Finite Group Databasebeta

← #31818201

Subgroups — #31818201

Order 17792.

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#22581756C212yesmaximal, complemented
2#22581755C211yesmaximal, characteristic, complemented
3#22581328C211yesmaximal, characteristic, complemented
4#22581223C211yesmaximal, characteristic, complemented
5#22581221C212yesmaximal, complemented
6#22581220C211yesmaximal, characteristic, complemented
7#22581121C212yesmaximal, complemented
8#22581119C212yesmaximal, complemented
9#22581070C211yesmaximal, characteristic, complemented
10#22581068C211yesmaximal, characteristic, complemented
11#576×C139C211yesmaximal, characteristic, complemented
12#17984888C2212yescomplemented
13#17984869—22no—
14#17984868—22no—
15#17984868C2212yescomplemented
16#17984867—22no—
17#17984867C2212yescomplemented
18#17984867C2212yescomplemented
19#17984865—21no—
20#17984862—22no—
21#17984862C2211yescharacteristic
22C2×D1112C2211yescharacteristic, complemented
23C2×#14555770—22no—
24C2×#14555770C2212yescomplemented
25D8×D139—21no—
26#17984849C2212yescomplemented
27#17984848C2211yescharacteristic, complemented
28#17984838C2212yescomplemented
29#17984778C2211yescharacteristic, complemented
30#17984777C2211yescharacteristic, complemented
31#17984769C2211yescharacteristic, complemented
32#17984769C2212yescomplemented
33#17984768C2211yescharacteristic, complemented
34#17984759C2212yescomplemented
35#17984758C2212yescomplemented
36C8×Dic139C2211yescharacteristic, complemented
37H(5,𝔽2)×C139C2212yescomplemented
38AGL(1,ℤ/8)×C139—22no—
39AGL(1,ℤ/8)×C139C2212yescomplemented
40#135×C139—21no—
41#135×C139C2211yescharacteristic
42C2×D8×C139—21no—
43C2×D8×C139C2211yescharacteristic, complemented
44#131×C139C2211yescharacteristic, complemented
45#14555786—22no—
46#14555785—22no—
47#14555785C2311yescharacteristic
48C2×D556—22no—
49C2×D556C2312yes—
50C2×C22⋊D139—42no—
51D4×D139—22no—
52D4×D139—42no—
53#14555775D411yescharacteristic, complemented
54#14555773—22no—
55#14555773D411yescharacteristic, complemented
56#14555772—22no—
57#14555772—22no—
58#14555772D411yescharacteristic, complemented
59D1112—21no—
60D1112—21no—
61#14555770—22no—
62#14555770—22no—
63#14555770—22no—
64#14555770—22no—
65#14555770—22no—
66#14555770D411yescharacteristic, complemented
67#14555769—22no—
68#14555769C2311yescharacteristic
69#14555769C2312yes—
70#14555768—21no—
71C2×C139⋊C8—22no—
72C2×C139⋊C8C2311yescharacteristic
73C8×D139—21no—
74C4×Dic139C2311yescharacteristic
75#41×C139—22no—
76#41×C139—22no—
77#41×C139C2312yes—
78#41×C139C2311yescharacteristic
79C2×D4×C139—22no—
80C2×D4×C139—22no—
81C2×D4×C139—42no—
82C2×D4×C139C2312yes—
83Q16×C139D411yescharacteristic, complemented
84SD16×C139—22no—
85SD16×C139D412yescomplemented
86D8×C139—22no—
87D8×C139—21no—
88D8×C139—21no—
89D8×C139D411yescharacteristic, complemented
90M16×C139—21no—
91M16×C139C2311yescharacteristic
92C2×C8×C139—21no—
93C2×C8×C139C2311yescharacteristic
94C22×D139—22no—
95C22×D139—42no—
96Dic278C2×D411yescharacteristic
97D556—22no—
98D556—22no—
99D556—42no—
100D556C2×D411yescharacteristic
101C22⋊D139—21no—
102C22⋊D139—42no—
103C22⋊D139—84no—
104C2×Dic139—21no—
105C4×D139—21no—
106C4×D139—42no—
107C139⋊C8—21no—
108C139⋊C8—22no—
109C139⋊C8C2×D411yescharacteristic
110C139⋊C8C2×D411yescharacteristic
111C23×C139—22no—
112C23×C139—42no—
113Q8×C139C2×D411yescharacteristic
114Q8×C139C2×D412yes—
115D4×C139—21no—
116D4×C139—22no—
117D4×C139—22no—
118D4×C139—22no—
119D4×C139—42no—
120D4×C139—42no—
121D4×C139C2×D411yescharacteristic
122D4×C139C2×D412yes—
123C2×C4×C139—21no—
124C2×C4×C139—22no—
125C2×C4×C139—42no—
126C2×C4×C139C2411yescharacteristic
127C8×C139—21no—
128C8×C139C2×D411yescharacteristic
129C8×C139C2×D411yescharacteristic
130C22×C139—21no—
131C22×C139—22no—
132C22×C139—42no—
133C22×C139—82no—
134C22×C139—162no—
135C22×C139H(5,𝔽2)11yescharacteristic
136C2×D139—21no—
137C2×D139—42no—
138C2×D139—82no—
139C2×D139—162no—
140C4×C139—21no—
141C4×C139—22no—
142C4×C139C22×D411yescharacteristic
143C4×C139C22×D411yescharacteristic
144Dic139—22no—
145Dic139—21no—
146C2×C139—21no—
147C2×C139—42no—
148C2×C139—41no—
149C2×C139—82no—
150C2×C139D4211yescharacteristic
151D139—41no—
152D139—82no—
153C139#329211yescharacteristic, complemented, Sylow
154#3292—1391nomaximal, Sylow
155#576—1392no—
156#576—1391no—
157#576D13911yescharacteristic, complemented
158#493—1391no—
159#472—1392no—
160#472—1391no—
161#472—1391no—
162#453—1392no—
163#453—1392no—
164#443—1391no—
165#443—1391no—
166H(5,𝔽2)—1392no—
167H(5,𝔽2)C2×D13912yescomplemented
168AGL(1,ℤ/8)—22no—
169AGL(1,ℤ/8)—2782no—
170AGL(1,ℤ/8)—2782no—
171AGL(1,ℤ/8)—2782no—
172AGL(1,ℤ/8)—1392no—
173AGL(1,ℤ/8)—1392no—
174AGL(1,ℤ/8)—1392no—
175AGL(1,ℤ/8)C2×D13912yescomplemented
176#135—21no—
177#135—2782no—
178#135—2781no—
179#135—1391no—
180#135C2×D13911yescharacteristic
181C2×D8—21no—
182C2×D8—2782no—
183C2×D8—2781no—
184C2×D8—1392no—
185C2×D8—1391no—
186C2×D8C2×D13911yescharacteristic, complemented
187#131—1392no—
188#131—1391no—
189#131C2×D13911yescharacteristic, complemented
190#127—1392no—
191#108—1391no—
192#108—1391no—
193#104—1391no—
194#104—1392no—
195#104—1391no—
196#100—1392no—
197#100—1392no—
198C4×C8—1391no—
199#41—22no—
200#41—22no—
201#41—2782no—
202#41—2782no—
203#41—1391no—
204#41C22×D13911yescharacteristic
205#41C22×D13912yes—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—42no—
209C2×D4—2782no—
210C2×D4—2782no—
211C2×D4—5562no—
212C2×D4—5562no—
213C2×D4—1392no—
214C2×D4C22×D13912yes—
215Q16—1391no—
216Q16C22⋊D13911yescharacteristic, complemented
217SD16—22no—
218SD16—2782no—
219SD16—2782no—
220SD16—2782no—
221SD16—1391no—
222SD16—1391no—
223SD16C22⋊D13912yescomplemented
224D8—22no—
225D8—21no—
226D8—21no—
227D8—2782no—
228D8—2782no—
229D8—2782no—
230D8—2782no—
231D8—2782no—
232D8—2781no—
233D8—2781no—
234D8—1391no—
235D8C22⋊D13911yescharacteristic, complemented
236M16—21no—
237M16—2782no—
238M16—2781no—
239M16—1392no—
240M16—1391no—
241M16C22×D13911yescharacteristic
242C2×C8—21no—
243C2×C8—2782no—
244C2×C8—2781no—
245C2×C8—1391no—
246C2×C8C22×D13911yescharacteristic
247C42—1391no—
248C23—22no—
249C23—42no—
250C23—2782no—
251C23—5562no—
252Q8—1391no—
253Q8D4×D13911yescharacteristic
254Q8C2×C22⋊D13912yes—
255D4—21no—
256D4—22no—
257D4—22no—
258D4—22no—
259D4—42no—
260D4—42no—
261D4—2781no—
262D4—2782no—
263D4—2782no—
264D4—5562no—
265D4—5562no—
266D4—11124no—
267D4—1391no—
268D4D4×D13911yescharacteristic
269D4C2×C22⋊D13912yes—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—42no—
273C2×C4—2781no—
274C2×C4—2781no—
275C2×C4—5562no—
276C2×C4C23×D13911yescharacteristic
277C8—21no—
278C8—2782no—
279C8—2781no—
280C8—1391no—
281C8—1391no—
282C8C2×C22⋊D13911yescharacteristic
283C8C2×C22⋊D13911yescharacteristic
284C22—21no—
285C22—22no—
286C22—42no—
287C22—82no—
288C22—162no—
289C22—2781no—
290C22—5562no—
291C22—11122no—
292C22—22242no—
293C22#1798488711yescharacteristic
294C4—21no—
295C4—22no—
296C4—2782no—
297C4—2781no—
298C4C2×D4×D13911yescharacteristic
299C4C22×C22⋊D13911yescharacteristic
300C2—21no—
301C2—41no—
302C2—42no—
303C2—82no—
304C2—5561no—
305C2—11122no—
306C2D4×C22⋊D13911yescharacteristic
subgroupG/H
centerC2D4×C22⋊D139
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
Fitting#576×C139C2
socleC2×C139D42
hypercenter#576D139
hypocenterC139#3292
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C2 ◁ C2×C4 ◁ #576

Complements

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

NH ≅ G/Ntype
1. #22581756C2semidirect
2. #22581755C2semidirect
3. #22581328C2semidirect
4. #22581223C2semidirect
5. #22581221C2semidirect
6. #22581220C2semidirect
7. #22581121C2semidirect
8. #22581119C2semidirect
9. #22581070C2semidirect
10. #22581068C2semidirect
11. #576×C139C2semidirect
12. #17984888C22semidirect
15. #17984868C22semidirect
17. #17984867C22semidirect
18. #17984867C22semidirect
22. C2×D1112C22semidirect
24. C2×#14555770C22semidirect
26. #17984849C22semidirect
27. #17984848C22semidirect
28. #17984838C22semidirect
29. #17984778C22semidirect
30. #17984777C22semidirect
31. #17984769C22semidirect
32. #17984769C22semidirect
33. #17984768C22semidirect
34. #17984759C22semidirect
35. #17984758C22semidirect
36. C8×Dic139C22semidirect
37. H(5,𝔽2)×C139C22semidirect
39. AGL(1,ℤ/8)×C139C22semidirect
43. C2×D8×C139C22semidirect
44. #131×C139C22semidirect
53. #14555775D4semidirect
55. #14555773D4semidirect
58. #14555772D4semidirect
66. #14555770D4semidirect
83. Q16×C139D4semidirect
85. SD16×C139D4semidirect
89. D8×C139D4semidirect
153. C139#3292semidirect
157. #576D139semidirect
167. H(5,𝔽2)C2×D139semidirect
175. AGL(1,ℤ/8)C2×D139semidirect
186. C2×D8C2×D139semidirect
189. #131C2×D139semidirect
216. Q16C22⋊D139semidirect
223. SD16C22⋊D139semidirect
235. D8C22⋊D139semidirect

Maximal subgroups

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

subgroupindex
#225817562
#225817552
#225813282
#225812232
#225812212
#225812202
#225811212
#225811192
#225810702
#225810682
#576×C1392
#3292139