Finite Group Databasebeta

← #35318689

Subgroups — #35318689

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#24909719C211yesmaximal, characteristic
2#24909371C211yesmaximal, characteristic, complemented
3#24906834C211yesmaximal, characteristic, complemented
4#24906824C211yesmaximal, characteristic
5#24905511C211yesmaximal, characteristic, complemented
6#24905508C211yesmaximal, characteristic
7C83×#1371C211yesmaximal, characteristic, complemented
8C2×#14581737—21no—
9C2×#14581737—21no—
10#19128081C2211yescharacteristic
11#19128078C2211yescharacteristic
12#19127987—21no—
13#19127902—21no—
14C8×Dic166C2211yescharacteristic
15C4⋊(C83⋊C16)C2211yescharacteristic
16Dic83⋊C16—21no—
17#19127476—21no—
18#19127446—21no—
19#19127444—21no—
20C4×C16×C83C2211yescharacteristic
21#335×C83C2211yescharacteristic
22#335×C83C2211yescharacteristic
23C2×Dic332—21no—
24C2×Dic332—21no—
25C2×#13050579—41no—
26C2×D664—21no—
27C2×D664—21no—
28#14581792—21no—
29#14581789—21no—
30C4×Dic166D411yescharacteristic
31#14581737—42no—
32#14581737—42no—
33C2×C83⋊C16—21no—
34#14581724—41no—
35C4⋊(C83⋊C8)D411yescharacteristic
36Dic83⋊C8—21no—
37#14581717—41no—
38#14581714—41no—
39C8×Dic83—21no—
40C2×C16×C83D411yescharacteristic
41C2×C16×C83D411yescharacteristic
42C4⋊C8×C83—21no—
43C4⋊C8×C83—21no—
44C4×C8×C83C2311yescharacteristic
45C2×Dic166—41no—
46C2×Dic166D811yescharacteristic
47C2×D332—41no—
48C2×D332—41no—
49C2×D332—41no—
50Dic332—41no—
51Dic332—41no—
52Dic332—81no—
53#13050579—82no—
54D664—41no—
55D664—41no—
56D664—81no—
57C2×C83⋊C8—21no—
58C4⋊Dic83—41no—
59C4⋊Dic83—41no—
60C4⋊Dic83D811yescharacteristic
61Dic83⋊C4—21no—
62C4×Dic83—21no—
63C83⋊C16—41no—
64C2×C8×C83—41no—
65C2×C8×C83—41no—
66C2×C8×C83C2×D411yescharacteristic
67C2×C8×C83#4111yescharacteristic
68C42×C83C2×D411yescharacteristic
69C16×C83—21no—
70C16×C83SD1612yescomplemented
71C22×D83—81no—
72C2×C4×C83#12111yescharacteristic
73C2×C4×C83C2×D811yescharacteristic
74C2×C4×C83#13511yescharacteristic
75C2×Dic83—21no—
76C2×Dic83—21no—
77C2×Dic83—81no—
78D332—81no—
79D332—81no—
80D332—81no—
81D332—81no—
82D332—161no—
83Dic166—21no—
84Dic166—21no—
85Dic166—81no—
86Dic166—81no—
87C8×C83—21no—
88C8×C83—81no—
89C8×C83—81no—
90C8×C83C2×SD1611yescharacteristic
91C8×C83AGL(1,ℤ/8)11yescharacteristic
92C83⋊C8—41no—
93C22×C83#46611yescharacteristic
94C2×D83—161no—
95C2×D83—161no—
96C2×D83—161no—
97C4×C83—21no—
98C4×C83#46011yescharacteristic
99C4×C83#47211yescharacteristic
100C4×C83#50911yescharacteristic
101C4×C83#51011yescharacteristic
102Dic83—41no—
103Dic83—41no—
104Dic83—161no—
105#12444—831nomaximal, Sylow
106C2×C83#166811yescharacteristic
107C2×C83#222011yescharacteristic
108C2×C83#222311yescharacteristic
109D83—321no—
110#2248—831no—
111#2183—831no—
112#1670—831no—
113#1668—831no—
114#1371D8311yescharacteristic, complemented
115#1362—831no—
116#1361—831no—
117C83#1244411yescharacteristic, complemented, Sylow
118C2×SD32—1661no—
119C2×SD32—1661no—
120#494—831no—
121#493—831no—
122#475—1661no—
123#460—1661no—
124C8×Q8—831no—
125C4⋊C16—831no—
126C4⋊C16—1661no—
127#346—1661no—
128C4×C16C2×D8311yescharacteristic
129#335C2×D8311yescharacteristic
130#335C2×D8311yescharacteristic
131#332—1661no—
132#331—1661no—
133C2×Q16—1661no—
134C2×Q16—1661no—
135C2×SD16—3321no—
136C2×D8—1661no—
137C2×D8—1661no—
138C4⋊Q8—1661no—
139#127—1661no—
140C4×Q8—831no—
141SD32—3322no—
142SD32—3322no—
143C2×C16—1661no—
144C2×C16D33211yescharacteristic
145C2×C16D33211yescharacteristic
146#106—3321no—
147C4⋊C8—21no—
148C4⋊C8—21no—
149C4⋊C8—831no—
150C4⋊C8—1661no—
151#103—3321no—
152#102—3321no—
153C4×C8—1661no—
154C4×C8C22×D8311yescharacteristic
155C2×Q8—3321no—
156C2×Q8—831no—
157C2×D4—3321no—
158C2×D4—3321no—
159C2×D4—3321no—
160Q16—3321no—
161Q16—3321no—
162Q16—6641no—
163SD16—6642no—
164D8—3321no—
165D8—3321no—
166D8—6641no—
167C2×C8—41no—
168C2×C8—41no—
169C2×C8—1661no—
170C2×C8C2×D33211yescharacteristic
171C2×C8#1305059111yescharacteristic
172C4⋊C4—3321no—
173C4⋊C4—3321no—
174C4⋊C4—831no—
175C4⋊C4—1661no—
176C42—1661no—
177C42D4×D8311yescharacteristic
178C16—21no—
179C16—3321no—
180C16#1305057912yescomplemented
181C23—6641no—
182Q8—6641no—
183Q8—6641no—
184Q8—1661no—
185Q8—1661no—
186D4—13281no—
187D4—6641no—
188D4—6641no—
189D4—6641no—
190D4—6641no—
191C2×C4—6641no—
192C2×C4—1661no—
193C2×C4—1661no—
194C2×C4#1458176711yescharacteristic
195C2×C4D8×D8311yescharacteristic
196C2×C4#1458181511yescharacteristic
197C8—21no—
198C8—81no—
199C8—81no—
200C8—3321no—
201C8C2×#1305057911yescharacteristic
202C8#1458182011yescharacteristic
203C22—13281no—
204C22—13281no—
205C22—13281no—
206C22#1912793811yescharacteristic
207C4—21no—
208C4—13281no—
209C4—3321no—
210C4—3321no—
211C4#1912790211yescharacteristic
212C4#1912797211yescharacteristic
213C4#1912813511yescharacteristic
214C4#1912814311yescharacteristic
215C2—26561no—
216C2#2490682411yescharacteristic
217C2#2490957711yescharacteristic
218C2#2490959411yescharacteristic
subgroupG/H
centerC22#19127938
commutator, cosocleC4×C8×C83C23
FrattiniC4×C8C22×D83
FittingC83×#1371C2
socleC22×C83#466
prime radical#19128078C22
hypercenter#1371D83
hypocenterC83#12444
LCSG ▷ C4×C8×C83 ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1371

Complements

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

NH ≅ G/Ntype
2. #24909371C2semidirect
3. #24906834C2semidirect
5. #24905511C2semidirect
7. C83×#1371C2semidirect
70. C16×C83SD16semidirect
114. #1371D83semidirect
117. C83#12444semidirect
180. C16#13050579semidirect

Maximal subgroups

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

subgroupindex
#249097192
#249093712
#249068342
#249068242
#249055112
#249055082
C83×#13712
#1244483