Finite Group Databasebeta

← #49168924

Subgroups — #49168924

Order 37504.

450 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 121 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#34036665C211yesmaximal, characteristic, complemented
2C2×D4×Dic293C211yesmaximal, characteristic, complemented
3#34036457C211yesmaximal, characteristic, complemented
4#34036443C211yesmaximal, characteristic, complemented
5#34036427C212yesmaximal, complemented
6#34036346C212yesmaximal, complemented
7#34036263C211yesmaximal, characteristic
8#34036238C212yesmaximal, complemented
9#34036230C212yesmaximal, complemented
10#34036183C211yesmaximal, characteristic, complemented
11#450×C293C211yesmaximal, characteristic, complemented
12C23×Dic293—21no—
13C2×#17989787C2212yes—
14#22627619—21no—
15#22627615C2211yescharacteristic, complemented
16#22627613C2211yescharacteristic, complemented
17#22627608—21no—
18#22627598C2211yescharacteristic
19Q8×Dic293C2211yescharacteristic, complemented
20D4×Dic293—21no—
21D4×Dic293—21no—
22D4×Dic293C2212yes—
23D4×Dic293C2212yes—
24#22627586C2211yescharacteristic, complemented
25#22627584—21no—
26#22627582—21no—
27C2×C4⋊Dic293C2211yescharacteristic
28C2×C22⋊Dic293—21no—
29C2×C4×Dic293C2211yescharacteristic
30#22627558C2212yescomplemented
31#22627555C2212yescomplemented
32#22627553C2212yescomplemented
33#22627548C2212yescomplemented
34#22627546C2212yescomplemented
35#22627545C2212yes—
36C22⋊#14556873C2211yescharacteristic
37#22627534C2212yescomplemented
38C22×D4×C293C2211yescharacteristic
39C2×SD16×C293C2212yes—
40C22⋊Q8×C293C2211yescharacteristic
41#102×C293C2212yes—
42C22⋊C8×C293C2211yescharacteristic
43C2×D1172C2311yescharacteristic
44C2×C22⋊D293—21no—
45C22×Dic293—22no—
46C22×Dic293—22no—
47C22×Dic293—21no—
48C22×Dic293—44no—
49C22×Dic293D411yescharacteristic
50C22×Dic293D411yescharacteristic
51C2×C4×D293—21no—
52#17989787—24no—
53C2×#14556873C2312yes—
54C4⋊Dic293—21no—
55C4⋊Dic293—21no—
56C4⋊Dic293—21no—
57C4⋊Dic293C2312yes—
58C4⋊Dic293C2311yescharacteristic
59#17989781—21no—
60C22⋊Dic293—22no—
61C22⋊Dic293—22no—
62#17989779—21no—
63#17989779—21no—
64#17989779—21no—
65#17989779—21no—
66C4×Dic293—21no—
67C4×Dic293D411yescharacteristic
68C4×Dic293D411yescharacteristic
69C4×Dic293C2311yescharacteristic
70C4×Dic293C2311yescharacteristic
71C24×C293—21no—
72C2×Q8×C293C2311yescharacteristic
73C2×D4×C293—24no—
74C2×D4×C293—21no—
75C2×D4×C293C2312yes—
76C22×C4×C293C2311yescharacteristic
77SD16×C293—24no—
78C2×C8×C293C2312yes—
79C4⋊C4×C293—21no—
80C4⋊C4×C293C2311yescharacteristic
81C22⋊C4×C293—21no—
82C22×D293—21no—
83D1172—22no—
84C22⋊D293—42no—
85C2×Dic293—22no—
86C2×Dic293—21no—
87C2×Dic293—22no—
88C2×Dic293—41no—
89C2×Dic293—44no—
90C2×Dic293—44no—
91C2×Dic293—42no—
92C2×Dic293C2×D411yescharacteristic
93C2×Dic293C2×D411yescharacteristic
94C2×Dic293C2×D411yescharacteristic
95C2×Dic293C2×D411yescharacteristic
96C4×D293—42no—
97#14556873—22no—
98C23×C293—24no—
99C23×C293—22no—
100C23×C293—42no—
101C23×C293—42no—
102C23×C293C2×D411yescharacteristic
103Q8×C293—22no—
104D4×C293—24no—
105D4×C293—22no—
106D4×C293#4114yes—
107C2×C4×C293—22no—
108C2×C4×C293—21no—
109C2×C4×C293—21no—
110C2×C4×C293C2×D411yescharacteristic
111C2×C4×C293C2411yescharacteristic
112C8×C293—22no—
113C22×C293—21no—
114C22×C293—21no—
115C22×C293—24no—
116C22×C293—48no—
117C22×C293—42no—
118C22×C293—42no—
119C22×C293AGL(1,ℤ/8)12yes—
120C22×C293C22×D411yescharacteristic
121C2×D293—41no—
122C2×D293—41no—
123C2×D293—41no—
124C4×C293—21no—
125C4×C293—41no—
126C4×C293—41no—
127C4×C293C2×#4112yes—
128Dic293—21no—
129Dic293—21no—
130Dic293—21no—
131Dic293—21no—
132Dic293—42no—
133Dic293—41no—
134C2×C293—22no—
135C2×C293—44no—
136C2×C293#52511yescharacteristic
137C2×C293C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C293#57511yescharacteristic
139D293—81no—
140C293#311411yescharacteristic, complemented, Sylow
141#3114—2931nomaximal, Sylow
142#529—2931no—
143C2×C4×D4—2931no—
144#483—2931no—
145#480—2931no—
146#478—2932no—
147#463—2932no—
148#450D29311yescharacteristic, complemented
149#447—2931no—
150#442—2932no—
151#440—2932no—
152#432—2931no—
153C22×D4C2×D29311yescharacteristic
154C23×C4—5861no—
155C2×SD16C2×D29312yes—
156C2×D8—2932no—
157#126—5861no—
158#125—2931no—
159#124—2931no—
160#123—5861no—
161C22⋊Q8C2×D29311yescharacteristic
162#121—2931no—
163C4×Q8—2931no—
164C4×D4—2932no—
165C4×D4—2932no—
166C4×D4—2931no—
167C4×D4—5861no—
168C4×D4—5861no—
169C4×D4—5861no—
170#117—5861no—
171C2×C4⋊C4—2931no—
172C2×C22⋊C4—5861no—
173C2×C42—2931no—
174#107—2932no—
175#106—2932no—
176C4⋊C8—2932no—
177#103—2932no—
178#102—2932no—
179#102—2932no—
180#102C2×D29312yes—
181C22⋊C8—2931no—
182C22⋊C8C2×D29311yescharacteristic
183#97—2932no—
184C24—21no—
185C2×Q8C22×D29311yescharacteristic
186C2×D4—24no—
187C2×D4—21no—
188C2×D4—2931no—
189C2×D4—5861no—
190C2×D4C22×D29312yes—
191C22×C4—2931no—
192C22×C4—2931no—
193C22×C4—5861no—
194C22×C4—5862no—
195C22×C4—5862no—
196C22×C4—5861no—
197C22×C4—11724no—
198C22×C4C22×D29311yescharacteristic
199SD16—24no—
200D8—5864no—
201C2×C8—2932no—
202C2×C8C22×D29312yes—
203C4⋊C4—21no—
204C4⋊C4—2931no—
205C4⋊C4—2932no—
206C4⋊C4—5861no—
207C4⋊C4—5861no—
208C4⋊C4—5861no—
209C4⋊C4—5861no—
210C4⋊C4C22×D29311yescharacteristic
211C22⋊C4—21no—
212C22⋊C4—5861no—
213C22⋊C4—5861no—
214C22⋊C4—5861no—
215C22⋊C4—5862no—
216C22⋊C4—5862no—
217C22⋊C4—5861no—
218C42—2931no—
219C42—2931no—
220C42—2931no—
221C42—2931no—
222C42—5861no—
223C23—24no—
224C23—22no—
225C23—42no—
226C23—42no—
227C23—5861no—
228C23D4×D29311yescharacteristic
229Q8—22no—
230D4—24no—
231D4—22no—
232D4—5862no—
233D4—11722no—
234D4#1798980114yes—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—2931no—
239C2×C4—2931no—
240C2×C4—2931no—
241C2×C4—2931no—
242C2×C4—5862no—
243C2×C4—5861no—
244C2×C4—5862no—
245C2×C4—11722no—
246C2×C4—11721no—
247C2×C4—11724no—
248C2×C4—11724no—
249C2×C4—11722no—
250C2×C4D4×D29311yescharacteristic
251C2×C4C23×D29311yescharacteristic
252C8—22no—
253C8—5862no—
254C22—21no—
255C22—21no—
256C22—24no—
257C22—48no—
258C22—42no—
259C22—42no—
260C22—11721no—
261C22—11721no—
262C22—11721no—
263C22#2262765112yes—
264C22C2×D4×D29311yescharacteristic
265C4—21no—
266C4—41no—
267C4—41no—
268C4—5861no—
269C4—5861no—
270C4—5861no—
271C4—5861no—
272C4—11722no—
273C4—11721no—
274C4C2×#1798980112yes—
275C2—22no—
276C2—44no—
277C2—23441no—
278C2#3403663511yescharacteristic
279C2C2×#2262765111yescharacteristic
280C2#3403692511yescharacteristic
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#450×C293C2
socleC22×C293C22×D4
prime radical#34036263C2
hypercenter#450D293
hypocenterC293#3114
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #34036665C2semidirect
2. C2×D4×Dic293C2semidirect
3. #34036457C2semidirect
4. #34036443C2semidirect
5. #34036427C2semidirect
6. #34036346C2semidirect
8. #34036238C2semidirect
9. #34036230C2semidirect
10. #34036183C2semidirect
11. #450×C293C2semidirect
15. #22627615C22semidirect
16. #22627613C22semidirect
19. Q8×Dic293C22semidirect
24. #22627586C22semidirect
30. #22627558C22semidirect
31. #22627555C22semidirect
32. #22627553C22semidirect
33. #22627548C22semidirect
34. #22627546C22semidirect
37. #22627534C22semidirect
140. C293#3114semidirect
148. #450D293semidirect

Maximal subgroups

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

subgroupindex
#340366652
C2×D4×Dic2932
#340364572
#340364432
#340364272
#340363462
#340362632
#340362382
#340362302
#340361832
#450×C2932
#3114293