Finite Group Databasebeta

← #49123215

Subgroups — #49123215

Order 36224.

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#31827267C211yesmaximal, characteristic, complemented
2C2×#22583551C211yesmaximal, characteristic, complemented
3#31826732C211yesmaximal, characteristic, complemented
4#31826279C211yesmaximal, characteristic, complemented
5#31826272C211yesmaximal, characteristic, complemented
6#31826250C211yesmaximal, characteristic, complemented
7#509×C283C211yesmaximal, characteristic, complemented
8#22583656—21no—
9#22583637—22no—
10#22583636—21no—
11#22583635—21no—
12#22583633—21no—
13C2×D2264C2211yescharacteristic
14D8×D283—21no—
15#22583616C2211yescharacteristic
16#22583551—22no—
17C2×C283⋊C16C2211yescharacteristic, complemented
18#22583546C2211yescharacteristic, complemented
19#22583527—21no—
20#135×C283C2211yescharacteristic
21C2×D8×C283C2211yescharacteristic, complemented
22SD32×C283—21no—
23D16×C283—21no—
24M32×C283C2211yescharacteristic, complemented
25#17985393—22no—
26#17985392—41no—
27#17985392D411yescharacteristic
28C2×D1132—21no—
29C2×D1132—41no—
30D4×D283—22no—
31D4×D283—42no—
32#17985380—41no—
33#17985379—22no—
34D2264—21no—
35D2264D412yes—
36#17985377—22no—
37#17985377—41no—
38#17985376—21no—
39#17985376D411yescharacteristic
40#17985375—21no—
41C8×D283—21no—
42C283⋊C16—21no—
43#41×C283—21no—
44C2×D4×C283—21no—
45Q16×C283D411yescharacteristic
46SD16×C283—21no—
47D8×C283—21no—
48D8×C283—21no—
49D8×C283D411yescharacteristic
50C2×C8×C283C2311yescharacteristic
51C16×C283—21no—
52C22×D283—42no—
53C22×D283—41no—
54Dic566D811yescharacteristic
55D1132—22no—
56D1132—42no—
57D1132—41no—
58D1132D811yescharacteristic
59C22⋊D283—21no—
60C22⋊D283—42no—
61C4×D283—21no—
62C4×D283—42no—
63C283⋊C8—21no—
64C283⋊C8—41no—
65C23×C283—41no—
66Q8×C283—21no—
67D4×C283—21no—
68D4×C283—41no—
69D4×C283—41no—
70D4×C283—41no—
71C2×C4×C283—41no—
72C2×C4×C283C2×D411yescharacteristic
73C8×C283C2×D411yescharacteristic
74C8×C283C2×D411yescharacteristic
75C22×C283—41no—
76C22×C283—81no—
77C22×C283—161no—
78C22×C283AGL(1,ℤ/8)11yescharacteristic
79C2×D283—21no—
80C2×D283—42no—
81C2×D283—162no—
82C2×D283—161no—
83C4×C283—41no—
84C4×C283#12011yescharacteristic
85C4×C283C2×D811yescharacteristic
86Dic283—21no—
87C2×C283—21no—
88C2×C283—81no—
89C2×C283—161no—
90C2×C283#44711yescharacteristic
91D283—41no—
92D283—82no—
93C283#219211yescharacteristic, complemented, Sylow
94#2192—2831nomaximal, Sylow
95#576—2831no—
96#509D28311yescharacteristic, complemented
97C2×D16—2831no—
98#472—2831no—
99#361—2831no—
100#359—2831no—
101#350—2831no—
102H(5,𝔽2)—5661no—
103AGL(1,ℤ/8)—5661no—
104AGL(1,ℤ/8)—5662no—
105AGL(1,ℤ/8)—5661no—
106#135—5661no—
107#135C2×D28311yescharacteristic
108C2×D8—2831no—
109C2×D8—5661no—
110C2×D8C2×D28311yescharacteristic, complemented
111#131—2831no—
112SD32—21no—
113D16—21no—
114D16—5662no—
115M32C2×D28311yescharacteristic, complemented
116C2×C16—2831no—
117#108—2831no—
118#100—5661no—
119#41—21no—
120#41—2831no—
121#41—5662no—
122#41—11321no—
123C2×D4—21no—
124C2×D4—5662no—
125C2×D4—5661no—
126C2×D4—11321no—
127C2×D4—11322no—
128Q16C22⋊D28311yescharacteristic
129SD16—21no—
130SD16—5662no—
131SD16—11321no—
132D8—21no—
133D8—21no—
134D8—2832no—
135D8—5662no—
136D8—5661no—
137D8—11321no—
138D8C22⋊D28311yescharacteristic
139M16—2831no—
140M16—5661no—
141M16—5661no—
142C2×C8—5661no—
143C2×C8C22×D28311yescharacteristic
144C16—21no—
145C16—5661no—
146C23—41no—
147C23—11322no—
148C23—11321no—
149Q8—21no—
150Q8—2831no—
151D4—21no—
152D4—41no—
153D4—41no—
154D4—41no—
155D4—2831no—
156D4—5661no—
157D4—5662no—
158D4—11322no—
159D4—11322no—
160D4—11321no—
161C2×C4—41no—
162C2×C4—5661no—
163C2×C4—11322no—
164C2×C4D4×D28311yescharacteristic
165C8—5661no—
166C8—11321no—
167C8D4×D28311yescharacteristic
168C8C2×C22⋊D28311yescharacteristic
169C22—41no—
170C22—81no—
171C22—161no—
172C22—5661no—
173C22—11322no—
174C22—45281no—
175C22—45282no—
176C22#2258363411yescharacteristic
177C4—41no—
178C4—5661no—
179C4#2258357811yescharacteristic
180C4D8×D28311yescharacteristic
181C2—21no—
182C2—81no—
183C2—161no—
184C2—11321no—
185C2—22642no—
186C2#3182659711yescharacteristic
subgroupG/H
centerC2#31826597
commutator, cosocleC2×C8×C283C23
FrattiniC2×C8C22×D283
Fitting#509×C283C2
socleC2×C283#447
hypercenter#509D283
hypocenterC283#2192
LCSG ▷ C2×C8×C283 ▷ C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #31827267C2semidirect
2. C2×#22583551C2semidirect
3. #31826732C2semidirect
4. #31826279C2semidirect
5. #31826272C2semidirect
6. #31826250C2semidirect
7. #509×C283C2semidirect
17. C2×C283⋊C16C22semidirect
18. #22583546C22semidirect
21. C2×D8×C283C22semidirect
24. M32×C283C22semidirect
93. C283#2192semidirect
96. #509D283semidirect
110. C2×D8C2×D283semidirect
115. M32C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318272672
C2×#225835512
#318267322
#318262792
#318262722
#318262502
#509×C2832
#2192283