Finite Group Databasebeta

← #24910732

Subgroups — #24910732

Order 10624.

528 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 155 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×#14581757C211yesmaximal, characteristic, complemented
2C2×#14581748C211yesmaximal, characteristic
3#19127639C211yesmaximal, characteristic, complemented
4#19127632C212yesmaximal, complemented
5#19127624C212yesmaximal
6#19127622C211yesmaximal, characteristic, complemented
7#19127613C211yesmaximal, characteristic
8C4⋊C4×Dic83C212yesmaximal, complemented
9#19127552C211yesmaximal, characteristic, complemented
10C22⋊C4×Dic83C212yesmaximal
11#399×C83C211yesmaximal, characteristic
12C22×Dic166C2211yescharacteristic
13C23×Dic83C2211yescharacteristic
14#14581757—24no—
15#14581748—24no—
16C2×C4⋊Dic83C2212yes—
17C2×Dic83⋊C4C2212yes—
18C2×Dic83⋊C4C2211yescharacteristic
19C2×Dic83⋊C4C2212yes—
20C2×Dic83⋊C4C2211yescharacteristic
21C2×C22⋊Dic83C2212yes—
22C2×C22⋊Dic83C2211yescharacteristic
23C2×C4×Dic83C2212yes—
24C2×C4×Dic83C2211yescharacteristic
25C2×C4×Dic83C2212yes—
26C2×C4×Dic83C2211yescharacteristic
27C83⋊#95C2211yescharacteristic
28C83⋊#95C2212yes—
29C83⋊#95C2212yes—
30C83⋊#95C2212yes—
31C83⋊#95C2212yes—
32C2×C4⋊C4×C83C2212yes—
33C2×C4⋊C4×C83C2211yescharacteristic
34C2×C22⋊C4×C83C2211yescharacteristic
35C2×C22⋊C4×C83C2212yes—
36#95×C83C2211yescharacteristic
37C2×Dic166—22no—
38C2×Dic166—24no—
39C22×Dic83—22no—
40C22×Dic83—22no—
41C22×Dic83Q814yescomplemented
42C22×Dic83C2312yes—
43C22×Dic83C2311yescharacteristic
44C22×Dic83C2311yescharacteristic
45C22×Dic83C2311yescharacteristic
46C22×Dic83C2312yes—
47C22×Dic83C2311yescharacteristic
48C4⋊Dic83—24no—
49Dic83⋊C4—22no—
50Dic83⋊C4—24no—
51Dic83⋊C4—22no—
52Dic83⋊C4—24no—
53C22⋊Dic83—22no—
54C22⋊Dic83—24no—
55C4×Dic83—22no—
56C4×Dic83—24no—
57C4×Dic83—24no—
58C4×Dic83D414yes—
59C24×C83C2311yescharacteristic
60C22×C4×C83C2311yescharacteristic
61C22×C4×C83C2312yes—
62C22×C4×C83C2311yescharacteristic
63C22×C4×C83C2312yes—
64C4⋊C4×C83—22no—
65C4⋊C4×C83—24no—
66C22⋊C4×C83—22no—
67C22⋊C4×C83—24no—
68C23×C83—21no—
69C23×C83—22no—
70C23×C83—22no—
71C23×C83—21no—
72C23×C83C2×Q812yes—
73C23×C83C2411yescharacteristic
74C2×C4×C83—22no—
75C2×C4×C83—22no—
76C2×C4×C83—22no—
77C2×C4×C83—21no—
78C2×C4×C83—22no—
79C2×C4×C83—22no—
80C2×C4×C83—22no—
81C2×C4×C83—22no—
82C2×C4×C83C2×D412yes—
83C2×C4×C83#4114yes—
84C2×Dic83—22no—
85C2×Dic83—21no—
86C2×Dic83—22no—
87C2×Dic83—21no—
88C2×Dic83—22no—
89C2×Dic83—28no—
90C2×Dic83—21no—
91C2×Dic83—22no—
92C2×Dic83—22no—
93C2×Dic83—22no—
94C2×Dic83—22no—
95C2×Dic83C2×D412yes—
96C2×Dic83C2×D412yes—
97C2×Dic83C2×Q812yes—
98C2×Dic83C2×Q812yes—
99C2×Dic83#4114yes—
100C2×Dic83#4112yes—
101Dic166—28no—
102C22×C83—22no—
103C22×C83—24no—
104C22×C83—41no—
105C22×C83—42no—
106C22×C83—41no—
107C22×C83C22×D411yescharacteristic
108C22×C83C22×Q811yescharacteristic
109C22×C83C2×#4111yescharacteristic
110C22×C83C2×#4112yes—
111C22×C83C2×#4112yes—
112C4×C83—22no—
113C4×C83—24no—
114C4×C83—41no—
115C4×C83—42no—
116Dic83—22no—
117Dic83—24no—
118Dic83—22no—
119Dic83—41no—
120Dic83—42no—
121Dic83C22⋊Q814yes—
122C2×C83—42no—
123C2×C83C2×C22⋊Q811yescharacteristic
124C2×C83#52511yescharacteristic
125C2×C83#52912yes—
126C2×C83#53211yescharacteristic
127C2×C83#53312yes—
128#2396—831nomaximal, Sylow
129C83#239611yescharacteristic, complemented, Sylow
130C2×C4×Q8—831no—
131C2×#117—831no—
132#400—831no—
133#399D8311yescharacteristic
134#398—832no—
135#396—832no—
136#395—831no—
137#393—831no—
138C4×C4⋊C4—832no—
139C4×C4⋊C4—831no—
140C4×C22⋊C4—832no—
141C22×Q8—831no—
142C23×C4—831no—
143C4×Q8—1664no—
144#117—1664no—
145C2×C4⋊C4—831no—
146C2×C4⋊C4—831no—
147C2×C4⋊C4—832no—
148C2×C4⋊C4—832no—
149C2×C4⋊C4—832no—
150C2×C4⋊C4C2×D8311yescharacteristic
151C2×C4⋊C4C2×D8312yes—
152C2×C22⋊C4—831no—
153C2×C22⋊C4—832no—
154C2×C22⋊C4C2×D8311yescharacteristic
155C2×C22⋊C4C2×D8312yes—
156C2×C42—831no—
157C2×C42—831no—
158C2×C42—832no—
159C2×C42—832no—
160#95—832no—
161#95—832no—
162#95—832no—
163#95—831no—
164#95—832no—
165#95C2×D8311yescharacteristic
166C24C22×D8311yescharacteristic
167C2×Q8—1664no—
168C2×Q8—1662no—
169C22×C4—831no—
170C22×C4—834no—
171C22×C4—831no—
172C22×C4—832no—
173C22×C4—831no—
174C22×C4—832no—
175C22×C4—831no—
176C22×C4—1662no—
177C22×C4—1662no—
178C22×C4C22×D8311yescharacteristic
179C22×C4C22×D8312yes—
180C22×C4C22×D8311yescharacteristic
181C22×C4C22×D8312yes—
182C4⋊C4—22no—
183C4⋊C4—24no—
184C4⋊C4—1662no—
185C4⋊C4—1664no—
186C4⋊C4—1664no—
187C4⋊C4—1662no—
188C4⋊C4—1664no—
189C22⋊C4—22no—
190C22⋊C4—24no—
191C22⋊C4—1662no—
192C22⋊C4—1664no—
193C42—834no—
194C42—1662no—
195C42—1664no—
196C42—1664no—
197C23—21no—
198C23—22no—
199C23—21no—
200C23—22no—
201C23Q8×D8312yes—
202C23C23×D8311yescharacteristic
203Q8—1668no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—832no—
213C2×C4—832no—
214C2×C4—832no—
215C2×C4—832no—
216C2×C4—832no—
217C2×C4—834no—
218C2×C4—1661no—
219C2×C4—1662no—
220C2×C4—1662no—
221C2×C4—1662no—
222C2×C4—1662no—
223C2×C4—1668no—
224C2×C4—1662no—
225C2×C4—1661no—
226C2×C4—1661no—
227C2×C4—1662no—
228C2×C4—1662no—
229C2×C4D4×D8312yes—
230C2×C4#1305058914yes—
231C22—22no—
232C22—24no—
233C22—41no—
234C22—42no—
235C22—41no—
236C22C2×D4×D8311yescharacteristic
237C22C2×Q8×D8311yescharacteristic
238C22#41×D8311yescharacteristic
239C22#41×D8312yes—
240C22C2×#1305058912yes—
241C4—22no—
242C4—24no—
243C4—41no—
244C4—42no—
245C4—3321no—
246C4—3322no—
247C4—834no—
248C4—1662no—
249C4—1664no—
250C4—1662no—
251C2—42no—
252C2C22⋊Q8×D8311yescharacteristic
253C2#1912820911yescharacteristic
254C2#1912822812yes—
255C2#1912824911yescharacteristic
256C2#1912825312yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#399×C83C2
prime radicalC24×C83C23
hypercenter#399D83
hypocenterC83#2396
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#14581757C2semidirect
3. #19127639C2semidirect
4. #19127632C2semidirect
6. #19127622C2semidirect
8. C4⋊C4×Dic83C2semidirect
9. #19127552C2semidirect
41. C22×Dic83Q8semidirect
129. C83#2396semidirect

Maximal subgroups

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

subgroupindex
C2×#145817572
C2×#145817482
#191276392
#191276322
#191276242
#191276222
#191276132
C4⋊C4×Dic832
#191275522
C22⋊C4×Dic832
#399×C832
#239683