Finite Group Databasebeta

← #49170388

Subgroups — #49170388

Order 37504.

546 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 125 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×#22627651C212yesmaximal, complemented
2#34036782C211yesmaximal, characteristic, complemented
3D4×D1172C211yesmaximal, characteristic, complemented
4#34036503C211yesmaximal, characteristic, complemented
5#34036378C212yesmaximal, complemented
6#34036325C211yesmaximal, characteristic, complemented
7#34036278C212yesmaximal, complemented
8#34036277C212yesmaximal, complemented
9C4×#17989787C211yesmaximal, characteristic, complemented
10#34036206C211yesmaximal, characteristic, complemented
11#461×C293C211yesmaximal, characteristic, complemented
12C2×#17989803C2212yescomplemented
13C22×D1172—21no—
14C2×D4×D293C2212yes—
15#22627651—28no—
16C2×#17989789C2212yescomplemented
17C2×#17989789C2212yescomplemented
18C2×D2344C2212yescomplemented
19C2×#17989787C2211yescharacteristic
20C2×#17989785C2212yescomplemented
21#22627621C2211yescharacteristic
22#22627609—22no—
23#22627600C2212yescomplemented
24#22627599—21no—
25#22627599C2212yes—
26#22627598—21no—
27#22627594—22no—
28C4×D1172C2211yescharacteristic
29C4×D1172C2211yescharacteristic, complemented
30C4⋊C4×D293—21no—
31#22627557C2211yescharacteristic, complemented
32#22627546C2212yescomplemented
33#22627545C2211yescharacteristic
34#22627544C2211yescharacteristic, complemented
35#22627536C2212yescomplemented
36C4×#14556873C2211yescharacteristic, complemented
37C2×SD16×C293C2212yes—
38C4⋊Q8×C293C2211yescharacteristic, complemented
39C4×D4×C293C2211yescharacteristic
40C4⋊C8×C293C2211yescharacteristic, complemented
41#103×C293C2212yescomplemented
42C23×D293—22no—
43#17989803—24no—
44#17989803—42no—
45C2×D1172—22no—
46C2×D1172—22no—
47C2×D1172—21no—
48C2×D1172—21no—
49C2×D1172—44no—
50C2×D1172D411yescharacteristic
51C2×D1172C2312yes—
52C2×D1172C2311yescharacteristic
53C2×C22⋊D293—22no—
54D4×D293—24no—
55D4×D293—42no—
56C2×C4×D293—22no—
57C2×C4×D293—21no—
58C2×C4×D293C2312yes—
59#17989789—24no—
60#17989789—42no—
61D2344—24no—
62#17989787D414yescomplemented
63#17989785—24no—
64C2×#14556873C2311yescharacteristic
65C2×#14556873C2311yescharacteristic
66C4⋊Dic293D411yescharacteristic
67C4⋊Dic293C2311yescharacteristic
68#17989781—22no—
69#17989779—22no—
70#17989779—22no—
71#17989779—21no—
72#17989779D412yes—
73C2×Q8×C293C2312yes—
74C2×D4×C293C2311yescharacteristic
75C22×C4×C293—21no—
76SD16×C293—24no—
77C2×C8×C293C2312yes—
78C4⋊C4×C293—21no—
79C4⋊C4×C293—21no—
80C4⋊C4×C293C2311yescharacteristic
81C22⋊C4×C293—21no—
82C42×C293C2311yescharacteristic
83C22×D293—21no—
84C22×D293—22no—
85C22×D293—44no—
86C22×D293—44no—
87C22×D293—82no—
88C22×D293C2×D412yes—
89D1172—22no—
90D1172—22no—
91D1172—21no—
92D1172—22no—
93D1172—44no—
94D1172—44no—
95D1172—42no—
96D1172—81no—
97D1172C2×D412yescomplemented
98C22⋊D293—44no—
99C2×Dic293—21no—
100C2×Dic293C2×D412yes—
101C4×D293—22no—
102C4×D293—22no—
103C4×D293—44no—
104C4×D293—42no—
105#14556873—21no—
106#14556873C2×D412yescomplemented
107C23×C293—21no—
108Q8×C293—22no—
109Q8×C293—22no—
110D4×C293—21no—
111D4×C293C2×D412yes—
112C2×C4×C293—21no—
113C2×C4×C293—22no—
114C2×C4×C293—42no—
115C2×C4×C293C2×D411yescharacteristic
116C2×C4×C293C2×D411yescharacteristic
117C2×C4×C293C2411yescharacteristic
118C8×C293—22no—
119C22×C293—22no—
120C22×C293—41no—
121C22×C293—41no—
122C22×C293C22×D411yescharacteristic
123C2×D293—22no—
124C2×D293—22no—
125C2×D293—22no—
126C2×D293—22no—
127C2×D293—41no—
128C2×D293—41no—
129C2×D293—42no—
130C2×D293—42no—
131C2×D293—42no—
132C2×D293—88no—
133C4×C293—21no—
134C4×C293—42no—
135C4×C293—41no—
136C4×C293AGL(1,ℤ/8)12yescomplemented
137C4×C293C22×D411yescharacteristic
138C4×C293H(5,𝔽2)11yescharacteristic
139Dic293—22no—
140Dic293—41no—
141C2×C293—42no—
142C2×C293D4211yescharacteristic
143C2×C293C2×AGL(1,ℤ/8)11yescharacteristic
144C2×C293#57711yescharacteristic
145D293—42no—
146D293—42no—
147D293—82no—
148C293#327211yescharacteristic, complemented, Sylow
149#3272—2931nomaximal, Sylow
150C2×AGL(1,ℤ/8)—2932no—
151#547—2931no—
152D42—2931no—
153#492—2931no—
154#468—2932no—
155#461D29311yescharacteristic, complemented
156#459—2931no—
157#450—2932no—
158#449—2932no—
159C4×D8—2931no—
160#436—2931no—
161C2×#41—2932no—
162C22×D4—2932no—
163C22×D4—5861no—
164AGL(1,ℤ/8)—5868no—
165C2×SD16—2932no—
166C2×SD16—2932no—
167C2×SD16C2×D29312yes—
168C2×D8—2931no—
169C2×D8—2932no—
170C2×M16—2932no—
171C4⋊Q8C2×D29311yescharacteristic, complemented
172#127—2931no—
173#123—5862no—
174C22⋊Q8—2932no—
175#121—2932no—
176#121—5861no—
177#121—5861no—
178#120—5862no—
179C4×D4—2931no—
180C4×D4—2931no—
181C4×D4C2×D29311yescharacteristic
182C2×C4⋊C4—5861no—
183#107—2931no—
184C4⋊C8C2×D29311yescharacteristic, complemented
185#103C2×D29312yescomplemented
186#102—2931no—
187#102—2932no—
188#102—2931no—
189C22⋊C8—2932no—
190C4×C8—2931no—
191C24—5862no—
192#41—5864no—
193#41—11722no—
194C2×Q8C22×D29312yes—
195C2×D4—2931no—
196C2×D4—2931no—
197C2×D4—2932no—
198C2×D4—5862no—
199C2×D4—5862no—
200C2×D4—5862no—
201C2×D4—5864no—
202C2×D4—5861no—
203C2×D4—5861no—
204C2×D4—11724no—
205C2×D4—11722no—
206C2×D4C22×D29311yescharacteristic
207C22×C4—21no—
208C22×C4—2932no—
209C22×C4—5862no—
210C22×C4—5861no—
211SD16—24no—
212SD16—5864no—
213SD16—11722no—
214D8—2934no—
215D8—5864no—
216M16—5864no—
217C2×C8—2931no—
218C2×C8—2931no—
219C2×C8C22×D29312yes—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—2931no—
223C4⋊C4—2931no—
224C4⋊C4—5862no—
225C4⋊C4C22×D29311yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—2932no—
228C22⋊C4—5862no—
229C22⋊C4—5862no—
230C22⋊C4—5861no—
231C42C22×D29311yescharacteristic
232C23—21no—
233C23—2932no—
234C23—23442no—
235C23—5861no—
236C23—5862no—
237C23—11724no—
238C23—11724no—
239Q8—22no—
240Q8—22no—
241D4—21no—
242D4—2932no—
243D4—23441no—
244D4—5862no—
245D4—5862no—
246D4—5861no—
247D4—5862no—
248D4—11724no—
249D4—11724no—
250D4—11724no—
251D4—11722no—
252D4D4×D29312yes—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—42no—
256C2×C4—2932no—
257C2×C4—5862no—
258C2×C4—5862no—
259C2×C4—5861no—
260C2×C4—11724no—
261C2×C4—11722no—
262C2×C4D4×D29311yescharacteristic
263C2×C4D4×D29311yescharacteristic
264C2×C4C23×D29311yescharacteristic
265C8—22no—
266C8—2932no—
267C8—5861no—
268C22—22no—
269C22—41no—
270C22—41no—
271C22—23448no—
272C22—5862no—
273C22—5862no—
274C22—5862no—
275C22—5862no—
276C22—11722no—
277C22—11722no—
278C22—11722no—
279C22—11721no—
280C22—11721no—
281C22C2×D4×D29311yescharacteristic
282C4—21no—
283C4—42no—
284C4—41no—
285C4—5862no—
286C4—11721no—
287C4#2262765112yescomplemented
288C4C2×D4×D29311yescharacteristic
289C4#2262767011yescharacteristic
290C2—42no—
291C2—23442no—
292C2—11722no—
293C2—11722no—
294C2#3403676311yescharacteristic
295C2C2×#2262765111yescharacteristic
296C2#3403693811yescharacteristic
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#461×C293C2
socleC22×C293C22×D4
hypercenter#461D293
hypocenterC293#3272
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. C2×#22627651C2semidirect
2. #34036782C2semidirect
3. D4×D1172C2semidirect
4. #34036503C2semidirect
5. #34036378C2semidirect
6. #34036325C2semidirect
7. #34036278C2semidirect
8. #34036277C2semidirect
9. C4×#17989787C2semidirect
10. #34036206C2semidirect
11. #461×C293C2semidirect
12. C2×#17989803C22semidirect
16. C2×#17989789C22semidirect
17. C2×#17989789C22semidirect
18. C2×D2344C22semidirect
20. C2×#17989785C22semidirect
23. #22627600C22semidirect
29. C4×D1172C22semidirect
31. #22627557C22semidirect
32. #22627546C22semidirect
34. #22627544C22semidirect
35. #22627536C22semidirect
36. C4×#14556873C22semidirect
38. C4⋊Q8×C293C22semidirect
40. C4⋊C8×C293C22semidirect
41. #103×C293C22semidirect
62. #17989787D4semidirect
97. D1172C2×D4semidirect
106. #14556873C2×D4semidirect
136. C4×C293AGL(1,ℤ/8)semidirect
148. C293#3272semidirect
155. #461D293semidirect
171. C4⋊Q8C2×D293semidirect
184. C4⋊C8C2×D293semidirect
185. #103C2×D293semidirect
287. C4#22627651semidirect

Maximal subgroups

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

subgroupindex
C2×#226276512
#340367822
D4×D11722
#340365032
#340363782
#340363252
#340362782
#340362772
C4×#179897872
#340362062
#461×C2932
#3272293