Finite Group Databasebeta

← #490×D137 #31791121

Subgroups — #490×D137 #31791121

Order 17536.

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

normal subgroups only

This group has 488 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#127×D137C211yesmaximal, characteristic
2#125×D137C211yesmaximal, characteristic, complemented
3#22577769C211yesmaximal, characteristic, complemented
4#22577768C211yesmaximal, characteristic, complemented
5#22577767C211yesmaximal, characteristic, complemented
6#22577766C214yesmaximal, complemented
7#102×D137C214yesmaximal, complemented
8#97×D137C211yesmaximal, characteristic, complemented
9#490×C137C211yesmaximal, characteristic, complemented
10C2×D4×D137—22no—
11C2×D4×D137C2212yes—
12C2×C8×D137C2212yes—
13#17984085C2211yescharacteristic
14#17984084—22no—
15#17984084C2212yes—
16#17984080C2211yescharacteristic, complemented
17#17984079C2212yescomplemented
18#17984078—22no—
19C4⋊C4×D137—22no—
20C4⋊C4×D137C2212yes—
21C42×D137C2211yescharacteristic
22#17984010C2214yescomplemented
23#17984009C2214yescomplemented
24#17984008C2214yescomplemented
25#17983999C2211yescharacteristic, complemented
26#17983998C2212yescomplemented
27#127×C137C2211yescharacteristic
28#125×C137C2211yescharacteristic, complemented
29#102×C137C2214yescomplemented
30#97×C137C2211yescharacteristic, complemented
31C23×D137—22no—
32C2×D548—22no—
33C2×D548C2312yes—
34C2×C22⋊D137—22no—
35C2×C22⋊D137—24no—
36D4×D137—28no—
37D4×D137—48no—
38C2×C4×D137—22no—
39C2×C4×D137D412yes—
40C2×C4×D137C2311yescharacteristic
41C2×#11764040C2312yes—
42C8×D137—24no—
43C4⋊Dic137—22no—
44C4⋊Dic137C2312yes—
45#14555489—22no—
46#14555489—22no—
47#14555489—22no—
48C4×Dic137D412yes—
49C4×Dic137C2311yescharacteristic
50C2×D4×C137—22no—
51C2×D4×C137C2312yes—
52C2×C8×C137C2312yes—
53C4⋊C4×C137—22no—
54C4⋊C4×C137C2312yes—
55C42×C137C2311yescharacteristic
56C22×D137—22no—
57C22×D137—44no—
58C22×D137—48no—
59C22×D137C2×D411yescharacteristic
60D548—24no—
61D548—44no—
62C22⋊D137—48no—
63C22⋊D137—44no—
64C2×Dic137—22no—
65C2×Dic137C2×D412yes—
66C2×Dic137C2×D411yescharacteristic
67C4×D137—24no—
68C4×D137—44no—
69C4×D137#4114yes—
70#11764040—22no—
71C23×C137—22no—
72D4×C137—24no—
73D4×C137—44no—
74C2×C4×C137—22no—
75C2×C4×C137C2×D412yes—
76C2×C4×C137C2411yescharacteristic
77C8×C137—22no—
78C22×C137—42no—
79C22×C137—44no—
80C22×C137C22×D411yescharacteristic
81C2×D137—42no—
82C2×D137—44no—
83C2×D137—88no—
84C2×D137#12412yes—
85C2×D137AGL(1,ℤ/8)14yes—
86C4×C137—22no—
87C4×C137—42no—
88C4×C137C2×#4112yes—
89Dic137—22no—
90Dic137—42no—
91Dic137#12412yes—
92C2×C137—82no—
93C2×C137C2×#12411yescharacteristic
94C2×C137C2×AGL(1,ℤ/8)12yes—
95D137—82no—
96D137#49014yesdirect product factor
97C137C2×#49011yescharacteristic, complemented, Sylow
98C2×#490—1371nomaximal, Sylow
99C2×#127—1371no—
100C2×#125—1371no—
101#490—1371no—
102#490—1371no—
103#490—1374no—
104#490—1371no—
105#490D13711yescharacteristic, direct product factor
106C2×#102—1374no—
107C2×#97—1371no—
108C22×D4—2742no—
109C22×D4—1372no—
110C22×C8—1372no—
111#127—2742no—
112#127—1372no—
113#127—1371no—
114#127C2×D13711yescharacteristic
115#125—2742no—
116#125—1372no—
117#125—1371no—
118#125C2×D13711yescharacteristic, complemented
119C2×C4⋊C4—2742no—
120C2×C4⋊C4—1372no—
121C2×C42—1371no—
122#102—1374no—
123#102—1374no—
124#102—1374no—
125#102C2×D13714yescomplemented
126#97—1372no—
127#97—1371no—
128#97C2×D13711yescharacteristic, complemented
129C24—2742no—
130C2×D4—22no—
131C2×D4—2748no—
132C2×D4—2742no—
133C2×D4—2744no—
134C2×D4—2742no—
135C2×D4—5488no—
136C2×D4—1372no—
137C2×D4C22×D13712yes—
138C22×C4—2742no—
139C22×C4—1372no—
140C22×C4—1371no—
141C2×C8—2744no—
142C2×C8—1372no—
143C2×C8C22×D13712yes—
144C4⋊C4—22no—
145C4⋊C4—2742no—
146C4⋊C4—2742no—
147C4⋊C4—2742no—
148C4⋊C4—2742no—
149C4⋊C4—1372no—
150C4⋊C4C22×D13712yes—
151C42—1372no—
152C42—1371no—
153C42C22×D13711yescharacteristic
154C23—22no—
155C23—2742no—
156C23—5484no—
157C23—5488no—
158C23—1371no—
159D4—24no—
160D4—44no—
161D4—2744no—
162D4—5484no—
163D4—5488no—
164D4—5484no—
165C2×C4—22no—
166C2×C4—2744no—
167C2×C4—2742no—
168C2×C4—5484no—
169C2×C4—1372no—
170C2×C4—1371no—
171C2×C4—1374no—
172C2×C4D4×D13712yes—
173C2×C4C23×D13711yescharacteristic
174C8—22no—
175C8—2742no—
176C22—44no—
177C22—42no—
178C22—5484no—
179C22—5482no—
180C22—10968no—
181C22—1374no—
182C22—1372no—
183C22C2×D4×D13711yescharacteristic
184C4—22no—
185C4—42no—
186C4—2742no—
187C4—5482no—
188C4—1372no—
189C4#41×D13712yes—
190C2—82no—
191C2—10962no—
192C2—1374no—
193C2#124×D13711yescharacteristic
194C2AGL(1,ℤ/8)×D13712yes—
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#490×C137C2
socleC22×C137C22×D4
prime radical#127×D137C2
hypercenter#490D137
hypocenterC137C2×#490
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
2. #125×D137C2semidirect
3. #22577769C2semidirect
4. #22577768C2semidirect
5. #22577767C2semidirect
6. #22577766C2semidirect
7. #102×D137C2semidirect
8. #97×D137C2semidirect
9. #490×C137C2semidirect
16. #17984080C22semidirect
17. #17984079C22semidirect
22. #17984010C22semidirect
23. #17984009C22semidirect
24. #17984008C22semidirect
25. #17983999C22semidirect
26. #17983998C22semidirect
28. #125×C137C22semidirect
29. #102×C137C22semidirect
30. #97×C137C22semidirect
96. D137#490direct (N × H)
97. C137C2×#490semidirect
105. #490D137direct (N × H)
118. #125C2×D137semidirect
125. #102C2×D137semidirect
128. #97C2×D137semidirect

Maximal subgroups

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

subgroupindex
#127×D1372
#125×D1372
#225777692
#225777682
#225777672
#225777662
#102×D1372
#97×D1372
#490×C1372
C2×#490137