Finite Group Databasebeta

← #24909107

Subgroups — #24909107

Order 10624.

280 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 65 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#19128260C211yesmaximal, characteristic
2C2×#14581709C211yesmaximal, characteristic, complemented
3#19127490C212yesmaximal, complemented
4#19127487C212yesmaximal, complemented
5C2×#99×C83C211yesmaximal, characteristic, complemented
6C22×C22⋊D83C411yescharacteristic, complemented
7C2×#13050574—21no—
8#14581775—21no—
9#14581773C411yescharacteristic, complemented
10#14581773C2211yescharacteristic
11#14581764—21no—
12#14581764—21no—
13#14581761—21no—
14#14581760C411yescharacteristic, complemented
15#14581760C2211yescharacteristic
16C2×C22⋊Dic83C411yescharacteristic, complemented
17#14581709—21no—
18#14581709C2212yes—
19C22×D4×C83C2211yescharacteristic
20C2×C22⋊C4×C83—21no—
21#99×C83—21no—
22#99×C83C2212yes—
23C23×D83C2×C411yescharacteristic, complemented
24C2×C22⋊D83—21no—
25C2×C22⋊D83—21no—
26C2×C22⋊D83—21no—
27C2×C22⋊D83—21no—
28C2×C22⋊D83—41no—
29C2×C22⋊D83—41no—
30C22×Dic83—21no—
31C22×Dic83C2×C411yescharacteristic, complemented
32#13050574—22no—
33#13050574—41no—
34C2×C83⋊C8—41no—
35Dic83⋊C4—21no—
36Dic83⋊C4—21no—
37C22⋊Dic83—21no—
38C22⋊Dic83—21no—
39C22⋊Dic83—21no—
40C22⋊Dic83C2×C411yescharacteristic, complemented
41C22⋊Dic83C2×C411yescharacteristic, complemented
42(C2×D83)⋊C4—21no—
43(C2×D83)⋊C4—21no—
44C24×C83C2×C411yescharacteristic
45C24×C83D411yescharacteristic
46C2×D4×C83—41no—
47C2×D4×C83—41no—
48C2×D4×C83C2×C411yescharacteristic
49C2×D4×C83D411yescharacteristic
50C2×D4×C83D411yescharacteristic
51C2×D4×C83C2311yescharacteristic
52C22×C4×C83—41no—
53C22×C4×C83D411yescharacteristic
54C22⋊C4×C83—22no—
55C22⋊C4×C83—41no—
56C23×C83—21no—
57C23×C83—21no—
58C23×C83—21no—
59C23×C83—21no—
60C23×C83—81no—
61C23×C83—81no—
62C23×C83C22⋊C411yescharacteristic
63C23×C83C22⋊C411yescharacteristic
64C23×C83C22⋊C411yescharacteristic
65C23×C83C22×C411yescharacteristic
66C23×C83C2×D411yescharacteristic
67C22×D83—21no—
68C22×D83—41no—
69C22×D83—81no—
70D4×C83—41no—
71D4×C83—41no—
72D4×C83—41no—
73D4×C83—41no—
74C2×C4×C83—21no—
75C2×C4×C83—21no—
76C2×C4×C83—42no—
77C2×C4×C83—82no—
78C2×C4×C83C22⋊C411yescharacteristic
79C2×C4×C83C2×D411yescharacteristic
80C22⋊D83—42no—
81C22⋊D83—41no—
82C22⋊D83—41no—
83C22⋊D83—81no—
84C22⋊D83—81no—
85C2×Dic83—21no—
86C2×Dic83—21no—
87C2×Dic83—21no—
88C2×Dic83—41no—
89C2×Dic83—81no—
90C83⋊C8—42no—
91C22×C83—21no—
92C22×C83—21no—
93C22×C83—21no—
94C22×C83—21no—
95C22×C83—41no—
96C22×C83—41no—
97C22×C83—41no—
98C22×C83—41no—
99C22×C83—41no—
100C22×C83—42no—
101C22×C83—81no—
102C22×C83—81no—
103C22×C83—81no—
104C22×C83#9911yescharacteristic
105C22×C83#9911yescharacteristic
106C22×C83C2×C22⋊C411yescharacteristic
107C2×D83—41no—
108C2×D83—41no—
109C2×D83—41no—
110C2×D83—82no—
111C4×C83—21no—
112C4×C83—21no—
113C4×C83—82no—
114Dic83—41no—
115Dic83—41no—
116Dic83—41no—
117C2×C83—21no—
118C2×C83—21no—
119C2×C83—21no—
120C2×C83—41no—
121C2×C83—41no—
122C2×C83—41no—
123C2×C83—41no—
124C2×C83C2×#9911yescharacteristic
125D83—81no—
126#2121—831nomaximal, Sylow
127C83#212111yescharacteristic, complemented, Sylow
128#534—831no—
129C2×#100—831no—
130C2×#99D8311yescharacteristic, complemented
131#352—832no—
132AΣL(1,𝔽16)—832no—
133C22×D4—831no—
134C22×D4C2×D8311yescharacteristic
135C2×M16—1661no—
136#123—831no—
137#123—831no—
138#123—1661no—
139#121—1661no—
140#121—1661no—
141#120—831no—
142#120—831no—
143#120—1661no—
144C2×C22⋊C4—21no—
145C2×C22⋊C4—831no—
146#100—832no—
147#100—1661no—
148#99—21no—
149#99C2×D8312yes—
150C24—831no—
151C24C4×D8311yescharacteristic
152C24C22⋊D8311yescharacteristic
153C2×D4—41no—
154C2×D4—41no—
155C2×D4—3321no—
156C2×D4—3321no—
157C2×D4—1661no—
158C2×D4—1661no—
159C2×D4—1661no—
160C2×D4—1661no—
161C2×D4C4×D8311yescharacteristic
162C2×D4D33211yescharacteristic
163C2×D4C22⋊D8311yescharacteristic
164C2×D4C22×D8311yescharacteristic
165C22×C4—41no—
166C22×C4—831no—
167C22×C4—1661no—
168C22×C4D33211yescharacteristic
169M16—3321no—
170M16—1662no—
171C2×C8—3321no—
172C4⋊C4—1661no—
173C4⋊C4—1661no—
174C22⋊C4—22no—
175C22⋊C4—41no—
176C22⋊C4—831no—
177C22⋊C4—831no—
178C22⋊C4—1661no—
179C22⋊C4—1661no—
180C22⋊C4—1661no—
181C22⋊C4—1661no—
182C22⋊C4—1661no—
183C23—21no—
184C23—21no—
185C23—21no—
186C23—21no—
187C23—81no—
188C23—81no—
189C23—3321no—
190C23—6641no—
191C23—1661no—
192C23(C2×D83)⋊C411yescharacteristic
193C23(C2×D83)⋊C411yescharacteristic
194C23(C2×D83)⋊C411yescharacteristic
195C23C2×C4×D8311yescharacteristic
196C23C2×C22⋊D8311yescharacteristic
197D4—41no—
198D4—41no—
199D4—41no—
200D4—41no—
201D4—3322no—
202D4—3321no—
203D4—3321no—
204D4—6641no—
205D4—6641no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—42no—
209C2×C4—82no—
210C2×C4—3321no—
211C2×C4—6641no—
212C2×C4—1661no—
213C2×C4—1661no—
214C2×C4—1661no—
215C2×C4(C2×D83)⋊C411yescharacteristic
216C2×C4C2×D33211yescharacteristic
217C8—3322no—
218C22—21no—
219C22—21no—
220C22—21no—
221C22—21no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—41no—
227C22—42no—
228C22—81no—
229C22—81no—
230C22—81no—
231C22—3321no—
232C22—3321no—
233C22—3321no—
234C22—6642no—
235C22#1458170611yescharacteristic
236C22#1458170611yescharacteristic
237C22C2×(C2×D83)⋊C411yescharacteristic
238C4—21no—
239C4—21no—
240C4—82no—
241C4—3321no—
242C4—3321no—
243C4—3321no—
244C2—21no—
245C2—21no—
246C2—21no—
247C2—41no—
248C2—41no—
249C2—41no—
250C2—41no—
251C2—6641no—
252C2C2×#1458170611yescharacteristic
subgroupG/H
centerC2C2×#14581706
commutatorC23×C83C22×C4
FrattiniC2×D4C22×D83
FittingC2×#99×C83C2
socleC2×C83C2×#99
prime radical#19128260C2
cosocleC2×D4×C83C23
hypercenterC2×#99D83
hypocenterC83#2121
LCSG ▷ C23×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C23 ◁ C22×D4 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
2. C2×#14581709C2semidirect
3. #19127490C2semidirect
4. #19127487C2semidirect
5. C2×#99×C83C2semidirect
6. C22×C22⋊D83C4semidirect
9. #14581773C4semidirect
14. #14581760C4semidirect
16. C2×C22⋊Dic83C4semidirect
23. C23×D83C2×C4semidirect
31. C22×Dic83C2×C4semidirect
40. C22⋊Dic83C2×C4semidirect
41. C22⋊Dic83C2×C4semidirect
127. C83#2121semidirect
130. C2×#99D83semidirect

Maximal subgroups

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

subgroupindex
#191282602
C2×#145817092
#191274902
#191274872
C2×#99×C832
#212183