Finite Group Databasebeta

← #35291946

Subgroups — #35291946

Order 21248.

198 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 37 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#24908750C211yesmaximal, characteristic
2#24905414C211yesmaximal, characteristic, complemented
3C83×#1309C211yesmaximal, characteristic, complemented
4#19128501C411yescharacteristic
5C2×#14581720—21no—
6#19127683—21no—
7#19127483—21no—
8C83⋊#338C411yescharacteristic
9#423×C83—21no—
10#423×C83C2211yescharacteristic
11C2×#13050590—21no—
12#14581825—41no—
13#14581820—41no—
14#14581814—41no—
15C2×#13050574—21no—
16#14581732—41no—
17#14581720—41no—
18#14581720—41no—
19#14581710—41no—
20#14581708—41no—
21C2×M16×C83—22no—
22C2×M16×C83C2×C411yescharacteristic
23C2×M16×C83D411yescharacteristic
24C2×C42×C83D411yescharacteristic
25C4⋊C8×C83—41no—
26C4⋊C8×C83—41no—
27C4⋊C8×C83—41no—
28#13050590—41no—
29#13050590—41no—
30#13050590—41no—
31#13050590—41no—
32C2×Dic166—21no—
33C2×D332—21no—
34C2×C22⋊D83—41no—
35C2×C4×D83—41no—
36Dic332—41no—
37#13050579—42no—
38D664—41no—
39#13050574—41no—
40#13050574—41no—
41C2×C83⋊C8—41no—
42C83⋊C16—81no—
43C22×C4×C83—21no—
44C22×C4×C83C22⋊C411yescharacteristic
45M16×C83—41no—
46M16×C83—41no—
47M16×C83—82no—
48C2×C8×C83—21no—
49C2×C8×C83—21no—
50C2×C8×C83—42no—
51C42×C83—21no—
52C42×C83D811yescharacteristic
53C42×C83SD1611yescharacteristic
54C23×C83#10411yescharacteristic
55C22×D83—41no—
56C2×C4×C83—21no—
57C2×C4×C83—21no—
58C2×C4×C83—21no—
59C2×C4×C83—21no—
60C2×C4×C83—41no—
61C2×C4×C83—41no—
62C2×C4×C83#9911yescharacteristic
63C2×C4×C83#10211yescharacteristic
64C22⋊D83—81no—
65C22⋊D83—81no—
66C2×Dic83—41no—
67D332—41no—
68D332—41no—
69C4×D83—81no—
70C4×D83—81no—
71Dic166—41no—
72Dic166—41no—
73C8×C83—41no—
74C8×C83—41no—
75C8×C83—82no—
76C83⋊C8—81no—
77C22×C83—21no—
78C22×C83—41no—
79C22×C83#32711yescharacteristic
80C2×D83—81no—
81C2×D83—161no—
82C4×C83—21no—
83C4×C83—21no—
84C4×C83—21no—
85C4×C83—41no—
86C4×C83AΣL(1,𝔽16)11yescharacteristic
87C4×C83#35511yescharacteristic
88Dic83—81no—
89#7414—831nomaximal, Sylow
90C2×C83—21no—
91C2×C83—41no—
92C2×C83#134511yescharacteristic
93D83—161no—
94#2050—831no—
95#1327—831no—
96#1309D8311yescharacteristic, complemented
97C83#741411yescharacteristic, complemented, Sylow
98#575—831no—
99#423—21no—
100#423C2×D8311yescharacteristic
101C2×#104—1661no—
102#413—1661no—
103#349—1661no—
104#338—831no—
105C2×#41—1661no—
106#137—3321no—
107AGL(1,ℤ/8)—3321no—
108#135—3321no—
109C2×M16—22no—
110C2×M16—1661no—
111C2×M16C4×D8311yescharacteristic
112C2×M16D33211yescharacteristic
113C2×C42C22⋊D8311yescharacteristic
114M32—3321no—
115C4⋊C8—41no—
116C4⋊C8—41no—
117C4⋊C8—41no—
118#104—3321no—
119#104—3321no—
120#101—3321no—
121#100—3321no—
122#41—3321no—
123#41—3321no—
124#41—3321no—
125#41—3321no—
126C2×Q8—1661no—
127C2×D4—3321no—
128C2×D4—1661no—
129C22×C4—21no—
130C22×C4—3321no—
131C22×C4(C2×D83)⋊C411yescharacteristic
132Q16—3321no—
133SD16—3322no—
134D8—3321no—
135M16—41no—
136M16—41no—
137M16—82no—
138M16—3321no—
139M16—3321no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—42no—
143C2×C8—3321no—
144C42—21no—
145C42#1305057511yescharacteristic
146C42#1305057711yescharacteristic
147C16—6641no—
148C23—3321no—
149C23#1458172011yescharacteristic
150Q8—3321no—
151Q8—3321no—
152D4—3321no—
153D4—3321no—
154D4—6641no—
155D4—6641no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—3321no—
163C2×C4—6641no—
164C2×C4—6641no—
165C2×C4#1458170611yescharacteristic
166C2×C4#1458171211yescharacteristic
167C8—41no—
168C8—41no—
169C8—82no—
170C8—6641no—
171C22—21no—
172C22—41no—
173C22—13281no—
174C22—6641no—
175C22#1912743411yescharacteristic
176C4—21no—
177C4—21no—
178C4—21no—
179C4—41no—
180C4—6641no—
181C4#1912748911yescharacteristic
182C4#1912749711yescharacteristic
183C2—21no—
184C2—41no—
185C2—13281no—
186C2#2490546511yescharacteristic
subgroupG/H
centerC2#24905465
commutatorC2×M16×C83C2×C4
Frattini#423C2×D83
FittingC83×#1309C2
socleC2×C83#1345
prime radical#19128501C4
cosocle#423×C83C22
hypercenter#1309D83
hypocenterC83#7414
LCSG ▷ C2×M16×C83 ▷ C22×C4×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #423 ◁ #1309

Complements

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

NH ≅ G/Ntype
2. #24905414C2semidirect
3. C83×#1309C2semidirect
96. #1309D83semidirect
97. C83#7414semidirect

Maximal subgroups

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

subgroupindex
#249087502
#249054142
C83×#13092
#741483