Finite Group Databasebeta

← #31789699

Subgroups — #31789699

Order 17536.

458 conjugacy classes of subgroups; only the 217 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 217 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#22578170C212yesmaximal, complemented
2#22577525C214yesmaximal, complemented
3#22577520C214yesmaximal, complemented
4#22577452C211yesmaximal, characteristic, complemented
5C2×#17984015C212yesmaximal, complemented
6M16×Dic137C211yesmaximal, characteristic, complemented
7#575×C137C211yesmaximal, characteristic
8#17984095—24no—
9#17984095C2218yescomplemented
10C2×#14555494—24no—
11C2×#14555494C2212yes—
12C2×#14555494C2211yescharacteristic
13C2×C4×Dic137C2211yescharacteristic
14#17984024C2214yescomplemented
15#17984015C2214yescomplemented
16#17984015C2214yescomplemented
17#17983998C2212yescomplemented
18C8×Dic137C2212yescomplemented
19C2×#41×C137C2212yes—
20#137×C137C414yescomplemented
21AGL(1,ℤ/8)×C137C414yescomplemented
22#135×C137C2214yes—
23C2×M16×C137C2211yescharacteristic
24C22×Dic137—21no—
25#14555494—28no—
26#14555494—22no—
27#14555494D414yes—
28#14555494C2314yes—
29C2×#11764040—28no—
30C2×#11764040—22no—
31C2×#11764040C2312yes—
32C4×Dic137—21no—
33C4×Dic137C2311yescharacteristic
34C4×Dic137C2311yescharacteristic
35#41×C137—22no—
36#41×C137—22no—
37#41×C137C2×C414yes—
38#41×C137C2314yes—
39C2×Q8×C137C2×C412yes—
40C2×D4×C137—22no—
41C2×D4×C137C2×C412yes—
42C22×C4×C137—22no—
43C22×C4×C137C2311yescharacteristic
44Q16×C137C2×C414yescomplemented
45SD16×C137C2×C418yescomplemented
46D8×C137C2×C414yescomplemented
47M16×C137C2×C414yes—
48C2×C8×C137C2312yes—
49C2×Dic137—21no—
50C2×Dic137—21no—
51C2×Dic137—42no—
52#11764040—24no—
53#11764040C2×D414yes—
54C23×C137—22no—
55C23×C137#4111yescharacteristic
56Q8×C137—22no—
57Q8×C137C22×C414yes—
58D4×C137—24no—
59D4×C137—24no—
60D4×C137—22no—
61D4×C137C22×C414yes—
62C2×C4×C137—22no—
63C2×C4×C137—24no—
64C2×C4×C137—24no—
65C2×C4×C137C22×C412yes—
66C2×C4×C137C2×D412yes—
67C2×C4×C137#4111yescharacteristic
68C2×C4×C137C2411yescharacteristic
69C8×C137C22×C414yes—
70C22×C137—24no—
71C22×C137—41no—
72C22×C137—82no—
73C22×C137C4×D412yes—
74C22×C137C2×#4111yescharacteristic
75C4×C137—24no—
76C4×C137C4×D412yes—
77C4×C137C23×C411yescharacteristic
78C4×C137C22×D411yescharacteristic
79Dic137—22no—
80Dic137—41no—
81C2×C137—21no—
82C2×C137—22no—
83C2×C137—44no—
84C2×C137C2×C4×D411yescharacteristic
85C137#295611yescharacteristic, complemented, Sylow
86#2956—1371nomaximal, Sylow
87#575D13711yescharacteristic
88#568—1372no—
89#444—1374no—
90#443—1374no—
91#430—1371no—
92C2×#104—1372no—
93C4×M16—1371no—
94C2×#41C2×D13712yes—
95#137Dic13714yescomplemented
96AGL(1,ℤ/8)Dic13714yescomplemented
97#135C2×D13714yes—
98#131—2744no—
99#131—1378no—
100C2×M16—2744no—
101C2×M16—1372no—
102C2×M16—1371no—
103C2×M16C2×D13711yescharacteristic
104C2×C42—1371no—
105#108—1374no—
106#104—1374no—
107#104—1374no—
108#97—1372no—
109C4×C8—1372no—
110#41—22no—
111#41—22no—
112#41C2×Dic13714yes—
113#41C22×D13714yes—
114C2×Q8C2×Dic13712yes—
115C2×D4—22no—
116C2×D4C2×Dic13712yes—
117C22×C4—22no—
118C22×C4—2741no—
119C22×C4C22×D13711yescharacteristic
120Q16C2×Dic13714yescomplemented
121SD16C2×Dic13718yescomplemented
122D8C2×Dic13714yescomplemented
123M16—2742no—
124M16—2748no—
125M16—1374no—
126M16—1374no—
127M16C2×Dic13714yes—
128C2×C8—2742no—
129C2×C8—2748no—
130C2×C8—1372no—
131C2×C8C22×D13712yes—
132C42—2741no—
133C42—1371no—
134C42—1371no—
135C23—22no—
136C23#1455550911yescharacteristic
137Q8—22no—
138Q8C22×Dic13714yes—
139D4—24no—
140D4—24no—
141D4—22no—
142D4C22×Dic13714yes—
143C2×C4—22no—
144C2×C4—24no—
145C2×C4—24no—
146C2×C4—2741no—
147C2×C4—2741no—
148C2×C4—5482no—
149C2×C4C22×Dic13712yes—
150C2×C4D4×D13712yes—
151C2×C4#1455550911yescharacteristic
152C2×C4C23×D13711yescharacteristic
153C8—2744no—
154C8—1374no—
155C8C22×Dic13714yes—
156C22—24no—
157C22—41no—
158C22—82no—
159C22D4×Dic13712yes—
160C22C2×#1455550911yescharacteristic
161C4—24no—
162C4—2742no—
163C4—5481no—
164C4D4×Dic13712yes—
165C4C23×Dic13711yescharacteristic
166C4C2×D4×D13711yescharacteristic
167C2—21no—
168C2—22no—
169C2—44no—
170C2C2×D4×Dic13711yescharacteristic
subgroupG/H
centerC4C2×D4×D137
commutatorC4×C137C23×C4
FrattiniC2×C4C23×D137
Fitting, prime radical#575×C137C2
socleC2×C137C2×C4×D4
cosocleC2×C4×C137C24
hypercenter#575D137
hypocenterC137#2956
LCSG ▷ C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C4 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #22578170C2semidirect
2. #22577525C2semidirect
3. #22577520C2semidirect
4. #22577452C2semidirect
5. C2×#17984015C2semidirect
6. M16×Dic137C2semidirect
9. #17984095C22semidirect
14. #17984024C22semidirect
15. #17984015C22semidirect
16. #17984015C22semidirect
17. #17983998C22semidirect
18. C8×Dic137C22semidirect
20. #137×C137C4semidirect
21. AGL(1,ℤ/8)×C137C4semidirect
44. Q16×C137C2×C4semidirect
45. SD16×C137C2×C4semidirect
46. D8×C137C2×C4semidirect
85. C137#2956semidirect
95. #137Dic137semidirect
96. AGL(1,ℤ/8)Dic137semidirect
120. Q16C2×Dic137semidirect
121. SD16C2×Dic137semidirect
122. D8C2×Dic137semidirect

Maximal subgroups

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

subgroupindex
#225781702
#225775252
#225775202
#225774522
C2×#179840152
M16×Dic1372
#575×C1372
#2956137