Finite Group Databasebeta

← #24913254

Subgroups — #24913254

Order 10624.

548 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 149 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
2C2×#14581751C211yesmaximal, characteristic, complemented
3#19127638C212yesmaximal, complemented
4#19127628C212yesmaximal, complemented
5#19127617C214yesmaximal, complemented
6#19127602C212yesmaximal, complemented
7#19127600C212yesmaximal, complemented
8#380×C83C211yesmaximal, characteristic, complemented
9C22×D332C2211yescharacteristic
10C22×C22⋊D83C2212yes—
11#14581766—24no—
12#14581751—24no—
13C2×C4⋊Dic83C2211yescharacteristic
14C2×C4⋊Dic83C2211yescharacteristic
15C2×C4⋊Dic83C2211yescharacteristic
16C2×C4⋊Dic83C2214yes—
17C2×C22⋊Dic83C2212yes—
18C2×(C2×D83)⋊C4C2212yes—
19C2×(C2×D83)⋊C4C2214yes—
20C2×(C2×D83)⋊C4C2214yes—
21C2×C4×Dic83C2212yes—
22C83⋊#95C2214yes—
23C23×C4×C83C2211yescharacteristic
24C2×C4⋊C4×C83C2212yes—
25C2×C22⋊C4×C83C2212yes—
26#95×C83C2212yes—
27C23×D83C2312yes—
28C2×D332—22no—
29C2×D332—24no—
30C2×C22⋊D83—24no—
31C2×C22⋊D83—44no—
32C22×Dic83C2312yes—
33C22×Dic83C2312yes—
34C22×Dic83C2312yes—
35C4⋊Dic83—22no—
36C4⋊Dic83—22no—
37C4⋊Dic83—22no—
38C4⋊Dic83—44no—
39C22⋊Dic83—24no—
40(C2×D83)⋊C4—24no—
41(C2×D83)⋊C4—44no—
42(C2×D83)⋊C4—44no—
43C4×Dic83—24no—
44C24×C83C2311yescharacteristic
45C22×C4×C83—24no—
46C22×C4×C83D414yes—
47C22×C4×C83C2314yes—
48C22×C4×C83C2311yescharacteristic
49C22×C4×C83C2311yescharacteristic
50C4⋊C4×C83—24no—
51C22⋊C4×C83—42no—
52C23×C83—22no—
53C23×C83—21no—
54C23×C83—22no—
55C23×C83—21no—
56C23×C83C2×D412yes—
57C23×C83C2411yescharacteristic
58C22×D83—22no—
59C22×D83—24no—
60C22×D83—22no—
61C22×D83—24no—
62C22×D83—22no—
63C2×C4×C83—22no—
64C2×C4×C83—28no—
65C2×C4×C83—24no—
66C2×C4×C83—24no—
67C2×C4×C83—24no—
68C2×C4×C83C2×D412yes—
69C2×C4×C83C2×D412yes—
70C2×C4×C83#4114yes—
71C22⋊D83—48no—
72C2×Dic83—22no—
73C2×Dic83—22no—
74C2×Dic83—22no—
75C2×Dic83—22no—
76C2×Dic83—22no—
77C2×Dic83—22no—
78C2×Dic83—24no—
79C2×Dic83—22no—
80D332—44no—
81C22×C83—22no—
82C22×C83—22no—
83C22×C83—22no—
84C22×C83—42no—
85C22×C83—42no—
86C22×C83C22×D411yescharacteristic
87C22×C83C2×#4112yes—
88C22×C83H(5,𝔽2)11yescharacteristic
89C22×C83H(5,𝔽2)12yes—
90C22×C832−1+411yescharacteristic
91C2×D83—42no—
92C2×D83—42no—
93C2×D83—42no—
94C2×D83—44no—
95C2×D83—44no—
96C4×C83—22no—
97C4×C83—44no—
98C4×C83#12314yes—
99Dic83—42no—
100Dic83—42no—
101Dic83—42no—
102C2×C83—42no—
103C2×C83C2×#12311yescharacteristic
104C2×C83#53511yescharacteristic
105C2×C83#53711yescharacteristic
106C2×C83#54012yes—
107C2×C83#54212yes—
108D83—82no—
109#2624—831nomaximal, Sylow
110C83#262411yescharacteristic, complemented, Sylow
111C2×#121—831no—
112C2×#117—831no—
113#399—832no—
114#397—832no—
115#394—834no—
116#390—832no—
117#389—832no—
118#380D8311yescharacteristic, complemented
119C22×D4—832no—
120C22×D4—831no—
121C23×C4C2×D8311yescharacteristic
122#121—1664no—
123#117—1664no—
124C2×C4⋊C4—834no—
125C2×C4⋊C4—831no—
126C2×C4⋊C4—831no—
127C2×C4⋊C4—831no—
128C2×C4⋊C4C2×D8312yes—
129C2×C22⋊C4—834no—
130C2×C22⋊C4—834no—
131C2×C22⋊C4—832no—
132C2×C22⋊C4—832no—
133C2×C22⋊C4C2×D8312yes—
134C2×C42—832no—
135#95—834no—
136#95C2×D8312yes—
137C24—832no—
138C24C22×D8311yescharacteristic
139C2×D4—3324no—
140C2×D4—1664no—
141C2×D4—1662no—
142C2×D4—1664no—
143C22×C4—24no—
144C22×C4—832no—
145C22×C4—832no—
146C22×C4—832no—
147C22×C4D33214yes—
148C22×C4C22×D8314yes—
149C22×C4C22×D8311yescharacteristic
150C22×C4C22×D8311yescharacteristic
151C4⋊C4—24no—
152C4⋊C4—3324no—
153C4⋊C4—1662no—
154C4⋊C4—1662no—
155C4⋊C4—1662no—
156C22⋊C4—42no—
157C22⋊C4—3324no—
158C22⋊C4—3324no—
159C22⋊C4—1664no—
160C22⋊C4—1664no—
161C42—1664no—
162C23—22no—
163C23—21no—
164C23—22no—
165C23—21no—
166C23—1662no—
167C23—1664no—
168C23—1662no—
169C23—1664no—
170C23—1662no—
171C23C2×D33212yes—
172C23C23×D8311yescharacteristic
173D4—3328no—
174D4—3324no—
175C2×C4—22no—
176C2×C4—28no—
177C2×C4—24no—
178C2×C4—24no—
179C2×C4—24no—
180C2×C4—1662no—
181C2×C4—1662no—
182C2×C4—1662no—
183C2×C4—1662no—
184C2×C4—1662no—
185C2×C4—1662no—
186C2×C4—1664no—
187C2×C4—1662no—
188C2×C4C2×D33212yes—
189C2×C4C2×D33212yes—
190C2×C4#1305058914yes—
191C22—22no—
192C22—22no—
193C22—22no—
194C22—42no—
195C22—42no—
196C22—3324no—
197C22—3322no—
198C22—3322no—
199C22—3322no—
200C22—3324no—
201C22C22×D33211yescharacteristic
202C22C2×#1305058912yes—
203C22#1458183711yescharacteristic
204C22#1458183812yes—
205C22#1458184011yescharacteristic
206C4—22no—
207C4—44no—
208C4—3322no—
209C4—3322no—
210C4—3322no—
211C4#1458177614yes—
212C2—42no—
213C2—6642no—
214C2C2×#1458177611yescharacteristic
215C2#1912827011yescharacteristic
216C2#1912828211yescharacteristic
217C2#1912830412yes—
218C2#1912832412yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#380×C83C2
prime radicalC2×#14581766C2
hypercenter#380D83
hypocenterC83#2624
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
2. C2×#14581751C2semidirect
3. #19127638C2semidirect
4. #19127628C2semidirect
5. #19127617C2semidirect
6. #19127602C2semidirect
7. #19127600C2semidirect
8. #380×C83C2semidirect
110. C83#2624semidirect
118. #380D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817662
C2×#145817512
#191276382
#191276282
#191276172
#191276022
#191276002
#380×C832
#262483