Finite Group Databasebeta

← #496×D293 #49169215

Subgroups — #496×D293 #49169215

Order 37504.

668 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 145 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×SD16×D293C212yesmaximal, complemented
2C2×D8×D293C212yesmaximal, complemented
3#127×D293C211yesmaximal, characteristic, complemented
4#124×D293C211yesmaximal, characteristic, complemented
5#34036519C211yesmaximal, characteristic, complemented
6#34036518C211yesmaximal, characteristic, complemented
7#34036517C212yesmaximal, complemented
8#34036516C211yesmaximal, characteristic, complemented
9#34036515C212yesmaximal, complemented
10#97×D293C211yesmaximal, characteristic, complemented
11#496×C293C211yesmaximal, characteristic, complemented
12C2×Q8×D293C2211yescharacteristic
13C2×D4×D293—22no—
14C2×D4×D293C2211yescharacteristic
15C2×D4×D293C2212yes—
16C2×#17989791C2212yescomplemented
17C2×#17989790C2212yescomplemented
18C2×#17989789C2212yescomplemented
19SD16×D293—28no—
20C2×D2344C2212yescomplemented
21C2×#17989787C2212yescomplemented
22C2×#17989787C2212yescomplemented
23D8×D293—28no—
24C2×C8×D293C2212yes—
25#22627621C2211yescharacteristic, complemented
26#22627620—22no—
27#22627620C2212yescomplemented
28#22627613C2211yescharacteristic, complemented
29#22627612C2211yescharacteristic, complemented
30#22627611C2211yescharacteristic, complemented
31#22627610—22no—
32C22⋊C4×D293—22no—
33C42×D293C2211yescharacteristic
34#22627535C2211yescharacteristic, complemented
35#22627534C2212yescomplemented
36C2×SD16×C293C2212yescomplemented
37C2×D8×C293C2212yescomplemented
38#127×C293C2211yescharacteristic, complemented
39#124×C293C2211yescharacteristic, complemented
40#97×C293C2211yescharacteristic, complemented
41C23×D293—22no—
42C23×D293—21no—
43C2×Dic586—21no—
44C2×Dic586C2311yescharacteristic
45Q8×D293—24no—
46C2×D1172—22no—
47C2×D1172C2311yescharacteristic
48C2×D1172C2312yes—
49C2×C22⋊D293—21no—
50C2×C22⋊D293—22no—
51C2×C22⋊D293—24no—
52D4×D293—24no—
53D4×D293—24no—
54D4×D293—24no—
55D4×D293—48no—
56C2×C4×D293—21no—
57C2×C4×D293D412yes—
58C2×C4×D293C2311yescharacteristic
59#17989791—24no—
60#17989790—24no—
61#17989789—24no—
62D2344—24no—
63#17989787—24no—
64#17989787—24no—
65C2×#14556873C2312yes—
66C8×D293D418yescomplemented
67C22⋊Dic293—22no—
68#17989779—22no—
69#17989779—22no—
70C4×Dic293D412yes—
71C4×Dic293C2311yescharacteristic
72C2×Q8×C293C2311yescharacteristic
73C2×D4×C293—22no—
74C2×D4×C293C2311yescharacteristic
75C2×D4×C293C2312yes—
76SD16×C293—24no—
77D8×C293—24no—
78C2×C8×C293C2312yes—
79C22⋊C4×C293—22no—
80C42×C293C2311yescharacteristic
81C22×D293—21no—
82C22×D293—22no—
83C22×D293—42no—
84C22×D293—44no—
85C22×D293—48no—
86C22×D293—44no—
87C22×D293C2×D411yescharacteristic
88Dic586—22no—
89Dic586—42no—
90D1172—22no—
91D1172—22no—
92D1172—22no—
93D1172—44no—
94C22⋊D293—48no—
95C22⋊D293—42no—
96C22⋊D293—44no—
97C2×Dic293—21no—
98C2×Dic293C2×D412yes—
99C2×Dic293C2×D411yescharacteristic
100C4×D293—24no—
101C4×D293—42no—
102C4×D293C2×D414yes—
103#14556873C2×D414yescomplemented
104C23×C293—21no—
105C23×C293—22no—
106Q8×C293—22no—
107D4×C293—22no—
108D4×C293—22no—
109D4×C293—22no—
110D4×C293—44no—
111C2×C4×C293—21no—
112C2×C4×C293C2×D412yes—
113C2×C4×C293C2411yescharacteristic
114C8×C293C2×D414yescomplemented
115C22×C293—41no—
116C22×C293—42no—
117C22×C293—44no—
118C22×C293—42no—
119C22×C293C22×D411yescharacteristic
120C2×D293—41no—
121C2×D293—42no—
122C2×D293—44no—
123C2×D293—42no—
124C2×D293—88no—
125C2×D293—84no—
126C2×D293#12712yes—
127C2×D293AGL(1,ℤ/8)14yes—
128C4×C293—22no—
129C4×C293—41no—
130C4×C293C22×D412yes—
131Dic293—22no—
132Dic293—41no—
133Dic293#12712yes—
134C2×C293—82no—
135C2×C293—81no—
136C2×C293C2×#12711yescharacteristic
137C2×C293C2×AGL(1,ℤ/8)12yes—
138D293—82no—
139D293—81no—
140D293#49614yesdirect product factor
141C293C2×#49611yescharacteristic, complemented, Sylow
142C2×#496—2931nomaximal, Sylow
143C22×SD16—2932no—
144C22×D8—2932no—
145C2×#127—2931no—
146C2×#124—2931no—
147#496—2932no—
148#496—2932no—
149#496—2931no—
150#496—2931no—
151#496—2931no—
152#496D29311yescharacteristic, direct product factor
153C2×#97—2931no—
154C22×Q8—2931no—
155C22×D4—2931no—
156C22×D4—2932no—
157C22×D4—5862no—
158C2×SD16—2932no—
159C2×SD16—2932no—
160C2×SD16—2932no—
161C2×SD16—5868no—
162C2×SD16C2×D29312yescomplemented
163C2×D8—2932no—
164C2×D8—2932no—
165C2×D8—2932no—
166C2×D8—5868no—
167C2×D8C2×D29312yescomplemented
168C22×C8—2932no—
169#127—2932no—
170#127—2931no—
171#127—5862no—
172#127C2×D29311yescharacteristic, complemented
173#124—2931no—
174#124—2931no—
175#124—2931no—
176#124—5862no—
177#124C2×D29311yescharacteristic, complemented
178C2×C22⋊C4—5862no—
179C2×C42—2931no—
180#97—2932no—
181#97—2931no—
182#97C2×D29311yescharacteristic, complemented
183C24—5861no—
184C24—5862no—
185C2×Q8—2931no—
186C2×Q8—5861no—
187C2×Q8—5864no—
188C2×Q8C22×D29311yescharacteristic
189C2×D4—22no—
190C2×D4—2931no—
191C2×D4—2932no—
192C2×D4—5864no—
193C2×D4—5864no—
194C2×D4—5864no—
195C2×D4—5861no—
196C2×D4—5864no—
197C2×D4—5862no—
198C2×D4—5862no—
199C2×D4—11728no—
200C2×D4C22×D29311yescharacteristic
201C2×D4C22×D29312yes—
202C22×C4—2931no—
203C22×C4—2932no—
204C22×C4—5861no—
205SD16—24no—
206SD16—5864no—
207SD16—5864no—
208SD16—5864no—
209D8—24no—
210D8—5864no—
211D8—5864no—
212D8—5864no—
213C2×C8—2932no—
214C2×C8—2938no—
215C2×C8C22×D29312yes—
216C22⋊C4—22no—
217C22⋊C4—5862no—
218C22⋊C4—5862no—
219C22⋊C4—5862no—
220C42—2932no—
221C42—2931no—
222C42C22×D29311yescharacteristic
223C23—21no—
224C23—22no—
225C23—2931no—
226C23—5861no—
227C23—5862no—
228C23—11722no—
229C23—11724no—
230C23—11728no—
231C23—11724no—
232Q8—22no—
233Q8—5862no—
234Q8—11722no—
235D4—22no—
236D4—22no—
237D4—22no—
238D4—44no—
239D4—5862no—
240D4—5862no—
241D4—5862no—
242D4—11724no—
243D4—11722no—
244D4—11724no—
245D4—11728no—
246C2×C4—21no—
247C2×C4—2931no—
248C2×C4—2934no—
249C2×C4—2932no—
250C2×C4—5864no—
251C2×C4—5861no—
252C2×C4—11722no—
253C2×C4D4×D29312yes—
254C2×C4C23×D29311yescharacteristic
255C8—2934no—
256C8D4×D29314yescomplemented
257C22—44no—
258C22—42no—
259C22—41no—
260C22—42no—
261C22—2934no—
262C22—2932no—
263C22—23448no—
264C22—23444no—
265C22—11724no—
266C22—11722no—
267C22—11721no—
268C22—11722no—
269C22C2×D4×D29311yescharacteristic
270C4—22no—
271C4—41no—
272C4—2932no—
273C4—5862no—
274C4—11721no—
275C4C2×D4×D29312yes—
276C2—82no—
277C2—81no—
278C2—2934no—
279C2—23442no—
280C2—23441no—
281C2#127×D29311yescharacteristic
282C2AGL(1,ℤ/8)×D29312yes—
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#496×C293C2
socleC22×C293C22×D4
hypercenter#496D293
hypocenterC293C2×#496
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D293C2semidirect
2. C2×D8×D293C2semidirect
3. #127×D293C2semidirect
4. #124×D293C2semidirect
5. #34036519C2semidirect
6. #34036518C2semidirect
7. #34036517C2semidirect
8. #34036516C2semidirect
9. #34036515C2semidirect
10. #97×D293C2semidirect
11. #496×C293C2semidirect
16. C2×#17989791C22semidirect
17. C2×#17989790C22semidirect
18. C2×#17989789C22semidirect
20. C2×D2344C22semidirect
21. C2×#17989787C22semidirect
22. C2×#17989787C22semidirect
25. #22627621C22semidirect
27. #22627620C22semidirect
28. #22627613C22semidirect
29. #22627612C22semidirect
30. #22627611C22semidirect
34. #22627535C22semidirect
35. #22627534C22semidirect
36. C2×SD16×C293C22semidirect
37. C2×D8×C293C22semidirect
38. #127×C293C22semidirect
39. #124×C293C22semidirect
40. #97×C293C22semidirect
66. C8×D293D4semidirect
103. #14556873C2×D4semidirect
114. C8×C293C2×D4semidirect
140. D293#496direct (N × H)
141. C293C2×#496semidirect
152. #496D293direct (N × H)
162. C2×SD16C2×D293semidirect
167. C2×D8C2×D293semidirect
172. #127C2×D293semidirect
177. #124C2×D293semidirect
182. #97C2×D293semidirect
256. C8D4×D293semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D2932
C2×D8×D2932
#127×D2932
#124×D2932
#340365192
#340365182
#340365172
#340365162
#340365152
#97×D2932
#496×C2932
C2×#496293