Finite Group Databasebeta

← #24912959

Subgroups — #24912959

Order 10624.

440 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 129 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#19127640C211yesmaximal, characteristic, complemented
2#19127631C211yesmaximal, characteristic, complemented
3#19127615C212yesmaximal
4#19127595C212yesmaximal, complemented
5#19127595C212yesmaximal, complemented
6#19127594C212yesmaximal, complemented
7#19127594C212yesmaximal, complemented
8C4×Dic83⋊C4C212yesmaximal, complemented
9#390×C83C211yesmaximal, characteristic
10C2×C4⋊Dic83C2211yescharacteristic
11C2×C4⋊Dic83C2211yescharacteristic
12C2×Dic83⋊C4C2212yes—
13C2×Dic83⋊C4C2212yes—
14C2×C22⋊Dic83C2212yes—
15C2×C22⋊Dic83C2212yes—
16C2×C22⋊Dic83C2212yes—
17C2×C22⋊Dic83C2212yes—
18C2×C4×Dic83C2212yes—
19C2×C4×Dic83C2212yes—
20C83⋊#95C2212yes—
21C83⋊#95C2212yes—
22C83⋊#95C2212yes—
23C83⋊#95C2212yes—
24C83⋊#95C2211yescharacteristic
25C83⋊#95C2211yescharacteristic
26C22×D4×C83C2211yescharacteristic
27C2×C4⋊C4×C83C2211yescharacteristic
28C2×C22⋊C4×C83C2212yes—
29C2×C22⋊C4×C83C2212yes—
30C2×C42×C83C2211yescharacteristic
31C22×Dic83C2312yes—
32C22×Dic83C2312yes—
33C22×Dic83C2312yes—
34C22×Dic83C2312yes—
35C4⋊Dic83—41no—
36C4⋊Dic83—41no—
37Dic83⋊C4—24no—
38Dic83⋊C4—24no—
39C22⋊Dic83—24no—
40C22⋊Dic83—24no—
41C22⋊Dic83—42no—
42C22⋊Dic83—42no—
43C4×Dic83—24no—
44C4×Dic83—42no—
45C24×C83C2312yes—
46C2×D4×C83—22no—
47C2×D4×C83—41no—
48C2×D4×C83—41no—
49C22×C4×C83C2311yescharacteristic
50C22×C4×C83C2311yescharacteristic
51C22×C4×C83C2312yes—
52C22×C4×C83C2311yescharacteristic
53C4⋊C4×C83—41no—
54C22⋊C4×C83—24no—
55C22⋊C4×C83—24no—
56C42×C83—41no—
57C23×C83—22no—
58C23×C83—22no—
59C23×C83—22no—
60C23×C83—22no—
61C23×C83—22no—
62C23×C83—22no—
63C23×C83—22no—
64C23×C83C2411yescharacteristic
65D4×C83—44no—
66C2×C4×C83—22no—
67C2×C4×C83—21no—
68C2×C4×C83—22no—
69C2×C4×C83—21no—
70C2×C4×C83—22no—
71C2×C4×C83—22no—
72C2×C4×C83—21no—
73C2×C4×C83—21no—
74C2×C4×C83#4114yes—
75C2×C4×C83#4112yes—
76C2×Dic83—22no—
77C2×Dic83—22no—
78C2×Dic83—22no—
79C2×Dic83—22no—
80C2×Dic83—22no—
81C2×Dic83—22no—
82C2×Dic83—22no—
83C2×Dic83—22no—
84C2×Dic83—22no—
85C2×Dic83—22no—
86C2×Dic83—22no—
87C2×Dic83#4114yes—
88C22×C83—42no—
89C22×C83—42no—
90C22×C83—42no—
91C22×C83—42no—
92C22×C83—42no—
93C22×C83—42no—
94C22×C83—42no—
95C22×C83C2×#4112yes—
96C22×C83C2×#4112yes—
97C22×C83C2×#4111yescharacteristic
98C22×C83H(5,𝔽2)11yescharacteristic
99C22×C83H(5,𝔽2)11yescharacteristic
100C4×C83—22no—
101C4×C83—24no—
102C4×C83—41no—
103C4×C83—41no—
104Dic83—24no—
105Dic83—42no—
106Dic83—42no—
107Dic83—42no—
108C2×C83—82no—
109C2×C83#52912yes—
110C2×C83#53811yescharacteristic
111C2×C83#55312yes—
112C2×C83#55511yescharacteristic
113C2×C83#55911yescharacteristic
114#2593—831nomaximal, Sylow
115C83#259311yescharacteristic, complemented, Sylow
116#400—831no—
117#398—831no—
118#394—832no—
119#390D8311yescharacteristic
120#388—832no—
121#388—832no—
122#388—832no—
123#388—832no—
124C4×C4⋊C4—832no—
125C22×D4C2×D8311yescharacteristic
126C2×C4⋊C4—831no—
127C2×C4⋊C4—831no—
128C2×C4⋊C4—832no—
129C2×C4⋊C4—832no—
130C2×C4⋊C4C2×D8311yescharacteristic
131C2×C22⋊C4—832no—
132C2×C22⋊C4—832no—
133C2×C22⋊C4—832no—
134C2×C22⋊C4—832no—
135C2×C22⋊C4C2×D8312yes—
136C2×C22⋊C4C2×D8312yes—
137C2×C42—832no—
138C2×C42—832no—
139C2×C42C2×D8311yescharacteristic
140#95—832no—
141#95—832no—
142#95—831no—
143#95—832no—
144#95—832no—
145#95—831no—
146C24C22×D8312yes—
147C2×D4—22no—
148C2×D4—41no—
149C2×D4—41no—
150C22×C4—832no—
151C22×C4—832no—
152C22×C4—832no—
153C22×C4—832no—
154C22×C4C22×D8311yescharacteristic
155C22×C4C22×D8311yescharacteristic
156C22×C4C22×D8312yes—
157C22×C4C22×D8311yescharacteristic
158C4⋊C4—41no—
159C4⋊C4—3321no—
160C4⋊C4—3321no—
161C4⋊C4—1664no—
162C4⋊C4—1664no—
163C22⋊C4—24no—
164C22⋊C4—24no—
165C22⋊C4—3322no—
166C22⋊C4—3322no—
167C22⋊C4—1664no—
168C22⋊C4—1664no—
169C42—41no—
170C42—3322no—
171C42—1664no—
172C23—22no—
173C23—22no—
174C23—22no—
175C23—22no—
176C23—22no—
177C23—22no—
178C23—22no—
179C23C23×D8311yescharacteristic
180D4—44no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—834no—
190C2×C4—1662no—
191C2×C4—1662no—
192C2×C4—1662no—
193C2×C4—1662no—
194C2×C4—1662no—
195C2×C4—1662no—
196C2×C4—1662no—
197C2×C4—1662no—
198C2×C4—1662no—
199C2×C4—1662no—
200C2×C4—1662no—
201C2×C4#1305058914yes—
202C2×C4#1305059012yes—
203C22—42no—
204C22—42no—
205C22—42no—
206C22—42no—
207C22—42no—
208C22—42no—
209C22—42no—
210C22#41×D8312yes—
211C22C2×#1305058912yes—
212C22C2×#1305059011yescharacteristic
213C22#1458183711yescharacteristic
214C22#1458183711yescharacteristic
215C4—24no—
216C4—22no—
217C4—41no—
218C4—41no—
219C4—3322no—
220C4—3322no—
221C4—3322no—
222C4—1664no—
223C2—82no—
224C2#1912823312yes—
225C2#1912828911yescharacteristic
226C2#1912839412yes—
227C2#1912841311yescharacteristic
228C2#1912843711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#390×C83C2
prime radicalC22×D4×C83C22
hypercenter#390D83
hypocenterC83#2593
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #19127640C2semidirect
2. #19127631C2semidirect
4. #19127595C2semidirect
5. #19127595C2semidirect
6. #19127594C2semidirect
7. #19127594C2semidirect
8. C4×Dic83⋊C4C2semidirect
115. C83#2593semidirect

Maximal subgroups

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

subgroupindex
#191276402
#191276312
#191276152
#191275952
#191275942
C4×Dic83⋊C42
#390×C832
#259383