Finite Group Databasebeta

← #35318656

Subgroups — #35318656

Order 21248.

224 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 59 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, complemented
2#24909539C211yesmaximal, characteristic
3#24909300C211yesmaximal, characteristic, complemented
4#24905538C211yesmaximal, characteristic, complemented
5#24905502C211yesmaximal, characteristic, complemented
6#24905499C211yesmaximal, characteristic, complemented
7C83×#2214C211yesmaximal, characteristic, complemented
8#19128136—21no—
9C2×#14581735—21no—
10C2×#14581735—21no—
11#19128118—21no—
12#19128078C2211yescharacteristic
13#19127522C2211yescharacteristic
14#19127522C2211yescharacteristic
15#19127513—21no—
16#19127485—21no—
17C4×C83⋊C16C2211yescharacteristic
18#510×C83—21no—
19#494×C83C2211yescharacteristic
20#443×C83C2211yescharacteristic
21#364×C83C2211yescharacteristic
22#362×C83—21no—
23C2×D664—21no—
24C2×D664—21no—
25#14581789—21no—
26#14581735—42no—
27#14581735—41no—
28#14581735—41no—
29#14581735—41no—
30#14581733—41no—
31#14581732—21no—
32#14581732—21no—
33#14581732—41no—
34C2×C83⋊C16—41no—
35C2×C83⋊C16D411yescharacteristic
36C2×C83⋊C16D411yescharacteristic
37#135×C83D411yescharacteristic
38C2×Q16×C83—21no—
39C2×Q16×C83—21no—
40#131×C83—21no—
41C4⋊Q8×C83—21no—
42Q32×C83—41no—
43SD32×C83—41no—
44M32×C83—21no—
45#108×C83D411yescharacteristic
46#104×C83—21no—
47C4×C8×C83C2311yescharacteristic
48C2×D332—41no—
49C2×D332—81no—
50D664—41no—
51D664—41no—
52D664—41no—
53D664—41no—
54C83⋊C16—21no—
55C83⋊C16—41no—
56C83⋊C16—41no—
57C83⋊C16D811yescharacteristic
58C83⋊C16D811yescharacteristic
59#41×C83—21no—
60C2×Q8×C83—41no—
61Q16×C83—41no—
62Q16×C83—41no—
63Q16×C83—41no—
64Q16×C83—41no—
65Q16×C83D811yescharacteristic
66SD16×C83—21no—
67D8×C83D811yescharacteristic
68M16×C83—21no—
69M16×C83—41no—
70C2×C8×C83—41no—
71C2×C8×C83C2×D411yescharacteristic
72C2×C8×C83C2×D411yescharacteristic
73C4⋊C4×C83—81no—
74C42×C83#4111yescharacteristic
75C16×C83—41no—
76C22×D83—81no—
77Q8×C83—21no—
78Q8×C83—81no—
79Q8×C83—81no—
80D4×C83—21no—
81D4×C83—41no—
82C2×C4×C83—21no—
83C2×C4×C83—41no—
84C2×C4×C83—81no—
85C2×C4×C83#12111yescharacteristic
86D332—81no—
87D332—81no—
88D332—162no—
89C8×C83—41no—
90C8×C83C2×D811yescharacteristic
91C8×C83C2×D811yescharacteristic
92C8×C83#13511yescharacteristic
93C8×C83AGL(1,ℤ/8)11yescharacteristic
94C22×C83—41no—
95C22×C83#47211yescharacteristic
96C2×D83—162no—
97C2×D83—161no—
98C4×C83—22no—
99C4×C83—41no—
100C4×C83—161no—
101C4×C83#45911yescharacteristic
102C4×C83#46611yescharacteristic
103#12438—831nomaximal, Sylow
104C2×C83—21no—
105C2×C83—81no—
106C2×C83#166711yescharacteristic
107D83—321no—
108#2248—831no—
109#2214D8311yescharacteristic, complemented
110#2213—831no—
111#2170—831no—
112#1373—831no—
113#1358—831no—
114#1357—831no—
115C83#1243811yescharacteristic, complemented, Sylow
116#510—21no—
117#509—1661no—
118C2×SD32—1661no—
119C2×SD32—1661no—
120#504—1661no—
121#494C2×D8311yescharacteristic
122#493—831no—
123#443C2×D8311yescharacteristic
124#364—831no—
125#364—831no—
126#364C2×D8311yescharacteristic
127#362—21no—
128#361—1661no—
129#350—1661no—
130C4×C16—831no—
131#135C22⋊D8311yescharacteristic
132C2×Q16—21no—
133C2×Q16—21no—
134C2×D8—1661no—
135C2×D8—1661no—
136#131—21no—
137C4⋊Q8—21no—
138#127—1661no—
139Q32—41no—
140SD32—41no—
141SD32—3321no—
142SD32—3321no—
143SD32—3321no—
144SD32—3322no—
145D16—3321no—
146M32—21no—
147M32—3321no—
148M32—1661no—
149M32—1661no—
150C2×C16—3321no—
151C2×C16—831no—
152C2×C16—831no—
153#108C22⋊D8311yescharacteristic
154#104—21no—
155C4×C8C22×D8311yescharacteristic
156#41—21no—
157C2×Q8—41no—
158C2×D4—3321no—
159C2×D4—6641no—
160Q16—41no—
161Q16—41no—
162Q16—41no—
163Q16—41no—
164Q16#1305057511yescharacteristic
165SD16—21no—
166D8—3321no—
167D8—3321no—
168D8—3321no—
169D8—3321no—
170D8#1305057511yescharacteristic
171M16—21no—
172M16—41no—
173C2×C8—41no—
174C2×C8D4×D8311yescharacteristic
175C2×C8C2×C22⋊D8311yescharacteristic
176C4⋊C4—81no—
177C42#1305058911yescharacteristic
178C16—41no—
179C16—3321no—
180C16—3321no—
181C16—831no—
182C16—831no—
183C16—1661no—
184C23—6641no—
185Q8—21no—
186Q8—81no—
187Q8—81no—
188D4—21no—
189D4—41no—
190D4—13282no—
191D4—6641no—
192D4—6641no—
193C2×C4—21no—
194C2×C4—41no—
195C2×C4—81no—
196C2×C4#1458176411yescharacteristic
197C8—41no—
198C8D8×D8311yescharacteristic
199C8C2×#1305057511yescharacteristic
200C8#1458181211yescharacteristic
201C8#1458181611yescharacteristic
202C22—41no—
203C22—13282no—
204C22—13281no—
205C22#1912796911yescharacteristic
206C4—22no—
207C4—41no—
208C4—161no—
209C4#1912789411yescharacteristic
210C4#1912793511yescharacteristic
211C2—21no—
212C2—81no—
213C2—26561no—
214C2#2490681611yescharacteristic
subgroupG/H
centerC2#24906816
commutator, cosocleC4×C8×C83C23
FrattiniC4×C8C22×D83
FittingC83×#2214C2
socleC2×C83#1667
prime radical#24909539C2
hypercenter#2214D83
hypocenterC83#12438
LCSG ▷ C4×C8×C83 ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #2214

Complements

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

NH ≅ G/Ntype
1. #24909720C2semidirect
3. #24909300C2semidirect
4. #24905538C2semidirect
5. #24905502C2semidirect
6. #24905499C2semidirect
7. C83×#2214C2semidirect
109. #2214D83semidirect
115. C83#12438semidirect

Maximal subgroups

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

subgroupindex
#249097202
#249095392
#249093002
#249055382
#249055022
#249054992
C83×#22142
#1243883