Finite Group Databasebeta

← #49170553

Subgroups — #49170553

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#34036937C212yesmaximal, complemented
2#34036933C212yesmaximal, complemented
3#34036519C211yesmaximal, characteristic, complemented
4#34036397C212yesmaximal, complemented
5#34036395C212yesmaximal, complemented
6#34036306C211yesmaximal, characteristic, complemented
7#34036298C212yesmaximal, complemented
8#34036248C212yesmaximal, complemented
9#454×C293C211yesmaximal, characteristic, complemented
10#22627670C2212yescomplemented
11#22627669C2211yescharacteristic
12#22627655—22no—
13#22627655C2212yescomplemented
14#22627652C2212yes—
15#22627651—22no—
16#22627649—22no—
17#22627649C2212yescomplemented
18#22627648—22no—
19#22627645—22no—
20#22627644—22no—
21#22627644C2212yes—
22C2×#17989789—22no—
23C2×#17989789C2212yescomplemented
24C2×#17989787C2212yescomplemented
25D8×D293—22no—
26#22627632C2212yescomplemented
27#22627631C2212yescomplemented
28#22627621C2211yescharacteristic, complemented
29#22627559C2212yescomplemented
30#22627552C2212yescomplemented
31#22627541C2211yescharacteristic, complemented
32#22627540C2212yescomplemented
33#22627535C2211yescharacteristic, complemented
342−1+4×C293C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C293C2212yescomplemented
36#124×C293C2211yescharacteristic, complemented
37#104×C293C2212yescomplemented
38#101×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#17989792—22no—
51#17989792D412yescomplemented
52#17989791—22no—
53#17989790D412yescomplemented
54#17989789—22no—
55#17989789—22no—
56#17989789—22no—
57#17989789—22no—
58#17989789D412yescomplemented
59D2344—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×C293—41no—
72#41×C293C2312yes—
73C2×Q8×C293—22no—
74C2×Q8×C293C2311yescharacteristic
75C2×D4×C293C2311yescharacteristic
76SD16×C293—22no—
77D8×C293—22no—
78M16×C293C2312yes—
79C22⋊C4×C293—41no—
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—
100Q8×C293—22no—
101Q8×C293—41no—
102Q8×C293C2×D412yes—
103D4×C293—22no—
104D4×C293—22no—
105D4×C293—41no—
106D4×C293C2×D412yes—
107C2×C4×C293—22no—
108C2×C4×C293—21no—
109C2×C4×C293—21no—
110C2×C4×C293—42no—
111C2×C4×C293C2411yescharacteristic
112C8×C293—22no—
113C22×C293—22no—
114C22×C293—41no—
115C22×C293—81no—
116C22×C293H(5,𝔽2)11yescharacteristic
117C2×D293—22no—
118C2×D293—42no—
119C2×D293—82no—
120C2×D293—161no—
121C2×D293—162no—
122C4×C293—22no—
123C4×C293—41no—
124C4×C293—41no—
125C4×C293C22×D412yes—
126Dic293—22no—
127C2×C293—21no—
128C2×C293—42no—
129C2×C293—81no—
130C2×C293D4211yescharacteristic
131D293—42no—
132D293—82no—
133C293#328811yescharacteristic, complemented, Sylow
134#3288—2931nomaximal, Sylow
135#577—2932no—
136#576—2932no—
137#496—2931no—
138#472—2932no—
139#471—2932no—
140#455—2931no—
141#454D29311yescharacteristic, complemented
142#453—2932no—
143#444—2932no—
1442−1+4C2×D29311yescharacteristic, complemented
145H(5,𝔽2)—2932no—
146H(5,𝔽2)—2931no—
147#137—2932no—
148#137—5862no—
149AGL(1,ℤ/8)—2932no—
150AGL(1,ℤ/8)—2932no—
151AGL(1,ℤ/8)—5862no—
152AGL(1,ℤ/8)—5862no—
153AGL(1,ℤ/8)—5862no—
154AGL(1,ℤ/8)C2×D29312yescomplemented
155#135—2932no—
156#135—5862no—
157#135—5862no—
158C2×SD16—2932no—
159C2×SD16—5862no—
160C2×D8—2932no—
161C2×D8—5862no—
162#131—2932no—
163#131—2932no—
164#127—2931no—
165#124C2×D29311yescharacteristic, complemented
166#108—2932no—
167#104—2932no—
168#104C2×D29312yescomplemented
169#101C2×D29311yescharacteristic, complemented
170#100—2931no—
171#100—2932no—
172#97—2931no—
173#41—22no—
174#41—41no—
175#41—2932no—
176#41—5862no—
177#41—5862no—
178#41—5862no—
179#41C22×D29312yes—
180C2×Q8—22no—
181C2×Q8C22×D29311yescharacteristic
182C2×D4—2932no—
183C2×D4—5862no—
184C2×D4—5862no—
185C2×D4—5862no—
186C2×D4—11721no—
187C2×D4—11722no—
188C2×D4—11721no—
189C2×D4C22×D29311yescharacteristic
190Q16—2932no—
191Q16—5862no—
192SD16—22no—
193SD16—2932no—
194SD16—2932no—
195SD16—5862no—
196SD16—5862no—
197SD16—5862no—
198SD16—5862no—
199SD16—5862no—
200D8—22no—
201D8—2932no—
202D8—5862no—
203D8—5862no—
204D8—5862no—
205D8—5862no—
206M16—2932no—
207M16—5862no—
208M16—5862no—
209M16C22×D29312yes—
210C2×C8—2932no—
211C2×C8—5862no—
212C2×C8—5862no—
213C22⋊C4—41no—
214C42C22×D29311yescharacteristic
215C23—21no—
216C23—5862no—
217C23—11722no—
218C23—11721no—
219Q8—22no—
220Q8—41no—
221Q8—2932no—
222Q8D4×D29312yes—
223D4—22no—
224D4—22no—
225D4—41no—
226D4—2932no—
227D4—23442no—
228D4—5862no—
229D4—5862no—
230D4—5862no—
231D4—11722no—
232D4—11722no—
233D4—11722no—
234D4D4×D29312yes—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—42no—
239C2×C4—5862no—
240C2×C4—11722no—
241C2×C4—11721no—
242C2×C4C23×D29311yescharacteristic
243C8—22no—
244C8—2932no—
245C8—2932no—
246C8—5862no—
247C22—22no—
248C22—41no—
249C22—81no—
250C22—23442no—
251C22—5862no—
252C22—46882no—
253C22—46881no—
254C22—11722no—
255C22#2262766911yescharacteristic
256C4—22no—
257C4—41no—
258C4—41no—
259C4—5862no—
260C4C2×D4×D29312yes—
261C2—21no—
262C2—42no—
263C2—81no—
264C2—23442no—
265C2—11722no—
266C2#3403676111yescharacteristic
subgroupG/H
centerC2#34036761
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#454×C293C2
socleC2×C293D42
hypercenter#454D293
hypocenterC293#3288
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C2 ◁ C2×C4 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #34036937C2semidirect
2. #34036933C2semidirect
3. #34036519C2semidirect
4. #34036397C2semidirect
5. #34036395C2semidirect
6. #34036306C2semidirect
7. #34036298C2semidirect
8. #34036248C2semidirect
9. #454×C293C2semidirect
10. #22627670C22semidirect
13. #22627655C22semidirect
17. #22627649C22semidirect
23. C2×#17989789C22semidirect
24. C2×#17989787C22semidirect
26. #22627632C22semidirect
27. #22627631C22semidirect
28. #22627621C22semidirect
29. #22627559C22semidirect
30. #22627552C22semidirect
31. #22627541C22semidirect
32. #22627540C22semidirect
33. #22627535C22semidirect
34. 2−1+4×C293C22semidirect
35. AGL(1,ℤ/8)×C293C22semidirect
36. #124×C293C22semidirect
37. #104×C293C22semidirect
38. #101×C293C22semidirect
51. #17989792D4semidirect
53. #17989790D4semidirect
58. #17989789D4semidirect
63. #17989787D4semidirect
133. C293#3288semidirect
141. #454D293semidirect
144. 2−1+4C2×D293semidirect
154. AGL(1,ℤ/8)C2×D293semidirect
165. #124C2×D293semidirect
168. #104C2×D293semidirect
169. #101C2×D293semidirect

Maximal subgroups

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

subgroupindex
#340369372
#340369332
#340365192
#340363972
#340363952
#340363062
#340362982
#340362482
#454×C2932
#3288293