Finite Group Databasebeta

← #24914732

Subgroups — #24914732

Order 10624.

484 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 121 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#19127630C212yesmaximal, complemented
2#19127618C212yesmaximal, complemented
3#19127610C211yesmaximal, characteristic
4#19127601C212yesmaximal, complemented
5#19127591C212yesmaximal, complemented
6#19127582C212yesmaximal, complemented
7#19127578C212yesmaximal, complemented
8#19127562C211yesmaximal, characteristic, complemented
9#399×C83C211yesmaximal, characteristic, complemented
10C22×D332C2211yescharacteristic
11C22×C22⋊D83C2212yes—
12C23×Dic83C2211yescharacteristic
13C2×Dic83⋊C4C2212yes—
14C2×C22⋊Dic83C2211yescharacteristic
15C2×(C2×D83)⋊C4C2212yes—
16C2×(C2×D83)⋊C4C2212yes—
17C2×(C2×D83)⋊C4C2212yes—
18C2×(C2×D83)⋊C4C2212yes—
19C2×(C2×D83)⋊C4C2212yes—
20C2×C4×Dic83C2212yes—
21C83⋊#95C2212yes—
22C83⋊#95C2212yes—
23C83⋊#95C2212yes—
24C83⋊#95C2212yes—
25C83⋊#95C2211yescharacteristic
26C2×C4⋊C4×C83C2212yes—
27C2×C4⋊C4×C83C2211yescharacteristic
28C2×C22⋊C4×C83C2211yescharacteristic
29C2×C22⋊C4×C83C2212yes—
30#95×C83C2211yescharacteristic
31C23×D83C2312yes—
32C2×D332—41no—
33C2×D332—42no—
34C2×C22⋊D83—24no—
35C2×C22⋊D83—24no—
36C2×C22⋊D83—42no—
37C22×Dic83—22no—
38C22×Dic83—41no—
39C22×Dic83—41no—
40C22×Dic83C2312yes—
41C22×Dic83C2312yes—
42C22×Dic83C2312yes—
43Dic83⋊C4—24no—
44C22⋊Dic83—41no—
45(C2×D83)⋊C4—24no—
46(C2×D83)⋊C4—42no—
47(C2×D83)⋊C4—42no—
48(C2×D83)⋊C4—42no—
49(C2×D83)⋊C4—42no—
50C4×Dic83—42no—
51C24×C83C2311yescharacteristic
52C22×C4×C83C2311yescharacteristic
53C22×C4×C83C2311yescharacteristic
54C22×C4×C83C2312yes—
55C22×C4×C83C2312yes—
56C4⋊C4×C83—42no—
57C4⋊C4×C83—41no—
58C22⋊C4×C83—42no—
59C22⋊C4×C83—41no—
60C23×C83—22no—
61C23×C83—21no—
62C23×C83—22no—
63C23×C83—21no—
64C23×C83#4112yes—
65C23×C83C2411yescharacteristic
66C22×D83—22no—
67C22×D83—22no—
68C22×D83—22no—
69C22×D83—22no—
70C22×D83—22no—
71C22×D83—22no—
72C22×D83—22no—
73C2×C4×C83—22no—
74C2×C4×C83—22no—
75C2×C4×C83—22no—
76C2×C4×C83—21no—
77C2×C4×C83—22no—
78C2×C4×C83—21no—
79C2×C4×C83—22no—
80C2×C4×C83—22no—
81C2×C4×C83—22no—
82C2×C4×C83—22no—
83C22⋊D83—48no—
84C2×Dic83—22no—
85C2×Dic83—22no—
86C2×Dic83—22no—
87C2×Dic83—22no—
88C2×Dic83—22no—
89C2×Dic83—22no—
90C2×Dic83—22no—
91C2×Dic83—22no—
92C2×Dic83—44no—
93C2×Dic83#4114yes—
94D332—44no—
95C22×C83—22no—
96C22×C83—24no—
97C22×C83—42no—
98C22×C83—41no—
99C22×C83—41no—
100C22×C83C2×#4112yes—
101C22×C83C2×#4111yescharacteristic
102C22×C83H(5,𝔽2)11yescharacteristic
103C22×C83H(5,𝔽2)12yes—
104C22×C83H(5,𝔽2)11yescharacteristic
105C2×D83—42no—
106C2×D83—42no—
107C2×D83—42no—
108C2×D83—42no—
109C2×D83—42no—
110C2×D83—42no—
111C2×D83—42no—
112C4×C83—42no—
113C4×C83—41no—
114C4×C83—41no—
115C4×C83—42no—
116Dic83—24no—
117Dic83—42no—
118Dic83—42no—
119C2×C83—42no—
120C2×C83#53811yescharacteristic
121C2×C83#54012yes—
122C2×C83#55112yes—
123C2×C83#55911yescharacteristic
124C2×C83#56011yescharacteristic
125D83—82no—
126#2752—831nomaximal, Sylow
127C83#275211yescharacteristic, complemented, Sylow
128#399D8311yescharacteristic, complemented
129#397—832no—
130#394—832no—
131#392—831no—
132#390—832no—
133#388—832no—
134#387—832no—
135#386—832no—
136#381—831no—
137C22×D4—832no—
138C22×D4—831no—
139C23×C4—831no—
140C2×C4⋊C4—832no—
141C2×C4⋊C4C2×D8312yes—
142C2×C4⋊C4C2×D8311yescharacteristic
143C2×C22⋊C4—831no—
144C2×C22⋊C4—832no—
145C2×C22⋊C4—832no—
146C2×C22⋊C4—832no—
147C2×C22⋊C4—832no—
148C2×C22⋊C4—832no—
149C2×C22⋊C4C2×D8311yescharacteristic
150C2×C22⋊C4C2×D8312yes—
151C2×C42—832no—
152#95—832no—
153#95—832no—
154#95—832no—
155#95—832no—
156#95—831no—
157#95C2×D8311yescharacteristic
158C24—832no—
159C24C22×D8311yescharacteristic
160C2×D4—3321no—
161C2×D4—3322no—
162C2×D4—3322no—
163C2×D4—1664no—
164C2×D4—1664no—
165C22×C4—3321no—
166C22×C4—3321no—
167C22×C4—832no—
168C22×C4—832no—
169C22×C4—832no—
170C22×C4—1662no—
171C22×C4C22×D8311yescharacteristic
172C22×C4C22×D8311yescharacteristic
173C22×C4C22×D8312yes—
174C22×C4C22×D8312yes—
175C4⋊C4—42no—
176C4⋊C4—41no—
177C4⋊C4—1664no—
178C22⋊C4—42no—
179C22⋊C4—41no—
180C22⋊C4—3322no—
181C22⋊C4—3322no—
182C22⋊C4—3322no—
183C22⋊C4—3322no—
184C22⋊C4—3321no—
185C22⋊C4—1664no—
186C42—3322no—
187C23—22no—
188C23—21no—
189C23—22no—
190C23—21no—
191C23—1662no—
192C23—1662no—
193C23—1662no—
194C23—1662no—
195C23—1662no—
196C23—1662no—
197C23—1662no—
198C23#1305059112yes—
199C23C23×D8311yescharacteristic
200D4—3324no—
201D4—3328no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—3324no—
213C2×C4—834no—
214C2×C4—1662no—
215C2×C4—1662no—
216C2×C4—1662no—
217C2×C4—1662no—
218C2×C4—1662no—
219C2×C4—1662no—
220C2×C4—1662no—
221C2×C4—1662no—
222C22—24no—
223C22—22no—
224C22—41no—
225C22—41no—
226C22—42no—
227C22—3322no—
228C22—3322no—
229C22—3322no—
230C22—3322no—
231C22—3322no—
232C22—3322no—
233C22—3322no—
234C22#41×D8312yes—
235C22C2×#1305059111yescharacteristic
236C22#1458183712yes—
237C22#1458183711yescharacteristic
238C22#1458183811yescharacteristic
239C4—41no—
240C4—41no—
241C4—42no—
242C4—42no—
243C4—3322no—
244C4—3322no—
245C4—1664no—
246C2—42no—
247C2—6642no—
248C2#1912829111yescharacteristic
249C2#1912830312yes—
250C2#1912837812yes—
251C2#1912843511yescharacteristic
252C2#1912843811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#399×C83C2
prime radical#19127610C2
hypercenter#399D83
hypocenterC83#2752
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. #19127630C2semidirect
2. #19127618C2semidirect
4. #19127601C2semidirect
5. #19127591C2semidirect
6. #19127582C2semidirect
7. #19127578C2semidirect
8. #19127562C2semidirect
9. #399×C83C2semidirect
127. C83#2752semidirect
128. #399D83semidirect

Maximal subgroups

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

subgroupindex
#191276302
#191276182
#191276102
#191276012
#191275912
#191275822
#191275782
#191275622
#399×C832
#275283