Finite Group Databasebeta

← #24917117

Subgroups — #24917117

Order 10624.

460 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 133 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×#14581800C211yesmaximal, characteristic
3#19128277C211yesmaximal, characteristic
4#19128275C211yesmaximal, characteristic
5#19127958C214yesmaximal, complemented
6#19127941C214yesmaximal, complemented
7#19127737C211yesmaximal, characteristic, complemented
8#19127711C211yesmaximal, characteristic, complemented
9#419×C83C211yesmaximal, characteristic, complemented
10C2×#13050590C2211yescharacteristic
11C22×Dic166C2211yescharacteristic
12C2×#13050580—24no—
13C2×#13050580C2212yes—
14C2×#13050580C2212yes—
15#14581800—24no—
16C2×#13050573C2212yes—
17C22×C83⋊C8C2211yescharacteristic
18C2×C8×D83C2212yes—
19#14581792—21no—
20#14581790—21no—
21#14581781—21no—
22#14581780—21no—
23#14581776—22no—
24#14581775—22no—
25#14581772C2214yes—
26#14581768C2214yes—
27#14581751C2211yescharacteristic
28#14581725C2212yes—
29#14581723C2212yes—
30#14581717C2214yes—
31C22×Q8×C83C2211yescharacteristic
32C2×M16×C83C2211yescharacteristic
33#117×C83C2211yescharacteristic
34#103×C83C2214yes—
35#13050590—22no—
36#13050590—22no—
37C2×Dic166—22no—
38C2×Dic166—22no—
39C2×Dic166—21no—
40C2×Dic166D411yescharacteristic
41C2×Dic166C2312yes—
42C2×D332D411yescharacteristic
43C2×C22⋊D83D412yes—
44C22×Dic83—21no—
45C2×C4×D83C2312yes—
46#13050580—24no—
47#13050580—24no—
48#13050573—24no—
49C2×C83⋊C8D414yes—
50C2×C83⋊C8C2311yescharacteristic
51C2×C83⋊C8C2311yescharacteristic
52C8×D83—24no—
53C4⋊Dic83—24no—
54C4⋊Dic83C2312yes—
55Dic83⋊C4—24no—
56C22⋊Dic83—21no—
57(C2×D83)⋊C4—24no—
58(C2×D83)⋊C4—24no—
59C4×Dic83—21no—
60C2×Q8×C83—21no—
61C2×Q8×C83—22no—
62C2×Q8×C83—22no—
63C2×Q8×C83C2312yes—
64C22×C4×C83—21no—
65C22×C4×C83C2311yescharacteristic
66M16×C83—22no—
67C2×C8×C83C2312yes—
68C4⋊C4×C83C2312yes—
69C22⋊C4×C83—21no—
70C42×C83—21no—
71C23×C83C2×D411yescharacteristic
72C22×D83C2×D412yes—
73Q8×C83—22no—
74Q8×C83—22no—
75Q8×C83—42no—
76C2×C4×C83—22no—
77C2×C4×C83—22no—
78C2×C4×C83—42no—
79C2×C4×C83C2×D411yescharacteristic
80C2×C4×C83C2×D412yes—
81C2×C4×C83#4112yes—
82C2×C4×C83C2411yescharacteristic
83C22⋊D83—24no—
84C2×Dic83—22no—
85C2×Dic83—22no—
86C2×Dic83—42no—
87C2×Dic83C2×D412yes—
88D332—22no—
89C4×D83—22no—
90C4×D83—22no—
91Dic166—22no—
92Dic166—22no—
93Dic166—22no—
94Dic166—42no—
95C8×C83—22no—
96C83⋊C8C2×D412yes—
97C83⋊C8C2×D412yes—
98C22×C83—22no—
99C22×C83#12112yes—
100C22×C83C22×D411yescharacteristic
101C2×D83—22no—
102C2×D83—24no—
103C4×C83—42no—
104C4×C83—42no—
105C4×C83#12112yes—
106C4×C83C22×D411yescharacteristic
107C4×C83C2×#4111yescharacteristic
108Dic83—22no—
109Dic83—42no—
110Dic83—42no—
111C2×C83—22no—
112C2×C83C2×#12111yescharacteristic
113C2×C83#57812yes—
114D83—42no—
115#3059—831nomaximal, Sylow
116C83#305911yescharacteristic, complemented, Sylow
117C22×Q16—831no—
118C2×#131—831no—
119#536—831no—
120#536—831no—
121#470—834no—
122#467—834no—
123#427—831no—
124#419—831no—
125#419D8311yescharacteristic, complemented
126C2×#41—831no—
127C22×Q8—831no—
128C22×Q8C2×D8311yescharacteristic
129C2×Q16—832no—
130C2×Q16—832no—
131C2×Q16—1664no—
132#131—1664no—
133C2×M16—832no—
134C2×M16C2×D8311yescharacteristic
135C22×C8—831no—
136C22×C8—832no—
137C4⋊Q8—1661no—
138C4⋊Q8—1661no—
139#124—1661no—
140#124—1661no—
141#123—1662no—
142#123—1662no—
143C22⋊Q8—834no—
144C22⋊Q8—834no—
145#117—831no—
146#117C2×D8311yescharacteristic
147#107—832no—
148#106—832no—
149#103—834no—
150#103C2×D8314yes—
151#41—1662no—
152#41—1662no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—831no—
157C2×Q8—832no—
158C2×Q8—1662no—
159C2×Q8—1662no—
160C2×Q8—1661no—
161C2×Q8C22×D8312yes—
162C2×D4—832no—
163C2×D4—831no—
164C22×C4—21no—
165C22×C4—832no—
166C22×C4—1661no—
167C22×C4C22×D8311yescharacteristic
168Q16—1664no—
169Q16—1664no—
170M16—22no—
171M16—1664no—
172C2×C8—834no—
173C2×C8—831no—
174C2×C8—831no—
175C2×C8—1664no—
176C2×C8C22×D8312yes—
177C4⋊C4—832no—
178C4⋊C4—1664no—
179C4⋊C4—1664no—
180C4⋊C4C22×D8312yes—
181C22⋊C4—21no—
182C22⋊C4—1664no—
183C22⋊C4—1664no—
184C22⋊C4—1661no—
185C42—21no—
186C42—1661no—
187C23—832no—
188C23D4×D8311yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—42no—
192Q8—3322no—
193Q8—1662no—
194Q8—1662no—
195Q8—1662no—
196D4—1664no—
197D4—1662no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—42no—
201C2×C4—3322no—
202C2×C4—832no—
203C2×C4—1662no—
204C2×C4—1662no—
205C2×C4—1662no—
206C2×C4—1662no—
207C2×C4D4×D8311yescharacteristic
208C2×C4D4×D8312yes—
209C2×C4#1305059012yes—
210C2×C4C23×D8311yescharacteristic
211C8—22no—
212C8—832no—
213C8—832no—
214C22—22no—
215C22—1662no—
216C22—1664no—
217C22#1458176512yes—
218C22C2×D4×D8311yescharacteristic
219C4—42no—
220C4—42no—
221C4—3322no—
222C4—3322no—
223C4—1662no—
224C4#1458176512yes—
225C4C2×D4×D8311yescharacteristic
226C4C2×#1305059011yescharacteristic
227C2—22no—
228C2—3322no—
229C2C2×#1458176511yescharacteristic
230C2#1912851712yes—
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#419×C83C2
socleC22×C83C22×D4
prime radicalC2×#13050590C22
hypercenter#419D83
hypocenterC83#3059
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. C22×#13050580C2semidirect
5. #19127958C2semidirect
6. #19127941C2semidirect
7. #19127737C2semidirect
8. #19127711C2semidirect
9. #419×C83C2semidirect
116. C83#3059semidirect
125. #419D83semidirect

Maximal subgroups

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

subgroupindex
C22×#130505802
C2×#145818002
#191282772
#191282752
#191279582
#191279412
#191277372
#191277112
#419×C832
#305983