Finite Group Databasebeta

← #31815752

Subgroups — #31815752

Order 17792.

724 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 145 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
1#22581789C211yesmaximal, characteristic, complemented
2#22581751C212yesmaximal, complemented
3#22581135C212yesmaximal, complemented
4#22581130C212yesmaximal, complemented
5#22581125C212yesmaximal, complemented
6#22581120C212yesmaximal, complemented
7C2×#17984770C212yesmaximal, complemented
8#22580937C211yesmaximal, characteristic, complemented
9#532×C139C211yesmaximal, characteristic, complemented
10#17984890C2212yes—
11#17984889—22no—
12#17984888—22no—
13#17984887—22no—
14#17984887C2212yes—
15C2×#14555785—22no—
16C2×#14555785C2212yes—
17#41×D139—24no—
18#17984873C2214yescomplemented
19#17984872—22no—
20#17984868C2214yescomplemented
21#17984867—22no—
22#17984862—24no—
23C2×#14555769C2212yescomplemented
24#17984770C2214yescomplemented
25#17984770C2214yescomplemented
26#17984761C2212yescomplemented
27#17984759C2212yescomplemented
28C2×#41×C139C2211yescharacteristic
29C2×#41×C139C2211yescharacteristic, complemented
30C4⋊Q8×C139C2211yescharacteristic, complemented
31#127×C139C2211yescharacteristic, complemented
32#124×C139C2211yescharacteristic, complemented
33#124×C139C2211yescharacteristic, complemented
34#121×C139—22no—
35C4×D4×C139—22no—
36C2×C42×C139C2211yescharacteristic, complemented
37#14555786—24no—
38#14555785—24no—
39#14555785—24no—
40#14555785—22no—
41#14555785—24no—
42#14555785—22no—
43#14555785D414yescomplemented
44#14555785C2314yes—
45#14555784—28no—
46#14555784—24no—
47C2×Dic278—22no—
48C2×Dic278D412yescomplemented
49Q8×D139—24no—
50C2×D556—22no—
51C2×D556D412yescomplemented
52C2×C22⋊D139—24no—
53C2×C22⋊D139—22no—
54D4×D139—28no—
55D4×D139—24no—
56C2×C4×D139—24no—
57C2×C4×D139—22no—
58#14555775—24no—
59#14555773—24no—
60#14555772—24no—
61#14555770—24no—
62#14555769—22no—
63#14555769C2314yes—
64C2×C139⋊C8—22no—
65#41×C139—22no—
66#41×C139—22no—
67#41×C139—22no—
68#41×C139—22no—
69C2×Q8×C139C2311yescharacteristic
70C2×Q8×C139C2311yescharacteristic
71C2×D4×C139—22no—
72C2×D4×C139—21no—
73C2×D4×C139D412yes—
74C2×D4×C139C2311yescharacteristic
75C2×D4×C139C2311yescharacteristic
76C22×C4×C139—21no—
77C22×C4×C139—21no—
78C22×C4×C139D412yes—
79C22×C4×C139C2311yescharacteristic
80C4⋊C4×C139—22no—
81C22⋊C4×C139—22no—
82C22⋊C4×C139—22no—
83C42×C139—21no—
84C42×C139C2311yescharacteristic
85C42×C139C2311yescharacteristic
86C22×D139—24no—
87C22×D139—22no—
88Dic278—24no—
89Dic278—22no—
90Dic278C2×D414yes—
91D556—24no—
92D556—22no—
93D556C2×D414yes—
94C22⋊D139—28no—
95C22⋊D139—24no—
96C22⋊D139—24no—
97C2×Dic139—24no—
98C2×Dic139—22no—
99C4×D139—28no—
100C4×D139—24no—
101C4×D139—24no—
102C139⋊C8—24no—
103C23×C139—21no—
104C23×C139C2×D412yes—
105C23×C139C2×D411yescharacteristic
106Q8×C139—22no—
107Q8×C139—22no—
108D4×C139—24no—
109D4×C139—22no—
110D4×C139—22no—
111D4×C139—44no—
112D4×C139—42no—
113C2×C4×C139—21no—
114C2×C4×C139—22no—
115C2×C4×C139—22no—
116C2×C4×C139—21no—
117C2×C4×C139—21no—
118C2×C4×C139—42no—
119C2×C4×C139—41no—
120C2×C4×C139—41no—
121C2×C4×C139C2×D412yes—
122C2×C4×C139C2×D412yes—
123C2×C4×C139C2×D412yes—
124C2×C4×C139C2×D411yescharacteristic
125C2×C4×C139C2411yescharacteristic
126C22×C139—22no—
127C22×C139—41no—
128C22×C139—42no—
129C22×C139—42no—
130C22×C139—41no—
131C22×C139#12012yes—
132C22×C139C22×D411yescharacteristic
133C2×D139—24no—
134C2×D139—82no—
135C2×D139—84no—
136C4×C139—22no—
137C4×C139—22no—
138C4×C139—41no—
139C4×C139—41no—
140C4×C139#12011yescharacteristic
141C4×C139#12011yescharacteristic
142C4×C139C22×D412yes—
143Dic139—24no—
144C2×C139—21no—
145C2×C139—22no—
146C2×C139—42no—
147C2×C139—81no—
148C2×C139C2×#12011yescharacteristic
149D139—44no—
150C139#301811yescharacteristic, complemented, Sylow
151#3018—1391nomaximal, Sylow
152#585—1391no—
153#575—1392no—
154#532D13911yescharacteristic, complemented
155#456—1392no—
156#455—1392no—
157#454—1392no—
158#453—1392no—
159C2×#104—1392no—
160#413—1391no—
1612−1+4—2782no—
1622−1+4—1392no—
163H(5,𝔽2)—2782no—
164H(5,𝔽2)—2782no—
165H(5,𝔽2)—1392no—
166C2×#41—2784no—
167C2×#41—2782no—
168C2×#41—1392no—
169C2×#41C2×D13911yescharacteristic
170C2×#41C2×D13911yescharacteristic, complemented
171#137—2782no—
172#137—1394no—
173AGL(1,ℤ/8)—2782no—
174AGL(1,ℤ/8)—1394no—
175#135—2784no—
176C2×M16—1392no—
177C4⋊Q8C2×D13911yescharacteristic, complemented
178#127C2×D13911yescharacteristic, complemented
179#124C2×D13911yescharacteristic, complemented
180#124C2×D13911yescharacteristic, complemented
181#121—22no—
182C4×D4—22no—
183C2×C42C2×D13911yescharacteristic, complemented
184#104—1394no—
185#104—1394no—
186#101—1392no—
187#100—1392no—
188#41—22no—
189#41—22no—
190#41—22no—
191#41—22no—
192#41—2784no—
193#41—2784no—
194#41—2788no—
195#41—2784no—
196#41—2784no—
197#41—2782no—
198#41—2782no—
199#41—2784no—
200#41—1394no—
201#41—1394no—
202C2×Q8—2782no—
203C2×Q8—2784no—
204C2×Q8—1392no—
205C2×Q8C22×D13911yescharacteristic
206C2×Q8C22×D13911yescharacteristic
207C2×D4—22no—
208C2×D4—21no—
209C2×D4—2782no—
210C2×D4—2788no—
211C2×D4—2784no—
212C2×D4—2784no—
213C2×D4—2782no—
214C2×D4—1392no—
215C2×D4C22⋊D13912yes—
216C2×D4C22×D13911yescharacteristic
217C2×D4C22×D13911yescharacteristic
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—2784no—
221C22×C4—2782no—
222C22×C4C22⋊D13912yes—
223C22×C4C22×D13911yescharacteristic
224Q16—2784no—
225SD16—2784no—
226SD16—2784no—
227D8—2784no—
228M16—2782no—
229M16—1394no—
230C2×C8—2782no—
231C4⋊C4—22no—
232C22⋊C4—22no—
233C22⋊C4—22no—
234C42—21no—
235C42C22×D13911yescharacteristic
236C42C22×D13911yescharacteristic
237C23—21no—
238C23—2784no—
239C23—2782no—
240C23C2×C22⋊D13912yes—
241C23C2×C22⋊D13911yescharacteristic
242Q8—22no—
243Q8—22no—
244Q8—2784no—
245Q8—2782no—
246Q8—1394no—
247D4—24no—
248D4—22no—
249D4—22no—
250D4—44no—
251D4—42no—
252D4—2784no—
253D4—2788no—
254D4—2784no—
255D4—2784no—
256D4—2782no—
257D4—1394no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—2788no—
267C2×C4—2784no—
268C2×C4—2782no—
269C2×C4—2784no—
270C2×C4—2784no—
271C2×C4D4×D13912yes—
272C2×C4D4×D13912yes—
273C2×C4C2×C22⋊D13912yes—
274C2×C4C2×C22⋊D13911yescharacteristic
275C2×C4C23×D13911yescharacteristic
276C8—2784no—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—42no—
281C22—41no—
282C22—2784no—
283C22—11122no—
284C22—11124no—
285C22#1798481012yes—
286C22C22×C22⋊D13911yescharacteristic
287C4—22no—
288C4—22no—
289C4—41no—
290C4—41no—
291C4—2784no—
292C4#1798481011yescharacteristic
293C4#1798481011yescharacteristic
294C4C2×D4×D13912yes—
295C2—21no—
296C2—22no—
297C2—42no—
298C2—81no—
299C2—5564no—
300C2C2×#1798481011yescharacteristic
subgroupG/H
centerC4#17984810
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
Fitting#532×C139C2
socleC2×C139C2×#120
hypercenter#532D139
hypocenterC139#3018
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C4 ◁ C22×C4 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #22581789C2semidirect
2. #22581751C2semidirect
3. #22581135C2semidirect
4. #22581130C2semidirect
5. #22581125C2semidirect
6. #22581120C2semidirect
7. C2×#17984770C2semidirect
8. #22580937C2semidirect
9. #532×C139C2semidirect
18. #17984873C22semidirect
20. #17984868C22semidirect
23. C2×#14555769C22semidirect
24. #17984770C22semidirect
25. #17984770C22semidirect
26. #17984761C22semidirect
27. #17984759C22semidirect
29. C2×#41×C139C22semidirect
30. C4⋊Q8×C139C22semidirect
31. #127×C139C22semidirect
32. #124×C139C22semidirect
33. #124×C139C22semidirect
36. C2×C42×C139C22semidirect
43. #14555785D4semidirect
48. C2×Dic278D4semidirect
51. C2×D556D4semidirect
150. C139#3018semidirect
154. #532D139semidirect
170. C2×#41C2×D139semidirect
177. C4⋊Q8C2×D139semidirect
178. #127C2×D139semidirect
179. #124C2×D139semidirect
180. #124C2×D139semidirect
183. C2×C42C2×D139semidirect

Maximal subgroups

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

subgroupindex
#225817892
#225817512
#225811352
#225811302
#225811252
#225811202
C2×#179847702
#225809372
#532×C1392
#3018139