Finite Group Databasebeta

← #24913245

Subgroups — #24913245

Order 10624.

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

normal subgroups only

This group has 526 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
1C22×C4⋊Dic83C211yesmaximal, characteristic, complemented
2#19127640C212yesmaximal, complemented
3#19127637C212yesmaximal, complemented
4#19127631C212yesmaximal, complemented
5#19127626C212yesmaximal, complemented
6#19127581C212yesmaximal
7#19127573C212yesmaximal, complemented
8#19127564C211yesmaximal, characteristic, complemented
9C2×#123×C83C211yesmaximal, characteristic
10C23×Dic83C2212yes—
11C2×C4⋊Dic83—24no—
12C2×C4⋊Dic83C2212yes—
13C2×C4⋊Dic83C2212yes—
14C2×Dic83⋊C4C2214yes—
15C2×C22⋊Dic83C2212yes—
16C2×C22⋊Dic83C2214yes—
17C83⋊#95C2212yes—
18C83⋊#95C2212yes—
19C83⋊#95C2214yes—
20C83⋊#95C2214yes—
21C22×D4×C83C2211yescharacteristic
22C23×C4×C83C2211yescharacteristic
23#123×C83—24no—
24C2×C4⋊C4×C83C2211yescharacteristic
25C2×C4⋊C4×C83C2211yescharacteristic
26C2×C22⋊C4×C83C2211yescharacteristic
27C2×C22⋊C4×C83C2211yescharacteristic
28C2×C22⋊C4×C83C2211yescharacteristic
29C22×Dic83—24no—
30C22×Dic83—44no—
31C22×Dic83C2314yes—
32C22×Dic83C2314yes—
33C4⋊Dic83—24no—
34C4⋊Dic83—24no—
35Dic83⋊C4—44no—
36C22⋊Dic83—24no—
37C22⋊Dic83—44no—
38C24×C83C2311yescharacteristic
39C24×C83C2311yescharacteristic
40C2×D4×C83—22no—
41C2×D4×C83—24no—
42C22×C4×C83—24no—
43C22×C4×C83D414yes—
44C22×C4×C83C2311yescharacteristic
45C22×C4×C83C2311yescharacteristic
46C22×C4×C83C2311yescharacteristic
47C22×C4×C83C2311yescharacteristic
48C22×C4×C83C2311yescharacteristic
49C4⋊C4×C83—22no—
50C4⋊C4×C83—22no—
51C22⋊C4×C83—22no—
52C22⋊C4×C83—22no—
53C22⋊C4×C83—22no—
54C23×C83—22no—
55C23×C83—22no—
56C23×C83—21no—
57C23×C83—22no—
58C23×C83—22no—
59C23×C83—22no—
60C23×C83C2×D412yes—
61C23×C83#4114yes—
62C23×C83C2411yescharacteristic
63D4×C83—44no—
64C2×C4×C83—28no—
65C2×C4×C83—22no—
66C2×C4×C83—22no—
67C2×C4×C83—21no—
68C2×C4×C83—21no—
69C2×C4×C83—21no—
70C2×C4×C83—22no—
71C2×C4×C83—22no—
72C2×C4×C83—22no—
73C2×C4×C83C2×D412yes—
74C2×C4×C83C2×D412yes—
75C2×Dic83—24no—
76C2×Dic83—24no—
77C2×Dic83—24no—
78C2×Dic83—24no—
79C2×Dic83—24no—
80C2×Dic83—24no—
81C2×Dic83—48no—
82C22×C83—22no—
83C22×C83—22no—
84C22×C83—28no—
85C22×C83—41no—
86C22×C83—41no—
87C22×C83—41no—
88C22×C83—44no—
89C22×C83#12314yes—
90C22×C83C22×D411yescharacteristic
91C22×C83C2×#4112yes—
92C22×C83H(5,𝔽2)12yes—
93C22×C832−1+412yes—
94C4×C83—22no—
95C4×C83—22no—
96C4×C83—41no—
97C4×C83—41no—
98C4×C83—41no—
99Dic83—44no—
100Dic83—44no—
101C2×C83—24no—
102C2×C83—81no—
103C2×C83C2×#12311yescharacteristic
104C2×C83#53511yescharacteristic
105C2×C83#53611yescharacteristic
106C2×C83#53914yes—
107#2623—831nomaximal, Sylow
108C83#262311yescharacteristic, complemented, Sylow
109C2×#123D8311yescharacteristic
110C22×C4⋊C4—831no—
111#400—832no—
112#399—832no—
113#398—832no—
114#397—832no—
115#386—832no—
116#385—832no—
117#381—831no—
118C22×D4C2×D8311yescharacteristic
119C23×C4—832no—
120C23×C4C2×D8311yescharacteristic
121#123—24no—
122C2×C4⋊C4—834no—
123C2×C4⋊C4—832no—
124C2×C4⋊C4—832no—
125C2×C4⋊C4—1664no—
126C2×C4⋊C4C2×D8311yescharacteristic
127C2×C4⋊C4C2×D8311yescharacteristic
128C2×C22⋊C4—834no—
129C2×C22⋊C4—832no—
130C2×C22⋊C4C2×D8311yescharacteristic
131C2×C22⋊C4C2×D8311yescharacteristic
132C2×C22⋊C4C2×D8311yescharacteristic
133#95—832no—
134#95—832no—
135#95—834no—
136#95—834no—
137C24C22×D8311yescharacteristic
138C24C22×D8311yescharacteristic
139C2×D4—22no—
140C2×D4—24no—
141C22×C4—24no—
142C22×C4—3324no—
143C22×C4—834no—
144C22×C4—834no—
145C22×C4—1664no—
146C22×C4C22⋊D8314yes—
147C22×C4C22×D8311yescharacteristic
148C22×C4C22×D8311yescharacteristic
149C22×C4C22×D8311yescharacteristic
150C22×C4C22×D8311yescharacteristic
151C22×C4C22×D8311yescharacteristic
152C4⋊C4—22no—
153C4⋊C4—22no—
154C4⋊C4—3324no—
155C4⋊C4—1664no—
156C4⋊C4—1664no—
157C22⋊C4—22no—
158C22⋊C4—22no—
159C22⋊C4—22no—
160C22⋊C4—3324no—
161C22⋊C4—1664no—
162C23—22no—
163C23—22no—
164C23—21no—
165C23—22no—
166C23—22no—
167C23—22no—
168C23C2×C22⋊D8312yes—
169C23#1305058914yes—
170C23C23×D8311yescharacteristic
171D4—44no—
172C2×C4—28no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—3328no—
182C2×C4—1664no—
183C2×C4—1664no—
184C2×C4—1664no—
185C2×C4—1664no—
186C2×C4—1664no—
187C2×C4—1664no—
188C2×C4C2×C22⋊D8312yes—
189C2×C4C2×C22⋊D8312yes—
190C22—22no—
191C22—22no—
192C22—28no—
193C22—41no—
194C22—41no—
195C22—41no—
196C22—44no—
197C22#1458177314yes—
198C22C22×C22⋊D8311yescharacteristic
199C22C2×#1305058912yes—
200C22#1458183812yes—
201C22#1458183912yes—
202C4—22no—
203C4—22no—
204C4—41no—
205C4—41no—
206C4—41no—
207C4—3324no—
208C4—3324no—
209C2—24no—
210C2—81no—
211C2C2×#1458177311yescharacteristic
212C2#1912826511yescharacteristic
213C2#1912827211yescharacteristic
214C2#1912829514yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC2×#123×C83C2
prime radicalC22×D4×C83C22
hypercenterC2×#123D83
hypocenterC83#2623
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C22×C4⋊Dic83C2semidirect
2. #19127640C2semidirect
3. #19127637C2semidirect
4. #19127631C2semidirect
5. #19127626C2semidirect
7. #19127573C2semidirect
8. #19127564C2semidirect
108. C83#2623semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊Dic832
#191276402
#191276372
#191276312
#191276262
#191275812
#191275732
#191275642
C2×#123×C832
#262383