Finite Group Databasebeta

← #31792579

Subgroups — #31792579

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#22578212C212yesmaximal, complemented
2#22578211C212yesmaximal, complemented
3#22577777C211yesmaximal, characteristic, complemented
4#22577669C212yesmaximal, complemented
5#22577668C212yesmaximal, complemented
6#22577585C211yesmaximal, characteristic, complemented
7#22577572C212yesmaximal, complemented
8#22577517C212yesmaximal, complemented
9#456×C137C211yesmaximal, characteristic, complemented
10#17984136C2211yescharacteristic
11#17984134C2212yescomplemented
12#17984120—22no—
13#17984119—22no—
14#17984119C2212yescomplemented
15#17984116C2212yes—
16#17984115—22no—
17#17984113—22no—
18#17984113C2212yes—
19#17984111—22no—
20#17984108—22no—
21#17984108C2212yes—
22C2×#14555497—22no—
23C2×#14555497C2212yescomplemented
24C2×#14555497C2212yescomplemented
25SD16×D137—22no—
26#17984096C2212yescomplemented
27#17984095C2212yescomplemented
28#17984085C2211yescharacteristic
29#17984023C2212yescomplemented
30#17984016C2212yescomplemented
31#17984007C2211yescharacteristic, complemented
32#17984004C2212yescomplemented
33C4×#11764040C2211yescharacteristic, complemented
342−1+4×C137C2211yescharacteristic
35#137×C137C2212yescomplemented
36C4⋊Q8×C137C2211yescharacteristic, complemented
37#104×C137C2212yescomplemented
38#101×C137C2211yescharacteristic, complemented
39#14555511—22no—
40#14555511—22no—
41#14555510—22no—
42#14555510—41no—
43#14555510C2312yes—
44Q8×D137—22no—
45C2×D548—22no—
46C2×D548—41no—
47C2×D548C2312yes—
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#14555497—22no—
59#14555497—22no—
60#14555497D412yescomplemented
61D1096—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×Q8×C137C2311yescharacteristic
76Q16×C137—22no—
77SD16×C137—22no—
78M16×C137C2312yes—
79C4⋊C4×C137—41no—
80C42×C137C2311yescharacteristic
81C22×D137—22no—
82C22×D137—42no—
83Dic274—41no—
84Dic274C2×D412yes—
85D548—22no—
86D548—22no—
87D548—42no—
88D548—41no—
89D548—82no—
90D548C2×D412yes—
91C22⋊D137—22no—
92C22⋊D137—42no—
93C4×D137—22no—
94C4×D137—42no—
95C4×D137—42no—
96#11764040—22no—
97#11764040C2×D412yes—
98#11764040C2×D412yes—
99Q8×C137—22no—
100Q8×C137—22no—
101Q8×C137—41no—
102Q8×C137C2×D412yes—
103D4×C137—22no—
104D4×C137—41no—
105D4×C137C2×D412yes—
106C2×C4×C137—21no—
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×C137H(5,𝔽2)11yescharacteristic
115C2×D137—22no—
116C2×D137—42no—
117C2×D137—82no—
118C2×D137—162no—
119C4×C137—22no—
120C4×C137—22no—
121C4×C137—41no—
122C4×C137—41no—
123C4×C137C22×D412yes—
124Dic137—22no—
125C2×C137—21no—
126C2×C137—42no—
127C2×C137D4211yescharacteristic
128D137—42no—
129D137—82no—
130C137#329111yescharacteristic, complemented, Sylow
131#3291—1371nomaximal, Sylow
132#577—1372no—
133#577—1372no—
134#492—1371no—
135#471—1372no—
136#471—1372no—
137#456—1371no—
138#456D13711yescharacteristic, complemented
139#453—1372no—
140#443—1372no—
1412−1+4—1371no—
1422−1+4C2×D13711yescharacteristic
143H(5,𝔽2)—1372no—
144#137—2742no—
145#137—2742no—
146#137—1372no—
147#137C2×D13712yescomplemented
148AGL(1,ℤ/8)—2742no—
149AGL(1,ℤ/8)—2742no—
150AGL(1,ℤ/8)—1372no—
151AGL(1,ℤ/8)—1372no—
152#135—2742no—
153#135—2742no—
154#135—1372no—
155C2×SD16—2742no—
156C2×SD16—2742no—
157C2×SD16—1372no—
158C2×SD16—1372no—
159#131—1372no—
160#131—1372no—
161C4⋊Q8C2×D13711yescharacteristic, complemented
162#127—1371no—
163#108—1372no—
164#104—1372no—
165#104C2×D13712yescomplemented
166#101—1371no—
167#101C2×D13711yescharacteristic, complemented
168#100—1372no—
169C4×C8—1371no—
170#41—22no—
171#41—41no—
172#41—2742no—
173#41—2742no—
174#41—2742no—
175#41—5481no—
176#41—1372no—
177#41C22×D13712yes—
178C2×Q8—22no—
179C2×Q8—2742no—
180C2×Q8C22×D13711yescharacteristic
181C2×Q8C22×D13711yescharacteristic
182C2×D4—2742no—
183C2×D4—2742no—
184C2×D4—5481no—
185C2×D4—5482no—
186C2×D4—1372no—
187Q16—22no—
188Q16—2742no—
189Q16—1372no—
190SD16—22no—
191SD16—2742no—
192SD16—2742no—
193SD16—2742no—
194SD16—2742no—
195SD16—2742no—
196SD16—2742no—
197SD16—2742no—
198SD16—1372no—
199SD16—1372no—
200D8—2742no—
201D8—2742no—
202D8—1372no—
203M16—2742no—
204M16—2742no—
205M16—1372no—
206M16C22×D13712yes—
207C2×C8—2742no—
208C2×C8—2742no—
209C2×C8—1372no—
210C4⋊C4—41no—
211C42C22×D13711yescharacteristic
212C23—2742no—
213C23—5482no—
214Q8—22no—
215Q8—22no—
216Q8—41no—
217Q8—5481no—
218Q8—1372no—
219Q8D4×D13712yes—
220D4—22no—
221D4—41no—
222D4—2742no—
223D4—2742no—
224D4—2742no—
225D4—5482no—
226D4—5481no—
227D4—5482no—
228D4—10962no—
229D4—1372no—
230D4D4×D13712yes—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—42no—
236C2×C4—2742no—
237C2×C4—5482no—
238C2×C4—5482no—
239C2×C4C23×D13711yescharacteristic
240C8—22no—
241C8—2742no—
242C8—1372no—
243C8—1372no—
244C22—22no—
245C22—2742no—
246C22—5482no—
247C22—10962no—
248C22—21922no—
249C22#1798413311yescharacteristic
250C4—22no—
251C4—22no—
252C4—41no—
253C4—41no—
254C4—2742no—
255C4C2×D4×D13712yes—
256C2—21no—
257C2—42no—
258C2—5482no—
259C2—10962no—
260C2#2257803511yescharacteristic
subgroupG/H
centerC2#22578035
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#456×C137C2
socleC2×C137D42
hypercenter#456D137
hypocenterC137#3291
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #22578212C2semidirect
2. #22578211C2semidirect
3. #22577777C2semidirect
4. #22577669C2semidirect
5. #22577668C2semidirect
6. #22577585C2semidirect
7. #22577572C2semidirect
8. #22577517C2semidirect
9. #456×C137C2semidirect
11. #17984134C22semidirect
14. #17984119C22semidirect
23. C2×#14555497C22semidirect
24. C2×#14555497C22semidirect
26. #17984096C22semidirect
27. #17984095C22semidirect
29. #17984023C22semidirect
30. #17984016C22semidirect
31. #17984007C22semidirect
32. #17984004C22semidirect
33. C4×#11764040C22semidirect
35. #137×C137C22semidirect
36. C4⋊Q8×C137C22semidirect
37. #104×C137C22semidirect
38. #101×C137C22semidirect
51. #14555500D4semidirect
53. #14555498D4semidirect
60. #14555497D4semidirect
63. #14555495D4semidirect
130. C137#3291semidirect
138. #456D137semidirect
147. #137C2×D137semidirect
161. C4⋊Q8C2×D137semidirect
165. #104C2×D137semidirect
167. #101C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225782122
#225782112
#225777772
#225776692
#225776682
#225775852
#225775722
#225775172
#456×C1372
#3291137