Finite Group Databasebeta

← #31818200

Subgroups — #31818200

Order 17792.

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#22581757C212yesmaximal, complemented
2#22581756C212yesmaximal, complemented
3#22581329C211yesmaximal, characteristic, complemented
4#22581222C212yesmaximal, complemented
5#22581221C212yesmaximal, complemented
6#22581122C212yesmaximal, complemented
7#22581120C211yesmaximal, characteristic, complemented
8#22581067C212yesmaximal, complemented
9#453×C139C211yesmaximal, characteristic, complemented
10#17984888C2212yescomplemented
11#17984887C2211yescharacteristic, complemented
12#17984870C2212yescomplemented
13#17984869—22no—
14#17984868—22no—
15#17984868C2212yescomplemented
16#17984867—22no—
17#17984867C2212yescomplemented
18#17984866—22no—
19#17984863—22no—
20#17984862—22no—
21#17984862C2212yes—
22C2×#14555770—22no—
23C2×#14555770C2212yescomplemented
24C2×#14555770C2212yescomplemented
25D8×D139—22no—
26#17984850C2212yescomplemented
27#17984849C2212yescomplemented
28#17984839C2211yescharacteristic, complemented
29#17984777C2212yescomplemented
30#17984770C2212yescomplemented
31#17984759C2211yescharacteristic, complemented
32#17984758C2212yescomplemented
33C4×C139⋊C8C2211yescharacteristic, complemented
34H(5,𝔽2)×C139C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C139C2212yescomplemented
36#127×C139C2211yescharacteristic, complemented
37#104×C139C2212yescomplemented
38#100×C139C2211yescharacteristic, complemented
39#14555786—22no—
40#14555785—22no—
41#14555785C2312yes—
42#14555784—22no—
43C2×D556—22no—
44C2×D556—41no—
45C2×D556C2312yes—
46C2×C22⋊D139—41no—
47D4×D139—22no—
48D4×D139—22no—
49D4×D139—42no—
50#14555775D412yescomplemented
51#14555774—22no—
52#14555773—22no—
53#14555773D412yescomplemented
54#14555772—22no—
55#14555772D412yescomplemented
56D1112—22no—
57#14555770—22no—
58#14555770—22no—
59#14555770—22no—
60#14555770—22no—
61#14555770—22no—
62#14555770—22no—
63#14555770D412yescomplemented
64#14555769—22no—
65#14555769C2312yes—
66#14555768—22no—
67C2×C139⋊C8—22no—
68C2×C139⋊C8C2312yes—
69C8×D139—22no—
70#41×C139—22no—
71#41×C139C2312yes—
72C2×D4×C139—22no—
73C2×D4×C139—41no—
74C2×D4×C139—41no—
75C2×D4×C139C2311yescharacteristic
76C2×D4×C139C2311yescharacteristic
77SD16×C139—22no—
78D8×C139—22no—
79M16×C139C2312yes—
80C42×C139C2311yescharacteristic
81C22×D139—22no—
82C22×D139—41no—
83C22×D139—42no—
84Dic278C2×D412yes—
85D556—22no—
86D556—22no—
87D556—42no—
88D556—82no—
89D556C2×D412yes—
90C22⋊D139—22no—
91C22⋊D139—42no—
92C22⋊D139—42no—
93C2×Dic139—41no—
94C4×D139—22no—
95C4×D139—42no—
96C139⋊C8—22no—
97C139⋊C8C2×D412yes—
98C139⋊C8C2×D412yes—
99C23×C139—21no—
100C23×C139—21no—
101C23×C139—41no—
102Q8×C139C2×D412yes—
103D4×C139—22no—
104D4×C139—22no—
105D4×C139—22no—
106D4×C139—42no—
107D4×C139—82no—
108D4×C139C2×D412yes—
109C2×C4×C139—21no—
110C2×C4×C139—22no—
111C2×C4×C139—41no—
112C2×C4×C139C2411yescharacteristic
113C8×C139—22no—
114C22×C139—22no—
115C22×C139—41no—
116C22×C139—41no—
117C22×C139—81no—
118C22×C139—81no—
119C22×C139—161no—
120C22×C139H(5,𝔽2)11yescharacteristic
121C2×D139—22no—
122C2×D139—42no—
123C2×D139—82no—
124C2×D139—161no—
125C2×D139—162no—
126C4×C139—22no—
127C4×C139—22no—
128C4×C139C22×D412yes—
129Dic139—22no—
130C2×C139—21no—
131C2×C139—42no—
132C2×C139—81no—
133C2×C139—81no—
134C2×C139D4211yescharacteristic
135D139—42no—
136D139—82no—
137C139#329211yescharacteristic, complemented, Sylow
138#3292—1391nomaximal, Sylow
139#576—1392no—
140#576—1392no—
141#493—1391no—
142#472—1392no—
143#472—1392no—
144#453—1391no—
145#453—1392no—
146#453D13911yescharacteristic, complemented
147#443—1392no—
148H(5,𝔽2)—1392no—
149H(5,𝔽2)—1391no—
150H(5,𝔽2)C2×D13911yescharacteristic, complemented
151AGL(1,ℤ/8)—2782no—
152AGL(1,ℤ/8)—2782no—
153AGL(1,ℤ/8)—2782no—
154AGL(1,ℤ/8)—2782no—
155AGL(1,ℤ/8)—1392no—
156AGL(1,ℤ/8)—1392no—
157AGL(1,ℤ/8)—1392no—
158AGL(1,ℤ/8)C2×D13912yescomplemented
159#135—2782no—
160#135—2782no—
161#135—1392no—
162C2×D8—2782no—
163C2×D8—2782no—
164C2×D8—1392no—
165C2×D8—1392no—
166#131—1392no—
167#131—1392no—
168#127—1391no—
169#127C2×D13911yescharacteristic, complemented
170#108—1392no—
171#104—1392no—
172#104C2×D13912yescomplemented
173#100—1391no—
174#100—1392no—
175#100C2×D13911yescharacteristic, complemented
176C4×C8—1391no—
177#41—22no—
178#41—2782no—
179#41—2782no—
180#41—2782no—
181#41—1392no—
182#41C22×D13912yes—
183C2×D4—22no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—2782no—
187C2×D4—2782no—
188C2×D4—2782no—
189C2×D4—5561no—
190C2×D4—5562no—
191C2×D4—5561no—
192C2×D4—1392no—
193C2×D4C22×D13911yescharacteristic
194C2×D4C22×D13911yescharacteristic
195Q16—1392no—
196SD16—22no—
197SD16—2782no—
198SD16—2782no—
199SD16—2782no—
200SD16—1392no—
201SD16—1392no—
202D8—22no—
203D8—2782no—
204D8—2782no—
205D8—2782no—
206D8—2782no—
207D8—2782no—
208D8—2782no—
209D8—2782no—
210D8—1392no—
211M16—2782no—
212M16—2782no—
213M16—1392no—
214M16C22×D13912yes—
215C2×C8—2782no—
216C2×C8—2782no—
217C2×C8—1392no—
218C42C22×D13911yescharacteristic
219C23—21no—
220C23—21no—
221C23—41no—
222C23—2782no—
223C23—5562no—
224C23—5561no—
225Q8—1392no—
226Q8D4×D13912yes—
227D4—22no—
228D4—22no—
229D4—22no—
230D4—42no—
231D4—82no—
232D4—2782no—
233D4—2782no—
234D4—2782no—
235D4—5562no—
236D4—5562no—
237D4—5562no—
238D4—11122no—
239D4—1392no—
240D4D4×D13912yes—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—2782no—
245C2×C4—5562no—
246C2×C4—5561no—
247C2×C4C23×D13911yescharacteristic
248C8—22no—
249C8—2782no—
250C8—1392no—
251C8—1392no—
252C22—22no—
253C22—41no—
254C22—41no—
255C22—81no—
256C22—81no—
257C22—161no—
258C22—2782no—
259C22—5562no—
260C22—11122no—
261C22—22242no—
262C22—22241no—
263C22#1798488711yescharacteristic
264C4—22no—
265C4—22no—
266C4—2782no—
267C4C2×D4×D13912yes—
268C2—21no—
269C2—42no—
270C2—81no—
271C2—81no—
272C2—5562no—
273C2—11122no—
274C2#2258158511yescharacteristic
subgroupG/H
centerC2#22581585
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
Fitting#453×C139C2
socleC2×C139D42
hypercenter#453D139
hypocenterC139#3292
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #22581757C2semidirect
2. #22581756C2semidirect
3. #22581329C2semidirect
4. #22581222C2semidirect
5. #22581221C2semidirect
6. #22581122C2semidirect
7. #22581120C2semidirect
8. #22581067C2semidirect
9. #453×C139C2semidirect
10. #17984888C22semidirect
11. #17984887C22semidirect
12. #17984870C22semidirect
15. #17984868C22semidirect
17. #17984867C22semidirect
23. C2×#14555770C22semidirect
24. C2×#14555770C22semidirect
26. #17984850C22semidirect
27. #17984849C22semidirect
28. #17984839C22semidirect
29. #17984777C22semidirect
30. #17984770C22semidirect
31. #17984759C22semidirect
32. #17984758C22semidirect
33. C4×C139⋊C8C22semidirect
34. H(5,𝔽2)×C139C22semidirect
35. AGL(1,ℤ/8)×C139C22semidirect
36. #127×C139C22semidirect
37. #104×C139C22semidirect
38. #100×C139C22semidirect
50. #14555775D4semidirect
53. #14555773D4semidirect
55. #14555772D4semidirect
63. #14555770D4semidirect
137. C139#3292semidirect
146. #453D139semidirect
150. H(5,𝔽2)C2×D139semidirect
158. AGL(1,ℤ/8)C2×D139semidirect
169. #127C2×D139semidirect
172. #104C2×D139semidirect
175. #100C2×D139semidirect

Maximal subgroups

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

subgroupindex
#225817572
#225817562
#225813292
#225812222
#225812212
#225811222
#225811202
#225810672
#453×C1392
#3292139