Finite Group Databasebeta

← #24906458

Subgroups — #24906458

Order 10624.

216 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 49 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#19128298C211yesmaximal, characteristic
2#19128031C211yesmaximal, characteristic, complemented
3#19128014C211yesmaximal, characteristic
4#19127437C211yesmaximal, characteristic, complemented
5#19127434C211yesmaximal, characteristic
6#19127427C211yesmaximal, characteristic, complemented
7#398×C83C211yesmaximal, characteristic, complemented
8#14581787—21no—
9#14581782—21no—
10#14581777—21no—
11#14581775—21no—
12#14581771C2211yescharacteristic
13#14581770—21no—
14#14581766C2211yescharacteristic
15C4×C22⋊D83—21no—
16#14581725—21no—
17#14581723—21no—
18#14581715—21no—
19#14581712—21no—
20C22⋊(C83⋊C8)C2211yescharacteristic
21C22⋊(C83⋊C8)C2211yescharacteristic
22C2×C4⋊C4×C83—21no—
23C2×C4⋊C4×C83C2211yescharacteristic
24C2×C4⋊C4×C83C2211yescharacteristic
25#95×C83—21no—
26#95×C83C2211yescharacteristic
27C2×Dic166D411yescharacteristic
28C2×D332D411yescharacteristic
29C2×C22⋊D83—21no—
30C2×C4×D83—21no—
31C2×C83⋊C8—21no—
32C2×C83⋊C8—21no—
33C4⋊Dic83D411yescharacteristic
34C4⋊Dic83D411yescharacteristic
35Dic83⋊C4—21no—
36Dic83⋊C4—21no—
37Dic83⋊C4—21no—
38Dic83⋊C4—21no—
39C22⋊Dic83—21no—
40(C2×D83)⋊C4—21no—
41(C2×D83)⋊C4—21no—
42(C2×D83)⋊C4—21no—
43(C2×D83)⋊C4—21no—
44C4×Dic83—21no—
45C22×C4×C83—21no—
46C22×C4×C83—21no—
47C22×C4×C83D411yescharacteristic
48C22×C4×C83D411yescharacteristic
49C22×C4×C83C2311yescharacteristic
50C4⋊C4×C83—21no—
51C4⋊C4×C83—21no—
52C4⋊C4×C83—21no—
53C4⋊C4×C83—21no—
54C4⋊C4×C83—42no—
55C23×C83C2×D411yescharacteristic
56C22×D83—21no—
57C2×C4×C83—22no—
58C2×C4×C83—21no—
59C2×C4×C83—21no—
60C2×C4×C83—21no—
61C2×C4×C83—21no—
62C2×C4×C83—41no—
63C2×C4×C83—41no—
64C2×C4×C83—41no—
65C2×C4×C83—41no—
66C2×C4×C83—41no—
67C2×C4×C83—41no—
68C2×C4×C83—41no—
69C2×C4×C83C2×D411yescharacteristic
70C2×C4×C83C2×D411yescharacteristic
71C22⋊D83—41no—
72C22⋊D83—41no—
73C2×Dic83—21no—
74C2×Dic83—21no—
75C2×Dic83—21no—
76D332—41no—
77C4×D83—81no—
78Dic166—41no—
79C83⋊C8—41no—
80C83⋊C8—41no—
81C22×C83—21no—
82C22×C83—21no—
83C22×C83#12011yescharacteristic
84C22×C83#13511yescharacteristic
85C22×C83AGL(1,ℤ/8)11yescharacteristic
86C2×D83—41no—
87C2×D83—41no—
88C2×D83—41no—
89C4×C83—21no—
90C4×C83—21no—
91C4×C83—42no—
92C4×C83—41no—
93C4×C83—41no—
94C4×C83—81no—
95Dic83—41no—
96Dic83—41no—
97Dic83—41no—
98C2×C83—21no—
99C2×C83—21no—
100C2×C83#44911yescharacteristic
101C2×C83#45611yescharacteristic
102C2×C83#45811yescharacteristic
103D83—81no—
104#1609—831nomaximal, Sylow
105C83#160911yescharacteristic, complemented, Sylow
106#539—831no—
107#483—831no—
108#480—831no—
109#398D8311yescharacteristic, complemented
110#328—831no—
111#327—831no—
112#324—831no—
113#126—1661no—
114#125—1661no—
115#123—1661no—
116#123—1661no—
117C22⋊Q8—831no—
118C22⋊Q8—1661no—
119#121—831no—
120C4×D4—1661no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4C2×D8311yescharacteristic
123C2×C4⋊C4C2×D8311yescharacteristic
124#107—1661no—
125#106—1661no—
126#103—1661no—
127#102—1661no—
128C22⋊C8—831no—
129C22⋊C8—831no—
130#95—21no—
131#95C2×D8311yescharacteristic
132C2×Q8—831no—
133C2×D4—831no—
134C2×D4—1661no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—1661no—
138C22×C4C22⋊D8311yescharacteristic
139C22×C4C22⋊D8311yescharacteristic
140C22×C4C22×D8311yescharacteristic
141C2×C8—1661no—
142C2×C8—1661no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—21no—
146C4⋊C4—21no—
147C4⋊C4—42no—
148C4⋊C4—831no—
149C4⋊C4—831no—
150C4⋊C4—1661no—
151C4⋊C4—1661no—
152C4⋊C4—1661no—
153C4⋊C4—1661no—
154C22⋊C4—1661no—
155C22⋊C4—1661no—
156C22⋊C4—1661no—
157C22⋊C4—1661no—
158C22⋊C4—1661no—
159C42—1661no—
160C23—1661no—
161C23C2×C22⋊D8311yescharacteristic
162Q8—3321no—
163D4—3321no—
164D4—3321no—
165D4—3321no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—6641no—
179C2×C4—1661no—
180C2×C4—1661no—
181C2×C4—1661no—
182C2×C4D4×D8311yescharacteristic
183C2×C4D4×D8311yescharacteristic
184C8—3321no—
185C8—3321no—
186C22—21no—
187C22—21no—
188C22—3321no—
189C22—3321no—
190C22—3321no—
191C22#1458176011yescharacteristic
192C22#1458181211yescharacteristic
193C22#1458181811yescharacteristic
194C4—21no—
195C4—21no—
196C4—42no—
197C4—41no—
198C4—41no—
199C4—81no—
200C4—3321no—
201C4—3321no—
202C4—3321no—
203C2—21no—
204C2—21no—
205C2—6641no—
206C2#1912784411yescharacteristic
207C2#1912788311yescharacteristic
208C2#1912789011yescharacteristic
subgroupG/H
centerC22#14581760
commutator, cosocleC22×C4×C83C23
FrattiniC22×C4C22×D83
Fitting#398×C83C2
socleC22×C83#120
prime radical#14581766C22
hypercenter#398D83
hypocenterC83#1609
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #398

Complements

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

NH ≅ G/Ntype
2. #19128031C2semidirect
4. #19127437C2semidirect
6. #19127427C2semidirect
7. #398×C83C2semidirect
105. C83#1609semidirect
109. #398D83semidirect

Maximal subgroups

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

subgroupindex
#191282982
#191280312
#191280142
#191274372
#191274342
#191274272
#398×C832
#160983