Finite Group Databasebeta

← #49123220

Subgroups — #49123220

Order 36224.

220 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 49 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#31827270C211yesmaximal, characteristic, complemented
2#31826903C211yesmaximal, characteristic, complemented
3C2×D4528C211yesmaximal, characteristic, complemented
4#31826735C211yesmaximal, characteristic, complemented
5#31826279C211yesmaximal, characteristic, complemented
6#31826273C211yesmaximal, characteristic, complemented
7#350×C283C211yesmaximal, characteristic, complemented
8#22583656—21no—
9#22583638—22no—
10#22583638—21no—
11#22583635—21no—
12#22583632—21no—
13#22583632C2211yescharacteristic
14C2×D2264—21no—
15C2×D2264C2211yescharacteristic
16C2×D2264C2211yescharacteristic, complemented
17#22583555—21no—
18D4528—22no—
19D4528—21no—
20#22583546C2211yescharacteristic, complemented
21#22583526—21no—
22#131×C283C2211yescharacteristic
23M32×C283C2211yescharacteristic, complemented
24C2×C16×C283C2211yescharacteristic, complemented
25#17985393—41no—
26#17985392—22no—
27#17985392—21no—
28C2×D1132—22no—
29C2×D1132—21no—
30C2×D1132—21no—
31D4×D283—42no—
32D4×D283—41no—
33Dic1132D411yescharacteristic
34#17985381—22no—
35#17985381—21no—
36#17985379—41no—
37D2264—22no—
38D2264—21no—
39D2264—21no—
40D2264—21no—
41D2264D411yescharacteristic
42D2264D412yes—
43#17985377—41no—
44#17985376—21no—
45#41×C283D411yescharacteristic
46M16×C283—21no—
47M16×C283D411yescharacteristic
48C2×C8×C283—21no—
49C2×C8×C283C2311yescharacteristic
50C16×C283—21no—
51C16×C283—21no—
52C22×D283—42no—
53C22×D283—41no—
54C22×D283—41no—
55Dic566—21no—
56D1132—21no—
57D1132—22no—
58D1132—42no—
59D1132—41no—
60D1132—41no—
61D1132—41no—
62C22⋊D283—42no—
63C22⋊D283—41no—
64C4×D283—42no—
65C4×D283—41no—
66C283⋊C8—41no—
67Q8×C283D811yescharacteristic
68D4×C283—21no—
69D4×C283D811yescharacteristic
70C2×C4×C283—21no—
71C2×C4×C283C2×D411yescharacteristic
72C8×C283—21no—
73C8×C283C2×D411yescharacteristic
74C8×C283C2×D411yescharacteristic
75C22×C283—21no—
76C22×C283AGL(1,ℤ/8)11yescharacteristic
77C2×D283—42no—
78C2×D283—41no—
79C2×D283—81no—
80C2×D283—161no—
81C2×D283—162no—
82C2×D283—161no—
83C4×C283—21no—
84C4×C283#12011yescharacteristic
85C4×C283C2×D811yescharacteristic
86Dic283—41no—
87C2×C283—21no—
88C2×C283—41no—
89C2×C283#44711yescharacteristic
90D283—82no—
91D283—81no—
92D283—161no—
93C283#219211yescharacteristic, complemented, Sylow
94#2192—2831nomaximal, Sylow
95#576—2831no—
96#509—2831no—
97C2×D16—2831no—
98#472—2831no—
99#361—2831no—
100#359—2831no—
101#350D28311yescharacteristic, complemented
102H(5,𝔽2)—5661no—
103AGL(1,ℤ/8)—5662no—
104AGL(1,ℤ/8)—5661no—
105AGL(1,ℤ/8)—5661no—
106#135—2831no—
107#135—5661no—
108C2×D8—2831no—
109C2×D8—2831no—
110C2×D8—5661no—
111#131C2×D28311yescharacteristic
112SD32—5661no—
113D16—5662no—
114D16—5661no—
115M32C2×D28311yescharacteristic, complemented
116C2×C16C2×D28311yescharacteristic, complemented
117#108—2831no—
118#100—5661no—
119#41—5662no—
120#41—5661no—
121#41—11321no—
122#41D113211yescharacteristic
123C2×D4—5662no—
124C2×D4—5661no—
125C2×D4—5661no—
126C2×D4—11322no—
127C2×D4—11321no—
128Q16—2831no—
129SD16—5662no—
130SD16—5661no—
131SD16—11321no—
132D8—2832no—
133D8—2831no—
134D8—5662no—
135D8—5661no—
136D8—5661no—
137D8—5661no—
138D8—11321no—
139M16—21no—
140M16—5661no—
141M16D113211yescharacteristic
142C2×C8—21no—
143C2×C8C22×D28311yescharacteristic
144C16—21no—
145C16—21no—
146C23—11322no—
147C23—11321no—
148C23—11321no—
149Q8—5661no—
150Q8D226411yescharacteristic
151D4—21no—
152D4—5662no—
153D4—5661no—
154D4—11322no—
155D4—11322no—
156D4—11321no—
157D4—11321no—
158D4—11321no—
159D4—11321no—
160D4D226411yescharacteristic
161C2×C4—21no—
162C2×C4—11322no—
163C2×C4—11321no—
164C2×C4C2×D113211yescharacteristic
165C8—21no—
166C8—11321no—
167C8D4×D28311yescharacteristic
168C8D4×D28311yescharacteristic
169C22—21no—
170C22—11322no—
171C22—11321no—
172C22—45281no—
173C22—45281no—
174C22—45282no—
175C22—22641no—
176C22#2258363811yescharacteristic
177C4—21no—
178C4—11321no—
179C4#2258358011yescharacteristic
180C4C2×D226411yescharacteristic
181C2—21no—
182C2—41no—
183C2—45281no—
184C2—22642no—
185C2—22641no—
186C2#3182660111yescharacteristic
subgroupG/H
centerC2#31826601
commutator, cosocleC2×C8×C283C23
FrattiniC2×C8C22×D283
Fitting#350×C283C2
socleC2×C283#447
hypercenter#350D283
hypocenterC283#2192
LCSG ▷ C2×C8×C283 ▷ C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C2×C4 ◁ #131 ◁ #350

Complements

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

NH ≅ G/Ntype
1. #31827270C2semidirect
2. #31826903C2semidirect
3. C2×D4528C2semidirect
4. #31826735C2semidirect
5. #31826279C2semidirect
6. #31826273C2semidirect
7. #350×C283C2semidirect
16. C2×D2264C22semidirect
20. #22583546C22semidirect
23. M32×C283C22semidirect
24. C2×C16×C283C22semidirect
93. C283#2192semidirect
101. #350D283semidirect
115. M32C2×D283semidirect
116. C2×C16C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318272702
#318269032
C2×D45282
#318267352
#318262792
#318262732
#350×C2832
#2192283