Finite Group Databasebeta

← #24906004

Subgroups — #24906004

Order 10624.

286 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 73 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#19128212C211yesmaximal, characteristic
2#19127665C211yesmaximal, characteristic, complemented
3#19127437C212yesmaximal, complemented
4#19127434C212yesmaximal, complemented
5#380×C83C211yesmaximal, characteristic, complemented
6C2×#13050590C411yescharacteristic, complemented
7C2×#13050574—21no—
8#14581775—21no—
9#14581771C411yescharacteristic, complemented
10#14581771C2211yescharacteristic
11#14581766C411yescharacteristic, complemented
12#14581766C2211yescharacteristic
13#14581761—21no—
14C4×C22⋊D83—21no—
15C4×C22⋊D83—21no—
16#14581751C411yescharacteristic, complemented
17C22⋊(C83⋊C8)—21no—
18C22⋊(C83⋊C8)C2212yes—
19C23×C4×C83C2211yescharacteristic
20C2×C4⋊C4×C83—21no—
21C2×C22⋊C4×C83—21no—
22#95×C83C2212yes—
23#13050590—42no—
24C2×Dic166C2×C411yescharacteristic
25C2×D332C2×C411yescharacteristic
26C2×C22⋊D83—21no—
27C2×C22⋊D83—21no—
28C2×C22⋊D83—21no—
29C2×C4×D83—21no—
30#13050574—42no—
31C2×C83⋊C8—22no—
32C4⋊Dic83C2×C411yescharacteristic
33C4⋊Dic83C2×C411yescharacteristic
34Dic83⋊C4—21no—
35Dic83⋊C4—21no—
36C22⋊Dic83—21no—
37C22⋊Dic83—21no—
38C22⋊Dic83—21no—
39(C2×D83)⋊C4—21no—
40(C2×D83)⋊C4—21no—
41C4×Dic83—21no—
42C24×C83D411yescharacteristic
43C22×C4×C83—22no—
44C22×C4×C83—42no—
45C22×C4×C83C2×C411yescharacteristic
46C22×C4×C83C2×C411yescharacteristic
47C22×C4×C83D411yescharacteristic
48C22×C4×C83D411yescharacteristic
49C22×C4×C83D411yescharacteristic
50C22×C4×C83C2311yescharacteristic
51C4⋊C4×C83—42no—
52C22⋊C4×C83—42no—
53C23×C83—21no—
54C23×C83—21no—
55C23×C83—41no—
56C23×C83—41no—
57C23×C83C22⋊C411yescharacteristic
58C23×C83C22⋊C411yescharacteristic
59C23×C83C2×D411yescharacteristic
60C22×D83—21no—
61C2×C4×C83—22no—
62C2×C4×C83—22no—
63C2×C4×C83—22no—
64C2×C4×C83—41no—
65C2×C4×C83—41no—
66C2×C4×C83—41no—
67C2×C4×C83—42no—
68C2×C4×C83—42no—
69C2×C4×C83—42no—
70C2×C4×C83C22⋊C411yescharacteristic
71C2×C4×C83C22⋊C411yescharacteristic
72C2×C4×C83C22×C411yescharacteristic
73C2×C4×C83C2×D411yescharacteristic
74C22⋊D83—42no—
75C22⋊D83—42no—
76C22⋊D83—41no—
77C22⋊D83—41no—
78C2×Dic83—21no—
79C2×Dic83—21no—
80C2×Dic83—21no—
81D332—41no—
82C4×D83—42no—
83Dic166—41no—
84C83⋊C8—42no—
85C22×C83—21no—
86C22×C83—21no—
87C22×C83—21no—
88C22×C83—21no—
89C22×C83—21no—
90C22×C83—21no—
91C22×C83—21no—
92C22×C83—21no—
93C22×C83—42no—
94C22×C83—42no—
95C22×C83C2×C22⋊C411yescharacteristic
96C22×C83AGL(1,ℤ/8)11yescharacteristic
97C22×C83#13711yescharacteristic
98C2×D83—41no—
99C2×D83—41no—
100C2×D83—41no—
101C4×C83—21no—
102C4×C83—21no—
103C4×C83—21no—
104C4×C83—82no—
105C4×C83#9912yes—
106Dic83—41no—
107Dic83—41no—
108Dic83—41no—
109C2×C83—21no—
110C2×C83—21no—
111C2×C83—41no—
112C2×C83—41no—
113C2×C83C2×#9911yescharacteristic
114C2×C83#41711yescharacteristic
115C2×C83#42111yescharacteristic
116D83—81no—
117#1513—831nomaximal, Sylow
118C83#151311yescharacteristic, complemented, Sylow
119#525—831no—
120#407—831no—
121#380D8311yescharacteristic, complemented
122#328—832no—
123#327—832no—
124C2×#41—831no—
125C23×C4C2×D8311yescharacteristic
126C2×M16—1661no—
127#123—1661no—
128C22⋊Q8—831no—
129C22⋊Q8—831no—
130#121—831no—
131#121—831no—
132#120—1661no—
133C4×D4—1661no—
134C4×D4—1661no—
135#117—831no—
136C2×C4⋊C4—21no—
137C2×C22⋊C4—21no—
138C22⋊C8—832no—
139C22⋊C8—1661no—
140#95C2×D8312yes—
141C24C22⋊D8311yescharacteristic
142#41—3322no—
143C2×Q8—831no—
144C2×D4—831no—
145C2×D4—1661no—
146C2×D4—1661no—
147C2×D4—1661no—
148C22×C4—22no—
149C22×C4—42no—
150C22×C4—1661no—
151C22×C4C4×D8311yescharacteristic
152C22×C4C4×D8311yescharacteristic
153C22×C4D33211yescharacteristic
154C22×C4D33211yescharacteristic
155C22×C4C22⋊D8311yescharacteristic
156C22×C4C22×D8311yescharacteristic
157M16—3322no—
158C2×C8—1662no—
159C4⋊C4—42no—
160C4⋊C4—831no—
161C4⋊C4—831no—
162C4⋊C4—1661no—
163C4⋊C4—1661no—
164C22⋊C4—42no—
165C22⋊C4—1661no—
166C22⋊C4—1661no—
167C22⋊C4—1661no—
168C22⋊C4—1661no—
169C22⋊C4—1661no—
170C42—1661no—
171C23—21no—
172C23—21no—
173C23—41no—
174C23—41no—
175C23—1661no—
176C23(C2×D83)⋊C411yescharacteristic
177C23(C2×D83)⋊C411yescharacteristic
178C23C2×C22⋊D8311yescharacteristic
179Q8—3321no—
180D4—3321no—
181D4—3322no—
182D4—3322no—
183D4—3321no—
184D4—3321no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—3322no—
195C2×C4—1661no—
196C2×C4—1661no—
197C2×C4—1661no—
198C2×C4(C2×D83)⋊C411yescharacteristic
199C2×C4(C2×D83)⋊C411yescharacteristic
200C2×C4C2×C4×D8311yescharacteristic
201C2×C4C2×D33211yescharacteristic
202C8—3322no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—42no—
212C22—42no—
213C22—3321no—
214C22—3321no—
215C22—3321no—
216C22C2×(C2×D83)⋊C411yescharacteristic
217C22#1458181811yescharacteristic
218C22#1458182311yescharacteristic
219C4—21no—
220C4—21no—
221C4—21no—
222C4—82no—
223C4—3321no—
224C4—3321no—
225C4—3321no—
226C4#1458170612yes—
227C2—21no—
228C2—21no—
229C2—41no—
230C2—41no—
231C2—6641no—
232C2C2×#1458170611yescharacteristic
233C2#1912770011yescharacteristic
234C2#1912771911yescharacteristic
subgroupG/H
centerC22C2×(C2×D83)⋊C4
commutatorC2×C4×C83C22×C4
FrattiniC22×C4C22×D83
Fitting#380×C83C2
socleC22×C83C2×C22⋊C4
prime radical#19128212C2
cosocleC22×C4×C83C23
hypercenter#380D83
hypocenterC83#1513
LCSG ▷ C2×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
2. #19127665C2semidirect
3. #19127437C2semidirect
4. #19127434C2semidirect
5. #380×C83C2semidirect
6. C2×#13050590C4semidirect
9. #14581771C4semidirect
11. #14581766C4semidirect
16. #14581751C4semidirect
118. C83#1513semidirect
121. #380D83semidirect