Finite Group Databasebeta

← #49123229

Subgroups — #49123229

Order 36224.

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#31827267C211yesmaximal, characteristic
2C2×#22583553C211yesmaximal, characteristic, complemented
3#31826737C211yesmaximal, characteristic, complemented
4#31826279C211yesmaximal, characteristic, complemented
5#31826272C211yesmaximal, characteristic, complemented
6#31826250C211yesmaximal, characteristic, complemented
7#510×C283C211yesmaximal, characteristic, complemented
8#22583656—21no—
9#22583641—21no—
10#22583637—22no—
11#22583635—21no—
12#22583633—21no—
13C2×D2264C2211yescharacteristic
14D8×D283—21no—
15#22583616C2211yescharacteristic
16#22583553—22no—
17C2×C283⋊C16C2211yescharacteristic, complemented
18#22583546C2211yescharacteristic, complemented
19#22583529—21no—
20#135×C283C2211yescharacteristic
21C2×Q16×C283C2211yescharacteristic, complemented
22Q32×C283—21no—
23SD32×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#17985382—41no—
33#17985379—22no—
34#17985379—41no—
35D2264—21no—
36D2264D412yes—
37#17985377—22no—
38#17985376—21no—
39#17985376D411yescharacteristic
40#17985375—21no—
41C8×D283—21no—
42C283⋊C16—21no—
43#41×C283—21no—
44C2×Q8×C283—21no—
45Q16×C283—21no—
46Q16×C283—21no—
47Q16×C283D411yescharacteristic
48SD16×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—
65Q8×C283—21no—
66Q8×C283—41no—
67Q8×C283—41no—
68D4×C283—21no—
69D4×C283—41no—
70C2×C4×C283—41no—
71C2×C4×C283—41no—
72C2×C4×C283C2×D411yescharacteristic
73C8×C283C2×D411yescharacteristic
74C8×C283C2×D411yescharacteristic
75C22×C283—41no—
76C22×C283AGL(1,ℤ/8)11yescharacteristic
77C2×D283—21no—
78C2×D283—42no—
79C2×D283—162no—
80C2×D283—161no—
81C4×C283—41no—
82C4×C283—81no—
83C4×C283#12011yescharacteristic
84C4×C283C2×D811yescharacteristic
85Dic283—21no—
86C2×C283—21no—
87C2×C283—81no—
88C2×C283#44711yescharacteristic
89D283—41no—
90D283—82no—
91C283#219411yescharacteristic, complemented, Sylow
92#2194—2831nomaximal, Sylow
93#576—2831no—
94#510D28311yescharacteristic, complemented
95C2×SD32—2831no—
96#473—2831no—
97#361—2831no—
98#359—2831no—
99#350—2831no—
100H(5,𝔽2)—5661no—
101#137—5661no—
102AGL(1,ℤ/8)—5661no—
103AGL(1,ℤ/8)—5662no—
104#135—5661no—
105#135C2×D28311yescharacteristic
106C2×Q16C2×D28311yescharacteristic, complemented
107C2×D8—2831no—
108C2×D8—5661no—
109#131—2831no—
110Q32—21no—
111SD32—21no—
112SD32—5662no—
113M32C2×D28311yescharacteristic, complemented
114C2×C16—2831no—
115#108—2831no—
116#101—5661no—
117#41—21no—
118#41—2831no—
119#41—5662no—
120#41—11321no—
121C2×Q8—21no—
122C2×D4—5662no—
123C2×D4—5661no—
124C2×D4—11321no—
125C2×D4—11322no—
126Q16—21no—
127Q16—21no—
128Q16—11321no—
129Q16C22⋊D28311yescharacteristic
130SD16—21no—
131SD16—5662no—
132SD16—11321no—
133D8—2832no—
134D8—5662no—
135D8—5661no—
136D8C22⋊D28311yescharacteristic
137M16—2831no—
138M16—5661no—
139M16—5661no—
140C2×C8—5661no—
141C2×C8C22×D28311yescharacteristic
142C16—21no—
143C16—5661no—
144C23—11322no—
145C23—11321no—
146Q8—21no—
147Q8—41no—
148Q8—41no—
149Q8—2831no—
150D4—21no—
151D4—41no—
152D4—2831no—
153D4—5661no—
154D4—5662no—
155D4—11322no—
156D4—11322no—
157D4—11321no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—5661no—
161C2×C4—11322no—
162C2×C4D4×D28311yescharacteristic
163C8—5661no—
164C8—11321no—
165C8D4×D28311yescharacteristic
166C8C2×C22⋊D28311yescharacteristic
167C22—41no—
168C22—5661no—
169C22—11322no—
170C22—45281no—
171C22—45282no—
172C22#2258363411yescharacteristic
173C4—41no—
174C4—81no—
175C4—5661no—
176C4#2258357811yescharacteristic
177C4D8×D28311yescharacteristic
178C2—21no—
179C2—81no—
180C2—11321no—
181C2—22642no—
182C2#3182659711yescharacteristic
subgroupG/H
centerC2#31826597
commutator, cosocleC2×C8×C283C23
FrattiniC2×C8C22×D283
Fitting#510×C283C2
socleC2×C283#447
prime radical#31827267C2
hypercenter#510D283
hypocenterC283#2194
LCSG ▷ C2×C8×C283 ▷ C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #510

Complements

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

NH ≅ G/Ntype
2. C2×#22583553C2semidirect
3. #31826737C2semidirect
4. #31826279C2semidirect
5. #31826272C2semidirect
6. #31826250C2semidirect
7. #510×C283C2semidirect
17. C2×C283⋊C16C22semidirect
18. #22583546C22semidirect
21. C2×Q16×C283C22semidirect
24. M32×C283C22semidirect
91. C283#2194semidirect
94. #510D283semidirect
106. C2×Q16C2×D283semidirect
113. M32C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318272672
C2×#225835532
#318267372
#318262792
#318262722
#318262502
#510×C2832
#2194283