Finite Group Databasebeta

← #24909412

Subgroups — #24909412

Order 10624.

216 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 47 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#19128498C211yesmaximal, characteristic
2#19128142C211yesmaximal, characteristic, complemented
3#19127956C211yesmaximal, characteristic, complemented
4#19127512C211yesmaximal, characteristic, complemented
5#19127511C211yesmaximal, characteristic, complemented
6#19127482C211yesmaximal, characteristic, complemented
7#510×C83C211yesmaximal, characteristic, complemented
8#41×D83—21no—
9#14581824—21no—
10#14581821—21no—
11#14581819—21no—
12#14581816—21no—
13#14581814C2211yescharacteristic
14#14581812—21no—
15C2×#13050573C2211yescharacteristic
16#14581768—21no—
17#14581738—21no—
18#14581735—21no—
19#14581732C2211yescharacteristic, complemented
20#14581724C2211yescharacteristic, complemented
21#14581716—21no—
22#135×C83C2211yescharacteristic
23C2×Q16×C83C2211yescharacteristic, complemented
24Q32×C83—21no—
25SD32×C83—21no—
26M32×C83C2211yescharacteristic, complemented
27#13050591—21no—
28#13050590—21no—
29#13050590—41no—
30#13050589—21no—
31#13050589—41no—
32Q8×D83—21no—
33D4×D83—21no—
34D4×D83—41no—
35C2×C4×D83—41no—
36C2×C4×D83D411yescharacteristic
37Dic332D411yescharacteristic
38#13050580—21no—
39#13050579—21no—
40#13050578—21no—
41#13050577—21no—
42D664D411yescharacteristic
43#13050575—21no—
44#13050573—21no—
45#13050573—21no—
46C2×C83⋊C8D411yescharacteristic
47C4⋊Dic83—21no—
48Dic83⋊C4—41no—
49(C2×D83)⋊C4—41no—
50C83⋊C16—21no—
51#41×C83—21no—
52C2×Q8×C83—21no—
53Q16×C83—21no—
54Q16×C83—21no—
55Q16×C83D411yescharacteristic
56SD16×C83—21no—
57D8×C83D411yescharacteristic
58C2×C8×C83C2311yescharacteristic
59C16×C83—21no—
60C22×D83—41no—
61C22×D83D811yescharacteristic
62Q8×C83—21no—
63Q8×C83—41no—
64Q8×C83—41no—
65D4×C83—21no—
66D4×C83—41no—
67C2×C4×C83—41no—
68C2×C4×C83—41no—
69C2×C4×C83C2×D411yescharacteristic
70C22⋊D83—41no—
71C22⋊D83—41no—
72C22⋊D83—41no—
73C2×Dic83—41no—
74C2×Dic83—41no—
75C2×Dic83D811yescharacteristic
76D332—21no—
77D332—41no—
78C4×D83—21no—
79C4×D83—21no—
80C4×D83—41no—
81C4×D83—41no—
82C4×D83—41no—
83Dic166—21no—
84Dic166—41no—
85C8×C83C2×D411yescharacteristic
86C8×C83C2×D411yescharacteristic
87C83⋊C8—21no—
88C22×C83—41no—
89C22×C83C2×D811yescharacteristic
90C2×D83—21no—
91C2×D83—41no—
92C2×D83—41no—
93C2×D83—82no—
94C4×C83—41no—
95C4×C83—81no—
96C4×C83#12011yescharacteristic
97C4×C83AGL(1,ℤ/8)11yescharacteristic
98Dic83—21no—
99Dic83—41no—
100Dic83—81no—
101C2×C83—21no—
102C2×C83—81no—
103C2×C83#44711yescharacteristic
104D83—41no—
105D83—81no—
106#2191—831nomaximal, Sylow
107C83#219111yescharacteristic, complemented, Sylow
108#575—831no—
109#510—831no—
110#510D8311yescharacteristic, complemented
111#470—831no—
112#360—831no—
113#360—831no—
114#349—831no—
115C2×#41—1661no—
116#137—1661no—
117#137—1661no—
118AGL(1,ℤ/8)—1661no—
119AGL(1,ℤ/8)—1661no—
120#135—831no—
121#135—1661no—
122#135C2×D8311yescharacteristic
123C2×Q16C2×D8311yescharacteristic, complemented
124C2×M16—831no—
125C22⋊Q8—1661no—
126Q32—21no—
127Q32—1661no—
128SD32—21no—
129SD32—1661no—
130M32—831no—
131M32C2×D8311yescharacteristic, complemented
132#106—831no—
133#103—1661no—
134#41—21no—
135#41—3321no—
136#41—3321no—
137#41—1661no—
138#41—1661no—
139#41—1661no—
140C2×Q8—21no—
141C2×Q8—1661no—
142C2×D4—3321no—
143C2×D4—1661no—
144C22×C4—3321no—
145C22×C4—831no—
146Q16—21no—
147Q16—21no—
148Q16—831no—
149Q16—1661no—
150Q16C22⋊D8311yescharacteristic
151SD16—21no—
152SD16—1661no—
153SD16—1661no—
154SD16—1661no—
155D8—831no—
156D8—1661no—
157D8C22⋊D8311yescharacteristic
158M16—1661no—
159M16—1661no—
160C2×C8—831no—
161C2×C8C22×D8311yescharacteristic
162C4⋊C4—3321no—
163C4⋊C4—1661no—
164C22⋊C4—3321no—
165C16—21no—
166C16—1661no—
167C23—3321no—
168C23—831no—
169Q8—21no—
170Q8—41no—
171Q8—41no—
172Q8—3321no—
173Q8—1661no—
174D4—21no—
175D4—41no—
176D4—3321no—
177D4—3321no—
178D4—3321no—
179D4—3321no—
180D4—1661no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—3321no—
184C2×C4—3321no—
185C2×C4—3321no—
186C2×C4—3321no—
187C2×C4—3321no—
188C2×C4—831no—
189C2×C4—1661no—
190C2×C4—1661no—
191C2×C4D4×D8311yescharacteristic
192C8—1661no—
193C8D4×D8311yescharacteristic
194C8C2×C22⋊D8311yescharacteristic
195C22—41no—
196C22—3321no—
197C22—3321no—
198C22—6642no—
199C22—1661no—
200C22D8×D8311yescharacteristic
201C4—41no—
202C4—81no—
203C4—3321no—
204C4—6641no—
205C4—1661no—
206C4#1458176011yescharacteristic
207C4#1458181611yescharacteristic
208C2—21no—
209C2—81no—
210C2—3321no—
211C2—6641no—
212C2#1912783311yescharacteristic
subgroupG/H
centerC2#19127833
commutator, cosocleC2×C8×C83C23
FrattiniC2×C8C22×D83
Fitting#510×C83C2
socleC2×C83#447
prime radical#19128498C2
hypercenter#510D83
hypocenterC83#2191
LCSG ▷ C2×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #510

Complements

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

NH ≅ G/Ntype
2. #19128142C2semidirect
3. #19127956C2semidirect
4. #19127512C2semidirect
5. #19127511C2semidirect
6. #19127482C2semidirect
7. #510×C83C2semidirect
19. #14581732C22semidirect
20. #14581724C22semidirect
23. C2×Q16×C83C22semidirect
26. M32×C83C22semidirect
107. C83#2191semidirect
110. #510D83semidirect
123. C2×Q16C2×D83semidirect
131. M32C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191284982
#191281422
#191279562
#191275122
#191275112
#191274822
#510×C832
#219183