Finite Group Databasebeta

← #31815659

Subgroups — #31815659

Order 17792.

754 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C22×#14555784C211yesmaximal, characteristic
2C2×#17984871C212yesmaximal, complemented
3C22⋊Q8×D139C211yesmaximal, characteristic, complemented
4#22581116C212yesmaximal, complemented
5#22581112C212yesmaximal, complemented
6#22581111C211yesmaximal, characteristic, complemented
7#22581108C211yesmaximal, characteristic, complemented
8#22581104C211yesmaximal, characteristic, complemented
9#22580951C212yesmaximal, complemented
10#22580915C211yesmaximal, characteristic, complemented
11#450×C139C211yesmaximal, characteristic, complemented
12C2×#14555784—28no—
13C2×#14555784—21no—
14C2×#14555784—21no—
15C2×#14555784C2212yes—
16C2×#14555784C2212yes—
17C22×Dic278C2211yescharacteristic
18C2×Q8×D139C2211yescharacteristic, complemented
19C22×C22⋊D139—21no—
20C23×Dic139—21no—
21C22×C4×D139C2211yescharacteristic
22#17984871—28no—
23C2×Dic556C2212yescomplemented
24C2×#14555775C2212yescomplemented
25C2×#14555773C2212yescomplemented
26C2×#14555768C2212yescomplemented
27#17984822C2211yescharacteristic, complemented
28#17984821C2211yescharacteristic
29#17984820—21no—
30#17984819—21no—
31#17984818C2211yescharacteristic, complemented
32C4⋊C4×D139—21no—
33C4⋊C4×D139C2211yescharacteristic, complemented
34C22⋊C4×D139—21no—
35#17984767C2212yescomplemented
36#17984765C2212yescomplemented
37#17984763C2212yescomplemented
38C22⋊(C139⋊C8)C2211yescharacteristic
39(C2×D139)⋊C8C2212yescomplemented
40C22×D4×C139C2211yescharacteristic
41C2×SD16×C139C2212yescomplemented
42C22⋊Q8×C139C2211yescharacteristic
43#102×C139C2212yescomplemented
44C22⋊C8×C139C2211yescharacteristic
45C23×D139D411yescharacteristic
46#14555784—28no—
47#14555784—28no—
48#14555784—24no—
49#14555784—24no—
50#14555784—24no—
51#14555784D418yes—
52C2×Dic278—24no—
53C2×Dic278—21no—
54C2×Dic278—21no—
55C2×Dic278C2311yescharacteristic
56C2×Dic278C2312yes—
57Q8×D139—24no—
58C2×C22⋊D139—28no—
59C2×C22⋊D139—22no—
60C2×C22⋊D139—22no—
61C22×Dic139—28no—
62C22×Dic139—22no—
63C22×Dic139—22no—
64C22×Dic139—21no—
65C22×Dic139D411yescharacteristic
66C2×C4×D139—24no—
67C2×C4×D139—21no—
68C2×C4×D139—21no—
69C2×C4×D139D411yescharacteristic
70C2×C4×D139D411yescharacteristic
71C2×C4×D139C2311yescharacteristic
72C2×C4×D139C2311yescharacteristic
73Dic556—24no—
74#14555775—24no—
75#14555773—24no—
76#14555768—24no—
77C2×C139⋊C8C2312yes—
78C4⋊Dic139—21no—
79C4⋊Dic139C2311yescharacteristic
80Dic139⋊C4—21no—
81Dic139⋊C4—21no—
82Dic139⋊C4—21no—
83C22⋊Dic139—21no—
84(C2×D139)⋊C4—21no—
85(C2×D139)⋊C4—21no—
86C24×C139—21no—
87C2×Q8×C139C2311yescharacteristic
88C2×D4×C139—24no—
89C2×D4×C139—21no—
90C2×D4×C139C2312yes—
91C22×C4×C139C2311yescharacteristic
92SD16×C139—24no—
93C2×C8×C139C2312yes—
94C4⋊C4×C139—21no—
95C4⋊C4×C139C2311yescharacteristic
96C22⋊C4×C139—21no—
97C22×D139—22no—
98C22×D139—22no—
99C22×D139—41no—
100C22×D139C2×D411yescharacteristic
101C22×D139C2×D411yescharacteristic
102Dic278—24no—
103Dic278—22no—
104Dic278—22no—
105Dic278—42no—
106Dic278C2×D414yes—
107C22⋊D139—28no—
108C22⋊D139—28no—
109C2×Dic139—24no—
110C2×Dic139—28no—
111C2×Dic139—22no—
112C2×Dic139—22no—
113C2×Dic139—28no—
114C2×Dic139—21no—
115C2×Dic139—21no—
116C2×Dic139C2×D411yescharacteristic
117C2×Dic139C2×D411yescharacteristic
118C4×D139—22no—
119C4×D139—22no—
120C4×D139—22no—
121C4×D139—42no—
122C4×D139—42no—
123C4×D139C2×D414yes—
124C139⋊C8—22no—
125C23×C139—24no—
126C23×C139—22no—
127C23×C139—42no—
128C23×C139—42no—
129C23×C139C2×D411yescharacteristic
130Q8×C139—22no—
131D4×C139—24no—
132D4×C139—22no—
133D4×C139C2×D414yes—
134C2×C4×C139—22no—
135C2×C4×C139—21no—
136C2×C4×C139—21no—
137C2×C4×C139C2×D411yescharacteristic
138C2×C4×C139C2411yescharacteristic
139C8×C139—22no—
140C22×C139—21no—
141C22×C139—21no—
142C22×C139—24no—
143C22×C139—48no—
144C22×C139—42no—
145C22×C139—42no—
146C22×C139#13712yescomplemented
147C22×C139C22×D411yescharacteristic
148C2×D139—21no—
149C2×D139—21no—
150C2×D139—21no—
151C2×D139—21no—
152C2×D139—21no—
153C2×D139—44no—
154C2×D139#12012yes—
155C4×C139—21no—
156C4×C139—41no—
157C4×C139—41no—
158C4×C139C22×D412yes—
159Dic139—21no—
160Dic139—24no—
161Dic139—41no—
162Dic139—41no—
163Dic139#12012yes—
164C2×C139—22no—
165C2×C139—44no—
166C2×C139C2×#12011yescharacteristic
167C2×C139C2×#13711yescharacteristic
168C2×C139#57511yescharacteristic
169D139—22no—
170D139—41no—
171C139#300711yescharacteristic, complemented, Sylow
172#3007—1391nomaximal, Sylow
173C22×#41—1391no—
174C2×#137—1392no—
175C2×C22⋊Q8—1391no—
176#452—1392no—
177#452—1392no—
178#451—1391no—
179#451—1391no—
180#450—1391no—
181#450D13911yescharacteristic, complemented
182#417—1392no—
183#407—1391no—
184C2×#41—2788no—
185C2×#41—2781no—
186C2×#41—2781no—
187C2×#41—1392no—
188C2×#41—1392no—
189C22×Q8—1391no—
190C22×Q8—1391no—
191C22×D4—2781no—
192C22×D4C2×D13911yescharacteristic
193C23×C4—2781no—
194C23×C4—1391no—
195#137—2788no—
196C2×Q16—1392no—
197C2×Q16—1392no—
198C2×SD16—1392no—
199C2×SD16C2×D13912yescomplemented
200C2×M16—1392no—
201C22⋊Q8—2781no—
202C22⋊Q8—2781no—
203C22⋊Q8—1391no—
204C22⋊Q8—1391no—
205C22⋊Q8—1391no—
206C22⋊Q8C2×D13911yescharacteristic
207C2×C4⋊C4—2781no—
208C2×C4⋊C4—1391no—
209C2×C22⋊C4—2781no—
210#103—1392no—
211#103—1392no—
212#102—1392no—
213#102C2×D13912yescomplemented
214C22⋊C8—1391no—
215C22⋊C8—1392no—
216C22⋊C8C2×D13911yescharacteristic
217C24—21no—
218C24—1391no—
219#41—2788no—
220#41—2788no—
221#41—2784no—
222#41—2784no—
223#41—2784no—
224#41—1398no—
225C2×Q8—2784no—
226C2×Q8—2781no—
227C2×Q8—2781no—
228C2×Q8—2784no—
229C2×Q8—1391no—
230C2×Q8—1392no—
231C2×Q8C22×D13911yescharacteristic
232C2×D4—24no—
233C2×D4—21no—
234C2×D4—2788no—
235C2×D4—2782no—
236C2×D4—2782no—
237C2×D4C22×D13912yes—
238C22×C4—2784no—
239C22×C4—2781no—
240C22×C4—2781no—
241C22×C4—2788no—
242C22×C4—2781no—
243C22×C4—2782no—
244C22×C4—2782no—
245C22×C4—1391no—
246C22×C4—1391no—
247C22×C4—1391no—
248C22×C4—1391no—
249C22×C4—1391no—
250C22×C4C22×D13911yescharacteristic
251Q16—2784no—
252Q16—2784no—
253SD16—24no—
254SD16—2784no—
255M16—2784no—
256C2×C8—1392no—
257C2×C8C22×D13912yes—
258C4⋊C4—21no—
259C4⋊C4—2781no—
260C4⋊C4—2781no—
261C4⋊C4—2781no—
262C4⋊C4—2781no—
263C4⋊C4—1391no—
264C4⋊C4C22×D13911yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—2781no—
267C22⋊C4—2781no—
268C22⋊C4—2781no—
269C23—24no—
270C23—22no—
271C23—42no—
272C23—42no—
273C23—2782no—
274C23—2782no—
275C23—5561no—
276C23—1391no—
277C23—1391no—
278C23D4×D13911yescharacteristic
279Q8—22no—
280Q8—2784no—
281Q8—2782no—
282Q8—2782no—
283Q8—5562no—
284Q8—1394no—
285D4—24no—
286D4—22no—
287D4—2788no—
288D4—2788no—
289D4D4×D13914yes—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—2782no—
294C2×C4—2782no—
295C2×C4—2784no—
296C2×C4—2788no—
297C2×C4—2782no—
298C2×C4—2782no—
299C2×C4—2788no—
300C2×C4—2782no—
301C2×C4—2781no—
302C2×C4—2781no—
303C2×C4—5562no—
304C2×C4—5562no—
305C2×C4—1391no—
306C2×C4—1391no—
307C2×C4—1394no—
308C2×C4D4×D13911yescharacteristic
309C2×C4C23×D13911yescharacteristic
310C8—22no—
311C8—2782no—
312C22—21no—
313C22—21no—
314C22—24no—
315C22—48no—
316C22—42no—
317C22—42no—
318C22—2781no—
319C22—2781no—
320C22—2781no—
321C22—2781no—
322C22—2781no—
323C22—5564no—
324C22—1392no—
325C22#1798487112yescomplemented
326C22C2×D4×D13911yescharacteristic
327C4—21no—
328C4—41no—
329C4—41no—
330C4—2781no—
331C4—2784no—
332C4—5561no—
333C4—5561no—
334C4—1392no—
335C4C2×D4×D13912yes—
336C2—22no—
337C2—44no—
338C2—2782no—
339C2—5561no—
340C2#120×D13911yescharacteristic
341C2C2×#1798487111yescharacteristic
342C2#2258174911yescharacteristic
subgroupG/H
centerC22C2×D4×D139
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
Fitting#450×C139C2
socleC22×C139C22×D4
prime radicalC22×#14555784C2
hypercenter#450D139
hypocenterC139#3007
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
2. C2×#17984871C2semidirect
3. C22⋊Q8×D139C2semidirect
4. #22581116C2semidirect
5. #22581112C2semidirect
6. #22581111C2semidirect
7. #22581108C2semidirect
8. #22581104C2semidirect
9. #22580951C2semidirect
10. #22580915C2semidirect
11. #450×C139C2semidirect
18. C2×Q8×D139C22semidirect
23. C2×Dic556C22semidirect
24. C2×#14555775C22semidirect
25. C2×#14555773C22semidirect
26. C2×#14555768C22semidirect
27. #17984822C22semidirect
31. #17984818C22semidirect
33. C4⋊C4×D139C22semidirect
35. #17984767C22semidirect
36. #17984765C22semidirect
37. #17984763C22semidirect
39. (C2×D139)⋊C8C22semidirect
41. C2×SD16×C139C22semidirect
43. #102×C139C22semidirect
146. C22×C139#137semidirect
171. C139#3007semidirect
181. #450D139semidirect
199. C2×SD16C2×D139semidirect
213. #102C2×D139semidirect
325. C22#17984871semidirect

Maximal subgroups

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

subgroupindex
C22×#145557842
C2×#179848712
C22⋊Q8×D1392
#225811162
#225811122
#225811112
#225811082
#225811042
#225809512
#225809152
#450×C1392
#3007139