Finite Group Databasebeta

← #44486502

Subgroups — #44486502

Order 30592.

220 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 47 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#29505574C211yesmaximal, characteristic, complemented
2C2×#21439901C211yesmaximal, characteristic, complemented
3#29505039C211yesmaximal, characteristic, complemented
4#29504586C211yesmaximal, characteristic, complemented
5#29504579C211yesmaximal, characteristic, complemented
6#29504557C211yesmaximal, characteristic, complemented
7#509×C239C211yesmaximal, characteristic, complemented
8#21440006—21no—
9#21439987—22no—
10#21439986—21no—
11#21439985—21no—
12#21439983—21no—
13C2×D1912C2211yescharacteristic
14D8×D239—21no—
15#21439966C2211yescharacteristic
16#21439901—22no—
17C2×C239⋊C16C2211yescharacteristic, complemented
18#21439896C2211yescharacteristic, complemented
19#21439877—21no—
20#135×C239C2211yescharacteristic
21C2×D8×C239C2211yescharacteristic, complemented
22SD32×C239—21no—
23D16×C239—21no—
24M32×C239C2211yescharacteristic, complemented
25#16838549—22no—
26#16838548—41no—
27#16838548D411yescharacteristic
28C2×D956—21no—
29C2×D956—41no—
30D4×D239—22no—
31D4×D239—42no—
32#16838536—41no—
33#16838535—22no—
34D1912—21no—
35D1912D412yes—
36#16838533—22no—
37#16838533—41no—
38#16838532—21no—
39#16838532D411yescharacteristic
40#16838531—21no—
41C8×D239—21no—
42C239⋊C16—21no—
43#41×C239—21no—
44C2×D4×C239—21no—
45Q16×C239D411yescharacteristic
46SD16×C239—21no—
47D8×C239—21no—
48D8×C239—21no—
49D8×C239D411yescharacteristic
50C2×C8×C239C2311yescharacteristic
51C16×C239—21no—
52C22×D239—42no—
53C22×D239—41no—
54Dic478D811yescharacteristic
55D956—22no—
56D956—42no—
57D956—41no—
58D956D811yescharacteristic
59C22⋊D239—21no—
60C22⋊D239—42no—
61C4×D239—21no—
62C4×D239—42no—
63C239⋊C8—21no—
64C239⋊C8—41no—
65C23×C239—41no—
66Q8×C239—21no—
67D4×C239—21no—
68D4×C239—41no—
69D4×C239—41no—
70D4×C239—41no—
71C2×C4×C239—41no—
72C2×C4×C239C2×D411yescharacteristic
73C8×C239C2×D411yescharacteristic
74C8×C239C2×D411yescharacteristic
75C22×C239—41no—
76C22×C239—81no—
77C22×C239—161no—
78C22×C239AGL(1,ℤ/8)11yescharacteristic
79C2×D239—21no—
80C2×D239—42no—
81C2×D239—162no—
82C2×D239—161no—
83C4×C239—41no—
84C4×C239#12011yescharacteristic
85C4×C239C2×D811yescharacteristic
86Dic239—21no—
87C2×C239—21no—
88C2×C239—81no—
89C2×C239—161no—
90C2×C239#44711yescharacteristic
91D239—41no—
92D239—82no—
93C239#219211yescharacteristic, complemented, Sylow
94#2192—2391nomaximal, Sylow
95#576—2391no—
96#509D23911yescharacteristic, complemented
97C2×D16—2391no—
98#472—2391no—
99#361—2391no—
100#359—2391no—
101#350—2391no—
102H(5,𝔽2)—4781no—
103AGL(1,ℤ/8)—4781no—
104AGL(1,ℤ/8)—4782no—
105AGL(1,ℤ/8)—4781no—
106#135—4781no—
107#135C2×D23911yescharacteristic
108C2×D8—4781no—
109C2×D8—2391no—
110C2×D8C2×D23911yescharacteristic, complemented
111#131—2391no—
112SD32—21no—
113D16—21no—
114D16—4782no—
115M32C2×D23911yescharacteristic, complemented
116C2×C16—2391no—
117#108—2391no—
118#100—4781no—
119#41—21no—
120#41—9561no—
121#41—4782no—
122#41—2391no—
123C2×D4—21no—
124C2×D4—9561no—
125C2×D4—9562no—
126C2×D4—4782no—
127C2×D4—4781no—
128Q16C22⋊D23911yescharacteristic
129SD16—21no—
130SD16—9561no—
131SD16—4782no—
132D8—21no—
133D8—21no—
134D8—9561no—
135D8—4782no—
136D8—4781no—
137D8—2392no—
138D8C22⋊D23911yescharacteristic
139M16—4781no—
140M16—4781no—
141M16—2391no—
142C2×C8—4781no—
143C2×C8C22×D23911yescharacteristic
144C16—21no—
145C16—4781no—
146C23—41no—
147C23—9562no—
148C23—9561no—
149Q8—21no—
150Q8—2391no—
151D4—21no—
152D4—41no—
153D4—41no—
154D4—41no—
155D4—9562no—
156D4—9562no—
157D4—9561no—
158D4—4781no—
159D4—4782no—
160D4—2391no—
161C2×C4—41no—
162C2×C4—9562no—
163C2×C4—4781no—
164C2×C4D4×D23911yescharacteristic
165C8—9561no—
166C8—4781no—
167C8D4×D23911yescharacteristic
168C8C2×C22⋊D23911yescharacteristic
169C22—41no—
170C22—81no—
171C22—161no—
172C22—9562no—
173C22—4781no—
174C22—38241no—
175C22—38242no—
176C22#2143998411yescharacteristic
177C4—41no—
178C4—4781no—
179C4#2143992811yescharacteristic
180C4D8×D23911yescharacteristic
181C2—21no—
182C2—81no—
183C2—161no—
184C2—19122no—
185C2—9561no—
186C2#2950490411yescharacteristic
subgroupG/H
centerC2#29504904
commutator, cosocleC2×C8×C239C23
FrattiniC2×C8C22×D239
Fitting#509×C239C2
socleC2×C239#447
hypercenter#509D239
hypocenterC239#2192
LCSG ▷ C2×C8×C239 ▷ C4×C239 ▷ C2×C239 ▷ C239
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #29505574C2semidirect
2. C2×#21439901C2semidirect
3. #29505039C2semidirect
4. #29504586C2semidirect
5. #29504579C2semidirect
6. #29504557C2semidirect
7. #509×C239C2semidirect
17. C2×C239⋊C16C22semidirect
18. #21439896C22semidirect
21. C2×D8×C239C22semidirect
24. M32×C239C22semidirect
93. C239#2192semidirect
96. #509D239semidirect
110. C2×D8C2×D239semidirect
115. M32C2×D239semidirect

Maximal subgroups

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

subgroupindex
#295055742
C2×#214399012
#295050392
#295045862
#295045792
#295045572
#509×C2392
#2192239