Finite Group Databasebeta

← #31792557

Subgroups — #31792557

Order 17536.

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

normal subgroups only

This group has 464 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#22578211C212yesmaximal, complemented
2#22578207C212yesmaximal, complemented
3#22577793C211yesmaximal, characteristic, complemented
4#22577671C212yesmaximal, complemented
5#22577669C212yesmaximal, complemented
6#22577580C211yesmaximal, characteristic, complemented
7#22577572C212yesmaximal, complemented
8#22577522C212yesmaximal, complemented
9#454×C137C211yesmaximal, characteristic, complemented
10#17984134C2212yescomplemented
11#17984133C2211yescharacteristic
12#17984119—22no—
13#17984119C2212yescomplemented
14#17984116C2212yes—
15#17984115—22no—
16#17984113—22no—
17#17984113C2212yescomplemented
18#17984112—22no—
19#17984109—22no—
20#17984108—22no—
21#17984108C2212yes—
22C2×#14555497—22no—
23C2×#14555497C2212yescomplemented
24C2×#14555495C2212yescomplemented
25D8×D137—22no—
26#17984096C2212yescomplemented
27#17984095C2212yescomplemented
28#17984085C2211yescharacteristic, complemented
29#17984023C2212yescomplemented
30#17984016C2212yescomplemented
31#17984005C2211yescharacteristic, complemented
32#17984004C2212yescomplemented
33#17983999C2211yescharacteristic, complemented
342−1+4×C137C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C137C2212yescomplemented
36#124×C137C2211yescharacteristic, complemented
37#104×C137C2212yescomplemented
38#101×C137C2211yescharacteristic, complemented
39#14555511—22no—
40#14555510—22no—
41#14555510C2312yes—
42#14555509—22no—
43C2×D548—22no—
44C2×D548—41no—
45C2×D548C2312yes—
46C2×C22⋊D137—41no—
47D4×D137—22no—
48D4×D137—22no—
49D4×D137—42no—
50#14555500—22no—
51#14555500D412yescomplemented
52#14555499—22no—
53#14555498D412yescomplemented
54#14555497—22no—
55#14555497—22no—
56#14555497—22no—
57#14555497—22no—
58#14555497D412yescomplemented
59D1096—22no—
60#14555495—22no—
61#14555495—22no—
62#14555495—22no—
63#14555495D412yescomplemented
64#14555494—22no—
65#14555494C2312yes—
66#14555493—22no—
67C2×#11764040—22no—
68C2×#11764040C2312yes—
69C8×D137—22no—
70#41×C137—22no—
71#41×C137—41no—
72#41×C137C2312yes—
73C2×Q8×C137—22no—
74C2×Q8×C137C2311yescharacteristic
75C2×D4×C137C2311yescharacteristic
76SD16×C137—22no—
77D8×C137—22no—
78M16×C137C2312yes—
79C22⋊C4×C137—41no—
80C42×C137C2311yescharacteristic
81C22×D137—22no—
82C22×D137—41no—
83C22×D137—42no—
84Dic274C2×D412yes—
85D548—22no—
86D548—22no—
87D548—42no—
88D548—82no—
89D548C2×D412yes—
90C22⋊D137—22no—
91C22⋊D137—42no—
92C22⋊D137—42no—
93C2×Dic137—41no—
94C4×D137—22no—
95C4×D137—42no—
96#11764040—22no—
97#11764040C2×D412yes—
98#11764040C2×D412yes—
99C23×C137—21no—
100Q8×C137—22no—
101Q8×C137—41no—
102Q8×C137C2×D412yes—
103D4×C137—22no—
104D4×C137—22no—
105D4×C137—41no—
106D4×C137C2×D412yes—
107C2×C4×C137—22no—
108C2×C4×C137—21no—
109C2×C4×C137—21no—
110C2×C4×C137—42no—
111C2×C4×C137C2411yescharacteristic
112C8×C137—22no—
113C22×C137—22no—
114C22×C137—41no—
115C22×C137—81no—
116C22×C137H(5,𝔽2)11yescharacteristic
117C2×D137—22no—
118C2×D137—42no—
119C2×D137—82no—
120C2×D137—161no—
121C2×D137—162no—
122C4×C137—22no—
123C4×C137—41no—
124C4×C137—41no—
125C4×C137C22×D412yes—
126Dic137—22no—
127C2×C137—21no—
128C2×C137—42no—
129C2×C137—81no—
130C2×C137D4211yescharacteristic
131D137—42no—
132D137—82no—
133C137#328811yescharacteristic, complemented, Sylow
134#3288—1371nomaximal, Sylow
135#577—1372no—
136#576—1372no—
137#496—1371no—
138#472—1372no—
139#471—1372no—
140#455—1371no—
141#454D13711yescharacteristic, complemented
142#453—1372no—
143#444—1372no—
1442−1+4C2×D13711yescharacteristic, complemented
145H(5,𝔽2)—1372no—
146H(5,𝔽2)—1371no—
147#137—2742no—
148#137—1372no—
149AGL(1,ℤ/8)—2742no—
150AGL(1,ℤ/8)—2742no—
151AGL(1,ℤ/8)—2742no—
152AGL(1,ℤ/8)—1372no—
153AGL(1,ℤ/8)—1372no—
154AGL(1,ℤ/8)C2×D13712yescomplemented
155#135—2742no—
156#135—2742no—
157#135—1372no—
158C2×SD16—2742no—
159C2×SD16—1372no—
160C2×D8—2742no—
161C2×D8—1372no—
162#131—1372no—
163#131—1372no—
164#127—1371no—
165#124C2×D13711yescharacteristic, complemented
166#108—1372no—
167#104—1372no—
168#104C2×D13712yescomplemented
169#101C2×D13711yescharacteristic, complemented
170#100—1371no—
171#100—1372no—
172#97—1371no—
173#41—22no—
174#41—41no—
175#41—2742no—
176#41—2742no—
177#41—2742no—
178#41—1372no—
179#41C22×D13712yes—
180C2×Q8—22no—
181C2×Q8C22×D13711yescharacteristic
182C2×D4—2742no—
183C2×D4—2742no—
184C2×D4—2742no—
185C2×D4—5481no—
186C2×D4—5482no—
187C2×D4—5481no—
188C2×D4—1372no—
189C2×D4C22×D13711yescharacteristic
190Q16—2742no—
191Q16—1372no—
192SD16—22no—
193SD16—2742no—
194SD16—2742no—
195SD16—2742no—
196SD16—2742no—
197SD16—2742no—
198SD16—1372no—
199SD16—1372no—
200D8—22no—
201D8—2742no—
202D8—2742no—
203D8—2742no—
204D8—2742no—
205D8—1372no—
206M16—2742no—
207M16—2742no—
208M16—1372no—
209M16C22×D13712yes—
210C2×C8—2742no—
211C2×C8—2742no—
212C2×C8—1372no—
213C22⋊C4—41no—
214C42C22×D13711yescharacteristic
215C23—21no—
216C23—2742no—
217C23—5482no—
218C23—5481no—
219Q8—22no—
220Q8—41no—
221Q8—1372no—
222Q8D4×D13712yes—
223D4—22no—
224D4—22no—
225D4—41no—
226D4—2742no—
227D4—2742no—
228D4—2742no—
229D4—5482no—
230D4—5482no—
231D4—5482no—
232D4—10962no—
233D4—1372no—
234D4D4×D13712yes—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—42no—
239C2×C4—2742no—
240C2×C4—5482no—
241C2×C4—5481no—
242C2×C4C23×D13711yescharacteristic
243C8—22no—
244C8—2742no—
245C8—1372no—
246C8—1372no—
247C22—22no—
248C22—41no—
249C22—81no—
250C22—2742no—
251C22—5482no—
252C22—10962no—
253C22—21922no—
254C22—21921no—
255C22#1798413311yescharacteristic
256C4—22no—
257C4—41no—
258C4—41no—
259C4—2742no—
260C4C2×D4×D13712yes—
261C2—21no—
262C2—42no—
263C2—81no—
264C2—5482no—
265C2—10962no—
266C2#2257803511yescharacteristic
subgroupG/H
centerC2#22578035
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#454×C137C2
socleC2×C137D42
hypercenter#454D137
hypocenterC137#3288
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #22578211C2semidirect
2. #22578207C2semidirect
3. #22577793C2semidirect
4. #22577671C2semidirect
5. #22577669C2semidirect
6. #22577580C2semidirect
7. #22577572C2semidirect
8. #22577522C2semidirect
9. #454×C137C2semidirect
10. #17984134C22semidirect
13. #17984119C22semidirect
17. #17984113C22semidirect
23. C2×#14555497C22semidirect
24. C2×#14555495C22semidirect
26. #17984096C22semidirect
27. #17984095C22semidirect
28. #17984085C22semidirect
29. #17984023C22semidirect
30. #17984016C22semidirect
31. #17984005C22semidirect
32. #17984004C22semidirect
33. #17983999C22semidirect
34. 2−1+4×C137C22semidirect
35. AGL(1,ℤ/8)×C137C22semidirect
36. #124×C137C22semidirect
37. #104×C137C22semidirect
38. #101×C137C22semidirect
51. #14555500D4semidirect
53. #14555498D4semidirect
58. #14555497D4semidirect
63. #14555495D4semidirect
133. C137#3288semidirect
141. #454D137semidirect
144. 2−1+4C2×D137semidirect
154. AGL(1,ℤ/8)C2×D137semidirect
165. #124C2×D137semidirect
168. #104C2×D137semidirect
169. #101C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225782112
#225782072
#225777932
#225776712
#225776692
#225775802
#225775722
#225775222
#454×C1372
#3288137