Finite Group Databasebeta

← #44486516

Subgroups — #44486516

Order 30592.

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

normal subgroups only

This group has 216 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
2C2×#21439903C211yesmaximal, characteristic, complemented
3#29505044C211yesmaximal, characteristic, complemented
4#29504586C211yesmaximal, characteristic, complemented
5#29504579C211yesmaximal, characteristic, complemented
6#29504557C211yesmaximal, characteristic, complemented
7#510×C239C211yesmaximal, characteristic, complemented
8#21440006—21no—
9#21439991—21no—
10#21439987—22no—
11#21439985—21no—
12#21439983—21no—
13C2×D1912C2211yescharacteristic
14D8×D239—21no—
15#21439966C2211yescharacteristic
16#21439903—22no—
17C2×C239⋊C16C2211yescharacteristic, complemented
18#21439896C2211yescharacteristic, complemented
19#21439879—21no—
20#135×C239C2211yescharacteristic
21C2×Q16×C239C2211yescharacteristic, complemented
22Q32×C239—21no—
23SD32×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#16838538—41no—
33#16838535—22no—
34#16838535—41no—
35D1912—21no—
36D1912D412yes—
37#16838533—22no—
38#16838532—21no—
39#16838532D411yescharacteristic
40#16838531—21no—
41C8×D239—21no—
42C239⋊C16—21no—
43#41×C239—21no—
44C2×Q8×C239—21no—
45Q16×C239—21no—
46Q16×C239—21no—
47Q16×C239D411yescharacteristic
48SD16×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—
65Q8×C239—21no—
66Q8×C239—41no—
67Q8×C239—41no—
68D4×C239—21no—
69D4×C239—41no—
70C2×C4×C239—41no—
71C2×C4×C239—41no—
72C2×C4×C239C2×D411yescharacteristic
73C8×C239C2×D411yescharacteristic
74C8×C239C2×D411yescharacteristic
75C22×C239—41no—
76C22×C239AGL(1,ℤ/8)11yescharacteristic
77C2×D239—21no—
78C2×D239—42no—
79C2×D239—162no—
80C2×D239—161no—
81C4×C239—41no—
82C4×C239—81no—
83C4×C239#12011yescharacteristic
84C4×C239C2×D811yescharacteristic
85Dic239—21no—
86C2×C239—21no—
87C2×C239—81no—
88C2×C239#44711yescharacteristic
89D239—41no—
90D239—82no—
91C239#219411yescharacteristic, complemented, Sylow
92#2194—2391nomaximal, Sylow
93#576—2391no—
94#510D23911yescharacteristic, complemented
95C2×SD32—2391no—
96#473—2391no—
97#361—2391no—
98#359—2391no—
99#350—2391no—
100H(5,𝔽2)—4781no—
101#137—4781no—
102AGL(1,ℤ/8)—4782no—
103AGL(1,ℤ/8)—4781no—
104#135—4781no—
105#135C2×D23911yescharacteristic
106C2×Q16C2×D23911yescharacteristic, complemented
107C2×D8—4781no—
108C2×D8—2391no—
109#131—2391no—
110Q32—21no—
111SD32—21no—
112SD32—4782no—
113M32C2×D23911yescharacteristic, complemented
114C2×C16—2391no—
115#108—2391no—
116#101—4781no—
117#41—21no—
118#41—9561no—
119#41—4782no—
120#41—2391no—
121C2×Q8—21no—
122C2×D4—9561no—
123C2×D4—9562no—
124C2×D4—4782no—
125C2×D4—4781no—
126Q16—21no—
127Q16—21no—
128Q16—9561no—
129Q16C22⋊D23911yescharacteristic
130SD16—21no—
131SD16—9561no—
132SD16—4782no—
133D8—4782no—
134D8—4781no—
135D8—2392no—
136D8C22⋊D23911yescharacteristic
137M16—4781no—
138M16—4781no—
139M16—2391no—
140C2×C8—4781no—
141C2×C8C22×D23911yescharacteristic
142C16—21no—
143C16—4781no—
144C23—9562no—
145C23—9561no—
146Q8—21no—
147Q8—41no—
148Q8—41no—
149Q8—2391no—
150D4—21no—
151D4—41no—
152D4—9562no—
153D4—9562no—
154D4—9561no—
155D4—4781no—
156D4—4782no—
157D4—2391no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—9562no—
161C2×C4—4781no—
162C2×C4D4×D23911yescharacteristic
163C8—9561no—
164C8—4781no—
165C8D4×D23911yescharacteristic
166C8C2×C22⋊D23911yescharacteristic
167C22—41no—
168C22—9562no—
169C22—4781no—
170C22—38241no—
171C22—38242no—
172C22#2143998411yescharacteristic
173C4—41no—
174C4—81no—
175C4—4781no—
176C4#2143992811yescharacteristic
177C4D8×D23911yescharacteristic
178C2—21no—
179C2—81no—
180C2—19122no—
181C2—9561no—
182C2#2950490411yescharacteristic
subgroupG/H
centerC2#29504904
commutator, cosocleC2×C8×C239C23
FrattiniC2×C8C22×D239
Fitting#510×C239C2
socleC2×C239#447
prime radical#29505574C2
hypercenter#510D239
hypocenterC239#2194
LCSG ▷ C2×C8×C239 ▷ C4×C239 ▷ C2×C239 ▷ C239
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #510

Complements

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

NH ≅ G/Ntype
2. C2×#21439903C2semidirect
3. #29505044C2semidirect
4. #29504586C2semidirect
5. #29504579C2semidirect
6. #29504557C2semidirect
7. #510×C239C2semidirect
17. C2×C239⋊C16C22semidirect
18. #21439896C22semidirect
21. C2×Q16×C239C22semidirect
24. M32×C239C22semidirect
91. C239#2194semidirect
94. #510D239semidirect
106. C2×Q16C2×D239semidirect
113. M32C2×D239semidirect

Maximal subgroups

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

subgroupindex
#295055742
C2×#214399032
#295050442
#295045862
#295045792
#295045572
#510×C2392
#2194239