Finite Group Databasebeta

← #24914045

Subgroups — #24914045

Order 10624.

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

normal subgroups only

This group has 612 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
1C2×#14581765C212yesmaximal, complemented
2#19127631C211yesmaximal, characteristic, complemented
3#19127617C212yesmaximal, complemented
4#19127609C212yesmaximal, complemented
5#19127602C211yesmaximal, characteristic, complemented
6#19127595C212yesmaximal, complemented
7#19127577C212yesmaximal, complemented
8#19127563C212yesmaximal, complemented
9#394×C83C211yesmaximal, characteristic, complemented
10C22×D332C2211yescharacteristic
11C22×C22⋊D83C2212yes—
12C22×C22⋊D83C2212yes—
13C22×C4×D83C2212yes—
14#14581765—28no—
15C2×C4⋊Dic83C2211yescharacteristic
16C2×Dic83⋊C4C2212yes—
17C2×C22⋊Dic83C2212yes—
18C2×C22⋊Dic83C2212yes—
19C2×(C2×D83)⋊C4C2212yes—
20C2×(C2×D83)⋊C4C2212yes—
21C2×(C2×D83)⋊C4C2212yes—
22C2×C4×Dic83C2211yescharacteristic
23C83⋊#95C2212yes—
24C83⋊#95C2212yes—
25C83⋊#95C2212yes—
26C83⋊#95C2211yescharacteristic
27C22×D4×C83C2211yescharacteristic
28C2×C4⋊C4×C83C2211yescharacteristic
29C2×C22⋊C4×C83C2212yes—
30C2×C22⋊C4×C83C2212yes—
31#95×C83C2211yescharacteristic
32C23×D83C2312yes—
33C2×D332—41no—
34C2×D332—41no—
35C2×D332D414yescomplemented
36C2×C22⋊D83—24no—
37C2×C22⋊D83—24no—
38C2×C22⋊D83—24no—
39C2×C22⋊D83—42no—
40C2×C22⋊D83—42no—
41C2×C22⋊D83—42no—
42C22×Dic83C2312yes—
43C22×Dic83C2312yes—
44C22×Dic83C2312yes—
45C2×C4×D83—24no—
46C2×C4×D83—24no—
47C2×C4×D83—42no—
48C4⋊Dic83—41no—
49Dic83⋊C4—24no—
50C22⋊Dic83—24no—
51C22⋊Dic83—42no—
52(C2×D83)⋊C4—24no—
53(C2×D83)⋊C4—42no—
54(C2×D83)⋊C4—42no—
55C4×Dic83—41no—
56C24×C83C2312yes—
57C2×D4×C83—41no—
58C2×D4×C83—42no—
59C22×C4×C83C2312yes—
60C22×C4×C83C2311yescharacteristic
61C22×C4×C83C2311yescharacteristic
62C22×C4×C83C2311yescharacteristic
63C4⋊C4×C83—41no—
64C22⋊C4×C83—24no—
65C22⋊C4×C83—42no—
66C23×C83—22no—
67C23×C83—22no—
68C23×C83—22no—
69C23×C83—22no—
70C23×C83—22no—
71C23×C83—22no—
72C23×C83—22no—
73C23×C83C2411yescharacteristic
74C22×D83—22no—
75C22×D83—22no—
76C22×D83—22no—
77C22×D83—22no—
78C22×D83—22no—
79C22×D83—22no—
80C22×D83C2×D414yes—
81D4×C83—44no—
82C2×C4×C83—21no—
83C2×C4×C83—21no—
84C2×C4×C83—22no—
85C2×C4×C83—22no—
86C2×C4×C83—22no—
87C2×C4×C83—22no—
88C2×C4×C83—21no—
89C2×C4×C83—21no—
90C2×C4×C83—22no—
91C2×C4×C83C2×D412yes—
92C22⋊D83—48no—
93C22⋊D83—48no—
94C2×Dic83—22no—
95C2×Dic83—22no—
96C2×Dic83—22no—
97C2×Dic83—22no—
98C2×Dic83—22no—
99C2×Dic83—22no—
100C2×Dic83—22no—
101C2×Dic83—22no—
102C2×Dic83#4114yes—
103D332—44no—
104C4×D83—48no—
105C22×C83—42no—
106C22×C83—42no—
107C22×C83—42no—
108C22×C83—42no—
109C22×C83—42no—
110C22×C83—42no—
111C22×C83—42no—
112C22×C83C22×D411yescharacteristic
113C22×C83C2×#4112yes—
114C22×C83H(5,𝔽2)11yescharacteristic
115C22×C83H(5,𝔽2)12yes—
116C22×C83H(5,𝔽2)11yescharacteristic
117C2×D83—24no—
118C2×D83—24no—
119C2×D83—24no—
120C2×D83—42no—
121C2×D83—42no—
122C2×D83—42no—
123C2×D83—42no—
124C4×C83—22no—
125C4×C83—42no—
126C4×C83—41no—
127C4×C83—41no—
128Dic83—24no—
129Dic83—42no—
130Dic83—42no—
131C2×C83—82no—
132C2×C83#53411yescharacteristic
133C2×C83#54012yes—
134C2×C83#54612yes—
135C2×C83#55511yescharacteristic
136C2×C83#56011yescharacteristic
137D83—44no—
138#2699—831nomaximal, Sylow
139C83#269911yescharacteristic, complemented, Sylow
140C2×#121—832no—
141#398—831no—
142#394—832no—
143#394D8311yescharacteristic, complemented
144#392—832no—
145#390—831no—
146#388—832no—
147#386—832no—
148#381—832no—
149C22×D4—832no—
150C22×D4—832no—
151C22×D4—831no—
152C22×D4C2×D8311yescharacteristic
153C23×C4—832no—
154#121—1668no—
155C2×C4⋊C4—831no—
156C2×C4⋊C4—832no—
157C2×C4⋊C4C2×D8311yescharacteristic
158C2×C22⋊C4—832no—
159C2×C22⋊C4—832no—
160C2×C22⋊C4—832no—
161C2×C22⋊C4—832no—
162C2×C22⋊C4—832no—
163C2×C22⋊C4C2×D8312yes—
164C2×C22⋊C4C2×D8312yes—
165C2×C42—831no—
166#95—832no—
167#95—832no—
168#95—832no—
169#95—831no—
170#95C2×D8311yescharacteristic
171C24—832no—
172C24C22×D8312yes—
173C2×D4—42no—
174C2×D4—41no—
175C2×D4—3321no—
176C2×D4—3321no—
177C2×D4—3322no—
178C2×D4—3322no—
179C2×D4—3322no—
180C2×D4—834no—
181C2×D4—1664no—
182C2×D4—1664no—
183C2×D4—1664no—
184C22×C4—3322no—
185C22×C4—832no—
186C22×C4—832no—
187C22×C4—832no—
188C22×C4—1664no—
189C22×C4—1664no—
190C22×C4C22×D8311yescharacteristic
191C22×C4C22×D8312yes—
192C22×C4C22×D8311yescharacteristic
193C22×C4C22×D8311yescharacteristic
194C4⋊C4—41no—
195C4⋊C4—3321no—
196C4⋊C4—1664no—
197C22⋊C4—24no—
198C22⋊C4—42no—
199C22⋊C4—3322no—
200C22⋊C4—3322no—
201C22⋊C4—3322no—
202C22⋊C4—1664no—
203C22⋊C4—1664no—
204C42—3321no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—834no—
213C23—1662no—
214C23—1662no—
215C23—1662no—
216C23—1662no—
217C23—1662no—
218C23—1662no—
219C23C23×D8311yescharacteristic
220D4—44no—
221D4—3324no—
222D4—3328no—
223D4—3328no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—3328no—
234C2×C4—834no—
235C2×C4—1662no—
236C2×C4—1662no—
237C2×C4—1662no—
238C2×C4—1662no—
239C2×C4—1662no—
240C2×C4—1662no—
241C2×C4—1662no—
242C2×C4—1662no—
243C2×C4D4×D8312yes—
244C22—42no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—42no—
251C22—3322no—
252C22—3322no—
253C22—3322no—
254C22—3322no—
255C22—1664no—
256C22—1664no—
257C22—1664no—
258C22C2×D4×D8311yescharacteristic
259C22#41×D8312yes—
260C22#1458183712yes—
261C22#1458183711yescharacteristic
262C22#1458183811yescharacteristic
263C4—22no—
264C4—42no—
265C4—41no—
266C4—41no—
267C4—3322no—
268C4—3322no—
269C4—1664no—
270C2—82no—
271C2—3324no—
272C2#1912825911yescharacteristic
273C2#1912830312yes—
274C2#1912834812yes—
275C2#1912841311yescharacteristic
276C2#1912844011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#394×C83C2
hypercenter#394D83
hypocenterC83#2699
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#14581765C2semidirect
2. #19127631C2semidirect
3. #19127617C2semidirect
4. #19127609C2semidirect
5. #19127602C2semidirect
6. #19127595C2semidirect
7. #19127577C2semidirect
8. #19127563C2semidirect
9. #394×C83C2semidirect
35. C2×D332D4semidirect
139. C83#2699semidirect
143. #394D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817652
#191276312
#191276172
#191276092
#191276022
#191275952
#191275772
#191275632
#394×C832
#269983