Finite Group Databasebeta

← #31815688

Subgroups — #31815688

Order 17792.

732 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 149 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×#17984889C211yesmaximal, characteristic, complemented
2C2×#17984868C211yesmaximal, characteristic, complemented
3C22×#14555773C211yesmaximal, characteristic, complemented
4#22581096C214yesmaximal, complemented
5#22581091C214yesmaximal, complemented
6#22580961C211yesmaximal, characteristic, complemented
7#22580947C211yesmaximal, characteristic, complemented
8#22580920C211yesmaximal, characteristic, complemented
9#535×C139C211yesmaximal, characteristic, complemented
10#17984889—24no—
11#17984889—24no—
12C2×#14555785C2211yescharacteristic
13C2×#14555785C2212yescomplemented
14C2×#14555784—22no—
15C2×#14555784—21no—
16C22×Dic278C2211yescharacteristic
17C22×Dic278C2212yescomplemented
18C2×Q8×D139—21no—
19#17984868—24no—
20C2×#14555773—24no—
21C2×#14555773C2212yes—
22C2×#14555773C2214yes—
23C2×#14555770C2212yes—
24C2×#14555769C2211yescharacteristic
25C22×C139⋊C8C2211yescharacteristic
26#17984765C2212yescomplemented
27#17984765C2214yescomplemented
28#17984762C2212yescomplemented
29C22⋊(C139⋊C8)C2214yescomplemented
30C2×#41×C139C2211yescharacteristic
31C22×D4×C139C2211yescharacteristic
32#127×C139—21no—
33#124×C139—21no—
34#121×C139C2214yes—
35#120×C139—22no—
36#117×C139C2211yescharacteristic
37#14555785—24no—
38#14555785—24no—
39#14555785—22no—
40#14555785D414yescomplemented
41#14555784—216no—
42#14555784—28no—
43C2×Dic278—24no—
44C2×Dic278—24no—
45C2×Dic278—21no—
46C2×Dic278—22no—
47C2×Dic278—22no—
48C2×Dic278D414yescomplemented
49C2×Dic278C2312yes—
50C2×Dic278C2311yescharacteristic
51Q8×D139—28no—
52C2×D556C2311yescharacteristic
53C2×C22⋊D139—21no—
54C2×C22⋊D139—22no—
55C22×Dic139—21no—
56C22×Dic139—22no—
57C2×C4×D139—21no—
58C2×C4×D139—22no—
59#14555773—28no—
60#14555773—24no—
61#14555770—24no—
62#14555769—22no—
63C2×C139⋊C8—22no—
64C2×C139⋊C8C2312yes—
65C2×C139⋊C8C2312yes—
66C24×C139—21no—
67#41×C139—22no—
68#41×C139—22no—
69C2×Q8×C139D411yescharacteristic
70C2×D4×C139—24no—
71C2×D4×C139—22no—
72C2×D4×C139—22no—
73C2×D4×C139—21no—
74C2×D4×C139D412yes—
75C2×D4×C139D411yescharacteristic
76C2×D4×C139C2312yes—
77C22×C4×C139C2312yes—
78C22×C4×C139C2311yescharacteristic
79C4⋊C4×C139C2312yes—
80C22⋊C4×C139—21no—
81C22⋊C4×C139—24no—
82C42×C139—21no—
83C22×D139—21no—
84Dic278—28no—
85Dic278—24no—
86Dic278—21no—
87Dic278—22no—
88Dic278C2×D412yes—
89Dic278C2×D414yes—
90D556—21no—
91D556C2×D412yes—
92C22⋊D139—28no—
93C22⋊D139—24no—
94C2×Dic139—28no—
95C2×Dic139—24no—
96C2×Dic139—21no—
97C2×Dic139—22no—
98C4×D139—28no—
99C4×D139—24no—
100C139⋊C8—22no—
101C139⋊C8—22no—
102C23×C139—22no—
103C23×C139—42no—
104C23×C139—82no—
105C23×C139C2×D412yes—
106C23×C139C2×D411yescharacteristic
107Q8×C139—22no—
108D4×C139—24no—
109D4×C139—22no—
110D4×C139—22no—
111D4×C139—22no—
112D4×C139—42no—
113D4×C139—84no—
114C2×C4×C139—22no—
115C2×C4×C139—22no—
116C2×C4×C139—22no—
117C2×C4×C139C2×D412yes—
118C2×C4×C139C2×D412yes—
119C2×C4×C139C2×D412yes—
120C2×C4×C139C2×D411yescharacteristic
121C2×C4×C139C2411yescharacteristic
122C22×C139—22no—
123C22×C139—22no—
124C22×C139—24no—
125C22×C139—42no—
126C22×C139—44no—
127C22×C139—84no—
128C22×C139#12012yes—
129C22×C139C22×D411yescharacteristic
130C2×D139—22no—
131C2×D139—42no—
132C4×C139—22no—
133C4×C139—42no—
134C4×C139#12012yes—
135C4×C139C22×D411yescharacteristic
136C4×C139C22×D411yescharacteristic
137Dic139—24no—
138Dic139—22no—
139C2×C139—22no—
140C2×C139—42no—
141C2×C139—82no—
142C2×C139C2×#12011yescharacteristic
143C2×C139#57712yes—
144D139—42no—
145C139#301011yescharacteristic, complemented, Sylow
146#3010—1391nomaximal, Sylow
147C2×2−1+4—1391no—
148C2×AGL(1,ℤ/8)—1391no—
149C22×SD16—1391no—
150#535D13911yescharacteristic, complemented
151#449—1394no—
152#448—1394no—
153#419—1391no—
154#416—1391no—
155#408—1391no—
1562−1+4—2784no—
1572−1+4—2784no—
158C2×#41—2782no—
159C2×#41—2781no—
160C2×#41—1392no—
161C2×#41—1391no—
162C2×#41C2×D13911yescharacteristic
163C22×Q8—2781no—
164C22×Q8—1392no—
165C22×Q8—1391no—
166C22×D4C2×D13911yescharacteristic
167AGL(1,ℤ/8)—2784no—
168C2×SD16—2784no—
169C2×SD16—1394no—
170C2×SD16—1392no—
171C2×D8—1392no—
172C2×M16—1391no—
173C22×C8—1391no—
174#127—21no—
175#124—21no—
176#121C2×D13914yes—
177#120—22no—
178#117C2×D13911yescharacteristic
179#103—1392no—
180#103—1394no—
181#102—1392no—
182C22⋊C8—1394no—
183C24—21no—
184#41—22no—
185#41—22no—
186#41—2784no—
187#41—2784no—
188#41—27816no—
189#41—2788no—
190#41—2782no—
191#41—1394no—
192C2×Q8—2784no—
193C2×Q8—2784no—
194C2×Q8—2788no—
195C2×Q8—2782no—
196C2×Q8—2782no—
197C2×Q8—2781no—
198C2×Q8—1394no—
199C2×Q8—1391no—
200C2×Q8—1392no—
201C2×Q8C22⋊D13911yescharacteristic
202C2×D4—24no—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—21no—
206C2×D4—2782no—
207C2×D4—2781no—
208C2×D4—1391no—
209C2×D4C22⋊D13912yes—
210C2×D4C22⋊D13911yescharacteristic
211C2×D4C22×D13912yes—
212C22×C4—2782no—
213C22×C4—2781no—
214C22×C4—2782no—
215C22×C4—2781no—
216C22×C4C22×D13912yes—
217C22×C4C22×D13911yescharacteristic
218SD16—2788no—
219SD16—2784no—
220D8—2784no—
221M16—2782no—
222C2×C8—2782no—
223C2×C8—1392no—
224C2×C8—1392no—
225C4⋊C4C22×D13912yes—
226C22⋊C4—21no—
227C22⋊C4—24no—
228C42—21no—
229C23—22no—
230C23—42no—
231C23—82no—
232C23—2781no—
233C23C2×C22⋊D13912yes—
234C23C2×C22⋊D13911yescharacteristic
235Q8—22no—
236Q8—2788no—
237Q8—2784no—
238Q8—2781no—
239Q8—2782no—
240Q8—1392no—
241Q8—1394no—
242D4—24no—
243D4—22no—
244D4—22no—
245D4—22no—
246D4—42no—
247D4—84no—
248D4—2788no—
249D4—2784no—
250D4—2781no—
251D4—1392no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—2788no—
256C2×C4—2788no—
257C2×C4—2784no—
258C2×C4—2782no—
259C2×C4—2781no—
260C2×C4—2784no—
261C2×C4D4×D13912yes—
262C2×C4D4×D13912yes—
263C2×C4C2×C22⋊D13912yes—
264C2×C4C2×C22⋊D13911yescharacteristic
265C2×C4C23×D13911yescharacteristic
266C8—2782no—
267C8—2782no—
268C22—22no—
269C22—24no—
270C22—22no—
271C22—44no—
272C22—42no—
273C22—84no—
274C22—2782no—
275C22—5562no—
276C22#1798481012yes—
277C22C22×C22⋊D13911yescharacteristic
278C4—22no—
279C4—42no—
280C4—2784no—
281C4—2782no—
282C4#1798481012yes—
283C4C2×D4×D13911yescharacteristic
284C4C2×D4×D13911yescharacteristic
285C2—22no—
286C2—42no—
287C2—82no—
288C2—5562no—
289C2C2×#1798481011yescharacteristic
290C2#2258176412yes—
subgroupG/H
centerC22C22×C22⋊D139
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
Fitting#535×C139C2
socleC22×C139C22×D4
hypercenter#535D139
hypocenterC139#3010
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#17984889C2semidirect
2. C2×#17984868C2semidirect
3. C22×#14555773C2semidirect
4. #22581096C2semidirect
5. #22581091C2semidirect
6. #22580961C2semidirect
7. #22580947C2semidirect
8. #22580920C2semidirect
9. #535×C139C2semidirect
13. C2×#14555785C22semidirect
17. C22×Dic278C22semidirect
26. #17984765C22semidirect
27. #17984765C22semidirect
28. #17984762C22semidirect
29. C22⋊(C139⋊C8)C22semidirect
40. #14555785D4semidirect
48. C2×Dic278D4semidirect
145. C139#3010semidirect
150. #535D139semidirect

Maximal subgroups

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

subgroupindex
C2×#179848892
C2×#179848682
C22×#145557732
#225810962
#225810912
#225809612
#225809472
#225809202
#535×C1392
#3010139