Finite Group Databasebeta

← #24917754

Subgroups — #24917754

Order 10624.

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

normal subgroups only

This group has 382 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#19128306C211yesmaximal, characteristic
2#19128025C212yesmaximal, complemented
3#19128008C212yesmaximal, complemented
4#19127901C212yesmaximal, complemented
5#19127840C212yesmaximal, complemented
6#19127799C212yesmaximal, complemented
7C4×#13050577C212yesmaximal, complemented
8#19127762C211yesmaximal, characteristic, complemented
9#519×C83C211yesmaximal, characteristic, complemented
10C2×#13050577C2212yes—
11C2×#13050577C2212yes—
12#14581789C2211yescharacteristic
13#14581784C2211yescharacteristic
14#14581777—22no—
15#14581766C2212yes—
16#14581765—22no—
17C4×D332C2212yes—
18#14581725C2212yes—
19#14581723C2212yes—
20C4⋊(C83⋊C8)C2212yes—
21#14581716C2212yes—
22#14581716C2212yes—
23#14581712C2212yes—
24#14581712C2212yes—
25C22⋊(C83⋊C8)C2212yes—
26#14581703C2211yescharacteristic
27C4×C83⋊C8C2211yescharacteristic
28C22×Q8×C83C2211yescharacteristic
29C4×Q8×C83—22no—
30C4×Q8×C83C2212yes—
31C4×Q8×C83C2212yes—
32#117×C83—22no—
33#117×C83C2211yescharacteristic
34#117×C83C2211yescharacteristic
35C2×D332—22no—
36C2×D332C2312yes—
37C2×C22⋊D83—22no—
38C2×C4×D83—22no—
39#13050577—24no—
40#13050577—24no—
41C2×C83⋊C8C2312yes—
42C2×C83⋊C8C2312yes—
43C4⋊Dic83C2312yes—
44Dic83⋊C4—22no—
45(C2×D83)⋊C4—22no—
46(C2×D83)⋊C4—22no—
47C2×Q8×C83—24no—
48C2×Q8×C83—21no—
49C2×Q8×C83C2312yes—
50C22×C4×C83—21no—
51C22×C4×C83C2311yescharacteristic
52C4⋊C4×C83—22no—
53C4⋊C4×C83—22no—
54C4⋊C4×C83D412yes—
55C4⋊C4×C83C2312yes—
56C22⋊C4×C83—21no—
57C22⋊C4×C83D412yes—
58C42×C83—22no—
59C42×C83—22no—
60C42×C83—21no—
61C42×C83C2311yescharacteristic
62C42×C83C2311yescharacteristic
63C23×C83C2×D411yescharacteristic
64C22×D83—22no—
65Q8×C83—24no—
66Q8×C83—22no—
67Q8×C83#4114yes—
68C2×C4×C83—24no—
69C2×C4×C83—22no—
70C2×C4×C83—22no—
71C2×C4×C83—22no—
72C2×C4×C83C2×D412yes—
73C2×C4×C83C2×D412yes—
74C2×C4×C83C2×D411yescharacteristic
75C2×C4×C83C2411yescharacteristic
76C22⋊D83—82no—
77C2×Dic83—22no—
78D332—22no—
79D332—22no—
80D332—82no—
81C4×D83—82no—
82C83⋊C8—22no—
83C83⋊C8—22no—
84C22×C83—21no—
85C22×C83—22no—
86C22×C83C22×D411yescharacteristic
87C2×D83—42no—
88C2×D83—42no—
89C2×D83—42no—
90C4×C83—24no—
91C4×C83—21no—
92C4×C83—22no—
93C4×C83—22no—
94C4×C83—42no—
95C4×C83C2×#4112yes—
96Dic83—42no—
97C2×C83—41no—
98C2×C83#52511yescharacteristic
99C2×C83#57712yes—
100D83—82no—
101#3126—831nomaximal, Sylow
102C83#312611yescharacteristic, complemented, Sylow
103#540—831no—
104#519D8311yescharacteristic, complemented
105#482—832no—
106#479—832no—
107#460—832no—
108#448—832no—
109#440—832no—
110C4×SD16—832no—
111#433—831no—
112C22×Q8C2×D8311yescharacteristic
113C2×SD16—832no—
114C2×SD16—832no—
115#127—831no—
116#125—831no—
117#123—1662no—
118#121—832no—
119#121—1662no—
120C4×Q8—22no—
121C4×Q8C2×D8312yes—
122C4×Q8C2×D8312yes—
123C4×D4—832no—
124#117—22no—
125#117C2×D8311yescharacteristic
126#117C2×D8311yescharacteristic
127#107—832no—
128#106—832no—
129C4⋊C8—832no—
130#103—832no—
131#103—832no—
132#102—832no—
133#102—832no—
134C22⋊C8—832no—
135#97—831no—
136C4×C8—831no—
137C2×Q8—24no—
138C2×Q8—21no—
139C2×Q8C22×D8312yes—
140C2×D4—832no—
141C2×D4—1662no—
142C2×D4—1662no—
143C22×C4—21no—
144C22×C4—1662no—
145C22×C4C22×D8311yescharacteristic
146SD16—1664no—
147SD16—1664no—
148C2×C8—832no—
149C2×C8—832no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—832no—
153C4⋊C4—1662no—
154C4⋊C4C22⋊D8312yes—
155C4⋊C4C22×D8312yes—
156C22⋊C4—21no—
157C22⋊C4—1662no—
158C22⋊C4—1662no—
159C22⋊C4C22⋊D8312yes—
160C42—22no—
161C42—22no—
162C42—21no—
163C42C22×D8311yescharacteristic
164C42C22×D8311yescharacteristic
165C23—1662no—
166C23C2×C22⋊D8311yescharacteristic
167Q8—24no—
168Q8—22no—
169Q8#1305059014yes—
170D4—6642no—
171D4—6642no—
172D4—1662no—
173D4—1662no—
174C2×C4—24no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—6642no—
179C2×C4—1662no—
180C2×C4C2×C22⋊D8312yes—
181C2×C4C2×C22⋊D8312yes—
182C2×C4C2×C22⋊D8311yescharacteristic
183C2×C4C23×D8311yescharacteristic
184C8—1662no—
185C8—1662no—
186C22—21no—
187C22—22no—
188C22—3322no—
189C22—3322no—
190C22—3322no—
191C22C22×C22⋊D8311yescharacteristic
192C4—24no—
193C4—21no—
194C4—22no—
195C4—22no—
196C4—42no—
197C4—3322no—
198C4C2×#1305059012yes—
199C2—41no—
200C2—6642no—
201C2#1912821211yescharacteristic
202C2#1912850912yes—
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#519×C83C2
socleC22×C83C22×D4
prime radical#19128306C2
hypercenter#519D83
hypocenterC83#3126
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ #117 ◁ #519

Complements

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

NH ≅ G/Ntype
2. #19128025C2semidirect
3. #19128008C2semidirect
4. #19127901C2semidirect
5. #19127840C2semidirect
6. #19127799C2semidirect
7. C4×#13050577C2semidirect
8. #19127762C2semidirect
9. #519×C83C2semidirect
102. C83#3126semidirect
104. #519D83semidirect

Maximal subgroups

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

subgroupindex
#191283062
#191280252
#191280082
#191279012
#191278402
#191277992
C4×#130505772
#191277622
#519×C832
#312683