Finite Group Databasebeta

← #31818172

Subgroups — #31818172

Order 17792.

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

normal subgroups only

This group has 464 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#22581761C212yesmaximal, complemented
2#22581757C212yesmaximal, complemented
3#22581343C211yesmaximal, characteristic, complemented
4#22581221C212yesmaximal, complemented
5#22581219C212yesmaximal, complemented
6#22581130C211yesmaximal, characteristic, complemented
7#22581122C212yesmaximal, complemented
8#22581072C212yesmaximal, complemented
9#454×C139C211yesmaximal, characteristic, complemented
10#17984888C2212yescomplemented
11#17984887C2211yescharacteristic
12#17984873—22no—
13#17984873C2212yescomplemented
14#17984870C2212yes—
15#17984869—22no—
16#17984867—22no—
17#17984867C2212yescomplemented
18#17984866—22no—
19#17984863—22no—
20#17984862—22no—
21#17984862C2212yes—
22C2×#14555772—22no—
23C2×#14555772C2212yescomplemented
24C2×#14555770C2212yescomplemented
25D8×D139—22no—
26#17984850C2212yescomplemented
27#17984849C2212yescomplemented
28#17984839C2211yescharacteristic, complemented
29#17984777C2212yescomplemented
30#17984770C2212yescomplemented
31#17984759C2211yescharacteristic, complemented
32#17984758C2212yescomplemented
33#17984753C2211yescharacteristic, complemented
342−1+4×C139C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C139C2212yescomplemented
36#124×C139C2211yescharacteristic, complemented
37#104×C139C2212yescomplemented
38#101×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#14555775—22no—
51#14555775D412yescomplemented
52#14555774—22no—
53#14555773D412yescomplemented
54#14555772—22no—
55#14555772—22no—
56#14555772—22no—
57#14555772—22no—
58#14555772D412yescomplemented
59D1112—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×C139—41no—
72#41×C139C2312yes—
73C2×Q8×C139—22no—
74C2×Q8×C139C2311yescharacteristic
75C2×D4×C139C2311yescharacteristic
76SD16×C139—22no—
77D8×C139—22no—
78M16×C139C2312yes—
79C22⋊C4×C139—41no—
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—
100Q8×C139—22no—
101Q8×C139—41no—
102Q8×C139C2×D412yes—
103D4×C139—22no—
104D4×C139—22no—
105D4×C139—41no—
106D4×C139C2×D412yes—
107C2×C4×C139—22no—
108C2×C4×C139—21no—
109C2×C4×C139—21no—
110C2×C4×C139—42no—
111C2×C4×C139C2411yescharacteristic
112C8×C139—22no—
113C22×C139—22no—
114C22×C139—41no—
115C22×C139—81no—
116C22×C139H(5,𝔽2)11yescharacteristic
117C2×D139—22no—
118C2×D139—42no—
119C2×D139—82no—
120C2×D139—161no—
121C2×D139—162no—
122C4×C139—22no—
123C4×C139—41no—
124C4×C139—41no—
125C4×C139C22×D412yes—
126Dic139—22no—
127C2×C139—21no—
128C2×C139—42no—
129C2×C139—81no—
130C2×C139D4211yescharacteristic
131D139—42no—
132D139—82no—
133C139#328811yescharacteristic, complemented, Sylow
134#3288—1391nomaximal, Sylow
135#577—1392no—
136#576—1392no—
137#496—1391no—
138#472—1392no—
139#471—1392no—
140#455—1391no—
141#454D13911yescharacteristic, complemented
142#453—1392no—
143#444—1392no—
1442−1+4C2×D13911yescharacteristic, complemented
145H(5,𝔽2)—1392no—
146H(5,𝔽2)—1391no—
147#137—2782no—
148#137—1392no—
149AGL(1,ℤ/8)—2782no—
150AGL(1,ℤ/8)—2782no—
151AGL(1,ℤ/8)—2782no—
152AGL(1,ℤ/8)—1392no—
153AGL(1,ℤ/8)—1392no—
154AGL(1,ℤ/8)C2×D13912yescomplemented
155#135—2782no—
156#135—2782no—
157#135—1392no—
158C2×SD16—2782no—
159C2×SD16—1392no—
160C2×D8—2782no—
161C2×D8—1392no—
162#131—1392no—
163#131—1392no—
164#127—1391no—
165#124C2×D13911yescharacteristic, complemented
166#108—1392no—
167#104—1392no—
168#104C2×D13912yescomplemented
169#101C2×D13911yescharacteristic, complemented
170#100—1391no—
171#100—1392no—
172#97—1391no—
173#41—22no—
174#41—41no—
175#41—2782no—
176#41—2782no—
177#41—2782no—
178#41—1392no—
179#41C22×D13912yes—
180C2×Q8—22no—
181C2×Q8C22×D13911yescharacteristic
182C2×D4—2782no—
183C2×D4—2782no—
184C2×D4—2782no—
185C2×D4—5561no—
186C2×D4—5562no—
187C2×D4—5561no—
188C2×D4—1392no—
189C2×D4C22×D13911yescharacteristic
190Q16—2782no—
191Q16—1392no—
192SD16—22no—
193SD16—2782no—
194SD16—2782no—
195SD16—2782no—
196SD16—2782no—
197SD16—2782no—
198SD16—1392no—
199SD16—1392no—
200D8—22no—
201D8—2782no—
202D8—2782no—
203D8—2782no—
204D8—2782no—
205D8—1392no—
206M16—2782no—
207M16—2782no—
208M16—1392no—
209M16C22×D13912yes—
210C2×C8—2782no—
211C2×C8—2782no—
212C2×C8—1392no—
213C22⋊C4—41no—
214C42C22×D13911yescharacteristic
215C23—21no—
216C23—2782no—
217C23—5562no—
218C23—5561no—
219Q8—22no—
220Q8—41no—
221Q8—1392no—
222Q8D4×D13912yes—
223D4—22no—
224D4—22no—
225D4—41no—
226D4—2782no—
227D4—2782no—
228D4—2782no—
229D4—5562no—
230D4—5562no—
231D4—5562no—
232D4—11122no—
233D4—1392no—
234D4D4×D13912yes—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—42no—
239C2×C4—2782no—
240C2×C4—5562no—
241C2×C4—5561no—
242C2×C4C23×D13911yescharacteristic
243C8—22no—
244C8—2782no—
245C8—1392no—
246C8—1392no—
247C22—22no—
248C22—41no—
249C22—81no—
250C22—2782no—
251C22—5562no—
252C22—11122no—
253C22—22242no—
254C22—22241no—
255C22#1798488711yescharacteristic
256C4—22no—
257C4—41no—
258C4—41no—
259C4—2782no—
260C4C2×D4×D13912yes—
261C2—21no—
262C2—42no—
263C2—81no—
264C2—5562no—
265C2—11122no—
266C2#2258158511yescharacteristic
subgroupG/H
centerC2#22581585
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
Fitting#454×C139C2
socleC2×C139D42
hypercenter#454D139
hypocenterC139#3288
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C2 ◁ C2×C4 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #22581761C2semidirect
2. #22581757C2semidirect
3. #22581343C2semidirect
4. #22581221C2semidirect
5. #22581219C2semidirect
6. #22581130C2semidirect
7. #22581122C2semidirect
8. #22581072C2semidirect
9. #454×C139C2semidirect
10. #17984888C22semidirect
13. #17984873C22semidirect
17. #17984867C22semidirect
23. C2×#14555772C22semidirect
24. C2×#14555770C22semidirect
26. #17984850C22semidirect
27. #17984849C22semidirect
28. #17984839C22semidirect
29. #17984777C22semidirect
30. #17984770C22semidirect
31. #17984759C22semidirect
32. #17984758C22semidirect
33. #17984753C22semidirect
34. 2−1+4×C139C22semidirect
35. AGL(1,ℤ/8)×C139C22semidirect
36. #124×C139C22semidirect
37. #104×C139C22semidirect
38. #101×C139C22semidirect
51. #14555775D4semidirect
53. #14555773D4semidirect
58. #14555772D4semidirect
63. #14555770D4semidirect
133. C139#3288semidirect
141. #454D139semidirect
144. 2−1+4C2×D139semidirect
154. AGL(1,ℤ/8)C2×D139semidirect
165. #124C2×D139semidirect
168. #104C2×D139semidirect
169. #101C2×D139semidirect

Maximal subgroups

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

subgroupindex
#225817612
#225817572
#225813432
#225812212
#225812192
#225811302
#225811222
#225810722
#454×C1392
#3288139