Finite Group Databasebeta

← #24914505

Subgroups — #24914505

Order 10624.

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

normal subgroups only

This group has 552 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#19127626C211yesmaximal, characteristic, complemented
3#19127623C212yesmaximal, complemented
4#19127591C212yesmaximal, complemented
5#19127591C212yesmaximal, complemented
6#19127589C212yesmaximal, complemented
7#19127556C212yesmaximal
8#19127556C211yesmaximal, characteristic
9#399×C83C211yesmaximal, characteristic, complemented
10C24×D83C2211yescharacteristic
11C2×C4⋊Dic83C2211yescharacteristic, complemented
12C2×Dic83⋊C4C2212yescomplemented
13C2×C22⋊Dic83C2212yes—
14C2×C22⋊Dic83C2211yescharacteristic
15C2×(C2×D83)⋊C4C2212yes—
16C2×(C2×D83)⋊C4C2212yes—
17C2×(C2×D83)⋊C4C2212yes—
18C2×(C2×D83)⋊C4C2212yes—
19C2×(C2×D83)⋊C4C2212yes—
20C2×(C2×D83)⋊C4C2212yes—
21C2×C4×Dic83C2212yescomplemented
22C83⋊#95C2212yescomplemented
23C83⋊#95C2212yescomplemented
24C83⋊#95C2211yescharacteristic, complemented
25C83⋊#95C2212yescomplemented
26C2×C4⋊C4×C83C2211yescharacteristic, complemented
27C2×C4⋊C4×C83C2212yescomplemented
28C2×C22⋊C4×C83C2212yes—
29C2×C22⋊C4×C83C2211yescharacteristic
30#95×C83C2211yescharacteristic, complemented
31C23×D83—22no—
32C23×D83—24no—
33C23×D83—41no—
34C23×D83—42no—
35C23×D83—41no—
36C23×D83C2312yes—
37C22×Dic83C2312yes—
38C22×Dic83C2312yes—
39C22×Dic83C2312yes—
40C4⋊Dic83—41no—
41Dic83⋊C4—42no—
42C22⋊Dic83—24no—
43C22⋊Dic83—41no—
44(C2×D83)⋊C4—24no—
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×C83C2312yes—
54C22×C4×C83C2312yes—
55C22×C4×C83C2311yescharacteristic
56C4⋊C4×C83—41no—
57C4⋊C4×C83—42no—
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—44no—
73C22×D83—48no—
74C22×D83—48no—
75C22×D83—44no—
76C22×D83—44no—
77C22×D83#4114yes—
78C2×C4×C83—22no—
79C2×C4×C83—22no—
80C2×C4×C83—22no—
81C2×C4×C83—21no—
82C2×C4×C83—22no—
83C2×C4×C83—21no—
84C2×C4×C83—22no—
85C2×C4×C83—22no—
86C2×C4×C83—22no—
87C2×C4×C83—22no—
88C2×Dic83—22no—
89C2×Dic83—22no—
90C2×Dic83—22no—
91C2×Dic83—22no—
92C2×Dic83—22no—
93C2×Dic83—22no—
94C2×Dic83—22no—
95C2×Dic83—22no—
96C2×Dic83—22no—
97C22×C83—22no—
98C22×C83—24no—
99C22×C83—42no—
100C22×C83—41no—
101C22×C83—41no—
102C22×C83C2×#4112yes—
103C22×C83C2×#4111yescharacteristic
104C22×C83H(5,𝔽2)12yes—
105C22×C83H(5,𝔽2)11yescharacteristic
106C22×C83H(5,𝔽2)11yescharacteristic
107C2×D83—24no—
108C2×D83—24no—
109C2×D83—24no—
110C2×D83—416no—
111C2×D83—42no—
112C2×D83—42no—
113C2×D83—42no—
114C2×D83—42no—
115C4×C83—42no—
116C4×C83—41no—
117C4×C83—42no—
118C4×C83—41no—
119Dic83—42no—
120Dic83—42no—
121Dic83—42no—
122C2×C83—42no—
123C2×C83#53812yes—
124C2×C83#53811yescharacteristic
125C2×C83#55112yes—
126C2×C83#55111yescharacteristic
127C2×C83#56011yescharacteristic
128D83—44no—
129#2735—831nomaximal, Sylow
130C83#273511yescharacteristic, complemented, Sylow
131#399D8311yescharacteristic, complemented
132#397—831no—
133#397—832no—
134#396—832no—
135#388—832no—
136#388—832no—
137#388—832no—
138#379—831no—
139#379—832no—
140C25—831no—
141C2×C4⋊C4—832no—
142C2×C4⋊C4—831no—
143C2×C4⋊C4C2×D8311yescharacteristic, complemented
144C2×C4⋊C4C2×D8312yescomplemented
145C2×C22⋊C4—831no—
146C2×C22⋊C4—832no—
147C2×C22⋊C4—832no—
148C2×C22⋊C4—832no—
149C2×C22⋊C4—832no—
150C2×C22⋊C4—832no—
151C2×C22⋊C4—832no—
152C2×C22⋊C4—832no—
153C2×C22⋊C4C2×D8312yes—
154C2×C22⋊C4C2×D8311yescharacteristic
155C2×C42—832no—
156#95—832no—
157#95—831no—
158#95—832no—
159#95—832no—
160#95C2×D8311yescharacteristic, complemented
161C24—3321no—
162C24—3322no—
163C24—3321no—
164C24—832no—
165C24—1664no—
166C24—1662no—
167C24C22×D8311yescharacteristic
168C22×C4—832no—
169C22×C4—832no—
170C22×C4—832no—
171C22×C4C22×D8311yescharacteristic
172C22×C4C22×D8312yes—
173C22×C4C22×D8312yes—
174C22×C4C22×D8311yescharacteristic
175C4⋊C4—41no—
176C4⋊C4—42no—
177C4⋊C4—3322no—
178C4⋊C4—3321no—
179C22⋊C4—42no—
180C22⋊C4—41no—
181C22⋊C4—3322no—
182C22⋊C4—3322no—
183C22⋊C4—3322no—
184C22⋊C4—3322no—
185C22⋊C4—3321no—
186C22⋊C4—1664no—
187C22⋊C4—1664no—
188C22⋊C4—1664no—
189C42—3322no—
190C23—22no—
191C23—22no—
192C23—21no—
193C23—21no—
194C23—3324no—
195C23—3328no—
196C23—3324no—
197C23—3328no—
198C23—3324no—
199C23—834no—
200C23—1662no—
201C23—1662no—
202C23—1662no—
203C23—1662no—
204C23—1662no—
205C23—1662no—
206C23#1305059112yes—
207C23C23×D8311yescharacteristic
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—1662no—
219C2×C4—1662no—
220C2×C4—1662no—
221C2×C4—1662no—
222C2×C4—1662no—
223C2×C4—1662no—
224C2×C4—1662no—
225C2×C4—1662no—
226C2×C4—1662no—
227C22—24no—
228C22—22no—
229C22—41no—
230C22—41no—
231C22—42no—
232C22—33216no—
233C22—3322no—
234C22—3322no—
235C22—3322no—
236C22—3322no—
237C22—1664no—
238C22—1664no—
239C22—1664no—
240C22#41×D8312yes—
241C22C2×#1305059111yescharacteristic
242C22#1458183712yes—
243C22#1458183711yescharacteristic
244C22#1458183811yescharacteristic
245C4—41no—
246C4—42no—
247C4—41no—
248C4—42no—
249C4—3322no—
250C4—3322no—
251C4—3322no—
252C2—42no—
253C2—3324no—
254C2#1912828712yes—
255C2#1912829111yescharacteristic
256C2#1912837812yes—
257C2#1912837911yescharacteristic
258C2#1912843811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#399×C83C2
prime radicalC24×D83C22
hypercenter#399D83
hypocenterC83#2735
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. #19127630C2semidirect
2. #19127626C2semidirect
3. #19127623C2semidirect
4. #19127591C2semidirect
5. #19127591C2semidirect
6. #19127589C2semidirect
9. #399×C83C2semidirect
11. C2×C4⋊Dic83C22semidirect
12. C2×Dic83⋊C4C22semidirect
21. C2×C4×Dic83C22semidirect
22. C83⋊#95C22semidirect
23. C83⋊#95C22semidirect
24. C83⋊#95C22semidirect
25. C83⋊#95C22semidirect
26. C2×C4⋊C4×C83C22semidirect
27. C2×C4⋊C4×C83C22semidirect
30. #95×C83C22semidirect
130. C83#2735semidirect
131. #399D83semidirect
143. C2×C4⋊C4C2×D83semidirect
144. C2×C4⋊C4C2×D83semidirect
160. #95C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191276302
#191276262
#191276232
#191275912
#191275892
#191275562
#399×C832
#273583