Finite Group Databasebeta

← #24907989

Subgroups — #24907989

Order 10624.

344 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 79 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#19128467C211yesmaximal, characteristic, complemented
2#19128270C211yesmaximal, characteristic
3#19127707C211yesmaximal, characteristic, complemented
4#19127705C211yesmaximal, characteristic, complemented
5#19127674C211yesmaximal, characteristic, complemented
6#19127457C211yesmaximal, characteristic, complemented
7H(3,ℤ/4)×C83C211yesmaximal, characteristic, complemented
8C2×#13050590C2211yescharacteristic
9C22×D332C2211yescharacteristic
10#14581800—22no—
11#14581798—22no—
12C2×#13050574C2211yescharacteristic
13M16×D83—22no—
14#14581789C412yescomplemented
15#14581780C412yescomplemented
16#14581767—22no—
17#14581762—22no—
18#14581751C2211yescharacteristic
19#14581714—22no—
20#14581712—22no—
21#14581706—22no—
22C2×M16×C83C2211yescharacteristic
23#117×C83C2211yescharacteristic
24#117×C83C2211yescharacteristic
25C23×D83—21no—
26#13050590—21no—
27#13050590—21no—
28#13050590—21no—
29#13050590—21no—
30C2×Dic166C2×C411yescharacteristic
31C2×D332—21no—
32C2×D332—21no—
33C2×D332—21no—
34C2×D332—21no—
35C2×D332—21no—
36C2×D332—24no—
37C2×D332C2×C412yes—
38C2×D332C2×C411yescharacteristic
39C2×C22⋊D83D412yes—
40C2×C4×D83D412yes—
41#13050574—21no—
42#13050574—21no—
43#13050573—24no—
44C2×C83⋊C8D412yescomplemented
45C8×D83—24no—
46C4⋊Dic83—21no—
47C22⋊Dic83—21no—
48(C2×D83)⋊C4—24no—
49C4×Dic83—21no—
50C22×C4×C83C2311yescharacteristic
51M16×C83—21no—
52M16×C83—21no—
53C2×C8×C83D412yescomplemented
54C4⋊C4×C83—21no—
55C4⋊C4×C83—21no—
56C22⋊C4×C83—21no—
57C22⋊C4×C83—21no—
58C42×C83—21no—
59C42×C83C2×C412yes—
60C23×C83C2×D411yescharacteristic
61C22×D83—22no—
62C22×D83—41no—
63C22×D83—41no—
64C22×D83—82no—
65C22×D83C22⋊C412yes—
66C2×C4×C83—22no—
67C2×C4×C83—41no—
68C2×C4×C83C22×C411yescharacteristic
69C2×C4×C83C2×D411yescharacteristic
70C2×C4×C83C2×D411yescharacteristic
71C2×C4×C83C2×D411yescharacteristic
72C2×C4×C83#4111yescharacteristic
73C2×C4×C83#4111yescharacteristic
74C22⋊D83—22no—
75C22⋊D83—22no—
76C2×Dic83—41no—
77C2×Dic83C22⋊C412yes—
78D332—21no—
79D332—21no—
80D332—22no—
81D332—22no—
82D332—41no—
83D332—41no—
84D332—84no—
85C4×D83—22no—
86C4×D83—22no—
87Dic166—21no—
88Dic166—21no—
89C8×C83—22no—
90C83⋊C8—22no—
91C22×C83—41no—
92C22×C83C2×C22⋊C411yescharacteristic
93C22×C83#12111yescharacteristic
94C22×C83#12111yescharacteristic
95C2×D83—22no—
96C2×D83—42no—
97C2×D83—44no—
98C2×D83—42no—
99C2×D83—82no—
100C2×D83—82no—
101C4×C83—42no—
102C4×C83—81no—
103C4×C83C4×D411yescharacteristic
104C4×C83C4×D411yescharacteristic
105C4×C83#12011yescharacteristic
106C4×C83#12311yescharacteristic
107Dic83—22no—
108Dic83—81no—
109C2×C83—21no—
110C2×C83—21no—
111C2×C83—21no—
112C2×C83#38611yescharacteristic
113D83—42no—
114D83—82no—
115#1910—831nomaximal, Sylow
116C83#191011yescharacteristic, complemented, Sylow
117#568—831no—
118#535—831no—
119#418—831no—
120#418—831no—
121#410—831no—
122H(3,ℤ/4)—831no—
123H(3,ℤ/4)D8311yescharacteristic, complemented
124C2×#41—831no—
125C22×D4—831no—
126#131—1662no—
127#131—1662no—
128C2×M16—831no—
129C2×M16—1662no—
130C2×M16C2×D8311yescharacteristic
131#127—832no—
132#124—832no—
133#121—1662no—
134#120—1662no—
135#117—831no—
136#117C2×D8311yescharacteristic
137#117C2×D8311yescharacteristic
138#102—1662no—
139#102—1662no—
140#99—1662no—
141C24—1661no—
142#41—1661no—
143#41—1661no—
144#41—1661no—
145#41—1661no—
146C2×Q8—831no—
147C2×D4—832no—
148C2×D4—831no—
149C2×D4—832no—
150C2×D4—1664no—
151C2×D4—1661no—
152C2×D4—1661no—
153C2×D4—1661no—
154C2×D4—1661no—
155C2×D4—1661no—
156C22×C4—832no—
157C22×C4C22×D8311yescharacteristic
158M16—21no—
159M16—21no—
160M16—1664no—
161M16—1661no—
162M16—1661no—
163C2×C8—832no—
164C2×C8—1664no—
165C2×C8D33212yescomplemented
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—1661no—
169C22⋊C4—21no—
170C22⋊C4—21no—
171C22⋊C4—1664no—
172C22⋊C4—1661no—
173C42—21no—
174C42—1661no—
175C42C4×D8312yes—
176C23—3321no—
177C23—3321no—
178C23—832no—
179C23—6642no—
180C23—1662no—
181C23D4×D8311yescharacteristic
182Q8—1661no—
183Q8—1661no—
184D4—3321no—
185D4—3321no—
186D4—6644no—
187D4—1662no—
188D4—1662no—
189D4—1662no—
190D4—1661no—
191D4—1662no—
192D4—1661no—
193C2×C4—22no—
194C2×C4—41no—
195C2×C4—3321no—
196C2×C4—832no—
197C2×C4—1662no—
198C2×C4—1662no—
199C2×C4C2×C4×D8311yescharacteristic
200C2×C4D4×D8311yescharacteristic
201C2×C4D4×D8311yescharacteristic
202C2×C4C2×D33211yescharacteristic
203C2×C4#1305059011yescharacteristic
204C2×C4#1305059111yescharacteristic
205C8—22no—
206C8—1662no—
207C22—41no—
208C22—3322no—
209C22—3324no—
210C22—3322no—
211C22—6642no—
212C22—6642no—
213C22—1662no—
214C22C22⋊C4×D8311yescharacteristic
215C22#1458176511yescharacteristic
216C22#1458176711yescharacteristic
217C4—42no—
218C4—81no—
219C4—6641no—
220C4—1662no—
221C4#1458175411yescharacteristic
222C4C4×D33211yescharacteristic
223C4#1458176211yescharacteristic
224C4#1458177711yescharacteristic
225C2—21no—
226C2—21no—
227C2—21no—
228C2—3322no—
229C2—6642no—
230C2#1912757911yescharacteristic
subgroupG/H
centerC2#19127579
commutatorC2×C4×C83C22×C4
FrattiniC22×C4C22×D83
FittingH(3,ℤ/4)×C83C2
socleC2×C83#386
prime radical#19128270C2
cosocleC22×C4×C83C23
hypercenterH(3,ℤ/4)D83
hypocenterC83#1910
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C22×C4 ◁ H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
1. #19128467C2semidirect
3. #19127707C2semidirect
4. #19127705C2semidirect
5. #19127674C2semidirect
6. #19127457C2semidirect
7. H(3,ℤ/4)×C83C2semidirect
14. #14581789C4semidirect
15. #14581780C4semidirect
44. C2×C83⋊C8D4semidirect
53. C2×C8×C83D4semidirect
116. C83#1910semidirect
123. H(3,ℤ/4)D83semidirect
165. C2×C8D332semidirect

Maximal subgroups

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

subgroupindex
#191284672
#191282702
#191277072
#191277052
#191276742
#191274572
H(3,ℤ/4)×C832
#191083