Finite Group Databasebeta

← #31816012

Subgroups — #31816012

Order 17792.

500 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 133 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×#17984862C211yesmaximal, characteristic, complemented
2C22×#14555768C211yesmaximal, characteristic, complemented
3#22581465C211yesmaximal, characteristic, complemented
4#22581460C211yesmaximal, characteristic, complemented
5#22581210C212yesmaximal, complemented
6#22581205C212yesmaximal, complemented
7#22581202C212yesmaximal, complemented
8#22581198C212yesmaximal, complemented
9#22580989C211yesmaximal, characteristic, complemented
10#22580950C211yesmaximal, characteristic, complemented
11#416×C139C211yesmaximal, characteristic, complemented
12C2×#14555785C2211yescharacteristic
13C22×C4×D139C2211yescharacteristic
14#17984862—24no—
15C2×#14555775C2211yescharacteristic, complemented
16C2×#14555773C2211yescharacteristic, complemented
17C2×#14555772C2211yescharacteristic, complemented
18C2×#14555770C2211yescharacteristic, complemented
19C2×#14555768—24no—
20C2×#14555768C2212yes—
21C2×#14555768C2212yes—
22C22×C139⋊C8C2211yescharacteristic
23#17984826—21no—
24#17984823—21no—
25#17984822C2212yes—
26#17984818C2212yes—
27#17984817C2212yes—
28#17984813C2212yes—
29#17984812—21no—
30#17984810—21no—
31C4×D556—22no—
32C4×C22⋊D139—22no—
33#17984801C2211yescharacteristic
34#17984775C2212yescomplemented
35#17984773C2212yescomplemented
36#17984767C2212yescomplemented
37#17984764C2212yescomplemented
38C2×#41×C139C2211yescharacteristic
39C22×C8×C139C2211yescharacteristic
40#117×C139C2211yescharacteristic
41#103×C139C2212yescomplemented
42#102×C139C2212yescomplemented
43C23×D139D411yescharacteristic
44#14555785—22no—
45#14555785—22no—
46C2×Dic278C2311yescharacteristic
47C2×D556—22no—
48C2×D556C2311yescharacteristic
49C2×C22⋊D139—22no—
50C2×C22⋊D139—21no—
51C22×Dic139D411yescharacteristic
52C2×C4×D139—22no—
53C2×C4×D139—21no—
54C2×C4×D139—22no—
55C2×C4×D139D412yes—
56C2×C4×D139C2312yes—
57#14555775—22no—
58#14555773—22no—
59#14555772—22no—
60#14555770—22no—
61#14555768—24no—
62#14555768—24no—
63C2×C139⋊C8D414yescomplemented
64C2×C139⋊C8C2311yescharacteristic
65C2×C139⋊C8C2311yescharacteristic
66C4⋊Dic139—22no—
67C4⋊Dic139C2312yes—
68Dic139⋊C4—22no—
69C22⋊Dic139—22no—
70C22⋊Dic139—21no—
71(C2×D139)⋊C4—22no—
72(C2×D139)⋊C4—22no—
73(C2×D139)⋊C4—22no—
74C4×Dic139—21no—
75#41×C139—22no—
76#41×C139—22no—
77C2×Q8×C139C2311yescharacteristic
78C2×D4×C139—21no—
79C2×D4×C139C2311yescharacteristic
80C22×C4×C139—21no—
81C22×C4×C139C2311yescharacteristic
82C2×C8×C139—22no—
83C2×C8×C139C2312yes—
84C4⋊C4×C139C2312yes—
85C22⋊C4×C139—21no—
86C42×C139—21no—
87C22×D139—22no—
88C22×D139—21no—
89C22×D139—42no—
90C22×D139C2×D412yes—
91Dic278—21no—
92Dic278—21no—
93D556—21no—
94D556—21no—
95D556—44no—
96C22⋊D139—44no—
97C22⋊D139—42no—
98C2×Dic139—22no—
99C2×Dic139—22no—
100C2×Dic139—21no—
101C2×Dic139C2×D412yes—
102C4×D139—24no—
103C4×D139—22no—
104C4×D139—22no—
105C4×D139—42no—
106C139⋊C8C2×D412yes—
107C139⋊C8C2×D412yes—
108C23×C139—21no—
109C23×C139C2×D411yescharacteristic
110Q8×C139—21no—
111Q8×C139—21no—
112D4×C139—21no—
113D4×C139—21no—
114D4×C139—42no—
115C2×C4×C139—22no—
116C2×C4×C139—21no—
117C2×C4×C139—42no—
118C2×C4×C139C2×D412yes—
119C2×C4×C139C2×D411yescharacteristic
120C2×C4×C139#4112yes—
121C2×C4×C139C2411yescharacteristic
122C8×C139—22no—
123C22×C139—22no—
124C22×C139—41no—
125C22×C139—42no—
126C22×C139#12112yes—
127C22×C139C22×D411yescharacteristic
128C2×D139—22no—
129C2×D139—22no—
130C2×D139—22no—
131C2×D139—44no—
132C2×D139—41no—
133C2×D139—42no—
134C4×C139—41no—
135C4×C139—42no—
136C4×C139#12112yes—
137C4×C139C22×D411yescharacteristic
138C4×C139C2×#4111yescharacteristic
139Dic139—22no—
140Dic139—41no—
141Dic139—42no—
142C2×C139—22no—
143C2×C139—81no—
144C2×C139C2×#12111yescharacteristic
145C2×C139#57512yes—
146D139—42no—
147D139—81no—
148C139#305411yescharacteristic, complemented, Sylow
149#3054—1391nomaximal, Sylow
150C2×#135—1391no—
151C22×M16—1391no—
152#525—1391no—
153#525—1391no—
154#470—1392no—
155#469—1392no—
156#468—1392no—
157#468—1392no—
158#427—1391no—
159#416—1391no—
160#416D13911yescharacteristic, complemented
161C2×#41—1391no—
162C2×#41C2×D13911yescharacteristic
163C23×C4—1391no—
164#135—2784no—
165C2×Q16—1391no—
166C2×SD16—1391no—
167C2×SD16—1391no—
168C2×D8—1391no—
169C2×M16—2784no—
170C2×M16—1392no—
171C2×M16—1392no—
172C22×C8—1391no—
173C22×C8C2×D13911yescharacteristic
174#123—2781no—
175#123—2781no—
176C22⋊Q8—1392no—
177C22⋊Q8—1392no—
178#121—1392no—
179#121—1392no—
180#120—2781no—
181#120—2781no—
182C4×D4—2782no—
183C4×D4—2782no—
184#117—1391no—
185#117C2×D13911yescharacteristic
186#107—1392no—
187#106—1392no—
188#103—1392no—
189#103C2×D13912yescomplemented
190#102—1392no—
191#102C2×D13912yescomplemented
192C24—1391no—
193#41—22no—
194#41—22no—
195#41—2782no—
196#41—2782no—
197C2×Q8—1391no—
198C2×Q8C22×D13911yescharacteristic
199C2×D4—21no—
200C2×D4—2782no—
201C2×D4—2782no—
202C2×D4—2781no—
203C2×D4—1391no—
204C2×D4C22×D13911yescharacteristic
205C22×C4—21no—
206C22×C4—2782no—
207C22×C4—2782no—
208C22×C4—2781no—
209C22×C4—1391no—
210C22×C4—1392no—
211C22×C4—1392no—
212C22×C4C22×D13911yescharacteristic
213Q16—2782no—
214SD16—2782no—
215SD16—2782no—
216D8—2782no—
217M16—2784no—
218M16—2784no—
219C2×C8—22no—
220C2×C8—1394no—
221C2×C8—1391no—
222C2×C8—1391no—
223C2×C8C22×D13912yes—
224C4⋊C4—2782no—
225C4⋊C4—2782no—
226C4⋊C4—1392no—
227C4⋊C4C22×D13912yes—
228C22⋊C4—21no—
229C22⋊C4—2782no—
230C22⋊C4—2782no—
231C22⋊C4—2782no—
232C22⋊C4—2781no—
233C22⋊C4—2782no—
234C42—21no—
235C42—2781no—
236C23—21no—
237C23—2782no—
238C23—2781no—
239C23—5562no—
240C23—1392no—
241C23D4×D13911yescharacteristic
242Q8—21no—
243Q8—21no—
244Q8—2781no—
245Q8—2781no—
246D4—21no—
247D4—21no—
248D4—42no—
249D4—2781no—
250D4—2781no—
251D4—5564no—
252D4—5564no—
253D4—5562no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—42no—
257C2×C4—2782no—
258C2×C4—2784no—
259C2×C4—2782no—
260C2×C4—2782no—
261C2×C4—2782no—
262C2×C4—2781no—
263C2×C4—5562no—
264C2×C4—1392no—
265C2×C4D4×D13912yes—
266C2×C4D4×D13911yescharacteristic
267C2×C4#1455578512yes—
268C2×C4C23×D13911yescharacteristic
269C8—22no—
270C8—1392no—
271C8—1392no—
272C22—22no—
273C22—41no—
274C22—42no—
275C22—2782no—
276C22—2782no—
277C22—2782no—
278C22—5564no—
279C22—5561no—
280C22—5562no—
281C22#1798481512yes—
282C22C2×D4×D13911yescharacteristic
283C4—41no—
284C4—42no—
285C4—2782no—
286C4—5561no—
287C4—5562no—
288C4#1798481512yes—
289C4C2×D4×D13911yescharacteristic
290C4C2×#1455578511yescharacteristic
291C2—22no—
292C2—81no—
293C2—5562no—
294C2—11121no—
295C2C2×#1798481511yescharacteristic
296C2#2258175012yes—
subgroupG/H
centerC2×C4D4×D139
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
Fitting#416×C139C2
socleC22×C139C22×D4
hypercenter#416D139
hypocenterC139#3054
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. C2×#17984862C2semidirect
2. C22×#14555768C2semidirect
3. #22581465C2semidirect
4. #22581460C2semidirect
5. #22581210C2semidirect
6. #22581205C2semidirect
7. #22581202C2semidirect
8. #22581198C2semidirect
9. #22580989C2semidirect
10. #22580950C2semidirect
11. #416×C139C2semidirect
15. C2×#14555775C22semidirect
16. C2×#14555773C22semidirect
17. C2×#14555772C22semidirect
18. C2×#14555770C22semidirect
34. #17984775C22semidirect
35. #17984773C22semidirect
36. #17984767C22semidirect
37. #17984764C22semidirect
41. #103×C139C22semidirect
42. #102×C139C22semidirect
63. C2×C139⋊C8D4semidirect
148. C139#3054semidirect
160. #416D139semidirect
189. #103C2×D139semidirect
191. #102C2×D139semidirect

Maximal subgroups

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

subgroupindex
C2×#179848622
C22×#145557682
#225814652
#225814602
#225812102
#225812052
#225812022
#225811982
#225809892
#225809502
#416×C1392
#3054139