Finite Group Databasebeta

← #15685186

Subgroups — #15685186

Order 2944.

454 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 117 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#13093230C211yesmaximal, characteristic, complemented
2#13093226C212yesmaximal, complemented
3#13093222C211yesmaximal, characteristic, complemented
4#13092802C211yesmaximal, characteristic, complemented
5#13092688C211yesmaximal, characteristic, complemented
6#13092683C212yesmaximal, complemented
7#13092593C212yesmaximal, complemented
8#13092592C212yesmaximal, complemented
9#13092533C211yesmaximal, characteristic, complemented
10#13092532C211yesmaximal, characteristic, complemented
11C23×#473C211yesmaximal, characteristic, complemented
12#10625160C2212yes—
13#10625158C2212yescomplemented
14C23×#137C2212yes—
15C2×Q16×C23C2211yescharacteristic, complemented
16C23×#131C2211yescharacteristic
17#10625121C2211yescharacteristic
18#10625118C2212yescomplemented
19#10625113C2212yescomplemented
20C2×#67274C2212yescomplemented
21#10625090—22no—
22#10625090—21no—
23#10625089—22no—
24#10625089—22no—
25Q16×D23—21no—
26#10625085—22no—
27SD16×D23—22no—
28#10625079—22no—
29#10625079C2212yes—
30#10625079C2212yescomplemented
31#10625078C2212yescomplemented
32#10625075—21no—
33#10625075C2211yescharacteristic
34C2×D184—21no—
35C2×D184C2211yescharacteristic
36#10625072C2211yescharacteristic, complemented
37C23×#108C2211yescharacteristic, complemented
38C23×#101C2212yescomplemented
39#10625019C2211yescharacteristic
40#10625007C2211yescharacteristic, complemented
41#10625007C2212yescomplemented
42#10625005C2212yescomplemented
43#10625004C2211yescharacteristic, complemented
44C8×Dic23C2211yescharacteristic, complemented
45#41×C23C2311yescharacteristic
46C2×Q8×C23C2312yes—
47#67292—22no—
48#67292—22no—
49#67292—22no—
50Q8×D23—22no—
51Q8×D23—22no—
52D4×D23—22no—
53D4×D23—42no—
54#67288—22no—
55#67288—42no—
56#67288C2311yescharacteristic
57#67288C2312yes—
58C2×D92—22no—
59C2×D92C2312yes—
60Q16×C23—22no—
61Q16×C23—21no—
62Q16×C23—21no—
63SD16×C23—22no—
64M16×C23—21no—
65M16×C23C2311yescharacteristic
66M16×C23C2312yes—
67C2×C8×C23—21no—
68C2×C8×C23C2311yescharacteristic
69#67275—22no—
70#67275D411yescharacteristic
71#67274—22no—
72#67274—22no—
73#67274—22no—
74#67274D411yescharacteristic
75#67273D411yescharacteristic
76#67272D411yescharacteristic
77(C2×D23)⋊C4—42no—
78C4×Dic23C2311yescharacteristic
79#67267C2311yescharacteristic
80C2×C23⋊C8C2311yescharacteristic
81Dic92D411yescharacteristic
82D184—22no—
83D184—22no—
84D184—21no—
85D184—21no—
86D184D411yescharacteristic
87#67263—22no—
88#67263—22no—
89#67263D412yes—
90#67262—21no—
91#67262—22no—
92C8×D23—21no—
93C8×D23—22no—
94C22×D23—22no—
95C22×D23—42no—
96Q8×C23—22no—
97Q8×C23—22no—
98Q8×C23C2×D411yescharacteristic
99D4×C23—21no—
100D4×C23C2×D411yescharacteristic
101C2×C4×C23—21no—
102C2×C4×C23—22no—
103C2×C4×C23C2411yescharacteristic
104C22⋊D23—21no—
105C22⋊D23—22no—
106C22⋊D23—42no—
107C2×Dic23—21no—
108D92—22no—
109D92—22no—
110D92—42no—
111D92—42no—
112D92C2×D412yes—
113D92C2×D411yescharacteristic
114C4×D23—22no—
115C4×D23—21no—
116C4×D23—42no—
117C4×D23—42no—
118C4×D23—42no—
119Dic46—42no—
120Dic46C2×D411yescharacteristic
121Dic46C2×D412yes—
122C8×C23—22no—
123C8×C23—21no—
124C8×C23C2×D411yescharacteristic
125C8×C23C2×D411yescharacteristic
126C23⋊C8—21no—
127C23⋊C8C2×D411yescharacteristic
128C23⋊C8C2×D411yescharacteristic
129#3290—231nomaximal, Sylow
130C22×C23—21no—
131C22×C23H(5,𝔽2)11yescharacteristic
132C2×D23—21no—
133C2×D23—22no—
134C2×D23—42no—
135C2×D23—82no—
136C2×D23—162no—
137C4×C23—21no—
138C4×C23—42no—
139C4×C23C22×D411yescharacteristic
140C4×C23C22×D411yescharacteristic
141Dic23—21no—
142Dic23—22no—
143Dic23—41no—
144#578—231no—
145#577—232no—
146#576—231no—
147#495—231no—
148#473D2311yescharacteristic, complemented
149#472—231no—
150#471—232no—
151#455—232no—
152#454—232no—
153#443—231no—
154#443—231no—
155C2×C23—21no—
156C2×C23—41no—
157C2×C23D4211yescharacteristic
158D23—41no—
159D23—42no—
160D23—82no—
1612−1+4—232no—
162H(5,𝔽2)—232no—
163#137—232no—
164#137—462no—
165#137—462no—
166#137C2×D2312yes—
167AGL(1,ℤ/8)—232no—
168AGL(1,ℤ/8)—232no—
169AGL(1,ℤ/8)—462no—
170AGL(1,ℤ/8)—462no—
171#135—231no—
172#135—231no—
173#135—461no—
174#135—461no—
175#135—462no—
176C2×Q16—461no—
177C2×Q16C2×D2311yescharacteristic, complemented
178C2×SD16—232no—
179C2×SD16—462no—
180C2×D8—231no—
181C2×D8—461no—
182#131—232no—
183#131—231no—
184#131C2×D2311yescharacteristic
185#124—232no—
186#108—231no—
187#108C2×D2311yescharacteristic, complemented
188#104—231no—
189#104—231no—
190#104—232no—
191#101C2×D2312yescomplemented
192#100—232no—
193C4×C8—231no—
194C23#329011yescharacteristic, complemented, Sylow
195#41—231no—
196#41—232no—
197#41—462no—
198#41—462no—
199#41—462no—
200#41—462no—
201#41—922no—
202#41C22×D2311yescharacteristic
203C2×Q8—462no—
204C2×Q8—462no—
205C2×Q8C22×D2312yes—
206C2×D4—232no—
207C2×D4—462no—
208C2×D4—462no—
209C2×D4—922no—
210Q16—22no—
211Q16—21no—
212Q16—21no—
213Q16—231no—
214Q16—231no—
215Q16—462no—
216SD16—22no—
217SD16—231no—
218SD16—231no—
219SD16—232no—
220SD16—462no—
221SD16—462no—
222SD16—462no—
223SD16—462no—
224SD16—462no—
225D8—231no—
226D8—231no—
227D8—462no—
228D8—462no—
229D8—461no—
230D8—461no—
231M16—21no—
232M16—231no—
233M16—462no—
234M16—461no—
235M16C22×D2312yes—
236M16C22×D2311yescharacteristic
237C2×C8—21no—
238C2×C8—231no—
239C2×C8—462no—
240C2×C8—461no—
241C2×C8C22×D2311yescharacteristic
242C22⋊C4—922no—
243C42—231no—
244C23—462no—
245C23—922no—
246Q8—22no—
247Q8—22no—
248Q8—231no—
249Q8—232no—
250Q8—922no—
251Q8D4×D2311yescharacteristic
252D4—21no—
253D4—231no—
254D4—232no—
255D4—462no—
256D4—461no—
257D4—462no—
258D4—462no—
259D4—922no—
260D4—922no—
261D4—922no—
262D4D4×D2311yescharacteristic
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—462no—
266C2×C4—461no—
267C2×C4—461no—
268C2×C4—922no—
269C2×C4—922no—
270C2×C4—922no—
271C2×C4C23×D2311yescharacteristic
272C8—21no—
273C8—22no—
274C8—231no—
275C8—231no—
276C8—461no—
277C8D4×D2311yescharacteristic
278C8D4×D2311yescharacteristic
279C22—21no—
280C22—461no—
281C22—462no—
282C22—922no—
283C22—3682no—
284C22—1842no—
285C22#1062516011yescharacteristic
286C4—21no—
287C4—42no—
288C4—462no—
289C4—461no—
290C4—921no—
291C4C2×D4×D2311yescharacteristic
292C4C2×D4×D2311yescharacteristic
293C2—21no—
294C2—41no—
295C2—922no—
296C2—921no—
297C2—1842no—
298C2#1309305111yescharacteristic
subgroupG/H
centerC2#13093051
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC23×#473C2
socleC2×C23D42
hypercenter#473D23
hypocenterC23#3290
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C2 ◁ C2×C4 ◁ #473

Complements

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

NH ≅ G/Ntype
1. #13093230C2semidirect
2. #13093226C2semidirect
3. #13093222C2semidirect
4. #13092802C2semidirect
5. #13092688C2semidirect
6. #13092683C2semidirect
7. #13092593C2semidirect
8. #13092592C2semidirect
9. #13092533C2semidirect
10. #13092532C2semidirect
11. C23×#473C2semidirect
13. #10625158C22semidirect
15. C2×Q16×C23C22semidirect
18. #10625118C22semidirect
19. #10625113C22semidirect
20. C2×#67274C22semidirect
30. #10625079C22semidirect
31. #10625078C22semidirect
36. #10625072C22semidirect
37. C23×#108C22semidirect
38. C23×#101C22semidirect
40. #10625007C22semidirect
41. #10625007C22semidirect
42. #10625005C22semidirect
43. #10625004C22semidirect
44. C8×Dic23C22semidirect
148. #473D23semidirect
177. C2×Q16C2×D23semidirect
187. #108C2×D23semidirect
191. #101C2×D23semidirect
194. C23#3290semidirect

Maximal subgroups

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

subgroupindex
#130932302
#130932262
#130932222
#130928022
#130926882
#130926832
#130925932
#130925922
#130925332
#130925322
C23×#4732
#329023