Finite Group Databasebeta

← #24911926

Subgroups — #24911926

Order 10624.

568 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 153 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×#14581751C211yesmaximal, characteristic, complemented
2#19127615C214yesmaximal, complemented
3#19127604C211yesmaximal, characteristic, complemented
4#19127604C211yesmaximal, characteristic, complemented
5#19127589C214yesmaximal, complemented
6#19127570C211yesmaximal, characteristic, complemented
7#19127566C211yesmaximal, characteristic, complemented
8C4×C22⋊Dic83C211yesmaximal, characteristic, complemented
9C2×#121×C83C211yesmaximal, characteristic
10#14581751—24no—
11C2×C4⋊Dic83C2211yescharacteristic
12C2×C4⋊Dic83C2211yescharacteristic
13C2×Dic83⋊C4C2214yes—
14C2×C22⋊Dic83C2214yes—
15C2×C22⋊Dic83C2212yes—
16C2×C22⋊Dic83C2212yes—
17C2×C22⋊Dic83C2214yes—
18C2×C4×Dic83C2211yescharacteristic
19C2×C4×Dic83C2211yescharacteristic
20C2×C4×Dic83C2212yes—
21C83⋊#95C2214yes—
22C83⋊#95C2212yes—
23C22×D4×C83C2211yescharacteristic
24C22×D4×C83C2212yes—
25C23×C4×C83C2211yescharacteristic
26#121×C83—24no—
27C2×C4⋊C4×C83C2211yescharacteristic
28C2×C22⋊C4×C83C2212yes—
29C22×Dic83C2312yes—
30C22×Dic83C2312yes—
31C22×Dic83C2312yes—
32C22×Dic83C2312yes—
33C4⋊Dic83—22no—
34C4⋊Dic83—22no—
35Dic83⋊C4—44no—
36C22⋊Dic83—24no—
37C22⋊Dic83—24no—
38C22⋊Dic83—28no—
39C22⋊Dic83—44no—
40C4×Dic83—24no—
41C4×Dic83—22no—
42C4×Dic83—22no—
43C24×C83C2312yes—
44C24×C83C2311yescharacteristic
45C2×D4×C83—24no—
46C2×D4×C83—22no—
47C2×D4×C83—22no—
48C2×D4×C83—22no—
49C2×D4×C83—44no—
50C22×C4×C83—22no—
51C22×C4×C83—22no—
52C22×C4×C83D414yes—
53C22×C4×C83C2312yes—
54C22×C4×C83C2311yescharacteristic
55C22×C4×C83C2311yescharacteristic
56C4⋊C4×C83—22no—
57C22⋊C4×C83—24no—
58C23×C83—21no—
59C23×C83—21no—
60C23×C83—22no—
61C23×C83—22no—
62C23×C83—22no—
63C23×C83—22no—
64C23×C83—22no—
65C23×C83—24no—
66C23×C83—24no—
67C23×C83C2×D412yes—
68C23×C83C2411yescharacteristic
69D4×C83—48no—
70D4×C83—44no—
71C2×C4×C83—21no—
72C2×C4×C83—21no—
73C2×C4×C83—28no—
74C2×C4×C83—22no—
75C2×C4×C83—24no—
76C2×C4×C83C2×D412yes—
77C2×C4×C83C2×D412yes—
78C2×C4×C83#4112yes—
79C2×C4×C83#4112yes—
80C2×Dic83—22no—
81C2×Dic83—22no—
82C2×Dic83—22no—
83C2×Dic83—22no—
84C2×Dic83—22no—
85C2×Dic83—24no—
86C2×Dic83—22no—
87C2×Dic83—22no—
88C2×Dic83—24no—
89C2×Dic83#4114yes—
90C22×C83—22no—
91C22×C83—22no—
92C22×C83—22no—
93C22×C83—42no—
94C22×C83—42no—
95C22×C83—42no—
96C22×C83—42no—
97C22×C83—42no—
98C22×C83—44no—
99C22×C83—44no—
100C22×C83C22×D411yescharacteristic
101C22×C83C2×#4112yes—
102C22×C83C2×#4111yescharacteristic
103C22×C83C2×#4111yescharacteristic
104C22×C83H(5,𝔽2)12yes—
105C4×C83—22no—
106C4×C83—42no—
107C4×C83#12314yes—
108Dic83—24no—
109Dic83—42no—
110Dic83—42no—
111Dic83—42no—
112C2×C83—42no—
113C2×C83—82no—
114C2×C83C2×#12311yescharacteristic
115C2×C83#53211yescharacteristic
116C2×C83#53511yescharacteristic
117C2×C83#55312yes—
118C2×C83#55512yes—
119#2500—831nomaximal, Sylow
120C83#250011yescharacteristic, complemented, Sylow
121C2×#121D8311yescharacteristic
122C2×#117—831no—
123#394—834no—
124#390—831no—
125#390—831no—
126#388—834no—
127#384—831no—
128#382—831no—
129C4×C22⋊C4—831no—
130C22×D4C2×D8311yescharacteristic
131C22×D4C2×D8312yes—
132C23×C4C2×D8311yescharacteristic
133#121—24no—
134#117—1664no—
135C2×C4⋊C4—834no—
136C2×C4⋊C4—831no—
137C2×C4⋊C4—831no—
138C2×C4⋊C4C2×D8311yescharacteristic
139C2×C22⋊C4—834no—
140C2×C22⋊C4—832no—
141C2×C22⋊C4—832no—
142C2×C22⋊C4—834no—
143C2×C22⋊C4C2×D8312yes—
144C2×C42—832no—
145C2×C42—831no—
146C2×C42—831no—
147#95—832no—
148#95—834no—
149C24C22×D8312yes—
150C24C22×D8311yescharacteristic
151C2×D4—24no—
152C2×D4—22no—
153C2×D4—22no—
154C2×D4—22no—
155C2×D4—44no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—832no—
159C22×C4—832no—
160C22×C4—832no—
161C22×C4—832no—
162C22×C4C22⋊D8314yes—
163C22×C4C22×D8312yes—
164C22×C4C22×D8311yescharacteristic
165C22×C4C22×D8311yescharacteristic
166C4⋊C4—22no—
167C4⋊C4—3324no—
168C4⋊C4—1662no—
169C4⋊C4—1662no—
170C22⋊C4—24no—
171C22⋊C4—3324no—
172C22⋊C4—1664no—
173C22⋊C4—1668no—
174C22⋊C4—1664no—
175C42—1664no—
176C42—1662no—
177C42—1662no—
178C23—22no—
179C23—21no—
180C23—21no—
181C23—22no—
182C23—22no—
183C23—24no—
184C23—22no—
185C23—22no—
186C23—24no—
187C23C2×C22⋊D8312yes—
188C23C23×D8311yescharacteristic
189D4—48no—
190D4—44no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—28no—
194C2×C4—22no—
195C2×C4—24no—
196C2×C4—834no—
197C2×C4—1662no—
198C2×C4—1662no—
199C2×C4—1662no—
200C2×C4—1664no—
201C2×C4—1662no—
202C2×C4—1662no—
203C2×C4—1664no—
204C2×C4—1662no—
205C2×C4—1662no—
206C2×C4C2×C22⋊D8312yes—
207C2×C4C2×C22⋊D8312yes—
208C2×C4#1305058912yes—
209C2×C4#1305058912yes—
210C22—22no—
211C22—22no—
212C22—22no—
213C22—42no—
214C22—42no—
215C22—44no—
216C22—44no—
217C22—42no—
218C22—42no—
219C22—42no—
220C22C22×C22⋊D8311yescharacteristic
221C22#41×D8312yes—
222C22C2×#1305058911yescharacteristic
223C22C2×#1305058911yescharacteristic
224C22#1458183712yes—
225C4—22no—
226C4—42no—
227C4—3322no—
228C4—3322no—
229C4—3322no—
230C4—1664no—
231C4#1458177314yes—
232C2—42no—
233C2—82no—
234C2C2×#1458177311yescharacteristic
235C2#1912824711yescharacteristic
236C2#1912826811yescharacteristic
237C2#1912839812yes—
238C2#1912841512yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting, prime radicalC2×#121×C83C2
hypercenterC2×#121D83
hypocenterC83#2500
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#14581751C2semidirect
2. #19127615C2semidirect
3. #19127604C2semidirect
4. #19127604C2semidirect
5. #19127589C2semidirect
6. #19127570C2semidirect
7. #19127566C2semidirect
8. C4×C22⋊Dic83C2semidirect
120. C83#2500semidirect

Maximal subgroups

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

subgroupindex
C2×#145817512
#191276152
#191276042
#191275892
#191275702
#191275662
C4×C22⋊Dic832
C2×#121×C832
#250083