Finite Group Databasebeta

← #24911921

Subgroups — #24911921

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×#14581766C211yesmaximal, characteristic
2#19127616C214yesmaximal, complemented
3#19127602C211yesmaximal, characteristic, complemented
4#19127602C211yesmaximal, characteristic, complemented
5#19127592C214yesmaximal, complemented
6#19127570C211yesmaximal, characteristic, complemented
7#19127566C211yesmaximal, characteristic, complemented
8C4×C22⋊Dic83C211yesmaximal, characteristic, complemented
9C2×#117×C83C211yesmaximal, characteristic, complemented
10C22×D332C2211yescharacteristic
11C22×C22⋊D83C2212yes—
12#14581766—24no—
13C2×C4⋊Dic83C2211yescharacteristic
14C2×Dic83⋊C4C2214yes—
15C2×C22⋊Dic83C2212yes—
16C2×(C2×D83)⋊C4C2212yes—
17C2×(C2×D83)⋊C4C2214yes—
18C2×(C2×D83)⋊C4C2214yes—
19C2×C4×Dic83C2211yescharacteristic
20C2×C4×Dic83C2211yescharacteristic
21C83⋊#95C2214yes—
22C83⋊#95C2212yes—
23C23×C4×C83C2211yescharacteristic
24#117×C83—24no—
25C2×C4⋊C4×C83C2211yescharacteristic
26C2×C4⋊C4×C83C2211yescharacteristic
27C2×C22⋊C4×C83C2212yes—
28C2×C42×C83C2212yes—
29C23×D83C2312yes—
30C2×D332—22no—
31C2×D332—22no—
32C2×D332—22no—
33C2×C22⋊D83—24no—
34C2×C22⋊D83—44no—
35C22×Dic83C2312yes—
36C22×Dic83C2312yes—
37C22×Dic83C2312yes—
38C4⋊Dic83—22no—
39Dic83⋊C4—44no—
40C22⋊Dic83—24no—
41(C2×D83)⋊C4—24no—
42(C2×D83)⋊C4—28no—
43(C2×D83)⋊C4—44no—
44C4×Dic83—22no—
45C4×Dic83—22no—
46C24×C83C2311yescharacteristic
47C22×C4×C83—22no—
48C22×C4×C83—22no—
49C22×C4×C83D414yes—
50C22×C4×C83C2312yes—
51C22×C4×C83C2312yes—
52C22×C4×C83C2311yescharacteristic
53C22×C4×C83C2311yescharacteristic
54C4⋊C4×C83—22no—
55C4⋊C4×C83—22no—
56C22⋊C4×C83—24no—
57C42×C83—24no—
58C23×C83—21no—
59C23×C83—21no—
60C23×C83—22no—
61C23×C83—22no—
62C23×C83C2×D412yes—
63C23×C83C2411yescharacteristic
64C22×D83—22no—
65C22×D83—24no—
66C22×D83—22no—
67C22×D83—22no—
68C22×D83—24no—
69C2×C4×C83—21no—
70C2×C4×C83—21no—
71C2×C4×C83—28no—
72C2×C4×C83—22no—
73C2×C4×C83—22no—
74C2×C4×C83—24no—
75C2×C4×C83—24no—
76C2×C4×C83C2×D412yes—
77C2×C4×C83C2×D412yes—
78C2×C4×C83#4112yes—
79C2×C4×C83#4112yes—
80C22⋊D83—48no—
81C2×Dic83—22no—
82C2×Dic83—22no—
83C2×Dic83—22no—
84C2×Dic83—22no—
85C2×Dic83—22no—
86C2×Dic83—22no—
87C2×Dic83—24no—
88C2×Dic83#4114yes—
89D332—44no—
90C22×C83—22no—
91C22×C83—22no—
92C22×C83—22no—
93C22×C83—42no—
94C22×C83—42no—
95C22×C83C22×D411yescharacteristic
96C22×C83C2×#4112yes—
97C22×C83C2×#4111yescharacteristic
98C22×C83C2×#4111yescharacteristic
99C22×C83H(5,𝔽2)12yes—
100C2×D83—42no—
101C2×D83—42no—
102C2×D83—42no—
103C2×D83—44no—
104C2×D83—44no—
105C4×C83—22no—
106C4×C83—42no—
107C4×C83—42no—
108C4×C83#12314yes—
109Dic83—24no—
110Dic83—42no—
111Dic83—42no—
112C2×C83—42no—
113C2×C83C2×#12311yescharacteristic
114C2×C83#53211yescharacteristic
115C2×C83#53511yescharacteristic
116C2×C83#55312yes—
117C2×C83#55512yes—
118D83—82no—
119#2500—831nomaximal, Sylow
120C83#250011yescharacteristic, complemented, Sylow
121C2×#121—831no—
122C2×#117D8311yescharacteristic, complemented
123#394—834no—
124#390—831no—
125#390—831no—
126#388—834no—
127#384—831no—
128#382—831no—
129C4×C22⋊C4—831no—
130C22×D4—832no—
131C22×D4—831no—
132C23×C4C2×D8311yescharacteristic
133#121—1664no—
134#117—24no—
135C2×C4⋊C4—834no—
136C2×C4⋊C4—831no—
137C2×C4⋊C4C2×D8311yescharacteristic
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—831no—
145C2×C42—831no—
146C2×C42C2×D8312yes—
147#95—832no—
148#95—834no—
149C24—832no—
150C24C22×D8311yescharacteristic
151C2×D4—3324no—
152C2×D4—1664no—
153C2×D4—1662no—
154C2×D4—1662no—
155C2×D4—1662no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—832no—
159C22×C4—832no—
160C22×C4—832no—
161C22×C4C22⋊D8314yes—
162C22×C4C22×D8312yes—
163C22×C4C22×D8312yes—
164C22×C4C22×D8311yescharacteristic
165C22×C4C22×D8311yescharacteristic
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—3324no—
169C4⋊C4—1662no—
170C22⋊C4—24no—
171C22⋊C4—3324no—
172C22⋊C4—1668no—
173C22⋊C4—1664no—
174C22⋊C4—1664no—
175C42—24no—
176C42—1662no—
177C42—1662no—
178C23—22no—
179C23—21no—
180C23—21no—
181C23—22no—
182C23—1662no—
183C23—1664no—
184C23—1662no—
185C23—1662no—
186C23—1664no—
187C23C2×C22⋊D8312yes—
188C23C23×D8311yescharacteristic
189D4—3328no—
190D4—3324no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—28no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—24no—
197C2×C4—24no—
198C2×C4—834no—
199C2×C4—1662no—
200C2×C4—1662no—
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#1305059012yes—
209C2×C4#1305059012yes—
210C22—22no—
211C22—22no—
212C22—22no—
213C22—42no—
214C22—42no—
215C22—3324no—
216C22—3324no—
217C22—3322no—
218C22—3322no—
219C22—3322no—
220C22C22×C22⋊D8311yescharacteristic
221C22#41×D8312yes—
222C22C2×#1305059011yescharacteristic
223C22C2×#1305059011yescharacteristic
224C22#1458183812yes—
225C4—22no—
226C4—42no—
227C4—42no—
228C4—3322no—
229C4—3322no—
230C4—1664no—
231C4#1458177514yes—
232C2—42no—
233C2—6642no—
234C2C2×#1458177511yescharacteristic
235C2#1912824711yescharacteristic
236C2#1912826511yescharacteristic
237C2#1912839512yes—
238C2#1912841412yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC2×#117×C83C2
prime radicalC2×#14581766C2
hypercenterC2×#117D83
hypocenterC83#2500
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #19127616C2semidirect
3. #19127602C2semidirect
4. #19127602C2semidirect
5. #19127592C2semidirect
6. #19127570C2semidirect
7. #19127566C2semidirect
8. C4×C22⋊Dic83C2semidirect
9. C2×#117×C83C2semidirect
120. C83#2500semidirect
122. C2×#117D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817662
#191276162
#191276022
#191275922
#191275702
#191275662
C4×C22⋊Dic832
C2×#117×C832
#250083