Finite Group Databasebeta

← #24916974

Subgroups — #24916974

Order 10624.

494 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 133 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×#14581820C211yesmaximal, characteristic
2C22×#13050577C211yesmaximal, characteristic, complemented
3#19128275C211yesmaximal, characteristic, complemented
4#19128175C211yesmaximal, characteristic, complemented
5#19127919C214yesmaximal, complemented
6#19127905C214yesmaximal, complemented
7#19127729C211yesmaximal, characteristic, complemented
8#19127695C211yesmaximal, characteristic, complemented
9#419×C83C211yesmaximal, characteristic, complemented
10C2×#13050590C2211yescharacteristic
11C22×D332C2211yescharacteristic
12#14581820—24no—
13C2×#13050579C2212yes—
14C2×#13050577—24no—
15C2×#13050577C2214yes—
16C2×D664C2212yes—
17C22×C83⋊C8C2211yescharacteristic
18#14581790C2212yescomplemented
19#14581781C2212yescomplemented
20#14581775—22no—
21#14581768—22no—
22C4×D332—22no—
23#14581754C2214yes—
24#14581751C2211yescharacteristic
25C22⋊C4×D83—22no—
26Dic83⋊C8C2214yescomplemented
27#14581715C2212yescomplemented
28#14581712C2212yescomplemented
29C22×Q8×C83C2211yescharacteristic
30C2×M16×C83C2211yescharacteristic
31#117×C83C2211yescharacteristic
32#103×C83C2214yescomplemented
33C23×D83—21no—
34#13050590—22no—
35#13050590—22no—
36C2×Dic166C2311yescharacteristic
37C2×D332—21no—
38C2×D332—22no—
39C2×D332D414yes—
40C2×D332C2311yescharacteristic
41C2×D332C2312yes—
42C2×C22⋊D83—21no—
43C2×C4×D83—21no—
44C2×C4×D83—24no—
45#13050579—24no—
46#13050577—28no—
47D664—24no—
48C2×C83⋊C8—22no—
49C2×C83⋊C8C2312yes—
50C4⋊Dic83D412yes—
51Dic83⋊C4—24no—
52C22⋊Dic83D412yes—
53(C2×D83)⋊C4—24no—
54(C2×D83)⋊C4—24no—
55C4×Dic83C2312yes—
56C2×Q8×C83—21no—
57C2×Q8×C83—22no—
58C2×Q8×C83—22no—
59C2×Q8×C83C2312yes—
60C22×C4×C83—21no—
61C22×C4×C83C2311yescharacteristic
62M16×C83—22no—
63C2×C8×C83C2312yes—
64C4⋊C4×C83C2312yes—
65C22⋊C4×C83—21no—
66C42×C83—21no—
67C23×C83C2×D411yescharacteristic
68C22×D83—22no—
69C22×D83—24no—
70C22×D83—21no—
71C22×D83—82no—
72Q8×C83—22no—
73Q8×C83—22no—
74Q8×C83—42no—
75C2×C4×C83—22no—
76C2×C4×C83—22no—
77C2×C4×C83—42no—
78C2×C4×C83C2×D411yescharacteristic
79C2×C4×C83C2×D412yes—
80C2×C4×C83#4112yes—
81C2×C4×C83C2411yescharacteristic
82C22⋊D83—42no—
83C2×Dic83—21no—
84C2×Dic83C2×D414yes—
85D332—24no—
86D332—21no—
87D332—22no—
88D332—21no—
89D332C2×D414yes—
90C4×D83—42no—
91C4×D83—84no—
92Dic166—21no—
93Dic166—21no—
94C8×C83—22no—
95C83⋊C8—22no—
96C22×C83—22no—
97C22×C83#12112yes—
98C22×C83C22×D411yescharacteristic
99C2×D83—24no—
100C2×D83—44no—
101C2×D83—41no—
102C2×D83—42no—
103C2×D83—84no—
104C4×C83—42no—
105C4×C83—42no—
106C4×C83#12112yes—
107C4×C83C22×D411yescharacteristic
108C4×C83C2×#4111yescharacteristic
109Dic83—24no—
110Dic83—41no—
111C2×C83—22no—
112C2×C83C2×#12111yescharacteristic
113C2×C83#57712yes—
114D83—44no—
115D83—81no—
116#3042—831nomaximal, Sylow
117C83#304211yescharacteristic, complemented, Sylow
118C2×AGL(1,ℤ/8)—831no—
119C22×SD16—831no—
120#536—831no—
121#518—831no—
122#463—834no—
123#461—834no—
124#424—831no—
125#419D8311yescharacteristic, complemented
126#416—831no—
127C2×#41—831no—
128C22×Q8C2×D8311yescharacteristic
129C22×D4—831no—
130AGL(1,ℤ/8)—1664no—
131C2×SD16—834no—
132C2×SD16—832no—
133C2×SD16—1664no—
134C2×D8—832no—
135C2×M16C2×D8311yescharacteristic
136C22×C8—831no—
137C4⋊Q8—832no—
138#124—832no—
139#123—1662no—
140C22⋊Q8—1662no—
141C4×D4—834no—
142C4×D4—1662no—
143#117—831no—
144#117C2×D8311yescharacteristic
145C2×C22⋊C4—1662no—
146C4⋊C8—834no—
147#103—832no—
148#103C2×D8314yescomplemented
149#102—832no—
150C24—1661no—
151#41—1662no—
152#41—1662no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—831no—
157C2×Q8C22×D8312yes—
158C2×D4—834no—
159C2×D4—832no—
160C2×D4—831no—
161C2×D4—1661no—
162C2×D4—1661no—
163C2×D4—1662no—
164C22×C4—21no—
165C22×C4—1664no—
166C22×C4—1661no—
167C22×C4C22×D8311yescharacteristic
168SD16—1664no—
169SD16—1668no—
170D8—1664no—
171M16—22no—
172C2×C8—832no—
173C2×C8—1662no—
174C2×C8C22×D8312yes—
175C4⋊C4—832no—
176C4⋊C4—1664no—
177C4⋊C4C22×D8312yes—
178C22⋊C4—21no—
179C22⋊C4—832no—
180C22⋊C4—1664no—
181C22⋊C4—1664no—
182C42—21no—
183C42—832no—
184C23—6642no—
185C23—1661no—
186C23—1662no—
187C23—1664no—
188C23D4×D8311yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—42no—
192Q8—1661no—
193Q8—1661no—
194D4—3322no—
195D4—834no—
196D4—1664no—
197D4—1661no—
198D4—1661no—
199D4—1662no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—42no—
203C2×C4—3322no—
204C2×C4—834no—
205C2×C4—6644no—
206C2×C4—1661no—
207C2×C4D4×D8311yescharacteristic
208C2×C4D4×D8312yes—
209C2×C4#1305059012yes—
210C2×C4C23×D8311yescharacteristic
211C8—22no—
212C8—1662no—
213C22—22no—
214C22—3324no—
215C22—3321no—
216C22—3322no—
217C22—6644no—
218C22—1664no—
219C22#1458176512yes—
220C22C2×D4×D8311yescharacteristic
221C4—42no—
222C4—42no—
223C4—3321no—
224C4—1664no—
225C4#1458176512yes—
226C4C2×D4×D8311yescharacteristic
227C4C2×#1305059011yescharacteristic
228C2—22no—
229C2—3324no—
230C2—6641no—
231C2C2×#1458176511yescharacteristic
232C2#1912851012yes—
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#419×C83C2
socleC22×C83C22×D4
prime radicalC2×#14581820C2
hypercenter#419D83
hypocenterC83#3042
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
2. C22×#13050577C2semidirect
3. #19128275C2semidirect
4. #19128175C2semidirect
5. #19127919C2semidirect
6. #19127905C2semidirect
7. #19127729C2semidirect
8. #19127695C2semidirect
9. #419×C83C2semidirect
18. #14581790C22semidirect
19. #14581781C22semidirect
26. Dic83⋊C8C22semidirect
27. #14581715C22semidirect
28. #14581712C22semidirect
32. #103×C83C22semidirect
117. C83#3042semidirect
125. #419D83semidirect
148. #103C2×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145818202
C22×#130505772
#191282752
#191281752
#191279192
#191279052
#191277292
#191276952
#419×C832
#304283