Finite Group Databasebeta

← #31815707

Subgroups — #31815707

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×#17984862C211yesmaximal, characteristic, complemented
3#22581531C211yesmaximal, characteristic, complemented
4#22581112C212yesmaximal, complemented
5#22581102C212yesmaximal, complemented
6#22581095C212yesmaximal, complemented
7#22581085C212yesmaximal, complemented
8#22580955C211yesmaximal, characteristic, complemented
9C2×#17984763C211yesmaximal, characteristic, complemented
10#22580921C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C139C211yesmaximal, characteristic, complemented
12#17984887—24no—
13#17984887—24no—
14C2×#14555785C2211yescharacteristic
15C2×#14555785C2211yescharacteristic
16C2×#14555784—21no—
17C2×#14555784C2212yes—
18C22×C22⋊D139—22no—
19C2×D4×D139—21no—
20C2×D4×D139C2212yes—
21#17984862—24no—
22C2×#14555775C2211yescharacteristic, complemented
23C2×#14555773C2211yescharacteristic, complemented
24C2×#14555772C2211yescharacteristic, complemented
25C2×#14555770C2211yescharacteristic, complemented
26C22×C139⋊C8C2211yescharacteristic, complemented
27#17984821—21no—
28#17984818C2212yescomplemented
29#17984816—21no—
30#17984814—22no—
31#17984813C2212yescomplemented
32C2×C4⋊Dic139C2211yescharacteristic, complemented
33#17984767C2212yescomplemented
34#17984764C2212yescomplemented
35#17984763C2212yescomplemented
36#17984763C2212yescomplemented
37(C2×D139)⋊C8C2212yescomplemented
38(C2×D139)⋊C8C2212yescomplemented
39C2×#41×C139C2211yescharacteristic, complemented
40C22×D4×C139C2211yescharacteristic
41AGL(1,ℤ/8)×C139—24no—
42C2×SD16×C139C2212yescomplemented
43C2×D8×C139C2212yescomplemented
44C2×M16×C139C2211yescharacteristic, complemented
45C23×D139—22no—
46#14555785—22no—
47#14555785—22no—
48#14555785—22no—
49#14555785D414yescomplemented
50#14555784—28no—
51#14555784—24no—
52#14555784—24no—
53C2×Dic278D411yescharacteristic
54C2×Dic278C2311yescharacteristic
55C2×D556D411yescharacteristic
56C2×D556C2311yescharacteristic
57C2×C22⋊D139—216no—
58C2×C22⋊D139—22no—
59C2×C22⋊D139—22no—
60C2×C22⋊D139—22no—
61C2×C22⋊D139—21no—
62C2×C22⋊D139D412yes—
63D4×D139—28no—
64D4×D139—24no—
65D4×D139—24no—
66C22×Dic139—22no—
67C22×Dic139—21no—
68C2×C4×D139—21no—
69C2×C4×D139C2312yes—
70#14555775—22no—
71#14555773—22no—
72#14555772—22no—
73#14555770—22no—
74C2×C139⋊C8—22no—
75C2×C139⋊C8C2311yescharacteristic
76C2×C139⋊C8C2311yescharacteristic
77C4⋊Dic139—21no—
78C4⋊Dic139C2312yes—
79Dic139⋊C4—22no—
80C22⋊Dic139—22no—
81(C2×D139)⋊C4—22no—
82C24×C139—21no—
83#41×C139—22no—
84#41×C139—22no—
85C2×Q8×C139C2311yescharacteristic
86C2×D4×C139—21no—
87C2×D4×C139—21no—
88C2×D4×C139—22no—
89C2×D4×C139D414yescomplemented
90C2×D4×C139C2311yescharacteristic
91C2×D4×C139C2312yes—
92C22×C4×C139—21no—
93C22×C4×C139C2311yescharacteristic
94SD16×C139—24no—
95D8×C139—24no—
96M16×C139—22no—
97C2×C8×C139C2312yes—
98C22×D139—21no—
99C22×D139—28no—
100C22×D139—82no—
101C22×D139—82no—
102C22×D139C2×D412yes—
103Dic278—22no—
104Dic278—21no—
105Dic278C2×D412yes—
106D556—22no—
107D556—21no—
108D556C2×D412yes—
109C22⋊D139—28no—
110C22⋊D139—28no—
111C22⋊D139—24no—
112C22⋊D139—24no—
113C22⋊D139—82no—
114C2×Dic139—28no—
115C2×Dic139—21no—
116C2×Dic139—22no—
117C2×Dic139—81no—
118C2×Dic139C2×D412yes—
119C4×D139—24no—
120C4×D139—22no—
121C4×D139—22no—
122C139⋊C8—21no—
123C139⋊C8—21no—
124C23×C139—22no—
125C23×C139—24no—
126C23×C139—21no—
127C23×C139—81no—
128C23×C139—81no—
129C23×C139C2×D411yescharacteristic
130Q8×C139—21no—
131Q8×C139—21no—
132D4×C139—24no—
133D4×C139—22no—
134D4×C139—21no—
135D4×C139—21no—
136D4×C139—42no—
137D4×C139C2×D414yes—
138C2×C4×C139—21no—
139C2×C4×C139—42no—
140C2×C4×C139C2×D411yescharacteristic
141C2×C4×C139C2×D412yes—
142C2×C4×C139C2×D412yes—
143C2×C4×C139C2411yescharacteristic
144C8×C139—22no—
145C22×C139—21no—
146C22×C139—21no—
147C22×C139—24no—
148C22×C139—42no—
149C22×C139—42no—
150C22×C139—41no—
151C22×C139—41no—
152C22×C139—41no—
153C22×C139—84no—
154C22×C139#12012yes—
155C22×C139C22×D411yescharacteristic
156C2×D139—22no—
157C2×D139—22no—
158C2×D139—22no—
159C2×D139—22no—
160C2×D139—41no—
161C2×D139—41no—
162C2×D139—88no—
163C4×C139—41no—
164C4×C139#12012yes—
165C4×C139C22×D411yescharacteristic
166C4×C139C22×D411yescharacteristic
167Dic139—22no—
168Dic139—22no—
169Dic139—42no—
170C2×C139—22no—
171C2×C139—44no—
172C2×C139—81no—
173C2×C139C2×#12011yescharacteristic
174C2×C139#57611yescharacteristic
175C2×C139#57711yescharacteristic
176D139—42no—
177D139—42no—
178C139#301211yescharacteristic, complemented, Sylow
179#3012—1391nomaximal, Sylow
180C2×H(5,𝔽2)—1391no—
181C2×AGL(1,ℤ/8)D13911yescharacteristic, complemented
182C2×#135—1391no—
183#537—1391no—
184#452—1392no—
185#450—1392no—
186#449—1392no—
187#447—1392no—
188#417—1391no—
189C2×#102—1391no—
190#408—1391no—
191H(5,𝔽2)—2784no—
192H(5,𝔽2)—2784no—
193C2×#41—2781no—
194C2×#41—1392no—
195C2×#41—1391no—
196C2×#41—1391no—
197C2×#41C2×D13911yescharacteristic, complemented
198C22×D4—2781no—
199C22×D4—2782no—
200C22×D4—1392no—
201C22×D4C2×D13911yescharacteristic
202AGL(1,ℤ/8)—24no—
203#135—2784no—
204C2×Q16—1391no—
205C2×SD16—1391no—
206C2×SD16—1391no—
207C2×SD16C2×D13912yescomplemented
208C2×D8—1391no—
209C2×D8C2×D13912yescomplemented
210C2×M16C2×D13911yescharacteristic, complemented
211C22×C8—1391no—
212C22⋊Q8—2781no—
213C22⋊Q8—1392no—
214#121—2781no—
215#121—2782no—
216#121—1392no—
217C2×C4⋊C4—1391no—
218#103—1392no—
219#102—1392no—
220#102—1392no—
221#102—1392no—
222C22⋊C8—1392no—
223C22⋊C8—1392no—
224C24—21no—
225C24—2782no—
226#41—22no—
227#41—22no—
228#41—2788no—
229#41—2784no—
230#41—2784no—
231#41—2782no—
232#41—2782no—
233#41—2782no—
234#41—1394no—
235C2×Q8—1391no—
236C2×Q8—1391no—
237C2×Q8C22×D13911yescharacteristic
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—22no—
241C2×D4—2788no—
242C2×D4—2784no—
243C2×D4—27816no—
244C2×D4—2782no—
245C2×D4—2781no—
246C2×D4—2784no—
247C2×D4—2782no—
248C2×D4—2782no—
249C2×D4—1392no—
250C2×D4—1391no—
251C2×D4—1391no—
252C2×D4C22⋊D13914yescomplemented
253C2×D4C22×D13911yescharacteristic
254C2×D4C22×D13912yes—
255C22×C4—21no—
256C22×C4—2782no—
257C22×C4—2781no—
258C22×C4—2781no—
259C22×C4—1392no—
260C22×C4C22×D13911yescharacteristic
261Q16—2782no—
262SD16—24no—
263SD16—2782no—
264SD16—2782no—
265D8—24no—
266D8—2782no—
267M16—22no—
268C2×C8—2782no—
269C2×C8—1391no—
270C2×C8—1391no—
271C2×C8C22×D13912yes—
272C4⋊C4—2782no—
273C4⋊C4—2781no—
274C4⋊C4—1392no—
275C22⋊C4—2782no—
276C22⋊C4—2782no—
277C23—22no—
278C23—24no—
279C23—21no—
280C23—81no—
281C23—81no—
282C23—2788no—
283C23—2781no—
284C23—11122no—
285C23—11122no—
286C23—1392no—
287C23D4×D13911yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—2782no—
291Q8—2781no—
292Q8—1392no—
293D4—24no—
294D4—22no—
295D4—21no—
296D4—21no—
297D4—42no—
298D4—2788no—
299D4—2788no—
300D4—2784no—
301D4—2784no—
302D4—2782no—
303D4—2781no—
304D4—11122no—
305D4—1392no—
306D4C2×C22⋊D13914yes—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—2788no—
310C2×C4—2781no—
311C2×C4—2784no—
312C2×C4—2782no—
313C2×C4—2782no—
314C2×C4—2782no—
315C2×C4—11121no—
316C2×C4—1392no—
317C2×C4D4×D13911yescharacteristic
318C2×C4D4×D13912yes—
319C2×C4C2×C22⋊D13912yes—
320C2×C4C23×D13911yescharacteristic
321C8—22no—
322C8—2781no—
323C8—2781no—
324C22—21no—
325C22—21no—
326C22—24no—
327C22—42no—
328C22—42no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—84no—
333C22—2782no—
334C22—2782no—
335C22—2782no—
336C22—2782no—
337C22—5561no—
338C22—5561no—
339C22—11128no—
340C22#1798481012yes—
341C22C2×D4×D13911yescharacteristic
342C4—41no—
343C4—2782no—
344C4—2782no—
345C4—5562no—
346C4#1798481012yes—
347C4C2×D4×D13911yescharacteristic
348C4C22×C22⋊D13911yescharacteristic
349C2—22no—
350C2—44no—
351C2—81no—
352C2—5562no—
353C2—5562no—
354C2C2×#1798481011yescharacteristic
355C2#2258175711yescharacteristic
356C2#2258176311yescharacteristic
subgroupG/H
centerC22C2×D4×D139
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
FittingC2×AGL(1,ℤ/8)×C139C2
socleC22×C139C22×D4
hypercenterC2×AGL(1,ℤ/8)D139
hypocenterC139#3012
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#17984887C2semidirect
2. C2×#17984862C2semidirect
3. #22581531C2semidirect
4. #22581112C2semidirect
5. #22581102C2semidirect
6. #22581095C2semidirect
7. #22581085C2semidirect
8. #22580955C2semidirect
9. C2×#17984763C2semidirect
10. #22580921C2semidirect
11. C2×AGL(1,ℤ/8)×C139C2semidirect
22. C2×#14555775C22semidirect
23. C2×#14555773C22semidirect
24. C2×#14555772C22semidirect
25. C2×#14555770C22semidirect
26. C22×C139⋊C8C22semidirect
28. #17984818C22semidirect
31. #17984813C22semidirect
32. C2×C4⋊Dic139C22semidirect
33. #17984767C22semidirect
34. #17984764C22semidirect
35. #17984763C22semidirect
36. #17984763C22semidirect
37. (C2×D139)⋊C8C22semidirect
38. (C2×D139)⋊C8C22semidirect
39. C2×#41×C139C22semidirect
42. C2×SD16×C139C22semidirect
43. C2×D8×C139C22semidirect
44. C2×M16×C139C22semidirect
49. #14555785D4semidirect
89. C2×D4×C139D4semidirect
178. C139#3012semidirect
181. C2×AGL(1,ℤ/8)D139semidirect
197. C2×#41C2×D139semidirect
207. C2×SD16C2×D139semidirect
209. C2×D8C2×D139semidirect
210. C2×M16C2×D139semidirect
252. C2×D4C22⋊D139semidirect

Maximal subgroups

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

subgroupindex
C2×#179848872
C2×#179848622
#225815312
#225811122
#225811022
#225810952
#225810852
#225809552
C2×#179847632
#225809212
C2×AGL(1,ℤ/8)×C1392
#3012139