Finite Group Databasebeta

← #35302019

Subgroups — #35302019

Order 21248.

452 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 111 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#24916565C211yesmaximal, characteristic, complemented
2#24906201C211yesmaximal, characteristic, complemented
3#24906131C212yesmaximal
4#24905353C212yesmaximal, complemented
5C83×#3138C211yesmaximal, characteristic
6#19127832—22no—
7Q8×C83⋊C8—22no—
8#19127762—22no—
9#19127754C2212yes—
10#19127725C2212yes—
11#19127447—22no—
12#19127445—22no—
13#19127445—22no—
14#533×C83C2211yescharacteristic
15C2×#127×C83C2211yescharacteristic
16#490×C83C414yes—
17#486×C83—22no—
18#418×C83—22no—
19C4×M16×C83C2211yescharacteristic
20C2×#13050574—22no—
21C4⋊(C83⋊C8)—24no—
22C4⋊(C83⋊C8)—22no—
23C4⋊(C83⋊C8)—22no—
24C4⋊(C83⋊C8)D412yes—
25C4⋊(C83⋊C8)D412yes—
26C22⋊(C83⋊C8)—22no—
27#14581703—22no—
28C4×C83⋊C8—22no—
29C4×C83⋊C8—22no—
30C22×D4×C83—21no—
31C22×D4×C83—22no—
32C2×M16×C83—21no—
33C4⋊Q8×C83D412yes—
34#127×C83—21no—
35#127×C83—22no—
36#127×C83C2×C412yes—
37#125×C83C2×C412yes—
38C22⋊Q8×C83—22no—
39C4×Q8×C83—22no—
40#117×C83D412yes—
41C2×C42×C83C2311yescharacteristic
42#102×C83—28no—
43#97×C83C2×C412yes—
44C4×C8×C83—21no—
45#13050574—82no—
46C2×C83⋊C8—22no—
47C2×C83⋊C8—22no—
48C2×C83⋊C8—42no—
49C24×C83—41no—
50C2×Q8×C83—22no—
51C2×D4×C83—22no—
52C2×D4×C83—24no—
53C2×D4×C83—44no—
54C2×D4×C83—41no—
55C2×D4×C83—42no—
56C2×D4×C83—82no—
57C2×D4×C83—82no—
58C22×C4×C83C2×D411yescharacteristic
59C22×C4×C83#4112yes—
60M16×C83—42no—
61C2×C8×C83—22no—
62C4⋊C4×C83—24no—
63C4⋊C4×C83—22no—
64C4⋊C4×C83C22⋊C414yes—
65C22⋊C4×C83—22no—
66C42×C83—22no—
67C42×C83C22×C411yescharacteristic
68C42×C83C2×D412yes—
69C42×C83C2×D411yescharacteristic
70C23×C83—42no—
71C23×C83—82no—
72C23×C83—84no—
73C23×C83#12311yescharacteristic
74Q8×C83—42no—
75Q8×C83—42no—
76D4×C83—44no—
77D4×C83—84no—
78D4×C83—82no—
79D4×C83—82no—
80D4×C83—82no—
81C2×C4×C83—22no—
82C2×C4×C83—22no—
83C2×C4×C83—22no—
84C2×C4×C83—24no—
85C2×C4×C83C2×C22⋊C411yescharacteristic
86C2×C4×C83C4×D412yes—
87C2×C4×C83#12011yescharacteristic
88C2×C4×C83#12112yes—
89C8×C83—42no—
90C83⋊C8—42no—
91C83⋊C8—42no—
92C83⋊C8—82no—
93C22×C83—21no—
94C22×C83—22no—
95C22×C83—82no—
96C22×C83—84no—
97C22×C83—164no—
98C22×C83#38611yescharacteristic
99C4×C83—21no—
100C4×C83—22no—
101C4×C83—44no—
102C4×C83#45412yes—
103C4×C83#47112yes—
104C4×C83#47212yes—
105#9438—831nomaximal, Sylow
106C2×C83—41no—
107C2×C83—162no—
108C2×C83#188811yescharacteristic
109C2×C83#190712yes—
110#3138D8311yescharacteristic
111#2986—831no—
112#1561—831no—
113#1545—832no—
114#1299—832no—
115C83#943811yescharacteristic, complemented, Sylow
116#533C2×D8311yescharacteristic
117C2×#127C2×D8311yescharacteristic
118#490Dic8314yes—
119#486—22no—
120#446—1662no—
121C8×Q8—1662no—
122#433—1662no—
123#431—832no—
124#423—832no—
125#418—22no—
126C4×M16C2×D8311yescharacteristic
127#332—1662no—
128#331—1662no—
129#331—1662no—
130C22×D4—21no—
131C22×D4—22no—
132C2×M16—21no—
133C2×M16—1662no—
134C4⋊Q8C22⋊D8312yes—
135#127—22no—
136#127—21no—
137#127C2×Dic8312yes—
138#125C2×Dic8312yes—
139C22⋊Q8—22no—
140C4×Q8—22no—
141#117C22⋊D8312yes—
142C2×C42C22×D8311yescharacteristic
143C4⋊C8—832no—
144C4⋊C8—832no—
145C4⋊C8—1664no—
146C4⋊C8—1662no—
147C4⋊C8—1662no—
148#102—28no—
149C22⋊C8—1662no—
150#97—1662no—
151#97C2×Dic8312yes—
152C4×C8—21no—
153C4×C8—1662no—
154C4×C8—1662no—
155C24—41no—
156C2×Q8—22no—
157C2×D4—22no—
158C2×D4—24no—
159C2×D4—44no—
160C2×D4—41no—
161C2×D4—42no—
162C2×D4—82no—
163C2×D4—82no—
164C22×C4C2×C22⋊D8311yescharacteristic
165C22×C4#1305058912yes—
166M16—42no—
167M16—6642no—
168C2×C8—22no—
169C2×C8—3322no—
170C2×C8—1662no—
171C2×C8—1662no—
172C4⋊C4—22no—
173C4⋊C4—24no—
174C4⋊C4C22⋊Dic8314yes—
175C22⋊C4—22no—
176C42—22no—
177C42C22×Dic8311yescharacteristic
178C42D4×D8312yes—
179C42C2×C22⋊D8311yescharacteristic
180C23—42no—
181C23—82no—
182C23—84no—
183C23#1458177311yescharacteristic
184Q8—42no—
185Q8—42no—
186D4—44no—
187D4—84no—
188D4—82no—
189D4—82no—
190D4—82no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4C2×C22⋊Dic8311yescharacteristic
196C2×C4D4×Dic8312yes—
197C2×C4#1458176011yescharacteristic
198C2×C4#1458176412yes—
199C8—42no—
200C8—3322no—
201C8—3322no—
202C8—6642no—
203C22—21no—
204C22—22no—
205C22—82no—
206C22—84no—
207C22—164no—
208C22#1912758111yescharacteristic
209C4—21no—
210C4—22no—
211C4—44no—
212C4#1912787312yes—
213C4#1912796212yes—
214C4#1912796912yes—
215C2—41no—
216C2—162no—
217C2#2490786711yescharacteristic
218C2#2490797312yes—
subgroupG/H
centerC22#19127581
commutatorC42×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#3138C2
socleC22×C83#386
prime radicalC2×#127×C83C22
cosocleC2×C42×C83C23
hypercenter#3138D83
hypocenterC83#9438
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3138

Complements

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

NH ≅ G/Ntype
1. #24916565C2semidirect
2. #24906201C2semidirect
4. #24905353C2semidirect
115. C83#9438semidirect

Maximal subgroups

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

subgroupindex
#249165652
#249062012
#249061312
#249053532
C83×#31382
#943883