Finite Group Databasebeta

← #49170384

Subgroups — #49170384

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
3#34036764C211yesmaximal, characteristic, complemented
4#34036372C212yesmaximal, complemented
5#34036331C211yesmaximal, characteristic, complemented
6#34036321C211yesmaximal, characteristic, complemented
7#34036278C212yesmaximal, complemented
8#34036273C212yesmaximal, complemented
9C4×D2344C211yesmaximal, characteristic, complemented
10#34036208C211yesmaximal, characteristic, complemented
11#492×C293C211yesmaximal, characteristic, complemented
12C2×#17989803C2212yescomplemented
13C2×D4×D293—21no—
14C2×D4×D293C2212yes—
15#22627651—28no—
16C2×#17989789C2212yes—
17C2×D2344C2211yescharacteristic
18C2×#17989787C2212yescomplemented
19C2×#17989785C2212yescomplemented
20#22627609—22no—
21#22627600C2212yescomplemented
22#22627599—21no—
23#22627595—21no—
24#22627595C2212yes—
25#22627592—22no—
26C4×D1172C2211yescharacteristic
27C4×D1172C2211yescharacteristic
28C4×D1172C2211yescharacteristic, complemented
29C4⋊C4×D293—21no—
30#22627558C2211yescharacteristic, complemented
31C4⋊#14556873C2211yescharacteristic, complemented
32#22627548C2212yescomplemented
33#22627546C2211yescharacteristic
34#22627546C2211yescharacteristic, complemented
35#22627545C2212yescomplemented
36#22627536C2212yescomplemented
37C2×SD16×C293C2212yescomplemented
38C2×SD16×C293C2212yescomplemented
39C4⋊Q8×C293C2211yescharacteristic, complemented
40#127×C293C2211yescharacteristic
41C4×C8×C293C2211yescharacteristic, complemented
42C23×D293—22no—
43#17989803—24no—
44#17989803—42no—
45C2×D1172—22no—
46C2×D1172D411yescharacteristic
47C2×D1172C2311yescharacteristic
48C2×D1172C2311yescharacteristic
49C2×C22⋊D293—22no—
50C2×C22⋊D293—22no—
51D4×D293—24no—
52D4×D293—42no—
53D4×D293—44no—
54C2×C4×D293—22no—
55C2×C4×D293—21no—
56C2×C4×D293—21no—
57C2×C4×D293C2312yes—
58#17989789—24no—
59D2344D414yescomplemented
60#17989787—24no—
61#17989785—24no—
62C2×#14556873C2312yes—
63C4⋊Dic293D411yescharacteristic
64C4⋊Dic293C2311yescharacteristic
65C4⋊Dic293C2311yescharacteristic
66#17989781—22no—
67C22⋊Dic293—22no—
68#17989779—22no—
69#17989779—21no—
70#17989779—21no—
71#17989779D412yes—
72C2×Q8×C293C2312yes—
73C2×D4×C293—21no—
74C2×D4×C293—21no—
75C2×D4×C293C2312yes—
76SD16×C293—24no—
77SD16×C293—42no—
78C2×C8×C293C2311yescharacteristic
79C2×C8×C293C2311yescharacteristic
80C4⋊C4×C293—21no—
81C4⋊C4×C293—21no—
82C42×C293C2311yescharacteristic
83C22×D293—21no—
84C22×D293—21no—
85C22×D293—44no—
86C22×D293—44no—
87C22×D293—82no—
88C22×D293C2×D412yes—
89D1172—22no—
90D1172—21no—
91D1172—21no—
92D1172—44no—
93D1172C2×D412yes—
94D1172C2×D412yescomplemented
95C22⋊D293—44no—
96C22⋊D293—44no—
97C2×Dic293—21no—
98C2×Dic293—21no—
99C2×Dic293C2×D412yes—
100C4×D293—22no—
101C4×D293—22no—
102C4×D293—44no—
103C4×D293—42no—
104C4×D293—42no—
105#14556873—22no—
106C23×C293—22no—
107Q8×C293—22no—
108Q8×C293—22no—
109D4×C293—22no—
110D4×C293—22no—
111D4×C293—42no—
112D4×C293—81no—
113C2×C4×C293—22no—
114C2×C4×C293C2×D411yescharacteristic
115C2×C4×C293C2×D411yescharacteristic
116C2×C4×C293C2411yescharacteristic
117C8×C293—21no—
118C8×C293C2×D412yescomplemented
119C22×C293—42no—
120C22×C293—42no—
121C22×C293—42no—
122C22×C293C22×D411yescharacteristic
123C2×D293—22no—
124C2×D293—22no—
125C2×D293—22no—
126C2×D293—22no—
127C2×D293—22no—
128C2×D293—41no—
129C2×D293—41no—
130C2×D293—41no—
131C2×D293—41no—
132C2×D293—88no—
133C4×C293—21no—
134C4×C293—42no—
135C4×C293AGL(1,ℤ/8)12yescomplemented
136C4×C293C22×D411yescharacteristic
137C4×C293H(5,𝔽2)11yescharacteristic
138Dic293—22no—
139Dic293—41no—
140Dic293—41no—
141C2×C293—82no—
142C2×C293D4211yescharacteristic
143C2×C293C2×AGL(1,ℤ/8)11yescharacteristic
144C2×C293#57711yescharacteristic
145D293—42no—
146D293—42no—
147D293—42no—
148C293#327211yescharacteristic, complemented, Sylow
149#3272—2931nomaximal, Sylow
150C2×AGL(1,ℤ/8)—2932no—
151#547—2931no—
152D42—2931no—
153#492D29311yescharacteristic, complemented
154#468—2932no—
155#461—2931no—
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×SD16C2×D29312yescomplemented
167C2×SD16C2×D29312yescomplemented
168C2×D8—2932no—
169C2×D8—2931no—
170C2×M16—2932no—
171C4⋊Q8C2×D29311yescharacteristic, complemented
172#127C2×D29311yescharacteristic
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×D4—2931no—
182C2×C4⋊C4—5861no—
183#107—2931no—
184C4⋊C8—2931no—
185#103—2932no—
186#102—2932no—
187#102—2931no—
188#102—2931no—
189C22⋊C8—2932no—
190C4×C8C2×D29311yescharacteristic, complemented
191C24—5862no—
192#41—5864no—
193#41—11722no—
194C2×Q8C22×D29312yes—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—2931no—
198C2×D4—2931no—
199C2×D4—2931no—
200C2×D4—5862no—
201C2×D4—5862no—
202C2×D4—5864no—
203C2×D4—5862no—
204C2×D4—11722no—
205C2×D4—11724no—
206C2×D4C22×D29312yes—
207C22×C4—2932no—
208C22×C4—5862no—
209C22×C4—5861no—
210C22×C4—5861no—
211SD16—24no—
212SD16—42no—
213SD16—5864no—
214D8—2934no—
215D8—5864no—
216M16—5864no—
217C2×C8—2932no—
218C2×C8C22×D29311yescharacteristic
219C2×C8C22×D29311yescharacteristic
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—2931no—
223C4⋊C4—2931no—
224C4⋊C4—2931no—
225C4⋊C4—5862no—
226C22⋊C4—2932no—
227C22⋊C4—5862no—
228C22⋊C4—5862no—
229C22⋊C4—5861no—
230C22⋊C4—5861no—
231C42C22×D29311yescharacteristic
232C23—22no—
233C23—2932no—
234C23—23442no—
235C23—5861no—
236C23—5861no—
237C23—11724no—
238C23—11724no—
239Q8—22no—
240Q8—22no—
241D4—22no—
242D4—22no—
243D4—42no—
244D4—81no—
245D4—2932no—
246D4—2932no—
247D4—5862no—
248D4—5861no—
249D4—5861no—
250D4—11724no—
251D4—11724no—
252D4—11724no—
253C2×C4—22no—
254C2×C4—2932no—
255C2×C4—5862no—
256C2×C4—5862no—
257C2×C4—5861no—
258C2×C4—5861no—
259C2×C4—11724no—
260C2×C4—11722no—
261C2×C4—11722no—
262C2×C4D4×D29311yescharacteristic
263C2×C4D4×D29311yescharacteristic
264C2×C4C23×D29311yescharacteristic
265C8—21no—
266C8—5862no—
267C8D4×D29312yescomplemented
268C22—42no—
269C22—42no—
270C22—42no—
271C22—23448no—
272C22—5862no—
273C22—5862no—
274C22—5862no—
275C22—5862no—
276C22—5862no—
277C22—11721no—
278C22—11721no—
279C22—11721no—
280C22—11721no—
281C22C2×D4×D29311yescharacteristic
282C4—21no—
283C4—42no—
284C4—5862no—
285C4—11721no—
286C4—11721no—
287C4#2262765112yescomplemented
288C4C2×D4×D29311yescharacteristic
289C4#2262766911yescharacteristic
290C2—82no—
291C2—11722no—
292C2—11722no—
293C2—11722no—
294C2#3403676411yescharacteristic
295C2C2×#2262765111yescharacteristic
296C2#3403693511yescharacteristic
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#492×C293C2
socleC22×C293C22×D4
hypercenter#492D293
hypocenterC293#3272
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. C2×#22627651C2semidirect
2. #34036782C2semidirect
3. #34036764C2semidirect
4. #34036372C2semidirect
5. #34036331C2semidirect
6. #34036321C2semidirect
7. #34036278C2semidirect
8. #34036273C2semidirect
9. C4×D2344C2semidirect
10. #34036208C2semidirect
11. #492×C293C2semidirect
12. C2×#17989803C22semidirect
18. C2×#17989787C22semidirect
19. C2×#17989785C22semidirect
21. #22627600C22semidirect
28. C4×D1172C22semidirect
30. #22627558C22semidirect
31. C4⋊#14556873C22semidirect
32. #22627548C22semidirect
34. #22627546C22semidirect
35. #22627545C22semidirect
36. #22627536C22semidirect
37. C2×SD16×C293C22semidirect
38. C2×SD16×C293C22semidirect
39. C4⋊Q8×C293C22semidirect
41. C4×C8×C293C22semidirect
59. D2344D4semidirect
94. D1172C2×D4semidirect
118. C8×C293C2×D4semidirect
135. C4×C293AGL(1,ℤ/8)semidirect
148. C293#3272semidirect
153. #492D293semidirect
166. C2×SD16C2×D293semidirect
167. C2×SD16C2×D293semidirect
171. C4⋊Q8C2×D293semidirect
190. C4×C8C2×D293semidirect
267. C8D4×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
#340367642
#340363722
#340363312
#340363212
#340362782
#340362732
C4×D23442
#340362082
#492×C2932
#3272293