Finite Group Databasebeta

← #35292007

Subgroups — #35292007

Order 21248.

224 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 43 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#24905588C211yesmaximal, characteristic
2#24905395C211yesmaximal, characteristic
3C83×#2093C211yesmaximal, characteristic
4C2×#14581707—21no—
5#19127427—21no—
6C2×#107×C83—21no—
7C2×#102×C83—21no—
8#390×C83C2211yescharacteristic
9#384×C83C411yescharacteristic, complemented
10#336×C83C411yescharacteristic, complemented
11C2×C22⋊Dic83—41no—
12#14581707—22no—
13#14581707—41no—
14C22⋊(C83⋊C8)—41no—
15C22×D4×C83D411yescharacteristic
16C22×C8×C83—21no—
17C2×C4⋊C4×C83—21no—
18C2×C4⋊C4×C83—41no—
19C2×C4⋊C4×C83D411yescharacteristic
20C2×C22⋊C4×C83—41no—
21C2×C42×C83C2×C411yescharacteristic
22#107×C83—42no—
23#102×C83—42no—
24C22×Dic83—81no—
25C2×C83⋊C8—81no—
26C22⋊Dic83—42no—
27C22⋊Dic83—81no—
28C24×C83—21no—
29C2×D4×C83—21no—
30C2×D4×C83—41no—
31C2×D4×C83—41no—
32C2×D4×C83SD1611yescharacteristic
33C2×D4×C83Q1611yescharacteristic
34C22×C4×C83—21no—
35C22×C4×C83—21no—
36C22×C4×C83—41no—
37C22×C4×C83C22⋊C411yescharacteristic
38C2×C8×C83—22no—
39C2×C8×C83—41no—
40C2×C8×C83—41no—
41C4⋊C4×C83—41no—
42C4⋊C4×C83—41no—
43C4⋊C4×C83—41no—
44C4⋊C4×C83—82no—
45C22⋊C4×C83—82no—
46C42×C83—41no—
47C23×C83—21no—
48C23×C83—21no—
49C23×C83—41no—
50C23×C83—41no—
51C23×C83—81no—
52C23×C83—81no—
53C23×C83#9911yescharacteristic
54D4×C83—41no—
55D4×C83—41no—
56D4×C83—81no—
57C2×C4×C83—21no—
58C2×C4×C83—21no—
59C2×C4×C83—22no—
60C2×C4×C83—41no—
61C2×C4×C83—41no—
62C2×C4×C83—41no—
63C2×C4×C83—81no—
64C2×C4×C83—82no—
65C2×C4×C83—81no—
66C2×C4×C83#10311yescharacteristic
67C2×C4×C83#10411yescharacteristic
68C2×Dic83—82no—
69C2×Dic83—161no—
70C2×Dic83—161no—
71C8×C83—42no—
72C83⋊C8—161no—
73C22×C83—21no—
74C22×C83—21no—
75C22×C83—41no—
76C22×C83—41no—
77C22×C83—81no—
78C22×C83—81no—
79C22×C83—81no—
80C22×C83—81no—
81C22×C83—161no—
82C22×C83#32811yescharacteristic
83C22×C83#35211yescharacteristic
84C22×C83#35311yescharacteristic
85C4×C83—21no—
86C4×C83—21no—
87C4×C83—42no—
88C4×C83—81no—
89C4×C83—161no—
90Dic83—162no—
91#7434—831nomaximal, Sylow
92C2×C83—21no—
93C2×C83—21no—
94C2×C83—81no—
95C2×C83—81no—
96C2×C83#135211yescharacteristic
97C2×C83#140711yescharacteristic
98C2×C83#140811yescharacteristic
99#2093D8311yescharacteristic
100#1393—831no—
101#1318—831no—
102C83#743411yescharacteristic, complemented, Sylow
103C2×#107—21no—
104C2×#102—21no—
105C2×#99—1661no—
106#390C2×D8311yescharacteristic
107#384Dic8311yescharacteristic, complemented
108#336Dic8311yescharacteristic, complemented
109#324—1661no—
110C22×D4C22⋊D8311yescharacteristic
111C22×C8—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4C22⋊D8311yescharacteristic
115C2×C22⋊C4—41no—
116C2×C22⋊C4—3321no—
117C2×C42C2×Dic8311yescharacteristic
118#107—42no—
119#102—42no—
120#99—3321no—
121#99—1662no—
122C22⋊C8—3321no—
123C24—21no—
124C2×D4—21no—
125C2×D4—41no—
126C2×D4—41no—
127C2×D4#1305057711yescharacteristic
128C2×D4#1305058011yescharacteristic
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—41no—
132C22×C4—6641no—
133C22×C4C22⋊Dic8311yescharacteristic
134C2×C8—22no—
135C2×C8—41no—
136C2×C8—41no—
137C2×C8—6641no—
138C4⋊C4—41no—
139C4⋊C4—41no—
140C4⋊C4—41no—
141C4⋊C4—82no—
142C22⋊C4—82no—
143C22⋊C4—3322no—
144C22⋊C4—6641no—
145C42—41no—
146C23—21no—
147C23—21no—
148C23—41no—
149C23—41no—
150C23—81no—
151C23—81no—
152C23#1458170711yescharacteristic
153D4—41no—
154D4—41no—
155D4—81no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—81no—
163C2×C4—82no—
164C2×C4—81no—
165C2×C4—13281no—
166C2×C4—13281no—
167C2×C4—6642no—
168C2×C4#1458171611yescharacteristic
169C2×C4#1458171911yescharacteristic
170C8—42no—
171C8—13281no—
172C22—21no—
173C22—21no—
174C22—41no—
175C22—41no—
176C22—81no—
177C22—81no—
178C22—81no—
179C22—81no—
180C22—161no—
181C22#1912743811yescharacteristic
182C22#1912749111yescharacteristic
183C22#1912749411yescharacteristic
184C4—21no—
185C4—21no—
186C4—42no—
187C4—81no—
188C4—161no—
189C4—13282no—
190C2—21no—
191C2—21no—
192C2—81no—
193C2—81no—
194C2#2490548711yescharacteristic
195C2#2490562311yescharacteristic
196C2#2490562611yescharacteristic
subgroupG/H
centerC22#19127494
commutatorC2×C42×C83C2×C4
Frattini#390C2×D83
FittingC83×#2093C2
socleC22×C83#353
prime radicalC22×D4×C83D4
cosocle#390×C83C22
hypercenter#2093D83
hypocenterC83#7434
LCSG ▷ C2×C42×C83 ▷ C22×C4×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #390 ◁ #2093

Complements

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

NH ≅ G/Ntype
9. #384×C83C4semidirect
10. #336×C83C4semidirect
102. C83#7434semidirect
107. #384Dic83semidirect
108. #336Dic83semidirect

Maximal subgroups

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

subgroupindex
#249055882
#249053952
C83×#20932
#743483