Finite Group Databasebeta

← #49170582

Subgroups — #49170582

Order 37504.

476 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 476 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
1#34036932C212yesmaximal, complemented
2#34036931C211yesmaximal, characteristic, complemented
3#34036504C211yesmaximal, characteristic, complemented
4#34036399C211yesmaximal, characteristic, complemented
5#34036397C212yesmaximal, complemented
6#34036396C211yesmaximal, characteristic, complemented
7#34036297C212yesmaximal, complemented
8#34036295C212yesmaximal, complemented
9#34036246C211yesmaximal, characteristic, complemented
10#34036244C211yesmaximal, characteristic, complemented
11#576×C293C211yesmaximal, characteristic, complemented
12#22627670C2212yescomplemented
13#22627651—22no—
14#22627650—22no—
15#22627650C2212yescomplemented
16#22627649—22no—
17#22627649C2212yescomplemented
18#22627649C2212yescomplemented
19#22627647—21no—
20#22627644—22no—
21#22627644C2211yescharacteristic
22C2×D2344C2211yescharacteristic, complemented
23C2×#17989787—22no—
24C2×#17989787C2212yescomplemented
25D8×D293—21no—
26#22627631C2212yescomplemented
27#22627630C2211yescharacteristic, complemented
28#22627620C2212yescomplemented
29#22627560C2211yescharacteristic, complemented
30#22627559C2211yescharacteristic, complemented
31#22627551C2211yescharacteristic, complemented
32#22627551C2212yescomplemented
33#22627550C2211yescharacteristic, complemented
34#22627541C2212yescomplemented
35#22627540C2212yescomplemented
36C8×Dic293C2211yescharacteristic, complemented
37H(5,𝔽2)×C293C2212yescomplemented
38AGL(1,ℤ/8)×C293—22no—
39AGL(1,ℤ/8)×C293C2212yescomplemented
40#135×C293—21no—
41#135×C293C2211yescharacteristic
42C2×D8×C293—21no—
43C2×D8×C293C2211yescharacteristic, complemented
44#131×C293C2211yescharacteristic, complemented
45#17989803—22no—
46#17989802—22no—
47#17989802C2311yescharacteristic
48C2×D1172—22no—
49C2×D1172C2312yes—
50C2×C22⋊D293—42no—
51D4×D293—22no—
52D4×D293—42no—
53#17989792D411yescharacteristic, complemented
54#17989790—22no—
55#17989790D411yescharacteristic, complemented
56#17989789—22no—
57#17989789—22no—
58#17989789D411yescharacteristic, complemented
59D2344—21no—
60D2344—21no—
61#17989787—22no—
62#17989787—22no—
63#17989787—22no—
64#17989787—22no—
65#17989787—22no—
66#17989787D411yescharacteristic, complemented
67#17989786—22no—
68#17989786C2311yescharacteristic
69#17989786C2312yes—
70#17989785—21no—
71C2×#14556873—22no—
72C2×#14556873C2311yescharacteristic
73C8×D293—21no—
74C4×Dic293C2311yescharacteristic
75#41×C293—22no—
76#41×C293—22no—
77#41×C293C2312yes—
78#41×C293C2311yescharacteristic
79C2×D4×C293—22no—
80C2×D4×C293—22no—
81C2×D4×C293—42no—
82C2×D4×C293C2312yes—
83Q16×C293D411yescharacteristic, complemented
84SD16×C293—22no—
85SD16×C293D412yescomplemented
86D8×C293—22no—
87D8×C293—21no—
88D8×C293—21no—
89D8×C293D411yescharacteristic, complemented
90M16×C293—21no—
91M16×C293C2311yescharacteristic
92C2×C8×C293—21no—
93C2×C8×C293C2311yescharacteristic
94C22×D293—22no—
95C22×D293—42no—
96Dic586C2×D411yescharacteristic
97D1172—22no—
98D1172—22no—
99D1172—42no—
100D1172C2×D411yescharacteristic
101C22⋊D293—21no—
102C22⋊D293—42no—
103C22⋊D293—84no—
104C2×Dic293—21no—
105C4×D293—21no—
106C4×D293—42no—
107#14556873—21no—
108#14556873—22no—
109#14556873C2×D411yescharacteristic
110#14556873C2×D411yescharacteristic
111C23×C293—22no—
112C23×C293—42no—
113Q8×C293C2×D411yescharacteristic
114Q8×C293C2×D412yes—
115D4×C293—21no—
116D4×C293—22no—
117D4×C293—22no—
118D4×C293—22no—
119D4×C293—42no—
120D4×C293—42no—
121D4×C293C2×D411yescharacteristic
122D4×C293C2×D412yes—
123C2×C4×C293—21no—
124C2×C4×C293—22no—
125C2×C4×C293—42no—
126C2×C4×C293C2411yescharacteristic
127C8×C293—21no—
128C8×C293C2×D411yescharacteristic
129C8×C293C2×D411yescharacteristic
130C22×C293—21no—
131C22×C293—22no—
132C22×C293—42no—
133C22×C293—82no—
134C22×C293—162no—
135C22×C293H(5,𝔽2)11yescharacteristic
136C2×D293—21no—
137C2×D293—42no—
138C2×D293—82no—
139C2×D293—162no—
140C4×C293—21no—
141C4×C293—22no—
142C4×C293C22×D411yescharacteristic
143C4×C293C22×D411yescharacteristic
144Dic293—22no—
145Dic293—21no—
146C2×C293—21no—
147C2×C293—42no—
148C2×C293—41no—
149C2×C293—82no—
150C2×C293D4211yescharacteristic
151D293—41no—
152D293—82no—
153C293#329211yescharacteristic, complemented, Sylow
154#3292—2931nomaximal, Sylow
155#576—2932no—
156#576—2931no—
157#576D29311yescharacteristic, complemented
158#493—2931no—
159#472—2932no—
160#472—2931no—
161#472—2931no—
162#453—2932no—
163#453—2932no—
164#443—2931no—
165#443—2931no—
166H(5,𝔽2)—2932no—
167H(5,𝔽2)C2×D29312yescomplemented
168AGL(1,ℤ/8)—22no—
169AGL(1,ℤ/8)—2932no—
170AGL(1,ℤ/8)—2932no—
171AGL(1,ℤ/8)—2932no—
172AGL(1,ℤ/8)—5862no—
173AGL(1,ℤ/8)—5862no—
174AGL(1,ℤ/8)—5862no—
175AGL(1,ℤ/8)C2×D29312yescomplemented
176#135—21no—
177#135—2931no—
178#135—5862no—
179#135—5861no—
180#135C2×D29311yescharacteristic
181C2×D8—21no—
182C2×D8—2932no—
183C2×D8—2931no—
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#108—2931no—
193#104—2931no—
194#104—2932no—
195#104—2931no—
196#100—2932no—
197#100—2932no—
198C4×C8—2931no—
199#41—22no—
200#41—22no—
201#41—2931no—
202#41—5862no—
203#41—5862no—
204#41C22×D29311yescharacteristic
205#41C22×D29312yes—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—42no—
209C2×D4—2932no—
210C2×D4—5862no—
211C2×D4—5862no—
212C2×D4—11722no—
213C2×D4—11722no—
214C2×D4C22×D29312yes—
215Q16—2931no—
216Q16C22⋊D29311yescharacteristic, complemented
217SD16—22no—
218SD16—2931no—
219SD16—2931no—
220SD16—5862no—
221SD16—5862no—
222SD16—5862no—
223SD16C22⋊D29312yescomplemented
224D8—22no—
225D8—21no—
226D8—21no—
227D8—2931no—
228D8—5862no—
229D8—5862no—
230D8—5862no—
231D8—5862no—
232D8—5862no—
233D8—5861no—
234D8—5861no—
235D8C22⋊D29311yescharacteristic, complemented
236M16—21no—
237M16—2932no—
238M16—2931no—
239M16—5862no—
240M16—5861no—
241M16C22×D29311yescharacteristic
242C2×C8—21no—
243C2×C8—2931no—
244C2×C8—5862no—
245C2×C8—5861no—
246C2×C8C22×D29311yescharacteristic
247C42—2931no—
248C23—22no—
249C23—42no—
250C23—5862no—
251C23—11722no—
252Q8—2931no—
253Q8D4×D29311yescharacteristic
254Q8C2×C22⋊D29312yes—
255D4—21no—
256D4—22no—
257D4—22no—
258D4—22no—
259D4—42no—
260D4—42no—
261D4—2931no—
262D4—23444no—
263D4—5861no—
264D4—5862no—
265D4—5862no—
266D4—11722no—
267D4—11722no—
268D4D4×D29311yescharacteristic
269D4C2×C22⋊D29312yes—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—42no—
273C2×C4—5861no—
274C2×C4—5861no—
275C2×C4—11722no—
276C2×C4C23×D29311yescharacteristic
277C8—21no—
278C8—2931no—
279C8—2931no—
280C8—5862no—
281C8—5861no—
282C8C2×C22⋊D29311yescharacteristic
283C8C2×C22⋊D29311yescharacteristic
284C22—21no—
285C22—22no—
286C22—42no—
287C22—82no—
288C22—162no—
289C22—23442no—
290C22—5861no—
291C22—46882no—
292C22—11722no—
293C22#2262766911yescharacteristic
294C4—21no—
295C4—22no—
296C4—5862no—
297C4—5861no—
298C4C2×D4×D29311yescharacteristic
299C4C22×C22⋊D29311yescharacteristic
300C2—21no—
301C2—41no—
302C2—42no—
303C2—82no—
304C2—23442no—
305C2—11721no—
306C2D4×C22⋊D29311yescharacteristic
subgroupG/H
centerC2D4×C22⋊D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#576×C293C2
socleC2×C293D42
hypercenter#576D293
hypocenterC293#3292
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C2 ◁ C2×C4 ◁ #576

Complements

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

NH ≅ G/Ntype
1. #34036932C2semidirect
2. #34036931C2semidirect
3. #34036504C2semidirect
4. #34036399C2semidirect
5. #34036397C2semidirect
6. #34036396C2semidirect
7. #34036297C2semidirect
8. #34036295C2semidirect
9. #34036246C2semidirect
10. #34036244C2semidirect
11. #576×C293C2semidirect
12. #22627670C22semidirect
15. #22627650C22semidirect
17. #22627649C22semidirect
18. #22627649C22semidirect
22. C2×D2344C22semidirect
24. C2×#17989787C22semidirect
26. #22627631C22semidirect
27. #22627630C22semidirect
28. #22627620C22semidirect
29. #22627560C22semidirect
30. #22627559C22semidirect
31. #22627551C22semidirect
32. #22627551C22semidirect
33. #22627550C22semidirect
34. #22627541C22semidirect
35. #22627540C22semidirect
36. C8×Dic293C22semidirect
37. H(5,𝔽2)×C293C22semidirect
39. AGL(1,ℤ/8)×C293C22semidirect
43. C2×D8×C293C22semidirect
44. #131×C293C22semidirect
53. #17989792D4semidirect
55. #17989790D4semidirect
58. #17989789D4semidirect
66. #17989787D4semidirect
83. Q16×C293D4semidirect
85. SD16×C293D4semidirect
89. D8×C293D4semidirect
153. C293#3292semidirect
157. #576D293semidirect
167. H(5,𝔽2)C2×D293semidirect
175. AGL(1,ℤ/8)C2×D293semidirect
186. C2×D8C2×D293semidirect
189. #131C2×D293semidirect
216. Q16C22⋊D293semidirect
223. SD16C22⋊D293semidirect
235. D8C22⋊D293semidirect

Maximal subgroups

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

subgroupindex
#340369322
#340369312
#340365042
#340363992
#340363972
#340363962
#340362972
#340362952
#340362462
#340362442
#576×C2932
#3292293