Finite Group Databasebeta

← #31815761

Subgroups — #31815761

Order 17792.

762 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 161 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#22581751C212yesmaximal, complemented
2#22581520C211yesmaximal, characteristic, complemented
3#22581131C214yesmaximal, complemented
4#22581121C214yesmaximal, complemented
5#22580970C212yesmaximal, complemented
6C2×#17984759C211yesmaximal, characteristic, complemented
7C2×H(5,𝔽2)×C139C211yesmaximal, characteristic, complemented
8C2×#14555785C2211yescharacteristic, complemented
9#17984873—22no—
10#17984872C2214yescomplemented
11#17984868—22no—
12#17984867C2214yescomplemented
13#17984862—24no—
14C2×#14555769C2212yescomplemented
15#17984838C2212yescomplemented
16#17984829C2212yescomplemented
17#17984813—22no—
18#17984811—22no—
19#17984801C2211yescharacteristic, complemented
20#17984769C2218yescomplemented
21#17984759C2214yescomplemented
22H(5,𝔽2)×C139—24no—
23H(5,𝔽2)×C139—22no—
24H(5,𝔽2)×C139C2214yes—
25C2×#41×C139—22no—
26C2×#41×C139C2212yes—
27C22×D4×C139—22no—
28C22×D4×C139—22no—
29C22×D4×C139C2211yescharacteristic
30#14555785—22no—
31#14555785—22no—
32C2×Dic278C2311yescharacteristic, complemented
33C2×D556C2311yescharacteristic, complemented
34C2×C22⋊D139—21no—
35C2×C22⋊D139—24no—
36C2×C4×D139—21no—
37#14555775—24no—
38#14555773—24no—
39#14555772—24no—
40#14555770—24no—
41#14555769—22no—
42#14555769C2314yescomplemented
43C2×C139⋊C8—22no—
44C4⋊Dic139—21no—
45C22⋊Dic139—21no—
46C22⋊Dic139—24no—
47C4×Dic139C2312yescomplemented
48C24×C139—22no—
49C24×C139D412yes—
50#41×C139—28no—
51#41×C139—28no—
52#41×C139—22no—
53#41×C139—22no—
54#41×C139D414yescomplemented
55#41×C139C2314yes—
56C2×Q8×C139D412yescomplemented
57C2×D4×C139—24no—
58C2×D4×C139—28no—
59C2×D4×C139—28no—
60C2×D4×C139—22no—
61C2×D4×C139—28no—
62C2×D4×C139—28no—
63C2×D4×C139—22no—
64C2×D4×C139—22no—
65C2×D4×C139D412yes—
66C2×D4×C139D412yescomplemented
67C2×D4×C139C2312yes—
68C22×C4×C139—22no—
69C22×C4×C139—22no—
70C22×C4×C139C2311yescharacteristic
71C22×D139—21no—
72Dic278—22no—
73D556—22no—
74C22⋊D139—41no—
75C22⋊D139—41no—
76C22⋊D139—84no—
77C2×Dic139—22no—
78C2×Dic139—21no—
79C4×D139—42no—
80C139⋊C8—24no—
81C23×C139—28no—
82C23×C139—22no—
83C23×C139—22no—
84C23×C139—22no—
85C23×C139—82no—
86C23×C139—84no—
87C23×C139C2×D414yes—
88C23×C139C2×D411yescharacteristic
89Q8×C139—22no—
90Q8×C139C2×D414yes—
91D4×C139—216no—
92D4×C139—24no—
93D4×C139—24no—
94D4×C139—24no—
95D4×C139—22no—
96D4×C139—24no—
97D4×C139C2×D414yes—
98C2×C4×C139—28no—
99C2×C4×C139—22no—
100C2×C4×C139—24no—
101C2×C4×C139—24no—
102C2×C4×C139C2×D411yescharacteristic
103C2×C4×C139C2×D412yes—
104C2×C4×C139C2×D412yes—
105C2×C4×C139C2411yescharacteristic
106C22×C139—24no—
107C22×C139—24no—
108C22×C139—41no—
109C22×C139—44no—
110C22×C139—44no—
111C22×C139—44no—
112C22×C139—82no—
113C22×C139—88no—
114C22×C139#12011yescharacteristic
115C22×C139#12011yescharacteristic
116C22×C139C22×D411yescharacteristic
117C2×D139—42no—
118C2×D139—41no—
119C4×C139—24no—
120C4×C139#12012yes—
121C4×C139C22×D412yes—
122Dic139—42no—
123Dic139—41no—
124C2×C139—21no—
125C2×C139—21no—
126C2×C139—21no—
127C2×C139—44no—
128C2×C139—44no—
129C2×C139C2×#12011yescharacteristic
130D139—81no—
131C139#301911yescharacteristic, complemented, Sylow
132#3019—1391nomaximal, Sylow
133C2×H(5,𝔽2)D13911yescharacteristic, complemented
134#575—1392no—
135#535—1391no—
136#455—1394no—
137#453—1394no—
138#421—1392no—
139C2×#100—1391no—
140H(5,𝔽2)—24no—
141H(5,𝔽2)—22no—
142H(5,𝔽2)C2×D13914yes—
143C2×#41—22no—
144C2×#41—1391no—
145C2×#41C2×D13912yes—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4C2×D13911yescharacteristic
149#137—2782no—
150#137—1394no—
151AGL(1,ℤ/8)—2782no—
152AGL(1,ℤ/8)—1394no—
153#135—2784no—
154C2×M16—1392no—
155#127—1392no—
156#124—1392no—
157#121—2782no—
158#120—2782no—
159#117—1391no—
160#104—1398no—
161#100—1394no—
162C24—22no—
163C24C22⋊D13912yes—
164#41—28no—
165#41—28no—
166#41—22no—
167#41—22no—
168#41—2782no—
169#41—2782no—
170#41C22⋊D13914yescomplemented
171#41C22×D13914yes—
172C2×Q8—1391no—
173C2×Q8C22⋊D13912yescomplemented
174C2×D4—24no—
175C2×D4—28no—
176C2×D4—28no—
177C2×D4—22no—
178C2×D4—28no—
179C2×D4—28no—
180C2×D4—22no—
181C2×D4—22no—
182C2×D4—2781no—
183C2×D4—2784no—
184C2×D4—1391no—
185C2×D4C22⋊D13912yes—
186C2×D4C22⋊D13912yescomplemented
187C2×D4C22×D13912yes—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—2781no—
191C22×C4C22×D13911yescharacteristic
192Q16—2784no—
193SD16—2784no—
194SD16—2784no—
195D8—2784no—
196M16—2782no—
197M16—1394no—
198C2×C8—2782no—
199C4⋊C4—2781no—
200C22⋊C4—2781no—
201C22⋊C4—2784no—
202C42—1392no—
203C23—28no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—82no—
208C23—84no—
209C23—2781no—
210C23C2×C22⋊D13914yes—
211C23C2×C22⋊D13911yescharacteristic
212Q8—22no—
213Q8—2782no—
214Q8C2×C22⋊D13914yes—
215D4—216no—
216D4—24no—
217D4—24no—
218D4—24no—
219D4—22no—
220D4—24no—
221D4—2782no—
222D4—5561no—
223D4—5561no—
224D4—11124no—
225D4C2×C22⋊D13914yes—
226C2×C4—28no—
227C2×C4—22no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—2782no—
231C2×C4—2781no—
232C2×C4—5562no—
233C2×C4C2×C22⋊D13911yescharacteristic
234C2×C4C2×C22⋊D13912yes—
235C2×C4C2×C22⋊D13912yes—
236C2×C4C23×D13911yescharacteristic
237C8—2784no—
238C22—24no—
239C22—24no—
240C22—41no—
241C22—44no—
242C22—44no—
243C22—44no—
244C22—82no—
245C22—88no—
246C22—5562no—
247C22—5561no—
248C22#1798481111yescharacteristic
249C22#1798481111yescharacteristic
250C22C22×C22⋊D13911yescharacteristic
251C4—24no—
252C4—5562no—
253C4—5561no—
254C4#1798481112yes—
255C4C22×C22⋊D13912yes—
256C2—21no—
257C2—21no—
258C2—21no—
259C2—44no—
260C2—44no—
261C2—11121no—
262C2C2×#1798481111yescharacteristic
subgroupG/H
centerC2C2×#17984811
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
FittingC2×H(5,𝔽2)×C139C2
socleC2×C139C2×#120
hypercenterC2×H(5,𝔽2)D139
hypocenterC139#3019
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #22581751C2semidirect
2. #22581520C2semidirect
3. #22581131C2semidirect
4. #22581121C2semidirect
5. #22580970C2semidirect
6. C2×#17984759C2semidirect
7. C2×H(5,𝔽2)×C139C2semidirect
8. C2×#14555785C22semidirect
10. #17984872C22semidirect
12. #17984867C22semidirect
14. C2×#14555769C22semidirect
15. #17984838C22semidirect
16. #17984829C22semidirect
19. #17984801C22semidirect
20. #17984769C22semidirect
21. #17984759C22semidirect
32. C2×Dic278C23semidirect
33. C2×D556C23semidirect
42. #14555769C23semidirect
47. C4×Dic139C23semidirect
54. #41×C139D4semidirect
56. C2×Q8×C139D4semidirect
66. C2×D4×C139D4semidirect
131. C139#3019semidirect
133. C2×H(5,𝔽2)D139semidirect
170. #41C22⋊D139semidirect
173. C2×Q8C22⋊D139semidirect
186. C2×D4C22⋊D139semidirect

Maximal subgroups

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

subgroupindex
#225817512
#225815202
#225811312
#225811212
#225809702
C2×#179847592
C2×H(5,𝔽2)×C1392
#3019139