Finite Group Databasebeta

← #24915141

Subgroups — #24915141

Order 10624.

448 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 125 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#19127641C212yesmaximal, complemented
2#19127640C211yesmaximal, characteristic, complemented
3#19127629C212yesmaximal, complemented
4#19127589C212yesmaximal, complemented
5#19127581C212yesmaximal
6#19127576C212yesmaximal, complemented
7#19127576C212yesmaximal, complemented
8#19127566C211yesmaximal, characteristic, complemented
9#394×C83C211yesmaximal, characteristic
10C23×Dic83C2212yes—
11C2×C4⋊Dic83C2211yescharacteristic
12C2×C4⋊Dic83C2211yescharacteristic
13C2×C4⋊Dic83C2212yes—
14C2×Dic83⋊C4C2212yes—
15C2×C22⋊Dic83C2212yes—
16C2×C22⋊Dic83C2212yes—
17C2×C22⋊Dic83C2212yes—
18C2×C4×Dic83C2211yescharacteristic
19C83⋊#95C2212yes—
20C83⋊#95C2212yes—
21C83⋊#95C2212yes—
22C83⋊#95C2212yes—
23C83⋊#95C2212yes—
24C83⋊#95C2212yes—
25C83⋊#95C2211yescharacteristic
26C22×D4×C83C2211yescharacteristic
27C2×C4⋊C4×C83C2211yescharacteristic
28C2×C22⋊C4×C83C2212yes—
29C2×C22⋊C4×C83C2212yes—
30#95×C83C2211yescharacteristic
31C22×Dic83—42no—
32C22×Dic83—42no—
33C22×Dic83—42no—
34C22×Dic83C2312yes—
35C22×Dic83C2312yes—
36C22×Dic83C2312yes—
37C22×Dic83C2312yes—
38C4⋊Dic83—41no—
39C4⋊Dic83—41no—
40C4⋊Dic83—42no—
41Dic83⋊C4—42no—
42C22⋊Dic83—24no—
43C22⋊Dic83—42no—
44C22⋊Dic83—42no—
45C4×Dic83—22no—
46C24×C83C2312yes—
47C2×D4×C83—24no—
48C2×D4×C83—41no—
49C22×C4×C83C2312yes—
50C22×C4×C83C2311yescharacteristic
51C22×C4×C83C2311yescharacteristic
52C22×C4×C83C2311yescharacteristic
53C4⋊C4×C83—41no—
54C22⋊C4×C83—24no—
55C22⋊C4×C83—24no—
56C23×C83—22no—
57C23×C83—22no—
58C23×C83—22no—
59C23×C83—22no—
60C23×C83—22no—
61C23×C83—22no—
62C23×C83#4114yes—
63C23×C83C2411yescharacteristic
64D4×C83—44no—
65C2×C4×C83—21no—
66C2×C4×C83—21no—
67C2×C4×C83—22no—
68C2×C4×C83—22no—
69C2×C4×C83—22no—
70C2×C4×C83—22no—
71C2×C4×C83—21no—
72C2×C4×C83—21no—
73C2×C4×C83—22no—
74C2×C4×C83#4112yes—
75C2×Dic83—22no—
76C2×Dic83—22no—
77C2×Dic83—22no—
78C2×Dic83—22no—
79C2×Dic83—22no—
80C2×Dic83—22no—
81C2×Dic83—22no—
82C2×Dic83—22no—
83C2×Dic83—22no—
84C2×Dic83—22no—
85C2×Dic83—22no—
86C2×Dic83—22no—
87C2×Dic83—48no—
88C22×C83—24no—
89C22×C83—24no—
90C22×C83—24no—
91C22×C83—42no—
92C22×C83—42no—
93C22×C83—42no—
94C22×C83—42no—
95C22×C83C2×#4112yes—
96C22×C83C2×#4111yescharacteristic
97C22×C83H(5,𝔽2)11yescharacteristic
98C22×C83H(5,𝔽2)12yes—
99C22×C832−1+411yescharacteristic
100C4×C83—22no—
101C4×C83—41no—
102C4×C83—41no—
103C4×C83—42no—
104Dic83—42no—
105Dic83—42no—
106Dic83—42no—
107Dic83—42no—
108C2×C83—44no—
109C2×C83#53912yes—
110C2×C83#53911yescharacteristic
111C2×C83#55111yescharacteristic
112C2×C83#55312yes—
113C2×C83H(3,𝔽4)11yescharacteristic
114#2782—831nomaximal, Sylow
115C83#278211yescharacteristic, complemented, Sylow
116#400—831no—
117#400—832no—
118#397—832no—
119#394D8311yescharacteristic
120#388—832no—
121#386—832no—
122#385—832no—
123#385—832no—
124#382—831no—
125C22×D4C2×D8311yescharacteristic
126C23×C4—832no—
127C2×C4⋊C4—831no—
128C2×C4⋊C4—831no—
129C2×C4⋊C4—832no—
130C2×C4⋊C4—832no—
131C2×C4⋊C4C2×D8311yescharacteristic
132C2×C22⋊C4—832no—
133C2×C22⋊C4—832no—
134C2×C22⋊C4—832no—
135C2×C22⋊C4C2×D8312yes—
136C2×C22⋊C4C2×D8312yes—
137C2×C42—831no—
138#95—832no—
139#95—832no—
140#95—832no—
141#95—832no—
142#95—832no—
143#95—832no—
144#95—831no—
145#95C2×D8311yescharacteristic
146C24C22×D8312yes—
147C2×D4—24no—
148C2×D4—41no—
149C22×C4—3322no—
150C22×C4—3322no—
151C22×C4—3322no—
152C22×C4—832no—
153C22×C4—832no—
154C22×C4—832no—
155C22×C4—832no—
156C22×C4C22×D8311yescharacteristic
157C22×C4C22×D8311yescharacteristic
158C22×C4C22×D8312yes—
159C22×C4C22×D8311yescharacteristic
160C4⋊C4—41no—
161C4⋊C4—3322no—
162C4⋊C4—3321no—
163C4⋊C4—3321no—
164C4⋊C4—3322no—
165C22⋊C4—24no—
166C22⋊C4—24no—
167C22⋊C4—3322no—
168C22⋊C4—3322no—
169C22⋊C4—1664no—
170C42—1662no—
171C23—22no—
172C23—22no—
173C23—22no—
174C23—22no—
175C23—22no—
176C23—22no—
177C23#1305058914yes—
178C23C23×D8311yescharacteristic
179D4—44no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—3328no—
190C2×C4—1662no—
191C2×C4—1662no—
192C2×C4—1662no—
193C2×C4—1662no—
194C2×C4—1662no—
195C2×C4—1662no—
196C2×C4—1662no—
197C2×C4—1662no—
198C2×C4—1662no—
199C2×C4—1662no—
200C2×C4—1662no—
201C2×C4—1662no—
202C2×C4#1305058912yes—
203C22—24no—
204C22—24no—
205C22—24no—
206C22—42no—
207C22—42no—
208C22—42no—
209C22—42no—
210C22C2×#1305058912yes—
211C22C2×#1305058911yescharacteristic
212C22#1458183711yescharacteristic
213C22#1458183812yes—
214C22#1458183911yescharacteristic
215C4—22no—
216C4—41no—
217C4—41no—
218C4—42no—
219C4—3322no—
220C4—3322no—
221C4—3322no—
222C4—3322no—
223C2—44no—
224C2#1912829512yes—
225C2#1912829611yescharacteristic
226C2#1912838211yescharacteristic
227C2#1912840312yes—
228C2#1912844311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#394×C83C2
prime radicalC22×D4×C83C22
hypercenter#394D83
hypocenterC83#2782
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. #19127641C2semidirect
2. #19127640C2semidirect
3. #19127629C2semidirect
4. #19127589C2semidirect
6. #19127576C2semidirect
7. #19127576C2semidirect
8. #19127566C2semidirect
115. C83#2782semidirect

Maximal subgroups

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

subgroupindex
#191276412
#191276402
#191276292
#191275892
#191275812
#191275762
#191275662
#394×C832
#278283