Finite Group Databasebeta

← #31815713

Subgroups — #31815713

Order 17792.

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

normal subgroups only

This group has 764 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×#17984887C211yesmaximal, characteristic, complemented
2C2×#17984868C211yesmaximal, characteristic, complemented
3C2×#17984864C211yesmaximal, characteristic, complemented
4#22581534C211yesmaximal, characteristic, complemented
5#22581116C212yesmaximal, complemented
6#22581105C212yesmaximal, complemented
7#22581099C212yesmaximal, complemented
8#22581088C212yesmaximal, complemented
9#22580952C211yesmaximal, characteristic, complemented
10#22580919C211yesmaximal, characteristic, complemented
11C2×#102×C139C211yesmaximal, characteristic, complemented
12#17984887—24no—
13#17984887—24no—
14C2×#14555785C2211yescharacteristic
15C2×#14555785C2211yescharacteristic
16C2×#14555785C2211yescharacteristic, complemented
17C2×#14555784—21no—
18C2×#14555784C2212yes—
19C22×C22⋊D139—22no—
20C2×D4×D139—21no—
21C2×D4×D139C2212yes—
22#17984868—24no—
23#17984864—24no—
24C2×Dic556C2211yescharacteristic, complemented
25C2×#14555774C2211yescharacteristic, complemented
26C2×#14555774C2211yescharacteristic, complemented
27C2×#14555773C2212yescomplemented
28C2×D1112C2211yescharacteristic, complemented
29C2×#14555770C2212yescomplemented
30C2×#14555769C2211yescharacteristic, complemented
31#17984822—21no—
32#17984822C2212yescomplemented
33#17984817—21no—
34#17984817C2212yescomplemented
35#17984816—22no—
36#17984765C2212yescomplemented
37#17984762C2212yescomplemented
38(C2×D139)⋊C8C2212yescomplemented
39(C2×D139)⋊C8C2212yescomplemented
40C22×D4×C139C2211yescharacteristic
41C22×C8×C139C2211yescharacteristic, complemented
42C2×C4⋊C4×C139C2211yescharacteristic, complemented
43#102×C139C2212yescomplemented
44#102×C139C2212yescomplemented
45C23×D139—22no—
46#14555785—22no—
47#14555785—22no—
48#14555785—22no—
49#14555785—22no—
50#14555785—22no—
51#14555785D414yescomplemented
52#14555784—28no—
53#14555784—24no—
54#14555784—24no—
55C2×Dic278D411yescharacteristic
56C2×Dic278C2311yescharacteristic
57C2×Dic278C2311yescharacteristic
58C2×D556—22no—
59C2×D556D411yescharacteristic
60C2×D556C2311yescharacteristic
61C2×D556C2311yescharacteristic
62C2×C22⋊D139—216no—
63C2×C22⋊D139—22no—
64C2×C22⋊D139—22no—
65C2×C22⋊D139—21no—
66C2×C22⋊D139—21no—
67C2×C22⋊D139D412yes—
68D4×D139—28no—
69D4×D139—24no—
70D4×D139—24no—
71C22×Dic139—22no—
72C2×C4×D139—21no—
73C2×C4×D139—21no—
74C2×C4×D139C2312yes—
75Dic556—22no—
76#14555774—22no—
77#14555774—22no—
78#14555773—24no—
79D1112—22no—
80#14555770—24no—
81#14555769—22no—
82C2×C139⋊C8C2312yes—
83C4⋊Dic139—22no—
84(C2×D139)⋊C4—22no—
85(C2×D139)⋊C4—22no—
86C24×C139—21no—
87C2×D4×C139—21no—
88C2×D4×C139—22no—
89C2×D4×C139D414yescomplemented
90C2×D4×C139C2312yes—
91C22×C4×C139—21no—
92C22×C4×C139C2311yescharacteristic
93C2×C8×C139—22no—
94C2×C8×C139C2311yescharacteristic
95C2×C8×C139C2311yescharacteristic
96C4⋊C4×C139—21no—
97C4⋊C4×C139C2312yes—
98C22×D139—21no—
99C22×D139—28no—
100C22×D139—21no—
101C22×D139—82no—
102C22×D139—82no—
103C22×D139C2×D412yes—
104Dic278—22no—
105Dic278—21no—
106Dic278—21no—
107Dic278—21no—
108Dic278C2×D412yes—
109D556—22no—
110D556—21no—
111D556—21no—
112D556—21no—
113D556—82no—
114D556C2×D412yes—
115C22⋊D139—28no—
116C22⋊D139—28no—
117C22⋊D139—24no—
118C22⋊D139—24no—
119C22⋊D139—42no—
120C2×Dic139—28no—
121C2×Dic139—21no—
122C2×Dic139—21no—
123C2×Dic139C2×D412yes—
124C4×D139—24no—
125C4×D139—22no—
126C4×D139—22no—
127C4×D139—42no—
128C139⋊C8—22no—
129C23×C139—22no—
130C23×C139—24no—
131C23×C139—81no—
132C23×C139—81no—
133C23×C139C2×D411yescharacteristic
134D4×C139—24no—
135D4×C139—22no—
136D4×C139C2×D414yes—
137C2×C4×C139—22no—
138C2×C4×C139—81no—
139C2×C4×C139C2×D411yescharacteristic
140C2×C4×C139C2×D412yes—
141C2×C4×C139C2×D412yes—
142C2×C4×C139C2411yescharacteristic
143C8×C139—21no—
144C8×C139—21no—
145C22×C139—21no—
146C22×C139—21no—
147C22×C139—24no—
148C22×C139—42no—
149C22×C139—42no—
150C22×C139—84no—
151C22×C139#12012yes—
152C22×C139C22×D411yescharacteristic
153C2×D139—22no—
154C2×D139—22no—
155C2×D139—22no—
156C2×D139—22no—
157C2×D139—41no—
158C2×D139—41no—
159C2×D139—41no—
160C2×D139—41no—
161C2×D139—41no—
162C2×D139—88no—
163C4×C139—42no—
164C4×C139#12012yes—
165C4×C139C22×D411yescharacteristic
166C4×C139C22×D411yescharacteristic
167Dic139—22no—
168Dic139—22no—
169Dic139—41no—
170C2×C139—22no—
171C2×C139—44no—
172C2×C139C2×#12011yescharacteristic
173C2×C139#57611yescharacteristic
174C2×C139#57711yescharacteristic
175D139—42no—
176D139—42no—
177D139—81no—
178C139#301211yescharacteristic, complemented, Sylow
179#3012—1391nomaximal, Sylow
180C2×H(5,𝔽2)—1391no—
181C2×AGL(1,ℤ/8)—1391no—
182C2×#135—1391no—
183#537—1391no—
184#452—1392no—
185#450—1392no—
186#449—1392no—
187#447—1392no—
188#417—1391no—
189C2×#102D13911yescharacteristic, complemented
190#408—1391no—
191H(5,𝔽2)—2784no—
192H(5,𝔽2)—2784no—
193C2×#41—2781no—
194C2×#41—1392no—
195C2×#41—1391no—
196C2×#41—1391no—
197C2×#41—1391no—
198C22×D4—2781no—
199C22×D4—2782no—
200C22×D4—1392no—
201C22×D4C2×D13911yescharacteristic
202AGL(1,ℤ/8)—2784no—
203#135—2784no—
204C2×Q16—1391no—
205C2×SD16—1392no—
206C2×SD16—1391no—
207C2×SD16—1391no—
208C2×D8—1392no—
209C2×D8—1391no—
210C2×M16—1391no—
211C22×C8C2×D13911yescharacteristic, complemented
212C22⋊Q8—2781no—
213C22⋊Q8—1392no—
214#121—2781no—
215#121—2782no—
216#121—1392no—
217C2×C4⋊C4C2×D13911yescharacteristic, complemented
218#103—1392no—
219#102—1392no—
220#102C2×D13912yescomplemented
221#102C2×D13912yescomplemented
222C22⋊C8—1392no—
223C22⋊C8—1392no—
224C24—21no—
225C24—2782no—
226#41—2788no—
227#41—2784no—
228#41—2784no—
229#41—2782no—
230#41—2782no—
231#41—2782no—
232#41—2782no—
233#41—2782no—
234#41—1394no—
235C2×Q8—1391no—
236C2×Q8—1391no—
237C2×Q8—1391no—
238C2×D4—21no—
239C2×D4—22no—
240C2×D4—2782no—
241C2×D4—2788no—
242C2×D4—2784no—
243C2×D4—27816no—
244C2×D4—2782no—
245C2×D4—2781no—
246C2×D4—2784no—
247C2×D4—2781no—
248C2×D4—2782no—
249C2×D4—1392no—
250C2×D4—1391no—
251C2×D4—1391no—
252C2×D4—1391no—
253C2×D4D55614yescomplemented
254C2×D4C22×D13912yes—
255C22×C4—21no—
256C22×C4—2782no—
257C22×C4—2781no—
258C22×C4—2781no—
259C22×C4—1392no—
260C22×C4C22×D13911yescharacteristic
261Q16—2782no—
262SD16—2782no—
263SD16—2782no—
264SD16—2784no—
265D8—2782no—
266D8—2784no—
267M16—2782no—
268C2×C8—22no—
269C2×C8—1392no—
270C2×C8C22×D13911yescharacteristic
271C2×C8C22×D13911yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—2782no—
274C4⋊C4C22×D13912yes—
275C22⋊C4—2782no—
276C22⋊C4—2782no—
277C23—22no—
278C23—24no—
279C23—81no—
280C23—81no—
281C23—2788no—
282C23—2781no—
283C23—2781no—
284C23—11122no—
285C23—11122no—
286C23—1392no—
287C23D4×D13911yescharacteristic
288Q8—2782no—
289Q8—2781no—
290Q8—2781no—
291Q8—2781no—
292Q8—1392no—
293D4—24no—
294D4—22no—
295D4—2788no—
296D4—2788no—
297D4—2784no—
298D4—2784no—
299D4—2782no—
300D4—2781no—
301D4—2781no—
302D4—2781no—
303D4—5562no—
304D4—11122no—
305D4—1392no—
306D4C2×D55614yes—
307C2×C4—22no—
308C2×C4—81no—
309C2×C4—2788no—
310C2×C4—2781no—
311C2×C4—2784no—
312C2×C4—2782no—
313C2×C4—2782no—
314C2×C4—2781no—
315C2×C4—5562no—
316C2×C4—1392no—
317C2×C4D4×D13911yescharacteristic
318C2×C4D4×D13912yes—
319C2×C4C2×D55612yes—
320C2×C4C23×D13911yescharacteristic
321C8—21no—
322C8—21no—
323C8—2782no—
324C22—21no—
325C22—21no—
326C22—24no—
327C22—42no—
328C22—42no—
329C22—84no—
330C22—2782no—
331C22—2782no—
332C22—2782no—
333C22—2782no—
334C22—5561no—
335C22—5561no—
336C22—5561no—
337C22—5561no—
338C22—5561no—
339C22—11128no—
340C22#1798481212yes—
341C22C2×D4×D13911yescharacteristic
342C4—42no—
343C4—2782no—
344C4—2782no—
345C4—5561no—
346C4#1798481212yes—
347C4C2×D4×D13911yescharacteristic
348C4C22×D55611yescharacteristic
349C2—22no—
350C2—44no—
351C2—5562no—
352C2—5562no—
353C2—11121no—
354C2C2×#1798481211yescharacteristic
355C2#2258175411yescharacteristic
356C2#2258175911yescharacteristic
subgroupG/H
centerC22C2×D4×D139
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
FittingC2×#102×C139C2
socleC22×C139C22×D4
hypercenterC2×#102D139
hypocenterC139#3012
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C2×#17984887C2semidirect
2. C2×#17984868C2semidirect
3. C2×#17984864C2semidirect
4. #22581534C2semidirect
5. #22581116C2semidirect
6. #22581105C2semidirect
7. #22581099C2semidirect
8. #22581088C2semidirect
9. #22580952C2semidirect
10. #22580919C2semidirect
11. C2×#102×C139C2semidirect
16. C2×#14555785C22semidirect
24. C2×Dic556C22semidirect
25. C2×#14555774C22semidirect
26. C2×#14555774C22semidirect
27. C2×#14555773C22semidirect
28. C2×D1112C22semidirect
29. C2×#14555770C22semidirect
30. C2×#14555769C22semidirect
32. #17984822C22semidirect
34. #17984817C22semidirect
36. #17984765C22semidirect
37. #17984762C22semidirect
38. (C2×D139)⋊C8C22semidirect
39. (C2×D139)⋊C8C22semidirect
41. C22×C8×C139C22semidirect
42. C2×C4⋊C4×C139C22semidirect
43. #102×C139C22semidirect
44. #102×C139C22semidirect
51. #14555785D4semidirect
89. C2×D4×C139D4semidirect
178. C139#3012semidirect
189. C2×#102D139semidirect
211. C22×C8C2×D139semidirect
217. C2×C4⋊C4C2×D139semidirect
220. #102C2×D139semidirect
221. #102C2×D139semidirect
253. C2×D4D556semidirect

Maximal subgroups

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

subgroupindex
C2×#179848872
C2×#179848682
C2×#179848642
#225815342
#225811162
#225811052
#225810992
#225810882
#225809522
#225809192
C2×#102×C1392
#3012139