Finite Group Databasebeta

← #49168919

Subgroups — #49168919

Order 37504.

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

normal subgroups only

This group has 450 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×C4×D1172C211yesmaximal, characteristic, complemented
2#34036457C211yesmaximal, characteristic, complemented
3#34036440C211yesmaximal, characteristic, complemented
4#34036426C212yesmaximal, complemented
5#34036341C212yesmaximal, complemented
6#34036279C211yesmaximal, characteristic, complemented
7#34036262C211yesmaximal, characteristic
8#34036239C212yesmaximal, complemented
9#34036227C212yesmaximal, complemented
10#34036186C211yesmaximal, characteristic, complemented
11#529×C293C211yesmaximal, characteristic, complemented
12C22×D1172C2211yescharacteristic
13C22×C4×D293—21no—
14C2×#17989789C2212yes—
15C2×#17989787C2212yes—
16C4×D1172—21no—
17C4×D1172—21no—
18C4×D1172C2212yes—
19C4×D1172C2212yes—
20C2×C4⋊Dic293C2211yescharacteristic
21C2×#17989779—21no—
22#22627557C2212yescomplemented
23#22627555C2212yescomplemented
24C4⋊#14556873C2212yescomplemented
25#22627548C2212yescomplemented
26#22627545C2212yescomplemented
27#22627544C2212yes—
28#22627544C2212yes—
29C22⋊#14556873C2211yescharacteristic
30C22⋊#14556873C2211yescharacteristic
31#22627535C2212yescomplemented
32#126×C293—21no—
33#125×C293C2211yescharacteristic, complemented
34#124×C293C2211yescharacteristic, complemented
35#123×C293—21no—
36C22⋊Q8×C293C2211yescharacteristic
37#121×C293C2211yescharacteristic
38C4×Q8×C293C2211yescharacteristic, complemented
39C4×D4×C293—21no—
40C4×D4×C293C2211yescharacteristic, complemented
41#117×C293—21no—
42C2×C42×C293C2211yescharacteristic
43C23×D293—21no—
44C2×D1172—24no—
45C2×D1172—21no—
46C2×D1172C2312yes—
47C22×Dic293—21no—
48C2×C4×D293—22no—
49C2×C4×D293—22no—
50C2×C4×D293—44no—
51#17989789—24no—
52#17989787—24no—
53C2×#14556873C2312yes—
54C2×#14556873C2312yes—
55C4⋊Dic293—21no—
56C4⋊Dic293C2312yes—
57#17989779—22no—
58#17989779—22no—
59C2×Q8×C293C2311yescharacteristic
60C2×D4×C293—21no—
61C2×D4×C293C2311yescharacteristic
62C22×C4×C293—21no—
63C22×C4×C293D411yescharacteristic
64C22×C4×C293D411yescharacteristic
65C22×C4×C293C2311yescharacteristic
66C4⋊C4×C293—21no—
67C4⋊C4×C293—21no—
68C4⋊C4×C293—21no—
69C4⋊C4×C293—21no—
70C4⋊C4×C293C2311yescharacteristic
71C4⋊C4×C293C2311yescharacteristic
72C22⋊C4×C293—21no—
73C22⋊C4×C293—21no—
74C22⋊C4×C293—21no—
75C22⋊C4×C293—21no—
76C22⋊C4×C293—21no—
77C42×C293—21no—
78C42×C293D411yescharacteristic
79C42×C293D411yescharacteristic
80C42×C293C2311yescharacteristic
81C42×C293C2311yescharacteristic
82C22×D293—24no—
83C22×D293—22no—
84C22×D293—42no—
85C22×D293—42no—
86D1172—24no—
87D1172—22no—
88D1172#4114yes—
89C2×Dic293—22no—
90C2×Dic293—42no—
91C4×D293—44no—
92C4×D293—44no—
93#14556873—22no—
94#14556873—22no—
95C23×C293—21no—
96C23×C293C2×D411yescharacteristic
97Q8×C293—22no—
98D4×C293—22no—
99D4×C293—42no—
100C2×C4×C293—22no—
101C2×C4×C293—22no—
102C2×C4×C293—21no—
103C2×C4×C293—21no—
104C2×C4×C293—21no—
105C2×C4×C293—41no—
106C2×C4×C293—42no—
107C2×C4×C293C2×D411yescharacteristic
108C2×C4×C293C2×D411yescharacteristic
109C2×C4×C293C2×D411yescharacteristic
110C2×C4×C293C2×D411yescharacteristic
111C2×C4×C293C2×D411yescharacteristic
112C2×C4×C293C2411yescharacteristic
113C22×C293—21no—
114C22×C293—21no—
115C22×C293—41no—
116C22×C293—41no—
117C22×C293—41no—
118C22×C293AGL(1,ℤ/8)12yes—
119C22×C293C22×D411yescharacteristic
120C2×D293—24no—
121C2×D293—48no—
122C2×D293—42no—
123C2×D293—42no—
124C4×C293—21no—
125C4×C293—21no—
126C4×C293—21no—
127C4×C293—21no—
128C4×C293—21no—
129C4×C293—41no—
130C4×C293—41no—
131C4×C293—41no—
132C4×C293C2×#4112yes—
133Dic293—42no—
134C2×C293—22no—
135C2×C293—81no—
136C2×C293#52511yescharacteristic
137C2×C293C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C293#57511yescharacteristic
139D293—44no—
140C293#311411yescharacteristic, complemented, Sylow
141#3114—2931nomaximal, Sylow
142#529D29311yescharacteristic, complemented
143C2×C4×D4—2931no—
144#483—2931no—
145#480—2931no—
146#478—2932no—
147#463—2932no—
148#450—2931no—
149#447—2931no—
150#442—2932no—
151#440—2932no—
152#432—2931no—
153C22×D4—2931no—
154C23×C4—5861no—
155C2×SD16—2932no—
156C2×D8—2932no—
157#126—21no—
158#125C2×D29311yescharacteristic, complemented
159#124C2×D29311yescharacteristic, complemented
160#123—21no—
161C22⋊Q8C2×D29311yescharacteristic
162#121C2×D29311yescharacteristic
163C4×Q8C2×D29311yescharacteristic, complemented
164C4×D4—21no—
165C4×D4—2932no—
166C4×D4—2932no—
167C4×D4—5861no—
168C4×D4—5861no—
169C4×D4C2×D29311yescharacteristic, complemented
170#117—21no—
171C2×C4⋊C4—2931no—
172C2×C22⋊C4—5861no—
173C2×C42C2×D29311yescharacteristic
174#107—2932no—
175#106—2932no—
176C4⋊C8—2932no—
177#103—2932no—
178#102—2932no—
179#102—2932no—
180#102—2932no—
181C22⋊C8—2931no—
182C22⋊C8—2931no—
183#97—2932no—
184C24—5861no—
185C2×Q8C22×D29311yescharacteristic
186C2×D4—21no—
187C2×D4—2932no—
188C2×D4—5864no—
189C2×D4—5861no—
190C2×D4C22×D29311yescharacteristic
191C22×C4—21no—
192C22×C4—5861no—
193C22×C4—5862no—
194C22×C4—5862no—
195C22×C4—11724no—
196C22×C4C22⋊D29311yescharacteristic
197C22×C4C22⋊D29311yescharacteristic
198C22×C4C22×D29311yescharacteristic
199SD16—5864no—
200D8—5864no—
201C2×C8—2932no—
202C2×C8—2932no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—2932no—
208C4⋊C4—5861no—
209C4⋊C4C22×D29311yescharacteristic
210C4⋊C4C22×D29311yescharacteristic
211C22⋊C4—21no—
212C22⋊C4—21no—
213C22⋊C4—21no—
214C22⋊C4—21no—
215C22⋊C4—21no—
216C22⋊C4—5862no—
217C22⋊C4—5862no—
218C42—21no—
219C42C22⋊D29311yescharacteristic
220C42C22⋊D29311yescharacteristic
221C42C22×D29311yescharacteristic
222C42C22×D29311yescharacteristic
223C23—21no—
224C23—5864no—
225C23—5862no—
226C23—11722no—
227C23—11722no—
228C23C2×C22⋊D29311yescharacteristic
229Q8—22no—
230D4—22no—
231D4—42no—
232D4—2934no—
233D4—5864no—
234D4—5862no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—5862no—
243C2×C4—11724no—
244C2×C4—11724no—
245C2×C4—11722no—
246C2×C4C2×C22⋊D29311yescharacteristic
247C2×C4C2×C22⋊D29311yescharacteristic
248C2×C4C2×C22⋊D29311yescharacteristic
249C2×C4C2×C22⋊D29311yescharacteristic
250C2×C4C2×C22⋊D29311yescharacteristic
251C2×C4C23×D29311yescharacteristic
252C8—5862no—
253C8—5862no—
254C22—21no—
255C22—21no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—5864no—
260C22—11728no—
261C22—11722no—
262C22—11722no—
263C22#2262764912yes—
264C22C22×C22⋊D29311yescharacteristic
265C4—21no—
266C4—21no—
267C4—21no—
268C4—21no—
269C4—21no—
270C4—41no—
271C4—41no—
272C4—41no—
273C4—11722no—
274C4#41×D29312yes—
275C2—22no—
276C2—81no—
277C2—11724no—
278C2#3403663611yescharacteristic
279C2C2×#2262764911yescharacteristic
280C2#3403692711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#529×C293C2
socleC22×C293C22×D4
prime radical#34036262C2
hypercenter#529D293
hypocenterC293#3114
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. C2×C4×D1172C2semidirect
2. #34036457C2semidirect
3. #34036440C2semidirect
4. #34036426C2semidirect
5. #34036341C2semidirect
6. #34036279C2semidirect
8. #34036239C2semidirect
9. #34036227C2semidirect
10. #34036186C2semidirect
11. #529×C293C2semidirect
22. #22627557C22semidirect
23. #22627555C22semidirect
24. C4⋊#14556873C22semidirect
25. #22627548C22semidirect
26. #22627545C22semidirect
31. #22627535C22semidirect
33. #125×C293C22semidirect
34. #124×C293C22semidirect
38. C4×Q8×C293C22semidirect
40. C4×D4×C293C22semidirect
140. C293#3114semidirect
142. #529D293semidirect
158. #125C2×D293semidirect
159. #124C2×D293semidirect
163. C4×Q8C2×D293semidirect
169. C4×D4C2×D293semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D11722
#340364572
#340364402
#340364262
#340363412
#340362792
#340362622
#340362392
#340362272
#340361862
#529×C2932
#3114293