Finite Group Databasebeta

← #24917045

Subgroups — #24917045

Order 10624.

520 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 145 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
1C2×#14581814C211yesmaximal, characteristic, complemented
2C22×C8×D83C211yesmaximal, characteristic, complemented
3#19128213C212yesmaximal, complemented
4#19127943C212yesmaximal, complemented
5#19127938C212yesmaximal, complemented
6#19127931C214yesmaximal, complemented
7#19127695C212yesmaximal, complemented
8#427×C83C211yesmaximal, characteristic, complemented
9C2×#13050590C2212yes—
10C22×C4×D83C2211yescharacteristic
11#14581814—24no—
12C2×Dic332C2211yescharacteristic, complemented
13C2×#13050579C2212yescomplemented
14C2×D664C2211yescharacteristic, complemented
15C22×C83⋊C8C2211yescharacteristic
16C2×C8×D83—24no—
17C2×C8×D83C2212yes—
18C2×C8×D83C2212yes—
19#14581776—22no—
20#14581772C2214yes—
21#14581767C2214yes—
22#14581762—22no—
23C4×D332—24no—
24#14581715C2214yescomplemented
25#14581712C2214yescomplemented
26C22×C8×C83C2211yescharacteristic
27#117×C83C2212yes—
28#107×C83C2212yescomplemented
29#106×C83C2212yescomplemented
30C23×D83D411yescharacteristic
31#13050590—24no—
32#13050590—24no—
33C2×Dic166C2312yes—
34C2×D332—24no—
35C2×D332C2312yes—
36C2×C22⋊D83—22no—
37C22×Dic83D411yescharacteristic
38C2×C4×D83—22no—
39C2×C4×D83—22no—
40C2×C4×D83—22no—
41C2×C4×D83D412yes—
42C2×C4×D83C2312yes—
43Dic332—22no—
44#13050579—24no—
45D664—22no—
46C2×C83⋊C8—22no—
47C2×C83⋊C8C2312yes—
48C8×D83—24no—
49C8×D83—24no—
50C4⋊Dic83—24no—
51(C2×D83)⋊C4—24no—
52(C2×D83)⋊C4—24no—
53C22×C4×C83C2311yescharacteristic
54C2×C8×C83D414yescomplemented
55C2×C8×C83C2311yescharacteristic
56C2×C8×C83C2311yescharacteristic
57C4⋊C4×C83C2314yes—
58C22⋊C4×C83—22no—
59C42×C83—22no—
60C23×C83C2×D411yescharacteristic
61C22×D83—22no—
62C22×D83—24no—
63C22×D83—22no—
64C22×D83C2×D412yes—
65C2×C4×C83—24no—
66C2×C4×C83C2×D411yescharacteristic
67C2×C4×C83C2×D412yes—
68C2×C4×C83#4112yes—
69C2×C4×C83C2411yescharacteristic
70C22⋊D83—44no—
71C2×Dic83—22no—
72C2×Dic83—22no—
73C2×Dic83C2×D412yes—
74D332—22no—
75D332—22no—
76D332—48no—
77C4×D83—24no—
78C4×D83—22no—
79C4×D83—22no—
80C4×D83—44no—
81Dic166—22no—
82Dic166—22no—
83C8×C83C2×D412yes—
84C8×C83C2×D412yes—
85C83⋊C8—22no—
86C22×C83—22no—
87C22×C83#12112yes—
88C22×C83C22×D411yescharacteristic
89C2×D83—28no—
90C2×D83—22no—
91C2×D83—22no—
92C2×D83—42no—
93C2×D83—44no—
94C2×D83#13514yescomplemented
95C4×C83—44no—
96C4×C83#12112yes—
97C4×C83C22×D411yescharacteristic
98C4×C83C2×#4111yescharacteristic
99Dic83—22no—
100Dic83—42no—
101C2×C83—22no—
102C2×C83C2×#12111yescharacteristic
103C2×C83C2×#13512yes—
104D83—24no—
105D83—82no—
106#3050—831nomaximal, Sylow
107C83#305011yescharacteristic, complemented, Sylow
108C2×#135—831no—
109C23×C8—831no—
110#525—832no—
111#467—832no—
112#466—832no—
113#465—834no—
114#427D8311yescharacteristic, complemented
115#416—832no—
116C2×#41—832no—
117C23×C4—831no—
118#135—1664no—
119C2×Q16—831no—
120C2×SD16—832no—
121C2×D8—831no—
122C22×C8—831no—
123C22×C8—832no—
124C22×C8—832no—
125C22×C8—1664no—
126C22×C8C2×D8311yescharacteristic
127#123—1662no—
128C22⋊Q8—834no—
129#121—834no—
130#120—1662no—
131C4×D4—1664no—
132#117C2×D8312yes—
133#107C2×D8312yescomplemented
134#106C2×D8312yescomplemented
135#103—834no—
136#102—834no—
137C24—831no—
138#41—1664no—
139#41—1664no—
140C2×Q8—832no—
141C2×D4—832no—
142C2×D4—1664no—
143C2×D4—1662no—
144C22×C4—831no—
145C22×C4—832no—
146C22×C4—832no—
147C22×C4—1662no—
148C22×C4—1662no—
149C22×C4—1662no—
150C22×C4C22×D8311yescharacteristic
151Q16—1662no—
152SD16—1664no—
153D8—1662no—
154C2×C8—832no—
155C2×C8—1662no—
156C2×C8—1664no—
157C2×C8—1664no—
158C2×C8D33214yescomplemented
159C2×C8C22×D8311yescharacteristic
160C2×C8C22×D8311yescharacteristic
161C4⋊C4—1664no—
162C4⋊C4C22×D8314yes—
163C22⋊C4—22no—
164C22⋊C4—1664no—
165C22⋊C4—1664no—
166C42—22no—
167C23—832no—
168C23—1664no—
169C23—1662no—
170C23—1662no—
171C23D4×D8311yescharacteristic
172Q8—1662no—
173Q8—1662no—
174D4—3328no—
175D4—3324no—
176D4—1662no—
177D4—1662no—
178C2×C4—24no—
179C2×C4—3324no—
180C2×C4—832no—
181C2×C4—1662no—
182C2×C4—1662no—
183C2×C4—1664no—
184C2×C4—1662no—
185C2×C4—1662no—
186C2×C4D4×D8311yescharacteristic
187C2×C4C2×D33212yes—
188C2×C4#1305059112yes—
189C2×C4C23×D8311yescharacteristic
190C8—1662no—
191C8C2×D33212yes—
192C8C2×D33212yes—
193C22—22no—
194C22—3322no—
195C22—3324no—
196C22—834no—
197C22—1668no—
198C22—1662no—
199C22—1662no—
200C22#1458176712yes—
201C22C2×D4×D8311yescharacteristic
202C4—44no—
203C4—3322no—
204C4—1662no—
205C4#1458176712yes—
206C4C22×D33211yescharacteristic
207C4C2×#1305059111yescharacteristic
208C2—22no—
209C2—6642no—
210C2—1664no—
211C2C2×#1458176711yescharacteristic
212C2#135×D8312yes—
subgroupG/H
centerC2×C4D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#427×C83C2
socleC22×C83C22×D4
hypercenter#427D83
hypocenterC83#3050
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #427

Complements

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

NH ≅ G/Ntype
1. C2×#14581814C2semidirect
2. C22×C8×D83C2semidirect
3. #19128213C2semidirect
4. #19127943C2semidirect
5. #19127938C2semidirect
6. #19127931C2semidirect
7. #19127695C2semidirect
8. #427×C83C2semidirect
12. C2×Dic332C22semidirect
13. C2×#13050579C22semidirect
14. C2×D664C22semidirect
24. #14581715C22semidirect
25. #14581712C22semidirect
28. #107×C83C22semidirect
29. #106×C83C22semidirect
54. C2×C8×C83D4semidirect
94. C2×D83#135semidirect
107. C83#3050semidirect
114. #427D83semidirect
133. #107C2×D83semidirect
134. #106C2×D83semidirect
158. C2×C8D332semidirect

Maximal subgroups

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

subgroupindex
C2×#145818142
C22×C8×D832
#191282132
#191279432
#191279382
#191279312
#191276952
#427×C832
#305083