Finite Group Databasebeta

← #24918661

Subgroups — #24918661

Order 10624.

366 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 121 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#19128111C212yesmaximal, complemented
3#19128067C212yesmaximal, complemented
4#19128025C214yesmaximal, complemented
5#19127988C212yesmaximal, complemented
6#19127978C212yesmaximal, complemented
7#19127758C211yesmaximal, characteristic, complemented
8#533×C83C211yesmaximal, characteristic, complemented
9C2×#13050590—21no—
10#14581792C2211yescharacteristic
11#14581789C2211yescharacteristic
12#14581780—21no—
13#14581766—21no—
14#14581766C2212yes—
15C4×D332—21no—
16C4×D332C2212yes—
17#14581725C2214yes—
18#14581723C2214yes—
19C4⋊(C83⋊C8)C2212yes—
20#14581712C2214yes—
21#14581712C2214yes—
22C22⋊(C83⋊C8)C2212yes—
23#14581703C2212yes—
24C4⋊Q8×C83C2211yescharacteristic
25C4⋊Q8×C83C2211yescharacteristic
26#125×C83C2212yes—
27C22⋊Q8×C83—21no—
28C22⋊Q8×C83—21no—
29C4×Q8×C83—21no—
30C4×Q8×C83—21no—
31#117×C83C2212yes—
32C2×C42×C83C2211yescharacteristic
33#13050590—44no—
34C2×Dic166—21no—
35C2×D332—21no—
36C2×D332—21no—
37C2×D332C2312yes—
38C2×C22⋊D83—22no—
39C2×C4×D83—22no—
40C2×C83⋊C8C2314yes—
41C4⋊Dic83—21no—
42C4⋊Dic83C2312yes—
43(C2×D83)⋊C4—22no—
44(C2×D83)⋊C4—22no—
45C2×Q8×C83—21no—
46C2×Q8×C83—21no—
47C22×C4×C83D411yescharacteristic
48C22×C4×C83D411yescharacteristic
49C22×C4×C83C2311yescharacteristic
50C4⋊C4×C83—22no—
51C4⋊C4×C83—22no—
52C4⋊C4×C83—21no—
53C4⋊C4×C83—21no—
54C4⋊C4×C83C2312yes—
55C4⋊C4×C83C2312yes—
56C22⋊C4×C83—22no—
57C42×C83—22no—
58C42×C83D411yescharacteristic
59C42×C83D411yescharacteristic
60C42×C83C2311yescharacteristic
61C42×C83C2311yescharacteristic
62C23×C83C2×D411yescharacteristic
63C22×D83—22no—
64Q8×C83—42no—
65Q8×C83—42no—
66C2×C4×C83—22no—
67C2×C4×C83—22no—
68C2×C4×C83—22no—
69C2×C4×C83—22no—
70C2×C4×C83—41no—
71C2×C4×C83C2×D411yescharacteristic
72C2×C4×C83C2×D411yescharacteristic
73C2×C4×C83C2×D411yescharacteristic
74C2×C4×C83C2×D411yescharacteristic
75C2×C4×C83C2×D411yescharacteristic
76C2×C4×C83C2411yescharacteristic
77C22⋊D83—44no—
78C2×Dic83—22no—
79D332—24no—
80D332—42no—
81D332—42no—
82C4×D83—44no—
83Dic166—42no—
84C83⋊C8—24no—
85C22×C83—21no—
86C22×C83—21no—
87C22×C83—21no—
88C22×C83C22×D411yescharacteristic
89C2×D83—42no—
90C2×D83—42no—
91C2×D83—42no—
92C4×C83—21no—
93C4×C83—21no—
94C4×C83—21no—
95C4×C83—21no—
96C4×C83—42no—
97C4×C83—42no—
98C4×C83AGL(1,ℤ/8)12yes—
99C4×C832−1+412yes—
100Dic83—42no—
101C2×C83—41no—
102C2×C83#53611yescharacteristic
103C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
104C2×C83#57511yescharacteristic
105D83—82no—
106#3235—831nomaximal, Sylow
107C83#323511yescharacteristic, complemented, Sylow
108#533D8311yescharacteristic, complemented
109#532—831no—
110#501—832no—
111#490—832no—
112#482—834no—
113#476—832no—
114#474—832no—
115#432—831no—
116C2×#41—1661no—
117C4⋊Q8—831no—
118C4⋊Q8C2×D8311yescharacteristic
119C4⋊Q8C2×D8311yescharacteristic
120#127—831no—
121#125C2×D8312yes—
122#124—1661no—
123C22⋊Q8—21no—
124C22⋊Q8—21no—
125#121—832no—
126#121—1661no—
127C4×Q8—21no—
128C4×Q8—21no—
129C4×D4—832no—
130C4×D4—1661no—
131#117C2×D8312yes—
132C2×C42C2×D8311yescharacteristic
133#107—834no—
134#106—834no—
135C4⋊C8—832no—
136#102—834no—
137#102—834no—
138C22⋊C8—832no—
139#97—832no—
140#41—3324no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—1661no—
144C2×D4—832no—
145C2×D4—1662no—
146C2×D4—1661no—
147C2×D4—1661no—
148C22×C4—1662no—
149C22×C4C22⋊D8311yescharacteristic
150C22×C4C22⋊D8311yescharacteristic
151C22×C4C22×D8311yescharacteristic
152C2×C8—834no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—832no—
158C4⋊C4—1661no—
159C4⋊C4C22×D8312yes—
160C4⋊C4C22×D8312yes—
161C22⋊C4—22no—
162C22⋊C4—1662no—
163C22⋊C4—1662no—
164C42—22no—
165C42C22⋊D8311yescharacteristic
166C42C22⋊D8311yescharacteristic
167C42C22×D8311yescharacteristic
168C42C22×D8311yescharacteristic
169C23—1662no—
170C23C2×C22⋊D8311yescharacteristic
171Q8—42no—
172Q8—42no—
173Q8—3322no—
174D4—3322no—
175D4—3322no—
176D4—3324no—
177D4—1664no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—41no—
183C2×C4—3324no—
184C2×C4—1662no—
185C2×C4C2×C22⋊D8311yescharacteristic
186C2×C4C2×C22⋊D8311yescharacteristic
187C2×C4C2×C22⋊D8311yescharacteristic
188C2×C4C2×C22⋊D8311yescharacteristic
189C2×C4C2×C22⋊D8311yescharacteristic
190C2×C4C23×D8311yescharacteristic
191C8—1664no—
192C22—21no—
193C22—21no—
194C22—21no—
195C22—3322no—
196C22—3322no—
197C22—3322no—
198C22C22×C22⋊D8311yescharacteristic
199C4—21no—
200C4—21no—
201C4—21no—
202C4—21no—
203C4—42no—
204C4—42no—
205C4—3322no—
206C4#1458181712yes—
207C4#1458184012yes—
208C2—41no—
209C2—6642no—
210C2#1912827511yescharacteristic
211C2C2×#1458181711yescharacteristic
212C2#1912849911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#533×C83C2
socleC22×C83C22×D4
prime radical#19128251C2
hypercenter#533D83
hypocenterC83#3235
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. #19128111C2semidirect
3. #19128067C2semidirect
4. #19128025C2semidirect
5. #19127988C2semidirect
6. #19127978C2semidirect
7. #19127758C2semidirect
8. #533×C83C2semidirect
107. C83#3235semidirect
108. #533D83semidirect

Maximal subgroups

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

subgroupindex
#191282512
#191281112
#191280672
#191280252
#191279882
#191279782
#191277582
#533×C832
#323583