Finite Group Databasebeta

← #49170581

Subgroups — #49170581

Order 37504.

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

normal subgroups only

This group has 476 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#34036933C212yesmaximal, complemented
2#34036932C212yesmaximal, complemented
3#34036505C211yesmaximal, characteristic, complemented
4#34036398C212yesmaximal, complemented
5#34036397C212yesmaximal, complemented
6#34036298C212yesmaximal, complemented
7#34036296C211yesmaximal, characteristic, complemented
8#34036243C212yesmaximal, complemented
9#453×C293C211yesmaximal, characteristic, complemented
10#22627670C2212yescomplemented
11#22627669C2211yescharacteristic, complemented
12#22627652C2212yescomplemented
13#22627651—22no—
14#22627650—22no—
15#22627650C2212yescomplemented
16#22627649—22no—
17#22627649C2212yescomplemented
18#22627648—22no—
19#22627645—22no—
20#22627644—22no—
21#22627644C2212yes—
22C2×#17989787—22no—
23C2×#17989787C2212yescomplemented
24C2×#17989787C2212yescomplemented
25D8×D293—22no—
26#22627632C2212yescomplemented
27#22627631C2212yescomplemented
28#22627621C2211yescharacteristic, complemented
29#22627559C2212yescomplemented
30#22627552C2212yescomplemented
31#22627541C2211yescharacteristic, complemented
32#22627540C2212yescomplemented
33C4×#14556873C2211yescharacteristic, complemented
34H(5,𝔽2)×C293C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C293C2212yescomplemented
36#127×C293C2211yescharacteristic, complemented
37#104×C293C2212yescomplemented
38#100×C293C2211yescharacteristic, complemented
39#17989803—22no—
40#17989802—22no—
41#17989802C2312yes—
42#17989801—22no—
43C2×D1172—22no—
44C2×D1172—41no—
45C2×D1172C2312yes—
46C2×C22⋊D293—41no—
47D4×D293—22no—
48D4×D293—22no—
49D4×D293—42no—
50#17989792D412yescomplemented
51#17989791—22no—
52#17989790—22no—
53#17989790D412yescomplemented
54#17989789—22no—
55#17989789D412yescomplemented
56D2344—22no—
57#17989787—22no—
58#17989787—22no—
59#17989787—22no—
60#17989787—22no—
61#17989787—22no—
62#17989787—22no—
63#17989787D412yescomplemented
64#17989786—22no—
65#17989786C2312yes—
66#17989785—22no—
67C2×#14556873—22no—
68C2×#14556873C2312yes—
69C8×D293—22no—
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—22no—
82C22×D293—41no—
83C22×D293—42no—
84Dic586C2×D412yes—
85D1172—22no—
86D1172—22no—
87D1172—42no—
88D1172—82no—
89D1172C2×D412yes—
90C22⋊D293—22no—
91C22⋊D293—42no—
92C22⋊D293—42no—
93C2×Dic293—41no—
94C4×D293—22no—
95C4×D293—42no—
96#14556873—22no—
97#14556873C2×D412yes—
98#14556873C2×D412yes—
99C23×C293—21no—
100C23×C293—21no—
101C23×C293—41no—
102Q8×C293C2×D412yes—
103D4×C293—22no—
104D4×C293—22no—
105D4×C293—22no—
106D4×C293—42no—
107D4×C293—82no—
108D4×C293C2×D412yes—
109C2×C4×C293—21no—
110C2×C4×C293—22no—
111C2×C4×C293—41no—
112C2×C4×C293C2411yescharacteristic
113C8×C293—22no—
114C22×C293—22no—
115C22×C293—41no—
116C22×C293—41no—
117C22×C293—81no—
118C22×C293—81no—
119C22×C293—161no—
120C22×C293H(5,𝔽2)11yescharacteristic
121C2×D293—22no—
122C2×D293—42no—
123C2×D293—82no—
124C2×D293—161no—
125C2×D293—162no—
126C4×C293—22no—
127C4×C293—22no—
128C4×C293C22×D412yes—
129Dic293—22no—
130C2×C293—21no—
131C2×C293—42no—
132C2×C293—81no—
133C2×C293—81no—
134C2×C293D4211yescharacteristic
135D293—42no—
136D293—82no—
137C293#329211yescharacteristic, complemented, Sylow
138#3292—2931nomaximal, Sylow
139#576—2932no—
140#576—2932no—
141#493—2931no—
142#472—2932no—
143#472—2932no—
144#453—2931no—
145#453—2932no—
146#453D29311yescharacteristic, complemented
147#443—2932no—
148H(5,𝔽2)—2932no—
149H(5,𝔽2)—2931no—
150H(5,𝔽2)C2×D29311yescharacteristic, complemented
151AGL(1,ℤ/8)—2932no—
152AGL(1,ℤ/8)—2932no—
153AGL(1,ℤ/8)—2932no—
154AGL(1,ℤ/8)—5862no—
155AGL(1,ℤ/8)—5862no—
156AGL(1,ℤ/8)—5862no—
157AGL(1,ℤ/8)—5862no—
158AGL(1,ℤ/8)C2×D29312yescomplemented
159#135—2932no—
160#135—5862no—
161#135—5862no—
162C2×D8—2932no—
163C2×D8—2932no—
164C2×D8—5862no—
165C2×D8—5862no—
166#131—2932no—
167#131—2932no—
168#127—2931no—
169#127C2×D29311yescharacteristic, complemented
170#108—2932no—
171#104—2932no—
172#104C2×D29312yescomplemented
173#100—2931no—
174#100—2932no—
175#100C2×D29311yescharacteristic, complemented
176C4×C8—2931no—
177#41—22no—
178#41—2932no—
179#41—5862no—
180#41—5862no—
181#41—5862no—
182#41C22×D29312yes—
183C2×D4—22no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—2932no—
187C2×D4—5862no—
188C2×D4—5862no—
189C2×D4—5862no—
190C2×D4—11721no—
191C2×D4—11722no—
192C2×D4—11721no—
193C2×D4C22×D29311yescharacteristic
194C2×D4C22×D29311yescharacteristic
195Q16—2932no—
196SD16—22no—
197SD16—2932no—
198SD16—2932no—
199SD16—5862no—
200SD16—5862no—
201SD16—5862no—
202D8—22no—
203D8—2932no—
204D8—5862no—
205D8—5862no—
206D8—5862no—
207D8—5862no—
208D8—5862no—
209D8—5862no—
210D8—5862no—
211M16—2932no—
212M16—5862no—
213M16—5862no—
214M16C22×D29312yes—
215C2×C8—2932no—
216C2×C8—5862no—
217C2×C8—5862no—
218C42C22×D29311yescharacteristic
219C23—21no—
220C23—21no—
221C23—41no—
222C23—5862no—
223C23—11722no—
224C23—11721no—
225Q8—2932no—
226Q8D4×D29312yes—
227D4—22no—
228D4—22no—
229D4—22no—
230D4—42no—
231D4—82no—
232D4—2932no—
233D4—23442no—
234D4—5862no—
235D4—5862no—
236D4—5862no—
237D4—11722no—
238D4—11722no—
239D4—11722no—
240D4D4×D29312yes—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—5862no—
245C2×C4—11722no—
246C2×C4—11721no—
247C2×C4C23×D29311yescharacteristic
248C8—22no—
249C8—2932no—
250C8—2932no—
251C8—5862no—
252C22—22no—
253C22—41no—
254C22—41no—
255C22—81no—
256C22—81no—
257C22—161no—
258C22—23442no—
259C22—5862no—
260C22—46882no—
261C22—46881no—
262C22—11722no—
263C22#2262766911yescharacteristic
264C4—22no—
265C4—22no—
266C4—5862no—
267C4C2×D4×D29312yes—
268C2—21no—
269C2—42no—
270C2—81no—
271C2—81no—
272C2—23442no—
273C2—11722no—
274C2#3403676111yescharacteristic
subgroupG/H
centerC2#34036761
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#453×C293C2
socleC2×C293D42
hypercenter#453D293
hypocenterC293#3292
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. #34036933C2semidirect
2. #34036932C2semidirect
3. #34036505C2semidirect
4. #34036398C2semidirect
5. #34036397C2semidirect
6. #34036298C2semidirect
7. #34036296C2semidirect
8. #34036243C2semidirect
9. #453×C293C2semidirect
10. #22627670C22semidirect
11. #22627669C22semidirect
12. #22627652C22semidirect
15. #22627650C22semidirect
17. #22627649C22semidirect
23. C2×#17989787C22semidirect
24. C2×#17989787C22semidirect
26. #22627632C22semidirect
27. #22627631C22semidirect
28. #22627621C22semidirect
29. #22627559C22semidirect
30. #22627552C22semidirect
31. #22627541C22semidirect
32. #22627540C22semidirect
33. C4×#14556873C22semidirect
34. H(5,𝔽2)×C293C22semidirect
35. AGL(1,ℤ/8)×C293C22semidirect
36. #127×C293C22semidirect
37. #104×C293C22semidirect
38. #100×C293C22semidirect
50. #17989792D4semidirect
53. #17989790D4semidirect
55. #17989789D4semidirect
63. #17989787D4semidirect
137. C293#3292semidirect
146. #453D293semidirect
150. H(5,𝔽2)C2×D293semidirect
158. AGL(1,ℤ/8)C2×D293semidirect
169. #127C2×D293semidirect
172. #104C2×D293semidirect
175. #100C2×D293semidirect

Maximal subgroups

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

subgroupindex
#340369332
#340369322
#340365052
#340363982
#340363972
#340362982
#340362962
#340362432
#453×C2932
#3292293