Finite Group Databasebeta

← #490×D251 #44560697

Subgroups — #490×D251 #44560697

Order 32128.

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×D251C211yesmaximal, characteristic
2#125×D251C211yesmaximal, characteristic, complemented
3#30632053C211yesmaximal, characteristic, complemented
4#30632052C211yesmaximal, characteristic, complemented
5#30632051C211yesmaximal, characteristic, complemented
6#30632050C214yesmaximal, complemented
7#102×D251C214yesmaximal, complemented
8#97×D251C211yesmaximal, characteristic, complemented
9#490×C251C211yesmaximal, characteristic, complemented
10C2×D4×D251—22no—
11C2×D4×D251C2212yes—
12C2×C8×D251C2212yes—
13#22548712C2211yescharacteristic
14#22548711—22no—
15#22548711C2212yes—
16#22548707C2211yescharacteristic, complemented
17#22548706C2212yescomplemented
18#22548705—22no—
19C4⋊C4×D251—22no—
20C4⋊C4×D251C2212yes—
21C42×D251C2211yescharacteristic
22#22548637C2214yescomplemented
23#22548636C2214yescomplemented
24#22548635C2214yescomplemented
25#22548626C2211yescharacteristic, complemented
26#22548625C2212yescomplemented
27#127×C251C2211yescharacteristic
28#125×C251C2211yescharacteristic, complemented
29#102×C251C2214yescomplemented
30#97×C251C2211yescharacteristic, complemented
31C23×D251—22no—
32C2×D1004—22no—
33C2×D1004C2312yes—
34C2×C22⋊D251—24no—
35C2×C22⋊D251—22no—
36D4×D251—28no—
37D4×D251—48no—
38C2×C4×D251—22no—
39C2×C4×D251D412yes—
40C2×C4×D251C2311yescharacteristic
41C2×C251⋊C8C2312yes—
42C8×D251—24no—
43C4⋊Dic251—22no—
44C4⋊Dic251C2312yes—
45Dic251⋊C4—22no—
46Dic251⋊C4—22no—
47Dic251⋊C4—22no—
48C4×Dic251D412yes—
49C4×Dic251C2311yescharacteristic
50C2×D4×C251—22no—
51C2×D4×C251C2312yes—
52C2×C8×C251C2312yes—
53C4⋊C4×C251—22no—
54C4⋊C4×C251C2312yes—
55C42×C251C2311yescharacteristic
56C22×D251—22no—
57C22×D251—44no—
58C22×D251—48no—
59C22×D251C2×D411yescharacteristic
60D1004—24no—
61D1004—44no—
62C22⋊D251—48no—
63C22⋊D251—44no—
64C2×Dic251—22no—
65C2×Dic251C2×D412yes—
66C2×Dic251C2×D411yescharacteristic
67C4×D251—24no—
68C4×D251—44no—
69C4×D251#4114yes—
70C251⋊C8—22no—
71C23×C251—22no—
72D4×C251—24no—
73D4×C251—44no—
74C2×C4×C251—22no—
75C2×C4×C251C2×D412yes—
76C2×C4×C251C2411yescharacteristic
77C8×C251—22no—
78C22×C251—42no—
79C22×C251—44no—
80C22×C251C22×D411yescharacteristic
81C2×D251—42no—
82C2×D251—44no—
83C2×D251—88no—
84C2×D251#12412yes—
85C2×D251AGL(1,ℤ/8)14yes—
86C4×C251—22no—
87C4×C251—42no—
88C4×C251C2×#4112yes—
89Dic251—22no—
90Dic251—42no—
91Dic251#12412yes—
92C2×C251—82no—
93C2×C251C2×#12411yescharacteristic
94C2×C251C2×AGL(1,ℤ/8)12yes—
95D251—82no—
96D251#49014yesdirect product factor
97C251C2×#49011yescharacteristic, complemented, Sylow
98C2×#490—2511nomaximal, Sylow
99C2×#127—2511no—
100C2×#125—2511no—
101#490—2511no—
102#490—2511no—
103#490—2514no—
104#490—2511no—
105#490D25111yescharacteristic, direct product factor
106C2×#102—2514no—
107C2×#97—2511no—
108C22×D4—5022no—
109C22×D4—2512no—
110C22×C8—2512no—
111#127—5022no—
112#127—2512no—
113#127—2511no—
114#127C2×D25111yescharacteristic
115#125—5022no—
116#125—2512no—
117#125—2511no—
118#125C2×D25111yescharacteristic, complemented
119C2×C4⋊C4—5022no—
120C2×C4⋊C4—2512no—
121C2×C42—2511no—
122#102—2514no—
123#102—2514no—
124#102—2514no—
125#102C2×D25114yescomplemented
126#97—2512no—
127#97—2511no—
128#97C2×D25111yescharacteristic, complemented
129C24—5022no—
130C2×D4—22no—
131C2×D4—10048no—
132C2×D4—5028no—
133C2×D4—5022no—
134C2×D4—5024no—
135C2×D4—5022no—
136C2×D4—2512no—
137C2×D4C22×D25112yes—
138C22×C4—5022no—
139C22×C4—2512no—
140C22×C4—2511no—
141C2×C8—5024no—
142C2×C8—2512no—
143C2×C8C22×D25112yes—
144C4⋊C4—22no—
145C4⋊C4—5022no—
146C4⋊C4—5022no—
147C4⋊C4—5022no—
148C4⋊C4—5022no—
149C4⋊C4—2512no—
150C4⋊C4C22×D25112yes—
151C42—2512no—
152C42—2511no—
153C42C22×D25111yescharacteristic
154C23—22no—
155C23—10048no—
156C23—10044no—
157C23—5022no—
158C23—2511no—
159D4—24no—
160D4—44no—
161D4—10044no—
162D4—10048no—
163D4—10044no—
164D4—5024no—
165C2×C4—22no—
166C2×C4—10044no—
167C2×C4—5024no—
168C2×C4—5022no—
169C2×C4—2512no—
170C2×C4—2511no—
171C2×C4—2514no—
172C2×C4D4×D25112yes—
173C2×C4C23×D25111yescharacteristic
174C8—22no—
175C8—5022no—
176C22—42no—
177C22—44no—
178C22—20088no—
179C22—10042no—
180C22—10044no—
181C22—2512no—
182C22—2514no—
183C22C2×D4×D25111yescharacteristic
184C4—22no—
185C4—42no—
186C4—10042no—
187C4—5022no—
188C4—2512no—
189C4#41×D25112yes—
190C2—82no—
191C2—20082no—
192C2—2514no—
193C2#124×D25111yescharacteristic
194C2AGL(1,ℤ/8)×D25112yes—
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#490×C251C2
socleC22×C251C22×D4
prime radical#127×D251C2
hypercenter#490D251
hypocenterC251C2×#490
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
2. #125×D251C2semidirect
3. #30632053C2semidirect
4. #30632052C2semidirect
5. #30632051C2semidirect
6. #30632050C2semidirect
7. #102×D251C2semidirect
8. #97×D251C2semidirect
9. #490×C251C2semidirect
16. #22548707C22semidirect
17. #22548706C22semidirect
22. #22548637C22semidirect
23. #22548636C22semidirect
24. #22548635C22semidirect
25. #22548626C22semidirect
26. #22548625C22semidirect
28. #125×C251C22semidirect
29. #102×C251C22semidirect
30. #97×C251C22semidirect
96. D251#490direct (N × H)
97. C251C2×#490semidirect
105. #490D251direct (N × H)
118. #125C2×D251semidirect
125. #102C2×D251semidirect
128. #97C2×D251semidirect

Maximal subgroups

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

subgroupindex
#127×D2512
#125×D2512
#306320532
#306320522
#306320512
#306320502
#102×D2512
#97×D2512
#490×C2512
C2×#490251