Finite Group Databasebeta

← #24916975

Subgroups — #24916975

Order 10624.

494 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 149 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×#14581818C211yesmaximal, characteristic
2C22×#13050578C211yesmaximal, characteristic, complemented
3#19128277C211yesmaximal, characteristic, complemented
4#19127916C214yesmaximal, complemented
5#19127906C214yesmaximal, complemented
6#19127728C211yesmaximal, characteristic, complemented
7#19127709C211yesmaximal, characteristic, complemented
8#19127695C211yesmaximal, characteristic, complemented
9#518×C83C211yesmaximal, characteristic, complemented
10C2×#13050590C2211yescharacteristic
11C22×Dic166C2211yescharacteristic
12#14581818—24no—
13C2×#13050578—24no—
14C2×#13050578C2212yes—
15C2×#13050578C2214yes—
16C2×#13050575C2212yes—
17C2×#13050574C2211yescharacteristic
18C22×C83⋊C8C2211yescharacteristic
19#14581792C2212yescomplemented
20#14581780C2212yescomplemented
21#14581776—22no—
22#14581772—22no—
23C4⋊(C83⋊C8)C2214yescomplemented
24#14581715C2212yescomplemented
25#14581715C2214yescomplemented
26#14581712C2212yescomplemented
27C22×D4×C83C2211yescharacteristic
28C4×D4×C83—22no—
29C4×D4×C83C2214yes—
30#117×C83C2211yescharacteristic
31#117×C83C2211yescharacteristic
32C2×C22⋊C4×C83—22no—
33#13050590—22no—
34#13050590—22no—
35C2×Dic166—22no—
36C2×Dic166—22no—
37C2×Dic166—21no—
38C2×Dic166C2312yes—
39C2×Dic166C2311yescharacteristic
40C2×D332C2311yescharacteristic
41C2×C22⋊D83—21no—
42C22×Dic83—21no—
43C2×C4×D83—21no—
44#13050578—28no—
45#13050578—24no—
46#13050575—24no—
47#13050574—22no—
48C2×C83⋊C8—22no—
49C2×C83⋊C8C2312yes—
50C2×C83⋊C8C2312yes—
51C4⋊Dic83—24no—
52(C2×D83)⋊C4—24no—
53C24×C83—21no—
54C2×D4×C83—21no—
55C2×D4×C83—22no—
56C2×D4×C83D414yes—
57C2×D4×C83C2312yes—
58C22×C4×C83—24no—
59C22×C4×C83C2311yescharacteristic
60C4⋊C4×C83D412yes—
61C4⋊C4×C83C2312yes—
62C22⋊C4×C83—21no—
63C22⋊C4×C83—24no—
64C22⋊C4×C83D412yes—
65C42×C83—21no—
66C42×C83C2312yes—
67C23×C83—22no—
68C23×C83—24no—
69C23×C83—82no—
70C23×C83C2×D411yescharacteristic
71C22×D83—21no—
72D4×C83—24no—
73D4×C83—22no—
74D4×C83C2×D414yes—
75C2×C4×C83—22no—
76C2×C4×C83—84no—
77C2×C4×C83C2×D414yes—
78C2×C4×C83C2×D411yescharacteristic
79C2×C4×C83C2×D412yes—
80C2×C4×C83#4112yes—
81C2×C4×C83C2411yescharacteristic
82C22⋊D83—42no—
83C2×Dic83—22no—
84C2×Dic83—21no—
85C2×Dic83—42no—
86D332—21no—
87D332—21no—
88C4×D83—42no—
89Dic166—22no—
90Dic166—21no—
91Dic166—22no—
92Dic166—21no—
93Dic166—42no—
94C83⋊C8—22no—
95C83⋊C8—22no—
96C22×C83—22no—
97C22×C83—24no—
98C22×C83—44no—
99C22×C83—84no—
100C22×C83#12112yes—
101C22×C83C22×D411yescharacteristic
102C2×D83—41no—
103C2×D83—42no—
104C4×C83—24no—
105C4×C83—42no—
106C4×C83#12112yes—
107C4×C83C22×D411yescharacteristic
108C4×C83C2×#4111yescharacteristic
109Dic83—42no—
110Dic83—41no—
111C2×C83—22no—
112C2×C83—44no—
113C2×C83C2×#12111yescharacteristic
114C2×C83#57712yes—
115D83—81no—
116#3042—831nomaximal, Sylow
117C83#304211yescharacteristic, complemented, Sylow
118C2×AGL(1,ℤ/8)—831no—
119C22×SD16—831no—
120#536—831no—
121#518D8311yescharacteristic, complemented
122#463—834no—
123#461—834no—
124#424—831no—
125#419—831no—
126#416—831no—
127C2×#41—831no—
128C22×Q8—831no—
129C22×D4C2×D8311yescharacteristic
130AGL(1,ℤ/8)—1664no—
131C2×SD16—834no—
132C2×SD16—832no—
133C2×SD16—1664no—
134C2×D8—832no—
135C2×M16—831no—
136C22×C8—831no—
137C4⋊Q8—832no—
138#124—832no—
139#123—1662no—
140C22⋊Q8—1662no—
141C4×D4—22no—
142C4×D4C2×D8314yes—
143#117C2×D8311yescharacteristic
144#117C2×D8311yescharacteristic
145C2×C22⋊C4—22no—
146C4⋊C8—834no—
147#103—832no—
148#103—834no—
149#102—832no—
150C24—21no—
151#41—1662no—
152#41—1662no—
153C2×Q8—832no—
154C2×Q8—831no—
155C2×Q8—1662no—
156C2×Q8—1662no—
157C2×Q8—1661no—
158C2×D4—21no—
159C2×D4—22no—
160C2×D4—831no—
161C2×D4—1661no—
162C2×D4D33214yes—
163C2×D4C22×D8312yes—
164C22×C4—24no—
165C22×C4—1661no—
166C22×C4—1661no—
167C22×C4C22×D8311yescharacteristic
168SD16—1668no—
169SD16—1664no—
170D8—1664no—
171M16—1662no—
172C2×C8—832no—
173C2×C8—832no—
174C2×C8—1662no—
175C4⋊C4—1664no—
176C4⋊C4C22⋊D8312yes—
177C4⋊C4C22×D8312yes—
178C22⋊C4—21no—
179C22⋊C4—24no—
180C22⋊C4—1664no—
181C22⋊C4C22⋊D8312yes—
182C42—21no—
183C42C22×D8312yes—
184C23—22no—
185C23—24no—
186C23—82no—
187C23—1661no—
188C23C2×C22⋊D8311yescharacteristic
189Q8—3322no—
190Q8—1662no—
191Q8—1661no—
192Q8—1662no—
193Q8—1661no—
194D4—24no—
195D4—22no—
196D4—3322no—
197D4—1661no—
198D4—1661no—
199D4C2×D33214yes—
200C2×C4—22no—
201C2×C4—84no—
202C2×C4—3322no—
203C2×C4—3322no—
204C2×C4—1662no—
205C2×C4—1661no—
206C2×C4C2×C22⋊D8314yes—
207C2×C4C2×C22⋊D8311yescharacteristic
208C2×C4C2×D33212yes—
209C2×C4#1305059012yes—
210C2×C4C23×D8311yescharacteristic
211C8—1662no—
212C8—1662no—
213C22—22no—
214C22—24no—
215C22—44no—
216C22—84no—
217C22—3321no—
218C22—3322no—
219C22#1458176612yes—
220C22C22×C22⋊D8311yescharacteristic
221C4—24no—
222C4—42no—
223C4—3321no—
224C4—3322no—
225C4#1458176612yes—
226C4C22×D33211yescharacteristic
227C4C2×#1305059011yescharacteristic
228C2—22no—
229C2—44no—
230C2—6641no—
231C2C2×#1458176611yescharacteristic
232C2#1912851212yes—
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#518×C83C2
socleC22×C83C22×D4
prime radicalC2×#14581818C2
hypercenter#518D83
hypocenterC83#3042
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
2. C22×#13050578C2semidirect
3. #19128277C2semidirect
4. #19127916C2semidirect
5. #19127906C2semidirect
6. #19127728C2semidirect
7. #19127709C2semidirect
8. #19127695C2semidirect
9. #518×C83C2semidirect
19. #14581792C22semidirect
20. #14581780C22semidirect
23. C4⋊(C83⋊C8)C22semidirect
24. #14581715C22semidirect
25. #14581715C22semidirect
26. #14581712C22semidirect
117. C83#3042semidirect
121. #518D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145818182
C22×#130505782
#191282772
#191279162
#191279062
#191277282
#191277092
#191276952
#518×C832
#304283