Finite Group Databasebeta

← #24916980

Subgroups — #24916980

Order 10624.

490 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C2×#14581822C211yesmaximal, characteristic, complemented
2C22×#13050578C211yesmaximal, characteristic, complemented
3#19128178C211yesmaximal, characteristic, complemented
4#19127920C214yesmaximal, complemented
5#19127898C214yesmaximal, complemented
6#19127728C211yesmaximal, characteristic, complemented
7#19127706C211yesmaximal, characteristic, complemented
8#19127697C211yesmaximal, characteristic, complemented
9#535×C83C211yesmaximal, characteristic
10C22×Dic166C2211yescharacteristic
11#14581822—24no—
12C2×#13050580C2212yescomplemented
13C2×#13050578—24no—
14C2×#13050578C2214yes—
15C2×#13050578C2212yescomplemented
16C2×#13050574C2211yescharacteristic
17C22×C83⋊C8C2211yescharacteristic
18C4×Dic166C2214yescomplemented
19#14581757—22no—
20#14581751C2211yescharacteristic
21#14581748—22no—
22C4⋊(C83⋊C8)C2214yescomplemented
23#14581716C2212yescomplemented
24#14581713C2214yes—
25#14581713C2212yescomplemented
26C2×#41×C83C2211yescharacteristic
27C22×D4×C83C2211yescharacteristic
28#127×C83C2212yes—
29#124×C83C2212yes—
30#121×C83—22no—
31#120×C83—22no—
32#117×C83C2211yescharacteristic
33C2×Dic166—21no—
34C2×Dic166—22no—
35C2×Dic166D414yescomplemented
36C2×Dic166C2312yes—
37C22×Dic83—21no—
38#13050580—24no—
39#13050578—28no—
40#13050578—24no—
41#13050574—22no—
42C2×C83⋊C8—22no—
43C2×C83⋊C8C2312yes—
44C2×C83⋊C8C2312yes—
45C4⋊Dic83C2312yes—
46Dic83⋊C4—24no—
47C22⋊Dic83—21no—
48C4×Dic83—24no—
49C4×Dic83—21no—
50C24×C83—21no—
51#41×C83—22no—
52#41×C83—22no—
53C2×Q8×C83C2311yescharacteristic
54C2×D4×C83—21no—
55C2×D4×C83—22no—
56C2×D4×C83—21no—
57C2×D4×C83—24no—
58C2×D4×C83—22no—
59C2×D4×C83C2312yes—
60C2×D4×C83C2311yescharacteristic
61C22×C4×C83—21no—
62C22×C4×C83C2311yescharacteristic
63C4⋊C4×C83D412yes—
64C22⋊C4×C83—24no—
65C22⋊C4×C83D412yes—
66C42×C83C2312yes—
67C23×C83—22no—
68C23×C83—21no—
69C23×C83—42no—
70C23×C83—82no—
71C23×C83C2×D411yescharacteristic
72Q8×C83—21no—
73Q8×C83—21no—
74D4×C83—22no—
75D4×C83—21no—
76D4×C83—22no—
77D4×C83—21no—
78D4×C83—42no—
79D4×C83—42no—
80D4×C83—84no—
81C2×C4×C83—21no—
82C2×C4×C83—42no—
83C2×C4×C83C2×D412yes—
84C2×C4×C83C2×D414yes—
85C2×C4×C83C2×D411yescharacteristic
86C2×C4×C83#4112yes—
87C2×C4×C83C2411yescharacteristic
88C2×Dic83—24no—
89C2×Dic83—22no—
90C2×Dic83—22no—
91Dic166—24no—
92Dic166—22no—
93Dic166C2×D414yes—
94C83⋊C8—22no—
95C83⋊C8—22no—
96C22×C83—22no—
97C22×C83—41no—
98C22×C83—42no—
99C22×C83—42no—
100C22×C83—44no—
101C22×C83—84no—
102C22×C83#12112yes—
103C22×C83C22×D411yescharacteristic
104C4×C83—24no—
105C4×C83—41no—
106C4×C83#12112yes—
107C4×C83C22×D411yescharacteristic
108C4×C83C2×#4111yescharacteristic
109Dic83—24no—
110Dic83—42no—
111C2×C83—22no—
112C2×C83—81no—
113C2×C83—82no—
114C2×C83C2×#12111yescharacteristic
115C2×C83#57712yes—
116#3043—831nomaximal, Sylow
117C83#304311yescharacteristic, complemented, Sylow
118C2×#137—831no—
119C22×SD16—831no—
120#535D8311yescharacteristic
121#519—831no—
122#464—834no—
123#460—834no—
124#424—831no—
125#418—831no—
126#416—831no—
127C2×#41C2×D8311yescharacteristic
128C22×Q8—831no—
129C22×D4C2×D8311yescharacteristic
130#137—1664no—
131C2×Q16—832no—
132C2×SD16—834no—
133C2×SD16—832no—
134C2×SD16—1664no—
135C2×M16—831no—
136C22×C8—831no—
137#127C2×D8312yes—
138#124C2×D8312yes—
139#121—22no—
140#120—22no—
141C4×Q8—834no—
142C4×Q8—1662no—
143#117—831no—
144#117—1662no—
145#117C2×D8311yescharacteristic
146C4⋊C8—834no—
147#103—832no—
148#102—832no—
149#102—834no—
150C24—21no—
151#41—22no—
152#41—22no—
153C2×Q8—834no—
154C2×Q8—832no—
155C2×Q8—1661no—
156C2×Q8—1662no—
157C2×Q8C22×D8311yescharacteristic
158C2×D4—21no—
159C2×D4—22no—
160C2×D4—21no—
161C2×D4—24no—
162C2×D4—22no—
163C2×D4C22×D8312yes—
164C2×D4C22×D8311yescharacteristic
165C22×C4—21no—
166C22×C4—1661no—
167C22×C4C22×D8311yescharacteristic
168Q16—1664no—
169SD16—1668no—
170SD16—1664no—
171M16—1662no—
172C2×C8—832no—
173C2×C8—832no—
174C2×C8—1662no—
175C4⋊C4—832no—
176C4⋊C4—1664no—
177C4⋊C4C22⋊D8312yes—
178C22⋊C4—24no—
179C22⋊C4—1661no—
180C22⋊C4C22⋊D8312yes—
181C42—1661no—
182C42—1664no—
183C42C22×D8312yes—
184C23—22no—
185C23—21no—
186C23—42no—
187C23—82no—
188C23C2×C22⋊D8311yescharacteristic
189Q8—21no—
190Q8—21no—
191Q8—834no—
192Q8—1664no—
193Q8—1662no—
194D4—22no—
195D4—21no—
196D4—22no—
197D4—21no—
198D4—42no—
199D4—42no—
200D4—84no—
201C2×C4—21no—
202C2×C4—42no—
203C2×C4—1662no—
204C2×C4—1664no—
205C2×C4—1662no—
206C2×C4D4×D8312yes—
207C2×C4C2×C22⋊D8314yes—
208C2×C4C2×C22⋊D8311yescharacteristic
209C2×C4#1305058912yes—
210C2×C4C23×D8311yescharacteristic
211C8—1662no—
212C8—1662no—
213C22—22no—
214C22—41no—
215C22—42no—
216C22—42no—
217C22—44no—
218C22—84no—
219C22#1458176312yes—
220C22C22×C22⋊D8311yescharacteristic
221C4—24no—
222C4—41no—
223C4—3322no—
224C4—1664no—
225C4#1458176312yes—
226C4C2×D4×D8311yescharacteristic
227C4C2×#1305058911yescharacteristic
228C2—22no—
229C2—82no—
230C2—81no—
231C2C2×#1458176311yescharacteristic
232C2#1912851212yes—
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting, prime radical#535×C83C2
socleC22×C83C22×D4
hypercenter#535D83
hypocenterC83#3043
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ #117 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#14581822C2semidirect
2. C22×#13050578C2semidirect
3. #19128178C2semidirect
4. #19127920C2semidirect
5. #19127898C2semidirect
6. #19127728C2semidirect
7. #19127706C2semidirect
8. #19127697C2semidirect
12. C2×#13050580C22semidirect
15. C2×#13050578C22semidirect
18. C4×Dic166C22semidirect
22. C4⋊(C83⋊C8)C22semidirect
23. #14581716C22semidirect
25. #14581713C22semidirect
35. C2×Dic166D4semidirect
117. C83#3043semidirect

Maximal subgroups

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

subgroupindex
C2×#145818222
C22×#130505782
#191281782
#191279202
#191278982
#191277282
#191277062
#191276972
#535×C832
#304383