Finite Group Databasebeta

← #49169265

Subgroups — #49169265

Order 37504.

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

normal subgroups only

This group has 596 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×#22627648C212yesmaximal, complemented
2C2×#22627643C211yesmaximal, characteristic, complemented
3C2×D8×D293C211yesmaximal, characteristic, complemented
4#34036692C212yesmaximal, complemented
5#34036517C212yesmaximal, complemented
6#34036516C212yesmaximal, complemented
7#34036513C211yesmaximal, characteristic, complemented
8#34036512C211yesmaximal, characteristic, complemented
9#34036504C211yesmaximal, characteristic, complemented
10#34036083C211yesmaximal, characteristic, complemented
11#493×C293C211yesmaximal, characteristic, complemented
12C2×#17989801C2212yes—
13C2×D4×D293C2212yes—
14#22627648—28no—
15#22627643—24no—
16C2×Dic1172C2211yescharacteristic, complemented
17C2×#17989791C2212yescomplemented
18C2×#17989790C2212yescomplemented
19C2×#17989790C2212yescomplemented
20C2×D2344C2211yescharacteristic
21C2×#17989787C2212yes—
22C2×#17989787C2212yescomplemented
23D8×D293—24no—
24C2×#17989785C2211yescharacteristic
25C2×C8×D293C2211yescharacteristic
26#22627620C2212yescomplemented
27#22627612C2212yescomplemented
28#22627611C2212yescomplemented
29#22627596—24no—
30#22627592—24no—
31#22627581C2211yescharacteristic
32#22627535C2211yescharacteristic, complemented
33#22627534C2211yescharacteristic, complemented
34C8×Dic293C2211yescharacteristic, complemented
35C2×D8×C293C2211yescharacteristic
36C2×D8×C293C2211yescharacteristic, complemented
37C2×D8×C293C2212yescomplemented
38#127×C293C2212yescomplemented
39C4×C8×C293C2211yescharacteristic, complemented
40C23×D293—22no—
41#17989801—24no—
42#17989801—24no—
43C2×Dic586C2312yes—
44C2×D1172C2312yes—
45C2×C22⋊D293—22no—
46C2×C22⋊D293—22no—
47C2×C22⋊D293—24no—
48D4×D293—24no—
49D4×D293—24no—
50C22×Dic293—22no—
51C2×C4×D293C2311yescharacteristic
52Dic1172—22no—
53#17989791—24no—
54#17989790—24no—
55#17989790—24no—
56D2344—22no—
57#17989787—24no—
58#17989787—24no—
59#17989785D414yescomplemented
60C2×#14556873C2311yescharacteristic
61C2×#14556873C2311yescharacteristic
62C8×D293D414yescomplemented
63#17989781D412yes—
64C22⋊Dic293—24no—
65#17989779—24no—
66#17989779D412yes—
67C4×Dic293C2311yescharacteristic
68C2×D4×C293—24no—
69C2×D4×C293C2312yes—
70C2×D4×C293C2312yes—
71D8×C293—22no—
72D8×C293—22no—
73D8×C293—24no—
74C2×C8×C293C2311yescharacteristic
75C2×C8×C293C2311yescharacteristic
76C42×C293C2311yescharacteristic
77C22×D293—22no—
78C22×D293—44no—
79C22×D293—84no—
80C22×D293C2×D411yescharacteristic
81Dic586—22no—
82Dic586—22no—
83D1172—22no—
84D1172—22no—
85C22⋊D293—44no—
86C22⋊D293—44no—
87C22⋊D293—84no—
88C2×Dic293—22no—
89C2×Dic293—44no—
90C2×Dic293C2×D412yes—
91C2×Dic293C2×D411yescharacteristic
92C4×D293C2×D412yes—
93C4×D293C2×D412yes—
94#14556873C2×D412yes—
95#14556873C2×D412yes—
96C23×C293—22no—
97C23×C293—22no—
98D4×C293—22no—
99D4×C293—22no—
100D4×C293—22no—
101D4×C293—22no—
102D4×C293—84no—
103C2×C4×C293C2×D412yes—
104C2×C4×C293C2411yescharacteristic
105C8×C293C2×D412yes—
106C8×C293C2×D412yes—
107C22×C293—42no—
108C22×C293—42no—
109C22×C293—44no—
110C22×C293—44no—
111C22×C293C22×D411yescharacteristic
112C2×D293—22no—
113C2×D293—42no—
114C2×D293—44no—
115C2×D293—88no—
116C2×D293#12712yes—
117C4×C293—22no—
118C4×C293C22×D411yescharacteristic
119C4×C293C22×D411yescharacteristic
120Dic293—22no—
121Dic293—42no—
122Dic293#12712yes—
123C2×C293—82no—
124C2×C293—82no—
125C2×C293C2×#12711yescharacteristic
126C2×C293#57612yes—
127D293—22no—
128D293—82no—
129C293#315511yescharacteristic, complemented, Sylow
130#3155—2931nomaximal, Sylow
131C2×AGL(1,ℤ/8)—2932no—
132C2×#135—2931no—
133C22×D8—2931no—
134#535—2932no—
135#496—2932no—
136#496—2932no—
137#495—2931no—
138#495—2931no—
139#493—2931no—
140#493D29311yescharacteristic, complemented
141#405—2931no—
142C2×#41—2932no—
143C22×D4—2932no—
144AGL(1,ℤ/8)—5868no—
145#135—5864no—
146C2×Q16—2931no—
147C2×SD16—2932no—
148C2×SD16—2932no—
149C2×SD16—2932no—
150C2×D8—2932no—
151C2×D8—2932no—
152C2×D8—2931no—
153C2×D8—5864no—
154C2×D8C2×D29311yescharacteristic
155C2×D8C2×D29311yescharacteristic, complemented
156C2×D8C2×D29312yescomplemented
157C2×M16—2931no—
158C22×C8—2931no—
159#127—2932no—
160#127C2×D29312yescomplemented
161#124—2932no—
162#124—2932no—
163#121—5864no—
164#120—5864no—
165#117—2931no—
166#97—2931no—
167#97—2931no—
168C4×C8—2931no—
169C4×C8C2×D29311yescharacteristic, complemented
170C24—5862no—
171#41—5864no—
172#41—5864no—
173C2×Q8—2932no—
174C2×D4—24no—
175C2×D4—2932no—
176C2×D4—5864no—
177C2×D4—5864no—
178C2×D4—5864no—
179C2×D4—5862no—
180C2×D4—5862no—
181C2×D4C22×D29312yes—
182C2×D4C22×D29312yes—
183C22×C4—2931no—
184C22×C4—5862no—
185Q16—5862no—
186SD16—5864no—
187SD16—5864no—
188SD16—5864no—
189D8—22no—
190D8—22no—
191D8—24no—
192D8—5864no—
193D8—5864no—
194D8—5862no—
195M16—2934no—
196C2×C8—2931no—
197C2×C8—2931no—
198C2×C8—2934no—
199C2×C8C22×D29311yescharacteristic
200C2×C8C22×D29311yescharacteristic
201C4⋊C4—2932no—
202C22⋊C4—2932no—
203C22⋊C4—5864no—
204C22⋊C4—5864no—
205C42—2931no—
206C42C22×D29311yescharacteristic
207C23—22no—
208C23—22no—
209C23—2931no—
210C23—23444no—
211C23—5862no—
212C23—11724no—
213Q8—5862no—
214Q8—5862no—
215D4—22no—
216D4—22no—
217D4—22no—
218D4—22no—
219D4—84no—
220D4—23444no—
221D4—5862no—
222D4—5862no—
223D4—11724no—
224D4—11724no—
225C2×C4—2931no—
226C2×C4—2932no—
227C2×C4—2932no—
228C2×C4—2932no—
229C2×C4—5862no—
230C2×C4—11724no—
231C2×C4D4×D29312yes—
232C2×C4C23×D29311yescharacteristic
233C8—2932no—
234C8—2932no—
235C8D4×D29312yes—
236C8D4×D29312yes—
237C22—42no—
238C22—42no—
239C22—44no—
240C22—44no—
241C22—2932no—
242C22—23448no—
243C22—5862no—
244C22—11722no—
245C22—11724no—
246C22C2×D4×D29311yescharacteristic
247C4—22no—
248C4—2932no—
249C4—5862no—
250C4—11722no—
251C4C2×D4×D29311yescharacteristic
252C4C2×D4×D29311yescharacteristic
253C2—82no—
254C2—82no—
255C2—23442no—
256C2—5862no—
257C2#127×D29311yescharacteristic
258C2#3403693012yes—
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#493×C293C2
socleC22×C293C22×D4
hypercenter#493D293
hypocenterC293#3155
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C42 ◁ #493

Complements

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

NH ≅ G/Ntype
1. C2×#22627648C2semidirect
2. C2×#22627643C2semidirect
3. C2×D8×D293C2semidirect
4. #34036692C2semidirect
5. #34036517C2semidirect
6. #34036516C2semidirect
7. #34036513C2semidirect
8. #34036512C2semidirect
9. #34036504C2semidirect
10. #34036083C2semidirect
11. #493×C293C2semidirect
16. C2×Dic1172C22semidirect
17. C2×#17989791C22semidirect
18. C2×#17989790C22semidirect
19. C2×#17989790C22semidirect
22. C2×#17989787C22semidirect
26. #22627620C22semidirect
27. #22627612C22semidirect
28. #22627611C22semidirect
32. #22627535C22semidirect
33. #22627534C22semidirect
34. C8×Dic293C22semidirect
36. C2×D8×C293C22semidirect
37. C2×D8×C293C22semidirect
38. #127×C293C22semidirect
39. C4×C8×C293C22semidirect
59. #17989785D4semidirect
62. C8×D293D4semidirect
129. C293#3155semidirect
140. #493D293semidirect
155. C2×D8C2×D293semidirect
156. C2×D8C2×D293semidirect
160. #127C2×D293semidirect
169. C4×C8C2×D293semidirect

Maximal subgroups

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

subgroupindex
C2×#226276482
C2×#226276432
C2×D8×D2932
#340366922
#340365172
#340365162
#340365132
#340365122
#340365042
#340360832
#493×C2932
#3155293