Finite Group Databasebeta

← #24909407

Subgroups — #24909407

Order 10624.

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

normal subgroups only

This group has 226 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, complemented
2#19128137C211yesmaximal, characteristic, complemented
3#19127951C211yesmaximal, characteristic, complemented
4#19127512C211yesmaximal, characteristic, complemented
5#19127511C211yesmaximal, characteristic, complemented
6#19127482C211yesmaximal, characteristic, complemented
7#509×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#14581763—21no—
17#14581736—21no—
18#14581733—21no—
19#14581732C2211yescharacteristic, complemented
20#14581724C2211yescharacteristic, complemented
21#14581713—21no—
22#135×C83C2211yescharacteristic
23C2×D8×C83C2211yescharacteristic, complemented
24SD32×C83—21no—
25D16×C83—21no—
26M32×C83C2211yescharacteristic, complemented
27#13050591—21no—
28#13050590—21no—
29#13050590—41no—
30#13050589—21no—
31#13050589—41no—
32Q8×D83—21no—
33C2×C22⋊D83—41no—
34D4×D83—21no—
35D4×D83—41no—
36C2×C4×D83—41no—
37C2×C4×D83D411yescharacteristic
38Dic332D411yescharacteristic, complemented
39#13050580—21no—
40#13050579—21no—
41#13050578—21no—
42#13050577—21no—
43D664D411yescharacteristic, complemented
44#13050575—21no—
45#13050573—21no—
46#13050573—21no—
47C2×C83⋊C8D411yescharacteristic
48C4⋊Dic83—21no—
49C22⋊Dic83—41no—
50C83⋊C16—21no—
51#41×C83—21no—
52C2×D4×C83—21no—
53Q16×C83D411yescharacteristic, complemented
54SD16×C83—21no—
55D8×C83—21no—
56D8×C83—21no—
57D8×C83D411yescharacteristic, complemented
58C2×C8×C83C2311yescharacteristic
59C16×C83—21no—
60C23×C83—41no—
61C22×D83—41no—
62C22×D83D811yescharacteristic
63Q8×C83—21no—
64D4×C83—21no—
65D4×C83—41no—
66D4×C83—41no—
67D4×C83—41no—
68C2×C4×C83—41no—
69C2×C4×C83C2×D411yescharacteristic
70C22⋊D83—41no—
71C22⋊D83—41no—
72C22⋊D83—41no—
73C22⋊D83—82no—
74C2×Dic83—41no—
75C2×Dic83—41no—
76C2×Dic83D811yescharacteristic
77D332—21no—
78D332—41no—
79C4×D83—21no—
80C4×D83—21no—
81C4×D83—41no—
82C4×D83—41no—
83C4×D83—41no—
84Dic166—21no—
85Dic166—41no—
86C8×C83C2×D411yescharacteristic
87C8×C83C2×D411yescharacteristic
88C83⋊C8—21no—
89C22×C83—41no—
90C22×C83—81no—
91C22×C83—81no—
92C22×C83—81no—
93C22×C83C2×D811yescharacteristic
94C2×D83—21no—
95C2×D83—41no—
96C2×D83—41no—
97C2×D83—82no—
98C4×C83—41no—
99C4×C83#12011yescharacteristic
100C4×C83AGL(1,ℤ/8)11yescharacteristic
101Dic83—21no—
102Dic83—41no—
103Dic83—81no—
104C2×C83—21no—
105C2×C83—81no—
106C2×C83—161no—
107C2×C83#44711yescharacteristic
108D83—41no—
109D83—81no—
110#2190—831nomaximal, Sylow
111C83#219011yescharacteristic, complemented, Sylow
112#575—831no—
113#509—831no—
114#509D8311yescharacteristic, complemented
115#469—831no—
116#360—831no—
117#360—831no—
118#349—831no—
119C2×#41—1661no—
120#137—1661no—
121#137—1661no—
122AGL(1,ℤ/8)—1661no—
123AGL(1,ℤ/8)—1661no—
124#135—831no—
125#135—1661no—
126#135C2×D8311yescharacteristic
127C2×D8C2×D8311yescharacteristic, complemented
128C2×M16—831no—
129#121—1661no—
130SD32—21no—
131SD32—1661no—
132D16—21no—
133D16—1661no—
134M32—831no—
135M32C2×D8311yescharacteristic, complemented
136#106—831no—
137#102—1661no—
138#41—21no—
139#41—3321no—
140#41—3321no—
141#41—1661no—
142#41—1661no—
143#41—1661no—
144C2×Q8—1661no—
145C2×D4—21no—
146C2×D4—3321no—
147C2×D4—3321no—
148C2×D4—1661no—
149C22×C4—3321no—
150C22×C4—831no—
151Q16—831no—
152Q16—1661no—
153Q16C22⋊D8311yescharacteristic, complemented
154SD16—21no—
155SD16—1661no—
156SD16—1661no—
157SD16—1661no—
158D8—21no—
159D8—21no—
160D8—831no—
161D8—1661no—
162D8C22⋊D8311yescharacteristic, complemented
163M16—1661no—
164M16—1661no—
165C2×C8—831no—
166C2×C8C22×D8311yescharacteristic
167C4⋊C4—1661no—
168C22⋊C4—3321no—
169C16—21no—
170C16—1661no—
171C23—41no—
172C23—3321no—
173C23—831no—
174Q8—21no—
175Q8—3321no—
176Q8—1661no—
177D4—21no—
178D4—41no—
179D4—41no—
180D4—41no—
181D4—3321no—
182D4—3321no—
183D4—3321no—
184D4—3321no—
185D4—6642no—
186D4—1661no—
187C2×C4—41no—
188C2×C4—3321no—
189C2×C4—3321no—
190C2×C4—3321no—
191C2×C4—3321no—
192C2×C4—3321no—
193C2×C4—831no—
194C2×C4—1661no—
195C2×C4—1661no—
196C2×C4D4×D8311yescharacteristic
197C8—1661no—
198C8D4×D8311yescharacteristic
199C8C2×C22⋊D8311yescharacteristic
200C22—41no—
201C22—81no—
202C22—81no—
203C22—81no—
204C22—3321no—
205C22—3321no—
206C22—6642no—
207C22—1661no—
208C22D8×D8311yescharacteristic
209C4—41no—
210C4—3321no—
211C4—6641no—
212C4—1661no—
213C4#1458176011yescharacteristic
214C4#1458181611yescharacteristic
215C2—21no—
216C2—81no—
217C2—161no—
218C2—3321no—
219C2—6641no—
220C2#1912783311yescharacteristic
subgroupG/H
centerC2#19127833
commutator, cosocleC2×C8×C83C23
FrattiniC2×C8C22×D83
Fitting#509×C83C2
socleC2×C83#447
hypercenter#509D83
hypocenterC83#2190
LCSG ▷ C2×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #19128498C2semidirect
2. #19128137C2semidirect
3. #19127951C2semidirect
4. #19127512C2semidirect
5. #19127511C2semidirect
6. #19127482C2semidirect
7. #509×C83C2semidirect
19. #14581732C22semidirect
20. #14581724C22semidirect
23. C2×D8×C83C22semidirect
26. M32×C83C22semidirect
38. Dic332D4semidirect
43. D664D4semidirect
53. Q16×C83D4semidirect
57. D8×C83D4semidirect
111. C83#2190semidirect
114. #509D83semidirect
127. C2×D8C2×D83semidirect
135. M32C2×D83semidirect
153. Q16C22⋊D83semidirect
162. D8C22⋊D83semidirect

Maximal subgroups

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

subgroupindex
#191284982
#191281372
#191279512
#191275122
#191275112
#191274822
#509×C832
#219083