Finite Group Databasebeta

← #49170583

Subgroups — #49170583

Order 37504.

476 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#34036934C211yesmaximal, characteristic, complemented
2#34036933C212yesmaximal, complemented
3#34036930C211yesmaximal, characteristic, complemented
4#34036504C211yesmaximal, characteristic, complemented
5#34036400C211yesmaximal, characteristic, complemented
6#34036398C212yesmaximal, complemented
7#34036299C212yesmaximal, complemented
8#34036295C212yesmaximal, complemented
9#34036245C211yesmaximal, characteristic, complemented
10#34036244C211yesmaximal, characteristic, complemented
11#472×C293C211yesmaximal, characteristic, complemented
12#22627670C2212yescomplemented
13#22627669C2212yescomplemented
14#22627652—22no—
15#22627652C2212yescomplemented
16#22627652C2212yescomplemented
17#22627651—22no—
18#22627650C2212yescomplemented
19#22627648—22no—
20#22627648—22no—
21#22627646—21no—
22#22627646C2211yescharacteristic
23#22627645—22no—
24#22627644C2211yescharacteristic
25#22627643—21no—
26C2×D2344—21no—
27C2×D2344C2211yescharacteristic, complemented
28C2×#17989787C2212yescomplemented
29D8×D293—21no—
30D8×D293—22no—
31#22627632C2212yescomplemented
32#22627630C2211yescharacteristic, complemented
33#22627620C2212yescomplemented
34#22627559C2211yescharacteristic, complemented
35#22627551C2211yescharacteristic, complemented
36#22627550C2211yescharacteristic, complemented
37#22627550C2212yescomplemented
38#22627540C2212yescomplemented
39C8×Dic293C2211yescharacteristic, complemented
40AGL(1,ℤ/8)×C293C2212yescomplemented
41C2×D8×C293C2211yescharacteristic, complemented
42#131×C293C2211yescharacteristic, complemented
43#108×C293C2211yescharacteristic, complemented
44#100×C293C2212yescomplemented
45#17989803—22no—
46#17989802—22no—
47#17989802C2311yescharacteristic
48#17989802C2312yes—
49#17989801—22no—
50#17989801—22no—
51C2×D1172—22no—
52C2×D1172C2312yes—
53C2×C22⋊D293—42no—
54C2×C22⋊D293—42no—
55D4×D293—22no—
56D4×D293—22no—
57D4×D293—22no—
58D4×D293—42no—
59Dic1172D411yescharacteristic, complemented
60#17989792D411yescharacteristic, complemented
61#17989791—22no—
62#17989791—22no—
63#17989791D412yescomplemented
64#17989790—22no—
65#17989790D411yescharacteristic, complemented
66#17989789D411yescharacteristic, complemented
67D2344—22no—
68D2344—22no—
69D2344—21no—
70D2344—21no—
71D2344D411yescharacteristic, complemented
72#17989787—22no—
73#17989787—22no—
74#17989787—22no—
75#17989787D411yescharacteristic, complemented
76#17989786C2311yescharacteristic
77#17989785—22no—
78#17989785—21no—
79C2×#14556873C2311yescharacteristic
80C8×D293—22no—
81C8×D293—21no—
82C4×Dic293C2311yescharacteristic
83#41×C293C2311yescharacteristic
84C2×D4×C293C2312yes—
85SD16×C293—22no—
86D8×C293—22no—
87D8×C293—21no—
88D8×C293—21no—
89M16×C293—21no—
90M16×C293C2312yes—
91M16×C293C2311yescharacteristic
92C2×C8×C293—21no—
93C2×C8×C293C2311yescharacteristic
94C22×D293—22no—
95C22×D293—42no—
96C22×D293—42no—
97Dic586C2×D412yes—
98Dic586C2×D411yescharacteristic
99D1172—22no—
100D1172—22no—
101D1172—42no—
102D1172C2×D412yes—
103D1172C2×D411yescharacteristic
104C22⋊D293—22no—
105C22⋊D293—21no—
106C22⋊D293—42no—
107C22⋊D293—42no—
108C22⋊D293—42no—
109C22⋊D293—84no—
110C2×Dic293—21no—
111C2×Dic293—42no—
112C4×D293—21no—
113C4×D293—22no—
114C4×D293—42no—
115#14556873—21no—
116#14556873C2×D411yescharacteristic
117#14556873C2×D411yescharacteristic
118C23×C293—22no—
119Q8×C293C2×D411yescharacteristic
120D4×C293—21no—
121D4×C293—22no—
122D4×C293—22no—
123D4×C293C2×D411yescharacteristic
124C2×C4×C293—21no—
125C2×C4×C293C2411yescharacteristic
126C8×C293—21no—
127C8×C293—22no—
128C8×C293C2×D411yescharacteristic
129C8×C293C2×D411yescharacteristic
130C22×C293—21no—
131C22×C293—42no—
132C22×C293—82no—
133C22×C293H(5,𝔽2)11yescharacteristic
134C2×D293—21no—
135C2×D293—22no—
136C2×D293—42no—
137C2×D293—82no—
138C2×D293—162no—
139C2×D293—162no—
140C4×C293—21no—
141C4×C293C22×D411yescharacteristic
142C4×C293C22×D411yescharacteristic
143Dic293—22no—
144Dic293—22no—
145Dic293—21no—
146C2×C293—21no—
147C2×C293—41no—
148C2×C293—82no—
149C2×C293D4211yescharacteristic
150D293—41no—
151D293—42no—
152D293—82no—
153C293#329211yescharacteristic, complemented, Sylow
154#3292—2931nomaximal, Sylow
155#576—2932no—
156#576—2931no—
157#576—2931no—
158#493—2931no—
159#472—2931no—
160#472—2932no—
161#472D29311yescharacteristic, complemented
162#453—2932no—
163#453—2932no—
164#443—2931no—
165#443—2931no—
166H(5,𝔽2)—2932no—
167H(5,𝔽2)—2932no—
168AGL(1,ℤ/8)—2932no—
169AGL(1,ℤ/8)—2932no—
170AGL(1,ℤ/8)—2932no—
171AGL(1,ℤ/8)—5862no—
172AGL(1,ℤ/8)—5862no—
173AGL(1,ℤ/8)—5862no—
174AGL(1,ℤ/8)—5862no—
175AGL(1,ℤ/8)C2×D29312yescomplemented
176#135—2931no—
177#135—2931no—
178#135—5861no—
179#135—5861no—
180#135—5862no—
181C2×D8—2932no—
182C2×D8—2931no—
183C2×D8—5861no—
184C2×D8—5862no—
185C2×D8—5861no—
186C2×D8C2×D29311yescharacteristic, complemented
187#131—2932no—
188#131—2931no—
189#131C2×D29311yescharacteristic, complemented
190#127—2932no—
191#108—2931no—
192#108C2×D29311yescharacteristic, complemented
193#104—2931no—
194#104—2932no—
195#104—2931no—
196#100—2932no—
197#100C2×D29312yescomplemented
198C4×C8—2931no—
199#41—2932no—
200#41—2931no—
201#41—5862no—
202#41—5862no—
203#41—5862no—
204#41—5862no—
205#41C22×D29311yescharacteristic
206C2×D4—2932no—
207C2×D4—5862no—
208C2×D4—5862no—
209C2×D4—5862no—
210C2×D4—5862no—
211C2×D4—11722no—
212C2×D4—11722no—
213C2×D4—11722no—
214C2×D4C22×D29312yes—
215Q16—2931no—
216Q16—2931no—
217SD16—22no—
218SD16—2931no—
219SD16—2931no—
220SD16—2932no—
221SD16—5862no—
222SD16—5862no—
223SD16—5862no—
224D8—22no—
225D8—21no—
226D8—21no—
227D8—2931no—
228D8—2931no—
229D8—5862no—
230D8—5862no—
231D8—5862no—
232D8—5862no—
233D8—5862no—
234D8—5861no—
235D8—5861no—
236M16—21no—
237M16—2931no—
238M16—5862no—
239M16—5861no—
240M16C22×D29312yes—
241M16C22×D29311yescharacteristic
242C2×C8—21no—
243C2×C8—2931no—
244C2×C8—5862no—
245C2×C8—5861no—
246C2×C8C22×D29311yescharacteristic
247C42—2931no—
248C23—22no—
249C23—5862no—
250C23—11722no—
251C23—11722no—
252Q8—2931no—
253Q8—2932no—
254Q8D4×D29311yescharacteristic
255D4—21no—
256D4—22no—
257D4—22no—
258D4—2931no—
259D4—2932no—
260D4—23444no—
261D4—5862no—
262D4—5861no—
263D4—5862no—
264D4—5862no—
265D4—11722no—
266D4—11722no—
267D4—11722no—
268D4—11722no—
269D4D4×D29311yescharacteristic
270C2×C4—21no—
271C2×C4—5861no—
272C2×C4—5862no—
273C2×C4—5861no—
274C2×C4—11722no—
275C2×C4—11722no—
276C2×C4C23×D29311yescharacteristic
277C8—21no—
278C8—22no—
279C8—2931no—
280C8—2931no—
281C8—5861no—
282C8D4×D29311yescharacteristic
283C8D4×D29311yescharacteristic
284C22—21no—
285C22—42no—
286C22—82no—
287C22—23442no—
288C22—5862no—
289C22—5861no—
290C22—46882no—
291C22—46882no—
292C22—11722no—
293C22#2262767011yescharacteristic
294C4—21no—
295C4—5862no—
296C4—5862no—
297C4—5861no—
298C4C2×D4×D29311yescharacteristic
299C4C2×D4×D29311yescharacteristic
300C2—21no—
301C2—41no—
302C2—82no—
303C2—23442no—
304C2—11722no—
305C2—11721no—
306C2#3403676311yescharacteristic
subgroupG/H
centerC2#34036763
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#472×C293C2
socleC2×C293D42
hypercenter#472D293
hypocenterC293#3292
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C2 ◁ C2×C4 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #34036934C2semidirect
2. #34036933C2semidirect
3. #34036930C2semidirect
4. #34036504C2semidirect
5. #34036400C2semidirect
6. #34036398C2semidirect
7. #34036299C2semidirect
8. #34036295C2semidirect
9. #34036245C2semidirect
10. #34036244C2semidirect
11. #472×C293C2semidirect
12. #22627670C22semidirect
13. #22627669C22semidirect
15. #22627652C22semidirect
16. #22627652C22semidirect
18. #22627650C22semidirect
27. C2×D2344C22semidirect
28. C2×#17989787C22semidirect
31. #22627632C22semidirect
32. #22627630C22semidirect
33. #22627620C22semidirect
34. #22627559C22semidirect
35. #22627551C22semidirect
36. #22627550C22semidirect
37. #22627550C22semidirect
38. #22627540C22semidirect
39. C8×Dic293C22semidirect
40. AGL(1,ℤ/8)×C293C22semidirect
41. C2×D8×C293C22semidirect
42. #131×C293C22semidirect
43. #108×C293C22semidirect
44. #100×C293C22semidirect
59. Dic1172D4semidirect
60. #17989792D4semidirect
63. #17989791D4semidirect
65. #17989790D4semidirect
66. #17989789D4semidirect
71. D2344D4semidirect
75. #17989787D4semidirect
153. C293#3292semidirect
161. #472D293semidirect
175. AGL(1,ℤ/8)C2×D293semidirect
186. C2×D8C2×D293semidirect
189. #131C2×D293semidirect
192. #108C2×D293semidirect
197. #100C2×D293semidirect

Maximal subgroups

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

subgroupindex
#340369342
#340369332
#340369302
#340365042
#340364002
#340363982
#340362992
#340362952
#340362452
#340362442
#472×C2932
#3292293