Finite Group Databasebeta

← #24912944

Subgroups — #24912944

Order 10624.

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

normal subgroups only

This group has 596 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
1C22×Dic83⋊C4C211yesmaximal, characteristic, complemented
2#19127639C212yesmaximal, complemented
3#19127615C212yesmaximal, complemented
4#19127595C214yesmaximal, complemented
5#19127581C212yesmaximal, complemented
6#19127581C212yesmaximal, complemented
7C4×C22⋊Dic83C211yesmaximal, characteristic, complemented
8C2×#121×C83C211yesmaximal, characteristic
9C23×Dic83C2211yescharacteristic
10C23×Dic83C2211yescharacteristic
11C2×Dic83⋊C4—24no—
12C2×Dic83⋊C4C2212yes—
13C2×Dic83⋊C4C2212yes—
14C2×C22⋊Dic83C2212yes—
15C2×C22⋊Dic83C2214yes—
16C2×C22⋊Dic83C2214yes—
17C2×C4×Dic83C2212yes—
18C83⋊#95C2214yes—
19C83⋊#95C2214yes—
20C83⋊#95C2212yes—
21C22×D4×C83C2212yes—
22C22×D4×C83C2211yescharacteristic
23C23×C4×C83C2211yescharacteristic
24#121×C83—24no—
25C2×C4⋊C4×C83C2211yescharacteristic
26C2×C22⋊C4×C83C2212yes—
27C22×Dic83—22no—
28C22×Dic83—24no—
29C22×Dic83—22no—
30C22×Dic83—42no—
31C22×Dic83C2312yes—
32C22×Dic83C2314yes—
33C22×Dic83C2312yes—
34Dic83⋊C4—24no—
35Dic83⋊C4—24no—
36C22⋊Dic83—24no—
37C22⋊Dic83—28no—
38C22⋊Dic83—44no—
39C4×Dic83—42no—
40C24×C83C2312yes—
41C24×C83C2311yescharacteristic
42C2×D4×C83—24no—
43C2×D4×C83—28no—
44C2×D4×C83—22no—
45C2×D4×C83—42no—
46C22×C4×C83—42no—
47C22×C4×C83D414yes—
48C22×C4×C83C2312yes—
49C22×C4×C83C2311yescharacteristic
50C22×C4×C83C2311yescharacteristic
51C4⋊C4×C83—22no—
52C22⋊C4×C83—24no—
53C23×C83—22no—
54C23×C83—22no—
55C23×C83—22no—
56C23×C83—24no—
57C23×C83—24no—
58C23×C83—24no—
59C23×C83C2×D412yes—
60C23×C83#4114yes—
61C23×C83C2411yescharacteristic
62D4×C83—44no—
63D4×C83—48no—
64C2×C4×C83—22no—
65C2×C4×C83—22no—
66C2×C4×C83—22no—
67C2×C4×C83—24no—
68C2×C4×C83—44no—
69C2×C4×C83C2×D412yes—
70C2×C4×C83C2×D412yes—
71C2×Dic83—28no—
72C2×Dic83—22no—
73C2×Dic83—22no—
74C2×Dic83—22no—
75C2×Dic83—24no—
76C2×Dic83—24no—
77C2×Dic83—24no—
78C2×Dic83—24no—
79C2×Dic83—44no—
80C2×Dic83#4114yes—
81C22×C83—22no—
82C22×C83—22no—
83C22×C83—28no—
84C22×C83—42no—
85C22×C83—42no—
86C22×C83—42no—
87C22×C83—48no—
88C22×C83#12414yes—
89C22×C83C22×D411yescharacteristic
90C22×C83C2×#4112yes—
91C22×C83C2×#4112yes—
92C22×C83H(5,𝔽2)12yes—
93C4×C83—22no—
94C4×C83—22no—
95C4×C83—42no—
96Dic83—24no—
97Dic83—42no—
98Dic83—44no—
99C2×C83—24no—
100C2×C83—82no—
101C2×C83#52511yescharacteristic
102C2×C83C2×#12411yescharacteristic
103C2×C83#53411yescharacteristic
104C2×C83#55314yes—
105#2591—831nomaximal, Sylow
106C83#259111yescharacteristic, complemented, Sylow
107C2×#121D8311yescharacteristic
108C22×C4⋊C4—831no—
109#400—832no—
110#394—832no—
111#388—834no—
112#386—832no—
113#386—832no—
114C4×C22⋊C4—831no—
115C22×D4C2×D8312yes—
116C22×D4C2×D8311yescharacteristic
117C23×C4—831no—
118C23×C4—831no—
119C23×C4C2×D8311yescharacteristic
120#121—24no—
121C2×C4⋊C4—832no—
122C2×C4⋊C4—832no—
123C2×C4⋊C4—1664no—
124C2×C4⋊C4C2×D8311yescharacteristic
125C2×C22⋊C4—834no—
126C2×C22⋊C4—832no—
127C2×C22⋊C4—834no—
128C2×C22⋊C4C2×D8312yes—
129C2×C42—832no—
130#95—832no—
131#95—834no—
132#95—834no—
133C24C22×D8312yes—
134C24C22×D8311yescharacteristic
135C2×D4—24no—
136C2×D4—28no—
137C2×D4—22no—
138C2×D4—42no—
139C22×C4—42no—
140C22×C4—3322no—
141C22×C4—832no—
142C22×C4—834no—
143C22×C4—832no—
144C22×C4—1662no—
145C22×C4—1664no—
146C22×C4—1662no—
147C22×C4C22⋊D8314yes—
148C22×C4C22×D8312yes—
149C22×C4C22×D8311yescharacteristic
150C22×C4C22×D8311yescharacteristic
151C4⋊C4—22no—
152C4⋊C4—1664no—
153C4⋊C4—1664no—
154C22⋊C4—24no—
155C22⋊C4—3324no—
156C22⋊C4—1668no—
157C22⋊C4—1664no—
158C42—3322no—
159C23—22no—
160C23—22no—
161C23—22no—
162C23—24no—
163C23—24no—
164C23—24no—
165C23C2×C22⋊D8312yes—
166C23#1305058914yes—
167C23C23×D8311yescharacteristic
168D4—44no—
169D4—48no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—24no—
174C2×C4—44no—
175C2×C4—3324no—
176C2×C4—834no—
177C2×C4—1662no—
178C2×C4—1664no—
179C2×C4—1664no—
180C2×C4—1664no—
181C2×C4—1664no—
182C2×C4—1668no—
183C2×C4—1662no—
184C2×C4—1662no—
185C2×C4C2×C22⋊D8312yes—
186C2×C4C2×C22⋊D8312yes—
187C22—28no—
188C22—22no—
189C22—22no—
190C22—48no—
191C22—42no—
192C22—42no—
193C22—42no—
194C22#1458177914yes—
195C22C22×C22⋊D8311yescharacteristic
196C22#41×D8312yes—
197C22C2×#1305058912yes—
198C22#1458183712yes—
199C4—22no—
200C4—22no—
201C4—42no—
202C4—3324no—
203C4—3322no—
204C4—1664no—
205C2—24no—
206C2—82no—
207C2#1912820811yescharacteristic
208C2C2×#1458177911yescharacteristic
209C2#1912826011yescharacteristic
210C2#1912839414yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting, prime radicalC2×#121×C83C2
hypercenterC2×#121D83
hypocenterC83#2591
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C22×Dic83⋊C4C2semidirect
2. #19127639C2semidirect
3. #19127615C2semidirect
4. #19127595C2semidirect
5. #19127581C2semidirect
6. #19127581C2semidirect
7. C4×C22⋊Dic83C2semidirect
106. C83#2591semidirect

Maximal subgroups

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

subgroupindex
C22×Dic83⋊C42
#191276392
#191276152
#191275952
#191275812
C4×C22⋊Dic832
C2×#121×C832
#259183