Finite Group Databasebeta

← #31792584

Subgroups — #31792584

Order 17536.

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

normal subgroups only

This group has 460 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#22578214C212yesmaximal, complemented
2#22578213C212yesmaximal, complemented
3#22577775C211yesmaximal, characteristic, complemented
4#22577665C212yesmaximal, complemented
5#22577664C212yesmaximal, complemented
6#22577587C212yesmaximal, complemented
7#22577570C211yesmaximal, characteristic, complemented
8#22577517C212yesmaximal, complemented
9#453×C137C211yesmaximal, characteristic, complemented
10#17984135C2212yes—
11#17984133C2211yescharacteristic, complemented
12#17984121C2212yescomplemented
13#17984118—22no—
14#17984118C2212yescomplemented
15#17984117—22no—
16#17984114—22no—
17#17984114C2212yes—
18#17984112—22no—
19#17984108—22no—
20#17984108C2212yes—
21#17984107—22no—
22C2×#14555498—22no—
23C2×#14555498C2212yescomplemented
24C2×#14555498C2212yescomplemented
25SD16×D137—22no—
26#17984096C2212yescomplemented
27#17984095C2212yescomplemented
28#17984088C2211yescharacteristic, complemented
29#17984023C2212yescomplemented
30#17984016C2212yescomplemented
31#17984006C2212yescomplemented
32#17984005C2211yescharacteristic, complemented
33C4×#11764040C2211yescharacteristic, complemented
34H(5,𝔽2)×C137C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C137C2212yes—
36#127×C137C2211yescharacteristic
37#104×C137C2212yescomplemented
38#100×C137C2211yescharacteristic, complemented
39#14555510—22no—
40#14555510C2312yes—
41#14555509—22no—
42#14555509—22no—
43#14555509—42no—
44C2×Dic274—22no—
45C2×Dic274C2312yes—
46Q8×D137—22no—
47C2×C22⋊D137—41no—
48D4×D137—22no—
49Dic548—22no—
50#14555500—22no—
51#14555500D412yescomplemented
52#14555499—22no—
53#14555498—22no—
54#14555498—22no—
55#14555498—22no—
56#14555498—22no—
57#14555498—22no—
58#14555498—22no—
59#14555498D412yescomplemented
60#14555497D412yescomplemented
61#14555495—22no—
62#14555495D412yescomplemented
63#14555494—22no—
64#14555494C2312yes—
65#14555493—22no—
66C2×#11764040—22no—
67C2×#11764040C2312yes—
68C8×D137—22no—
69C4⋊Dic137—41no—
70#41×C137—22no—
71#41×C137C2312yes—
72C2×D4×C137—22no—
73C2×D4×C137—41no—
74C2×D4×C137—41no—
75C2×D4×C137C2311yescharacteristic
76C2×D4×C137C2311yescharacteristic
77SD16×C137—22no—
78D8×C137—22no—
79M16×C137C2312yes—
80C42×C137C2311yescharacteristic
81C22×D137—41no—
82Dic274—22no—
83Dic274—22no—
84Dic274—42no—
85Dic274C2×D412yes—
86D548C2×D412yes—
87C22⋊D137—22no—
88C22⋊D137—42no—
89C22⋊D137—42no—
90C2×Dic137—22no—
91C2×Dic137—41no—
92C2×Dic137—42no—
93C4×D137—22no—
94C4×D137—42no—
95#11764040—22no—
96#11764040C2×D412yes—
97#11764040C2×D412yes—
98C23×C137—21no—
99C23×C137—21no—
100C23×C137—41no—
101Q8×C137C2×D412yes—
102D4×C137—22no—
103D4×C137—22no—
104D4×C137—22no—
105D4×C137—42no—
106D4×C137—82no—
107D4×C137C2×D412yes—
108C2×C4×C137—21no—
109C2×C4×C137—22no—
110C2×C4×C137—41no—
111C2×C4×C137C2411yescharacteristic
112C8×C137—22no—
113C22×C137—22no—
114C22×C137—41no—
115C22×C137—41no—
116C22×C137—81no—
117C22×C137—81no—
118C22×C137—161no—
119C22×C137H(5,𝔽2)11yescharacteristic
120C2×D137—22no—
121C2×D137—161no—
122C4×C137—22no—
123C4×C137—22no—
124C4×C137C22×D412yes—
125Dic137—22no—
126Dic137—42no—
127C2×C137—21no—
128C2×C137—42no—
129C2×C137—81no—
130C2×C137—81no—
131C2×C137D4211yescharacteristic
132D137—42no—
133C137#329111yescharacteristic, complemented, Sylow
134#3291—1371nomaximal, Sylow
135#577—1372no—
136#577—1372no—
137#492—1371no—
138#471—1372no—
139#471—1372no—
140#456—1372no—
141#453—1371no—
142#453D13711yescharacteristic, complemented
143#443—1372no—
1442−1+4—1372no—
145H(5,𝔽2)—1371no—
146H(5,𝔽2)C2×D13711yescharacteristic, complemented
147#137—2742no—
148#137—2742no—
149#137—1372no—
150#137—1372no—
151AGL(1,ℤ/8)—2742no—
152AGL(1,ℤ/8)—2742no—
153AGL(1,ℤ/8)—1372no—
154AGL(1,ℤ/8)C2×D13712yes—
155#135—2742no—
156#135—2742no—
157#135—1372no—
158C2×SD16—2742no—
159C2×SD16—2742no—
160C2×SD16—1372no—
161C2×SD16—1372no—
162#131—1372no—
163#131—1372no—
164C4⋊Q8—1371no—
165#127C2×D13711yescharacteristic
166#108—1372no—
167#104—1372no—
168#104C2×D13712yescomplemented
169#101—1372no—
170#100—1371no—
171#100C2×D13711yescharacteristic, complemented
172C4×C8—1371no—
173#41—22no—
174#41—2742no—
175#41—2742no—
176#41—2742no—
177#41—5482no—
178#41—1372no—
179#41C22×D13712yes—
180C2×Q8—2742no—
181C2×Q8—2742no—
182C2×Q8—1372no—
183C2×D4—22no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—2742no—
187C2×D4—5481no—
188C2×D4C22×D13711yescharacteristic
189C2×D4C22×D13711yescharacteristic
190Q16—2742no—
191Q16—2742no—
192Q16—1372no—
193SD16—22no—
194SD16—2742no—
195SD16—2742no—
196SD16—2742no—
197SD16—2742no—
198SD16—2742no—
199SD16—2742no—
200SD16—2742no—
201SD16—1372no—
202SD16—1372no—
203D8—22no—
204D8—2742no—
205D8—1372no—
206M16—2742no—
207M16—2742no—
208M16—1372no—
209M16C22×D13712yes—
210C2×C8—2742no—
211C2×C8—2742no—
212C2×C8—1372no—
213C4⋊C4—5481no—
214C42C22×D13711yescharacteristic
215C23—21no—
216C23—21no—
217C23—41no—
218C23—5481no—
219Q8—2742no—
220Q8—2742no—
221Q8—5482no—
222Q8—1372no—
223Q8D4×D13712yes—
224D4—22no—
225D4—22no—
226D4—22no—
227D4—42no—
228D4—82no—
229D4—2742no—
230D4—5482no—
231D4—5482no—
232D4—1372no—
233D4D4×D13712yes—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—41no—
237C2×C4—2742no—
238C2×C4—2742no—
239C2×C4—5482no—
240C2×C4—5481no—
241C2×C4—5482no—
242C2×C4C23×D13711yescharacteristic
243C8—22no—
244C8—2742no—
245C8—1372no—
246C8—1372no—
247C22—22no—
248C22—41no—
249C22—41no—
250C22—81no—
251C22—81no—
252C22—161no—
253C22—2742no—
254C22—21921no—
255C22#1798413311yescharacteristic
256C4—22no—
257C4—22no—
258C4—2742no—
259C4—5482no—
260C4C2×D4×D13712yes—
261C2—21no—
262C2—42no—
263C2—81no—
264C2—81no—
265C2—5482no—
266C2#2257803511yescharacteristic
subgroupG/H
centerC2#22578035
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#453×C137C2
socleC2×C137D42
hypercenter#453D137
hypocenterC137#3291
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #22578214C2semidirect
2. #22578213C2semidirect
3. #22577775C2semidirect
4. #22577665C2semidirect
5. #22577664C2semidirect
6. #22577587C2semidirect
7. #22577570C2semidirect
8. #22577517C2semidirect
9. #453×C137C2semidirect
11. #17984133C22semidirect
12. #17984121C22semidirect
14. #17984118C22semidirect
23. C2×#14555498C22semidirect
24. C2×#14555498C22semidirect
26. #17984096C22semidirect
27. #17984095C22semidirect
28. #17984088C22semidirect
29. #17984023C22semidirect
30. #17984016C22semidirect
31. #17984006C22semidirect
32. #17984005C22semidirect
33. C4×#11764040C22semidirect
34. H(5,𝔽2)×C137C22semidirect
37. #104×C137C22semidirect
38. #100×C137C22semidirect
51. #14555500D4semidirect
59. #14555498D4semidirect
60. #14555497D4semidirect
62. #14555495D4semidirect
133. C137#3291semidirect
142. #453D137semidirect
146. H(5,𝔽2)C2×D137semidirect
168. #104C2×D137semidirect
171. #100C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225782142
#225782132
#225777752
#225776652
#225776642
#225775872
#225775702
#225775172
#453×C1372
#3291137