Finite Group Databasebeta

← #24918536

Subgroups — #24918536

Order 10624.

396 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 109 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#19128314C211yesmaximal, characteristic, complemented
2#19128264C211yesmaximal, characteristic
3#19128095C211yesmaximal, characteristic, complemented
4#19128073C211yesmaximal, characteristic, complemented
5#19128058C211yesmaximal, characteristic, complemented
6#19127922C212yesmaximal, complemented
7#19127900C212yesmaximal, complemented
8#19127864C212yesmaximal, complemented
9#19127853C212yesmaximal, complemented
10#19127763C211yesmaximal, characteristic, complemented
11#490×C83C211yesmaximal, characteristic, complemented
12C2×#13050589C2211yescharacteristic
13C2×D4×D83C2211yescharacteristic
14C2×Dic332C2212yescomplemented
15C2×#13050579C2212yes—
16C2×#13050578C2212yes—
17C2×#13050578C2212yescomplemented
18#14581792C2211yescharacteristic, complemented
19#14581788C2211yescharacteristic
20#14581787—22no—
21#14581783—22no—
22#14581780C2211yescharacteristic
23#14581779C2211yescharacteristic
24#14581772C2212yes—
25#14581764—22no—
26#14581760—22no—
27#14581757C2212yescomplemented
28#14581749C2211yescharacteristic
29Dic83⋊C8C2212yescomplemented
30#14581715C2212yescomplemented
31#14581712C2212yes—
32(C2×D83)⋊C8C2212yes—
33C4×C83⋊C8C2211yescharacteristic, complemented
34#127×C83C2211yescharacteristic
35#125×C83C2211yescharacteristic, complemented
36#102×C83C2212yes—
37#102×C83C2212yescomplemented
38#97×C83C2211yescharacteristic, complemented
39C23×D83—21no—
40#13050589—42no—
41C2×Dic166C2312yes—
42C2×Dic166C2311yescharacteristic
43C2×D332C2311yescharacteristic
44C2×C22⋊D83—21no—
45C2×C22⋊D83—21no—
46C2×C22⋊D83—22no—
47D4×D83—42no—
48C22×Dic83—21no—
49C2×C4×D83C2311yescharacteristic
50Dic332—42no—
51#13050579—42no—
52#13050578—42no—
53#13050578—42no—
54C2×C83⋊C8C2312yes—
55C4⋊Dic83—22no—
56C4⋊Dic83—22no—
57Dic83⋊C4—22no—
58Dic83⋊C4D412yes—
59C22⋊Dic83—22no—
60(C2×D83)⋊C4—22no—
61(C2×D83)⋊C4—22no—
62(C2×D83)⋊C4D412yes—
63C4×Dic83—22no—
64C4×Dic83C2311yescharacteristic
65C2×D4×C83—22no—
66C2×D4×C83C2311yescharacteristic
67C2×D4×C83C2311yescharacteristic
68C2×C8×C83C2312yes—
69C4⋊C4×C83—22no—
70C4⋊C4×C83C2312yes—
71C42×C83C2311yescharacteristic
72C23×C83—21no—
73C23×C83—21no—
74C22×D83—21no—
75C22×D83—42no—
76C22×D83—82no—
77C22×D83C2×D411yescharacteristic
78D4×C83—22no—
79D4×C83—22no—
80D4×C83—82no—
81C2×C4×C83—22no—
82C2×C4×C83C2×D412yes—
83C2×C4×C83C2411yescharacteristic
84C22⋊D83—42no—
85C22⋊D83—42no—
86C22⋊D83—82no—
87C2×Dic83—22no—
88C2×Dic83—21no—
89C2×Dic83—42no—
90C2×Dic83C2×D412yes—
91C2×Dic83C2×D411yescharacteristic
92D332—22no—
93C4×D83—22no—
94Dic166—22no—
95Dic166—22no—
96Dic166—22no—
97C8×C83—22no—
98C83⋊C8—22no—
99C22×C83—41no—
100C22×C83—41no—
101C22×C83—42no—
102C22×C83—42no—
103C22×C83C22×D411yescharacteristic
104C2×D83—21no—
105C2×D83—22no—
106C2×D83—41no—
107C2×D83—42no—
108C2×D83—84no—
109C4×C83—22no—
110C4×C83—42no—
111C4×C83H(5,𝔽2)12yes—
112Dic83—21no—
113Dic83—22no—
114Dic83—42no—
115Dic83—41no—
116C2×C83—81no—
117C2×C83—81no—
118C2×C83#53511yescharacteristic
119C2×C83#57712yes—
120D83—41no—
121D83—81no—
122#3222—831nomaximal, Sylow
123C83#322211yescharacteristic, complemented, Sylow
124#541—831no—
125#535—831no—
126#497—831no—
127#492—831no—
128#490D8311yescharacteristic, complemented
129#488—831no—
130#464—832no—
131#460—832no—
132#452—832no—
133#450—832no—
134#433—831no—
135C2×#41—831no—
136C22×D4—831no—
137C2×Q16—832no—
138C2×SD16—832no—
139C2×SD16—832no—
140C2×SD16—832no—
141C4⋊Q8—831no—
142#127—831no—
143#127C2×D8311yescharacteristic
144#126—1662no—
145#125—1662no—
146#125C2×D8311yescharacteristic, complemented
147#124—831no—
148#124—831no—
149C22⋊Q8—832no—
150#121—1662no—
151#120—1662no—
152C4×Q8—832no—
153#117—831no—
154C4⋊C8—832no—
155#103—832no—
156#102—832no—
157#102C2×D8312yes—
158#102C2×D8312yescomplemented
159C22⋊C8—832no—
160#97C2×D8311yescharacteristic, complemented
161C4×C8—831no—
162C24—1661no—
163#41—3322no—
164C2×Q8—831no—
165C2×Q8—832no—
166C2×D4—22no—
167C2×D4—3322no—
168C2×D4—831no—
169C2×D4—1661no—
170C2×D4—1662no—
171C2×D4—1661no—
172C2×D4C22×D8311yescharacteristic
173C2×D4C22×D8311yescharacteristic
174C22×C4—831no—
175C22×C4—1661no—
176Q16—3322no—
177SD16—3322no—
178SD16—3322no—
179SD16—3322no—
180C2×C8—832no—
181C2×C8C22×D8312yes—
182C4⋊C4—22no—
183C4⋊C4—832no—
184C4⋊C4—1662no—
185C4⋊C4—1662no—
186C4⋊C4—1662no—
187C4⋊C4C22×D8312yes—
188C22⋊C4—832no—
189C22⋊C4—1662no—
190C22⋊C4—1662no—
191C22⋊C4—1662no—
192C42—831no—
193C42—1662no—
194C42C22×D8311yescharacteristic
195C23—21no—
196C23—21no—
197C23—3322no—
198C23—831no—
199C23—6642no—
200C23—1661no—
201Q8—1662no—
202Q8—1662no—
203Q8—1662no—
204D4—22no—
205D4—22no—
206D4—82no—
207D4—3322no—
208D4—3322no—
209D4—6642no—
210D4—1662no—
211C2×C4—22no—
212C2×C4—3322no—
213C2×C4—832no—
214C2×C4—831no—
215C2×C4—1662no—
216C2×C4—1661no—
217C2×C4—1662no—
218C2×C4D4×D8312yes—
219C2×C4C23×D8311yescharacteristic
220C8—22no—
221C8—1662no—
222C22—42no—
223C22—42no—
224C22—41no—
225C22—41no—
226C22—3322no—
227C22—3321no—
228C22—6644no—
229C22—1662no—
230C22—1661no—
231C22C2×D4×D8311yescharacteristic
232C4—22no—
233C4—42no—
234C4—3321no—
235C4—3322no—
236C4—1662no—
237C4—1661no—
238C4#1458183812yes—
239C2—81no—
240C2—81no—
241C2—3321no—
242C2—6641no—
243C2#1912826911yescharacteristic
244C2#1912851112yes—
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#490×C83C2
socleC22×C83C22×D4
prime radical#19128264C2
hypercenter#490D83
hypocenterC83#3222
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. #19128314C2semidirect
3. #19128095C2semidirect
4. #19128073C2semidirect
5. #19128058C2semidirect
6. #19127922C2semidirect
7. #19127900C2semidirect
8. #19127864C2semidirect
9. #19127853C2semidirect
10. #19127763C2semidirect
11. #490×C83C2semidirect
14. C2×Dic332C22semidirect
17. C2×#13050578C22semidirect
18. #14581792C22semidirect
27. #14581757C22semidirect
29. Dic83⋊C8C22semidirect
30. #14581715C22semidirect
33. C4×C83⋊C8C22semidirect
35. #125×C83C22semidirect
37. #102×C83C22semidirect
38. #97×C83C22semidirect
123. C83#3222semidirect
128. #490D83semidirect
146. #125C2×D83semidirect
158. #102C2×D83semidirect
160. #97C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191283142
#191282642
#191280952
#191280732
#191280582
#191279222
#191279002
#191278642
#191278532
#191277632
#490×C832
#322283