Finite Group Databasebeta

← #24918599

Subgroups — #24918599

Order 10624.

370 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 125 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#19128251C211yesmaximal, characteristic
2#19128104C211yesmaximal, characteristic, complemented
3#19128097C211yesmaximal, characteristic, complemented
4#19128049C211yesmaximal, characteristic, complemented
5#19128049C211yesmaximal, characteristic, complemented
6#19128025C214yesmaximal, complemented
7#19127998C214yesmaximal, complemented
8#19127754C211yesmaximal, characteristic, complemented
9#533×C83C211yesmaximal, characteristic, complemented
10C2×#13050590—21no—
11#14581792C2211yescharacteristic
12#14581789C2211yescharacteristic
13#14581780—21no—
14#14581766—21no—
15#14581766C2212yes—
16C4×D332—21no—
17C4×D332C2212yes—
18#14581725C2214yes—
19#14581723C2214yes—
20C4⋊(C83⋊C8)C2212yes—
21#14581712C2214yes—
22#14581712C2214yes—
23C22⋊(C83⋊C8)C2212yes—
24C4×C83⋊C8C2211yescharacteristic
25C4×C83⋊C8C2211yescharacteristic
26C4⋊Q8×C83C2211yescharacteristic
27C4⋊Q8×C83C2211yescharacteristic
28#125×C83C2212yes—
29C22⋊Q8×C83—21no—
30C22⋊Q8×C83—21no—
31C4×Q8×C83—21no—
32C4×Q8×C83—21no—
33#117×C83C2212yes—
34C2×C42×C83C2211yescharacteristic
35#13050590—44no—
36C2×Dic166—21no—
37C2×D332—21no—
38C2×D332—21no—
39C2×D332C2312yes—
40C2×C22⋊D83—22no—
41C2×C4×D83—22no—
42C2×C83⋊C8C2312yes—
43C2×C83⋊C8C2312yes—
44C4⋊Dic83—21no—
45C4⋊Dic83C2312yes—
46(C2×D83)⋊C4—22no—
47(C2×D83)⋊C4—22no—
48C2×Q8×C83—21no—
49C2×Q8×C83—21no—
50C22×C4×C83D411yescharacteristic
51C22×C4×C83D411yescharacteristic
52C22×C4×C83C2311yescharacteristic
53C4⋊C4×C83—22no—
54C4⋊C4×C83—22no—
55C4⋊C4×C83—21no—
56C4⋊C4×C83—21no—
57C4⋊C4×C83C2312yes—
58C4⋊C4×C83C2312yes—
59C22⋊C4×C83—22no—
60C42×C83—22no—
61C42×C83D411yescharacteristic
62C42×C83D411yescharacteristic
63C42×C83C2311yescharacteristic
64C42×C83C2311yescharacteristic
65C23×C83C2×D411yescharacteristic
66C22×D83—22no—
67Q8×C83—42no—
68Q8×C83—42no—
69C2×C4×C83—22no—
70C2×C4×C83—22no—
71C2×C4×C83—22no—
72C2×C4×C83—22no—
73C2×C4×C83—22no—
74C2×C4×C83C2×D411yescharacteristic
75C2×C4×C83C2×D411yescharacteristic
76C2×C4×C83C2×D411yescharacteristic
77C2×C4×C83C2×D411yescharacteristic
78C2×C4×C83C2×D411yescharacteristic
79C2×C4×C83C2411yescharacteristic
80C22⋊D83—44no—
81C2×Dic83—22no—
82D332—24no—
83D332—42no—
84D332—42no—
85C4×D83—44no—
86Dic166—42no—
87C83⋊C8—22no—
88C83⋊C8—22no—
89C22×C83—21no—
90C22×C83—21no—
91C22×C83—21no—
92C22×C83C22×D411yescharacteristic
93C2×D83—42no—
94C2×D83—42no—
95C2×D83—42no—
96C4×C83—21no—
97C4×C83—21no—
98C4×C83—21no—
99C4×C83—42no—
100C4×C83—42no—
101C4×C83#13512yes—
102C4×C83AGL(1,ℤ/8)12yes—
103C4×C832−1+412yes—
104Dic83—42no—
105C2×C83—41no—
106C2×C83#53611yescharacteristic
107C2×C83C2×#13511yescharacteristic
108C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
109D83—82no—
110#3228—831nomaximal, Sylow
111C83#322811yescharacteristic, complemented, Sylow
112#533D8311yescharacteristic, complemented
113#532—831no—
114#500—831no—
115#498—831no—
116#486—831no—
117#486—831no—
118#482—834no—
119#478—834no—
120#431—831no—
121C2×#41—1661no—
122C4⋊Q8—831no—
123C4⋊Q8C2×D8311yescharacteristic
124C4⋊Q8C2×D8311yescharacteristic
125#127—831no—
126#125C2×D8312yes—
127#124—1661no—
128C22⋊Q8—21no—
129C22⋊Q8—21no—
130#121—832no—
131#121—1661no—
132C4×Q8—21no—
133C4×Q8—21no—
134C4×D4—832no—
135C4×D4—1661no—
136#117C2×D8312yes—
137C2×C42C2×D8311yescharacteristic
138#107—834no—
139#106—834no—
140C4⋊C8—832no—
141#102—834no—
142#102—834no—
143C22⋊C8—832no—
144C4×C8—831no—
145C4×C8—831no—
146#41—3324no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—1661no—
150C2×D4—832no—
151C2×D4—1662no—
152C2×D4—1661no—
153C2×D4—1661no—
154C22×C4—1662no—
155C22×C4C22⋊D8311yescharacteristic
156C22×C4C22⋊D8311yescharacteristic
157C22×C4C22×D8311yescharacteristic
158C2×C8—832no—
159C2×C8—832no—
160C4⋊C4—22no—
161C4⋊C4—22no—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—832no—
165C4⋊C4—1661no—
166C4⋊C4C22×D8312yes—
167C4⋊C4C22×D8312yes—
168C22⋊C4—22no—
169C22⋊C4—1662no—
170C22⋊C4—1662no—
171C42—22no—
172C42C22⋊D8311yescharacteristic
173C42C22⋊D8311yescharacteristic
174C42C22×D8311yescharacteristic
175C42C22×D8311yescharacteristic
176C23—1662no—
177C23C2×C22⋊D8311yescharacteristic
178Q8—42no—
179Q8—42no—
180Q8—3322no—
181D4—3324no—
182D4—3322no—
183D4—3322no—
184D4—1664no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—3324no—
191C2×C4—1662no—
192C2×C4C2×C22⋊D8311yescharacteristic
193C2×C4C2×C22⋊D8311yescharacteristic
194C2×C4C2×C22⋊D8311yescharacteristic
195C2×C4C2×C22⋊D8311yescharacteristic
196C2×C4C2×C22⋊D8311yescharacteristic
197C2×C4C23×D8311yescharacteristic
198C8—1662no—
199C8—1662no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—3322no—
204C22—3322no—
205C22—3322no—
206C22C22×C22⋊D8311yescharacteristic
207C4—21no—
208C4—21no—
209C4—21no—
210C4—42no—
211C4—42no—
212C4—3322no—
213C4#1458181212yes—
214C4#1458181712yes—
215C4#1458184012yes—
216C2—41no—
217C2—6642no—
218C2#1912827511yescharacteristic
219C2C2×#1458181211yescharacteristic
220C2C2×#1458181711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#533×C83C2
socleC22×C83C22×D4
prime radical#19128251C2
hypercenter#533D83
hypocenterC83#3228
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
2. #19128104C2semidirect
3. #19128097C2semidirect
4. #19128049C2semidirect
5. #19128049C2semidirect
6. #19128025C2semidirect
7. #19127998C2semidirect
8. #19127754C2semidirect
9. #533×C83C2semidirect
111. C83#3228semidirect
112. #533D83semidirect

Maximal subgroups

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

subgroupindex
#191282512
#191281042
#191280972
#191280492
#191280252
#191279982
#191277542
#533×C832
#322883