Finite Group Databasebeta

← #24918422

Subgroups — #24918422

Order 10624.

394 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 117 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#19128256C211yesmaximal, characteristic, complemented
2#19128252C211yesmaximal, characteristic
3#19128080C211yesmaximal, characteristic, complemented
4#19128074C211yesmaximal, characteristic, complemented
5#19128049C211yesmaximal, characteristic, complemented
6#19127926C214yesmaximal, complemented
7#19127866C214yesmaximal, complemented
8#19127752C211yesmaximal, characteristic, complemented
9#487×C83C211yesmaximal, characteristic, complemented
10C2×#13050591C2212yes—
11C2×#13050580C2214yes—
12C2×#13050579C2214yes—
13#14581792C2211yescharacteristic
14#14581789C2211yescharacteristic
15#14581783—21no—
16#14581781C2212yes—
17#14581772C2212yes—
18#14581768—21no—
19#14581767—21no—
20#14581767—21no—
21Q8×Dic83—21no—
22#14581757C2212yes—
23#14581754—21no—
24#14581754—21no—
25#14581750—21no—
26C42×D83C2211yescharacteristic
27Dic83⋊C8C2212yes—
28#14581712C2214yes—
29(C2×D83)⋊C8C2212yes—
30C4×C83⋊C8C2211yescharacteristic
31C4⋊Q8×C83C2211yescharacteristic
32C4⋊Q8×C83C2211yescharacteristic
33#103×C83C2214yes—
34C4×C8×C83C2211yescharacteristic
35#13050591—44no—
36C2×Dic166C2312yes—
37C2×D332—22no—
38C2×D332—21no—
39C2×D332—21no—
40C2×D332C2312yes—
41C2×C4×D83—22no—
42C2×C4×D83D411yescharacteristic
43C2×C4×D83D411yescharacteristic
44C2×C4×D83C2311yescharacteristic
45#13050580—44no—
46#13050579—44no—
47C2×C83⋊C8C2312yes—
48C4⋊Dic83—21no—
49C4⋊Dic83—21no—
50C4⋊Dic83—22no—
51Dic83⋊C4—22no—
52(C2×D83)⋊C4—22no—
53(C2×D83)⋊C4—22no—
54(C2×D83)⋊C4—22no—
55C4×Dic83—22no—
56C4×Dic83D411yescharacteristic
57C4×Dic83D411yescharacteristic
58C4×Dic83C2311yescharacteristic
59C2×Q8×C83—21no—
60C2×Q8×C83C2312yes—
61C2×C8×C83C2312yes—
62C4⋊C4×C83—21no—
63C4⋊C4×C83—21no—
64C4⋊C4×C83—21no—
65C4⋊C4×C83C2312yes—
66C42×C83C2311yescharacteristic
67C22×D83—22no—
68C22×D83C2×D411yescharacteristic
69Q8×C83—24no—
70Q8×C83—42no—
71C2×C4×C83—22no—
72C2×C4×C83—22no—
73C2×C4×C83C2×D411yescharacteristic
74C2×C4×C83C2×D411yescharacteristic
75C2×C4×C83C2411yescharacteristic
76C2×Dic83—22no—
77C2×Dic83C2×D411yescharacteristic
78C2×Dic83C2×D411yescharacteristic
79C2×Dic83C2×D411yescharacteristic
80D332—24no—
81D332—44no—
82D332—42no—
83D332—42no—
84C4×D83—22no—
85C4×D83—22no—
86C4×D83—22no—
87C4×D83—44no—
88Dic166—24no—
89C8×C83—22no—
90C83⋊C8—22no—
91C22×C83C22×D411yescharacteristic
92C2×D83—21no—
93C2×D83—21no—
94C2×D83—21no—
95C2×D83—42no—
96C2×D83—42no—
97C2×D83—42no—
98C4×C83—42no—
99C4×C83—42no—
100C4×C83#13512yes—
101C4×C83#13712yes—
102C4×C83H(5,𝔽2)12yes—
103Dic83—21no—
104Dic83—21no—
105Dic83—21no—
106Dic83—42no—
107C2×C83#53511yescharacteristic
108C2×C83C2×#13511yescharacteristic
109C2×C83C2×#13711yescharacteristic
110D83—41no—
111D83—82no—
112#3210—831nomaximal, Sylow
113C83#321011yescharacteristic, complemented, Sylow
114#533—831no—
115#532—831no—
116#494—831no—
117#492—831no—
118#487D8311yescharacteristic, complemented
119#486—831no—
120#464—834no—
121#452—834no—
122#431—831no—
123C2×#41—832no—
124C2×Q16—834no—
125C2×SD16—834no—
126C4⋊Q8—831no—
127C4⋊Q8C2×D8311yescharacteristic
128C4⋊Q8C2×D8311yescharacteristic
129#127—831no—
130#125—1661no—
131#124—832no—
132C22⋊Q8—832no—
133C22⋊Q8—1661no—
134#121—1661no—
135#121—1661no—
136C4×Q8—832no—
137C4×Q8—1661no—
138C4×D4—1661no—
139C4×D4—1661no—
140#117—1661no—
141C2×C42—831no—
142C4⋊C8—832no—
143#103C2×D8314yes—
144#102—834no—
145C22⋊C8—832no—
146C4×C8—831no—
147C4×C8C2×D8311yescharacteristic
148#41—3324no—
149C2×Q8—21no—
150C2×Q8—832no—
151C2×Q8C22×D8312yes—
152C2×D4—832no—
153C2×D4—1662no—
154C2×D4—1661no—
155C2×D4—1661no—
156C22×C4—831no—
157C22×C4—831no—
158C22×C4—831no—
159C22×C4—1662no—
160Q16—3324no—
161SD16—3324no—
162C2×C8—832no—
163C2×C8C22×D8312yes—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—1662no—
168C4⋊C4—1662no—
169C4⋊C4—1661no—
170C4⋊C4—1661no—
171C4⋊C4C22×D8312yes—
172C22⋊C4—1662no—
173C22⋊C4—1662no—
174C22⋊C4—1662no—
175C42—831no—
176C42—831no—
177C42—831no—
178C42—1662no—
179C42C22×D8311yescharacteristic
180C23—831no—
181C23—1662no—
182Q8—24no—
183Q8—42no—
184Q8—1664no—
185D4—3324no—
186D4—3322no—
187D4—3322no—
188D4—1664no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—3324no—
192C2×C4—831no—
193C2×C4—831no—
194C2×C4—831no—
195C2×C4—1662no—
196C2×C4—1662no—
197C2×C4—1662no—
198C2×C4—1662no—
199C2×C4D4×D8311yescharacteristic
200C2×C4D4×D8311yescharacteristic
201C2×C4C23×D8311yescharacteristic
202C8—22no—
203C8—1662no—
204C22—3322no—
205C22—3322no—
206C22—3322no—
207C22—1661no—
208C22—1661no—
209C22—1661no—
210C22C2×D4×D8311yescharacteristic
211C4—42no—
212C4—42no—
213C4—3322no—
214C4—1661no—
215C4—1661no—
216C4—1661no—
217C4#1458181312yes—
218C4#1458182412yes—
219C4#1458183812yes—
220C2—3321no—
221C2—6642no—
222C2#1912826911yescharacteristic
223C2C2×#1458181311yescharacteristic
224C2C2×#1458182411yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#487×C83C2
socleC22×C83C22×D4
prime radical#19128252C2
hypercenter#487D83
hypocenterC83#3210
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
1. #19128256C2semidirect
3. #19128080C2semidirect
4. #19128074C2semidirect
5. #19128049C2semidirect
6. #19127926C2semidirect
7. #19127866C2semidirect
8. #19127752C2semidirect
9. #487×C83C2semidirect
113. C83#3210semidirect
118. #487D83semidirect

Maximal subgroups

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

subgroupindex
#191282562
#191282522
#191280802
#191280742
#191280492
#191279262
#191278662
#191277522
#487×C832
#321083