Finite Group Databasebeta

← #31793922

Subgroups — #31793922

Order 17536.

364 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 113 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#135×D137C212yesmaximal, complemented
2#22577845C212yesmaximal, complemented
3#22577842C212yesmaximal, complemented
4#22577840C212yesmaximal, complemented
5#22577837C212yesmaximal, complemented
6#22577835C211yesmaximal, characteristic, complemented
7#22577833C212yesmaximal, complemented
8C2×#17984027C211yesmaximal, characteristic, complemented
9#508×C137C211yesmaximal, characteristic, complemented
10#41×D137—22no—
11#17984111C2212yes—
12#17984110C2212yescomplemented
13#17984109—22no—
14#17984108—22no—
15#17984107C2212yes—
16Q16×D137C2212yescomplemented
17SD16×D137—22no—
18D8×D137C2212yescomplemented
19C2×C8×D137C2211yescharacteristic
20Dic1096C2211yescharacteristic, complemented
21#17984034C2212yescomplemented
22#17984033C2212yescomplemented
23#17984032C2212yescomplemented
24#17984031C2212yescomplemented
25D2192C2211yescharacteristic, complemented
26#17984029C2212yescomplemented
27#17984027C2212yescomplemented
28#17984027C2212yescomplemented
29C2×#14555485C2211yescharacteristic, complemented
30#135×C137C2212yescomplemented
31Q32×C137C2211yescharacteristic, complemented
32SD32×C137C2212yescomplemented
33D16×C137C2211yescharacteristic, complemented
34C2×C16×C137C2211yescharacteristic, complemented
35#14555511—22no—
36#14555510—22no—
37#14555510—42no—
38#14555509—22no—
39#14555509—42no—
40Q8×D137—22no—
41D4×D137—22no—
42D4×D137—42no—
43C2×C4×D137—42no—
44C2×C4×D137D411yescharacteristic
45Dic548C2312yes—
46#14555500—22no—
47#14555499—22no—
48#14555498—22no—
49#14555497—22no—
50D1096C2312yes—
51#14555495—22no—
52C2×#11764040D411yescharacteristic
53C8×D137D412yes—
54C8×D137C2312yes—
55#14555485C2312yes—
56#41×C137—22no—
57Q16×C137C2312yes—
58SD16×C137—22no—
59D8×C137C2312yes—
60C2×C8×C137C2311yescharacteristic
61C16×C137C2311yescharacteristic
62C16×C137C2311yescharacteristic
63C22×D137—42no—
64C22×D137D811yescharacteristic
65Dic274—22no—
66Dic274—42no—
67D548—22no—
68D548—42no—
69C22⋊D137—42no—
70C22⋊D137—42no—
71C22⋊D137—42no—
72C2×Dic137—42no—
73C2×Dic137D811yescharacteristic
74C4×D137—42no—
75C4×D137—42no—
76C4×D137—42no—
77C4×D137D812yes—
78C4×D137C2×D412yes—
79#11764040C2×D412yes—
80Q8×C137—22no—
81D4×C137—22no—
82D4×C137—42no—
83C2×C4×C137—42no—
84C2×C4×C137C2×D411yescharacteristic
85C8×C137C2×D411yescharacteristic
86C8×C137C2411yescharacteristic
87C22×C137—42no—
88C22×C137C2×D811yescharacteristic
89C2×D137—41no—
90C2×D137—42no—
91C2×D137—84no—
92C2×D137C2×D812yes—
93C4×C137—42no—
94C4×C137C2×D811yescharacteristic
95C4×C137C22×D411yescharacteristic
96Dic137—42no—
97Dic137C2×D812yes—
98C2×C137—21no—
99C2×C137—82no—
100C2×C137C22×D811yescharacteristic
101D137—22no—
102D137—82no—
103C137#341411yescharacteristic, complemented, Sylow
104#3414—1371nomaximal, Sylow
105C2×#135—1372no—
106#510—1372no—
107#510—1372no—
108#509—1372no—
109#509—1372no—
110#508—1372no—
111#508—1371no—
112#508D13711yescharacteristic, complemented
113C2×M32—1371no—
114C2×#41—2742no—
115#135—2742no—
116#135—2742no—
117#135—1372no—
118#135—1372no—
119#135—1372no—
120#135C2×D13712yescomplemented
121C2×Q16—1372no—
122C2×SD16—2742no—
123C2×D8—1372no—
124C22×C8—1371no—
125Q32—1372no—
126Q32—1371no—
127Q32C2×D13711yescharacteristic, complemented
128SD32—1372no—
129SD32—1372no—
130SD32—1372no—
131SD32C2×D13712yescomplemented
132D16—1372no—
133D16—1371no—
134D16C2×D13711yescharacteristic, complemented
135M32—1372no—
136M32—1372no—
137C2×C16—1371no—
138C2×C16C2×D13711yescharacteristic, complemented
139#41—22no—
140#41—2742no—
141#41—2742no—
142#41—2742no—
143#41—5482no—
144#41—5482no—
145C2×Q8—2742no—
146C2×D4—2742no—
147C2×D4—5482no—
148C22×C4—5482no—
149C22×C4—1371no—
150Q16—2742no—
151Q16—1372no—
152Q16C22×D13712yes—
153SD16—22no—
154SD16—2742no—
155SD16—2742no—
156SD16—2742no—
157D8—2742no—
158D8—1372no—
159D8C22×D13712yes—
160C2×C8—1371no—
161C2×C8—1372no—
162C2×C8—1372no—
163C2×C8C22×D13711yescharacteristic
164C16—1372no—
165C16C22×D13711yescharacteristic
166C16C22×D13711yescharacteristic
167C23—5482no—
168C23—1371no—
169Q8—22no—
170Q8—2742no—
171Q8—5482no—
172D4—22no—
173D4—42no—
174D4—2742no—
175D4—5482no—
176D4—5482no—
177D4—5482no—
178D4—5482no—
179C2×C4—42no—
180C2×C4—5482no—
181C2×C4—5482no—
182C2×C4—5482no—
183C2×C4—5482no—
184C2×C4—1371no—
185C2×C4—1372no—
186C2×C4—1372no—
187C2×C4D4×D13711yescharacteristic
188C8—1372no—
189C8D4×D13711yescharacteristic
190C8C23×D13711yescharacteristic
191C22—42no—
192C22—5481no—
193C22—5482no—
194C22—10964no—
195C22—1372no—
196C22D8×D13711yescharacteristic
197C4—42no—
198C4—5482no—
199C4—1372no—
200C4D8×D13711yescharacteristic
201C4C2×D4×D13711yescharacteristic
202C2—21no—
203C2—82no—
204C2—2742no—
205C2—10962no—
206C2C2×D8×D13711yescharacteristic
subgroupG/H
centerC4D8×D137
commutator, cosocleC8×C137C24
FrattiniC8C23×D137
Fitting#508×C137C2
socleC2×C137C22×D8
hypercenter#508D137
hypocenterC137#3414
LCSG ▷ C8×C137 ▷ C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. #135×D137C2semidirect
2. #22577845C2semidirect
3. #22577842C2semidirect
4. #22577840C2semidirect
5. #22577837C2semidirect
6. #22577835C2semidirect
7. #22577833C2semidirect
8. C2×#17984027C2semidirect
9. #508×C137C2semidirect
12. #17984110C22semidirect
16. Q16×D137C22semidirect
18. D8×D137C22semidirect
20. Dic1096C22semidirect
21. #17984034C22semidirect
22. #17984033C22semidirect
23. #17984032C22semidirect
24. #17984031C22semidirect
25. D2192C22semidirect
26. #17984029C22semidirect
27. #17984027C22semidirect
28. #17984027C22semidirect
29. C2×#14555485C22semidirect
30. #135×C137C22semidirect
31. Q32×C137C22semidirect
32. SD32×C137C22semidirect
33. D16×C137C22semidirect
34. C2×C16×C137C22semidirect
103. C137#3414semidirect
112. #508D137semidirect
120. #135C2×D137semidirect
127. Q32C2×D137semidirect
131. SD32C2×D137semidirect
134. D16C2×D137semidirect
138. C2×C16C2×D137semidirect

Maximal subgroups

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

subgroupindex
#135×D1372
#225778452
#225778422
#225778402
#225778372
#225778352
#225778332
C2×#179840272
#508×C1372
#3414137