect
32. #22627541C22semidirect
33. C4×#14556873C22semidirect
34. H(5,𝔽2)×C293C22semidirect
37. #104×C293C22semidirect
38. #100×C293C22semidirect
51. #17989792D4semidirect
59. #17989790D4semidirect
60. #17989789D4semidirect
62. #17989787D4semidirect
133. C293#3291semidirect
142. #453D293semidirect
146. H(5,𝔽2)C2×D293semidirect
168. #104C2×D293semidirect
171. #100C2×D293semidirect

Maximal subgroups

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

subgroupindex
#340369402
#340369392
#340365012
#340363912
#340363902
#340363132
#340362962
#340362432
#453×C2932
#3291293