Finite Group Databasebeta

← #24916739

Subgroups — #24916739

Order 10624.

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

normal subgroups only

This group has 752 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×#14581838C211yesmaximal, characteristic
2C22×#13050577C211yesmaximal, characteristic, complemented
3#19128275C211yesmaximal, characteristic, complemented
4#19127862C214yesmaximal, complemented
5#19127850C214yesmaximal, complemented
6#19127707C211yesmaximal, characteristic, complemented
7#19127697C211yesmaximal, characteristic, complemented
8#19127669C211yesmaximal, characteristic, complemented
9C2×#137×C83C211yesmaximal, characteristic, complemented
10#14581838—24no—
11#14581838—24no—
12C2×#13050591C2212yes—
13C2×#13050590—21no—
14C2×#13050590C2211yescharacteristic
15C22×D332—21no—
16C22×D332C2211yescharacteristic
17C2×D4×D83—22no—
18C2×D4×D83C2212yes—
19C2×#13050577—24no—
20C2×#13050577C2214yescomplemented
21C22×C83⋊C8C2211yescharacteristic, complemented
22#14581790—21no—
23#14581781—21no—
24#14581775—22no—
25#14581768C2214yescomplemented
26#14581751C2211yescharacteristic, complemented
27#14581716C2212yescomplemented
28#14581714C2214yescomplemented
29#14581713C2212yescomplemented
30(C2×D83)⋊C8C2214yescomplemented
31C2×#41×C83C2211yescharacteristic
32C22×Q8×C83C2211yescharacteristic, complemented
33#137×C83—24no—
34C2×Q16×C83C2212yescomplemented
35C2×SD16×C83C2212yescomplemented
36C2×M16×C83C2211yescharacteristic, complemented
37C23×D83—21no—
38C23×D83—22no—
39#13050591—24no—
40#13050591—24no—
41#13050590—28no—
42#13050590—22no—
43#13050590—22no—
44C2×Dic166D411yescharacteristic
45C2×D332—28no—
46C2×D332—22no—
47C2×D332—21no—
48C2×D332—22no—
49C2×D332D411yescharacteristic
50C2×D332D414yes—
51C2×D332C2312yes—
52C2×C22⋊D83—22no—
53C2×C22⋊D83D412yes—
54D4×D83—216no—
55D4×D83—24no—
56D4×D83—24no—
57C2×C4×D83—22no—
58C2×C4×D83C2312yes—
59#13050577—28no—
60C2×C83⋊C8—22no—
61C2×C83⋊C8C2312yes—
62C4⋊Dic83C2312yes—
63Dic83⋊C4—24no—
64C22⋊Dic83—21no—
65(C2×D83)⋊C4—24no—
66C4×Dic83—21no—
67#41×C83—22no—
68#41×C83D414yes—
69C2×Q8×C83—21no—
70C2×Q8×C83—22no—
71C2×Q8×C83—22no—
72C2×Q8×C83C2312yes—
73C2×Q8×C83C2311yescharacteristic
74C2×D4×C83—21no—
75C2×D4×C83C2311yescharacteristic
76C22×C4×C83—21no—
77C22×C4×C83—21no—
78C22×C4×C83C2311yescharacteristic
79Q16×C83—24no—
80SD16×C83—24no—
81M16×C83—22no—
82C2×C8×C83C2312yes—
83C23×C83—21no—
84C23×C83C2×D411yescharacteristic
85C22×D83—22no—
86C22×D83—28no—
87C22×D83—24no—
88C22×D83—84no—
89C22×D83—82no—
90C22×D83C2×D412yes—
91Q8×C83—21no—
92Q8×C83—22no—
93Q8×C83—22no—
94Q8×C83—42no—
95Q8×C83C2×D412yes—
96D4×C83—24no—
97D4×C83—21no—
98D4×C83C2×D412yes—
99C2×C4×C83—21no—
100C2×C4×C83—24no—
101C2×C4×C83—22no—
102C2×C4×C83—42no—
103C2×C4×C83C2×D411yescharacteristic
104C2×C4×C83C2×D412yes—
105C2×C4×C83C2×D412yes—
106C2×C4×C83C2411yescharacteristic
107C22⋊D83—28no—
108C22⋊D83—24no—
109C2×Dic83—22no—
110C2×Dic83C2×D412yes—
111D332—28no—
112D332—22no—
113D332—24no—
114D332—22no—
115D332C2×D414yes—
116C4×D83—28no—
117C4×D83—22no—
118C4×D83—22no—
119Dic166—22no—
120C8×C83—22no—
121C83⋊C8—22no—
122C22×C83—22no—
123C22×C83—22no—
124C22×C83—42no—
125C22×C83#12012yes—
126C22×C83C22×D411yescharacteristic
127C2×D83—22no—
128C2×D83—24no—
129C2×D83—24no—
130C2×D83—44no—
131C2×D83—84no—
132C2×D83—88no—
133C4×C83—22no—
134C4×C83—42no—
135C4×C83#12012yes—
136C4×C83C22×D411yescharacteristic
137C4×C83C22×D411yescharacteristic
138Dic83—22no—
139Dic83—42no—
140C2×C83—22no—
141C2×C83—42no—
142C2×C83C2×#12011yescharacteristic
143C2×C83#57712yes—
144D83—42no—
145D83—44no—
146#3009—831nomaximal, Sylow
147C83#300911yescharacteristic, complemented, Sylow
148C2×H(5,𝔽2)—831no—
149C2×#137D8311yescharacteristic, complemented
150C22×SD16—831no—
151#536—831no—
152#452—834no—
153#450—834no—
154#418—831no—
155#416—831no—
156#408—831no—
157H(5,𝔽2)—1664no—
158H(5,𝔽2)—1664no—
159C2×#41—831no—
160C2×#41—832no—
161C2×#41—1661no—
162C2×#41C2×D8311yescharacteristic
163C22×Q8C2×D8311yescharacteristic, complemented
164C22×D4—832no—
165C22×D4—831no—
166C22×D4—1661no—
167C22×D4—1662no—
168#137—24no—
169C2×Q16C2×D8312yescomplemented
170C2×SD16—834no—
171C2×SD16—1664no—
172C2×SD16C2×D8312yescomplemented
173C2×M16C2×D8311yescharacteristic, complemented
174C22×C8—831no—
175C4⋊Q8—1661no—
176#124—1661no—
177#123—1662no—
178C22⋊Q8—834no—
179#117—831no—
180#103—832no—
181#102—832no—
182#102—834no—
183C22⋊C8—834no—
184C24—1662no—
185C24—1661no—
186#41—22no—
187#41—1668no—
188#41—1664no—
189#41—1664no—
190#41—1662no—
191#41—1662no—
192#41C22⋊D8314yes—
193C2×Q8—21no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—831no—
197C2×Q8C22×D8312yes—
198C2×Q8C22×D8311yescharacteristic
199C2×D4—21no—
200C2×D4—832no—
201C2×D4—834no—
202C2×D4—831no—
203C2×D4—832no—
204C2×D4—1662no—
205C2×D4—1668no—
206C2×D4—1662no—
207C2×D4—1664no—
208C2×D4—16616no—
209C2×D4—1664no—
210C2×D4—1661no—
211C2×D4—1662no—
212C2×D4C22×D8311yescharacteristic
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—832no—
216C22×C4—1662no—
217C22×C4C22×D8311yescharacteristic
218Q16—24no—
219SD16—24no—
220SD16—1668no—
221M16—22no—
222C2×C8—832no—
223C2×C8—1662no—
224C2×C8C22×D8312yes—
225C4⋊C4—832no—
226C4⋊C4—1664no—
227C22⋊C4—1664no—
228C22⋊C4—1661no—
229C42—1661no—
230C23—21no—
231C23—832no—
232C23—6642no—
233C23—6644no—
234C23—1668no—
235C23—1662no—
236C23—1664no—
237C23D4×D8311yescharacteristic
238Q8—21no—
239Q8—22no—
240Q8—22no—
241Q8—42no—
242Q8—1662no—
243Q8C2×C22⋊D8312yes—
244D4—24no—
245D4—21no—
246D4—834no—
247D4—1668no—
248D4—1668no—
249D4—1664no—
250D4—1664no—
251D4—1662no—
252D4—1662no—
253D4C2×C22⋊D8312yes—
254C2×C4—21no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—832no—
259C2×C4—1668no—
260C2×C4—1662no—
261C2×C4—1662no—
262C2×C4—1662no—
263C2×C4D4×D8311yescharacteristic
264C2×C4D4×D8312yes—
265C2×C4C2×C22⋊D8312yes—
266C2×C4C23×D8311yescharacteristic
267C8—22no—
268C8—1662no—
269C22—22no—
270C22—22no—
271C22—42no—
272C22—3324no—
273C22—6644no—
274C22—6648no—
275C22—1664no—
276C22—1662no—
277C22—1664no—
278C22#1458176012yes—
279C22C2×D4×D8311yescharacteristic
280C4—22no—
281C4—42no—
282C4—3322no—
283C4—1662no—
284C4#1458176012yes—
285C4C2×D4×D8311yescharacteristic
286C4C22×C22⋊D8311yescharacteristic
287C2—22no—
288C2—42no—
289C2—3324no—
290C2—3322no—
291C2C2×#1458176011yescharacteristic
292C2#1912851012yes—
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingC2×#137×C83C2
socleC22×C83C22×D4
prime radicalC2×#14581838C2
hypercenterC2×#137D83
hypocenterC83#3009
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C22×#13050577C2semidirect
3. #19128275C2semidirect
4. #19127862C2semidirect
5. #19127850C2semidirect
6. #19127707C2semidirect
7. #19127697C2semidirect
8. #19127669C2semidirect
9. C2×#137×C83C2semidirect
20. C2×#13050577C22semidirect
21. C22×C83⋊C8C22semidirect
25. #14581768C22semidirect
26. #14581751C22semidirect
27. #14581716C22semidirect
28. #14581714C22semidirect
29. #14581713C22semidirect
30. (C2×D83)⋊C8C22semidirect
32. C22×Q8×C83C22semidirect
34. C2×Q16×C83C22semidirect
35. C2×SD16×C83C22semidirect
36. C2×M16×C83C22semidirect
147. C83#3009semidirect
149. C2×#137D83semidirect
163. C22×Q8C2×D83semidirect
169. C2×Q16C2×D83semidirect
172. C2×SD16C2×D83semidirect
173. C2×M16C2×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145818382
C22×#130505772
#191282752
#191278622
#191278502
#191277072
#191276972
#191276692
C2×#137×C832
#300983