Finite Group Databasebeta

← #35318690

Subgroups — #35318690

Order 21248.

228 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 67 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#24909720C211yesmaximal, characteristic
2#24909372C211yesmaximal, characteristic, complemented
3#24906823C211yesmaximal, characteristic
4#24905534C211yesmaximal, characteristic, complemented
5#24905512C211yesmaximal, characteristic, complemented
6#24905508C211yesmaximal, characteristic
7C83×#1670C211yesmaximal, characteristic, complemented
8C2×#14581735—21no—
9C2×#14581735—21no—
10#19128078C2211yescharacteristic
11#19127901—21no—
12C4⋊(C83⋊C16)—21no—
13C4⋊(C83⋊C16)C2211yescharacteristic
14#19127477—21no—
15C4×C83⋊C16C2211yescharacteristic
16#19127454C2211yescharacteristic
17#19127444—21no—
18#494×C83C2211yescharacteristic
19#475×C83—21no—
20C8×Q8×C83C2211yescharacteristic
21#335×C83C2211yescharacteristic
22#332×C83—21no—
23C2×#13050577—41no—
24C2×D664—21no—
25C2×D664—21no—
26#14581789—21no—
27#14581735—42no—
28#14581735—42no—
29C2×C83⋊C16—21no—
30C2×C83⋊C16D411yescharacteristic
31C2×C83⋊C16D411yescharacteristic
32C4⋊(C83⋊C8)—21no—
33#14581712—41no—
34C2×Q16×C83—21no—
35C2×Q16×C83—21no—
36C4⋊Q8×C83—21no—
37C4×Q8×C83D411yescharacteristic
38#106×C83—41no—
39C4⋊C8×C83—21no—
40C4⋊C8×C83—21no—
41C4⋊C8×C83D411yescharacteristic
42#103×C83—41no—
43C4×C8×C83—21no—
44C4×C8×C83C2311yescharacteristic
45C2×D332—41no—
46C2×D332—41no—
47C2×D332—41no—
48#13050577—82no—
49D664—41no—
50D664—41no—
51D664—81no—
52C2×C83⋊C8—41no—
53C83⋊C16—21no—
54C83⋊C16—41no—
55C83⋊C16SD1612yescomplemented
56C2×Q8×C83—41no—
57C2×Q8×C83D811yescharacteristic
58Q16×C83—41no—
59Q16×C83—41no—
60Q16×C83—81no—
61C2×C8×C83—21no—
62C2×C8×C83—41no—
63C2×C8×C83C2×D411yescharacteristic
64C2×C8×C83#4111yescharacteristic
65C4⋊C4×C83—21no—
66C4⋊C4×C83—41no—
67C4⋊C4×C83—41no—
68C4⋊C4×C83D811yescharacteristic
69C42×C83—21no—
70C42×C83C2×D411yescharacteristic
71C22×D83—81no—
72Q8×C83—21no—
73Q8×C83—21no—
74Q8×C83—81no—
75Q8×C83—81no—
76C2×C4×C83—21no—
77C2×C4×C83—21no—
78C2×C4×C83—81no—
79C2×C4×C83#12111yescharacteristic
80C2×C4×C83C2×D811yescharacteristic
81C2×C4×C83#13511yescharacteristic
82D332—81no—
83D332—81no—
84D332—81no—
85D332—81no—
86D332—161no—
87C8×C83—21no—
88C8×C83—41no—
89C8×C83—81no—
90C8×C83C2×SD1611yescharacteristic
91C8×C83AGL(1,ℤ/8)11yescharacteristic
92C83⋊C8—81no—
93C22×C83#46611yescharacteristic
94C2×D83—161no—
95C2×D83—161no—
96C2×D83—161no—
97C4×C83—21no—
98C4×C83—41no—
99C4×C83—41no—
100C4×C83—161no—
101C4×C83#46011yescharacteristic
102C4×C83#47211yescharacteristic
103C4×C83#50911yescharacteristic
104C4×C83#51011yescharacteristic
105#12444—831nomaximal, Sylow
106C2×C83#166811yescharacteristic
107C2×C83#222011yescharacteristic
108C2×C83#222311yescharacteristic
109D83—321no—
110#2248—831no—
111#2183—831no—
112#1670D8311yescharacteristic, complemented
113#1668—831no—
114#1371—831no—
115#1362—831no—
116#1361—831no—
117C83#1244411yescharacteristic, complemented, Sylow
118C2×SD32—1661no—
119C2×SD32—1661no—
120#494C2×D8311yescharacteristic
121#493—831no—
122#475—21no—
123#460—1661no—
124C8×Q8C2×D8311yescharacteristic
125C4⋊C16—831no—
126C4⋊C16—1661no—
127#346—1661no—
128C4×C16—831no—
129#335—831no—
130#335C2×D8311yescharacteristic
131#332—21no—
132#331—1661no—
133C2×Q16—21no—
134C2×Q16—21no—
135C2×SD16—3321no—
136C2×D8—1661no—
137C2×D8—1661no—
138C4⋊Q8—21no—
139#127—1661no—
140C4×Q8C22⋊D8311yescharacteristic
141SD32—3322no—
142SD32—3322no—
143C2×C16—831no—
144C2×C16—831no—
145C2×C16—1661no—
146#106—41no—
147C4⋊C8—21no—
148C4⋊C8—21no—
149C4⋊C8—1661no—
150C4⋊C8C22⋊D8311yescharacteristic
151#103—41no—
152#102—3321no—
153C4×C8—21no—
154C4×C8C22×D8311yescharacteristic
155C2×Q8—41no—
156C2×Q8#1305057511yescharacteristic
157C2×D4—3321no—
158C2×D4—3321no—
159C2×D4—3321no—
160Q16—41no—
161Q16—41no—
162Q16—81no—
163SD16—6642no—
164D8—3321no—
165D8—3321no—
166D8—6641no—
167C2×C8—21no—
168C2×C8—41no—
169C2×C8—3321no—
170C2×C8D4×D8311yescharacteristic
171C2×C8#1305058911yescharacteristic
172C4⋊C4—21no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4#1305057511yescharacteristic
176C42—21no—
177C42C2×C22⋊D8311yescharacteristic
178C16—3321no—
179C16—832no—
180C16—1661no—
181C23—6641no—
182Q8—21no—
183Q8—21no—
184Q8—81no—
185Q8—81no—
186D4—13281no—
187D4—6641no—
188D4—6641no—
189D4—6641no—
190D4—6641no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—81no—
194C2×C4#1458176411yescharacteristic
195C2×C4C2×#1305057511yescharacteristic
196C2×C4#1458181211yescharacteristic
197C8—21no—
198C8—41no—
199C8—81no—
200C8—6641no—
201C8SD16×D8311yescharacteristic
202C8#1458181911yescharacteristic
203C22—13281no—
204C22—13281no—
205C22—13281no—
206C22#1912793511yescharacteristic
207C4—21no—
208C4—41no—
209C4—41no—
210C4—161no—
211C4#1912789911yescharacteristic
212C4#1912796911yescharacteristic
213C4#1912813611yescharacteristic
214C4#1912814211yescharacteristic
215C2—26561no—
216C2#2490682111yescharacteristic
217C2#2490957811yescharacteristic
218C2#2490959511yescharacteristic
subgroupG/H
centerC22#19127935
commutator, cosocleC4×C8×C83C23
FrattiniC4×C8C22×D83
FittingC83×#1670C2
socleC22×C83#466
prime radical#19128078C22
hypercenter#1670D83
hypocenterC83#12444
LCSG ▷ C4×C8×C83 ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ #1670

Complements

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

NH ≅ G/Ntype
2. #24909372C2semidirect
4. #24905534C2semidirect
5. #24905512C2semidirect
7. C83×#1670C2semidirect
55. C83⋊C16SD16semidirect
112. #1670D83semidirect
117. C83#12444semidirect

Maximal subgroups

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

subgroupindex
#249097202
#249093722
#249068232
#249055342
#249055122
#249055082
C83×#16702
#1244483