Finite Group Databasebeta

← #31815679

Subgroups — #31815679

Order 17792.

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

normal subgroups only

This group has 752 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×#17984888C211yesmaximal, characteristic
2C2×#17984873C211yesmaximal, characteristic, complemented
3C22×#14555772C211yesmaximal, characteristic, complemented
4#22581113C214yesmaximal, complemented
5#22581103C214yesmaximal, complemented
6#22580957C211yesmaximal, characteristic, complemented
7#22580947C211yesmaximal, characteristic, complemented
8#22580920C211yesmaximal, characteristic, complemented
9#536×C139C211yesmaximal, characteristic, complemented
10#17984888—24no—
11#17984888—24no—
12C2×#14555786—21no—
13C2×#14555785C2212yes—
14C2×#14555785C2211yescharacteristic
15C22×D556C2212yes—
16C22×D556C2211yescharacteristic
17C2×D4×D139—22no—
18C2×D4×D139—21no—
19#17984873—24no—
20C2×#14555775C2212yescomplemented
21C2×#14555772—24no—
22C2×#14555772C2212yescomplemented
23C2×#14555772C2214yescomplemented
24C2×#14555769C2211yescharacteristic, complemented
25C22×C139⋊C8C2211yescharacteristic, complemented
26#17984765C2212yescomplemented
27#17984762C2212yescomplemented
28#17984762C2214yescomplemented
29C22⋊(C139⋊C8)C2214yescomplemented
30C2×#41×C139C2211yescharacteristic
31C22×Q8×C139C2211yescharacteristic, complemented
32C4⋊Q8×C139—21no—
33#124×C139—21no—
34#123×C139—22no—
35C22⋊Q8×C139C2214yescomplemented
36#117×C139C2211yescharacteristic, complemented
37C23×D139—22no—
38C23×D139—21no—
39#14555786—28no—
40#14555785—24no—
41#14555785—24no—
42#14555785—22no—
43#14555785D414yes—
44C2×Dic278C2311yescharacteristic
45C2×D556—24no—
46C2×D556—24no—
47C2×D556—22no—
48C2×D556—21no—
49C2×D556—22no—
50C2×D556D414yes—
51C2×D556C2311yescharacteristic
52C2×D556C2312yes—
53C2×C22⋊D139—22no—
54C2×C22⋊D139—21no—
55D4×D139—216no—
56D4×D139—28no—
57C2×C4×D139—22no—
58C2×C4×D139—21no—
59#14555775—24no—
60#14555772—28no—
61#14555772—24no—
62#14555769—22no—
63C2×C139⋊C8—22no—
64C2×C139⋊C8C2312yes—
65C2×C139⋊C8C2312yes—
66#41×C139—22no—
67#41×C139—22no—
68C2×Q8×C139—21no—
69C2×Q8×C139—22no—
70C2×Q8×C139—22no—
71C2×Q8×C139D411yescharacteristic
72C2×Q8×C139C2312yes—
73C2×D4×C139D412yes—
74C2×D4×C139D411yescharacteristic
75C22×C4×C139—21no—
76C22×C4×C139C2312yes—
77C22×C4×C139C2311yescharacteristic
78C4⋊C4×C139—24no—
79C4⋊C4×C139C2312yes—
80C22⋊C4×C139—21no—
81C22⋊C4×C139—24no—
82C42×C139—21no—
83C22×D139—22no—
84C22×D139—21no—
85C22×D139—28no—
86C22×D139—24no—
87C22×D139—82no—
88C22×D139—84no—
89Dic278—21no—
90Dic278C2×D412yes—
91D556—28no—
92D556—24no—
93D556—22no—
94D556—21no—
95D556C2×D412yes—
96D556C2×D414yes—
97C22⋊D139—28no—
98C22⋊D139—24no—
99C2×Dic139—21no—
100C4×D139—28no—
101C4×D139—24no—
102C139⋊C8—22no—
103C139⋊C8—22no—
104C23×C139C2×D412yes—
105C23×C139C2×D411yescharacteristic
106Q8×C139—22no—
107Q8×C139—22no—
108Q8×C139—22no—
109Q8×C139—42no—
110D4×C139—24no—
111D4×C139—22no—
112C2×C4×C139—22no—
113C2×C4×C139—22no—
114C2×C4×C139—22no—
115C2×C4×C139—22no—
116C2×C4×C139—42no—
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#12012yes—
126C22×C139C22×D411yescharacteristic
127C2×D139—24no—
128C2×D139—22no—
129C2×D139—44no—
130C2×D139—42no—
131C2×D139—84no—
132C2×D139—88no—
133C4×C139—22no—
134C4×C139—42no—
135C4×C139—42no—
136C4×C139#12012yes—
137C4×C139C22×D411yescharacteristic
138C4×C139C22×D411yescharacteristic
139Dic139—22no—
140C2×C139—22no—
141C2×C139—42no—
142C2×C139C2×#12011yescharacteristic
143C2×C139#57712yes—
144D139—44no—
145D139—42no—
146C139#300911yescharacteristic, complemented, Sylow
147#3009—1391nomaximal, Sylow
148C2×H(5,𝔽2)—1391no—
149C2×#137—1391no—
150C22×SD16—1391no—
151#536D13911yescharacteristic, complemented
152#452—1394no—
153#450—1394no—
154#418—1391no—
155#416—1391no—
156#408—1391no—
157H(5,𝔽2)—2784no—
158H(5,𝔽2)—2784no—
159C2×#41—2781no—
160C2×#41—1392no—
161C2×#41—1391no—
162C2×#41C2×D13911yescharacteristic
163C22×Q8C2×D13911yescharacteristic, complemented
164C22×D4—2782no—
165C22×D4—2781no—
166C22×D4—1392no—
167C22×D4—1391no—
168#137—2784no—
169C2×Q16—1392no—
170C2×SD16—2784no—
171C2×SD16—1394no—
172C2×SD16—1392no—
173C2×M16—1391no—
174C22×C8—1391no—
175C4⋊Q8—21no—
176#124—21no—
177#123—22no—
178C22⋊Q8C2×D13914yescomplemented
179#117C2×D13911yescharacteristic, complemented
180#103—1392no—
181#102—1392no—
182#102—1394no—
183C22⋊C8—1394no—
184C24—2782no—
185C24—2781no—
186#41—22no—
187#41—22no—
188#41—2784no—
189#41—2784no—
190#41—2788no—
191#41—2782no—
192#41—1394no—
193C2×Q8—21no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—1391no—
197C2×Q8C22⋊D13911yescharacteristic
198C2×Q8C22×D13912yes—
199C2×D4—2784no—
200C2×D4—2784no—
201C2×D4—27816no—
202C2×D4—2788no—
203C2×D4—2782no—
204C2×D4—2781no—
205C2×D4—2782no—
206C2×D4—2782no—
207C2×D4—2781no—
208C2×D4—1394no—
209C2×D4—1391no—
210C2×D4—1392no—
211C2×D4C22⋊D13912yes—
212C2×D4C22⋊D13911yescharacteristic
213C22×C4—21no—
214C22×C4—2782no—
215C22×C4—2781no—
216C22×C4C22×D13912yes—
217C22×C4C22×D13911yescharacteristic
218Q16—2784no—
219SD16—2788no—
220SD16—2784no—
221M16—2782no—
222C2×C8—2782no—
223C2×C8—1392no—
224C2×C8—1392no—
225C4⋊C4—24no—
226C4⋊C4C22×D13912yes—
227C22⋊C4—21no—
228C22⋊C4—24no—
229C42—21no—
230C23—2788no—
231C23—2782no—
232C23—2781no—
233C23—2784no—
234C23—11122no—
235C23—11124no—
236C23C2×C22⋊D13912yes—
237C23C2×C22⋊D13911yescharacteristic
238Q8—22no—
239Q8—22no—
240Q8—22no—
241Q8—42no—
242Q8—2781no—
243Q8—1392no—
244D4—24no—
245D4—22no—
246D4—2788no—
247D4—2788no—
248D4—2784no—
249D4—2784no—
250D4—2782no—
251D4—2781no—
252D4—1392no—
253D4—1394no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—42no—
259C2×C4—2788no—
260C2×C4—2781no—
261C2×C4—2784no—
262C2×C4D4×D13912yes—
263C2×C4D4×D13912yes—
264C2×C4C2×C22⋊D13912yes—
265C2×C4C2×C22⋊D13911yescharacteristic
266C2×C4C23×D13911yescharacteristic
267C8—2782no—
268C8—2782no—
269C22—22no—
270C22—24no—
271C22—22no—
272C22—2784no—
273C22—2782no—
274C22—5564no—
275C22—5562no—
276C22—11124no—
277C22—11128no—
278C22#1798481012yes—
279C22C22×C22⋊D13911yescharacteristic
280C4—22no—
281C4—42no—
282C4—42no—
283C4—2782no—
284C4#1798481012yes—
285C4C2×D4×D13911yescharacteristic
286C4C2×D4×D13911yescharacteristic
287C2—22no—
288C2—42no—
289C2—5564no—
290C2—5562no—
291C2C2×#1798481011yescharacteristic
292C2#2258176112yes—
subgroupG/H
centerC22C22×C22⋊D139
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
Fitting#536×C139C2
socleC22×C139C22×D4
prime radicalC2×#17984888C2
hypercenter#536D139
hypocenterC139#3009
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
2. C2×#17984873C2semidirect
3. C22×#14555772C2semidirect
4. #22581113C2semidirect
5. #22581103C2semidirect
6. #22580957C2semidirect
7. #22580947C2semidirect
8. #22580920C2semidirect
9. #536×C139C2semidirect
20. C2×#14555775C22semidirect
22. C2×#14555772C22semidirect
23. C2×#14555772C22semidirect
24. C2×#14555769C22semidirect
25. C22×C139⋊C8C22semidirect
26. #17984765C22semidirect
27. #17984762C22semidirect
28. #17984762C22semidirect
29. C22⋊(C139⋊C8)C22semidirect
31. C22×Q8×C139C22semidirect
35. C22⋊Q8×C139C22semidirect
36. #117×C139C22semidirect
146. C139#3009semidirect
151. #536D139semidirect
163. C22×Q8C2×D139semidirect
178. C22⋊Q8C2×D139semidirect
179. #117C2×D139semidirect

Maximal subgroups

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

subgroupindex
C2×#179848882
C2×#179848732
C22×#145557722
#225811132
#225811032
#225809572
#225809472
#225809202
#536×C1392
#3009139