Finite Group Databasebeta

← #24918092

Subgroups — #24918092

Order 10624.

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

normal subgroups only

This group has 362 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#19128306C211yesmaximal, characteristic
2#19128100C211yesmaximal, characteristic, complemented
3#19128075C211yesmaximal, characteristic, complemented
4#19128069C211yesmaximal, characteristic, complemented
5#19128067C211yesmaximal, characteristic, complemented
6#19127949C212yesmaximal, complemented
7#19127932C212yesmaximal, complemented
8#19127799C212yesmaximal, complemented
9C4×#13050577C212yesmaximal, complemented
10#19127657C211yesmaximal, characteristic, complemented
11#541×C83C211yesmaximal, characteristic, complemented
12C2×#13050577C2212yes—
13C2×#13050577C2212yes—
14C2×#13050574C2211yescharacteristic
15C22×C83⋊C8C2211yescharacteristic
16#14581789C2211yescharacteristic
17#14581784C2211yescharacteristic
18#14581777—22no—
19#14581766C2212yes—
20#14581765—22no—
21C4×D332C2212yes—
22#14581725C2212yes—
23#14581723C2212yes—
24#14581716C2212yes—
25#14581716C2212yes—
26#14581712C2212yes—
27#14581712C2212yes—
28#14581703C2211yescharacteristic
29#14581703C2211yescharacteristic
30C4×C83⋊C8C2211yescharacteristic
31C4×C83⋊C8C2211yescharacteristic
32C4⋊Q8×C83C2211yescharacteristic
33#126×C83—22no—
34#125×C83—22no—
35#125×C83C2211yescharacteristic
36C22⋊Q8×C83C2212yes—
37C4×Q8×C83C2212yes—
38#117×C83C2211yescharacteristic
39C2×D332—22no—
40C2×D332C2312yes—
41C2×C22⋊D83—22no—
42C2×C4×D83—22no—
43#13050577—24no—
44#13050577—24no—
45#13050574—22no—
46C2×C83⋊C8—22no—
47C2×C83⋊C8C2311yescharacteristic
48C2×C83⋊C8C2311yescharacteristic
49C2×C83⋊C8C2311yescharacteristic
50C2×C83⋊C8C2311yescharacteristic
51C4⋊Dic83C2312yes—
52Dic83⋊C4—22no—
53(C2×D83)⋊C4—22no—
54(C2×D83)⋊C4—22no—
55C2×Q8×C83C2312yes—
56C22×C4×C83C2311yescharacteristic
57C4⋊C4×C83—22no—
58C4⋊C4×C83—22no—
59C4⋊C4×C83—22no—
60C4⋊C4×C83—22no—
61C4⋊C4×C83D412yes—
62C4⋊C4×C83C2312yes—
63C22⋊C4×C83—22no—
64C22⋊C4×C83D412yes—
65C42×C83—22no—
66C42×C83C2311yescharacteristic
67C42×C83C2311yescharacteristic
68C23×C83C2×D411yescharacteristic
69C22×D83—22no—
70Q8×C83—22no—
71Q8×C83—22no—
72C2×C4×C83—21no—
73C2×C4×C83—21no—
74C2×C4×C83—22no—
75C2×C4×C83—22no—
76C2×C4×C83C2×D412yes—
77C2×C4×C83C2×D412yes—
78C2×C4×C83C2×D411yescharacteristic
79C2×C4×C83C2411yescharacteristic
80C22⋊D83—82no—
81C2×Dic83—22no—
82D332—22no—
83D332—22no—
84D332—82no—
85C4×D83—82no—
86C83⋊C8—21no—
87C83⋊C8—21no—
88C83⋊C8#4112yes—
89C83⋊C8#4112yes—
90C22×C83—21no—
91C22×C83—22no—
92C22×C83C22×D411yescharacteristic
93C2×D83—42no—
94C2×D83—42no—
95C2×D83—42no—
96C4×C83—21no—
97C4×C83—22no—
98C4×C83—22no—
99C4×C83—42no—
100C4×C83—42no—
101C4×C83C2×#4111yescharacteristic
102C4×C83C2×#4111yescharacteristic
103Dic83—42no—
104C2×C83—41no—
105C2×C83#53211yescharacteristic
106C2×C83#57712yes—
107D83—82no—
108#3174—831nomaximal, Sylow
109C83#317411yescharacteristic, complemented, Sylow
110#541D8311yescharacteristic, complemented
111#540—831no—
112#499—831no—
113#492—831no—
114#491—831no—
115#490—831no—
116#468—832no—
117#465—832no—
118#440—832no—
119C4×SD16—832no—
120#405—831no—
121C2×SD16—832no—
122C2×SD16—832no—
123C2×M16—831no—
124C22×C8—831no—
125C4⋊Q8C2×D8311yescharacteristic
126#127—831no—
127#126—22no—
128#125—22no—
129#125—831no—
130#125C2×D8311yescharacteristic
131#123—1662no—
132C22⋊Q8C2×D8312yes—
133#121—832no—
134#121—1662no—
135C4×Q8C2×D8312yes—
136C4×D4—832no—
137#117C2×D8311yescharacteristic
138#107—832no—
139#106—832no—
140#103—832no—
141#103—832no—
142#102—832no—
143#102—832no—
144#97—831no—
145#97—831no—
146C4×C8—831no—
147C4×C8—831no—
148C2×Q8C22×D8312yes—
149C2×D4—832no—
150C2×D4—1662no—
151C2×D4—1662no—
152C22×C4—1662no—
153C22×C4C22×D8311yescharacteristic
154SD16—1664no—
155SD16—1664no—
156M16—1662no—
157C2×C8—831no—
158C2×C8—831no—
159C2×C8—831no—
160C2×C8—831no—
161C2×C8—1662no—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—832no—
167C4⋊C4—1662no—
168C4⋊C4C22⋊D8312yes—
169C4⋊C4C22×D8312yes—
170C22⋊C4—22no—
171C22⋊C4—1662no—
172C22⋊C4—1662no—
173C22⋊C4C22⋊D8312yes—
174C42—22no—
175C42C22×D8311yescharacteristic
176C42C22×D8311yescharacteristic
177C23—1662no—
178C23C2×C22⋊D8311yescharacteristic
179Q8—22no—
180Q8—22no—
181D4—6642no—
182D4—6642no—
183D4—1662no—
184D4—1662no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—6642no—
190C2×C4—1662no—
191C2×C4C2×C22⋊D8312yes—
192C2×C4C2×C22⋊D8312yes—
193C2×C4C2×C22⋊D8311yescharacteristic
194C2×C4C23×D8311yescharacteristic
195C8—832no—
196C8—832no—
197C8—1661no—
198C8—1661no—
199C22—21no—
200C22—22no—
201C22—3322no—
202C22—3322no—
203C22—3322no—
204C22C22×C22⋊D8311yescharacteristic
205C4—21no—
206C4—22no—
207C4—22no—
208C4—42no—
209C4—42no—
210C4—3322no—
211C4#41×D8311yescharacteristic
212C4#41×D8311yescharacteristic
213C2—41no—
214C2—6642no—
215C2#1912824711yescharacteristic
216C2#1912850912yes—
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#541×C83C2
socleC22×C83C22×D4
prime radical#19128306C2
hypercenter#541D83
hypocenterC83#3174
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ #117 ◁ #541

Complements

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

NH ≅ G/Ntype
2. #19128100C2semidirect
3. #19128075C2semidirect
4. #19128069C2semidirect
5. #19128067C2semidirect
6. #19127949C2semidirect
7. #19127932C2semidirect
8. #19127799C2semidirect
9. C4×#13050577C2semidirect
10. #19127657C2semidirect
11. #541×C83C2semidirect
109. C83#3174semidirect
110. #541D83semidirect

Maximal subgroups

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

subgroupindex
#191283062
#191281002
#191280752
#191280692
#191280672
#191279492
#191279322
#191277992
C4×#130505772
#191276572
#541×C832
#317483