Finite Group Databasebeta

← #24917115

Subgroups — #24917115

Order 10624.

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

normal subgroups only

This group has 460 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
1C22×#13050580C211yesmaximal, characteristic, complemented
2C2×#14581799C211yesmaximal, characteristic
3#19128272C211yesmaximal, characteristic
4#19127957C214yesmaximal, complemented
5#19127940C214yesmaximal, complemented
6#19127737C211yesmaximal, characteristic, complemented
7#19127710C211yesmaximal, characteristic, complemented
8#19127709C211yesmaximal, characteristic, complemented
9#536×C83C211yesmaximal, characteristic
10C22×Dic166C2211yescharacteristic
11C2×#13050580—24no—
12C2×#13050580C2212yes—
13C2×#13050580C2212yes—
14#14581799—24no—
15C2×#13050574C2212yes—
16C2×#13050574C2211yescharacteristic
17C22×C83⋊C8C2212yes—
18C22×C83⋊C8C2211yescharacteristic
19#14581790—21no—
20#14581779—21no—
21#14581773—22no—
22#14581771C2214yes—
23#14581751C2211yescharacteristic
24#14581725C2212yes—
25#14581723C2212yes—
26#14581716C2214yes—
27#14581715C2214yes—
28C2×#41×C83C2211yescharacteristic
29C22×Q8×C83C2211yescharacteristic
30C4⋊Q8×C83—21no—
31#124×C83—21no—
32#123×C83—22no—
33C22⋊Q8×C83C2214yes—
34#117×C83C2211yescharacteristic
35C2×Dic166—22no—
36C2×Dic166—22no—
37C2×Dic166—21no—
38C2×Dic166C2312yes—
39C22×Dic83—21no—
40#13050580—24no—
41#13050580—24no—
42#13050574—22no—
43#13050574—24no—
44C2×C83⋊C8—24no—
45C2×C83⋊C8D414yes—
46C2×C83⋊C8C2312yes—
47C2×C83⋊C8C2311yescharacteristic
48C2×C83⋊C8C2311yescharacteristic
49C4⋊Dic83C2312yes—
50Dic83⋊C4—24no—
51C22⋊Dic83—24no—
52C22⋊Dic83—21no—
53C4×Dic83—21no—
54#41×C83—22no—
55#41×C83—22no—
56C2×Q8×C83—21no—
57C2×Q8×C83—22no—
58C2×Q8×C83—22no—
59C2×Q8×C83D411yescharacteristic
60C2×Q8×C83C2312yes—
61C2×D4×C83D412yes—
62C2×D4×C83D411yescharacteristic
63C22×C4×C83—21no—
64C22×C4×C83C2312yes—
65C22×C4×C83C2311yescharacteristic
66C4⋊C4×C83—24no—
67C4⋊C4×C83C2312yes—
68C22⋊C4×C83—21no—
69C22⋊C4×C83—24no—
70C42×C83—21no—
71C23×C83C2×D412yes—
72C23×C83C2×D411yescharacteristic
73Q8×C83—22no—
74Q8×C83—22no—
75Q8×C83—22no—
76Q8×C83—42no—
77D4×C83—24no—
78D4×C83—22no—
79C2×C4×C83—22no—
80C2×C4×C83—22no—
81C2×C4×C83—22no—
82C2×C4×C83—22no—
83C2×C4×C83—42no—
84C2×C4×C83C2×D412yes—
85C2×C4×C83C2×D412yes—
86C2×C4×C83C2×D411yescharacteristic
87C2×C4×C83#4112yes—
88C2×C4×C83C2411yescharacteristic
89C2×Dic83—22no—
90C2×Dic83—22no—
91C2×Dic83—42no—
92Dic166—22no—
93Dic166—22no—
94Dic166—42no—
95C83⋊C8—22no—
96C83⋊C8C2×D412yes—
97C83⋊C8C2×D412yes—
98C22×C83—22no—
99C22×C83—22no—
100C22×C83—24no—
101C22×C83#12112yes—
102C22×C83C22×D411yescharacteristic
103C4×C83—22no—
104C4×C83—42no—
105C4×C83—42no—
106C4×C83#12112yes—
107C4×C83C22×D411yescharacteristic
108C4×C83C2×#4111yescharacteristic
109Dic83—42no—
110Dic83—42no—
111C2×C83—22no—
112C2×C83—42no—
113C2×C83C2×#12111yescharacteristic
114C2×C83#57812yes—
115#3059—831nomaximal, Sylow
116C83#305911yescharacteristic, complemented, Sylow
117C22×Q16—831no—
118C2×#131—831no—
119#536—831no—
120#536D8311yescharacteristic
121#470—834no—
122#467—834no—
123#427—831no—
124#419—831no—
125#419—831no—
126C2×#41C2×D8311yescharacteristic
127C22×Q8—831no—
128C22×Q8C2×D8311yescharacteristic
129C2×Q16—832no—
130C2×Q16—832no—
131C2×Q16—1664no—
132#131—1664no—
133C2×M16—831no—
134C2×M16—832no—
135C22×C8—831no—
136C22×C8—832no—
137C4⋊Q8—21no—
138C4⋊Q8—1661no—
139#124—21no—
140#124—1661no—
141#123—22no—
142#123—1662no—
143C22⋊Q8—834no—
144C22⋊Q8C2×D8314yes—
145#117—831no—
146#117C2×D8311yescharacteristic
147#107—832no—
148#106—832no—
149#103—834no—
150#103—834no—
151#41—22no—
152#41—22no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—832no—
157C2×Q8—1662no—
158C2×Q8—1662no—
159C2×Q8—1661no—
160C2×Q8C22⋊D8311yescharacteristic
161C2×Q8C22×D8312yes—
162C2×D4C22⋊D8312yes—
163C2×D4C22⋊D8311yescharacteristic
164C22×C4—21no—
165C22×C4—1661no—
166C22×C4C22×D8312yes—
167C22×C4C22×D8311yescharacteristic
168Q16—1664no—
169Q16—1664no—
170M16—1664no—
171M16—1662no—
172C2×C8—834no—
173C2×C8—831no—
174C2×C8—831no—
175C2×C8—832no—
176C2×C8—1664no—
177C4⋊C4—24no—
178C4⋊C4—832no—
179C4⋊C4—1664no—
180C4⋊C4C22×D8312yes—
181C22⋊C4—21no—
182C22⋊C4—24no—
183C22⋊C4—1661no—
184C22⋊C4—1664no—
185C42—21no—
186C42—1661no—
187C23C2×C22⋊D8312yes—
188C23C2×C22⋊D8311yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—22no—
192Q8—42no—
193Q8—3322no—
194Q8—1662no—
195Q8—1662no—
196D4—24no—
197D4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—42no—
203C2×C4—3322no—
204C2×C4—1662no—
205C2×C4—1662no—
206C2×C4D4×D8312yes—
207C2×C4C2×C22⋊D8312yes—
208C2×C4C2×C22⋊D8311yescharacteristic
209C2×C4#1305058912yes—
210C2×C4C23×D8311yescharacteristic
211C8—832no—
212C8—832no—
213C8—1662no—
214C22—22no—
215C22—22no—
216C22—24no—
217C22#1458176412yes—
218C22C22×C22⋊D8311yescharacteristic
219C4—22no—
220C4—42no—
221C4—42no—
222C4—3322no—
223C4—3322no—
224C4#1458176412yes—
225C4C2×D4×D8311yescharacteristic
226C4C2×#1305058911yescharacteristic
227C2—22no—
228C2—42no—
229C2C2×#1458176411yescharacteristic
230C2#1912851612yes—
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#536×C83C2
socleC22×C83C22×D4
prime radicalC2×#41×C83C22
hypercenter#536D83
hypocenterC83#3059
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C22×#13050580C2semidirect
4. #19127957C2semidirect
5. #19127940C2semidirect
6. #19127737C2semidirect
7. #19127710C2semidirect
8. #19127709C2semidirect
116. C83#3059semidirect

Maximal subgroups

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

subgroupindex
C22×#130505802
C2×#145817992
#191282722
#191279572
#191279402
#191277372
#191277102
#191277092
#536×C832
#305983