Finite Group Databasebeta

← #31815760

Subgroups — #31815760

Order 17792.

762 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 145 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
1C2×#17984887C211yesmaximal, characteristic, complemented
2#22581751C212yesmaximal, complemented
3#22581130C214yesmaximal, complemented
4#22581120C214yesmaximal, complemented
5#22580971C212yesmaximal, complemented
6C2×#17984759C211yesmaximal, characteristic, complemented
7#535×C139C211yesmaximal, characteristic, complemented
8#17984887—24no—
9#17984887—22no—
10#17984887C2214yes—
11C2×#14555785C2212yes—
12C2×#14555784—22no—
13C22×C22⋊D139—22no—
14C2×D4×D139—22no—
15#17984873C2214yescomplemented
16#17984872—22no—
17#17984868C2214yescomplemented
18#17984867—22no—
19#17984862—24no—
20C2×#14555769C2212yescomplemented
21#17984770C2218yescomplemented
22#17984759C2214yescomplemented
23C2×#41×C139C2211yescharacteristic, complemented
24C22×D4×C139C2211yescharacteristic
25#127×C139C2212yescomplemented
26#124×C139C2212yescomplemented
27#121×C139—22no—
28#120×C139—22no—
29#117×C139C2211yescharacteristic, complemented
30C23×D139—22no—
31#14555785—22no—
32#14555785—22no—
33#14555785D414yescomplemented
34#14555785C2314yes—
35#14555784—28no—
36#14555784—28no—
37C2×Dic278D412yescomplemented
38C2×D556D412yescomplemented
39C2×C22⋊D139—24no—
40C2×C22⋊D139—28no—
41C2×C22⋊D139—22no—
42C2×C22⋊D139—28no—
43D4×D139—28no—
44D4×D139—28no—
45C22×Dic139—22no—
46C2×C4×D139—22no—
47#14555775—24no—
48#14555773—24no—
49#14555772—24no—
50#14555770—24no—
51#14555769—22no—
52#14555769C2314yescomplemented
53C2×C139⋊C8—22no—
54C24×C139D412yes—
55#41×C139—22no—
56#41×C139—22no—
57C2×Q8×C139C2311yescharacteristic, complemented
58C2×D4×C139—21no—
59C2×D4×C139—24no—
60C2×D4×C139—22no—
61C2×D4×C139—22no—
62C2×D4×C139D412yes—
63C2×D4×C139C2312yes—
64C2×D4×C139C2311yescharacteristic, complemented
65C22×C4×C139—21no—
66C22×C4×C139C2311yescharacteristic
67C4⋊C4×C139—21no—
68C22⋊C4×C139—21no—
69C22⋊C4×C139—24no—
70C42×C139C2312yescomplemented
71C22×D139—22no—
72C22×D139—28no—
73C22×D139—84no—
74Dic278—22no—
75Dic278C2×D414yes—
76D556—22no—
77D556C2×D414yes—
78C22⋊D139—216no—
79C22⋊D139—24no—
80C22⋊D139—24no—
81C2×Dic139—28no—
82C2×Dic139—22no—
83C4×D139—24no—
84C4×D139—24no—
85C139⋊C8—24no—
86C23×C139—22no—
87C23×C139—22no—
88C23×C139—21no—
89C23×C139—82no—
90C23×C139C2×D414yes—
91C23×C139C2×D411yescharacteristic
92Q8×C139—22no—
93D4×C139—24no—
94D4×C139—24no—
95D4×C139—22no—
96D4×C139—41no—
97D4×C139—41no—
98D4×C139—84no—
99C2×C4×C139—22no—
100C2×C4×C139—21no—
101C2×C4×C139—42no—
102C2×C4×C139C2×D412yes—
103C2×C4×C139C2×D412yes—
104C2×C4×C139C2×D411yescharacteristic
105C2×C4×C139C2411yescharacteristic
106C22×C139—24no—
107C22×C139—41no—
108C22×C139—44no—
109C22×C139—44no—
110C22×C139—42no—
111C22×C139—41no—
112C22×C139—82no—
113C22×C139#12011yescharacteristic
114C22×C139#12011yescharacteristic
115C22×C139C22×D411yescharacteristic
116C2×D139—24no—
117C2×D139—44no—
118C2×D139—88no—
119C4×C139—42no—
120C4×C139—41no—
121C4×C139#12012yes—
122C4×C139C22×D412yes—
123Dic139—24no—
124C2×C139—21no—
125C2×C139—21no—
126C2×C139—21no—
127C2×C139—44no—
128C2×C139—81no—
129C2×C139C2×#12011yescharacteristic
130D139—44no—
131C139#301911yescharacteristic, complemented, Sylow
132#3019—1391nomaximal, Sylow
133C2×H(5,𝔽2)—1391no—
134#575—1392no—
135#535D13911yescharacteristic, complemented
136#455—1394no—
137#453—1394no—
138#421—1392no—
139C2×#100—1391no—
140H(5,𝔽2)—2784no—
141H(5,𝔽2)—2782no—
142H(5,𝔽2)—1394no—
143C2×#41—2782no—
144C2×#41—1392no—
145C2×#41C2×D13911yescharacteristic, complemented
146C22×D4—2782no—
147C22×D4—2782no—
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#127C2×D13912yescomplemented
156#124C2×D13912yescomplemented
157#121—22no—
158#120—22no—
159#117C2×D13911yescharacteristic, complemented
160#104—1398no—
161#100—1394no—
162C24—2782no—
163C24C22⋊D13912yes—
164#41—22no—
165#41—22no—
166#41—2788no—
167#41—2788no—
168#41—2782no—
169#41—2782no—
170#41—1394no—
171#41—1394no—
172C2×Q8—1392no—
173C2×Q8C22×D13911yescharacteristic, complemented
174C2×D4—21no—
175C2×D4—24no—
176C2×D4—22no—
177C2×D4—22no—
178C2×D4—2788no—
179C2×D4—2788no—
180C2×D4—2784no—
181C2×D4—2788no—
182C2×D4—2782no—
183C2×D4—2788no—
184C2×D4—1392no—
185C2×D4C22⋊D13912yes—
186C2×D4C22×D13912yes—
187C2×D4C22×D13911yescharacteristic, complemented
188C22×C4—21no—
189C22×C4—2782no—
190C22×C4—2782no—
191C22×C4C22×D13911yescharacteristic
192Q16—2784no—
193SD16—2784no—
194SD16—2784no—
195D8—2784no—
196M16—2782no—
197M16—1394no—
198C2×C8—2782no—
199C4⋊C4—21no—
200C22⋊C4—21no—
201C22⋊C4—24no—
202C42C22×D13912yescomplemented
203C23—22no—
204C23—22no—
205C23—21no—
206C23—82no—
207C23—2788no—
208C23—2782no—
209C23—11124no—
210C23C2×C22⋊D13914yes—
211C23C2×C22⋊D13911yescharacteristic
212Q8—22no—
213Q8—2782no—
214Q8—1394no—
215D4—24no—
216D4—24no—
217D4—22no—
218D4—41no—
219D4—41no—
220D4—84no—
221D4—27816no—
222D4—2784no—
223D4—2784no—
224D4—2782no—
225D4—1394no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—42no—
229C2×C4—2788no—
230C2×C4—2782no—
231C2×C4—2784no—
232C2×C4—2784no—
233C2×C4D4×D13912yes—
234C2×C4D4×D13912yes—
235C2×C4C2×C22⋊D13911yescharacteristic
236C2×C4C23×D13911yescharacteristic
237C8—2784no—
238C22—24no—
239C22—41no—
240C22—44no—
241C22—44no—
242C22—42no—
243C22—41no—
244C22—82no—
245C22—2784no—
246C22—5564no—
247C22—11128no—
248C22#1798481011yescharacteristic
249C22#1798481011yescharacteristic
250C22C22×C22⋊D13911yescharacteristic
251C4—42no—
252C4—41no—
253C4—2784no—
254C4#1798481012yes—
255C4C2×D4×D13912yes—
256C2—21no—
257C2—21no—
258C2—21no—
259C2—44no—
260C2—81no—
261C2—5564no—
262C2C2×#1798481011yescharacteristic
subgroupG/H
centerC2C2×#17984810
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
Fitting#535×C139C2
socleC2×C139C2×#120
hypercenter#535D139
hypocenterC139#3019
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C2 ◁ C22×C4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#17984887C2semidirect
2. #22581751C2semidirect
3. #22581130C2semidirect
4. #22581120C2semidirect
5. #22580971C2semidirect
6. C2×#17984759C2semidirect
7. #535×C139C2semidirect
15. #17984873C22semidirect
17. #17984868C22semidirect
20. C2×#14555769C22semidirect
21. #17984770C22semidirect
22. #17984759C22semidirect
23. C2×#41×C139C22semidirect
25. #127×C139C22semidirect
26. #124×C139C22semidirect
29. #117×C139C22semidirect
33. #14555785D4semidirect
37. C2×Dic278D4semidirect
38. C2×D556D4semidirect
52. #14555769C23semidirect
57. C2×Q8×C139C23semidirect
64. C2×D4×C139C23semidirect
70. C42×C139C23semidirect
131. C139#3019semidirect
135. #535D139semidirect
145. C2×#41C2×D139semidirect
155. #127C2×D139semidirect
156. #124C2×D139semidirect
159. #117C2×D139semidirect
173. C2×Q8C22×D139semidirect
187. C2×D4C22×D139semidirect
202. C42C22×D139semidirect

Maximal subgroups

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

subgroupindex
C2×#179848872
#225817512
#225811302
#225811202
#225809712
C2×#179847592
#535×C1392
#3019139