Finite Group Databasebeta

← #24918187

Subgroups — #24918187

Order 10624.

370 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 115 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#19128232C212yesmaximal, complemented
2#19128101C211yesmaximal, characteristic, complemented
3#19128084C211yesmaximal, characteristic, complemented
4#19128054C212yesmaximal, complemented
5#19127959C211yesmaximal, characteristic, complemented
6#19127945C212yesmaximal, complemented
7#19127813C211yesmaximal, characteristic, complemented
8#19127807C211yesmaximal, characteristic, complemented
9#19127798C212yesmaximal, complemented
10M16×Dic83C211yesmaximal, characteristic, complemented
11#469×C83C211yesmaximal, characteristic
12C2×Dic332C2211yescharacteristic
13C2×#13050578C2212yes—
14C2×#13050574C2211yescharacteristic
15#14581785—22no—
16#14581783C2212yes—
17#14581779C2212yes—
18#14581773—22no—
19#14581771C2212yes—
20#14581757C2212yes—
21D4×Dic83—22no—
22D4×Dic83C2212yes—
23#14581748—22no—
24C2×C4×Dic83C2211yescharacteristic
25#14581725C2212yes—
26#14581724C2211yescharacteristic
27#14581717C2212yes—
28#14581715C2212yes—
29#14581713C2212yes—
30#14581702C2211yescharacteristic
31#14581702C2211yescharacteristic
32C8×Dic83C2211yescharacteristic
33C8×Dic83C2211yescharacteristic
34C2×D8×C83C2211yescharacteristic
35C2×M16×C83C2211yescharacteristic
36#121×C83C2212yes—
37#106×C83C2211yescharacteristic
38#102×C83C2212yes—
39C2×Dic166C2312yes—
40C22×Dic83—22no—
41C22×Dic83D412yes—
42Dic332—22no—
43#13050578—24no—
44#13050574—22no—
45C2×C83⋊C8C2311yescharacteristic
46C2×C83⋊C8C2311yescharacteristic
47C4⋊Dic83—22no—
48C4⋊Dic83C2312yes—
49Dic83⋊C4—22no—
50Dic83⋊C4—22no—
51Dic83⋊C4—22no—
52C22⋊Dic83—22no—
53C22⋊Dic83—22no—
54C22⋊Dic83—22no—
55C22⋊Dic83—22no—
56C4×Dic83—22no—
57C4×Dic83D412yes—
58C4×Dic83C2311yescharacteristic
59C4×Dic83C2311yescharacteristic
60C2×D4×C83—22no—
61C2×D4×C83C2312yes—
62C22×C4×C83C2311yescharacteristic
63D8×C83—22no—
64M16×C83—22no—
65C2×C8×C83C2311yescharacteristic
66C2×C8×C83C2311yescharacteristic
67C4⋊C4×C83C2312yes—
68C22⋊C4×C83—22no—
69C23×C83—22no—
70C23×C83C2×D411yescharacteristic
71D4×C83—22no—
72D4×C83—22no—
73D4×C83—44no—
74C2×C4×C83—21no—
75C2×C4×C83—21no—
76C2×C4×C83—22no—
77C2×C4×C83C2×D411yescharacteristic
78C2×C4×C83C2411yescharacteristic
79C2×Dic83—22no—
80C2×Dic83—22no—
81C2×Dic83—42no—
82C2×Dic83—44no—
83C2×Dic83C2×D412yes—
84C2×Dic83C2×D412yes—
85Dic166—22no—
86Dic166—22no—
87C8×C83—21no—
88C8×C83#4112yes—
89C83⋊C8—21no—
90C83⋊C8#4112yes—
91C22×C83—21no—
92C22×C83—22no—
93C22×C83—42no—
94C22×C83—42no—
95C22×C83—42no—
96C22×C83C22×D411yescharacteristic
97C4×C83—21no—
98C4×C83—42no—
99C4×C83C2×#4111yescharacteristic
100C4×C83C2×#4111yescharacteristic
101Dic83—22no—
102Dic83—22no—
103Dic83—42no—
104Dic83—42no—
105C2×C83—41no—
106C2×C83—82no—
107C2×C83#53211yescharacteristic
108C2×C83#57512yes—
109#3183—831nomaximal, Sylow
110C83#318311yescharacteristic, complemented, Sylow
111#529—832no—
112#499—831no—
113#495—831no—
114#488—832no—
115#470—831no—
116#469D8311yescharacteristic
117#468—832no—
118#442—831no—
119#441—831no—
120#440—832no—
121C4×M16—831no—
122C2×Q16—831no—
123C2×SD16—832no—
124C2×D8C2×D8311yescharacteristic
125C2×M16—831no—
126C2×M16C2×D8311yescharacteristic
127#126—1662no—
128#125—832no—
129#124—832no—
130#123—1662no—
131C22⋊Q8—832no—
132#121C2×D8312yes—
133C4×Q8—832no—
134C4×D4—832no—
135C4×D4—1662no—
136#117—1662no—
137C2×C42—831no—
138#107—832no—
139#106—831no—
140#106C2×D8311yescharacteristic
141#103—832no—
142#103—832no—
143#102—832no—
144#102C2×D8312yes—
145#97—831no—
146#97—831no—
147C4×C8—831no—
148C4×C8—831no—
149C2×Q8—832no—
150C2×D4—22no—
151C2×D4C22×D8312yes—
152C22×C4—832no—
153C22×C4—1662no—
154C22×C4C22×D8311yescharacteristic
155Q16—1662no—
156SD16—1664no—
157D8—22no—
158M16—22no—
159M16—1662no—
160C2×C8—831no—
161C2×C8—831no—
162C2×C8C22×D8311yescharacteristic
163C2×C8C22×D8311yescharacteristic
164C4⋊C4—832no—
165C4⋊C4—1662no—
166C4⋊C4—1662no—
167C4⋊C4—1662no—
168C4⋊C4—1662no—
169C4⋊C4C22×D8312yes—
170C22⋊C4—22no—
171C22⋊C4—1662no—
172C22⋊C4—1662no—
173C22⋊C4—1662no—
174C22⋊C4—1662no—
175C42—832no—
176C42—831no—
177C42—831no—
178C42—1662no—
179C23—22no—
180C23D4×D8311yescharacteristic
181Q8—1662no—
182Q8—1662no—
183D4—22no—
184D4—22no—
185D4—44no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—3324no—
190C2×C4—3322no—
191C2×C4—832no—
192C2×C4—832no—
193C2×C4—1662no—
194C2×C4—1662no—
195C2×C4D4×D8311yescharacteristic
196C2×C4C23×D8311yescharacteristic
197C8—21no—
198C8—832no—
199C8—1661no—
200C8#1305058912yes—
201C22—22no—
202C22—21no—
203C22—42no—
204C22—42no—
205C22—42no—
206C22C2×D4×D8311yescharacteristic
207C4—21no—
208C4—42no—
209C4—3322no—
210C4—3322no—
211C4—1662no—
212C4—1662no—
213C4#41×D8311yescharacteristic
214C4C2×#1305058911yescharacteristic
215C2—41no—
216C2—82no—
217C2#1912824611yescharacteristic
218C2#1912849712yes—
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting, prime radical#469×C83C2
socleC22×C83C22×D4
hypercenter#469D83
hypocenterC83#3183
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #19128232C2semidirect
2. #19128101C2semidirect
3. #19128084C2semidirect
4. #19128054C2semidirect
5. #19127959C2semidirect
6. #19127945C2semidirect
7. #19127813C2semidirect
8. #19127807C2semidirect
9. #19127798C2semidirect
10. M16×Dic83C2semidirect
110. C83#3183semidirect

Maximal subgroups

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

subgroupindex
#191282322
#191281012
#191280842
#191280542
#191279592
#191279452
#191278132
#191278072
#191277982
M16×Dic832
#469×C832
#318383