Finite Group Databasebeta

← #31792583

Subgroups — #31792583

Order 17536.

460 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 125 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#22578215C212yesmaximal, complemented
2#22578210C212yesmaximal, complemented
3#22577776C211yesmaximal, characteristic, complemented
4#22577667C212yesmaximal, complemented
5#22577666C212yesmaximal, complemented
6#22577587C212yesmaximal, complemented
7#22577572C212yesmaximal, complemented
8#22577521C211yesmaximal, characteristic, complemented
9#443×C137C211yesmaximal, characteristic, complemented
10#17984135C2212yes—
11#17984134C2212yescomplemented
12#17984121—22no—
13#17984121C2212yescomplemented
14#17984120—22no—
15#17984118C2212yescomplemented
16#17984116—22no—
17#17984116C2212yes—
18#17984113C2212yes—
19#17984112—22no—
20#17984110—22no—
21#17984110C2211yescharacteristic
22#17984109—22no—
23C2×#14555499—22no—
24C2×#14555499C2212yescomplemented
25C2×#14555499C2212yescomplemented
26SD16×D137—22no—
27#17984094C2212yescomplemented
28#17984088C2211yescharacteristic, complemented
29#17984085C2211yescharacteristic
30#17984024C2211yescharacteristic, complemented
31#17984016C2212yescomplemented
32#17984006C2212yescomplemented
33#17984004C2212yescomplemented
34#135×C137C2211yescharacteristic
35#131×C137C2212yescomplemented
36#108×C137C2211yescharacteristic, complemented
37#104×C137C2212yescomplemented
38C4×C8×C137C2211yescharacteristic, complemented
39#14555511—22no—
40#14555510—22no—
41#14555510—22no—
42#14555510C2312yes—
43#14555509—22no—
44#14555509—42no—
45C2×Dic274—22no—
46C2×Dic274C2312yes—
47Q8×D137—22no—
48C2×D548—22no—
49C2×D548—41no—
50C2×D548C2312yes—
51D4×D137—22no—
52D4×D137—42no—
53Dic548—22no—
54Dic548D411yescharacteristic, complemented
55#14555500—22no—
56#14555499—22no—
57#14555499—22no—
58#14555499—22no—
59#14555499—22no—
60#14555499—22no—
61#14555499—22no—
62#14555499D412yescomplemented
63#14555498—22no—
64#14555497—22no—
65D1096—22no—
66D1096D411yescharacteristic, complemented
67#14555495—22no—
68#14555494C2312yes—
69#14555493—22no—
70C8×D137—22no—
71C4⋊Dic137—41no—
72#41×C137C2312yes—
73Q16×C137D411yescharacteristic, complemented
74SD16×C137D412yescomplemented
75D8×C137D411yescharacteristic, complemented
76M16×C137—22no—
77M16×C137C2312yes—
78C2×C8×C137—22no—
79C2×C8×C137C2311yescharacteristic
80C2×C8×C137C2311yescharacteristic
81C42×C137C2311yescharacteristic
82C22×D137—22no—
83C22×D137—42no—
84Dic274—22no—
85Dic274—22no—
86Dic274—42no—
87Dic274C2×D412yes—
88D548—22no—
89D548—22no—
90D548—42no—
91D548—82no—
92D548C2×D412yes—
93C22⋊D137—22no—
94C22⋊D137—42no—
95C22⋊D137—42no—
96C2×Dic137—22no—
97C2×Dic137—42no—
98C4×D137—22no—
99C4×D137—42no—
100C4×D137—42no—
101#11764040—22no—
102Q8×C137C2×D412yes—
103D4×C137—22no—
104D4×C137C2×D412yes—
105C2×C4×C137—21no—
106C2×C4×C137—22no—
107C2×C4×C137C2411yescharacteristic
108C8×C137—22no—
109C8×C137C2×D411yescharacteristic
110C8×C137C2×D411yescharacteristic
111C8×C137C2×D411yescharacteristic
112C8×C137C2×D411yescharacteristic
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×C137C22×D411yescharacteristic
122C4×C137C22×D411yescharacteristic
123Dic137—22no—
124Dic137—42no—
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—1372no—
138#453—1372no—
139#443—1371no—
140#443D13711yescharacteristic, complemented
1412−1+4—1372no—
142H(5,𝔽2)—1372no—
143#137—2742no—
144#137—2742no—
145#137—1372no—
146#137—1372no—
147AGL(1,ℤ/8)—2742no—
148AGL(1,ℤ/8)—2742no—
149AGL(1,ℤ/8)—1372no—
150AGL(1,ℤ/8)—1372no—
151#135—2742no—
152#135—2742no—
153#135—1371no—
154#135C2×D13711yescharacteristic
155C2×SD16—2742no—
156C2×SD16—2742no—
157C2×SD16—1372no—
158C2×SD16—1372no—
159#131—1372no—
160#131C2×D13712yescomplemented
161C4⋊Q8—1371no—
162#127—1371no—
163#108—1371no—
164#108C2×D13711yescharacteristic, complemented
165#104—1372no—
166#104C2×D13712yescomplemented
167#101—1372no—
168#100—1372no—
169C4×C8C2×D13711yescharacteristic, complemented
170#41—2742no—
171#41—2742no—
172#41—2742no—
173#41—2742no—
174#41—5482no—
175#41—1372no—
176#41C22×D13712yes—
177C2×Q8—2742no—
178C2×Q8—2742no—
179C2×Q8—1372no—
180C2×D4—2742no—
181C2×D4—2742no—
182C2×D4—5481no—
183C2×D4—5482no—
184C2×D4—1372no—
185Q16—2742no—
186Q16—2742no—
187Q16—1371no—
188Q16D54811yescharacteristic, complemented
189SD16—2742no—
190SD16—2742no—
191SD16—2742no—
192SD16—2742no—
193SD16—2742no—
194SD16—2742no—
195SD16—2742no—
196SD16—2742no—
197SD16—1372no—
198SD16D54812yescomplemented
199D8—2742no—
200D8—2742no—
201D8—1371no—
202D8D54811yescharacteristic, complemented
203M16—22no—
204M16—2742no—
205M16—1372no—
206M16C22×D13712yes—
207C2×C8—22no—
208C2×C8—2742no—
209C2×C8C22×D13711yescharacteristic
210C2×C8C22×D13711yescharacteristic
211C4⋊C4—5481no—
212C42C22×D13711yescharacteristic
213C23—2742no—
214C23—5482no—
215Q8—2742no—
216Q8—2742no—
217Q8—5482no—
218Q8—1372no—
219Q8C2×D54812yes—
220D4—22no—
221D4—2742no—
222D4—2742no—
223D4—2742no—
224D4—5482no—
225D4—5482no—
226D4—5482no—
227D4—10962no—
228D4—1372no—
229D4C2×D54812yes—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—2742no—
233C2×C4—2742no—
234C2×C4—5482no—
235C2×C4—5482no—
236C2×C4—5482no—
237C2×C4C23×D13711yescharacteristic
238C8—22no—
239C8—2742no—
240C8D4×D13711yescharacteristic
241C8D4×D13711yescharacteristic
242C8C2×D54811yescharacteristic
243C8C2×D54811yescharacteristic
244C22—22no—
245C22—2742no—
246C22—5482no—
247C22—10962no—
248C22—21922no—
249C22#1798413411yescharacteristic
250C4—22no—
251C4—22no—
252C4—2742no—
253C4—5482no—
254C4C2×D4×D13711yescharacteristic
255C4C22×D54811yescharacteristic
256C2—21no—
257C2—42no—
258C2—5482no—
259C2—10962no—
260C2D4×D54811yescharacteristic
subgroupG/H
centerC2D4×D548
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#443×C137C2
socleC2×C137D42
hypercenter#443D137
hypocenterC137#3291
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ #443

Complements

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

NH ≅ G/Ntype
1. #22578215C2semidirect
2. #22578210C2semidirect
3. #22577776C2semidirect
4. #22577667C2semidirect
5. #22577666C2semidirect
6. #22577587C2semidirect
7. #22577572C2semidirect
8. #22577521C2semidirect
9. #443×C137C2semidirect
11. #17984134C22semidirect
13. #17984121C22semidirect
15. #17984118C22semidirect
24. C2×#14555499C22semidirect
25. C2×#14555499C22semidirect
27. #17984094C22semidirect
28. #17984088C22semidirect
30. #17984024C22semidirect
31. #17984016C22semidirect
32. #17984006C22semidirect
33. #17984004C22semidirect
35. #131×C137C22semidirect
36. #108×C137C22semidirect
37. #104×C137C22semidirect
38. C4×C8×C137C22semidirect
54. Dic548D4semidirect
62. #14555499D4semidirect
66. D1096D4semidirect
73. Q16×C137D4semidirect
74. SD16×C137D4semidirect
75. D8×C137D4semidirect
130. C137#3291semidirect
140. #443D137semidirect
160. #131C2×D137semidirect
164. #108C2×D137semidirect
166. #104C2×D137semidirect
169. C4×C8C2×D137semidirect
188. Q16D548semidirect
198. SD16D548semidirect
202. D8D548semidirect

Maximal subgroups

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

subgroupindex
#225782152
#225782102
#225777762
#225776672
#225776662
#225775872
#225775722
#225775212
#443×C1372
#3291137