Finite Group Databasebeta

← #24917847

Subgroups — #24917847

Order 10624.

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

normal subgroups only

This group has 400 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#19128254C211yesmaximal, characteristic, complemented
2C2×#14581779C211yesmaximal, characteristic
3#19128059C214yesmaximal, complemented
4#19128054C214yesmaximal, complemented
5#19127711C211yesmaximal, characteristic, complemented
6#19127709C211yesmaximal, characteristic, complemented
7#19127706C211yesmaximal, characteristic
8M16×Dic83C211yesmaximal, characteristic, complemented
9#418×C83C211yesmaximal, characteristic
10C22×Dic166C2211yescharacteristic
11C2×#13050574C2211yescharacteristic
12#14581791C2212yescomplemented
13#14581783C2212yescomplemented
14#14581779—22no—
15#14581779C2212yes—
16#14581779C2212yes—
17#14581769—22no—
18C4×Dic166—22no—
19#14581751C2211yescharacteristic
20C2×C22⋊Dic83—22no—
21C2×C4×Dic83C2211yescharacteristic
22#14581717C2214yescomplemented
23#14581715C2214yescomplemented
24#14581713C2214yes—
25#14581702C2212yescomplemented
26C8×Dic83C2212yescomplemented
27C22×D4×C83C2211yescharacteristic
28C2×M16×C83C2211yescharacteristic
29#117×C83C2211yescharacteristic
30#102×C83C2214yes—
31C2×Dic166—22no—
32C2×Dic166—22no—
33C2×Dic166—21no—
34C2×Dic166—22no—
35C2×Dic166C2312yes—
36C22×Dic83—21no—
37C22×Dic83D412yes—
38#13050574—22no—
39C2×C83⋊C8C2312yes—
40C4⋊Dic83—22no—
41C4⋊Dic83C2312yes—
42Dic83⋊C4—22no—
43Dic83⋊C4—22no—
44C22⋊Dic83—21no—
45C22⋊Dic83—24no—
46C22⋊Dic83—24no—
47C4×Dic83—21no—
48C4×Dic83D412yes—
49C4×Dic83C2311yescharacteristic
50C4×Dic83C2311yescharacteristic
51C24×C83—21no—
52C2×D4×C83—21no—
53C2×D4×C83—22no—
54C2×D4×C83—22no—
55C2×D4×C83C2312yes—
56C22×C4×C83C2311yescharacteristic
57M16×C83—22no—
58C2×C8×C83C2312yes—
59C4⋊C4×C83C2312yes—
60C22⋊C4×C83—21no—
61C42×C83—21no—
62C23×C83—22no—
63C23×C83—44no—
64C23×C83—42no—
65C23×C83C2×D411yescharacteristic
66D4×C83—22no—
67D4×C83—22no—
68D4×C83—42no—
69C2×C4×C83—22no—
70C2×C4×C83C2×D411yescharacteristic
71C2×C4×C83#4112yes—
72C2×C4×C83#4112yes—
73C2×C4×C83C2411yescharacteristic
74C2×Dic83—22no—
75C2×Dic83—22no—
76C2×Dic83—42no—
77C2×Dic83—42no—
78C2×Dic83C2×D412yes—
79C2×Dic83C2×D412yes—
80Dic166—22no—
81Dic166—22no—
82Dic166—44no—
83Dic166—42no—
84C8×C83—22no—
85C83⋊C8—22no—
86C22×C83—22no—
87C22×C83—42no—
88C22×C83—44no—
89C22×C83—84no—
90C22×C83#12412yes—
91C22×C83C22×D411yescharacteristic
92C4×C83—42no—
93C4×C83#12412yes—
94C4×C83C2×#4111yescharacteristic
95C4×C83C2×#4111yescharacteristic
96Dic83—22no—
97Dic83—22no—
98Dic83—42no—
99Dic83—42no—
100C2×C83—22no—
101C2×C83—82no—
102C2×C83C2×#12411yescharacteristic
103C2×C83#57512yes—
104#3140—831nomaximal, Sylow
105C83#314011yescharacteristic, complemented, Sylow
106#533—831no—
107C2×#124—831no—
108#489—834no—
109#488—834no—
110#419—831no—
111#419—831no—
112#418—831no—
113#418D8311yescharacteristic
114C4×M16—831no—
115C22×Q8—831no—
116C22×D4C2×D8311yescharacteristic
117C2×M16—831no—
118C2×M16C2×D8311yescharacteristic
119C4⋊Q8—832no—
120#125—832no—
121#124—832no—
122#124—832no—
123#124—1662no—
124C22⋊Q8—1662no—
125C4×Q8—1662no—
126#117—831no—
127#117C2×D8311yescharacteristic
128C2×C22⋊C4—1662no—
129C2×C42—831no—
130#103—834no—
131#103—834no—
132#102—834no—
133#102C2×D8314yes—
134#97—832no—
135C4×C8—832no—
136C24—21no—
137C2×Q8—832no—
138C2×Q8—1662no—
139C2×Q8—1662no—
140C2×Q8—1662no—
141C2×Q8—1661no—
142C2×D4—21no—
143C2×D4—22no—
144C2×D4—22no—
145C2×D4C22×D8312yes—
146C22×C4—832no—
147C22×C4—1661no—
148C22×C4C22×D8311yescharacteristic
149M16—22no—
150M16—1662no—
151C2×C8—832no—
152C2×C8C22×D8312yes—
153C4⋊C4—832no—
154C4⋊C4—1662no—
155C4⋊C4—1662no—
156C4⋊C4—1662no—
157C4⋊C4C22×D8312yes—
158C22⋊C4—21no—
159C22⋊C4—1664no—
160C22⋊C4—1664no—
161C22⋊C4—1661no—
162C42—21no—
163C42—832no—
164C42—831no—
165C42—831no—
166C42—1661no—
167C23—22no—
168C23—44no—
169C23—42no—
170C23D4×D8311yescharacteristic
171Q8—3324no—
172Q8—3322no—
173Q8—1662no—
174Q8—1662no—
175D4—22no—
176D4—22no—
177D4—42no—
178C2×C4—22no—
179C2×C4—3322no—
180C2×C4—3322no—
181C2×C4—832no—
182C2×C4—832no—
183C2×C4—1662no—
184C2×C4—1662no—
185C2×C4D4×D8311yescharacteristic
186C2×C4#1305058912yes—
187C2×C4#1305059012yes—
188C2×C4C23×D8311yescharacteristic
189C8—22no—
190C8—1662no—
191C22—22no—
192C22—42no—
193C22—44no—
194C22—84no—
195C22#1458177812yes—
196C22C2×D4×D8311yescharacteristic
197C4—42no—
198C4—3322no—
199C4—3322no—
200C4—1662no—
201C4—1662no—
202C4#1458177812yes—
203C4C2×#1305058911yescharacteristic
204C4C2×#1305059011yescharacteristic
205C2—22no—
206C2—82no—
207C2C2×#1458177811yescharacteristic
208C2#1912849712yes—
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#418×C83C2
socleC22×C83C22×D4
prime radicalC22×D4×C83C22
hypercenter#418D83
hypocenterC83#3140
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. #19128254C2semidirect
3. #19128059C2semidirect
4. #19128054C2semidirect
5. #19127711C2semidirect
6. #19127709C2semidirect
8. M16×Dic83C2semidirect
12. #14581791C22semidirect
13. #14581783C22semidirect
22. #14581717C22semidirect
23. #14581715C22semidirect
25. #14581702C22semidirect
26. C8×Dic83C22semidirect
105. C83#3140semidirect

Maximal subgroups

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

subgroupindex
#191282542
C2×#145817792
#191280592
#191280542
#191277112
#191277092
#191277062
M16×Dic832
#418×C832
#314083