Finite Group Databasebeta

← #31815982

Subgroups — #31815982

Order 17792.

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

normal subgroups only

This group has 520 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×#17984862C211yesmaximal, characteristic, complemented
2C23×C139⋊C8C211yesmaximal, characteristic, complemented
3#22581464C211yesmaximal, characteristic, complemented
4#22581192C212yesmaximal, complemented
5#22581187C212yesmaximal, complemented
6#22581184C212yesmaximal, complemented
7#22581180C212yesmaximal, complemented
8#22580989C211yesmaximal, characteristic, complemented
9#22580949C211yesmaximal, characteristic, complemented
10#22580947C211yesmaximal, characteristic, complemented
11#525×C139C211yesmaximal, characteristic, complemented
12C2×#14555785C2211yescharacteristic
13#17984862—24no—
14C2×#14555775C2211yescharacteristic, complemented
15C2×#14555773C2211yescharacteristic, complemented
16C2×#14555772C2211yescharacteristic, complemented
17C2×#14555770C2211yescharacteristic, complemented
18C22×C139⋊C8—24no—
19C22×C139⋊C8C2212yes—
20C22×C139⋊C8C2212yes—
21C22×C139⋊C8C2211yescharacteristic
22C22×C139⋊C8C2211yescharacteristic
23#17984825—21no—
24#17984821C2212yes—
25#17984816C2212yes—
26#17984811—21no—
27C4×C22⋊D139—22no—
28#17984801C2211yescharacteristic
29#17984775C2212yescomplemented
30#17984773C2212yescomplemented
31#17984766C2212yescomplemented
32#17984765C2212yescomplemented
33#17984763C2212yescomplemented
34#17984762C2212yescomplemented
35C2×#41×C139C2211yescharacteristic
36C23×C4×C139C2211yescharacteristic
37#123×C139—21no—
38C22⋊Q8×C139C2212yes—
39#121×C139C2212yes—
40#120×C139—21no—
41C4×D4×C139—22no—
42#117×C139C2211yescharacteristic
43#14555785—22no—
44#14555785—22no—
45C2×Dic278C2311yescharacteristic
46C2×D556C2311yescharacteristic
47C2×C22⋊D139—22no—
48C2×C22⋊D139—21no—
49C2×C4×D139—21no—
50#14555775—22no—
51#14555773—22no—
52#14555772—22no—
53#14555770—22no—
54C2×C139⋊C8—24no—
55C2×C139⋊C8—24no—
56C2×C139⋊C8—22no—
57C2×C139⋊C8D414yescomplemented
58C2×C139⋊C8C2312yes—
59C2×C139⋊C8C2311yescharacteristic
60C2×C139⋊C8C2311yescharacteristic
61C4⋊Dic139C2312yes—
62Dic139⋊C4—22no—
63C22⋊Dic139—21no—
64C22⋊Dic139—22no—
65(C2×D139)⋊C4—22no—
66C4×Dic139—21no—
67C24×C139D411yescharacteristic
68#41×C139—22no—
69#41×C139—22no—
70C2×Q8×C139C2311yescharacteristic
71C2×D4×C139—22no—
72C2×D4×C139—21no—
73C2×D4×C139C2311yescharacteristic
74C22×C4×C139—22no—
75C22×C4×C139—22no—
76C22×C4×C139—21no—
77C22×C4×C139D412yes—
78C22×C4×C139D411yescharacteristic
79C22×C4×C139C2312yes—
80C22×C4×C139C2311yescharacteristic
81C4⋊C4×C139—22no—
82C4⋊C4×C139C2312yes—
83C22⋊C4×C139—21no—
84C22⋊C4×C139—22no—
85C22⋊C4×C139—22no—
86C42×C139—21no—
87C22×D139—21no—
88Dic278—21no—
89Dic278—21no—
90D556—21no—
91D556—21no—
92C22⋊D139—42no—
93C22⋊D139—44no—
94C2×Dic139—22no—
95C2×Dic139—21no—
96C4×D139—42no—
97C139⋊C8—22no—
98C139⋊C8C2×D412yes—
99C139⋊C8C2×D412yes—
100C23×C139—22no—
101C23×C139—24no—
102C23×C139—21no—
103C23×C139C2×D411yescharacteristic
104C23×C139C2×D412yes—
105Q8×C139—21no—
106Q8×C139—21no—
107D4×C139—21no—
108D4×C139—21no—
109D4×C139—42no—
110D4×C139—44no—
111C2×C4×C139—22no—
112C2×C4×C139—24no—
113C2×C4×C139—22no—
114C2×C4×C139—22no—
115C2×C4×C139—22no—
116C2×C4×C139—21no—
117C2×C4×C139—42no—
118C2×C4×C139C2×D412yes—
119C2×C4×C139C2×D412yes—
120C2×C4×C139C2×D411yescharacteristic
121C2×C4×C139#4112yes—
122C2×C4×C139C2411yescharacteristic
123C22×C139—28no—
124C22×C139—22no—
125C22×C139—22no—
126C22×C139—22no—
127C22×C139—41no—
128C22×C139—42no—
129C22×C139#12112yes—
130C22×C139#13514yescomplemented
131C22×C139C22×D411yescharacteristic
132C2×D139—41no—
133C2×D139—42no—
134C4×C139—22no—
135C4×C139—41no—
136C4×C139—42no—
137C4×C139#12112yes—
138C4×C139C22×D411yescharacteristic
139C4×C139C2×#4111yescharacteristic
140Dic139—41no—
141Dic139—42no—
142C2×C139—22no—
143C2×C139—24no—
144C2×C139—81no—
145C2×C139C2×#12111yescharacteristic
146C2×C139C2×#13512yes—
147D139—81no—
148C139#305011yescharacteristic, complemented, Sylow
149#3050—1391nomaximal, Sylow
150C2×#135—1391no—
151C23×C8—1391no—
152#525—1391no—
153#525D13911yescharacteristic, complemented
154#467—1392no—
155#466—1392no—
156#465—1392no—
157#465—1392no—
158#427—1391no—
159#416—1391no—
160#416—1391no—
161C2×#41—1391no—
162C2×#41C2×D13911yescharacteristic
163C23×C4C2×D13911yescharacteristic
164#135—2784no—
165C2×Q16—1391no—
166C2×SD16—1391no—
167C2×SD16—1391no—
168C2×D8—1391no—
169C22×C8—2784no—
170C22×C8—1391no—
171C22×C8—1392no—
172C22×C8—1392no—
173C22×C8—1391no—
174#123—21no—
175#123—2781no—
176C22⋊Q8—1392no—
177C22⋊Q8C2×D13912yes—
178#121—1392no—
179#121C2×D13912yes—
180#120—21no—
181#120—2781no—
182C4×D4—22no—
183C4×D4—2782no—
184#117—1391no—
185#117C2×D13911yescharacteristic
186#107—1392no—
187#106—1392no—
188#103—1392no—
189#103—1392no—
190#102—1392no—
191#102—1392no—
192C24C22⋊D13911yescharacteristic
193#41—22no—
194#41—22no—
195#41—2782no—
196#41—2782no—
197C2×Q8—1391no—
198C2×Q8C22×D13911yescharacteristic
199C2×D4—22no—
200C2×D4—21no—
201C2×D4—2782no—
202C2×D4—2781no—
203C2×D4—1391no—
204C2×D4C22×D13911yescharacteristic
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—2781no—
209C22×C4C22⋊D13911yescharacteristic
210C22×C4C22⋊D13912yes—
211C22×C4C22×D13912yes—
212C22×C4C22×D13911yescharacteristic
213Q16—2782no—
214SD16—2782no—
215SD16—2782no—
216D8—2782no—
217C2×C8—2782no—
218C2×C8—2784no—
219C2×C8—2784no—
220C2×C8—1392no—
221C2×C8—1394no—
222C2×C8—1391no—
223C2×C8—1391no—
224C4⋊C4—22no—
225C4⋊C4—2782no—
226C4⋊C4—1392no—
227C4⋊C4C22×D13912yes—
228C22⋊C4—21no—
229C22⋊C4—22no—
230C22⋊C4—22no—
231C22⋊C4—2781no—
232C22⋊C4—2782no—
233C22⋊C4—2782no—
234C42—21no—
235C42—2781no—
236C23—22no—
237C23—24no—
238C23—21no—
239C23—2781no—
240C23C2×C22⋊D13911yescharacteristic
241C23C2×C22⋊D13912yes—
242Q8—21no—
243Q8—21no—
244Q8—2781no—
245Q8—2781no—
246D4—21no—
247D4—21no—
248D4—42no—
249D4—44no—
250D4—2781no—
251D4—2781no—
252D4—5562no—
253D4—5564no—
254C2×C4—22no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—42no—
261C2×C4—2782no—
262C2×C4—2781no—
263C2×C4—5562no—
264C2×C4D4×D13912yes—
265C2×C4C2×C22⋊D13912yes—
266C2×C4C2×C22⋊D13911yescharacteristic
267C2×C4#1455578412yes—
268C2×C4C23×D13911yescharacteristic
269C8—2782no—
270C8—1392no—
271C8—1392no—
272C22—28no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—41no—
277C22—42no—
278C22—5561no—
279C22—5562no—
280C22#1798481412yes—
281C22#1798486214yescomplemented
282C22C22×C22⋊D13911yescharacteristic
283C4—22no—
284C4—41no—
285C4—42no—
286C4—5561no—
287C4—5562no—
288C4#1798481412yes—
289C4C2×D4×D13911yescharacteristic
290C4C2×#1455578411yescharacteristic
291C2—22no—
292C2—24no—
293C2—81no—
294C2—11121no—
295C2C2×#1798481411yescharacteristic
296C2C2×#1798486212yes—
subgroupG/H
centerC2×C4C2×C22⋊D139
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
Fitting#525×C139C2
socleC22×C139C22×D4
hypercenter#525D139
hypocenterC139#3050
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C2×#17984862C2semidirect
2. C23×C139⋊C8C2semidirect
3. #22581464C2semidirect
4. #22581192C2semidirect
5. #22581187C2semidirect
6. #22581184C2semidirect
7. #22581180C2semidirect
8. #22580989C2semidirect
9. #22580949C2semidirect
10. #22580947C2semidirect
11. #525×C139C2semidirect
14. C2×#14555775C22semidirect
15. C2×#14555773C22semidirect
16. C2×#14555772C22semidirect
17. C2×#14555770C22semidirect
29. #17984775C22semidirect
30. #17984773C22semidirect
31. #17984766C22semidirect
32. #17984765C22semidirect
33. #17984763C22semidirect
34. #17984762C22semidirect
57. C2×C139⋊C8D4semidirect
130. C22×C139#135semidirect
148. C139#3050semidirect
153. #525D139semidirect
281. C22#17984862semidirect

Maximal subgroups

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

subgroupindex
C2×#179848622
C23×C139⋊C82
#225814642
#225811922
#225811872
#225811842
#225811802
#225809892
#225809492
#225809472
#525×C1392
#3050139