Finite Group Databasebeta

← #49169250

Subgroups — #49169250

Order 37504.

572 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 137 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×#22627653C212yesmaximal, complemented
2C2×#22627648C212yesmaximal, complemented
3#34036677C211yesmaximal, characteristic, complemented
4#124×D293C211yesmaximal, characteristic, complemented
5#34036524C211yesmaximal, characteristic, complemented
6#34036523C211yesmaximal, characteristic, complemented
7#34036516C211yesmaximal, characteristic, complemented
8#34036512C212yesmaximal, complemented
9#34036510C212yesmaximal, complemented
10#34036081C211yesmaximal, characteristic, complemented
11#496×C293C211yesmaximal, characteristic, complemented
12C2×#17989801C2212yes—
13C2×Q8×D293C2211yescharacteristic
14C2×D4×D293C2211yescharacteristic
15#22627653—28no—
16#22627648—28no—
17C2×Dic1172C2212yescomplemented
18C2×#17989792C2212yescomplemented
19C2×#17989791C2212yes—
20C2×#17989790C2212yes—
21C2×#17989790C2212yescomplemented
22C2×#17989787C2212yes—
23C2×#17989785C2212yes—
24#22627624C2211yescharacteristic, complemented
25#22627613C2211yescharacteristic
26#22627612C2211yescharacteristic
27#22627611C2211yescharacteristic
28#22627611C2212yescomplemented
29#22627610—22no—
30#22627595—22no—
31D4×Dic293—22no—
32C22⋊C4×D293—22no—
33C42×D293C2211yescharacteristic
34#22627535C2211yescharacteristic, complemented
35C8×Dic293C2212yescomplemented
36C2×SD16×C293C2212yescomplemented
37C2×D8×C293C2212yes—
38#127×C293C2211yescharacteristic, complemented
39#124×C293C2211yescharacteristic
40#97×C293C2211yescharacteristic, complemented
41C23×D293—21no—
42#17989801—24no—
43#17989801—24no—
44C2×Dic586—21no—
45C2×Dic586C2311yescharacteristic
46C2×Dic586C2312yes—
47Q8×D293—24no—
48C2×D1172C2311yescharacteristic
49C2×C22⋊D293—22no—
50C2×C22⋊D293—21no—
51D4×D293—24no—
52C22×Dic293—22no—
53C2×C4×D293—21no—
54C2×C4×D293D412yes—
55C2×C4×D293C2311yescharacteristic
56Dic1172—24no—
57#17989792—24no—
58#17989791—24no—
59#17989790—24no—
60#17989790—24no—
61#17989787—24no—
62#17989785D418yescomplemented
63C2×#14556873C2312yes—
64C4⋊Dic293—22no—
65C22⋊Dic293—24no—
66C22⋊Dic293—22no—
67#17989779—22no—
68#17989779—22no—
69C4×Dic293D412yes—
70C4×Dic293C2311yescharacteristic
71C2×Q8×C293C2311yescharacteristic
72C2×D4×C293—22no—
73C2×D4×C293C2311yescharacteristic
74C2×D4×C293C2312yes—
75SD16×C293—24no—
76D8×C293—24no—
77C2×C8×C293C2312yes—
78C22⋊C4×C293—22no—
79C42×C293C2311yescharacteristic
80C22×D293—21no—
81C22×D293—44no—
82C22×D293—42no—
83C22×D293C2×D411yescharacteristic
84Dic586—22no—
85Dic586—22no—
86Dic586—22no—
87Dic586—42no—
88D1172—22no—
89C22⋊D293—44no—
90C22⋊D293—42no—
91C2×Dic293—21no—
92C2×Dic293—22no—
93C2×Dic293—44no—
94C2×Dic293C2×D412yes—
95C2×Dic293C2×D411yescharacteristic
96C4×D293—42no—
97C4×D293—42no—
98C4×D293C2×D414yes—
99#14556873C2×D414yes—
100C23×C293—21no—
101C23×C293—22no—
102Q8×C293—22no—
103D4×C293—22no—
104D4×C293—22no—
105D4×C293—22no—
106D4×C293—44no—
107C2×C4×C293—21no—
108C2×C4×C293C2×D412yes—
109C2×C4×C293C2411yescharacteristic
110C8×C293C2×D414yes—
111C22×C293—42no—
112C22×C293—41no—
113C22×C293—44no—
114C22×C293—42no—
115C22×C293C22×D411yescharacteristic
116C2×D293—22no—
117C2×D293—41no—
118C2×D293—42no—
119C2×D293—84no—
120C2×D293#12712yes—
121C4×C293—22no—
122C4×C293—41no—
123C4×C293C22×D412yes—
124Dic293—22no—
125Dic293—41no—
126Dic293—42no—
127Dic293#12712yes—
128C2×C293—81no—
129C2×C293—82no—
130C2×C293C2×#12711yescharacteristic
131C2×C293#57512yes—
132D293—22no—
133D293—81no—
134C293#315311yescharacteristic, complemented, Sylow
135#3153—2931nomaximal, Sylow
136C2×#137—2932no—
137C2×AGL(1,ℤ/8)—2932no—
138#532—2931no—
139C2×#124—2931no—
140#497—2931no—
141#497—2931no—
142#496—2931no—
143#496D29311yescharacteristic, complemented
144#495—2932no—
145#495—2932no—
146C4×M16—2931no—
147C2×#41—2932no—
148C22×Q8—2931no—
149C22×D4—2931no—
150#137—5868no—
151AGL(1,ℤ/8)—5868no—
152C2×Q16—2932no—
153C2×Q16—2932no—
154C2×SD16—2932no—
155C2×SD16—2932no—
156C2×SD16—2932no—
157C2×SD16C2×D29312yescomplemented
158C2×D8—2932no—
159C2×D8C2×D29312yes—
160C2×M16—2932no—
161C4⋊Q8—2931no—
162#127C2×D29311yescharacteristic, complemented
163#124—2931no—
164#124—2931no—
165#124—2932no—
166#124—2931no—
167#124—5862no—
168#124C2×D29311yescharacteristic
169#121—5862no—
170C4×D4—5862no—
171C2×C22⋊C4—5862no—
172C2×C42—2931no—
173#97—2931no—
174#97C2×D29311yescharacteristic, complemented
175C4×C8—2932no—
176C24—5861no—
177#41—5864no—
178#41—5864no—
179C2×Q8—2931no—
180C2×Q8—2932no—
181C2×Q8—5861no—
182C2×Q8—5864no—
183C2×Q8C22×D29311yescharacteristic
184C2×D4—22no—
185C2×D4—2931no—
186C2×D4—5864no—
187C2×D4—5861no—
188C2×D4—5862no—
189C2×D4C22×D29311yescharacteristic
190C2×D4C22×D29312yes—
191C22×C4—2931no—
192C22×C4—2932no—
193C22×C4—5861no—
194C22×C4—5862no—
195Q16—5864no—
196Q16—5864no—
197SD16—24no—
198SD16—5864no—
199SD16—5864no—
200SD16—5864no—
201D8—24no—
202D8—5864no—
203M16—2938no—
204C2×C8—2932no—
205C2×C8C22×D29312yes—
206C4⋊C4—5862no—
207C22⋊C4—22no—
208C22⋊C4—5862no—
209C22⋊C4—5862no—
210C22⋊C4—5864no—
211C22⋊C4—5862no—
212C42—2932no—
213C42—2931no—
214C42C22×D29311yescharacteristic
215C23—21no—
216C23—22no—
217C23—2931no—
218C23—5861no—
219C23—11722no—
220C23—11724no—
221Q8—22no—
222Q8—5862no—
223Q8—5862no—
224Q8—5862no—
225Q8—11722no—
226D4—22no—
227D4—22no—
228D4—22no—
229D4—44no—
230D4—5862no—
231D4—11722no—
232D4—11724no—
233C2×C4—21no—
234C2×C4—2931no—
235C2×C4—2934no—
236C2×C4—2932no—
237C2×C4—5861no—
238C2×C4—5862no—
239C2×C4—11722no—
240C2×C4—11722no—
241C2×C4—11724no—
242C2×C4D4×D29312yes—
243C2×C4C23×D29311yescharacteristic
244C8—2934no—
245C8D4×D29314yes—
246C22—42no—
247C22—41no—
248C22—44no—
249C22—42no—
250C22—2932no—
251C22—23444no—
252C22—5862no—
253C22—11721no—
254C22—11722no—
255C22C2×D4×D29311yescharacteristic
256C4—22no—
257C4—41no—
258C4—2932no—
259C4—5862no—
260C4—11722no—
261C4—11721no—
262C4C2×D4×D29312yes—
263C2—81no—
264C2—82no—
265C2—23441no—
266C2—5862no—
267C2#127×D29311yescharacteristic
268C2#3403692512yes—
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#496×C293C2
socleC22×C293C22×D4
hypercenter#496D293
hypocenterC293#3153
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×#22627653C2semidirect
2. C2×#22627648C2semidirect
3. #34036677C2semidirect
4. #124×D293C2semidirect
5. #34036524C2semidirect
6. #34036523C2semidirect
7. #34036516C2semidirect
8. #34036512C2semidirect
9. #34036510C2semidirect
10. #34036081C2semidirect
11. #496×C293C2semidirect
17. C2×Dic1172C22semidirect
18. C2×#17989792C22semidirect
21. C2×#17989790C22semidirect
24. #22627624C22semidirect
28. #22627611C22semidirect
34. #22627535C22semidirect
35. C8×Dic293C22semidirect
36. C2×SD16×C293C22semidirect
38. #127×C293C22semidirect
40. #97×C293C22semidirect
62. #17989785D4semidirect
134. C293#3153semidirect
143. #496D293semidirect
157. C2×SD16C2×D293semidirect
162. #127C2×D293semidirect
174. #97C2×D293semidirect

Maximal subgroups

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

subgroupindex
C2×#226276532
C2×#226276482
#340366772
#124×D2932
#340365242
#340365232
#340365162
#340365122
#340365102
#340360812
#496×C2932
#3153293